NEW YORK UNIVERSITY COLLEGE OF ENGINEERING RESEARCH DIVISION Department of Meteorology and Oceanography : and Engineering Siatistics Group The DIRECTIONAL SPECTRUM of a WIND GENERATED SEA As determined from data obtained by the STEREO WAVE OBSERVATION PROJECT by \ Joseph Chase RE-ET PR4 q Louis J. Cote we oH oy Wilbur Marks n S: We Emanuel Mehr BY r] Willard J. Pierson, Jr. a4 Ls | a r’ ns F. Claude Roénne L Bitehs xt George Stephenson Richard C. Vetier Robert G. Walden with the assistance of personnel from the following organizations: David Taylor Model Basin George Washington University Logistics Research Project Naval Air Development Unit, N.A.S. South Weymouth, Mass. Office of Naval Research U.S. Navy Hydrographic Office U.S. Naval Photographic Interpretation Center Woods Hole Oceanographic Institution Prepared for THE OFFICE OF NAVAL RESEARCH UNDER CONTRACT NONR 285(03) 8 Uo Taeidooag é FWhS€hoOoO TOEO O NIV IOHM/18iN NEW YORK UNIVERSITY COLLEGE OF ENGINEERING RESEARCH DIVISION Department of Meteorology and Oceanography and Engineering Statistics Group The DIRECTIONAL SPECTRUM ofa WIND GENERATED SEA As determined from data obtained by the STEREO WAVE OBSERVATION PROJECT by Joseph Chase Louis J. Cote Wilbur Marks Emanuel Mehr Willard J. Pierson, Jr. F. Claude Roénne George Stephenson Richard C. Vetter Robert G. Walden with the assistance of personnel from the following organizations; David Taylor Model Basin George Washington University Logistics Research Project Naval Air Development Unit, N,A.S, South Weymouth, Mass. Office of Naval Research U.S. Navy Hydrographic Office U.S. Naval Photographic Interpretation Center Woods Hole Oceanographic Institution Prepared for The Office of Naval Research under Contract Nonr 285(03) MS aie, ay. F i + 1 t is 1 sanotisnlecnte! pala tiol 2 atonS at 100 295 a ete a .REEM' Atposye W atsgtec THE DIRECTIONAL SPECTRUM OF A WIND GENERATED SEA AS DETERMINED FROM DATA OBTAINED BY THE STEREO WAVE OBSERVATION PROJECT By Joseph Chase... . - . Woods Hole Oceanographic Institution Louis J. Cote. . .. . . Department of Meteorology and Oceanography, College of Engineering, New York University (now at Syracuse University, Dept. of Mathematics) Woods Hole Oceanographic Institution (now at David Taylor Model Basin) Emanuel Mehr... . . Engineering Statistics Group, Research Division, College of Engineering, New York University Dept. of Meteorology and Oceanography, College of Engineering, New York University Woods Hole Oceanographic Institution The George Washington University, Logistics Research Project, Washington, D.C. (now at Com- putation Laboratory, Research Division, Math. Dept., New York University, College of Engineering) Richard C. Vetter. . . . Geophysics Branch, Office of Naval Research Robert G. Walden, . . . Woods Hole Oceanographic Institution Wilbur Marks .. Willard J. Pierson, Jr. F, Claud Rinne... =. George Stephenson .. ° and members of the staff of the U.S. Navy Hydrographic Office. JULY 1957 This report is a technical report prepared for limited distribution for the Office of Naval Research under contract Nonr 285(03) at the Research Division of the College of Engineering of New York University. Reproduction in whole or in part for any purpose of the United States Government is permitted. li 4 Age GUTARARS aUIW A TO MUAT DE LAMOI | | SHAT YS QAMIATAO ATAG MOAT as a TOLOAT MOITAV ARCO BVAW ya ro solintiten] qhiqarygouss5O ofoN ahow W's 1” Ns WANES ialloD .vttrenygoniseoO bus yygolotosiol lo seomtompel . . 4. | ate $8 Won} viletevinU dvoY wet ork ree ontha wel ty" ae) {eoiemedieM to qo0 ~ytieravidd saree. | is wen) #olindisenl oidgetgoace sD Shot abaoW . (ntes& laboM. zope tt | otehy ia Aothess ,quotd esitaitht gaitoee yay | ytiarsvinl aAroY welt ,yairovalgnt to ame D)) ‘egeliod ydiergensesO bis ygolotostent Tei gh a i, Yitsxevind) AxcY well .gtttresnty aS nottuiitent! sidqetgonsesO slot aby a an apitdiged ~YlatevinU soigaidesW sg vos of] =tneD ta wom) .O.0 .cotgaisasW GosterT doseetel er sage wis yoke briG iotses2A ,yroistods) noteIng ) - . (gittrepaiged Io egellod .ytierevint pen wes ei e200 ev ei %9 oO donee 3 ao Leyse Part l. Part 2. Part 3. Part 4. Part 5. Part 6. Part 7. Part 8. Part 9. Partl0. Part 11. Part 12. Appendix; THE DIMENSIONAL STABILITY OF PHOTOGRAPHIC FILMS. Table of Contents INTRODUCTION , ° e e e e e ° ® e ° e ° e oe ® o s . Willard J, Pierson, od COOPERATING AGENCIES AND ORGANIZATION .... . Richard C. Vetter HISTORY e e e ° e e e e 9 e e e s e e es e e e ° e ° e e e Wilbur Marks OPERATIONAL PROCEDURE e e e se s es e e e eo s e s e 2 Wilbur Marks and F, Claude Ronne WAVE FORECASTS. e e ° e e . e ° e e e e e s e e e e U. S. Navy Hydrographic Office PHOTOGRAMMETRIC EVALUATION OF PROJECT SWOP .. U.S. Navy Hydrographic Office PRELIMINARY ANALYSIS, CHOICE OF GRID SPACING, AND DISCUSSION OF ALJASING ..... At) ccpeatoniac Willard J. Pierson, Jr,, Wilbur Marks, and Jos eh Gtsee EQUATIONS FOR LEVELING THE DATA, ESTIMATING THE SPECTRA, AND CORRECTING THE WAVE POLE RECORDS . Louis J. Cote and Willard J. Pierson, Jr. THE LEVELED DATA, THE NUMERICAL ANALYSIS, AND Tie NOME RIGCAT RESULTS. 6 = © 8 6 © = ee 8 8 ws el Emanuel Mehr and George Stephenson ANALYSIS OF WAVE POLE DATA. .s.. « « « » » « «© « oe Willard J. Pierson, Jr. THE STEREO PAIRS, AND THE INTERPRETATION AND ANALYSIS OF THE DIRECTIONAL SPECTRUM IN TERMS OE PWEAW ES TDR OUR NE” a) tell fou eh Vee a pales! rete, ce er: le Willard J. Pierson, Jr. CONCLUSIONS AND RECOMMENDATIONS ..... +... Willard J. Pierson, Jr. Simeon Braunstein iii 18 26 31 40 a7) 58 82 134 148 253 Caryl BNO, UN «s,s, 4d alga en nartouaoaron i | ; 2 FG dasine cou aa i ead ae i" Be THbo PR AE RaT, tH | | sie os eae j os. ape ie. r : x Ne A AEE oe ee Pou A a ee here 4 i, | ; ey ora ue bbe! THES ny) * ine eee | po ie ‘ Bane ers’ a Nha i ne i ie we iis aa rl secasme shih | we cada xp sleet yet a oe RONG pate Aor Rac Oe Le As | | 8 oat aig tae evil i ou). Caachi wees ch tt, Be) aaa gee eater te a * he 7% : a = & - un | ya ‘gene ee aed” uipts tee dud. ove tint f ae cope BIT af ibe 5 ee: easobas Bos tp WOO LIST OF ABBREVIATIONS Bu Aer Bureau of Aeronautics, Washington, D.C. DTMB David Taylor Model Basin, Washington, D.C. HYDRO U.S. Navy Hydrographic Office, Washington, D.C. NADC Naval Air Development Center, Johnsville, Pa. NADU Naval Air Development Unit, South Weymouth, Mass. NPC — Naval Photographic Center, Washington, D.C. NYU New York University, New York ONR Office of Naval Research, Washington, D. C- SIO Scripps Institute of Oceanography, La Jolla, Calif. WHOI Woods Hole Oceanographic Institution, Woods Hole, Mass. (Oe sorgainae® ae ‘iusnoroA to oso: haa {DAL aomnidesW sate lotioht votye? iva Bemis nosgaiiien® (20 R- oisigengorbylt ore B y! | 9 jattevoasdtot coed: taomnqelerrod aA levalt - : } ‘Tid gdiate nanny dao® * didld indeingeleved EA. level | 2.0 ,notgeidonW cvodaia® aldaexgorodst Leva stor all eethexoviad Aset walt “y siieqaeuae doexoasdt baw b0 amid ils etlnt al ~vdawsgonsei0 Be 6 eto shooW wetting oi ERRATA Page 33, line 11: For 74" read 74°. Page 104, Table heading: For Data Set 2 read Data Set 3. Page 180, line 16: Following and 11.3 should be a parenthetical remark: (as drawn for the full set of data) of elie sed blyode £11 bas arbwollot {sisb lo tos Link esft ot awaxh ast ‘atranaes avis mlte INTRODUCTION On November 25, 1954, two aircraft rendezvoused with the R. V. ATLANTIS at a point in the North Atlantic. The aircraft made a sequence of passes as the ATLANTIS and flew back to base. Months of preparation went into the flight, months of thought went into the problem of what to do with the aes ob- tained on the flight and by the ATLANTIS, and months of work went into the numerical processing of the daia. The results of the flight were stereo pairs of photographs of the sea sur- face. The ATLANTIS provided base line calibration and wave pole and visual observations. Two of the best pairs of photos were reduced to 5400 numbers on a rectangular grid, The wave pole records were reduced to a time series of discrete points of about 1,800 numbers each. The purpose was to take the two sets of 5400 numbers, estimate the directional spectrum of the waves on the sea surface and compare it with the frequency spectrum as estimated for the three sets of 1800 points read from the wave pole records as a check. To go from the 10,800 stereo numbers and the 5,400 wave pole numbers to the desired spectra required a total of about 9,000,000 multiplications and an equal number of additions. After the directional spectra were computed, the results obtained were inconsistent with the theoretical models, and the st.ereo data had to be carefully re-analyzed with the result that part of the data had to be discarded. The 5400 numbers were reduced to about 3500 numbers, and the computations were done over again. This task has just been accomplished, and the purpose of this report is to tell how the operation was planned, how the data were obtained, and how the computations were made. Finally, the results obtained will be analyzed and interpreted. The original data, the reduced data, and the results of all computations are included in both pictorial and tabular form. As this report is studied by its readers, it will become apparent that it would never have been written were it not for the combined efforts of a very large number of people with diverse talents and abilities. They represent a wide variety of U.S. Navy organizations and civilian research organizations. As many as possible have been mentioned and thanked in this report, but some who have helped immensely in this work remain anonymous because it 1s not possible to list them all. Our thanks are extended to all individuais who helped in this work and to all cooperating groups, and the hope is expressed that the final analysis of the results will prove of sufficient value to justify the tremendous effort expended on this task. Part 2 COOPERATING AGENCIES AND ORGANIZATION This part concerns the arrangements, meetings, official letters, dis- cussioms amd exchanges of ideas that went into bringing together the many di- verse people and agencies who contributed to SWOP and without whose assist- ance SWOP wild not have succeeded. The problem as first presented im the fall of 1953 was to cbhtain am accu- rate representation of the directional properties =e real ocean waves. The first job wes te try to find cut whether or not the propesed plan was feasible and te get and review the critical opinion ef others as to whether or mot it needed doing. Marks had been studying waves by stereo techniques, but om a much reduced scale. By taking photegraphs from a bridge, he was able te get useful data on the two-dimensional wave spectrum for a rather limited fetch. This, however, was quite a different thing from taking stereo-photos from airplames far out in the open ocean. Letters were written to people doing wave research asking their opinior. Ir the replies there was gemeral agreement that a good statistical treatment of a whole area of the sea surface was mecez- gary beforethe art of understanding waves could be much advanced. Ima letter to en author im February 1954, Walter Munk, of the SIO, stated that ".... the two-dimensional analysis is certainly the essential problem now. Im fact, IThave some serious doubts as to whether further extensive work on frequency analysis of records taken at a single point is even worthwiitle. If ome uses a spectral presentation of waves, one should really go all the way or motatall.” 3 Assured that the study was needed, the first of a long series of letters, official and otherwise, which helped bring together all the necessary people and components which were needed to make the plan succeed were written. In addition to the actual task of taking the photos under suitable wave con- ditions the data had to be analyzed on a stereo planigraph or similar instru- ment and facilities for the immense task of computing the required quantities with high speed digital computors had to be obtained. The first piece of official correspondence on SWOP in the files is dated November 9, 1953. It was a formal letter from the Chief of Naval Research te the Chief of Naval Operations outlining the reasons for the SWOP project and asking for certain services. Cameras, airplanes and a radio link for firing the cameras were requested. The letter went via the Bureau of Aero- nautics for comment. It picked up a favorable endorsement recommending that the project be assigned to the Photographic Squadron VJ-62 in Sanford, “lorida. To the practiced eye of our friends in CNO it was obvious that the proposed job was much more complicated than the letter indicated. The Naval Photo Interpretation Center was asked by CNO to study the proposal and comment. Asaresult of the review, a number of critical points were raised. There were problems of control of aircraft height, of control of distance between aircraft, of tilt and of simultaneous firing of the cameras. Establishing a pattern that was to become a routine method of solving the problems which arose, a conference of all concerned was called to discuss each point in detail. This conference which was held at the Naval Photo- 4 graphic Interpretation Center, Anacostia, Maryland, on March 2, 1954 is described by Marks in Part 4 of this report. It would have been useless, however, to proceed with plans and the con~ Eeeuction of equipment for SWOP without some assurance that a vessel could be made available. The vessel would have to go to the target area some place in the North Atlantic and wait, no one knew exactly how long, for favorable meteorological conditions to occur. This assurance was given by WHOL. The ATLANTIS could be put at the disposal of SWOP given sufficient advance notice. Getting an oceanographic research vessel with a very heavy schedule of other "equally important" projects under such circumstances would have been extremely difficult without the enthusiastic and understanding co- operation of Dr. Columbus Iselin of WHOI. The next item on our critical list was the weather. The Division of Oceanography, U.S. Navy Hydrographic Office was asked for advice con- cerning the best time and place for finding the desired wave conditions. A report on this aspect of the work is given in Part 6 of this report. The errors which could be anticipated in the data had been estimated and it had been shown that significant results could be obtained in spite of these errors. By May of 1954 enough arguments had been mustered to permit another try through official channels to get the airplanes and cameras we had to have. During a conversation with Cdr. James* about the possibility of using blimps to do the job, he suggested that NADU was the place to go for help. *ONR Air Branch Fortunately, Cdr. Robert H. Woods, Commanding Officer of the Naval Air Development Unit, and Cdr. Hoel, also of NADU, were in Washington on some other business and the problem was discussed with them. They were both interested, even enthusiastic about our project. We talked about using two of NADU's blimps to do the job. Their stability, slow motion and free- dom from vibration were particularly appealing. At last, operational people were interested in helping us. In the next few months the officers and men at NADU accepted each problem in the series of many to be overcome in our job as a challenge and made it a point to find the best answer in each case. It is impossible to give NADU too much credit for the magnificent job they did for us. On 16 July 1954, an interoffice memorandum was written explaining the necessity of assigning a priority of ''B'' to our project with NADU. The stringent weather requirements we had to meet and the necessity of being able to plan well in advance in order to have the WHOI R/V ATLANTIS, the NADU airships, and the weather all be at the right place at the same time were outlined. The priority was granted and on 19 July a letter was sent from ONR to NADU setting up a project directive for the accomplishment of SWOP. An abstract of the project contained in this letter will provide some idea of the plans at this point: "The Naval Air Development Unit will make one flight consisting of two aircraft (equipped with trimetrogon cameras and an FM radio link for the purpose of triggering the two cameras simultaneously) to a target area approximately 300 to 400 miles out over the North Atlantic. The Woods Hole Oceanographic Institution research vessel ''Atlantis'' will be 6 im the center of the target area making mumerous and continuous wave observations ami providing a "ground control" for the aerial photography by determining accurate distances between the "Atlartis" and a buoy. Pers cere from the Woods Hole Oceanographic Institution will assist in installation of the cameras and construct the F i radio link, " The stabilization of the airships in the rough weather they were apt to en- counter while ilying our missinn was a preblem. We expected to lick this by @ The Camera MOUNtS. On 19 July, arrangements were made through our Property Branch for the loam of the following equipment from the Phetagraphic Division in the Bureau of Aeronautics: 4 GA-8 aerial reconnaissance cameras @ Gyrostabilizing mounts 8 rolls of Topographic Base, Panchromatic film 91/2" x 200°! BuAer was most cooperative and helpful im loaning us this much needed equipment. On 21 July another letter was t te NADU from ONR. designating Wilbur Marks of WHOI as ONR field representative for the SWOP project. Two weeks later a conference was held at South Weymouth to set up de- tailed specifications and plans for SWOP. Lt.jg. Chandler was assigned to organize NADU's participation In SWOP. He, LGdr Champlin, Cdr Heel and Marks were able to clarify ideas abaut some of the equipment require- ments. The author went te work trying to get some of the items not avail- able at Woods Hole or NADU which they thought they would meed. The results of the conference are described by Marks in Part 4. A most important 7 result was the decision to change over from blimps to PZV's. A meeting on the 19th of August marked another milestone in the step by step progress we were making. Marks,. Ronne, Whitney and Walden from Woods Hole, Cdr. Leffen, LCdrs. Finlayson, Docktor, and Price from NADC, two representatives from Hydro, Pierson from NYU, and my- self from ONR attended, and Cdr. Wood, Cdr. Hoel, LCdr. Champlin, LCdr. Hollingshead and Lt. jg Chandler acted as hosts at NADU. Thanks mainly to the staff at NADU, a detailed program for the accomplishment of the operation was worked out. The participants were assigned various tasks and dates were set up for tentative completion of various phases of the project. As a result of the conference my tasks were to arrange fer the devo ment of the film at Bermuda and Anacostia, to get official sitinaa ties through BuAer for the installation of the cameras on the P2V's by the NADC and to get the necessary films and magazines (also through BuAer) sent to WHOL. A date was set for the test flight, 27 September, and a target date for the actual photographs at sea, 1 October. Considering the many things that still needed doing and the arrangements that had to be made, this was push- ing things just a bit. But to delay longer would have put us into the winter months with less chance of getting just the right meteorological conditions. On 25 October a letter was written to the Naval Photographic Center, NAS Anacostia, asking them to develop our black and white and color film 8 obtained both during the test flights for SWOP and the actual project flights. We wanted immediate development of the test flight film in order to be able to check out the photographic system. The P2V's were to land at Anacostia immediately after flying the test hop so that the films could be developed at the NPC and inspected by someone from the Photogrammetry Division of the U.S. Navy Hydrographic Office for accuracy with a minimum of delay. The test flight was planned for the third week in September. The results of the analysis of the test data and of the data finally obtained are described by members of the staff of the Photogrammetry Division of the U.S. Navy Hydrographic Office in Part 6. The flight check showed that one of the two cameras sent to Johnsville for installation was defective. Luckily we had originally asked for four cameras so we had spares to fall back on. The Photogrammetry Division at Hydro checked these cameras for us and were able to find two good cameras for us out of the four BuAer had originally sent. On September 16th a letter was sent from ONR to Hydro asking for their assistance in providing the required wave forecast. This request was made supplementary to the request for services from the Photogram- metry Branch and was intended to be part of the same project. Hydro, of course, agreed to provide these additional services. This aspect of the work is also discussed in Part 5. On the 29th the test flight was made. Iwas waiting at the field at Ana- costia for the P2V's to arrive. It was slightly after the desk workers quit- ting time when both planes touched down. The magazines were removed from both cameras and taken to the Navy Photo Center for development. Later that night the films had been inspected. Everything was fine except that the corners of the pictures taken from both cameras were blurred. The cameras had been mounted too high up in the fuselage of the aircraft so that part of the field of view was being cut off. Some minor surgery on the mounts was all that was required. By the 12th of October everything was ready to roll and Marks issued detailed instructions to all parties outlining exactly what and how each was to do his part. On the 15th the Woods Hole Port Captain issued letter instructions to the ATLANTIS Captain to depart Woods Hole on or about the 17th for latitude 39N, longitude 63.5W to the rendezvous with the waves, the weather and the P2V's. It was hoped that SWOP would be over and done with by the 25th so that the ATLANTIS could be back at Woods Hole on the 27th as early in the day as possible. When the right weather and cloud conditions finally occurred, the ATLANTIS was there after waiting for one week; the planes were there; the cameras were functioning; and on the 25th of October the pictures were taken. There remained the unexciting task of cleaning up after the operation. All the various items of borrowed equipment had to be returned. There was more correspondence back and forth piecing together some of the loose ends. Marks sent down a data sheet for the distance between the raft and the ATLANTIS in each pair of photographs. Prints of the best stereo pair ac- cording to the judgment of Hydro's Photogrammetrists were sent to him. 10 Details of the grid arrangement were worked out. BuAer was notified that the operational part of SWOP was over and all comcerned were thanked for § their help and cooperation. A similar letter was sent to NADU via their boss, the Commander, U.S. Navy Air Bases, ist Naval District. The films exposed over the North Atlantic were developed and sent to the Hydrographic Office. The low light intensity available during the SWOP flight threatened to produce negatives of marginal yalue, but careful develop- ment by the NPC at Anacostia saved the day. Hydro looked ever the stereo pairs, picked one of the best and began contouring. Further werk came te 2 temporary halt while Hydro's presentation of the contours were sent to Pierson and Marks for study. As was expected there was some tilt in the contouring. The Photogrammetry people at Hydro did their best to level the stereo-pair before contouring, but we all realized that with no established reference plane from which to work, perfect level- ing would be impossible. The contours showed a range of heights from lone foot to 24 feet while actual wave pole measurements taken on the spot from the ATLANTIS gave a2 significant wave height of about 7 feet. In addition to this tilt a somewhat closer look showed a ridge running al- most exactly down the center of the 2,000° x 4,000’ rectangle caused by the stereo-photes paralleled by a trough about 500' away. Im order for this feature to be real, a wave with a height of at least 9 feet and a period of about 14 seconds would have been in the area photographed. No such wave 1} could have been generated by any known meteorological disturbance in the Atlantic area. We were forced to conclude that the contoured surface was not only tilted but warped in some sort of "barrel'' shape with axis parallel to the long sides of the contoured rectangle. Fortunately, this barrel type of distortion was not present in other stereo pairs and the only real problem turned out to be that of removing tilt from the analysis and determining a zero reference plane, The tilt was not too serious. Mathematical analysis could take most of the tilt out of the data. After a discussion with Hydro, it was suggested by me that we have Hydro give us the data in the form of discrete elevations — ona grid system. They felt that such data would be more accurate and take less time than contouring. It looked as if it were about time for another small conference to see where we stood and determine just what should be done, Dr. Pierson came down from NYU on the first of March and we had a very profitable discussion with the people at Hydro's Photogrammetry Branch. He was finally able to decide on a simple 30 x 30 foot grid sys- tem, with sides parallel to the photographs. Unfortunately, the sides of the photographs did not line up with the direction of the surface waves as ex- pected. The preliminary analysis of the data is described in Part 7 of this report. i The important thing, however, was to have the grid system point in the same geometrical direction in each of the three pairs of photographs 12 which were to be processed for spot heights. By careful selection from the available prints, Hydro was able to come up with two good pairs with iden- tical orientation and another which was only 5° different in direction from the other two. They set up the grid system in the third so that it was aligned in the same direction as in the first two. A grid system of 60 x 90 points was finally settled upon, and Hydro began the laborious task of grinding out 5,400 spot heights for each of three pairs of stereo photogyaphs of the sea surface, With the data soon to be pouring out of Hydro, the next important prob- lem was to get it analyzed. We turned again to the DTMB UNIVAC. By this time the demands for time on their computer had grown tremendously. They would, however, be willing to rum the analysis if it were first programmed. Dr. Pierson investigated the possibility of having the programming done at NYU. The Engineering Statistics group of the NYU Research Division, under the direction of Mr. Leo Tick undertook to do the task in about two months. So, while Hydro was amassing the tables of numbers that were so import- ant, Mr. Emanuel Mehr was working out the problems of telling a mass of vacuum tubes and wires (the UNIVAC) what to do with the numbers we in- tended to feed it. By the first of April, Hydro was beginning to grind owt the data and I went out to see how they were doing and talk with them about the project. Iwas very much impressed with the careful and accurate job they were doing. They felt that their observations were accurate to plus or minus one foot and reproducibility to about 2/5ths of a foot. Asa check, I 13 picked out two spot heights which they had already determined andasked for a check. Im both cases the operator came within 1/5 foot of the previously recorded reading. By the 2Zist of April the first set of 5,400 spot heights had been completed and Hydro was procfreading the typed tables. These were forwarded to ONR on May 6, 1955. The second set of 5,400 SWOP numbers came into my office from Hydro. By this time they were turning out work at a good rate and doing an ever more accurate job. Their esti- mate was that the second batch of datawas accurateto within plus or minus a 3) HEHE, By the first of July Hydro had forwarded the last set of the three sets of data. This also was accurate to within plus or minus .5 feet. Thus one phase of the SWOP operation came toaclose. The extraction of the raw data from the stereo photographs had been completed. The question of whether or not we should have Hydro conteur another set of photos in order to show up some of the fine structure which would be eliminated from data taken from spot heights alone arose at this time. It was agreed that this should bedone but at a later date when SWOP was farther | along. It remained to analyze this data carefully to eliminate the tilt which wag known to be present in each model and finally to subject the data to analysis on an electronic computer. Our attentions were now focused to the problem of getting a firm commitment from DTMB concerning use of their UNIVAC. It looked as ifour original estimate of the time required for the 14 { analysis had been an order of magnitude too small, The workers at NYU worked out a method for leveling the data that would take about one and one- half hours of UNIVAC time. We hoped to get this done on the DTMB UNIVAC. Later the job was done commercially at New York. While working out the details of the programming of the data Mehr upped the estimate of total UNIVAC time to over 20 hours.” This made things look a little dark for us as far as getting the job done at DIMB was concerned. On 26th August cur letter to DIMB asking them for UNIVAC time was answered, The Model Basin wanted to program the analysis instead of having it done at NYU and wanted to examine the data to see whether or not it patie be practical to do the work on their UNIVAC. Their desire to do their own programming was understandable since an improperly yrogram- med operation could run up the total time used by the UNIVAC considerably. However, Mehr had already done most of the programming at NYU. An- other conference appeared to be in order, so on 19 September Dr. Polachek and Mr. Shapiro and Mr. St. Denis of DTMB and Mr. Mehr of NYU and my- self met at DTMB to discuss the problem. Mr. Mehr had finished program- ming the analysis. All that remained to be done was to ''de-bug"' his pro- gramming setup and then rum the analysis. He estimated that about 20 good UNIVAC hours would be required for each of the three sets of data. Con- sidering their other high priority commitments for the UNIVAC this was way out of line with what we hoped to get. lagreed to try to find funds to de-bug and run the first set of data commercially, --the plan being then to *For each set of data. 15 turn the rest of the data over to DTMB to have it run in bits and pieces as time became available. Additional funds were made available to NYU by ONR to perform the first part of this analysis, and I wrote an official letter to DTMB outlining our plan. We would have the analysis on the first set of data done commercially to provide an absolute check on the program- ming and then turn the remaining work over to DITMB. From the beginning it had oar our expectation that we would be able to use the UNIVAC computer at DIMB to perform the analysis of the SWOP data. That things did not turn out this way should not be taken as a reflection against DIT MB or any of the people on its staff. Without the encouragement from DTMB, in particular, from Manley St. Denis, early in our planning stage concerning possible use of their computer, we might not have gone ahead. Arrangements were finally made to have the work done on the ‘Logistics Computer", ONR Logistics Branch. Thanks are due to Dr. Max Woodbury and Dr. Fred D, Rigby of the ONR Logistics Branch for assist- ance in making the Logistics Computer available and to Dr. William Mar- low, Principal Investigator of the Logistics Research Project, and Mr. George Stephenson, Head of the LRP Computation Laboratory in Washington, D.C. Fortunately the Logistics Computer was able to make use of some of the programming already worked out for the UNIVAC. The theory for level- ing the data, and determining the spectrum is described in Part 8 of this report and the numerical procedures and the actual data obtained are given 16 in Part 9. As one last check on the quality of our SWOP data we had one more contouring job done by Hydro. This time the reference level was deter- mined by selecting points that had been leveled statistically. This leveled contoured model was forwarded in May 1956 along with an evaluation of the photogrammetric work. As mentioned before, the work done by the Photo- grammetry Division of the U.S. Navy Hydrographic Office is described in Part 6 of this report. And so, after 33 months and a correspondence file at. ONR two and one- half inches thick, as of June 1956 SWOP has been completed except for the task of interpreting results, drawing conclusions from them, and pre- paring this report. 7 Part 3 HISTORY For some years now the desirability of obtaining the two-dimensional sea spectrum has been explained in the literature (Pierson [1952] and St. Denis and Pierson [1953]), When most of the methods considered were found lacking, the technique of stereo-photography of the sea surface seemed ‘most amenable to possible methods of amalysis (Marks [1954a]). Representatives of the Woods Hole Oceanographic Institution initiated the first steps necessary to convert the idea of aerial stereo-photography of ocean waves into fact. At a meeting in Washington, the requirements for such an undertaking were established, and, equally important, the Photo- grammetry Division of the U.S. Navy Hydrographic Office expressed an interest in the job of reducing the photos to numerical data form. As a re- sult of this discussion, the first formal plan for obtaining the stereo-pairs was set down (Von Arx[1952]). The basic requirements were as follows: 1. Two aircraft to fly parallel, 600 feet apart, at 1000 feet; 2. Each to have a CA-8 metrogon camera aimed vertically down and one long focus 35 mm camera aimed at companion aircraft to determine the length of the stereo-base line; 3. Cameras to be triggered within 10 milliseconds of each other by an FM pulse; 4, Smoke bomb pattern to eee ground control; 5, Upwind flight of planes with flaps down to reduce plane speed and 18 ive better results; 6. Preliminary flight to determine best height and baseline suggested. The stereo-photos were made three years after the date of this first meet. ing. Much thought, discussion, planning and revision took place in that in- terval, and yet the final operational plan differed little in essence from that set forth in the Yon Arx note which is outlined above. At this point, work on the problem ceased, and almost a year passed before interest was revived. Wave theory was advancing ata rapid rate, and W.J. Pierson, convinced that basic theoretical conclusions should be substantiated by experimental work, persuaded the Office of Naval Research to begin a project of aerial stereo-photography. Shortly afterward, the Woods Hole Oceanographic Institution agreed to help onthe problem. Dr. Iselin, Senior Oceanographer, offered the services of the RV ATLANTIS to provide a horizontal scale factor and to obtain with a capacitance wave pole recorder a record of the sea surface as a function of time at a fixed point in order to determine the sea surface spectrum as a function of tre- quency. Since this frequency spectrum is the integral (with respect to direction) of a transformation of the two-dimensional spectrum to be obtain- ed by stereo-photography, it is the only method of testing the validity of the directional spectrum. As the mechanics of the project began to crystallize, the multitude of "minor problems" associated with an air-sea venture of this sort became evident, and a meeting was scheduled to coordinate the facilities available 19 at the moment and to provide general cognizance of the problems of the various groups involved. The U.S. Navy Photographic Interpretation Cen- ter (NPIC) was the host of a meeting at Washington, D.C. which was attended by representatives of New York University (NYU), Woods Hole Oceanographic Institution (WHOJ), the Office of Naval Research (ONR), David Taylor Model Basin (DTMB) and the U.S. Navy Hydrographic Office (HYDRO). The discussion was presided over by Mr. Richard C. Vetter (ONR). The results and conclusions of this conference provided the first link in the chain of events which ended in the successful aerial stereo-photography of the sea surface on October 25, 1954. The highlights of this meeting are listed: il. A justification for the mission was given by Professor Pierson through a description of the basic theory of sea spectra and a step- by-step elimination of other possible techniques. 2. HYDRO expressed ability and willingmess to contour the stereo- photes if they met certain photogrammetric specifications. 3, The stereo-baseline (distance between planes) is a vital factor in the contowring of the photoes and will have to be resolved. Photo- graphy of one plane from the other was ruled out on grounds of measuring inaccuracy. 4, The necessity for a horizontal unit of measurement four or five times the length of the ATLANTIS provided another unanswered question. 20 5. Mr. Vetter (ONR) agreed to act as administrator for the project and to try to initiate the next link in the chain, which was to obtain permission for the use of two suitable aircraft. As a result of this meeting the first operational plan was put om pape: (Pierson [1954]), and the project achieved an air of respectability exhances by the eagerness exhibited by the participants. Altitudes and baselines were defined. Sources of stereo-analysis error were given, and error magnitudes were estimated. 10 = (o) Sipe ww > 6 S = 4 > 2 (0) 0000 0200 0400 0600 0800 1000 1200 1400 1600 1800 2000 GGT. wave pole observations = sue stereo observations x S 10, i se ld g re Fa © op) 0000 0200 0400 0600 0800 1000 1200 1400 1600 1800 2000 G.C.T. FIGURE 7-1 WIND AND VISUAL WAVE OBSERVATION DATA OBTAINED BY THE RV. ATLANTIS - 50- winds had died down. Note that the visual estimate of the significant height was 7.5 feet at 1800Z, Stereo Contour Data The first data prepared by the Photogrammetry Division was in the form of a contour analysis of one of the stereo pairs. It was somewhat disconcert- ing because the expected waves with lengths of from 100 to 300 feet or so could not be seenin the De AO and the range of contoured heights was far in excess of anything to be expected from a 20 knot wind. The first hint of where the difficulty lay came from Woods Hole where line sections of the contoured surface were drawn. These showed alrnost a straight line tilt along a givensection with the waves we were lockingfor superimposed thereon, A line section with arbitrary scale units from the lower left to the upper right of the contoured surface is shown in figure 7.2. The untoreseen difficulty of determining atrue mean zero reference plane on the open ocean with no known reference points had arisen, It was also pointed out at this time that spot heights could be deter- mined by Hydro with far greater accuracy than the contowrs could be drawn due to the nature of the techniques involved. The original plan had been to choose an appropriate grid and read spot heights from the contoured data. This now had to be revised, and it was now necessary to find a way to deter- mine the true zero reference plane and to choose a grid, the desired number of points to be read, and the desired resolution and statistical reliability, all on the basis of the data then on hand. 51 Gee O lel ‘391440 =SIHDVYSONGAH SHL AS G3AGIAOYd JDVSYNS GAYNOLNOD JO NOILDSS 3NN -52- Decision on Grid Interval and Nus mber of Points to be Read Fortunately the theoretical aspects of the problem had béen carried out tothe point where the methods for the eacgafonent ae analyses of time series developed by Tukey [1949] ha d beaa exgs ndod to the twe -Cimensional wave wt number analysis desired in this preble he hus tae formulas for regolution, aliasing, and degress fos freedom Were ayedla te, - EL Ney aelt ib devived in Part 6. It was also realized that # was not esse vue to have the dominant wave direction roughly parallel to the sides of the rectangular grid of points to se used, The leveling problem was then studied and formulas were derived for Hee maining the true zero reference plane, ft was assumed thai the spot heights would be reported with reference to some unknown ar rhitra ary refer- ence plane of the form = = ax + by tc, “If Y is the reported value, then * = ie ax si by - c is the valuc with reference to a true xex 'Q re eroten ceric! and B2(H*)* will be a minimum, iy standard least squares techniques, the values for a, b and c can be determined, and the data can be leveled, The derivation and the procedures used are des crjoed Sa the fallowing two parts of this report, Since leyeling was no longer a problem, tue problems of stati stical re- liability, aliasing, and resoluwtian were skadied, Suppose that the spot heights are read at an interval of Az feet on a square vid, Then, due to the nature of the methods of analysis, sorae spectral components shorter than 2Ax and all components shorter than Ve ax: will be ajgased so that they appear as 53 longer waves than they actually are. Suppose that m lags are to be used in the x and y directions of the rectangular grid. Then the E value contributed by the waves with lengths from infinity to 4mAx will all be concentrated at the j zero wave number of the spectral coordinate system. The next wave nurmber will correspond to a wavelength of 2mAx and will actually cover a range fro 1 4mAx to 4mAx/3. Ona line at 45° to the grid system of the spectrum, it is necessary to shorten the above wavelengths by V2 /2. Finally, if N, and Ny are the number of points on the grid system in the x and y directions, then | the number of degrees of freedom is given by N N Pan eehe =e Wifes L my, my 2 where for purposes of symmetry it was decided to let m, = my. The significant wave heights reported by the wave pole observations cor-— responded to a wind of about 17 knots, and an attempt was made to choose a method of analysis such that a theoretical Neumann spectrum (Neumann [1954]). for 17 knots would be adequately resolved. Also since winds of 20 knots had occurred previously, it was decided to guard against wavelengths due to a wind of 20 knots in addition to those due to 17 knots. It was estimated that periods from 2.25 to 10 seconds would be present and that approximately 10 eeeces of the energy would be at frequencies above 0.29 cycles per second (or a period of 3.45 seconds). A wavelength of 60 ft corresponds to this period, and hence a spacing of 30 feet between points would be needed to insure no more than 10 percent aliasing. 54 A grid of 30 feet was therefore chosen. Smaller values of Ax would require a much greater m than that actually chosen and many more spot heights, The winds at the surface varied from 17 to 20 knots just prior to the time of the observations and hence the values seemed consistent, Spectral periods as high as ll seconds might have been present in the waves due to the 20 knot winds, This period would correspond to a wavelength of about 600 feet, With a grid spacing of 30 feet, the area of the stereo analysis for one pair of photographs was found to contain about 60 points on the short side and more than 90 points on the long side. This would imply the determination of 5400 spot heights from each stereo pair. Various lags were then tested and a value of m equal to 20 was chosen for two reasons, The first was that there would be adequate resolution, and the second was that there would be enough statistical reliability. With respect to resolution, wavelengths greater than 2400 feet would then show up at the origin and since this corresponds to a period of over 20 seconds the energy at zero wave number should be entirely due to aliasing and white noise reading error on the assumption that the Neumann spectrum was roughly correct. The next wave number would cover a range in lengths from 2400 feet to 800 feet, and it would also not be expected to show any appreciable wave energy. These values were also checked on the assumption that the peak of the spectrum would fall at an angle of 45 degrees to the coordinates of the spec- 55 trum, A wavelength of 1680 feet would still not be expected, and a wave- length of 560 feet would just barely be beginning to show up. On the basis of the transformation needed to go from the theoretical Neurnann frequency spectrum to the wave number spectrum, it was estimated that the peak in the spectrum would fall four or five wave numbers away from | the origin, and that the range covered would adequately trace out the details of the shape of the spectrum. The full consequences of these decisions will be discussed in a later section. With respect to statistical reliability, there are 16 degrees of freedom for each point estimated on the spectrum for each set of data. Fifty degrees of freedom are desirable so it was decided to do three pairs of stereo photos, since, with the same grid alignment of all three, the estimates for each set of data could be averaged to obtain final estimates with 48 degrees of freedom. | One of the stereo pairs spot heighted by Hydro turned out to have serious "barrel" distortion in addition to tilt, and it had to be discarded so the final results will be based on 32 degrees of freedom. A choice of a 60 by 90 grid and 20 lags (really 20 to the left, 20 up and 20 down plus all combinations such as, say, 5 to the left and 17 up) implies 861 points to be deterrmined for the co-variance surface and 861 points for the final spectrum. About 4,000,000 multiplications and an equal number of additions are needed to get each of the co-variance surfaces, and about 720,000 multiplications andadditions are needed to get each of the raw spectra. H Much the same censiderations entered in the above choices as enter in 56 the choice of time interval, number of lags, resolution and degrees of freedom in the analysis of a wave record as a function of time at a fixed point (Pierson and Marks [1952]) except that far more data processing and numerical computation is necessary. As an example, to double the resolution with the same grid spacing and same number of degrees of freedom would require 40 lags and four times the number of spot heights. The covariance surface would then require about 48 million multipli- cations and additions. The time required would be more than twelve times greater than was actually used. To have reduced the aliasing by halving Ax, would have required four times the number of spot heights, 40 lags, and the above number of multiplications. Moreover, the total energy over 3/4 of the area of the spectrum would have been only 10 percent of the total energy of the sea surface. For these reasons, Hydro was requested to read spot heights ona square grid with 30 feet between intersections. Essentially all of the details of the analysis were decided by this one choice of grid interval. References Marks, W., and J. Chase [1955]: Observation of the growth and decay of a wave spectrum. Contribution No. 769, from the Woods Hole Oceanographic Institution. Si, Part 8 EQUATIONS FOR LEVELING THE DATA, ESTIMATING THE DIRECTIONAL SPECTRUM, AND CORRECTING THE WAVE POLE SPECTRUM Leveling the Data The original spot height data reported by Hydro was reassigned a position code for computational purposes such that the points would fall in the first quad- rant of a Cartesian coordinate system and such that the first column of 90 points would fall on the y-axis and the bottom row of 60 points would fall on the x-axis. For simplicity in writing the following equations, let the free surface, 1, be represented by an N whena spot height is considered. The pattern of the points was as follows: 0° 89 ° . ° ° o e ° e ° ° e ° ° ° e ° e N59, 89 : ; 5 ° aos : 59, 0 58 and the general element will be designated by Nic: These 5400 points cover an extensive area of the sea surface such that quite a few waves are involved. Were they measured with respect to a zero determined by the level of the water in the absence of the waves, they would average to zero and the sum of their squares would be a minimum. However, they were read with reference to an arbitrary tilted plane instead of with refer- ence to the sea level. The values desired with respect to zero level are given by equation (8. !) where the unknown constants a, b, and c absorb the effects of the grid spacing. 6.1 Nix = Nj > aj > bk- © = Wa acan pA ell KeeiOkiens 4 ae me: a = G0) on = SIO) Consider equation (8. 2). m-] n-l R m-1 n-1 2 (8. 2} Va (Nii) =z Be NCE = (oe a8) k=0 j=0 k=0) j=0 The value of V should be a minimum with reference to true sea level (if the area covered by the points is large enough), and this can be accomplished if ONE 9a ; (8. 3) ye oe and a. ou Equations (8. 3) lead to equations (8. 4). 59 m-1 n-l 2; Zz (N., - aj - bk-c)j =0 k=0 j=0 JK d : m-1 n-1 (8. 4) = =f (Ny -aj-bk-c)k = 0 k=0 j=0 J m-1 n-l IN eye eo Ge) 0 <=0 j=0 JK The last equation simply states that the average of all the points in the plane when truly leveled should be zero. Points on a tilted surface could still average to zero, but V would not be a minimum; and thus the other two equa- tions assure that V will be a minimum. The indicated summations can be carried out, and the result is three simultaneous linear equations in the three unknowns, a, b, and c. (8. 25) mn(n - 1)(2n - 1) n(n-1)m(m-=-1) ma(n- 1) 5) met net jN: 6 4 2 k=0 yZ0 n(n - 1)m(m - 1 nm(m-1)(2m-1) am(m- 1) io & et at kN. 4 6 2 jk : ; m-1 n-1 ma(n - 1) nm(m_-_1) nm : 2 = i c Zz Nix k=0 j=0 | For m= 90, n= 60, the determinant is known, and the indicated sum- mations on the right hand side, when performed on the data, then permit the values of a, b and c to be found. Estimating the Directional Spectrum In theoretical discussions of wind generated meee waves, it has been shown by Cox and Munk [1954] and by Pierson [1955] that 60 @ ena) Q(x", y',t') = lim ot n(x, y, t) n(xtx', yty', ttt') dx dy dt T? 0 ae ik RM ee 2 28 Wp T fore) al ( 2 ee ! ! + ! = 5 [A(p, 6) ]~ cos[— (x'cos@ + y'sin®) - pt']dudé otce 1/4 =< [} [A*(a, B) ]2 cos(ax' + By' - /B(a-+ B*) f t')dp] da L658) ies vy In equation (8.6), a = ye cos@/g, B = p7sin@/g, = Ye (a7 7p B*) ee and @= tan™!(p/a). Also A ae 2 Jelal ye (a? + p2)"*; tano} B/a)] (8.7) [a*(a, p)]* = == Coe 6 hc Le ane fe ae (a If x' and y' are chosen to be zero, then an average over time can re- place an average over space and time, and the result is (8. 8) Q{t')=lim Ff nl y> t) n(x. y, ttt')dt T+ 0©o T ; Z Tt foe) Jt 2 1 AG [A(u, 6)]~ cos pt'dydé -T 0 =>) [AW I* cospt'dp 61 The above is equivalent to observing the waves at a fixed point as a function of time, and all knowledge of the direction of travel of the waves is lost since T [A(u, 0)]°de = [A(p)]? -T The procedures for analyzing waves as a function of time at a fixed point have been described by Pierson and Marks [1952], and Ijima [1956] has carried out quite a number of such analyses in Japan with very interesting results. The same techniques are being used by Lewis [1955] to analyze the spectra of model waves and ship motions in a towing tank. The wave pole records will be analyzed using the methods described by Pierson and Marks [1952]. If t' is chosen to be zero, then an average over space can replace an average over time and space and the result is mY Ze (62 (8.9) OYE 57) Ss than cate at n(x y) nlxtx', yty') dxdy X00 N IK [0 @) ah [Ax(a, B)]# cos (ax! + By’) dBda [o'@) In equation (8.9) the same right hand side results if -x' and -y' are substi- tuted for x' and y', and therefore Q(x', y') = Q(-x',-y'). The above is equivalent to observing the waves at an instant of time over anarea. Some knowledge of the direction of travel of the waves is lost. Con- sider, for example, a progressive simple sine wave observed at an instant 62 of time. Aline parallel to the crests can be determined, and the direction of travel of the wave will be perpendicular to this line, but the direction can be either one of two directions, one the opposite of the other. This indeterminancy is avoided in this analysis by considering a positive direction, x', to be the dominant direction of the wind and by assuming that the spectral components of the waves being studied are all traveling with an angle of +90° to fhe cin. Then [A(p', 9')]* would be zero for 1/2< §'< 7 and for -1/2<9'<-n, and [A(a', B')]* would be zero for -wm = N(k) N(k+p) k=1 eH OM wl haveneysikeles Q3 = Qo, Q = 2Q,, (p=1tom-1), and QU=Q,. (8. 16) il m e mph =e 2 Oo cas =e p=0 P AON Ms eG eirankaas ES 3 Let L = Li): and Lyy,-1 = Lm41- (8. 17) Wy = 0123 Ly 40.54 1p, + 0.23 bypass Hite 0; Uy) asco SB Define U* = U,/2, Up, = Up (n= 1 to 19), UX = U,/2, In the above equations (8.15) is the discrete approximation to equation (8.13). Also in equation (8.13), Q(t!) equals Q(-t') and to obtain the 65 discrete approximation to (8.14), Q(p) is expanded as a periodic even function about p equal to zero. Thus Q, received a weight of one, Q) through on receive double weight, and Q. received a weight of one. The values of L are the discrete estimates of the Fourier coefficients of the even expansion of Q. Due to the fact that the L's are only estimates of the spectrum since the series of readings is finite, they have to be filtered to recover a smoothed estimate of the spectrum in terms of the U's. There is, of course, another way to estimate the spectrum. The original series of points could be expanded in a Fourier series. Sine and cosine co- efficients a, and b,, for periods of nAt/1, nAt/2, nAt/3, etc. would then be computed. The quantity ae Baste oe is then a very unstable estimate of the energy at that particular frequency. A proper running weighted average of the values of cf would then recover the spectrum as determined by the Tukey method. The number of degrees of freedom (f) is a measure of the num- ber of values of °” weighted in the average and of the shape of the weighting process. The U's have a Chi Square distribution with f degrees of freedom. The values of U have the dimensions of (length), and U; as given above is an estimate of the contribution to the total variance made by frequen- cies in the range from 2m(h - 4) /Atm to 2n(h +4) /Atm.— The theoretical equations in this two-variable problem are given by + For h= 0 and h=m the values of U must be halved since one of the frequencies defined above is not applicable. 66 tv [b< Nd (8. 18) Q(x'y') = lim N(x y) Ux +x', y+ y')dxdy panied SY NE Y200 ae 2 and 00 oO (8.19) [A¥(a*, B%)]* + [A*(-0%,-8#)]? = Q(x!,y')cos(ax! + By") dx! dy! T =00 “-00 The analogous summation formula for the covariance surface over the set of leveled readings N5, is given by m-Il-Jql n-l-p y*_ yy* (8.20) Opa) =) = = ik itp: k+q k=0 j=0 (n-p)(m- (ql) qi -Z20, =LOsanGe a. ao ly OF Fo coe ye O This determines the estimates of the covariance surface for the first and fourth quadrants of the q,p plane (really x',y'). Since Q(p,q) = Q(-p,-q), the results can be extended into all four quadrants of the q,p plane. The function must now be extended into the entire q,p plane so that its Fourier coefficients can be determined, and the property that Q(p,q) = Q(-p,-q) must be preserved. This is accomplished by simply translating the covari- ance surface parallel to itself to fill the whole plane. As a consequence, the Q's have to be redefined slightly in order to weight them properly. The definitions are that 67 Q*(p, q) = 20(p, q) for p= 1 to 19, q=-19 to +19 that Q*(0, q) = Q(0, q) q =-19 to +19 Q*(20, q) = Q(20,q) q=-19 to +19 yi Q#(p, 20) = Q(p, 20) p=1 to 19 Q*(p, -20) = Q(p,-20) p=1 to 19 and that Qx(0, 20) = 5.Q(0, 20) Q#(0, -20) = 5 Q(0, -20) Q*(20, 20) = : Q(20,20) Q*(20,-20) =3 Q(20,-20) Thus points on the q-axis have unit weight (but since Q(0,q) = Q(0,-q), they could be considered as one set of values weighted twice). Points off the p-axis in the first and fourth quadrants are weighted twice due to extension a the -p quadrants, points on the sides are weighted once, (really 1/2 on four sides of the full expansion) and corner points are weighted one half (really 1/4 on the four corners). The raw estimates of the spectrum are then found from equation (8.21). 420 «20 (8.21) L(r,s)=—- = = Q*(p, q) cos le (rp + sq)] 800 q=-20 p=0 where pee MN Agee CO) s = =20, -19; 22.55 420. Note that L(r,s) = L(-r,-s) and that the spectral estimates have the same property as equation (8. 12). p 8 A check of the computations can be made at this point by defining the quantities, L*, as below. The sum of the 861 values of L* thus obtained should equal Q(0,0). Thus L*(r, s) = L(r, s) tos ae fly GA Gre, ile) s=-19 to +19 and L¥(0, s) = 4.L(0, s) s=-19 to +19 L¥(20,s)=41(20,8) 5s =-19 to +19 L¥(r, 20) = 4 L(x, 20) r= 1 to 19 Le(r,-20)=5 L(x,-20) x = 1 to 19 and L*(0, 20) = +18, 20) L*(0,-20) = 4 L(2, 20) L*(20,20) = + L(20, 20) L*(20,-20) = 4.1 (20.-20) : Also, in order to smooth on the line r = 0 and on the edges, the values of L are continued by the following equations. L(-1, b) = L(1, b) a 0, Ie, eeeo 3 +20 L(a, 21) -20, -19, eoeey 0, ooee gy +20, Hi] L(a, 19) b if L(a, -21) = L{a, <9) L(21,b) = L(19, b) L(-21,-21) = L(-19, -19) L(21, 21) = L(19, F9) The smoothing filter is a #traightforward extension of the smoothing 69 filter used in the one-dimensional case as shown by the following scheme where the product of the two one-dimensional smoothing filters give the filter values over a square grid of nine points. Table 8.1. The Smoothing Filter On2a3 0. 54 Oe 2g} O23 0.053 0.124 0.053 0. 54 0. 124 Onaee 0. 124 Onis 0.053 0. 124 0.053 The smoothed spectral estimates are finally obtained from equation (8.23). (8.22) U(r, s) = 0.053[L(rt1, stl) + L(r41, s-1) + L(r-1, stl) + L(r-1, s-1)] + 0.124[L(r, s¢1) + L(r, s-1) + L(r4#1, s) + L(r-1, s)] + 0.292[ L(r, s)] where again U(r,s) = U(-r,-s) and r= 1, 2,...., 20 The U's are estimates of that contribution tothe total variance of the sea sur- face made by waves with frequency components between 2n(r - 5/40 Ax and 2u(r + 5) /40 Ax inthe r direction, and between 2m(s - 5) /40 Ax and 20(s ee Ax inthe s direction.t One difficulty with estimating power spectra by these techniques is chat the operations on the original data described by the above equations do not guarantee that the spectral estimates will be positive, and yet in the thecry they should be. This is because a term of the form sinagX f sin BY x W operates on the spectrum in the complete derivation when X and Y are kept Gog hac ee a ae Except at the borders--see Part 11. 70 finite in equation (8,9), This term can have negative values which can make the L's and the U's come out negative, The L's in particular can be nega- tive quite frequently because of the operation of the above term on the esti- mated spectrum, The purpose of the smoothing filter is in part to eliminate as much as possible some of the negative values, Usually the negative values are quite small and do not materially affect the analysis. In time series theory in general, the spectrum is usually defined so that an integral over a given frequency band represents that contribution to the total variance of the process being studied made by the frequencies in that band. In ocean wave theory, another convenient way to define the spec- trum is so that an integral over a given frequency band represents the sum of the squares of the amplitudes of those simple harmonic progressive waves which lie in that frequency band. This is the definition used by Pierson [1955] and Pierson, Neumann and James [1956]. The E value thus defined is equal to twice the variance of the process under study. Equations (8, 6) through (8.14) and equations (8,18) and (8,19) are de- rived with the definition of the spectrum used in ocean wave theory. Equations (8.15), (8.16), (8.17), (8.20), (8.21) and (8. 22),have been derived in terms of variance. To place all equations in terms of ocean wave theory equations (8.15) and (8.12) should be multiplied by 2 on the right hand side and then all results would be obtained in terms of E values. In what follows, all results will be discussed in terms of variances and covariances as far as the directional spectra are concerned except that il when the U values for the two independently obtained estimates of the spec- trum are added together to get the best final estimate, the results will be in terms of E values. Degrees of Freedom In the single variable case, each of the final spectral estimates has a Chi Square distribution with f degrees of freedom where f as given by Tukey is determined by equation (8, 23). a ph Ne (8. 23) fa eae) This result is obtained from rather complex considerations of all of the operations on the original time series which have led to the final values of the U's. Inthe case of an electronic analogue analyzer, the procedure is described by Pierson [1954], and the results depend on the shape of the smoothing filter, In the two variable case under consideration here, the smoothing filter is known only at 9 points as given in Table 8.1. The values of U(r, s) for a particular r and s is a random variable with a Chi Square distribution with an as yet to be determined number of de- grees of freedom. When U is considered as a random variable the degrees of freedom can be found from the following equation. _ 2(E(U))2 _ (2 Wx)? M,.My (8. 24) f E(U2) 4(ZWy,)* mm y “I Pe) In equation (8. 24), the W's are given by Table 8.1, M, is the average number of x points, and My is the average number of y points used in com- puting a value of Q. It can be shown that the average number of x points used in computing the Q values is N,~ (m/2) and the average number of y points is Ny - (m,/2). All of the Q's enter in each value of U. The number of x points used for the individual Q's ranges by integer steps from N, to N, -m_ with an average value of N, - (m,,/2), and similarly for the y points. The value of (= W,)"/4(Z W,“) is equal to 1.58, and hence the final expression for the number of degrees of freedom is given by equation (8.25). Nx] Ny i (8. 25) ii = ll, Sys el ES Equation (8.25) may underestimate the number of degrees of freedom. Instead of (N - 4) as in equation (8.23), it has the product of two terms m Bd yd é, ! mips) and { ~3) and instead of a factor of 2 ithasa 1.58. The values My 2 my of Q near Q(0,0) are much larger than the values of Q on the edges, and therefore values of 1/4 instead of 1/2 might weight them more properly in equation (8. 25). A Correction for the Wave Pole Spectrum The wave pole used by the R. V. ATLANTIS was free floating, and its dimensions are shown in figure 8.1. Therefore it probably underwent a rather complex non-linear motion in heave, pitch and surge. If the motions can be linearized, the heaving motion is the most important, and the pitch 13 and surge can be neglected. The heaving motion in response to simple harmonic waves of amplitude a, can be described by equation (8. 26). (8. 26) Mz +f£z+ pgAjz = pgA; [D(u)]a, cos wt where Diy) is determined from the wave pressures on the horizontal areas of the wave pole, and M includes the added mass of the water set in motion by the moving wave pole. di) is given by equation (8. 27). Beda 28 = A le 30p2 ha iceeres Remi ee (8. 27) Diu) = e +a e F Teh aa(are ae where Ay: A, and A. are the cross sectional areas of the top, middle, and 3 bottom portions of the wave pole respectively. Note that as yp approaches infinity d(.) approaches zero and there is no force on the wave sable, As ~ approaches zero O(u) approaches one and the wave pole follows the wave profile exactly. For this particular wave pole as shown in figure 8.1, (Az/A, = (6/22 5) ; and (A3/Aj) = (i2f25)> 2 The function Q() is graphed in figure 8.2. Because of the greater magnitude of the areas of the larger submerged tanks and the rate of change of the wave pressure with depth, a wave crest actually produces a downward force instead of an upward force for most wave frequencies and this force is bs times greater than that which would have been produced by a very long wave acting on a pole of constant diameter Ay: The wave pole was calibrated in still water by measuring the period of 74 STILL WATER -p———_¥ REST POSITION FIGURE 8.1 CAPACITANCE WAVE POLE USED BY THE R.V. ATLANTIS -75- $(#) SGNO 24S € SGNOOD]3S ¥ 18 20 22 24 26 28 30 EXTRAPOLATED 6 S@NOD3S9 14 12 SGNO9049S 8 10 SGNO04S 2 FIG. 82 THE FUNCTION 4%) oscillation and the damping. The resonant frequency was between Ko = 2n/41 and Moe 27/42 and the ratio of observed damping to critical damping was 0.16. When these calibration values are used, equation (8.26) can be put in the form given by equation (8, 28). (8. 28) a + Ste + z= 25 Mle) cos pt Ho ° in which », is 24/41, The true resonant frequency works out to be 217/41.5 (a period half way between the two observed values) and the damping is 0.16 of critical damping. The motion of the wave pole under the above conditions is given by equation (8. 29). (8. 29) Pi [1 - (w/t ae Qi) cos pt i K(n/p,) 2. Olu) sin pt [1 - (ele )2]% + KA)? [1 = e/g? 14 + Kg)? in which K equals 0. 32. For ». corresponding to a period of ten seconds, the coefficient of the cosine is positive, and the sine term is small compared to the cosine term. Therefore the wave pole will move up as the crest of a wave passes and the height of the recorded wave will be less than the height of the actual wave. The forcing function tends to force the pole downward in a wave crest, but the left hand side of the equation is so far past resonance at the high frequency end that an additional 180 degree phase shift is introduced, and the wave pole moves up in a passing wave crest. The height of the water on the moving wave pole was recorded, and the “Spectrum of this function is to be obtained. What is desired is the spectrum of 77 the function that would have been obtained had the wave pole ween stationary. Let N(%) be the recorded wave neight and let | (t) be the true wave height. Then equation (8. 30) can be obtained. 8. 30) MW) = Me - 26) It states that if the pole were stationary, (z(t)= 0), Ue) would equal N (é) and that if the wave pole followed the wave proiile exactly, (Nit pe “| *(¢) would be zero. From equations (8.29) and (8. 30) the result is that a fea) patente abel Ns ams sets neta mY es seoe ao ey where D(p) is the denominator of the terms in (8. 28). Ii the wave pole response 2) presented by a stationary Gaussian process with the spectrum [ A(i:) ] 5 then follows that the spectrum of the recorded function is related to the spectrum of the waves by equation (8.32). The term in >rackets is 4 sum of the squares of the two coefficients in (8. 31). [(u/n)*- 1+ dw" + K*(u/p, itwtee)” 2 te +K Hie) (8. 32) [A *()]* = [A()]2 e of frequencies expected in the wave record, the numer- ator of this expression is always less than the denominator. Therefore the waves recorded sy the instrument will be too low and the spectrum computed from the wave record will have to be amplified to get the correct spectrum. 1 ¢: 1.2 : 14 a Se Dae ty tae oh tot es wl Si i539 2 The final equation given that [A*()]“ is known permits one to find [ A(z) | 78 ‘WNYULO3dS 310d 3AVM [OHM JHL HOS Vb cy 28 dls Ov 98S 9E HWE CE OF BS 92 He 22 OF BI YOLOV4 ) a 2 er | NOILOAYYOO Ol 8 9 v -79- Zee 2 UL yeaa fo )- 1] +K (y/o ) E(u /p ] a ass TneeNTe (8.33) [Au]? = [iw /io)” -1 + o(y)] 7+ K*(w/ito)* The form of the correction curve is shown in figure 8.3. The spectrum actually determined will be discussed in a later papt ox the report. The authors are indebted to Professor B. V. Korvin-Kroukovsky of the Experimental Towing Tank at Stevens Institute of Technology, and Harlow Farmer of Woods Hole Oceanographic Institution for the derivation, discussion and clarification of equation (8.26) and to Mr, Farmer again for calibration data, Prof. Korvin-Kroukovsky found from purely theoretical considerations using an added mass of unity that the resonant frequency should be near 38 seconds. Tucker [1956] has discussed the calibration of essentially the same wave pole except that the depth of submergence of the tanks is different, and the derivation follows his results. Tucker's results show that a change in the depth of submergence of the tanks by a few feet one way or the other changes the results markedly. 80 References Cox, C. andW.H. Munk [1954]: Statistics of the sea surface derived from sun glitter. Journal of Marine Research, v. 13, pp. 198-227. Ijima, T, T. Takahashi, and K. Nakamura [1956]: On the results of wave obser- vations at the Port of Onahama in August, September, and October 1955. (Measurements of Ocean Waves VIII. Transportation Technical Research Institute, Ministry of Transportation. In Japanese--summary in English. ) Lewis, E. V. [1955]: Ship model tests to determine bending moments in waves. TSNAME, vol. 62. Nakagowa, M. and T. Tjima [1956]: "On Waves at Sakata Harbor.'' The First Sakata-Harbor Construction Office, Regional Harbor Construction Bureau and Transportation Technics Research Institute, Ministry of Transportation, Tokyo. (In Japanese. ) Pierson, W. J. [1954]: An electronic wave spectrum analyzer and its use in engineering problems. Tech. Memo. No. 56, Beach Erosion Board, Washington, D.C. Pierson, W.J. [1955]: Wind Generated Gravity Waves in Advances in Geophysics, vol), 2. Academic Press, Inc. Pierson, W.J. and W. Marks [1952]: The power spectrum analysis of ocean wave mecerds, Trans. Amer. Geophys. Union, 33, no. 6. Pierson, W.J., G. Neumann, and R. W. James [1955]: Practical methods for obser- ving and forecasting ocean waves by means of wave spectra and statistics. H.O, Pub. No. 603, U.S. Navy Hydrographic Office. Tucker, M.J. [1956]: Comparison of wave spectra as measured by the NIO ship- borne wave recorder installed on the R. V. "Atlantis'' and the Woods Hole Oceanographic Institution wave pole. NIJIO Internal Report No. A6. Tukey, J.W. [1949]: The sampling theory of power spectrum estimates. Sym- poOsium on Applications of Autocorrelation Analysis to Physical Problems. Woods Hole, Mass.(Office of Naval Research, Washington, D.C.) 81 Part 9 THE LEVELED DATA, THE NUMERICAL ANALYSIS AND THE NUMERICAL RESULTS The Leveled Data The first numerical task was to level the data using the equations derived in Part 8. Missing data was a complicating factor. For the most part this was caused by the presence of the ATLANTIS. The missing coordinates are given below: Data Sei 2 Data Set 3 eS is (Seen iials (39, 45) (2. ais) (40, 45) (39, 50) (41, 45) (44, 63) (42, 45) (45, 63) (46, 62) wnere j is the index running from 0 to 59 and k is the index running from Q to 89. With missing points in the data, there are two possible ways to level the data. One would be to minimize the sum of the squares of the deviations of the known values of the spot heights from an unknown tilted plane. The other would be to interpolate the unknown values from the known data; use them in the equations given in Part 8; and level the data. The first 82 of the two procedures described above was employed, but it will also be shown that the second procedure is simpler and that it gives substantially the same resuits. The original set of equations for the assumed complete set of data is given by the matrix equation (see Part 8); (9. 1) AX=B where A is the matrix: ae Dijk Sj (9. 2) Djk Zk =k Sj Zk Di and where 2 represents the double sum: 89 59 (9. 3) Sa 2 k=0 j=0 A calculation shows A to be: 6, 318, 900 7,088, 850 159, 300 (9. 4) 7, 088, 850 14, 337, 900 240, 300 159, 300 240, 300 5, 400 B is defined as the vector: zy Nik (9. 5) Dk Njk 2Nix where 2 is defined by (9.3). The column vector, X, is given by 83 ip : (9. 6) X =|b where a, b, and c determine the coefficients of the unknown tilted refer- ence plans. The equation for the leveled data is Nie = jk 7 aj-vke-c The modified equations which take into account the missing data are given by eqn. (9.7) where the summations omit the missing points. (9. 7) AUX = Bl! The numerical results for sets 2 and 3 are as follows: Set 2 Cee Ss Ain ce Oem sen ise), nee 6,307,157 7,072,663 15900mm A' [7,081,560 14,329,800 240, 120 7,072,663 14, 313, 780 239, 0a 159, 138 240, 120 Bal 159,033 239,892 5,393 767, 640.680 847,416,000 B! |1,184,012.640 1,261,367.290 2ey ales 738 28, 650.330 -.010744 + 001575 x -. 000140 -.003584 $5.256144 +5.425500 84 The final equations for the leveled data, Ni are given by: (9. 8) Ni = Ny, + 010744) + .000140k - 5.256144 ae (9. 9) Nic = Nj - -001575j + .003584k + 5.425500 The leveled data are givenin Tables 9.1 and 9.2. Data at the missing points were determined by interpolating the leveled data at the neighboring points. The preceding can be simplified by interpolating the data at the start for the missing points. It will be shown that this method yields the same re- sults to at least five significant figures. From equation (9.8) the missing data in the second run are given by: (39, 45) : 4. 830825 (40, 45) : 4. 820081 (41, 45) : 4. 809338 (42,45) : 4. 789596 if 1B is assumed to be zero at the missing points. By interpolation at the neighboring points, one could assume the missing data to be given by: (39, 45) : 5.02 (40,45) : 5.03 (41,45) : 4.92 (42,45) : 4.77 If one were to start the leveling over again, with the missing data tabulated above under the assumption that Nic were zero at the missing points thus using the entire 5400 points and the matrix A, there would be no change in the results. This is a fundamental property of a least square 85 solution. However, if one were to level with the interpolated data, the re- sults would be almost indistinguishable, because the vectors EB for both ures. . I ree to et least 5 signifi jence the solutions agree. Thus in a repetition of this problem, missing data could simply be averaged at the start. The column vector 5 for Set 2 for the case in which Nix is assumed to be zero is given by 768, 420. 609 B= | 1, 184, 879. 288 26, 637. 989 and if the interpolated points are used, B for Set 2 is given by 3 J wt B= | 1, 184,901. 39 26, 638. 49 The missing data for Set 3 under the assumption that Ni is zero at each of the missing points is given by ( 2, 45) +: 5.26735 (39,50 ) : 5.30770 (44, 63) + 5.26898 (45, 63) +: 5.27056 (46, 63) ; 5.27213 (45, 62) +: 5.27414 (46, 62) : 5.27572 86 a By interpolation at the neighboring points one could assume the miss- ing data to be given by cz hayes (prea: 5) Viasat os ta) (697 OD) se EE (445, (63) 25.28 (45, 63). = 5.39 (46; 63): 5.59 (45, “6Z) "+ “5.43 (465 "(G2)" 2 75.53 The column vector B for Set 3 for the case in which Nix is assumed to be zero is given by 848, 825.093 | Bee ees. obo. ose 28, 687. 267 j ; and if the interpolated points are used, B for Set 3 is given by 848, 856. 34 B= jl, 263, 558. 44 28, 687. 81 The leveling equations for the two different ways of leveling each set of data were actually obtained. The greatest difference in the two sets of data between the two methods was -0.02 ft which was far below the level of accuracy in the original spot heights. The preceding calculations were carried out on the Univac, a large 87 digital computer. The actual computation took only a few minutes; however the clerical workinvolved in correcting errors in the data and in the program consumed almost two hours. Were the problem to be done for 4 new set of data, only a few minutes of Univac time would beneeded, as the program al- ready exists, and the data can be made ready by a card to tape converter. Spe Ginn Competeon bye iat te Tae. The computation of the covariance surface is an extremely long compu- tation, involving many millions of multiplications. Thus it could be most speedily handled by the IBM 704, the NORC, or the LARC, Programming the covariance surface on the Uniyac was especially diffi- cult because of its limited memory. Since many numbers must be available almost simultaneously, it was deemed inefficient to store the data on tape, as tape time would add considerably to the program's running time, The way out of this dilemma was to break up the 90 by 60 array info three arrays: 44 x 60, 2x 60, and 44x 60. There is considerable overlap. Since we want a lag of 20, the middle group must contain 20 rows above and below; thus it contains 42 rows in all, These sare were packed 4 onaline, In this way, it was _ possible to pack an entire section in the memory, and still have enough in- structions for the program, Since no room was left for sign, a constant was added to all the data to make them positive. The reader mav perceive how these factors added materially to the length of the computation run. Every two numbers had to be isolated by an ingenious system of shifts; then the same constant had toa be subtracted out, Only then could the numbers be multiplied and the product accumulated in a counter, Then the calculation, 88 i & ’ t { of the covariance surface is accomplished by means of three programs yield- ing three 21x41 matrices, The sum of these three Paeeriieae canal Q(p, q)(90 - ta] )(60 - p), and a division will obtain the required values for the covariance surface. Since the problem is quite long, procedures have been established in case of machine trouble. The program can be restarted by typing on supervisory control the initial desired two-dimensional lag. The computer will pick it up from that point. A flow chart for this program is given in figure 9.1. The spectrum program did not present such difficulties because the en- tire data could be easily written in the memory. These programs were compiled by means of Generalized Programming, a particular system of automatic programming developed by the Univac Divi- sion of the Sperry Rand Corp. These programs for a general array can be found in the G. P. Library under the call letters AUC2, and COSM. They are available at the Univac Division of Sperry Rand, Inc., 19th and Allegheny Avenue, Philadelphia, Pennsylvania. To complete the program, an input-output routine must be added. This program has been ein It exists on tape at the College of Engineering, New York University. Further checking is deemed desirable before a pro- duction run is attempted. The complete Univac procedure for determining the spectrum from leveled data has thus been set up. After further checking, it could be used given about twenty hours of Univac time for each set of data. 89 ACCUMULATE PRODUCT IF jep=n > OUT ON TAPE IF p>T (max lag) IF Q>T (max Jaq) Fig. 9.1 FLOW CHART FOR UNNORMALIZED COVARIANT SURFACE -90- Spectrum Computation by Means of the Logistics Computer The problem was eventually run on the Logistics Computer, owned by the Office of Naval Research and operated by the George Washington Univer- sity Logistics Research Project. Time was made available by ONR. This computer is a plugboard controlled electronic digital computer with a large internal drum memory of approximately 175,000 decimal digits. For this com- putation a word length of 12 decimal digits (in reality 11 1/2, since negative humbers are represented by 9's complements) was used, providing over 14,000 words of memory, more than enough to store all the necessary data ait each stage of computation. The leveled data Nic were provided on punched cards for both data sets 2 and 3. Alater computation was made on Data Set 2A derived from Data Set by the deletion of all j from 50 to 59 and all k from O to 19 inclusive, pro- viding a 50 x 70 array; and on Data Set 3C derived from Data Set 3 by the de- letion of all j from 0 to 9 and 50 to 59, inclusive, providing a 40 x 90 array. The Nik were converted from cards to paper tape. Conversion and input were checked by comparing it Nix on the drum with a check sum of the punched cards. As each value at Q(p, q)(90 -[al )(60 - p) was computed, it was punched out on paper tape, ready for further input. Total computation time for Data Sets 2 and 3 was about 30 hours apiece, each computation involving 3,433,500 multiplications and 6,867,000 drum references. For Sets 2A and 3C the computation time was about 18 hours each. The computer was allowed to run Overnight unattended without encountering too many difficulties. Due to lack oT of time no check on the above computation was made for Sets 2 and 3 other than visual observation of the results for reasonableness. For Sets 2A and 3C, however, a check computation of 2 Q(p, q)(90 - [dco - p) was made, two P>q minor errors being discovered and corrected in the results for Set 3C. This check computation required about 6 1/2 hours for each set and would have re- quired about 11 hours for Data Sets 2 and 3. Since the Logistics Computer does not include division as a basic operation, this had to be subroutined in order to find the values of Q(p, q), a matter of a few minutes. This division was checked by repétition of the program. The values of Q(p,q) were converted from tape to punched cards for listing, the conversion being checked in the case of Data Sets 2A and 3C by comparing the sum of the Q(p,q) on tape with the corresponding card sum. The Q(p, q) were fed back into the computer, being doubled before being stored. Each side of the resulting matrix was then multiplied by 1/2, the cor- ner elemenis belonging to two sides, being multiplied twice. In this manner the covariance surface Q*(p,q) was stored onthe drum. 1.21095 cos 2 (j= 0, 1, -*+, 39) were also stored on the drum. The factor 1.21095 x 10-7 was introduced to divide by 800, to locate the decimal point (since, say, 0.311 was entered as 311), and to convert from the scale of the stereo planigraph to feet (0.1016 mm = 1 foot). During the computation of the spectrum rp + sq was reduced modulo 40 to the least non-negative residue. Because of the ranges of p,q,r, and s, resetting of rp + sq at the limit of the range of any variable was quite easy. The total computation time for each spectrum was 92 : about 7 hours. Inthe case of Data Sets 2A and 3C the L(r,s) were punched into cards, the conversion from tape to cards being checked by summation. In the earlier computation of Data Sets 2 and 3, due to an error made with derivation of the equations in Part 8, the values of L*(r,s) were computed and punched into cards. Asa check ZL*(r,s) was compared with Q(0,0) /(1.016)4 to which it should be equal. This check was made by hand for Data Sets 2 and 3 and by machine for 2A and 3C. In all cases there was agreement to four signi- ficant figures, the values for Data Sets 2, 3, 2A, and 3C being 4.614, 4.299, 4.105, and 4.049, respectively. For Data Sets 2 and 3 the final smoothing was performed incorrectly, due to the above mentioned error, onthe L*(r, s) matrix rather than the L(r, s) matrix. This error was corrected by the time Data Sets 2A and 3C were run, and correct procedures are described in Part 8. The L(r,s) matrix on the drum was bordered to provide the proper values for r=-1,21 and s = -21, 21. A final computation of approximately 15 minutes per data set provides the U(r,s). For Data Sets 2 and 3 U(r,s) (incorrect in the two outer columns and rows because of the use of L*(r,s)) was punched on tape and converted to cards. Many of these values were checked by hand, and no errors were discovered. For Data Sets 2A and 3C U(r,s) (correctly computed from L(r, s)) was used to vee U*(r,s) (the borders being multiplied by 1/2), punched out on the tape, and converted to cards. For both of these sets as a check 2L*(r,s) was compared to ZU*(r,s), agreement being obtained to seven significant figures. Ole Although computation time was greater than it would have been on the Univac or other large machine, the problem as done on the Logistics Computer was conceptually simpler, because of the ability to store ultimately all data needed at any computation siage, and more economical (even if the problem had been chargedfor) due to the smaller cost per hour of this machine. In conclusion, the authors would like to thank Louis Grey, Anatole Holt and William Turanski for the help and advice they have given in the Univac pro- gramming, Gordon J. Morgan of the Logistics Research Project, William W. Ellis and Bernard Chasin who helped with the IBM card operations needed to provide the tables in this report. The original spot height data furnished by the U.S. Navy Hydrographic Office, the leveled data, the values of Q(p,q), L*(r,s), and U(r,s) for the ori- ginal computations, and the values of Q(p,q), L(r,s), and U(r,s) for the re- duced data, are given in the following tables. (Note that in order to compute the U(r, s) values, the L(r,s) values must be used, and not the L*(r,s) values.) The tabulated values of L*(r,s) should be doubled on all borders except at the corners where they should te quadrupled to obtain the L(r,s) values. These values are also available in a deck of IBM punched cards at the Research Divi- sion of the College of Engineering, New York University. The raw data and the leveled data are given in 540 cards per run, for Data Sets 2 and 3 and 350 and 360 cards for Data Sets 2A and 3C, respectively; and the covariance surface, the spectrum, and the smoothed spectrum are given on 841 cards per run. Thus 11,882 cards are available in all. All Logistics Computer prov grams used are in the possession of George Stephenson. 94 Table 9.1 Table 9.2 Table 9. 3 Table 9.4 Mable 9.5 Table 9.6 Table 9.7 Table 9.8 Gable 9.9 Gable 9; 10 Table 9.11 Table 9,12 Table Jo ks Hable 9. 14 Mable Yo 15 Table 9.16 Table 9.17 maple 9. Us Table 9.19 Table 9. 20 Index to Tables Leveled Spot Height Data for Data Set No. 2. Leveled Spot Height Data for Data Set No. 3. Covariance Surface for Data Set No. 2. Covariance Surface for Data Set No. 3. Table 9.3 plus Table 9.4. L*(r,s) Values for Data Set No. 2. L*(r,s) Values for Data Set No. 3. Spectral Estimates U(r,s) for Data Set No. 2. Spectral Estimates U(r,s) for Data Set No. 3. Table 9.8 plus Table 9.9. Unleveled Raw Data for Data Set No. 2. Unleveled Raw Data for Data Set No. 3. Errata Sheet for Table 9.8. Errata Sheet for Table 9.9. Covariance Surface for Data Set No. 2A. Covariance Surface for Data Set No. 3C. L(r, s) Values for Data Set No. 2A. L(r, s) Values for Data Set No. 3C. Spectral Estimates U(r, s) for Data Set No. 2A. Spectral Estimates U(r,s) for DataSet No. 3C. 95. Explanation of Tables The decimal point is not shown in any of the tables. The units for each table are given below. Negative numbers in all tables are shown by an asterisk (*). Tables 9.1 and 9.2. The value of N(00,00) in Table 9.1 (the number in the upper left corner of the first page) can be read as -2.56 feet (approximately). To get feet exactly divide 2.56 by 1.016. All other numbers can be similarly interpreted. Tables 9.3, 9.4, 9.15, and 9.16. The value of Q(0,0) in Table 9.3 is 4.763 (ft)* (approximately). To get “RIE sxactly diidle L762 Ie (016) Tables9.6, 9.7, 9.17, and 9.18. The value of L(0,20) in Table 9.17 is 0.0013 (ft)*. All other L(r,s) values can be interpreted similarly. To convert the L*(r,s) values in Table 9.6 and 9.7 to L(r,s) values, double all entries that have 20 as a coordinate, double the first column, and redouble the entries at the four corners. Tables 9.8, 9.9, 9.19, and 9.20. The value of U(1, 00) a Table 9.19 is 0.0302 (£t)?. All other values can be interpreted the same way for Tables 9.19 and 9.20 in which the values of U are bordered. Note Tables 9.13 and 9.14 in connection with Tables 9.8 and 9.9 96 in which the corrected values of U(r,s) are not bordered. Tables 9,11 and 9.12 N(0,00) is 50.0 feet approximately. To convert to feet exactly, divide by 1.016. 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DCNONDACMNANOLEODNMCCHAHAMDOEHAAMMMIAD ID AHONNAHHNAHOCOOGNONN AM HOHAODANCOOOOOHOHOOON HONOAAAMNOM HAMA ANON 43 bis e snp eee # * Sean BF en Rt ont een CAHAQMTTNOOMNDAHODADOM ONNOFONHOTADE ODN TNE NACOCHOMM TON TE DOAONADE NOTTOTHARMATNONMCOMNMNMANATd TOMDDHNNOONATALNANDAHDODNAUNOOVAMADI Tt HAHONANNNHOOCOCOCOCHOOKOO OHOOKHVHAHNHOHHOOCOOHNHOOOONOHOMTAHOONNNONTON Leveled Spot Height Data for Data Set No. 2 (k 42 eee « * * oe *fe TBH HEAT * ee oe ee bee COAUVMN THT END AHOTONM ONN TOMUVHOOHKOLL ON TTOOAAOCAODM THN TMNDDAOAAOL + ATREANATONONNAMNs OF TH HN TAOTAHOMOL OLN NH TOHAOCOVHONANTTNNNd AAT UTE O CODONONHHOOCOCHO TK NOHOO AHHH OUVOCOVIMVAHONUANVHAANQMANANNAHOANNO A AMON Table 9.1(4). 41 ee eeae on « es 8 gene oe ee eeenhee ene COAKAM TON TNDDAOnANM OTN TEMNODHOONOKE ONT TE OAHOHADEOON TEMNOHOOAOM MD 2 MDMA TOANTOOANNLOATADAMOVOONODMATOOADOAAMOAAAANONONS OND ene SCHOD FH HOH HODHHOTHNOON HOH OVHTHHROTFTHHOAMMAVAHONAMANOOMNUNHOOUNAANMT HO 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HMAANANOONOOHHAMUNAMAOCOCOMO nHHONAHNNOONOCOHONOCONOO e S28 Been eeoen ee oe eeeea eon e ft # * ae WOM TMNNRVACOCHAMMN TOTEM AAAOTOAM TTWOMMAAOCHAE EON OOM OAROTHOAM THNNOr AN ROAOMWNOTOTNANGOROTTEATANANEHONNDOFNCNATNO RN ODODANONMTENATHAOAMHAOLO ONNNNMOHNONHOOCONOHO OKH HOOK THA THHOHNOOKHMHAHMOTHHOOCOOOMOnKAMHOONON #teteeene eee et # * ee eee oe ene « * OR TITNVHHOTAQAN TNH TNUDAOKHAMOTNHOMMNRAOCHAME ON OOM DDHOAANMTTHOreA BRroovaneeMNrOnrnwvOONOOATNOEAMTNTOLNMONOHOMOTNOTONOATHYOL HOTMNANNA AN TINA HHH AVAANOVE RH HOTMNNHOOKMODCOCOHOOHOOOONNOOONNOKRANOMNAHONOO eee Bene Ey tee * erenee oe #2 pene * TJOTINNATOARANTNN TMD ODHOOCAANESCONMOT HE DAHOHDODMONN ONADHODANMTTNOrE NA Reonme tor dOUNEMOCANNWAT NCDODNOOWHMOATAMEOTONADOTATCNOHRONAOMTAL CY ONTMNNHOOHAMMAAHHNODOOCOOMMNOHONDOHOOHOHOH OOO TR THOOCOMMRANMNOHOHO Ree nee aan hee * fT APRA HR # 2 enne aan TNOTMNNDHAOTDAUM TON TEE VUHOOTHAVELONOTEDAHDOHDDOLTONOMECOBHOOANNM TOTO Se mHnoMNrdnOoONVdHNOONA TON OAT TOHAUVMNNMNOHP ALOE OUTANADAROHONONM MOL Oo =) OMNMMNNANOOHANNHHMHODOHOOMNNDONAHHOOCOHONMOHOOOHNOHOMMNVHORNONDOOHO ee ee fF HR RH ee eee Ree Ree nee OO NTENDAOCKCANM TON T TM DAHOANAUEOCNHNH OC DDAOHDAODEOON OEE DHOOKAMMSNTTE oO HOF OTOH OHONWATDOTDE MO CHDOVATATOE UMN DAN AH TDODANAAAANOOOMAAM AEH OD AM TOOONOARNTMUAMANHHOOOOHNOHOOCOONHODOANODAHHONHONMATNONHOATNOON hee ne a ee # 2 Hn RR ne ee on ee ee ft TOW TANDANOCOHRQMNMNONTTNVAHOHNOLTNONWOECNDAOCODOLK TNT Tr DAHOAANAMONHHTE AV LRdqroana2nnroDdOTHANRAMOMOATE OF ONAHATHONEAMOOCADNOOTOLEMONANOOMO 67 68 69 187% 2578 014 66 65 63 61 OMNNAHOONHOANMN THHOMMT VNOOHOHNONDHOOHOHOONOHVNHOHOOMTMMNOTOHOMANOS eee e eee ee tae ee fe PERE RHEE * feet # Fee Ff WNOTNMNDAOCHANMMNONMNTNNVAHODNAOLTTNTEMNNAOOKOLETHTOLDAHHORNOMCONTEM HHODMATEMNMUNODAOHONONOOMNAADAATUONENOTUNODAMNAOONNANNNHAMAOMMO MNNOONNAAGMMVNAAMNTTOOONOHNHOMNODCOCHODCONOMNOONOHMOOCONNOOOOANNHO een et Ree eee eee HHP RE ET = ete Fe FH WNTTMDAHOHD OS ONNTMNNNAHODHHAUMNTTOTIENNAHOHDOL TONOEFDDAHOHNOL TONOME OAM NOL OOTNOL AHA DNNNDNUNDEWODONANK HAN THEE AHAAOOOHNANUMOTOOTEOFAtHHH MANHOMMUNNOOHOAHAUTMN HAA HOONMHOMHOOCOHANOT HANA AHHONNMMONNOOOOCHOOOKd * Ce eee * HHH HR HHH RE eee * WNWOTNMDHAOAHAOL OOM GCMMDHOCOCHAUMM TNHOTINNAHOHDDOSONNOFDDHAOCCHAMMTHOOM DHOMNM TN AT TOWN TOOMNHADOGAAVELOMAHAOOCEEMNADONDOLL ADEM OKETMOOWONONNHOO ROHNONNOOCKHH HH HONOMTHR OK ONNRAHAROOHAUAMUNOONNOHONNMOONAUNHONOOOHO 7 Po PRET Be te Heche eer eee eang a CTMONUNNDAOAHOL TONOM MODHODCAANNEWONNTANNNHOKDHADEOTNOEENHOOHAMM THO Th SOMDADHHROTNDNNMATOM TANTIOHNODOANN ANN ANENDAADDODOHHAOMOMMNHOUNADOM DCODKCCOOCHOOONOKHOHONS OANHHHOMOCHOTHNAHHOHOHOHOKHONTHVANUNATOMOOOd EERE HEH BHT ee fF te eee seen e * MTOWOTMMDHOOKDOME ONT TE DHAHAOHAANE ON NHOMNDHAOADAHODETTNOONANAHOANAM THNOM MNN ANS ONDOOAHAAALME ACM NOTMOMNTOAMNATOIMNGTTHOODONDTORHONNMNNM AT DOAN CODCHONHHONHHOOCHMOOO NOONHNOHONDHOHOOCONHAUNHOHONOOHNOMHONNOOONK oe ae een 2% HeRRE HEHEHE FH HT * gene * #8 MOON CECMMDNOOANMNO ONO OF DDHOAADE WOON TMMNANHAOCOKHOEMNTOOTNAURHOTNAM TON Tm OTDOHANANANAANMNAT HO TTT ATT NT TINONODEOMNMNDDOODOMTANHOOMNNATNNOTMAMOAAVE COON OCHOHOMMNHOANNON A HON THORONKMNHRORNUONHOOOCONOONANHNMOMNTOONON eon Pee ER RT eee ERE Bt an peer ae e ae CHRON OTMNAHODANDH-VNNO TF DDHDAHDOLCODNOMMNDHAOOHAANTONW TANNA ON AAMT OWT PNOTNCWOHANOLMNNECMNANSCOTDNMNOCATETOMCME TTTAHIMTONOANTODDOTONE TATOO DODCSOON DON TNOOON GANT MAMMHAONOSCGNAONNIOTHHONDANNOMAMNANAMNAUAAOMNONNOO e f f aT nant eee EEO nae * * Rt He on ME TNOTENAHORHVNOEFOTNOCEDNAOOHOEM ONT TMDHAAOTNOMNTNONTONNVHONANM TTI) TT ADOWONNANTTMNMNOAMHAMNAL HOLOMNATHAAMNTOOMNOHODOON TOONA TOMNVNOWHHMODADO HOODOLHOOCOKHOMMA HAN AHMMNTAHONNNOHOOKHAHHONNODOOCOHHOOOAMM dA NOOO ee ee RRR Hee ee ee ae £ * eee a ene NEON HOC MDHOTAMLOTHMOLCODAOOHOOELONOTMNODANHROHAANKTIHOMMNNHOOHUME ONTO WONOONTAANNAMNMOK AAA DHOOMNHOUWONT TON TOTONTON Ts ODNNNEENOMNHOMANUd Ae ADODOODDDCOONONDAAMNdHOO NMAMAHONMNAUNAHOOHHMOOH HHO HONK HONMT UAH NOAANO eee eee te ee eet He Re e Fo RH eroge Renee NE OWN TMNDDHOOHOLL ONY OL DHAAONDOLONHONDVAHOAADEVUTNOTNMNVNHOOHUNE ONIN T WADE UNAM VATANAVONNMNDATORHHHMNAMTHONO TN TNONTATMN AMO TO AMNE Ee AOeh AOHOODDOHNOOONTNOON TANHHANMMOHOOHHHOOOHOMOONMONNHONATTTANMUNNO CANM TNOLDHAOANMNTNOED DAOANMTNOLFDHAOKANTNHOFDAOKNMN THO ODHAOANMN TNO OH SCOCCKOCOCSCSCOCRM ddd ddA AUAUNANANANAAANMMNMMNNMNMMN Se oe TTT INN InInIn nin 101 Leveled Spot Height Data for Data Set No. 2 (k = 80-89) Table 9.1(8). 102 88 87 85 83 e eeeenene 28 aon enane * 0088 ane THOATKODDOLONN CTMNDDHOCAOMMON ST TMNDAHOCHNAMNTNONOMANHODRANMTTNOEEUNHOO CANDO TH CTOOHAONGTOTINNAOLODAHHOOCAMM TH TAOS OMOWONAHNNAAAMAODOOAADOUAd COAMMO HNANN AO HAOCOMANAMAMOATMANOOONHOAMAMAOOAVNAHOCKHMNTOWOMAOOANOO * enennee ee eenee eennee aeeeee THODDAOAADL WOON OLE DACOANMEONTTEODAHOANAMTTNOTMMNVNHOOHAMNMTNTINUAHO MWUWOL NMOLOLONBHATENODNAME TAHIMHOMOL TANT TONOMEOMNODMNAMOMODDODNNDA ONNNS OCANNTNOHONMMANOANANONTNNONOAAHOTHONOONHONAHOTANNOMNTOOOTNHO ee eeenne eeee se eee enee eeennee 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WT AODMMMNOANTONAADONMAOANMNE DOAMNTODHANONMHOUMNNWDOHM TOL AADONM WO NHOWATEOONLMAVUHHOTHDOTMHANEONTATTRHODOAMOP ALR E TE NEEL Or HE ANorMHdH ANHUMMANOUMNOHOHOHOANNOOANTNOHTINAHAHHHOONDOOCOMNOGAUNTMNANANUNOCHONHO ° eee ee ee Ree Bee se * eeeeeeetane 2 NAADONMAODENTDOAE TONAHHANONEHODMNODOHEOTMNAHADONEDONMNNODAAE OTM ATAMNEAOVNOAENONODOTTEOODANHOONNNAUMONNMDNHNNOWKDAHHONHHOHTDAOTAMHOAMNM AOCOCOMAUNNOTMUNUMAANOHOOCOOCKHAUMAHOTMHOONOHOHOOCOTHAMAMNITNANOOHTMNO * * 88 ae ef hee ee feeeeeeene eee ene ® AS TITMAADONMDONMN TOON TOMAAUTNMNDOODLNTAAAMTOLHAONTHOMOOAMNTANRALY St TODODODTOTTONTOMMAONEMNTDAVNAME ES ONANTDNDHAAHOAMNMOCDOTNOUTAL NHL ENS NHAONNONNOMNOHOMMMAHNOCCOHNAHOOHNOHONHONNONOMNONOAUNNHORMORTTOM o he ot Ad * eeee & * eneee Bee ee eee fe oo OTNAAEH OOM HOVONME NONE NTONAMOOMHONDNMNOAMNODAANTOLHONMNE-DOArHNOM MEV AND TTODAMNODCDOMNANRMMNAMONUMOL AL ANHHKONNT THO TOHMNMMHNNNALANATONOCN TMANDONHAAMAOOHM HOH HOTHOOKN HON HOODOO OO KR AMNMNOTON NAAR OOKHOTMMMN Sra ae, O ee) RE OR een Ree eane eee 88 oe NOALMNONAAEOONHODONENOAMNNONHDADOTMHODE NK DOHMTODHAHDOOE AOOMNMNAOK a YAUMAHOTUNMONAN THOMANDUUVUOTNAAHANONMDAVNANA AE TOONHOOCLOAAUTENOAD AMNMNVOONTHANOOKHONTOHOOOK HH HOnK MH HHOOMORTMNMNNNHON nH HOHOORHHOOOKHONG ee « en fe Pee eee « eee eee Ree ee eee * fae DENENOCAMHTNADDOTMHAODMNOE DOA LT TNHHDONMAONMNHODOAMNTTAUAANTNONAOUMN ATOAENAMNMNONMHVOOONON A AATMNAL OTHE MNAMNTTONHADOAEDHDOOMNNHONNWONT HAI ATNOOHOHOONMOOCOMNONONOCOHOHOHMHOOOCKHNO CTE TOANHVOHHMOAMMONCOMY en 8 eee Fee ene en eee een ae on eee * NMAODMUMDONEOTUAANTOEHAOUMNODOAMNTTMNADDTNENDONMNONOAMT TEHANONEY NAAUKANI KS MAUMNDNNME-DOMNONTODHODONODDTAMNAMEOTTAHVHODHOTHOMNAHMNNOLHAUMO TITOAHOOCHNOHHOONTHHODVNHMONOHOHAUMUNNOOOTNOMNDNUNNHONHOOCOMNOHOHOO SAVMITMWOFDHAOHAUMTOOFDAOHAUMTNOLDAORANN THOLDHAOKAMNTNOLFOHORAMNTMOr OD SOKOKKCCSCSCSO Md ddd HHA UUNAUANUNAANAMMMMMNMA MMM Cee TOM in 103 Table 9.2(2) Leveled Spot Height Data for Data Set No. 2 (k = 20-39) e @ ae e ee eaoeeaaa e e a 2 eeaeae 8 888 eee oe DMDOAL WH TODAANTOLAONONENOALNTUAADOTS AODMNE DOALTODAAOT ee RR COUT AON NANT DON AN ONOTOOMOTAVNONANTHOONMNNMAATNOOHAROMNY NOONO AA dOA TA HOAMOCOHAOTAMAD TH AAAOTOMAMOOHAAHOOHANOOOCOONONMOMOT eee @ ° ee a8 © eo @ ee aoe eo oon HAVE SE AOALMNMAOAMMTDAANE TNAODMNMAVOALOTAAADONMACALMTVOAE OTA DH AACW dT ONDOALOCOLAAMALOMACHAOMADAANLALHL TOMOEOAMTAAMOAVAAM UNA MOM AOO MH AANTUOAANAHOCOOOOMNOMAONAAAMMANOONOOHOMAMAMAOMO AUIS rir een « one of ee ene oon ae oe Ne COAHDONM RODE NE DORMOTANADTOMNHODEMTDOTMO CMAANTOMANOUMN TUOAL To SOE ROOT ONDANN NNT TAAVOTNANAMALONMNNDANDMMCDAKOOCOUR AINE EOAIUAS OCHMHOOU AO HOC HOCOCOCSCOONHASHONOOMNONAOHOOCOCONHONNNONANUNOOR ANNO MW eenaae . ee of 2 aeae # 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of a! oueor ca woncsanaronsnesndaoalonecnnimanecwuanearwananserrond 22 ATH UNTOHNHAHOMOOCDDDLOATNNTNHANDHAAON TENN dd ME ONAANE ME NoATMANOrEOM AHONCAHONVNVANTOONO HO HHANAOOKMNHAHNMNAHOHOOOKHHONMANHONNd dH ONVNHOONHANO * Seeeeereeeee se eee ¢ eee eee w ae eee ) COD NM DOALOTNHANTFNOEAONNN TDOAE GEMAAUNONMACAUNNTODOAMOTEAANTOL NOD DADOONONNOMAATMNMOONAMMOHMOMOMTHOTMOTAHNODNMAENANNTNONNOOAMOTN NNTUMORNTATTNOOCOO HAVO HOO HOOT OM AHOHHHOTOMMUHOOTHVHUMNVAAAHANAHOOO 21 * eee ne oe He HHRER HEH HEH HH ee eee % WON MAONENTNODLUOTOHANTOMNOOMNN ST VAAN TOLCHDANTNE DONMHONAAEOOrHODON DAHDANAHNMANOHODOODOL AL TTANAEMNTAHONOMNAHHOTVNOTONTMNOMNAL AAS TNEO NOD HNOHMATAUNTOHOOMNMNOMHONIOOHTUNAHANAOOOCOCOMMOOCOC HA HOTNOOHOTNHOOHN 20 CANMPTNOEFDHAOHAUMIT NOL DAOAUMT NOL DHAOAAMIHOLDHOANMTNOFDHAOHNUM TNO Do COCCOOC COCO HAHAHAHAHAHA AVAVAANANANANMMMMMNMMMNMNAT eT TTT TT TIN ININWININIn 104 eeeee GO @ ee e eeeaeoae ee eo J eee 6 FSO OT COE TE EDS IESE IG Tan at IGS EOP he RS AO IDE att lg ie eat ane pay he Ae VANE NAIMDOMOOAARONOOTNOEMNDOMNACEMEMNNDE OE NHNOHOTCNOOTHNAMA- TI TOR HOOHNNTFTOTHOOCOCOM OHH HHOHANMNONOMHMOOCCOON rH HOOCOCOCOOHOMMNO Orr MO ® e088 0808 ee ee eee8 ® 806 ee e008 @ eee NAANTERAOCNLNE DOAEOODMHANTNNAODENTDOHEOFTEAANTOMNAOCOMNONOAPOTM Ae Rea owOer nnn se 0OOT TN FOAEOAIME TIOMOTOO MEA TONOAAAANM ON DAAALO HOCOTMANNNOTOONN THOONVHAHAHOOMHOHOOHOONMOOCOMHAOCONOCONHHOOMHOONMHO MN @eeenecaene e ee e eeoeed @eeeeoeeeco 8 e a AP TODA RNENMHODEN TNOMEOOMNAADCOMODONMN TNH RM TOLAMNTNE DOOM TNO WMNTONOEONOT HAE DOMNAODMNOOAMEANMNMNTOME HON TNHNNNVADNNNOMEFONACOAAC Te HOCOTNHNHOTFONOCOMNAMMHOOHOOCOMNNNOHOOHHOTMNOMOCOHORHOOMAMMOONHOMONe eveeean oe one 8 @ eee eooenenenoen e Sp] 6 Gael 7 0 SCOLIOSIS EO AA ASE FO NO SS DNOTF OT HODOMDOOO TMOOEMOHOATHIMANOEMMIAOMOAHNOM OW TNANMOMMNAE dew o NMAHON THAN HANON THHOMOMOTHOOCSCOHMMAONOV MH AMNN ANAM MHOOCOCKHONMAHANHOOMO me eeen 88 eee eee ee eeeeeoen ee @8@ @08 800 enee WHNOAMNMONAAE OTE AODOME DORMN TNAANTOEHODONMDOAMN TOAADTOh HOD NEDO AMAOCNMAMDAUOANM ODNOCOMM TNDEOANNANONDOTNANNNAMAAMN THT TH AOMME Ud Oe AMATMHONUNHNONTHAHONMO DAA TNAAHANMMHOOHMNONHOOCOOCONNMOHOOCONHOHNOON eeee eeeaenee oe eeeaeeee eo *® @ @@ e eeene DOME DOAK MN TANHANTOPAOCUSMNENOAMNTODAANTIMHODMNENOCAMTTNRANONEAOCNMWN Row onr Done CHME MOO ODDO TOD HHHOHOODOMOTMTOHANTANHOTVOODOPMNATOTEE ODT HON MOHRONN HORN MOROHOOON TNOTMANMOOMH nH MOHOMOOHOOCOCOOHMOM MANN Or! eee ae eee ee * eoeeeen eoee eeoe e e tw eeane | MOPAONMNM NODE VODHAANY OE HODENMODO MMT IONAUNONEAODMNONOAMOOM AH UTD Oi WM QADANOMM SE TMCATNOAUNQMOATNDAOMOMAMONCWOCCACANOHT TATONNUMDONCNOLD ONANNMOOT RMN HOON COOH OORHIMNAONM ARH HOOOCOO TON OOM HOOnHHOOMHONHONMMOOO eeeneeen @ eeoan ee * * eeeane 88 eon ° e ef ene @ DANIO TPODENODOALYT ON ADANONMNHAONMNODOMMOTMNHANINENODEMONAMEOOFHOD, Bore aD tT DON NAAMAME MOO TM OONNAHODNAOTOWROE MTN TAMMON UMMM INO ot, NI AMHOHOHVNONTHOOHOHOHHONNOOCOONNHOMANNONHHOOONONUMNMNHOOONANHOnHO WO eee of @ eee Oe ° eeene ef eeeeen @ enee * MEOMAADITMNMDONENODRAR OTH HADONEDONENONHAMNET OS HODONMDOHMNCNAHOOTS BR monmon Wor vs ONC OD TTAMNVALEOMADANVOOMMANE AMIN T TOTOR HNO TMAOF NE ARNOOCOHHHHHONHOOHOOCOCOCO nA HNOCOHNHNOHOHNHOMNOOCOCOHMNANOHONORHOHOC ee eee vee eee eee eee ne eee oe 8 on eenae eeee e ee WDADAMT OF AONTIEDONMN FAAAMOTMNAOCDTOMNOTENONAHDOTMHONMNNE VON OTOH NAMM AS DHAANTNNOAUNTHAANOANADDOEMAVTTMOWVDAHAOCDNEADNVDONDAOTWOWHOMI HANNAOMMACNNMAAMNOCOSOHMNOT FMNHONMHOOCOOHNHNONHONVAAN AM MONOOMONOS ee ee eeeeennane eenane eee ° eee e eeoae ae ae 6274 FANN 0D NL S00) WY 6 PE NSA FD) Rh SN NAN > 0 FUE. PN AVNOFAMMNANOVAEMEONN FVOOHhHHNAAYN TAMA MEN M dA A AANA EME OME NAOKI OW OCNHOOMITMMANOONNHONCOCOMANNTNONNONHNOOCOHOOMHHHOOMHOOHNNOONOOKOOO ee a8 * of eee ee * ee @ eee e ee esne8 ee ° POENO TRE OONDAADTOPHONMNMNOALOTODAANOMMAHONMNGT DOAMNTTHEAANTOMACOLY GS DANANN AE ITMANOCMH HAN TN AEA TNMN FONE NMAANANAEEAMAATIASPMMMMANDON THON SCHNOM A COCOCHNANMFHHANOTFHOHOHNMNOMHOOCHNO NH MOOCOHOONOCOCOHOWMOnANOOnHON 48 eee @8 enee eeenene ee ee eeneonr ° eeeenean e MEACOMN MODOMMOODAHNOME KODE MN TNOAMNTIMAANTOEDONENMODONMTOMMRADOMMDOS FOTONO DEAE ANA TOF OCOFOHOMOMTODNAMOODOHOTMMME HNO THOHOAAOMMOAMACUA NUNVHON MOHOCCOCOOTONOOROMOHAMMNOONOOMNAHOOCOOCOOCOOHMAMHONNONNONONOMm ® * ene eeeeeane ee . eeeeon . eee eeeaeoeodr 21300 2. P AD ND 2), YP 6.69 NN. SND P60 A OB Dl I CHW HONONTHAOCNHOEME ODODE ODHAODOOPAHHOMUVANNENONANOFHONMNDOOArPMOnNM A HHOONHHNONANNANAMMOONOHHONMNOOMOOCHONO TH HOOCONNHONNOONMNN MON dr ries enveneeee ee * e een @ eeeee 8 88 eeseeneen PFONAANTNMDODKNTCDAAMOOEAHAONONME DOHMNONAADOTMHODTNEDOAMMNFVHADO TM NNMAMAANTAHENUNMONTHNOM ATT TNOTNVOVFDUFOADTAMAMNAMO™ OOADAMNDOMAM OU A NAAHANNTOTHNUNOCOCCOMNOCHONDOOHOOHMOOCCOHHHOOHNUNMMANTOONTHHOOHONMO e e ee een * eee * eene a8 eene ee ee e DAME COS MON TNMANOREMNODAHDOOMHONMNENORMMN TVAADOTEHONEMEDOAE TON ZeQVTOQOMMMMT AUPE HONNOATN HON AAMANNOT ¥MARONOMONM YANNAGCHNOAS NAROONAMARNANOONOOMMOHHOOHRODOONOONOCOOH A HOOONMNONTAHNONOMHOMNOOOD ee een eeene eeaneene . . oe ae se eff eeeeene AMM TODA ADTOEHODMMMNNOAMNTODAADONM HOD NE DOHMOTORNANTMMAHONMNODOMM O Dot e AHN VOMATE RAMON OT VUCHOVCHNDHOPANNCODEMHArWANONNOOCCCANONING COOHATNNAHNOOHNOANOOCONHHOOHOHOOOHOMN AH HON FANT ONOMOMOOHONHOOO Leveled Spot Height Data for Data Set Ho. 3} (k = 40-59) epen nee eee eee & 8H HOR RHE ® ee 8 eneae eeaneee a De DOAN TFODAHD ONE AONE N TDOMMTOEAAVONMHONMNONOMMTOr HAVEN E MONEY Sr MAANODANMOO DANN TOOONOOM MD OTE TH HHONTONAATNOMONVANME EO HMOM MNAD wn CHAHONNNOO THO HOOHOOHOONMNNNGOOCOCONDOHNNOOCOOHOHOn AAA AHNONONONOONOS . a eeeeenne e sn eene ee ee ° ee eenee 2 08 * a MAONMNODOAEOTMAAD TNE DONMN ENC HE OTE AADTNMNONENODANEOOMAONCWMNO SS FORNHOOMNNONHOATDONAN AM ODFNDANHDONVEDAOHVNOANONTOOVOAATNAM AA TOMNNON Li AQOMAAMMNOCOHHNANOOCHOOCANHOOCHNOHNOHHONONONHOONNKHOON NAN HOOOOCOMAN AH, * e e888 * ee ee seennee e NEM TNADEOETMNAONONE DOODLE NC NHHEOTMHON THOMNDOAMNNVOAHVORMHODEY WE MMANODANDOTODNHNDNMAAOOMNN FTNNTHNOHHODA FAAIAOMOODMANOMOMAT ert MPEANONHFNOOCOCOHOOONHOMO NH Onn ONTHT OOCOHOOANHONHOHOONAHOOOM 40 o928 1648 o35e 2178 066s of SANVMINOFDACHAMFINHOEDACHAUM INOS DACHAMFNOKDACHAMFNOFDACHAUMINOrF OA SCOSCOSCOSCHSSCORM HMA MAMA AAANNANAANAAKAMNMMMMMMN MMM eee TTL 105 Leveled Spot Height Data for Data Set No. 3. 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ANANANVNAHOOCOOAMTTHHAON ecsco9909C00C0CC0C0O000000 AOMNDNDOFONVDAHAE VON WOMNAMHDANTNAKROVONS AOS NAMNMMMAMNAUAANAHNAUMNTTOMNAHO ec9eooccoooocoooCcooCcoocoe NONDONMMoONrAODoOTMOMHAI ONVAMNDAMOAMMNNNANHAALD NANT Tee TMM MMMM tH coocoococcooooccoooo IMNODANHOA- TONE DOroDoN WT EE AANMMNAOADAARATON MMA THN NH NNT eT THHE HAAN S9000DD00CC COCO COCO Odd WATE THOAEHAMAMENAAAD AS DONAIME THONDNATDONK werewNoOoooOr Orr oOooraAmMom ecoooooc CoCo oCoCoCoCCoCoO AKAN ATMONDOOL TIMNNDOOCTOS OOD CNHOnRKOHONOFHOFTOTHOwOO NEMO EE ESE OOF OHNE ecooc0oCcCCoCoCoCCoCoCoCodANM CADB-ONMITMNAAOAD™ OMEN Addddddd tt HOOOCOOCOOCOSO 109 NTR THOTOUHOMEEANNARWO OTNNEATOANAOAADOMOHONE MANOOCHNNAUMNMNMNAOOCOCOddaea e9000000c00000000000000 aeatene aan OOMNADNOAOALHAOTHOOTME NAOMON TOA ANE OVOMANNO WHTNNHAOOKNHANHOOCOCOdeen lolo Koko fo lofololoKo lolol holo kololololol~) ereaanene aon WTONNOAMDOMNANDOAE AND OMWANOUNUMTALANHNDODHOOr NVTNNMTTNHOOCHAARHOOOOCOOO cooocoocoocooooooCcoooK0oe eeeeanad eneaaoe DAOONALYOONAAAHMNNADANO WNALFOTOTHAONNTOTONTHOM NHMeTTNNAOOKNAHHOOCOMHOO cocoocoocococooococooooo enenee onee WM TADAAOHAAMHANAAARDOSOO CH OEMNONNH TEATAQMAVTE AM ON HONNNHOCOOMAHNOOCOde HOODOO OCOOOCOCOCOOCOOCOCOCCCOO e TIANHTONOANATHEOOMNTOOO CO-MNONADODNMODTNAOANAAAM TORE LCVNANNMMAMMMMMEMNOO NHADOOOOOCOCOCOOCOCOCRANCNoNO QDDOO- DTOODUMNTrOOOMMWTH MOUDdATOTEOHDNOHORHOAMO TANHAOE oC MOCeeerroneN WNANAHHODCOOOOCOOCCOOSCOCOS AAMT NOE OAS TITTIES CANQMPTHOL-DA ecooo00c0o0oo pererrenve -10 fable 9.3(0) plus Table 9.4(0). Table 9.5 19 18 17 16 15 14 13 12 10 eenenve Ate SOR ORRO@ AL NH OLKLAADANOLPMOAE TOM TOHOHROELKALONENTHAK Gave CKHCOMODNNDOAGCTAO™ ONO AVIVA ANCOKAAAMNMHANDATANHSCGA AA AKAOONMMNT TOWN TT HUM TAHOAMNOOMNAHIUM TRA HAMS COCODDDDCCOOCOO CSCO OOO SS ODOC AD0000000000009 eeeneenn eaneeeonre AONANK ANOLE VADYOCHBDWCIVADNMONODOLKDOOT YC CUA RNOMMANMNADEKOANTMMAMNUOTHOLHOVANYNATHDGHAOD 90% MMA AQMMADOAAAIMMAAANMNOGNSTAHIAMOL LOM AHS Od efet of GO COdDDDDTDOCOCOODOOOCOOSGOONS SCORN SDOCOO OC oCOOOSSG OOH eeeonae ® 82eansaere0ea AHOWODNONVDH HADTONDOOULRNADNOOCURRHOMADOCALRMMAN TOR NOAM €ATONOTAHMNUIMMNNDATDANCHM ME AAA OMAQH TN TMM GNA HOCCOCOCOOKMMNT TINAONTOOAMOTNORAAVKAARA COCKCDDDCOOCOCOOOCOOOOCOCCOOOCOSCOCSCOOC CCC CoOO OSS eeeoneataneoneaaead eeoenoeneaeng OE COMAMOHEOTAUOAMAAOADOAAONKRODNAMAAALAL AMS ASTANE NE ADOTTEHAANMN TE HONK EMAOTNEAONAATNNS AT EMAKAANCKAMIMMAANHOAMNMNMNVOATOHAAHALPNMNAOOOOO ecoocecocccecocococcoecocoececeririgooocseooceo aoeee e2enean ae ao eneseoeecenes sen MME ANAM TAADANE EM ANAUNATTY TOPOOMNAEOONAVAUMAD MAOCANTALEAN OWE NOTULFOFOMNANDNDAAOCNNE TMAWADOWW CAND OOCOOCOAAMMMNNHHOONTOONTHOCTKAADOM-TNADOOO COCOOOCOCODOCOOOCO SOO COOOOSCOCO COO OOO OCCOCOO oOo eeaeneeaes e9eoeaneeonecegeae PTOSKDAVONTOVYODANOOMNVOE Edt HONMOTAMHAOMNMOHOLO NUOT OTOL AMM TTRVOHAOVMOOCATALANMODNOM OK VONHOOT AO COCOANKHHOHTHUMMMAHOOMODAArMNMOMOREEOOEMU HAN 00000000000000000000000000000000000000000 oeooe eoeensnneene AAMNE OHRAKRROOMMATOAADOCNOMNONNADAANADAUOMEAOO O-ENDAMOTTNODN ANE ATMOS TNDNOMHOTHAMENAMANNANA MAMATNOMNM TVNIOONNACDOHHMUOEACHHOTHMMNOFOrCMTMNe Se COKDKDDDOGCOCDOCSCOOCSC SOOO OCOOCCOOHWnOG OCC OCOCoOC OB ooo o0o e@oaw eananrnneove WON DO~ FANNDOTNVONE LS FHDAADASOMNDONODANMNANTTS ON NAMONNWNNONE TADNTODONHDOPENADNTOONL MMT OOM MANNAMTNTNODOOCOCHOANTHEAAADONANTNOLCOLNIN]S eoo000000 C00 0C000CCCCC CCC 0C0CCSCCCO COC OCCCSCocCo00 oe aeoerneene eeecaeaevonoe AONAMNDADOMNVOO-NTOCTTHNONHOENAOAAHONDOMOS ONO MOCOOMNNAOCH-NHNANDADANNONNNDOMWAR™ OE NOONTANOLD CODKCOCAHAANMNHOOCOCSCOHANHONTOECEEOOMAHOMTOrOONM| foKoKoloho koko loko No fol olokoKo ko ololoololola loko loko loo foko Nolo feokololo fo kol—) neon eneenane aunt onee WOATNNTODIDHOOKDHEANTTNONE TONMOOT OTE NOVI dd AW NAME ADOADOMNHONONROEMATAOFOMUTOODNAAHMEDONNA SCOCSCKOSCOH RAH HOOOKAANMMNTHONNHEDANOAOTHOUMMTMAN SOCOKDDCCOOCOCOOCCOOCOCOCOOC COC OCOCCOO NM OOCOOC OOOO ooo eee eeee eaean ADAMEMANDOMONEATNAO-NATDOOUANDA-NOTAMOWTANO OTEK NOHMNMNTTAMNNOOLOMAMNTMNANTHOTNDANNOrFOr NED SCOCOCKCOOHAANTTNNNOOCMnDAHOMOHAANAMNODNMNAGCOOOOd SCOCCOCCOCOOCCOCOCCOCOOCCOCOCHHnHn AHO OCOOCOCCOSG gee enae enone DRMAADMNNAKAKHOTDOMNOMNEHOLCHDOVDOTTHNONATAUALPONOMO NOU NDDOOTONTHOOMHONOCOMOAN Er ANTAOD-NVOME OW HOD CHOCO THOOTMNNHN TNOANTTTNONOOAANNHOALMTMMNVHOS SOCOCDDODDOOCOCCOOCOCOOCCOCOCCOCCCOd nnn AH OOODOOOCO0CG 2 seonenant seen ageee WDDADKHATHOMNEFOHAANDEVAHANNDNAESENNODONNMNTNOW PTONATADDANMODADNNOTANATANTHETANODL ANTE ONAOT COSCCOHAUNNONTNNNNANDHAALETANODAROCAHADLOTMMMANAN CODQDDDOD ODDO OO OOO OOO OOOO OOOSCOnHOGOOCO9900050 one eaeneue ne AL NDANNAANNDODNANAMNME ONE MMNNONM DONANATTRHCTOD AVDOLPO FOR RMEAMNANANTANHOTHMATTAIOORODAAL ACH TON ANMANAOODCONWDADONNEOAAL TdT Or DODDONMMM AMM CODDDODOCOSDOCOCODCOOCOCOnHOOOO OOOO COC OO COOoC ooo feo ot eanneaacene NOK DNCCDNAAGNODNNEKOAOWANNOMAOTONTIMMNAANS dE TAN ONTO TNTONHOTDOMM NH AANKARM ATO ODM TRAD BOANOOCOAHHOTEAADT HWOUNAALTVONTNONTMANUUA COCDDDDDDOD DOOD OO COON nHHHOOOOOOD ODN 9DD90000000 eae eepeeeoren eae eoneane AMONDO OAM FOONRN AN MOAMNAOTNEONOLATNTOAMA TAR TON ADOMNONNOLE ADOMATARNOTHAHOANMNDAMANATTO NANNNNOOCKHHAAOMUOV DADO AMAMNNTNAOL TNO HAOMA AA COCOO DOO 0OD0OOCOO COO ddd nHHAHOOOOOSDOOOOO000 eepeoueea eae eeaneee WONT MMATNOVMNOOMONOEDTDAQMMAMOMO TMM A & OAM DN MOT DADAANAIMNACOLEAENOANTEMAMOCONDOOVOWMN Od HM T TMANUANNDORAADTOMAMNM TS VOL TNOSKCOCORAMMAN CODCCOOCO OOO OO dd HH OOO Odd Hn A AHOOCOOOCOOOOOCO eeeeeeeaeneeeeereeeee OF DAOMKOMTODNVHAOM OMT OMNME ANTE AN TOD OAM OF AAMATEONANMELANONAOATDONOADATAARHANOAMUM TOO NOOMN TEE ETTOLAANNONATNAAMNAMNADUMAOOMAMT TN COCCCOCC OOO OC ONAN AHH COCHHHAAOOCOCOOCCOCOC eeneeeeeeenegeeeon MOKANNHINTHAAN ANOTHDOMNAMNALATVATANAMOLMONM AADOTHOTAT TOMAR TH TOOAMDOTTOMANN TOMMACT HOW ONNHONN ET TET TODAVOANNADOTARTAODUTANOOCOCOUMM YS COCCOOCCOCOOOOOOKnMHANNHOCSOOHHADOOOOOCOOOOCOO eanee e ANHEDOFOMEE EHDA HOON TOLOUMANAOTNDAAATDONOTOS AN dE OAD TONN OR ODOLREAMVANUMACDRAKNADMOENIOANO PO OODDOLL OPS DOAOMOLOL OOUINANAAOOMAAMM TM AOd COSCDCDOOOCCODOCOMHAANY CMANHODOOCBOOSGOOO000900 AANDNANNOMAOSEACALUMAMAMNMNEATAS TOAMOMIAN OAK MADOTMATANAD-OANOMODODOMOM AVE DANATAMTDOAM LODO ddan AAA A TAAQUAVNONQNONMN AKAMA OAD COOddddddddddddAANTOAOINAN Md OOO CASK ON ETMAHOS AO™ ON ENATOMAMTMOLOACAAMTNOL OAS Kader eter ie tOOCOOCOCOSCOCOSCOCCOO Mad aaaddeal dag (a0 abt.) Anni teeny L* values for Data Set No, 2 Table 9.6 eiconinn anise Dee : oonn aS AAOHM ANE MNANOMHONAK OM oe 9°°900000 AAVAMMTOMNA THN TANNEMMANddoooo OOS, MOO00900D00 DOO RB0CCC0COC0000 sses Se°0c000000 92000000 0000000000 00000000000000000008000 peels aeiee eeneen e#eeeesen nwowWw YAy OO NAD-NNODONOOt TE NOF VOR EA SOS) HT OONDOOEUTOAOMANAr TY ONTNOMANTOOS 20000000 00000000000HD00000000000000000008 el29900090®9O000000C00000CCDO DCG G GG GGGCOCC000 eeee. on ee @@f AMONG HAN OLNDONTOHATUAHHAN abit hethd bat -NOUOMA OnOoHAHO TNE WNNENOOTEMEANNALNT HOT OONNOOeTS o rp SOSOLVODDO OOO CCCCOHOONO CHOCO CO0G ooo ©0000000 ccCCCCO COC COCCOCCOOOO OOOO OOOO O OS eee eee ee of ArTITNAANE NATONOWMOMANAGCANDANAMMOMD i SCACOCOTNOMN TE HONE OLANNHOMNTMNHOOTOOHNd A 99000000 COO CDDD0COCOHOOO00 ecccecccccccooccecoococccce eeee e006 AANUEMNA HoddtoOoOnO S°O090000000000000 eccccecococoocce eonetaennne eeeee A~NCOATME OMELLFOAROTNE OL oNrvaanm © COCOANNA THN MNOTTODOMNNDAANONORRH FA o0000000 ecooooooocoooooddHOoHO00 S99000000 COCKCOCCCGCGCGCGCCOCCOO00G eeeneae HOTONHOOTAMAA MOTOTNONHOOCOHO Soococ00000000 SO99000000000C0O eee 08 eenn CONTONENTURONEONATEMTNORARONKHS AHOCONNONTNHOOE OME AVOTOdTOAOHHNONY a ecoocoocc coo 0D00CCC ODO OCOOCOGCOG06 ©e0000000 Sl[Ss990099090G00C CCC COGGoGa000 eeeeeeon AAOVNAMAO AANDOOKG eooocooscoe 90000000 #e8e sa08 ee nee Otvd- oor ANNOMOMAOAMMATONONOE nMwnno COCCCHNO MN OEE TNOARUMHHOOCOOMRI GION oeo°o0000D0 SSOV9SSCHOCOCOCHOnHHOHOOGO0000 eo9000000 SS90000000D0D0000GCGCADGGG000 eeeneonn wANTHONSTO OnHOOCOnHO eooo0c000 oeooco0c0cne a epee Boe oF * eee nen SANNNOON DE DATOONTANDARHOUTORNTOMNONTANMNOOCdAr Ad AHODONNONTONNDODAAMNNGNAAHOL ON ONNAAANOOOOR lofofolololovolojolojoyolojeoy lolololosos lolojoyoy lolololojololosolojolojojojojeo} SAPDDDDDDDDD DODO ODDOCDGODD GOD D DOOD OOD D 00000000 ia) a eee eee dg * + aene fee ANNWOT-HOWOMNTTEFOTONONHONOMNDATATHOCMNTOHN TON MW CNODCONOANATONHNOTANFOHDNATHATOODUTANNOONNDDOOHKHO FH CDDDDDDDOCOCOOHOHOOOTOHOOHOODODOODOOOOD0C0C0O fofoKofololofolovololoveololojojosojojolojoseojoloyojoveojolofeojojolololojojloloyeojoye) * eee ae een gee COMM ADANTAADOCNOAMDNOMNEHODADNKAON SMAHOdATODO AQCOCOCOCTHNOAMNN NN OLMNTORHHNTHOTHODMWOTNYNOMNNAOOOd SODSDOCOCDDCOOCSCSCO HAO MAMOOCOHHOOOCOOUOOOCOCOCOCOO SCOOSCKCOCCOCCOCCOCCOOOCOCCOOCOCOC COC OOOO COCCCCCoO eee eee e oe eeeeene TANAAAAMNANNOMNMNNUOOHHE ROME ANAATANTOONONATNTHOT © COSSCOCKAAMN TOL NOAAHATAMNOMMAAHDACH THOOHHOOOO fa moto hosololosofololovovovojojlosovojosovojoy fed jek jojosolosojososojojojosojoje) ccccccccocccecccoccceoccoceocccccocccocccecccce oe geen e at eee ee WONAAVNHOMDAMDAMNTNDAMNHONNOANNH ANH AOTMNAAHOHE OO CNOCOANONATONTOOMNE TNONTHNOHADOTOTNHAHONOCOOSO B® CODDDDCCODOOCCOCOOCOnK Onn OHONOHHHOOOOOOCDOOCO000O SOOO CKCODCOCCOCOCOCOCOCOCOCOCOCCOOCOCCOO COCO OCCOCCOCCoO eeeennee ee eeee eee DMAONNTMNMNNDNOMON OLE MOKHDAOMNAADAHOMNAHAHNOTUNOD SCAHOSCHHOONDONENONNONDDOHODNEUHAODEMNNONNDDOGO SCOOCOOCOCOOCCOCOCSOKX nH OCOMOCd AAO OOCOCOOOCOCOCoOCoOOSO lofo Ko fo fo Foleo folofololojofofojolosolololokofololojosolojopojojojokokofolojofojloy~) oe ones eee eennenee DANMASODOCHADTDOAENNHNOCAHMTONOMOOWOCDONANNHOANMN SCODSCSCKHHONTOONEVDOHNADDOLCENOMEOSOMOMNOHHOHOO ™ GODDDDDDOCOCO OOOOH HOOOnHONHANNHODODODODDODOCOD00000 SPOSDDODSDODDDCDODDODDOOCOOCCOCOOCOC OOOO OCOCCoCoo one ane one see oe ONNOMODEOOANONMWOMNATONOOOL NE NDMNANNAOHOODMAO HODCOOKHHANME TOONKHONTMNODHNNONOHOMAHONHONOMNE CODCCOCOCD COCO OO nd HWONONNMAHHOODBOOOOOOOSCOO 00000000 D90DDD00DD 000000000000 90090999090000090 :) eeeeneade o* eon 28 COM TTNMOY DON TANOLCAAADNAN TANONOHHONWDDANMNNO WW HOCOONAO AH TONNE ONE NHNHMNODOWOAMNEONMONHHOOOK COCCODDODDODCSCOOCO HO OM NOMOTNTNAHOOOOOOOOOSCCS COCOCDOSCO ODOC OCOOCOO SOOO OCOOC OOO OOO OOOO OSCCOoCo0o eone & * * 2 * o* # @ © PE ANNTANT NATOON TONOANUADMMODANK ANAT TAN DONE COCKCONAN MHOAVHMOTAA-NOHROLEHMTONANMNONAAOO TF 60000000 COROT OAKNA TARE NE MADNONDDODOOOOOCOO CooCCCDCO CODOCOOOCOCOOHOO OOO COO SCOO COCO OOCoO aeee ee * eee ae ONTOMNNEM HOON OT HONE TMANTNNOEMOVANAMODAEOS ” CHOCHOOO HN OATENHMNTNDO THO TTODNHOCAMNGDHOOOOSO 00000000 DODOCOHOMNHONDOMOAAMAAHOHODOOOOOOOOS ecceococc ecoecoccececcconnoccococceecoocoocece oe on # een 88 wonMnmMmonor UNE OTHAOTONALAMEMT HOODS TH HONUNTEHOOMO CHOHOMOA MEE NODVODHAMOTAMANOMONNEOATORHHONAOO N oo900000 COCCCOCOK HE NHORMODOAMHAHOOCOOOOOOO00OO eoooceo0oo SOCCCDDOOOCOCOOnAHOOOOOOOOOSOOOOOCCCO oe 8 of on 88 CN OS MMNMONDANNOONOHOADHANOAH TOL NTORALANEUDOOHAM HHOTHQONA FOLK EDOOTE CT TNOTHATE ODE OAODOOOONOOOHN 00000000 DOSCOCOHNTONHMINMANMOOCOOOOODOOOCOOCCO 00000000 DOD00000000000000009090909099900900000 e @ * @ MONMAONT TCHONTEN THM AM AEM TTA ONT TT AOAMAOM ecooooo0e ANN ATMOLOTONMHOTOrOoNTANMAOOOOOOOS ° ©0000000 DODDODOOCO OH NONHODOOOODOSOSOOONONONO ecooooeoco SCoCCDDOOOCDDOOOO OOO OOO OOO OOOO OOS0O onaronTrTs NAHOKOLOMTNAHOAAMN TN OF DACHAMTNHOr OAS Rdideddeteied Hrd ODDOCOOOSOOO OFC OCS Sra taadan ag tit 11 \U* values for Data Set No. 3 Table 9.7) 18 AI 17 - 15 14 13 10 4 © o5C6Rr-.. ee Nee ae aaa EE = Sees ondcunennn@anbncnddnnna cabinasacnme COHOASCOCOTONVAUNA TATONOAMONOFTANTRMARGHGDOOAGOHAOG: SCOKKCKDDOOHDCOSCOSCOSONIOSCAGDSOSGOOHCR0000G, 2000000000000 0000089090 FEGRATDAI9G09C09000G) @ 202 20 C) oe 8 s aenee a CADE NHKHOMOTMANNOME SOUS OAD HUE ANOGE ANDAR HE NG. AAADIOCOCOANNNHUO TUM AMOCANTORM HAND Yr IOO MAO dried: CODD CODOODD COO HOO COONS OOGOMGA9DA00CC00GCOS) ©000060000000000000060G00000990550500089909 | @ueeegve eo on e098 uf ATAADOVODAL TNNADODHDOVNOMTHOVOAIDOTROOMOMAI COSCOCK HHO NAN COTANDOOM VOTAULMATDS OMNADHONGNROG SPOSCDDDDDOOOOCOMOCOSCOOMCSOOOCSOSOO CODD SC005D000 COSGDODODOCDOOOCOCOCOOOCOOOODOCOOOE OOSOOOSD OOS ® eeno8 oon @ oe @80 @ FOMNOTMEAAMOTOTOTOOHAOOTTOMNAONT OH UEOOh Get CONONOHOHNOTATONS CVO TNO MHNAMING HONOOOHOO COKDKDSEDODOCCO COOK OSCOOOTO0000nHD0000000000000 CocecocooecooCoOoCeECOocCScCocooccococccoccoooS ® e808 eo a eee 8 @ ROVME TONEAUMN ETE TNTDDOHROMMMOSCEN TH ODEOMARQUNHME ADCHOHOHOANQHTANATEOTONNEFOOMATHTOUHOnDHAWO HO SEoccndgccoCCoOOCCOCoCSCSOSCSCoOOMOSNSGrHIG COSCO OCCCOaACoCoO CoS SPQ0DDOCOOCCOCOCOCOCOOCOCSCOCOCSCSCOoCOCoCOSoACoCCOCOoCoOoOoOCSCS e @ #808 eo e ee 8089 @® ANVHATMONDAMNAHNEOMNMANMANTHATAROHNALIN TOOAMAA BDANOMNODONAMN FOTANDOVFOATNMOTOFONWNHHOOCONONKAS SCooscooMooCooSOMnOCOCOCOOMOOOOoOoCOHOOOC OO OCoOOSoS ee9pcnoCDCCOCcCoCOCOOCOOCOSCO COO COCOCSCSC NOS OoC OOOO ooo 2 en e088 eee eo 288 e NOAOMANNENDOCOTEANTODONAMEANODONOTUMNATAN RODAMHOHONMN TOW OMNDONANNEONAMHOLATOINNOOONOOCOd SC§OCOCDCOCCOCOOHOHMOCOCOCOWTOOCOOnWGOOCOCNOOCOCCOSCo090 SC§OKOHSCSOCCCOSCOF“OCSONOSG SOO SCOCOCOCOOSO COON COO OoOoOOoS aoe 2 ae aenoee © ONNNMN TN & DANNOMNANAAMNHDAAMHOLANIDANAACADOMOMNO COHONOHAN ALP NDNOTHOATANTMODNOLANOMAHOONONOS CODDDDOCD CODD D OO HOODOO NHODOO HOODOO OOOO OO0O90090 CODDDDDOD DDG ODDOODCOODO OOD OCCOGOO GO O00000000 2 8 880 oe 2 8 ® ©® ANDMOMN AA DUTNONMOLAHAOCIMEOMNONOMNE TMOMOMODOMA COO ndndHO HAE TOWOODNHONEVOTANATOE MH THOHONONHO COCDDDDDO ODDO HOHOOCOCODO HOOOnHOHOOOODODOOOOCC00O CODDKDDDD DODD OCOCOCNOOC OCC CCC OOCOOCOCCOOOCOoo0000 e eee © @ a #enen © HO TDONAN HW ODANNDOHE HAN TTDOOMAN VOM MOME AOMM OD CHNONOHA T NNTATHOTEONAAHDUNMODGONHNOOONONHO ©0000000 0 DODO OHOHOOOMOOOCOOHOHOOSOOOOOCO00SO CCOOCOCOO DODODCOOCOCCOCOCODO GOOD OOCOCOCOOCOCOCOOCOOCOOO e eff een eae enveue @ ROE AMOMNAT DADAMNCONAMOTANTNOLOCAMMNAOMNT AMM Or COHONCHOMNUMONANATNHES ONHOTOPOWUMNANAHOMONDS 00000000 OOOO HOHOCOOONOHOHOnHOHOODOGNOOSO000 ccooococeoccoooccoocococoooccoecceceooocooge e saree * 2 eff @ POTNALVHAAOCWNEDODAON T TOON EF ODNNAHAONNOLNAOOAE COACKHOKHO AHN ATHMNADONNMNAONOHOLFOOMNAANNONHAAS ecocoo0o00K0 OC COCKCOF OH OTOH AHOOHOOHOOOOOOOCOOOO0o ecoo0000o0 © COCODDDDDDODCOOCOOOCOOCSCOOOOCCOCOCCO0o e een ee geese 8 @ RPAMRAMADA LS MODOWNTHTOMNOL OATH ATOTUDOMANUAT HOD CHA AOHHOO ANON ANKHOLNMNAOMOADMMN AN OMAONONHNOS ecoooceoooo COCKCCOKOOn HOFF OH ORK OnHOOOOOOOOSCOOS ©CO0000000 0D ODGD00000D0DDOOODODOGO9D0090000909000 e ss enne eo * eeoaee © Ar eNTNOM Ss COMADOMNNHE TNDANNADAMNAOOVOON TIME Od AONONHOO AMON OTTEMARO TH AHMDDOHOMONNOnHHANONOr eo0oo0c000 0 CODDOH On HHHNANHOOOHHAOOOOOOOOSN00SO COCOOOOS 5D DOGCOODDOSCAOD GOGO GO DOGOAANINOOCSCOSNS e see eae * * 2 @ 8 © OM OCLONNAMANM TN TOM OTUOONM TOLEOAANANAOANODOTO CANON HHOMOD TON TOOWTALDAMONOKHOMNANOAARNOMAOS CODDDDDD OD OOO HO THOT ONHIMNANANHHAHO HOOOOOOOOOOO O0COOODO00O0O0D000 COD DOO000000 000000000000 2 @# a8 * eeeeee & DAODAMNONMAAN THE OATADOL AL OOOVNOTAMNOADAMOL ARNO COHONTHHOAMNNOTODANNAOMANNWOH TE AE NAAANOONAO CODDDDOOOOOOCDOC OOO dHNNGT TOOMOHHOGOOOOSOOO09 COOOOCOCOODOOCOSOOD ODO OCOOSOSOOGO DOG HG0090000 eanaee 2 * o * @ MACHOMNNLNMAANE AO TOOALATADDOOMAOMOAAN TOE ID COAKGHNOOTNDOMNAGNTNFODTAMOAAMMOATTOANOMNANOOS COCOCOCOCOCOONHO HOH TNOON TE MMH OOOOSDOOOOCCO OOCOOCOOOOOCCOOCSCOCOO COO ndHOODOOOO OOO OGOOOO C0 oe 80 8e « ee @ @ MONNDANAANATEONMAOMANE MA dMODONANADAM Tw HOM CORSCHODAAMOUATONA ALE AAMOAMEATDOWOMAANOMAS COCOCSCOCODCOOCOCO MAM OMAMNANEMAUAOHONOOOOOOOOCOS eeccoocoococcoccecocconcocooecocecoooooece e een @ oee 8 @ OUGTTNHOVHDOODTNATEMOAGOAEEODMNOAAMAOMMNO TAD OCAHONOTHARVNOTNDOATAATTITOROMATTFOMTAAOMANHOS CODD CCOOSCOOOHOONT HOSCHONTMANHHCOOSCOOOOSOSS COCOOCOCOCOCOCOOCOOCOOOCO COOOCOHHOOOOOCOOOOOOOOCOO {eb et ier kD * * @® e e @ ® @ AOOTOMADOMAAMMN ADO UAH TODAAATNOLES OMNNOOODE AA CAAONONOMOOMMMOMNANDONANOHNOM OPW THAMOMAOS COCCOCOCOCCOCOCOCHOANUOMATOANNHrIHOMOOOOCOOOOOOS DOCOOOOODOOCODD COD GOS AHDOVOGOGO OOO BOG G09000 eo #08 * * eee @ TNE TAONAODOMOTMDOS MNO DOMME OOMTOMOOANOATE IG COCOHOCOPOMN TE ANT OMOWOMOWVNALTNNOHOOOHOCOS | ecoo0c0000000000000 COtATOOODDODNOOCOCOOCOOO0CS6S ecooc00gcocOo000000o Coo nOOC OOOO OCOCOCOOCOCOOSS .. CADMONTNNAORDOe ON TMAGS AedcdddadeiadeddHOOOOCOOCSOOS lz Spectral Estimates, (U(r,s)), far Data Set No.2 Table 9.8 20 19 18 17 16 15 14 13 12 10 pevenene eee Peeevone CON ATAMNTMAMANANENOOKM A THOR ATOANNDOLOUMNORMAM OR CODDDnhHOaMNTTTTIONKMNOMNEN ANON TM TMNMNHOnNHOOCOO lololofolololololoko loko ko lolohoko holo fololololololelolololololololololololololole) (oJolosolololololololosolololofolololoololeololololololelelolololololololololeolole} eveoeeee ee hee eetennnn ADTH ON TAADNNONOMNATAAOAUMOAHANHMNDOOOEATADONMEA SCOKSKCOKHOARMNNNNN OF OATOMN THO OTT NEMNHOnHHHOOOO (-Folokololololosololokololosohosoholohokolosososeolololeolokololololololololoh-lole) lolololoholololololoho kolo lokoso hood soloi-jolololololololeololololololololokololo) eeeaeane eae ee en ee 2 ene CODVMOE TH DE ADOHNONANDOFTAONHNAOMNHONMUL ANTON AAOCOHHONTONNODDE TN ONNNM THE OrNNHNNTHOWHOOOnA eceoocooocococococecseococococoocococecoccc Cocco sUOCOCCO COCO ecocecccocoocooccoocoocoococococooccouccocoo eeeanetae eenee eeoaenee SOANMOEEAOANANOODE AA HAAEOLE ANTONE TMAMNeE Te ATOd AOOCOOHHONTONNOUMIONTETHHRHOOEFNNNNTHOnHHOOOKnA ec9coeo°oo99c0900000 00000 0000000000 CODD 0O C0000 000 eccccccccecceccccccceceococeccccccccceccccce eeeeononeeg eeane eeneaeeee ADT TONOWHOHAEADONNONODAONMNMTANEOMOMEOLFNTAO COCOA ANONTNHONN OF ONTEEFONYrOrNNNNTNOnHHOOOCOKn ecooosooocoagcose9CoCCNO COCO ODOSCOO COO OOCCCC000 SOOO OCOSCOGCGCOC COO COCO C OCC OOCOCOCOCO OOOO OOCCCO00 onenenen eonne eeeeenae DE MNAHONMNNOE- NOH TOME NADANOMNMTONAOTONTETEMNNAD SOSCSCSOAHNOATNOOOEDONMODDONAN-ODrONOOTNOHHOOOOd eccoccoococoqcocoqcqqqc coo cCcCCCCCoCSocCoCcococeccoo eceoec9900cC90 C0000 0000 C900 DOCOCC CCC OCOD CCOG000C000 ee enone ee apeee eeeeenee CeTNE DONE DONODOOOAMMHAOCDE NE ANOMNOADNONNONMOOK COOCSCOCAHNONTNNHNODOEMNANN HMMA OroNNNTNOnHOOOdd ecoc9c9c09c000000000000 0900000000 CA 00 C0000 000 fofopololovololosolololololelosolololalolololokolo lo kololokololokololokololok-loko) ~f RRR RAD 2 2 ee ee eonn HOUOMEEAT THON TF ONUOTNAHVNOM EP ONODORnEMONN TT HOOKd PHOOCOKAHONTNNNONALMNTHOTRMNDHDDODONTHOHROOOKnH oho Nofofololo Noho fohofofo ole fohojoy lolol olovololololololo holo loko loko N-l- io} Jofofokofofolo to lololojololololofolololohololokolololo loko holo lolololololohovele) eee eee * + eene see MNOMNE EEN ANODDDONOOMHAOLEANNANANOTONADNOTON AAHOOOHHO ATNNNODAAHTOTNONNDADOODONHNTHOHHOOOKdHA SS99DDSCODOD ODOC GDODCOOOCOO HOODOO DO DCDCDC O0G000000 ofofofololofololovolovojojolololololololololololololololololeolo lolol oholololololo) eee foe * ~ eee enee D-AANNOT DHADOOLP EEN AHDDNNIAMMNEOUWUTDONNADAGMOA SOCSCSCHHO HTNODODDOOTNOTOAMAODMOONTNOHHODOOO SSCOCCCOO CCOCCOOnrHOOOnOVUOOnCOOCODOOC OOOO D0CC00 SOSOCCOCCSC COCOCGOOOCOOCOCUCOCOOCOCOCOCGCOCCCCCOCO Hee HAT ee * eeeoene TMONYVOETTUTANDEONTOMOATMOMNATT TOONHOMMNOAMER SOSCSCOKHHONTHNONODOATAUNTHOTOHODEOOTNOHHOODOO S99900000 DOD DCCOHODD00DO Onn HOODOODDCDDDD00000 eccccceccccecccccecccocceccceocceccceccecoccccce eR ORE RH * eeaom eee OOTFAPOONNENHNODNAHMOONMOEMNMNEADATATONETNMORe AAOCOCHHONTNHONOKHNHOOTNATTDANOMEKONTNONHOOCOCOO SSSOSCOSOSO OOD COCO nH OOO COO Ona HOOOCODOOCOODO000 S9D0900O DOO COOD OCC OOCOOCODDOOCOOC OOOO OCOG0000 ee enon £08 SATMAMNAO TOHAHOHANANNUNHODONRTHNOTHOANMNeTAMOO ADOOOCKTHO NHODOD0HHANONFNONTMODLEOTNORHHOOD00 SSSSCCSSS SCCOSCOONnHHOOCOCKOCKHHAHOCOOCOOOOCOOCOOSO S999090005 DO0D0D0DDDOG00OCDDG OGD D00DR0000000000 ee eee nee 2eee see DE VASMMM OMAMHAANNENONAMDTTOCTOCAMNTNAHNOMOO SOSCSCOHHONTOOOL DHONAME NE MANMADE SE NNORHHOOOOO S9900000 DO0C0CCOO FHF OO0C00COn HH HOOOODODOCOCOO00000 oJohofofofofoloofolofo lolol olololololo lolol kool koko lolol koloko lol lo lolol - i>) eRe Ree eae ee OE NKE NN OODNEFONMNALMNOOO DONC ONMNAOOHNONANAM ONE oO SOSCSCOHHOATNOOFARNODNNNENXOTOAMNADENNOHHODOCOO SOSCOCOSCSCOCOCOOCCOCSCOCn HOO Mn NddtANNAHHOOOOOOCOSOOCOOO ceo0000D0000CCOCCOCC CSC 000000000 000000000000 hee ee eee 8 Oe PONWMNE OFF HT TVOF AAO AHTOMNODMNNOLHANMNDHONN TONG CODCSCOCHHAONTNNDDOMNNGE TRAN ARNOONONAONNOHHOOCOOO SCODOKCKCCOCOCOCOOCCOCOC RM HH OOH TIMMMMNAHOOOOOCOOCOCOOO eoooooocooCoCCoCCCCSooCCCoCoCCCSCCSCoSeSeeococoooCOCd enene nee eee e886 OEE TTONTOOWNIMOWDAMNHADNVOTATHONOOM THE NNARAAMM SOCSCOCHHONN VNOOHOT ONAHHMNASDHASCTMNOHHOOCOO CSCoCOCKCOCOCCOVOOCOKr Kr nn OnRE AON TNA MOOCOOCOOCOCCOCOO ecccoococovucoocooceococooecoococececoocococ nee eae one e806 OM TORE AMATDHOOOEAODEOF AL OOREENOMNTEOrFAATANdeM SCOTDSOTCOOCOMNUEFVYVOFONMNEEAONNATNOMNDDEOMODODODCOC0C0O0 SCODKCKCKCOCOCOCCOCOCOCMANHOOMVUOTNAHOOCOOOCSCOOSCCCOCD eccccecccccccceccecccecCrHnnccccecocccccocccce een HAD een nae AM OMHODTNNAONHONHOODHHOTOFOMNTHAMODODONADONMIN SCOSCOCOCTHOCOMVE WOE NOKAOCTHOENVFMNTONEEEEMOOCOCOOCOOCO SOSCKSCOCOCOSCOCOCOCOCOANNNOOTNOHTNHHOOCOOSCOSCOOCOCSO eceoeoooeooCoooCooCooCoC CoCo nrnHOOCOOOOCOOCOCoCCCCoOO ane oe * eenene DAOAUTNEONOMMIMNATUDONDEOCOCHONTOHEEMONHMNONATH SOCOOKSCSCONM OOO HHO NONOMAMODNDESSENNOSCOOSCOCOS SCOSCSCOOOCCOCOCCOANNHOCHNYTTNNHHOODOOOOCOOCOCOCCOSO eccc9cc00CCCoCCCCCOCOC CCC COCO OCOCOCCCOCSCOCCCCCSCCSCCSD anne af * * ee eof WTHOCOOOVATHAONSC HOM ONMAMNONOHONMOHAFAOOCOCOOHTH SCOCKOOCSCOCOATNNNOANODOTMNTODONADVNNNFTHOOOOCOOOO ec0oooCoOCoOOCOCOCOCOoCOnM NOOO nFOOOMHOOOOCOOCOCSCCSCSESCSCSCSO eccocooocCCo COC CC oC oCCCCCCOCOCSCOSCC COCO SCSC CoCo ooo CADE ONTMNHOADE ON TMANHOMAMPNHOEDACHAM ao Madara et HOCDDOOOQOSCORPCOQCOOd adda dd ae 113 Spectral Estimates, (U(r,s)), for Data Set No. 3 Table 9.9 eoone #oe eacseoen AOL OKT TANMNTNEMONMONATOTONMOCNAHAANNHDAOK At CODKCDOSCOCO KUM THNHTMAQHNOOHANTHNN TMAH OOOSCOOOO CODCDCOCD ODDO SCOCOOOOOO HOODOO SCOSOSBO ODOC 00000000 COODDDDDDDDDOOOODDOOD OOO COO SOO SS0000000005 20 aeeenee ena oven ne2 COM ANHDDO TONTUNNOMO TOADTAOVNIMOM aes TNOAHAORV ® COCOCSCOSHATMOOE IN TANALOSHATNOOONWTUYHOOOHOOOS A GoDDC CODCOD O OOOO OCOCOONOCOODNOOCOCNO CCN NDC0D0G0000 CODDDDOOCDDDOOCOCOO CO OCOSOOOOLSSOSGOCOO COSCO OCoCoSo ooo ee oe. oe 2ae eeneane ATDAGADHANOMODADAAANDANHONDANDOVOONDANDOOOWM COCOHHOO TMT OUOFOTMNADALAMINEOOMINAOOCOdHHOO cs} CODOVOCOSCOCODDDSOOCSOOOGNO OHO FDOCSCOSOGOCOOCCOCOOOoD COCOCCCCOCCOCOCOCOCOOOSCOOCOC OOO COSC CoOOCOeSoCoOOoO aeeoaae ane ee ezesee ME KHON AOMNALATDCATENMMMOMNVDOHNOWALTONMWOHODAAHS Rh COCHHHOOHATHEEE OM NMNDAANMTOM OOM TIN ADOOOSOSCOS A CODCOD ODDO DOO OOD OOO ODONDNOONSCNSS CONGO oCoC00GO céecceccccceccececececeoceccecceccececccccccecce enaanas nee aoa2e aes AN DOMOLANOAKHOMAMANMUATODNONAMNOOODAMODHAAOANTM WOW CODOTHHOOCAN STOLE EUNNMNOONANOH EE HYTMNAOOCOAHOOO A CD OC CC OC OCOD OCOCOOOOCONSCOSCOCOOCOSCOCSCoCoO0SCS coo CODDDTCOCOODDOOC OO OOCOCOOOCOOO SOO CCN GOOOo0000 eeeaeae “20 eeoeanee BORG GR NOAM RENAE EAH OTT OHATADAEATAUMOMNOLHAN COO T HOOT MN TOF EEN EME AONNHNHWE OE OTNHOOCONHOOO ea ecocooooc ooo ooo oOeCcCoOOCONoOCCOSCSOSOOSSCSOSSCoCoCoSooSo CODDDDDDDODSCOCO OC COOOOOO OOO SOOO SOG OD0G000000 eeonene one eeeoone MME OTADMOTAM~ TOMNTONNOTTOACTANOANHAMEMOMON CODOKAHHOONMN TOL EE OONMDAEMNTNHE OS OWMAOOOHOOOO t+ CDDDDGDDODO DOO ODOC OOOO OONDOONOONNO COCO OSNN000000 CODDDDDDCODCOOCDDOOCOGOOOSDOCOONDND COO GDO00CO0SoO Pree Rena ane eeaceone NO ODE-NRNODONNAMNOECNTONMNALCDOOVOOCHOIMODOOAM OLN COSCO HHH OAMNMOLFE ET OONTOMATTOODNDDrONMNHOOOHOOOO M as000000 000000000 000nRDD000000C00000000000 9990000 D0DD00CCDODO CODCOD 0000000000b00000000 fe eeone a ee weoge NTAAMNNODTHANQMEDDOADOKGE AL ANN AODNHNATODNNOANDOOKN gw COSCOCHHOOCHMH rer Or NNRNOATT00rEr0NM HOOOHOOOO A CODDDDDD OOD GCOOCOD OOOO HOOOOODDDGOA0GG0000000 lJoVololoVololoKoloko lolol olohoko lol ololololololololo Kolo olokooloko Nolo lolokolo) pene ae ~ae eeare AOANTOONDOC HME DOAHOLINOANDNONAMANHDOMNAVNOAN COOCMHHOnDH ANH OF Ee DDOrTITINHTNODDOrNHMNAOOCOHHOOO ececoocoocoocoocoooenceooocoocooococoocooceoo0K0 SCCOOCOOCOOCOOCCCOCOCUCOCOCOOCOOCCOCCOCOOCCOO 11 efeene eee eepone HOCATOFONNNAONHONOMOCK MOL nhATOrAnRTOANTORNANdY Oo COSHH HOOAM TOPE ADDO TNNHONDE DADE NNAHONAAAOO Modo Ro soho hosolofovososolojojojovosolose® leolosojolojejojelojolosoyolojojojojojoje) ecccceccccccccocccccccoccccccccecccccccecccc eee nae 2 eanreaa SOWDOOr TOAALEDWOOMNDOE TMTODOOONNONHANODACTNOMdA Hm COCOHHOOANTOFE DDE O0DAMNNALTOAAHDLOMNVNAOCHHAHAOO ecocoocoooc0cccoqccooco9o9CConHOCOCC COO oCOCoOSCOoCoOo SOCOOCCOCOCOCOCOCOOCCOKCCOCOOCOCCOCOCOOCOCCOCOCCCOCCoOOO eens aeeeeae COC DANON DAME NT OMNNORDOOANQANODTOrErDAMNOMTOONG SODOHHOOHMH OF EANDAONNONDOAADHADONMNAOCHAHOSCOO ececooccococooooccornkoCcrHnOHCOCoOCoCoccocecccoe SOOSCHDDADCSOOODDOD OOOO DOGO COCO OOOO GOOG O00G00000 eeeenr ee eaeeeeee ANOANMOMDNNNYHOCAEAAOPNMNOEMACATODNOEAMNADNS nh COMPMNHHOOMMN OLE AARON TNE NACCCODONMNHAOCOAHAOCOO SCOOCKOKCKCCCOOCOCCCOC COCO KN OOK nt Onn nH HOOOCOCOOOCOCOOO lofolofolofofofovololololosolovolovoleololojololojololoyovolololokojososojojososolo) peo ae eenone HODHONKdA OFYNTTAEHOOKMOOMNEHHNOMONTETNHODONOKNG © COSdnAHO ANN OL T Orr ANTON NTT VANE OMNACOONONOO 99090000 DODDDDDOCO On HANNAH AAHHOOOODODODOOOO000 S89000000 DODGDGDDDDDG0GDG00 0000000 00000000000 eee ae eeaeen NHAOAL LOFT DAHHNOOLHAANAHN TE TMAMOLCAUTHAHOAEOAN wn COCOONS AMM OOK EE OADVDAAMNTANTNALCHMAOOHOOHOO 209000000 DADC CCCOCO OK ATHTAUNNHAHHOOOOOSDOCOCCOO O09000000 DOO0DGDGCODDCCOO COO OOOOOO0090 900000 eee ee eneoe TADHOTMI AL HOOTHOHANMOONMAOCDOrONAHOTDONDOON COSMO ANTOLEAAMMOUDNNAGCOLPNADOCTNOOAHOAOO FT CODD 0COS COOCOOOn Fe RANCHO TYMAAHHOOCOCCOOCOOCOOSO SOCOCO00S DOOCOCOCOOCOOCOCOOGCOOOOOGOSCOOCOOCCO ee eee eeeae MODKIMOOM AAA UTETHAAALODNDNOOKHONTALANTE AL ANE 9 COAAHOOK ANN WDDAMWOOMEDAMADDTNALTNMNHOOAHOOSS 299900000 DOD00CCKhAAMNANHTEELE TVA AHAHOOCOOOCOOOO000 eccccece cocccecccceoccoceoecooccecececcocecccc eoae ereae HOP OMOAD UMNAHODTNOMWOMODNDODAMEMNDOANMAN TOM WW COOKHHOOO MMH OVHONUNAMMAH AMAHAONDNHADONMNAOOAHOOCOO 29000000 DODO COOHRANNHAMNEETNUdNAHAHOOOOOOOCOOCSCSO SO0900000 DODDCDOCDCODCOCOCOCOCOOOOCOOCOO OOOO OOO0O0O eenee eanen COMM OOTH THOTATNON MN ATE ADOAANMTONTE Ode MOTOO S000HO0O ANHUODDANALAALOMNAHNTMNAOSOTVNHOOHOOSO SOS0OCSCO DOCCOSOKMHHOONONMMNAAHOOSCOSCOOSOSCOCOSS 20000000 SO0000000G000D00D00000000000000000 eseeoe eeneae COTP ATK TATMNOTEDDDNN HNN ODODE AON Tdddd TAS TOO COOCCOCOOd ANN OOMOHOWOr TE OWDOHODODNMNAHOOSOOOSO © 00000000 CO0COOdndsnhHnH On TOTHOHAHOOOOCOOCOOOODSCOO 290000000 DA0DGDCOCOD DOD COOC DCO C OOO OCOoOoC Oooo oo CADMON TM NHOAO™ONEMNHOAAUMTMOFDACHAMTNHOL DAS Ndddddde dit tOOO COO OCOOCGOCOGG GS aeteeinintate aay 0 114 Table 9.8 Plus. fable 9.9 Table 9.10. 20 19 18 17 16 15 4 13 12 10 eeseonece eenee aeneeoeen WWOKAAOEMNMNE-DHDOMMNNOONE EEE HONNAArKHOMOdOKRK SOMO HH MONNEOANKODMN AE AAHMDOHAHADOTKCONNHHHOO SA9DKDSCCKOOOCCOCOCCO aH OOO Kn nHOOOHHOO ODDO DO0D00000 bate Ko doko kolo kao lo kolo lo loko loloNolololololololololololoh=lololololohelol-l-l-1-) eeeneeen eonan eenenenoe ADHONNOATEOENVEMNNAVOTTNNNHOANHOOLANANDOMNAKO SOAAANNOMNOAOKTHAMNO TATEMNANTOUMNAHOODOMNONANHHON SS0CKCKCKCC CCC Odd dt tt HOOKNHOOdn AHH HOOOCOCO00000 fo To fo fo Ko lola lo lolololo lol ololokolokelololololol-KololoholololoKolololol-i-1-1-) eeenene eevee eeeevece MAMNADUEMNEDOTODHAHANOENODMODANMAMOVTE Te TOONS AAAIANNNOVEOHAMNTTNNOUNEOMANAT TNA HORMONA SCOSCC99CCOCOC dd nd tnt HOOKVHOOMH dete HHOOOOOOD000 bofoko lo fokoTo ko folololol olla lok Nol lok olololol-Lolololol-loNololol-l-i-lol-) <1) eeenvaene eeaae eeenecoe MMMM ADU NNDAD- ANUNNNTONATEOADONHONONOHOOMNN SAA ANUNUNOTROTUMN TT NN GONE TNTATTAUNHORMONVdAddea S999 KCKCKCCCC ddd nt dnt AOOnHNHOOdd dH HA AA OOODODODD0000 cococcooccoocccccccoccoceecccoococcccoocoo eeeanennen enna aeaneeee ANN TAN NH ddA ADDNNTODNOTTE THN MATHANNMOATNS AOAFANUVNOMEOAUMMIMANOMDONNATTMNNHOLMONNdd ddd SOSCSCSCSCCCCO ddd st HW HOOKVHOOKdKH dH dH HHOOOOODO000 bofofofolote cao lololofololoKol fol ololololeolojololololololol-loleololoKol-lokeolol>) eoeeeaee aonan eeneaenn AS OTNOHONH AMEE NOTDAONMHAOTRAAMNALMNTHEOAMANON SCODAUMMNNOTEFONMMTMNNOOTONHNATTTMNNOKMNOAQddddd SCOSCCKCCCCSCSC dd dtd dH HOOKTNHOOKd dd dd HH OOCOCOCOCOCOCOOO fodo Rao fn fo folo he foo ho lao ko kolo koko fololofolojofolofololololoholololololololololo) eeeeane eann eeeeenee NOANAADDOSNMOANMOANDOEATMANOTMNOTADNNOTL DAAOM AA AANUNOTODONMNT ONMNDOMNMNHOFMTNTNNOLMOd ddd Odd SCOSCSCKCCOCCCCd dt dt dt nt HHOOCOKNNHOOKdn dd dt rHHOOOOOO0O00D fofoRofoko foo fo folokofololofofolofojololojolojolojolololokolokolololololol-lololo) ee one ae ee eee enan POTANHNONOAHROANEEONMONMNHATHONEOADDVODTONMrOMAM SHA AAANMOTHOUNNTON TOMNITMNUNDNNOTMYAE NOUN Added SOKOCKKCCC OCC d FHF HH HOOCOCOCOOCO ddd HAHAHA OOOODRD00000 loo Kofofo loo lo Nolo lolol lo lo lo holo lolololo lolol loloKololololololokoloko lolol lo) eevee eae eenaeeane NOW AODNADTODANODTHNDOMNADNOOHNH TMNOOMNNNODMOOKd AAAANNNOMEONMTOODOTNDOAE TOOTMMNADMONNHOdHA foKofoKololokolololoe tt tanaka tahalokonolololok hata h kkk lekeloleloleleleoleole) lofofofokololololololojololololoyolojolojlolololololololololololololololololokololo} aenee an * eeneaene DEOHONWAL EKO HNINATODOOHNANHDDNOVOLP ON AR Th ah COAANNNOTMOAMVEDANHMONTONOONNMHE TONNNHHOO SCOSCKCCCCCCOdM dn Henn nn OOCCOCOCOnRMHA A AROOOCOOOOCOCOO lofofoKofolololololololololefololololol of olololol ol el ol ol ol ololokolol ol ololol el oko) eeteen a Rae een ane WOME AOOTNMHAODNTOM ST THNOKHANNMEONAOEAd Ae NN NHN Ow SOSCONANOTEPOUNNTE OM HNNEOUOCDAOMONAE THANNHHOO SOKCKCKCDD0C0COd dnd dn AHH OOCOCOO ddd dH HAN OOOOOCO0C00 ecccococcoocoooccccccooccccoccoccccccocce eee ean oe een eone SCOOADOTNANANMN AE AOLOMANHAMNDAODNTHMNAL AYE OHDO AAMT OANNOTHPONNTEAL-NUENNOTNdHALOMNDTONNHO HOO SCOSCKCCCCCOC On dtd nt H nH HH OOCOnK HANNAH HHOOOOCOOOOOO lofofokofolololotofofokololololofolololofolololofololololokolololololol-lolokokole} ORR eae eennhaae SOANONDN COT DOCH NADHNMNOANLHODHAASHMNOHNODDOOE SATA UNVHOTODANMNTOAATHOMODHNVNOLNMNDTONNHOHOO SCOOCSCOCOCCOCOCH NTH HAN AHHH HA HNNNNA A HH OOOOCOCOCCOSC ec9c9cc900C CCC CCC COOCCCCNOoOOCOCOCCOCC COO CCOC00000 ennennan aeennaee DAANMANSHODOODHAONM AE TEE AO DHONTOEATNOMMDAADAOe SOAANNNOMDANMNDOHAOMEMUNUMNAOTHAOTNODTORHVHOHOO SOKOKCKCCCCOCOCNF NFA HN HHHOTNANVNAMNAAAHAHOOOCOOOOO0DO eccocecoccococcoocococCcooCSCCCoCoCScCcoCCSCCCCoNCOCCDCD eo e een en eoenene CE UNVHODNOONTANOONAORHNDONTODHATOTANOLOMMNAOr CORR AMNNOME RB AMN DOF ERONKALPHEOTHVTANNOTONANHOOOO COKDCKCOCOCOC CO nt nen HAHAH ANVNEMMMNE MVNA RAH OOOOOOOSC0O eooCooCOOOCOCOCAODOOODC OOO OCOCOCOOC COCO OOO ooOCoC00000 eee eee He een enee NN ONMMEMANNONESOFMNATNENTOOMNHAOCHMOMON TONNE w) CORA NMNOMEORMMDHOUOEATHTORMEANENDTONVNHOHOO SCOCKCDOCOCCOCOM HHH ANVNAHAMNAOSCNNMANANAHOOOOOOO00O eeoooCoCoOOCCOCOOCOOOOCCOCOCoOMOCOCOO CCC oCOoOoCoOooOoCooo eeeenee eneenaen NOMNOTNNNEATNNNMOSTDONTOONONM FHOOONMAODE AMY COPANNHOTE OAM THEA THDAMATMHE ODMOMATANNHOHOO COSCKOCOCCOCOCOF MAB ANQMANMMDNODNMNIMMOOOOOOOCOO eoocoooCoCoCoOCoOoOoOoCoOCoCoOomndnnOOCOUOOOCOoOoCoCoScoo eneenne eenene SRMINTONNTTNNOMNTNTNONATATE MOE TOOTOATANONNANdT COMP TFR HOR NANNTHANMOEE ME TTOMNMN EF OS THON OOOO CODD CODO COONAN AA NMNMTNNNGTTEANMNANHHHOOCOOCOOOOS eccoococeccooocoeoccococrnnNnceccoooccooceocoooccec eaeeenae eeanaenen OE EE TOOMTOMNOTEMONNNTDOCT ET TTNOTOAMNDODATOCN COSCO THOHNANMNOTNOMNOENMNN HOM OFANTNONONHHOOCOO COCCCOCOCOC COM NHANETN FHM OEMAATMNVNKM MMH OOOOOOCOO cocococoooooocoooooooooOnNHWOCOOCOO COCO OCoCoCCoCooS anenenen eoanen Boo S OOM RIN NUE HAT NCAR AMNOTHONANANHOL TH CODCCOMM ATH ANTM TOMVONEENADNMNNAN T€FONNOHNOOCOSO COSCOCCOOCOCOOCHAAHANTTNOMOEEE HNN dre OOOOOOSOOO eoocccccocooc ooo oooO CSCO SCSCoCoCSCoSSSooSSoSeosssesscs anennne eanveanee POLAT HE ONMDIAEDATMMDOOMMCAE AA AMNOF MTA OTM oo oo OOO MeN eT AUP N OHM HOMTOATN rh tHOOCOCCO CODCCOCOCO COHAN ANNAN HMANKANNKAHr A MOOOOOOCCOSO Y-9-1-N--N--) ao CODON EMUNAOCADE ONT MVNMHOMUMTHOFOACHAMTNO-D Setetet co Neldeldeead red OOCOOCOSOOCOCOCOCOOC COSCO nada add dana ds Unleveled Raw Data for Data Set No. 2» (k =.0-19) Table 9.11(0). 19 -~ 16 15 4 13 12 11 10 TAIMEMMONACO KO KNAMNNMDONRNNHOLOTANAHACONN KH TON DNONTALOOMONGTNHWONNW ONNAOONMOOCNAONMMTMAHADADADADAAADAOMONOCE FTNTMNODONWOLOHOMNNOAMM MONDNNNHNHNNNONNNNNNNHNNTTITNTOTIFTIGTITOTWHMNVT er TdTVIITITHNeCIe TET WNAOK AT THELONIE TOKE OE TN THATNOTIOOMAINe WTOME—TANAMANMTN “ ADH AANRNLMNNMOMAANAEDAAVADADOOAOG- AO AMOK MON TR OAeOMAtOONO worres CATION NNHHHHNHHNNNNT ET TONWTIVIT CIVIC ONMerTer Ewe E HNN TECES AMTODIUNKHMOOCOUWUTNEONONNINTHANOAENANTDODAOHNNNE ANON Caran RACNAMMAAMATM HH EMAAALNAAADAAACOAAK AADAC OUND MDODCORNGOReauIEE OTNNNNNHHNHHHNNNNNNNNTT TTT TTY NNT ITN TTT ITT Tere TNADAMOOCATAUELC AMON TTONNEMNOOTATENONDAOHNNODHODDNHOANAMNAMNNTAMN DE ADO DONTINANAAAMNNONMMNOOCANDAARAHODCONDONHOCOODAHOERNUONALE-NO-HNONTNNNONYS TONNHNNNTNONNNMNNNNNTeTTTONHNTTOWCNNNNVe eC erewreeC CTE TITS DMNMOMNTNDTAOMANUEAMNONDAMNALODTONNHADH-ONDANNONSNTNOLONAAN TOCA ADNO CANMAAMMAMOMHUVAOAEAARNDONOLOODDDDARNODONENNODADOTEEOFOTNODAHNA TION HNN HTNHNNNHNNNHHNTTeTNTNHNNTT TTT NTNOVTT Tee TINS AVMOMNAUNATOLUAHONNNDODON ST TONOTN THODNOORMOONTANTTTIOONNOARNOTNEE ND SOMME EMAMMOMAOVANHOOCODANAANOCHOODDAAHMMOLE TNNOUOLOCHOMDAMDANOUONUAG MUNN HNNHNNNNHHNNNNNTNTTHNNNN TT THNONHMNYI Tee TTT HNNOe Teer TTC MAANTMNANDNMANANOM ACK MN HOAMNHAHOMDNOANANOENMNDOMWVODAKd wNNWONDoNoronn TANTO MOMAUMNTAHOAHOVAAANDONAHAONAHHOOFADANANDONMNHOOL-ODOdTO ETE OrONTHEHAO NN NNNNNNNNHNNNNNTHOTOTTOHONNNNN TT HNNNNet errr MMT YET WNOAMNANNESTOYAULFADOHNACONCADOMOCONTINMAOMMAAEEONTHOOMOONAMETOMNON AAACTAMMMNONTONNE DOCONATODOMAANMNOODNONE OAD DANATOANNTMOO TOA NM NN NNNHNNHNNHNHHNTTONNN TON THHNTTNHHNHOH TdT TT TNTeNeTECTeT TIT MEER EMMAGHHAOHENANNNANO DON FNNHNANDHAOTNOWONON HANES OTNATONTOTONOOHnHOOSO SK DOWOMNTMNOTNTNOHVOHDHADTANDDOMNANMNMNNADOTE OOF VAANAAGEENNNALELANAS TION NNHNNNHNNNHNNHNTTNTTHONH se THNTMHHNNNHT Teer eT TNMNMNNTe steer Tee MOUMONDAMNOMELMAHDNOROHNHOKHADADDOOHNNHHE OMNOONHOEMTODNDNODOONE AONE NNOMAMNNATHONSDOOCHAMADDONTOOCOCTTOWOFWONDOOSCrHArTOONNOKEONre TOON NNN NN HNN NHN TTHNHHNN TTT HNNNMN CONN eT THON COI VvesII teres MANE NADNNANERTNDNOOOMN DOE 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NDOMN 0 ODAHOMNN AON OCODOMNEEAOOTOHAHTANOOCDNAAANTNNOAHANADONNNOTTO DAAMAADA ANSE TTHAANKHVAADARNNONAOCDHAAMTMODADOT TNO TMNN TODOCARN TCE NNN NNT ST TNNNNNNHH THNHNH HN THHHNHNTOTNTTHNHNNTNTTTT TIT ITITIII TS OC NWRAOTEAMACAYMATMMAADTAAMOMAOMON ALM OWN TMTMADNNUOEMMNDONOOADD @HOCOKAO ON MNNM dAAVHAAVOMACAUNUDODAAHHAAMANNOAOT TONAN TOAOOHOMAWO DNDN NHN NNHNNNNIN ST THNHNNNHNNHNHHOTTTOHNHNHNNNT TTT Tere TTI TONS Toe DNOAMOAOKMDACODOMOLES DW OTOOADSO ONS DOL ATALODVDAHNOMTH MANGO AE OTADO OmO AAAHOAL ANUNOAAAM ST AOMONORDALAOMANOHOAAAHOAOMO THOME IANCOC TMA DIN NHHS CNNNHH ee HNHHHHHHHTE TC TNONHNHHNNINIIIT Iter TT TNT ETS CORMARNOL HOAOMO T NOTNN ONCOL THONANATOAOMOLAOMO]TOOONAMEAUADOANOOTO CODARANQHAAMMNO OL OANHOONKTANODOLAMOOL HOOHALOAN OMAN A AN AAIADOM HO PUN MINH €HNNHKHN ST 4 TNHNHNNHNHTHNS eT THHTTNHMNNNT TTT TTT TMNT INIT Ieee COO STE A OE MOTAOMA TINE HNNANE MANE VOONNAACOMOMDTODTOAMENDOOAMOMO COONTNTAD CETMODDOPAANNAAAVAAODAAOCADAOHAMACOT OVE ANMOADOOEALAE DN HNN TH NNNNN HS NDONAOAEPNAMONDOAMANOHOTORALPANM SCADONOTANNATAYONOOTT TO ONOOHONTANVNTHNOHONSO SS9DDGDGDD DCC OSCOOCONOMONHONTEHAOHOHOOCOOODODOCOO sssssgssa5c0c00000000000000000000000000059N 000 * * * # #8 8 8 @ o @ eo 8 @ DNTAADOD ONDTONTHAOODCOTMATOTUTNAANTONAMNNONH NAP OVTHATHNUATDHDAAHOATOTCOMNNATTONANDADHN SosoosOoSSSOOSODIOONONTOSENANOHOAHODSSOSSONCSO esossseccccos009000DONTHODOSCOSNNSSNNNDSCSOC00 e * * * 2 8 © TIVONNNAHOMNENANMON DOP MAE ODONAGNACENDNOWTOOVAT NIVADOTAHUMADONAADIMANOMNINHAS TOTHEMNMNANSAHAN SSSSSOSCOSOSCSOSOCOSCOOOMNANLHANAMANDOOOODOOSSOOOCSO SS990gNSFOCOOCOCO ODO DO OCOD OONANOYVSDOODOOOCOOOCSCOO5S e ° e 8 * 8 ee 8 2 * 8 @ NAAHTOONODHADNOMNOHMOTT TOT TCNNDCONrATOTINADNA DADANNIOTIODONIONAOTTOANNNODAMONALSMANONANONDS SCOSDSDSCSDSCSCODDOOCOCODOONMOnDAWONTANOOOCOOOCOOOCSD0CCSO S9000DDDD DODO CODCOD ODD COCOOCOOOOOO0GGG00000000 * ° * ° AANAANK EC ONDNNOOL MONT TTNONEFOONNOM ore Nnananan HONDNANNTHOOTNOMNEONANMOLNMON TOWHTNYNNANONOH 3933995999959595959595 F9 F0N0T9H9999N5N95N9599595N5555 So09000DDDCCCOCOOCOCOCCCOCOOCOCCOCOCCOSCOCoOCoCOSCoCCCOoC0D SDALTKONIEMNAIDHADKCHIM*NGADANATNO>~DADSANT™ ET HE>~ DAD Nadddddd ddd SSSSSOSSOSSOSOOC SSS SSS Sd ddd ddd L(r,s8) Values for Data Set 3C Table 9.18 20 19 18 17 16 15 14 13 12 11 10 ° * e o SCNAHANONALNOAN HT DANANANADTAVAONENNHONAAANO AADOSOOHAOH SOOO FTHMNNONOFNAONONMNATOONDnHOHOOOOKdH! Ses99gggCooSoSCOOCOOC OOOO OODDODOOOOCOOD Do D000000 es9ceCesceSSCoOsSSSooOSDSDAS ODD ODSDDSODbOODDOSD000Gg ® 2 ANNO TARANMEMOUMN GT THHMHONNAOOATANNDOHNOL OAM AAOMOMAOAUNH MON THORHHOUOVTOHOHHANOHOOAHNHOROKdAA 3999999999999559959990859909959009009050009095200 S9990900000C9D 0005000000000 00000000000000000 ® C) o ATDOACAMNVANMNOOTOOMAE MOC DANAnOHMUTONDDONALA SOMONDOSSCSCOCKHAOHOONNKHOMAMNOCOHOOdAHHOOMHOOO00 eooeoooosesocococ C9000 COODOD DOC OOOCORDGOO0KgG0 es9o99gssscseo ODO DO COO OOOO CO COO OOOOOaDDb000 * ME STODNAOOTMOME AL NONTINMMOTALOWNYTuTTONNM ADOOAHOKOONSOONAMOMNAMN T TFOONNIHOHOHOSHHNHOd 99999999999990050905005500900cC500005000000 @OSS9S99G9S9FZB99F209S99990900009000005050000500 ® 2 * % ONOAUNAOOWONTAAUMNOOMANS Or NTHAOMOORY TAMAYO SSASADIOMFADIANNONHNOMANNHVOMOODSHMIODHODOWDO Seog oooCoCOOOSOC OO COKd HAO SOOOODDODD DODD OCO000 S9SOD9DGOOOCNOSCOCOADSCOOCOCOO OOOO OODDODD DOOD D00000 2 ry 2 DAMMAM OMNUANMODTAHMNDOMAMGDEMNAANALCMALHODOMNOA SASCTOMPAHONNOAHHOOSMNMAOTN HANMONTODODO RH HOODOO sooooccc0090000000°00000g0000g00000000000000 Loko Ko Kolo Kolo Kolo lola lolol elofolololalo lolol loko Nolol-loloko Nolo helolololo hone) e & * * a * AVNOTONHMNTONAONANOODHOOHTOONTArTONFATRNODN Nd DONONAHONHNONANAAAMHAOMNETNNNOKHANODAHOHOMNOOMAO 993999999959599955xH999599959999559559599555995 Foe foo fo faolololojololololololololmlololo lol oho lolol holo loko lolololololololol-) 2 ry t} @ ANNNADNANTNNTANAMNONAANAMOOMNODFWOTNTEeOAORORODA PIAHOTHOOHONIONMNTNONMAHDOANOMNANNNAONHOHOOOODS esococ90000000C000000000RF 0000000000 000000000 9993999039999399999999999909990909995359909990905390 * * ° ee © AVOOCODMIMADDOAHOMTUXAHADOHDHAODANHLOLANMT TIE SOAANDKTHOONNT AMA AME ANANTODOMOONONA A ANVHANNO 9993939093939939999093999"8N9999959959999599995990599059 eessoesoscscoosfoesasaosc0qesacooco09nsessesssce9s0cSesso * ° an 2 MOAAMOTONADAAONNOMMANMODOMIMAMHNNANNON TNS NNN AAOCOCOTHONTNIOONN9AN TOON > DN THNONNHHONOOHOOCOdA escc0c0c00o0D000CCO OC OC ONDOCOOSGCO COORD COO000000 esosc000sg90q00000z5T000000000090°_00000009000000000 oe ® MATANAOTOANION- ONAL AONE NMOTITT AE Hr ODOMTALCOM ADNAHOADIMNINIMNIN DAMN AOTNOOO TUN HH ANOMONOHOOd loo ho lololokofololohoholojolojosojovoloy. jek lok. losohohokojovolkojokojojojoyajojq) eos00cg0zc000z0000N000000000000000000000000°09090 2 2 oe e 8 * ° MAAN TE OS ONAN ODTOAEONNDOTOLTMNON THE HOLPONANOKGM AONONDIOIITFANNOOF OP NAUATHATNHNAOMTUNOTHMASd dd SCCKOOCCOCOOCOCOCOOCOOCOCOd HA tHOOOOCOnHOOOOOOCOSC9000 S8s0090000000 000 COCO COC OOO G0GCCOS99000000000000 e 2 * » MOT NOHOOCNOLANMHAOMNARALTDOWOHONHAHHROCHTAMATOM FAA DANOVVATANAHHOWOOOWOLCODAMNTNDOMOMNOCOOONVKORN SAODDDDDDDDOODOOOOCOOONNHOnHHOOOOCOOOSDSOSCOO0000 oVoloKolololoKolololololololololok ooo lololoso ly olokokolosolojoloyojojolojojoje) * @ 8 2 * . 8 © * CAMTAODAE TONAVDAHDOH DOC ONAUNNNOFOTODOONANMNS COSCSCHONDHOTOTORHOOOWNEATNOMALOHONNNOHONOHO 9999999999999 0RF90 FAFA HT9O999009099995595595959 ©99000000090000000000000000000000000000000 * * of o 8 @ * @ ATNUNDAOS TONHAONDAALDAODEAANE SNE OTNONE AD NAT ANAHADOCODONONOMHHNNATENDOTHODOFTONAMONANIOOOdA SS9D90DDG0DGDD ODOC OD OS ANHDDANORHHOODGOODGGN9D00N0 3990999595950909999959599595599599995999590599900 Ce Re en) ee e ¢ # 8 NOMNMNMAMODALTHAODTOTTDOONO NOC AARTNNOTTHORHAHONN AAANOTHONANAUYAM TAN TAHNVAOONTHAVNOTIUVUIVAVNOdA 99999990909995999549N9TNINNNSAIdASD93999595959599 3999909000950590909000909059099099999059095500 2 A als St, oF aM DONATARDATOOMDN ODDTNDAMONANN TAMANMNIODOMNEW NOOCOHHUNHMNOUN~NDNONANDNDWOCONONMOTANNMONSOOM SAPDDDCDDDOCOC OOD OO nT ANHNTNONNHOHOOOOOOOOOD000O S9990000000D9000059D0 0009 nF nHOFHOOSSD0000909090900005050 * * DMNOMNNE DHHOTHAORHATH ODNDAMNE SOM AANETTTOOTINITD DADTHONANANATNONTOTAATDOANOOMANANTHMNANANONS S9909D9D9D0D09D5D2D9000 FONE MANO nHOTHOSDO5SASISDS59N5905N0 9090099009099059000090000n00990009900905959050905 * * es © NNGTOCHMONONIONNDONAME TNODAVAMNANDAVNTAAAHCUDON PAH HODHONVNIOVNNTSY~ADO~DOAVUHMAHROTUNTISITIIIANGONG SA9DDDDODDOODOOO ON TOON THNOVOH HAD OSCOODO0SCS00 CODDDDDOOCODODOOOCOOOCOCOSOrRAHOOCOCOOCOOOSSCOINSDOOSo ° * * * . * AHHOMNMH AL TAVADOHODOMMNTHAAEOACTAMAOCNHAOCHOOK OFFA NONN AMANDA ATONDOON NAN Ode OFTTONSd dAdo CODCOD DDODCOSOOMMOOMOLFHOWO nH HOODODODODDDDO0D000 e000D0D0DDOCOCOOOCOCOCOOCCCCOC COCO OCOCoCCOoOOoO0OG00000 * os ee on @ * TNOMW~ NE OONNDE NNNKAHANONHAAANNDHEF ONNOOF NE NON NAHOOHHOTHOONTOMNONONEOONONSOMOTMOOKHOnHHOOnHAN 999939599999595955959955990xH995995959955599995999595959 eon0p000D0C OOOO OODOOCO OOOOH OOCONDOOD OOOO OCOCOCC00090 DND~ONTANNADAD~ONI*M*NADAN DADAN*ITHLO>~ DAS Neaddddddddd = Sdddddddd dd Spectral Estimates (U(r, s)) for Data Set 2A_ Table 9.19 19 18 17 16 15 14 13 12 il 10 MOU TTOENCHTTDOOOATNOMNATAM A OF OTNMOTHEMNMOM S8OSCSCSSSCSCSSSCSOSCKMK OOK dANMNNAGATHOOODDOCOOODRDCOD0000 ecoocoescococcescoccsoccsccooooooSoSoSoSoSCOOCSCSOD05D00 ecssoossoscecoscocscscsoscosoSCSoSooCSoSoSSSBOSCOSCSDSSSS MAAN-OATHDAAOOHOALDAMOMTADHODAA-OHDONe TOON SOCSCSCO Md O dd ddd dOddATOTUNANHOOCONHOOOCOOnHO 239999909099090009099090959999999055990909059005955090 ecesecoeccsecsococoscscoooS oso SCSCSCOCSoOOCOSeCDo009000 NNOCTNMDNOONMOCTOOCAHATMNDOOONDTOOOVUDDHOLFOOTNADN SOCCSCOd ddd dd ANA ddd ANO Td ddA AIOOCOCONHOOOOO000 CT TT Tn Nn nN a nN a N= kk --) ecsococecocsecoccsescosscoescooseoooosoSoosooSoSeSCOSCOSSCO NNTMNE SE OATONMDORHOLDAODMOKOMAHOLOFALEOOTOOM SCO90F99 CO dd ddd dtd dN HMO ddd ddd OnHOOOOD000 eoocoocoocescoooococosscsccsesspoooSoSoSoOOCODODOSD000 ecesecooceocosocescosoocoseoo ooo oOeo So oDDD00000 MOTANTONEONNMMNDDNOCODANAUMNETNNUMNVANHALLONMLAT SCOSCSCOSCSOS ddd HA HOOCHNANMNN MN dt ddd On AH DOOODOSD ecosecscoooooeosscococoesccosesseesoesoscsosec9esese9c900900 ecocsoocooceocococococococscocoscooSCoCoooCOCOCOC BOB 00000 PMOTTTOOMAADMOEVNOADOCOPTE TOOHEMN CT eNHONNOAS SCOSSCSCSCSMTHOOCKd nd ANUMNONN ddd ddd HOODOOO ececcooococococoococccscoocooesscoccoc9c9cccooCCSDC050000 ecoesceococoqcqcoqcooc9cococeoscooocooocC CoCo oOCO DOOD 005o TADOMNTCNADTANTEONAMNNEOTNHNONETNNODRNANALER YS SCOSSCSCSCOCMNNOn nA AANMNMN TOO NN ddd ddd tOnRAHOOOOO 99999395999999999559355359959959559555999995999595 eocooeqcqooocoensococcococococceceococsccaco90c9C0oONADO0o NOOTT TNE TONANTNNODAATHENDNNDDARAOONODONOON SOSCSCSOCTOSOK ddd dt nN ODALPANMNN GR ANNA HOOSOK HH OOS ecoococoocoseccococooosocosesesoscoesscosessesssseossS sooseocooocococoocossosssesssssessesesoessesoseocos NOW TMM EONRFFOONUMNET TONE DDNOMONHRHOLOMOE-ODHANTN COSKSCKCSCSC MKF RM ANUNMNDOLNNANNAUNNANNAEHOOCOORHOOOO 93993939399999999999399593959590599993990599995999559995 eso messeocoesoooocoessosossocsecsessesoessseesoses TOA™ DDO TDOKDINAANTNOAMMNONOCSOOMTDOTHARE OES OOT SCHOSCCOCH THOR A ANAUMNTMNNEOMEMMMNUNAE HOM OOCOOOOO cecososcocoocoocococoocoo coon HOCOCCCCCCSCCCSCSCSCoCoOoooSo escocoecscooscosesococscsososscscsecososesoesecsecseces ONODDONMTHNOD-DOOFNNO-ONNOMAMMMNDDOADOODON SAMOS Od NN ANON THOTOMNTMNNRHOONHSOOOSONS ecesoesscoosecsososcoornnsocoecescescooesscsesooses3e ecescooococoecscsescseossescocsesesoecsececsecseoesessessecosS MADE DANTTANE NS TTNODDDATNNADDOMANHODBAOCOnRAN CO OS SAO NNN OMA EMA OES TMNN AANA OOR HOODOO ecocoosocoosoooseoeocooornd#rAWOOSCSSCSoCooCSCSCSeseeesSa ecoscoscocoecscoocoesosooososeosesc“e“osessesosoeoe WADODTON THOTTOWSCOCONTHDONEANAROOCOWNFONNOES S CPSSOKd ded ANTOOOMOTTNRHONNNNNHHOOMANHOOO escoscoooooecoecoscoooddndndrnrOoCoSoooooeeoeooose scooesoocococoocoosooocoseceoooocesseeoeeoeoeoeos NOSCDOOMMF RN AMNANNNNTMAOMAOCNANEOORENONHOOW DAMON AA ANMNORK ACN TEMOCMNUNARM On AAA O 3399555595959 9090 95 Sdn AMAR HOOOOOSOSOSSNSSSCSS ecscocosecoecoooescocoooocosceooooesooseooaeaaaeogan MOTNAOOAATODONSONNDOGS COOP NNE ATONE OA 700 COCO OOCOORAMNNOTANADNITNOONMNAUN ddd dade Secoooooeooe SC OOOO OOK NNdnddRHOOOCOOOSCOooSSSSeo sesoocososoSoCoooSooSooSSSoSSSsSeSeSeses9esosssoes TOMOCOAHOMOMAEMTNNOTMOTNTANONE TNOMOOAAMNOT CHOC THOOCCOCOCORM ARNO ENOMAAETODHONENd AN AOOnAAS DODD. O SOS ODOCOOOO SC SOSA NNANANNAHS999955999595550 coco coCooCooSSCSSSCSOSCOSSOSOSSSSSSeseesesosoosseeS ONMOMODNE AE ADTONAMDAONNNONTOHIOTNONOARE ES AO Ooo ddd6 HOOONdE de Ado Ot dese OROEMdddOHOOOO SCOSCCOCOCOCOCOOCOOCOSCOM AORN AU TNAHOSCOOOCOOCOOOOOSCS ecoscoeoscooosessoossoscococsessoooeseseseosssescfe DTONNMOAOLCNOPFOANDTOOCANACNNAONAOMNOOMONN AD CAFO RNA OMAHA HAN ADNNOOMEOMNDOCMANANNHAHOOOOD SCOSCOSCSCOOSOOCOCOO On A MANNO CHMHADOSOCODOOSONSO900 eccoscocooosposoSoscooooonNHSoooCSSoSoSSoSSESeSoooocS HM TMONOLDALOMMEANMOOAOTOOTNTNNNMANTOONME OS SAAN RAR ANN ST OAMONATNONOMNAUME MAA SOSOS SOOSCCOCOSCSSCOCOSOSH ANH HNANMOTAHHOOOOCOOOCOOSCOOSO eescocooeseoeoeeooooooCoSrAASoOOSCSSOCOSOSC Oooo oSo WAUAMNTOMODEOCHON DS OFOANUMNHONNMOMOS EMNNONDOANNH CHAAR UNANAANNNMN MN RGAE OSNODAONACNAT ENA ANON SscoococoocoscoooeonNnoAMMerMNAoCoOCOSooooooooSo ecoeoeoooeosccoososeoosescosoooooSa Sooo ooooeo MOTRON FN ONMN ARH TNONDNONN SAH dA ONONHANOROLEEM Cd ddA ANMOSNODODONEONAKMdR nde ARMOOoS ssosssecososesossHesssosesssessossosses SsecceoecococococoosscSeoocescsessescoooeeoeoeeS SADMCHEMNASADM-CMHEMNAS Neadddddd dato OScCoceoces Spectral Estimates (U(r, s)) for Data Set 3C Table 9.20 = 16 15 14 13 12 11 10 AT ITANNEOTOOTHEAMNDOCOANHHDOOODOMHANMNCOONE Tene SSOSOSSCHSSOSSCSOCOOSCK A COSGNMNHOOOOnHHOSDSDOOCOSOOSDN000 ecoecco90coSCooC eC CNOOCOoCOoCOC OOO ONCSCSCOOCNO COS OSCOS eoosecocsesocsocsoocsocoescsoenassocCoCeSSsScCoeoeSeSED TARDNMADAVOOCOWTEOMNAOTTOMHANNHOMNARAOCMHOOLAOT COSCO KMdnH AO ddA NANG Ite Gd ddd ddd HOd dd HOOKO 399999939992999999399999059090999599999090059530099 ess0cccoo00c0o°00°0000°00000E0900000C0C0700000000000O000 FORDSOMNOOVDANADADTHDAONE YT VOHHAALPANMMNDONOOOOW SOCHHHOOCOCOH THOT NAAM MTNA NOOKdHAAOOnHHOOOO ceooo00ooco CoC CoCCCoCSC0GCgCSCCO0CCCOCCODCCCCC0C0C000 escoocoococoococsscocoecoocessoscoocoCoCcC OCOD SCND0R000 TOS DAADADAANAMODOOOIOAON WOTVOUR MT HODONNAUM SCOOSHHODOON NF nH AUNT HTN add Ht HOOH RM HOOKHHHOOO Ssoesc000ccc0SogSSC90000q0 C00 S900 S90000 500000000500 sssoocoocooscocoscoocesccocoessoocsosoescceeos9cp90e0sc—9e90000 MOT DADO HNOANONAOAHNOOMNAMANDAMOOMNALDOODDOOCN S209 Sd ddA NNN DAC Hdd dO HH OODODOOHOOSD Se0000o0CC CSCC CCoCCSCCSCONSSCSOSDCCOCOCOOOCCoOCOCe000 esco0c0cc0g0Gc 900 90OSCOCCoOCC CS oOeESCoCNGSGSGOSCOCOCOCCO0000 TDAADAAAHOMATOADITATONADODNODOOMHDDODMNNDACOM SOODO Md dH TOKAIDO AINAN TOM AHA HNMNNHOODOOHHOOHOO eccococ0o0cc0909C00000000000000000000 000000000 loool Tole lo fojolololokojosololojojofololo kolo fololalololololololo lo fololololoN-) NOdAHIMOOTNONTONENDOODSNONHONNDAROAArPOKRO CHM COd dtd ddd ddd dd AIAN NUTT UNVVUNA HOH OOn ROH AHO 933935959935959595539959959595935559599935999599959999995 Seo9g000000000000000000000000 000 000000000000 ATOAANDODAOTTANOODIT~OOANODAUL TANOHTANMDOANHH SOM NTH OOCONF TH ANUNAMNT ODOMMNNUNA ddA On OOKHO S900g0g00C CODD C CODCOD D ODDO DDOOOOCODCOO ODO 0000000 99999999099059599995959050090999099995900000905900 ROEAAHOHONALCAOMNTODSONDNONAANMNOWOOAAMANE ODO SOCOM ddd dtd HANAURQMNOATONMNMNNA ddd ddd ddd AHO 399999090909995905959599599x4d999999995995999550000 SSscq909905D00000D00059 00000050090 590000900000000 TOOODOADTOMNTATDMONNNOP DOONANMATHNITIMANDOAMO SOOSCSCKMT AANA TANNA T TTONATAUNUANN ddd ddA HOOHHO S990000COCOCOCOCCOC OO CCOn HOODOO OCOCOODCO460000000 Sec90ggg0gDoCoCSCCCOC OCOD OOOOCODDDOOODDOCOO00059000 DOMAHODTANNNAHOTOTARNHNOONOOCOOHODONANMANYTOO SAAN AAMT NOP NOTNENNNd ddd HHOOOOO Seeggogoooooo oo oCoCos odd HOOD ODCDOOSDDOSDOOC0000 sesc00900000000KCog0KdKgg)00000000000000009000000000 WATTORHDANDOTOOMNANALANTNODTCONOLCAONHAnTUL OAH SAH On HOG VAAAUMMNADHOOTOTAMN AUN AAA HOOOO SOODDOCOOOOCOOOOOCOCOHNHOOOOODDOOC OOOO 00000 —ofoloolololololoholololofololololololololololololololokolololololokolololoheo ke) MAANTAANANAAGEOTHOHNOOANNHAAHADTNOTHNMEOAAHONW SCOOK MFO nn TANNA AMT TOODDAHDOCNINNNUNAHOOOKNHHO SCCoOCoOCCOOCOOCOCOCOCOCOCORANNODOOOCCOCCCCCCCCCCOCCCO eccococo909cc09CCCOCCCOCCCCCOCCCCCCCCCCCCOCCOCOCO TOOODOAAONTHDTNOATIATNOANANANHE DE NOOODAOHN SOSCKOSCCOCCKHANAHATMNNOKRRADOLON TANNA HANH OOOHO 999909990959 0900090 dee TNAI9059995999590909909905 Sec09c00c00cCoCoCCCCCOCOCCCCCCOCOCCOCCCOCCOCOCOCCOCCOCOC0Ooo MITADN OM ADANONNNANHOONAOCDNNOOLADADYNMI TT ANONE SAAC OOCOSKT NAA AANMNNNDTOTNAHONDONNM dd AAA HOOOKD SPC9DDDDNDOCD OOO OOOO SDORNNN HNNADDODODDOOOONDDSD0D000 00999999999090959599595990090990909909909099999095 DNMNOODOMANANAL HOOT ON ALC NMHONNNDNARODAAOOMD On dH CON Md ddd TONHOOORHNONDDEVANHONHOOHOOKNO 999929595999999999595990490TVHAID9999099999999905 990959999999999595353990990959950590909900909099900900 AMADLDLEATAMOMNTOO DOIN T ONHIIMAAHAAANNEONNNANDA CHODOSTHNHOTRANNHOSONTDATNTMONDNOMNAVNA AA AAHOnO SCODCSOSCSCSCSODOOO COSCO nH HNDHAONTNAHODOOOOODOD9D000 ess900000005N09000500000009N00090009090909099009090 MADDDTOHNOVO~NADOTNOHDA~NANDOMME ON MOWNNOOAD> DADDIAVAHANANUNOTNHHADOAOOSTINOTNNANANd ddA DOOD DIIID ONO DIOS NAMA Or oOMONNHDODODOOSDSIOSCONNSO eses000z0090005N5N09N00000 FS9D000D05D000000000000090 MNANDAA ONY ONONTTTINATOKME OH ONANAUNTAAANEMNO DADA ODHAAVAAUYAMDMNANTAANAOGNDIAN TNAUNA NOMAD SCOSCSODDSCODOONCOONNAKAAMNT OE TdHHODOODOSCOOODNDSD ecs9009000000000000D00000 0000000000 0000000990 MWNNORADALMN TAP NNHMANOWCHOME OM TE DOMNANHAVA TAH Od ddA IAVAUNATONNAOTOMNE A TOALMMNMNVANHOOKnd AHO SOSCKOODSCOSCOCOSCO nn HOHMNITNTNHODDOOCOODODD0OGOOO eo00000000000000FT000000000000000000000000 NOOTTOP ADAAADAANOHD THAD TDONAADAAROACOTTOON SODDDCODOO nN AUNT TTANONMAK TENN HHOOODODOCDOO 9999959595995959595959595959%4N959595999959959995959955559 eoD0g0D0DCODDOOD00C0COCOCNOOOD COO OO OSC OOO CoOOCC000500 DAD~ CH IMNADAL~ON IAN GADANATNCEH DASANAFNCEH DAS Nddddddddd Ad SSCSSCOSSSCSSCSSOSCOSCOS CMH AANA NAAN 133 PART 106 ANALYSIS OF WAVE POLE DATA Determination of spectra and correction for wave pole motion Three wave pole records were taken at the times shown in Part 7. The records had been read at an 0.2 second interval at Woods Hole for another pur- pose, and the numbers were made available to us by Harlow Farmer of Woods Hole. Every fourth point in the sequence was used in the determination of the spectrum, and the result was that the first series had 1,758 points, the second had 1,686 points and the third had 1,764 points. Sixty points were estimated for each of the spectra. The formulas given by Tukey and described in Part 8 were used to compute the spectra. Since the points on the record were 0.8 seconds apart, and since 60 lags were used, the result was that the AE values of the spectrum for frequencies between p = 2n(k 26 and yp = an(k + 4)/96 would be estimated and plotted at the point p = 21(k)/96 as k ranges from 0 to 60. Frequencies above 27/1.6 would be aliased. The values for the three spectra which were obtained were averaged after a study of the individual values showed no statistically significant variation at the 5 percent level from record to record. Nevertheless, there may have been fluctuations from record to record due to variations in the wind field which would show up ata lower level of significance. The average was multiplied by the correction factor derived in Part 8 in order to obtain the true spectrum of the waves. The result is shown in figure 10.1. The solid curve is the estimate of 134 WNYLOSDS 31d 3AVM G3LOZYYOD Ol a2nb14 OS BY OF HY cy Ov SE 9E HE BE OF BE OS HZ ZZ OZ GI 91 vi 2 Of 8 9 & 2 O SGNO93aS 9 — SGNO9aS 8 — @GNO0aS g — SqNO938 ¢€ —— SsGNOd3S 21 — SGNO93S 9) —— SGNOD3AS ve — SGNO93S 8» — WNYULIAdS GALDAMYOINN oe ceecseseeree GNWG JONFGISNOD %S Y3MO1 --—— -—- WNULISdS G3ILIINNOD GNVG JON3GISNOD %S6 YaddN --—-—-- 3(L4) sIS9= the true wave spectrum. From the above lengths of record, one can compute that each spectral estimate has 174 degrees of freedom so that the dashed curves above and below the solid curve show the upper 95 percent and the lower 5 percent confidence bands for portions of the spectrum where it is not rapidly varying. The bounds are prob- . ably quite a bit broader at the point k= 10. Stated another way, the true spectrum would lie between the bounds shown fer nine points out of ten, where it is not varying too rapidly, given that it could be determined from a much larger record under which conditions were stationary. “ Comparison with the Neumann Spectrum This spectrum was compared with the theoretical spectrum de- rived by Neumann [1954] in two different ways. The first was by plotting the theoretical Neumann spectrum against the observed spectrum, and the second was by computing the co-cumulative spectrum. The comparison of the spectrum was obtained by evaluating the Neumann spectrum with dimensions of ft@-sec for a set of dif- ferent wind speeds at the frequencies given by ». = 27k/96 and multi- plying by 27/96 with dimensions of sec’! to get an estimate of the AE value with dimensions of ft* between w= 2n(k = 3)/96 and pp = 2n(k +5)/96 . The quantities are thus directly comparable. 136 a From Figure 10.1, one can see that the energy for frequencies less than 217/96 is negligible and is probably due to such effects as a slow drift of the recording instrument and a tilting back and forth of the wave pole due to the varying pressures of the wind act- ing on it. The total E value for the spectrum (E equals twice the variance of the wave record, and it also equals the sum of the squares of the amplitudes of the spectral components) for fre- quencies equal to or greater than 279/96 is 4.94 ft”. When the upper and lower confidence bounds are taken into consideration, as will be explained shortly, one can conclude that the true value probably lies between 5.28 ft? and 4.59 fe (See also Table 10.1.) Since (10. 1) E = 0.242 45)? as given in Pierson, Neumann and James [1955], where E is in ft and v is in knots, an E value of 4.94 st? implies a wind speed of 18.25 knots, and E value of 5.28 ft implies a wind of about 18.5 knots, and an E value of 4.59 £2 implies a wind speed of about 18.0 knots. The Neumann spectra for 19.00. 18.5. 18.25, 18,00 and 17.5 knots were computed in order to cover the above range, and a little extra, and plotted against the observed spectrum. The results are HSA7e shown in figure 10.2. Figure 10.2 shows that no single theoretical curve for a parti- cular wind velocity lies completely within the bounds of the 90 percent confidence bands. In general the values for the observed spectrum are too high for wp = 21(11)/96 and 217(12)/96 and too low near pe = 20(15)/96. However, it is also evident that at least one of the five points plotted for the five different theoretical spectra falls within the 90 percent confidence bands on the observed spectrum for all values of k between 10 and 30. A variation in wind velocity of +5 per- cent about a value of 18.25 knots is more than sufficient to explain the observed spectrum at each of these points. At values Shave k = 30, the observed spectrum is a little above the theoretical spectrum. This may in part be due toa small amount of white noise. An appeal to the meteorological turbulent variation of the wind speed and to the theory of wave generation and propagation © must be made in order to clarify this point. The observations of the ATLANTIS as plotted in 138 -6£1 - WNYHL93dS G3ANSSGO SNSHRA VHLOAdS NNWANSAN 40 LOW GNVS 3ONIGISNOD ZS Y3M071 (370d BAVA dO 103433 YOS NOILO3ZYNOD Y314¥) WNYLOAdS G3AN3SEO GNV8 3ON3GI4NOD %S6 Yaddn 0S 8) 98 + sb Ob ef of be ze Of 82 921 be 22 02 $+ ——_ + + ——_ + _+_ + +++ — +$— $$ +} +} + 1 — + 1 +> 1 +4 " ae Sn ~~ SLONW Os'2I SLONY OO'8! SLONN S28! SLONW OSB! SLONW OO'6! 8! 91 AJ 301 B14 02 of os 09 afi4) figure 7.1 show that the wind speed have values of 22 knots, 17 : a ig -@—? 9 Sg) | 19 bi ell 95 Oe Ve-Ozee ‘a Looe Ol psi evi 5809) ca adiouiy ol es &| fe oF 0 08 447 957 on v-€01 613 142- FIG.10.3 WEATHER OBSERVATIONS FOR OCTOBER 25,1954 Z O¢8l bS6l 190 SZ Z O£€2@1 pS6I 190 SZ H-¢ 01 614 9-€ 01614 -143- Z O€OO ¥S6I 190 SZ =m sz 1, ‘2IS05 8 $92: SS Si 3-01 614 A-€°0! B14 treated as if every other one was independent, and since they are distributed according to Chi Square with f degrees of freedom, the point on the CCS curve is approximately distributed according to Chi Square with 60, i i 2 Maik ie (10. 2) Nees) == 5) ANE, z | n=k By a degrees of freedom (see Pierson [1954]). Then the confidence bands accord- ing to Tukey for large N are approximately given by multiplying the point on the CCS curve by 1ot WIN 1g YN and The observed CCS curve is shown in figure 10.4 as plotted on the theo- retical family of curves given by Pierson, Neumann and James [1955]. The agreement for some CCS curve slightly in excess of 18 knots is quite strik- ing although the agreement is not perfect. The CCS curve is too high at the frequency side of the scale. This may in part be due to the summation of white noise errors at high frequencies, Also if the pole surges back and forth in the long period waves, it may en- counter shorter period chop while moving back in a trough in such a way as to falsely assign their height contribution to a higher frequency. 144 FETCH GRAPH CO-CUMULATIVE SPECTRA FOR WIND SPEEDS FROM tO TO 20 KNOTS AS A FUNCTION OF FETCH MULTIPLY TO FIND JE BY SIGNIFICANT WAVE HEIGHT AVERAGE 177 WAVE HEIGHT YoHIGHEST WAVE HEIGHT Fig!0.4 OBSERVED C.C.S. CURVE DASHED LINES ARE 5% AND 95% CONFIDENCE BANDS “145° Tabulation of Data The spectral data on which the above figures are based is given in Table 10.1. The frequency is determined by the formula }» = 27k/96, and the the entries are given in terms of k. The first column gives the period (96/k) which corresponds to the appropriate frequency. The next three columns give the three spectra actually obtained in terms of the contribution to the total variance in (£t)° of the er made by frequencies within the band. (The values should be doubled: to get AE values.) The fifth column is the average of the three observed spectra. The sixth column gives the function o() as derived in Part 8, and the seventh gives the function (designated by H(t) by which the observed spectrum must be multiplied to obtain the corrected spectrum. The next column gives the AE values in (zt)? which are the estimates of the area under the spectrum from p = 2n(k - 5/96 to p= 2n(k + 5196. The column fourth from the right gives the sum of the AE values in the previous column from the given value of k to 60 and this estimates the point on the CCS curve given by p= 2m(k - 51/96. The column third from the right gives the value of ZN. (eqn. 10.2) for that point on the CCS curve just ob- tained. The last two columns give the upper 95 percent and the lower 5 percent confidence bounds on the AE values. The original series of points from which the spectra were computed are not reproduced in this report. They can be made available on requestto the Department of Meteorology and Oceanography at N. Y. U. [For references see Part 11.] 146 k PERIOD 0 a 96 2 48 3 32 4 24 5 619.2 6 16 7 13.7 8. 12 9 10.7 10 9.6 11 8.7 12 8 13 704 14 6.86 15 6.4 16 6 17 5.6 18 5.3 19 5.05 20 4.8 21 4.57 22 4.36 23 4.17 24 4 25 3.84 26 3.6 27 3055 28 3.42 29 3-3 30 3.2 31 3 32 3 33 2.9 34 2.82 35 2.74 36 2.66 37 2.59 38 2.52 39 2.46 40 2.40 41 2.34 42 2.28 43 2.23 44 2.18 45 2.13 46 2.08 47 2.04 48 2 49 ‘1.96 50 1.94 51 1.88 52 1.85 53 1.81 54 1.78 55 1.75 56 1.71 57 1.68 58 1.66 59 1.63 60 1.60 RECORD #21 0.02054 0.0242 0.0260 0.0142 0.0116 0.0097 0.0074 0.0047 0.0068 0.0187 0.1045 0.2004 0.1826 0.1534 0.1311 0.1108 0.1012 0.0989 0.0793 0.0705 0.0648 0.0596 0.0568 0.0568 0.0580 0.0575 0.0498 0.0325 0.0236 0.0201 0.0184 0.0151 0.0166 0.0163 0.0131 0.0118 0.0113 0.0107 0.0095 0.0082 0.0053 0.0035 0.0036 0.0039 0.0035 0.0038 0.0036 0.0041 0.0044 0.0036 0.0038 0.0038 0.0032 0.0030 0.0026 0.0026 0.0023 0.0018 0.0020 0.0018 0.0015 RECORD #2 0.0399 0.0309 0.0178 0.0089 0.0068 0.0068 0.0057 0.0042 0.0041 0.0107 0.0713 0.1853 0.1954 0.1563 0.1340 0.1029 0.1042 0.1062 0.0937 0.0722 0.0508 9.0552 0.0731 0.0651 0.0484 0.0403 0.0391 0.0316 0.0250 0.0230 0.0197 0.0190 0.0212 0.0172 0.0101 0.0078 0.0083 0.0083 0.0086 0.0083 0.0071 0.0054 0.0051 0.0060 0.0062 0.0050 0.0042 0.0045 0.0042 0.0036 0.0029 0.0027 0.0018 0.0016 0.0023 0.0029 0.0026 0.0027 0.0029 0.0024 0.0021 RECORD #3 0.0118 0.0160 0.0191 0.0103 0.0055 0.0051 0.0073 0.0060 0.0057 0.0060 0.0444 0.1570 0.2292 0.1704 0.1102 0.1033 0.0977 0.1078 0.1194 0.1024 Q.0812 0.0519 0.0492 0.0622 0.0522 0.0401 0.0271 0.0209 0.0225 0.0213 0.0195 0.0233 0.0240 0.0148 0.0082 0.0091 0.0103 0.0072 0.0054 0.0061 0.0082 0.0084 0.0064 0.0043 0.0043 0.0049 0.0049 0.0037 0.0031 0.0033 0.0027 0.0025 0.0025 0.0027 0.0028 0.0027 0.0024 0.0024 0.0025 0.0025 0.0021 AVERAGE 0.0241 0.0237 0.0293 0.0111 0.0080 0.0072 0.0068 0.0050 0.0055 0.0118 0.0734 0.1809 0.2024. 0.1600 0.1251 0.1057 0.1010 0.1026 0.0975 0.0817 0.0656 0.0556 0.0597 0.0614 0.0529 0.0460 0.0387 0.0283 0.0237 0.0215 0.0192 0.0191 0.0206 0.0161 0.0105 0.0096 0.0100 0.0087 0.0078 0.0075 0.0068 0.0057 0.0050 0.0048 0.0046 0.0046 0.0042 0.0041 0.0039 0.0035 0.0031 0.0030 0.0025 0.0024 0.0026 0.0027 0.0024 0.0024 0.0024 0.0022 0.0019 Table 10.1. $(n) =2.050 -2.719 =30357 =3.890 4.443 4. 864 -5.168 -52380 5.487 -5-500 52433 50274 5.094 4,857 =4.585 4.292 =3.993 =3.696 =3.449 -3.128 -2.863 -2.620 =2.391 -2.181 H() 1.8119 1.7974 1.7470 1.6660 1.6044 1.5321 1.4629 1.4110 1.3430 1.2921 1.2483 1.2094 1.1775 1.1500 1.1264 1.1060 1.0891 1.0751 1.0640 1.0532 1¥0448 1.0379 1.0321 1.0273 1.0264 1.0300 1.0180 1.0140 1.0110 1.0090 1.0080 1.0070 1.0050 1.0020 SE 0.018 0.020 0.041 0.245 0.580 0.620 0.468 0.353 0.284 0.261 0.256 0.236 0.192 0.151 0.125 0.132 0.134 0.114 0.098 0.082 0.059 0.049 0.044 0.040 0.039 0.042 0.033 0.021 0.019 0.020 0.018 0.016 0.015 0.014 0.011 0.010 0.010 0.009 0.009 0.008 0.008 0.008 0.007 0.006 0.006 0.005 0.005 +0005 0.005 0.005 0.005 0.005 0.004 0.004 147 Co- Spectram values from the wave pole data. Twice the Degrees of Freedom fo: r Cumulative Cumulative E E 4.98 4.96 4.94 4.90 4.65 4.07 3.45 2.98 2.63 2035 2.09 1.83 1.59 1.40 1.25 1.12 0.99 0.86 0.75 0.65 0.57 0.51 0.46 0.41 0.37 0.33 0.29 0.26 0.24 0.22 0.20 0.18 0.17 0.15 0.15 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.06 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0.004 2773 2754 2732 2690 2525 2499 2692 2811 2823 2769 2723 2743 2819 2852 2804 2707 2683 2786 2911 3083 3272 3300 3273 3240 3189 3197 3439 3609 3533 3447 3419 3361 3287 3223 3156 3035 2891 2745 2603 2466 2324 2182 2039 1884 1719 1553 1379 1206 1033 863 690 516 345 174 Upper 9 oop ol, Band on AE 0.021 0.23 0.049 0.289 0.687 0.734 | 0.554 0.418 0.336 0.309 0.303 0.279 0.238 0.179 0.148 0.156 0.158 0.134 0.116 0.096 0.070 0.058 0.052 0.047 0.046 0.050 0.039 0.025 0.023 0.024 0.021 0.019 0.018 0.016 0.014 0.012 0.011 0.011 0.011 0.010 0.010 0.010 0.008 0.007 0.007 0.006 0.006 0.006 0.006 0.006 0.005 0.006 0.005 0.004 Lower 5% Confidence Band on 4E 0.015 0.016 0.034 0.203 0.482 0.515 0.389 0.293 0.236 0.217 0.213 0.196 0.159 0.125 0.104 0.110 0.111 0.094 0.081 0.068 0.049 0.041 0.037 0.033 0.033 0.035 0.027 0.018 0.016 0.017 0.015 0.013 0.013 0.011 0.010 0.008 0.008 0.008 0.008 0.007 0.007 0.006 0.006 0.005 0.005 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.003 Part Ii THE STEREO;PAIRS, AND THE INTERPRETATION AND ANALYSIS OF THE DIRECTIONAL SPECTRUM IN TERMS OF WAVE THEORY Introduction Of the one hundred stereoxpairs of photographs taken by the two aircraft, three were selected by the Photogrammetry Division foe spot height readings on the basis of picture quality and lack of cloud shadow areas. After leveling, Dat Set 1 was found to have a serious barrel distortion so it had to be abandoned, | The original numerical analysis of the two remaining sets and the numerical analysis of the reduced data were described in Part 9, In this part, the diffis | culties which were encountered in analyzing the original results, the way the decision was reached to use a smaller area of points, and the results of the analysis of the modified data will be described, The stereo pairs The two sets of stereo pairs chosen for analysis are shown in figures 11.1(A), 11.1(B), 11.2(A), and 11.2(B) where the lead plane picture is the first one of the pair. In order to be sure that the photographs chosen were not chosen, | say, for high waves in the vicinity of the ATLANTIS, 10 photographs as taken from one of the planes were picked at random from the 100 photographs available and the wave patterns in the vicinity of the ATLANTIS were com- pared qualitatively with each other and the two photographs chosen for analy» sis, There was no apparent difference in wave heights or wave patterns, so that it seemed safe to assume that the two pairs chosen for numerical analy- sis were representative within usual sampling variation of the sea state. 148 FIG. I. STEREO PHOTOGRAPH FOR DATA SET No 2 ( LEAD PLANE.) 149 FIG. II. STEREO PHOTOGRAPH FOR DATA SET No 2 (FOLLOWING PLANE.) 150 FIG kz STEREO PHOTOGRAPH FOR DATA SET Ne 3 (LEAD PLANE.) 151 3 STEREO PHOTOGRAPH FOR DATA SET Ne (FOLLOWING PLANE) 152 It should be noted that the exact pattern of the waves shown in figures 11,1 and 11.2 will never occur again and never occurred previous to the time of the photographs. However, patterns with the same statistical properties should Mreurlevery time the gross meteorological conditions are the same, The leveled data The spot height data after leveling according to the procedure described previously was plotted on a grid 90 points high by 60 points wide. The values as given in mm (x 10) were then contoured. The contouring was done by interpo- lating to the contour value along the lines joining the points where the data were plotted and connecting a interpolated points by straight line segments, Figure 11.3 illustrates the procedure employed. The contours can be roughly inter- preted in feet. To convert to feet exactly the values shown should be divided by 016. The contouring procedure illustrates the effect of the spot height readings. Any irregularities in the sea surface of shorter wave length than 60 feet are essentially undetectable. The exact position of the height contours cannot be determined, but if they could, they would wiggle all around about the straight line segments shown, break off into little closed contour patterns, and show a fine structure all the way down to the capillary level. The contours for Data Set 2 are shown in figure 11.4. The contours for Data Set 3 are shown in figure 11.5. The spot height readings are inaccurate by the very nature of the stereo process just as any system of obtaining data has inaccuracies init. Thecon- toured values and the values tabulated in the tables given before should be P56 du at ji Ol ro my pt ol + ob hie + OF W450 = +t; (oe) oO @ Ne Oot+t M+ sete on Ripe i+. A eats () a Se Ot ~ —_— or t + w+ be Ab at, nt. ol b NF oO + dP Sar at + N n7 @ ° ° Por) aor, ogy g et ear. on ) os +a 1 nm at -@ ot+ \ (B+ B+ Gt St at Bt OF By 7 7 SS ae Wee . 20 ~14" 22 ny Es 15 at o ho + tt 4 =20°-26 -29 -IT., 3 wis, Sy tbo + an fia -4 34 6 +o. + “bit 4 oe a PBS BS GEHBEZ =) lO =e -24 -4 Sp RP gk ak caer Oe: 744-39 -22 -14 -14 =13 X 720 - 19 ar a ; -- + ack) FER) =e SS yc) nN eS £2 1-22. -5 + HS oF + oe y pe ot =35 2p lo SG f 16-30 -16, : Ne EKG | bee eles 21, -1 18 \25 : 32° -19 -27 -i3. ° ° NY ° c+ +\+ +4 4 2 733 : 21) 26) 1G) er i20)38) aia eye ia : . Pree ae yy ees . "28 . 22 25 IR 15 33-327 22 10 ! } ew eteoceet * + pas + + + -26 -31 322 -27 2L\ 31 2 22 x -8 -12 -10 Oxudine2 293" 45.55 G6 yak! piSecSi lOP aie GIZe Spel sae Su maliGeealirg FIGURE 11.3 EXAMPLE OF CONTOURING PROCEDURE I54 QRX Lp VY 7 * “lx 5 C= Soe Gp - = Ne . IGG Wy . LU WH Ns ‘, LZ. : ‘ CP Uo ee Vyijp-~< ) Che SN > / wi ~ \ ~~ ae, QP < 4 Y y \ ~ LY Uy, ZB yy, Z Y Yj J Y UY WW XK ERS Z Y GL eye Y if UG py 8 ZI Ly © d < tf on —— GY aC SSS WUGY Yl. “yf _ Bye CZ . Ne y; \ yy ? a / So é (ae ier, ms a 2 E y SS) CU IK Se Zi aie 7, ¢ a IY Zi . ENS Yj or X 5 Sai Si ON NS \ : = ; Sig 7 ‘ SS. \ > om \ ° \ . s es . = - Roa ee " . 7 Y 7, 4 \ Z H =a Neal \ a \ N, > aw 2 VO Y Yj UY Yo 7 j Y /j CYL “ Le YW) QLLLA Ta roe SHADED AREAS BELOW SEA LEVEL CLEAR AREAS ABOVE SEA LEVEL LEVELED CONTOUR ANALYSIS SET No.2 155 AREA COVERED IS 2670 by 1770 FEET CONTOUR INTERVAL 0.2mm. 0.1016 mm. =| foot Fig. 11.4 v; =e m/e ~. Z> 1G Za \ GY “< ea (is IN als) Qy/ “A — / | i iN. ‘%y NO} ° OX ay i) & | ; { (@ 4) NS Wes E Us 156 \ Less LEVELED CONTOUR ANALYSIS SET No.3 FIGUR! considered to be of the form AS es (11.1) jk = Ajk(true) + €jk where Nik is the tabulated value; Wyic(true) is the true value and € jk is a random error picked according to some probability scheme to be discussed in detail later. The values of ¢jk will turn out to be appreciable, and they have the effect of making the pattern shown fuzzy in detail. Due to the size of E ike statistical evaluations of the patterns shown should be interpreted with considerable caution. There are other errors of a more serious nature in the data as shown in figures 11.4 and 11.5. These errors will be analyzed in me following para- graphs of this part of the report. The covariance surfaces The covariances were computed according to the equations given in Part 8 and plotted on a square grid of points 4] points ona side. The covariance sur- face for Data Set 2 is shown in figure 11.6, the surface for Data Set 3 is shown in figure 11.7, and the average of the two is shown in figure 11.8. The units of the contours are (mam) * x 100. Shaded areas are negative. The figures show an estimate of the correlation (when each value is divided by the value at the center) of the sea surface with itself over distances of the order of 600 feet in any direction. Roughly the correlation is less than +0.10 in any direction at a distance of 600 feet. Again the effects of errors in the data have distorted the pattern. The covariances would have the dimensions of (ét)*, if each number shown were divided by (1.016)2. WI 7 INUNITS OF TENTHS OF men SQUARED SSS ———+4,3,21 ——==+05 COVARIANCE SURFACE Q.)0%6 rm. | FOOT. LENGTH OF SIDE OF SQUARE = (200 FEET SET No 3 ——--05 FIG. I.7 ——— 08) IN UNTS OF TENTHS OF mm SQUARED 159 AU COVARIANCE SURFACE a LENGTH OF SDE OF Sour E= ZOU FEET SSS L 7a Zz Z a= Z ===/= - = 2 L . = : / U/ Ts Jaan Ww \ ————\ \ \ AS N SS — a/ / y/ 7 { | Po 7 ies Les, \ | 7) 160 Fic. 1.8 SETS 2 AND 3 AVERAGED The spe ctral estimates The spectral estimates U(r, s) are shown in figures 11.9, 11.10, and 11.11. Figure 11.11 is the average of figures 11.9 and 11,10. The values plotted at the grid intersections should be divided by 1000 to put them in units of Cae. The contours are correctly labeled in units of (ft). As described in Part 8, the spectra have the property that the same value is obtained at Uf{er,-s) as was obtained at U(r,s). If these figures are cut in half by a line through the origin, the sum of the U(r,s) values on one side of the line will equal the vari- ance of the spot height data. The contours do not give a true representation of the shape of the spectrum, As drawn, they represent an estimate of the volume under the true spectrum when integrated over a square of the size shown in the figure and centered at the contour position. Thus steep slopes in the spectral surface tend to be smoothed out. These spectra due to the errors hinted at above also have errors inthem. The region of analysis should be exactly square, The area shown is rectangular and in actuality the area analyzed should be as high as it is wide. Seven rows of numbers have been omitted from the top and bottom of the figures. At the top of the figure above the dash dot line and at the bottom of the figure below the dash dot line the omitted numbers were all slightly negative. Near the bottom and top edges they were of the order of 3002 (ft) 2, Moreover, near the left and right edges along the r axis of the figures there are considerable areas of negative values with some values of -0.024 (ft) 2. 161 @ YASWNN Las Viva 404 WhYLOSdS IVNOILOSYIC 6ll Old ee ee ee one” Td panos eae ™) CaN ae / Sed -~ 7 Ben Cae note ~------- -_ act + + S+------—-— oe 6 O-—si———r—— tT aan t a + + + + ‘7 W i’) u ri " + + + + + 1 L 4 L aN + + L L L + + + 9 9 + + 9 + 9 + a + 2 & & Zine ee + + + + 162 z ——— =< wo + . + . . + + 4 ¢ + ) 4 —— —" + + oo + “€ YaEWNN 13S Wivd YOS WNYLI3dS IWNOILO3UIC OM 9l4 —-_. z z z + > + ot te a4 See ee el fof —_—-— - —_——- 163 — eas a fie 0 Sn ea mm --* z + y 4 3 L eee Cenc nme | een _—.— my z = es aS ee pee eee ENS —_ _— —_— —— -/_ — ee” 64 Although it is not impossible to obtain a negative value in a power spec- trum computed according to the techniques described, it is highly improbable that such consistent patterns of negative numbers should occur, Given that the computations are correct, one possible explanation of what occurred is that the original data have been distorted by some unknown and undetected source of error to such an extent that they no longer represent a nny from a stationary Gaussian process in two variables. (Another very disturbing possible conclusion is that the ocean waves cannot be satisfactorily approximated by a stationary Pe cccian process in three variables. ) Moreover since the sum of all the values of U(r, s) must add up to the vari- ance of the original data (in these figures), the negative values have the effect of adding erroneous positive values to the already positive estimates in the other parts of the figure. A study of the figures and the data shows that the gross features of the analy- Sis appear to be correct but that there seems to be a background distortion in the pattern which is difficult to define precisely. Analysis of original results Various tests of the results were made at this point, and it soon became evident that there were serious discrepancies between the wave pole frequency spectrum and the average of the two directional spectra. The average of the sums of the values shown in the directional spectra (which in turn equals [2(00)> + Q(00)3]/[2(1.016)]) should give a number which when corrected for possible sources of error should be nearly the same as one half the E value 165 for the wave pole spectrum. There were two possible sources of error considered in the stereo data. Even after their removal, there was still a considerable discrepancy. The first possible source of error was what is called white noise reading error by Tukey [1949]* for the one dimensional case. It can be easily generalized to the two dimensional case. Let s XK % : (11.2) Vk = "ik (true) ar €; + €, + eff where 1; ik is the actual reading, Njk(true) is the reading that would be obtained from the stereo data with the stereo planigraph if there were absolutely no sources of photographic, machine or human error, and es ae and ejk are random errors. More precisely, let ej be numbers picked at random from a normal popu- lation with zero mean with an unknown variance and added to every value ofa column of Njk(true) } let oa be similar number with perhaps a different variance added to every row, and let jk be numbers picked at random from still a third different normal population with a zero mean and a different variance and added to the appropriate value of Wie(true) fe + as The errors just de- J J scribed will be referred to as column noise, row noise, and white noise, re- spectively. x For a more recent and more readily available reference, see Press and Tukey [1957]. 166 The effect of the random errors on Qo can then be determined under the assumption that the different types of errors are small and independent, elt :3) Qn i Q5q(true) = Soq v Spo + Soo where S,, = E(<;")¢ if p= 0 for any q and is zero if p #0; q id) Il E(6,*)* if q=0 for any p and is zero if q #0; dp) I = Ele)? if p=0 and q=0 and is zeroif p#0 and q#0. The effect of a random error along a column of the data is thus to cause a constant error to be added to every value on the vertical axis of the coordinate system of the covariance surface; an error along a row adds a constant error to each value on the horizontal axis; and a random error over the whole plane is concentrated as a spike at the origin. The values of L(r,s) can then be found from the values of Q(pq) (11.4) L(r, s) = L(r, 5) (true) tiWoaak Wrest Wes where Wos = 45 E(e;*) if r = 0 for any s and zero if r # 0; Zz, Wo = = E(<;) if s = 0 for any r and zero if s #0; ue and Ye = l E( es )2 » for every value of r and s. Thus, random errors along columns in the original data show up as a constant error along the horizontal coordinate axis in the L(r, s) plane; er- rors along rows show up as a constant error along the vertical axis, and ran- dom errors show up as a constant error at each point in the spectral plane, Of course, since the data are really a finite sample, there will be fluctu- ations from point to point in the L(r,s) plane. 167 Finally the computation of U(r, s) smooths the values of Wes and Wee into three rows or columns and assigns weights of 0.54 to the values given along to the axes and 0.23 to the row or column on either side of the axis. Random fluctuations in Wee are ale out so that U(r,s) is more nearly a constant at every point and equal to 1/800 of the white noise variance. The other source of error lies in the possibility of background curvature of the plane of the stereo data. It will be recalled that one set of data was so severely distorted by background curvature that it had to be abandoned. Al- though no curvature is detectable in figures 11.4 and 11.5, a very slight amount of curvature would produce high values for the spectral estimates near the origin. The effect of pure white noise can be estimated from the information given in Part 7. The accuracy of the spot height readings is considered to be 40.5 feet. Under the assumption that the errors are normally distributed this can be interpreted to mean that (11.5) P(-0.5 2 (33) 92 °0 95 °0 tyr = [(00T *) ,(Ur=2)] £(00)0 GG LOY coe 00 °0 5 “0 19h 7 (33) 000 95 *0 92°% — [(00T x) p(w) ] (00) SoULTICA pey921LLOD uo onTeA 39INJeEAINI VOURTICA 2VIUSpTFuUOS pue sostou peyIItTLOD UO quodzed G6 O}IYM TOF ONTeA sdUapTyuOS teddy peyerii0g yusored Gg IomOT aNyeAInD estou o}IyYM OENTeA TeUTSTIO *eJEP OII94S [TEUTSTIIO OY} FO SOOULTIVA 0} SUOTIIEITIOD “TTT eqGeL i=) ~~ Lea be equal, and yet the estimates obtained from the samples are not. The vari- ance of the stereo data is 1.54 times the estimated variance of the wave pole data and 1.48 times the upper confidence bound of the estimated variance of the wave pole data. This result is not necessarily highly improbable. If the number of prCenwe degrees of freedom of the 10,800 points in the stereo data is very low due to their correlation with each other, the result would be possible. Thus it is neces- sary to obtain an estimate of the degrees of freedom of the estimated variances of the stereo data. This can be done by applying a formula similar to the one used on the wave ‘pole spectrum in Part 10 except that now every fourth point is truly independent and there are 16 degrees of freedom per point for each of the original spectra and 32 degrees of freedom per point for the average spectrum. The total variance was found to have at least 800 degrees of freedom by means of a computation using the average spectrum and grouping data so as al- ways to decrease the computed degrees of freedom. The variances of the indi- vidual data sets as a consequence have about 400 degrees of freedom. Additional entries in Table 11.1 give the upper 95 percent and lower 5 percent confidence bounds on the estimates of the variance based on the above degrees of freedom. The lower 5 percent confidence bounds for the stereo data are greater than the upper 95 percent confidence bounds for the wave pole data. The hypo- thesis that the wave pole data and the stereo data are samples (free from any sources of additional error) from the population with the same variance must 171 therefore be rejected at least at the 5 percent significance level, and of course the probability that either variance would be obtained, given that the other is correct, is much less than 0,05. An application of the F test to the ratio of the two variances, that is, 1.54, with 1000 degrees of freedom for the wave pole data and 500 degrees of freedom for the stereo data, yields a rejection of the hypothesis that the vari- ances are from the same population at the 1 percent significance level. The directional spectra given in figures 11.9, 11.10, and 11.11 therefore do not have gross properties which agree with independently determined data from the wave pole. If the wave pole data are assumed ta be correct since in the original planning the wave pole data were thought of as a primary source of calibration, it must then be concluded that the directional spectra are in error, Moreover, the directional spectra have negative values which is a definite indication of something wrong. Of course, there would be one way to force the two spectra to agree. It would be to assume that the estimate of the white noise error was too low ap- proximately by a factor of 4. It would then be necessary to subtract about 0.0025 (£t)* from each spectra estimate. The effect would be to increase the size of the negative areas. Such a solution would only serve to increase the error in the result due to the negative areas of the spectrum. Various attempts were made'to correct the results by making changes in the covariance surfaceand calculating their effect on the spectrum and making changes inthe spectrum and calculating their effect on the covariance surface, 172 For example, quasi column noise whose effect would disappear at plus or minus ten lags in the vertical direction on the covariance surface could produce the negative areas in the spectra found on the horizontal axis, However the attempts were in general unsatisfactory as the different types of corrections propagated very oddly from one system to another. No notable success was achieved by these attempts. Detailed analysis of leveled spot height data The analysis of the data had reached an impasse,. After a number of con- ferences with Leo Tick and Prof, Max Woodbury, Prof, Woodbury suggested that the original data be studied to see if they could be corrected. Such a procedure would involve re-computation of the results, but the use of the Logistics computer at George Washington University was assured, and the problem was deemed so important that the added effort to obtain a satisfactory solution should be made. The ridge along the vertical axes of the covariance surfaces suggested some source of error in the vertical direction of the stereo data, Figures 11.4 and 11.5 and the leveled spot height values as tabulated were then studied very carefully to see if any discrepancies could be found, In fig. 11.4, it had been noted that the diagonally oriented wave crest- wave trough pattern on the left side and in the center of the figure changed to a vertical orientation on the right hand edge of the pattern. Very strong verti- cally oriented crests are especially pronounced in the lower right corner. This variation had been thought to be a possible perfectly natural variation in the data, but now this assumption was checked. 173 The ten columns of numbers on the far right of the figure and the twenty rows of numbers on the bottom of the figure were set apart from the main part of the figure, because of this tendency toward a vertical distortion, and divi- ded into three groups with the rest of the data comprising a fourth group. The breakdown was as follows: mi)-D)5 IIL Sages 89, 49 ° AREA A 3500 points 21,0 CAEN ADSI aes C 20,49 rare Vy be een in GTO) Yen 19,49 AREA D C 1000 points 2,0 1,0 OF 0 Oe Opa acher 0, 49 0,50, 0,51 . apis 89,59 AREA B 700 points : 20, 59 ° 19, 59 AREA C 200 points 0, 59 The variances of the sub-areas were computed, and probability histo- grams were drawn. The variances of areas B, Cand D were all greater than the variance of A, and since it was known that the total variance of Data Set 2 had about 400 degrees of freedom, these 400 degrees of freedom were apportioned in the ratio of the total number of points in each area. It was then possible to apply the F test to the ratios of the variances. results which were obtained. 174 Table 11.2 shows the 175 CT tals at sii iii UGE AAW) Oe lawn lie G7T OFI prea! USS, 006T T8Z‘OI a+0+0 « 2 e . 002 e s & CTh GY ip wigs Bele © (HOM he. Te 88 6€°T G6°S 00zT Giglas O+d we SSS 5 : A Cie SL Pe Sail SSE“ WADE aA Ee) a) ivael Jae) 006 izo Ss O+d ae 007 Mii 0090S POL'SZ TeIOL . . e « ° e 002 . . ¢ Oa) eee one Wek 29 O ZW Wir =O €L MEZA Sh eS OOOr ODEs a ° e e e e o 002 ; ° e ¢ SEE Ws BS Ach?“ O) Cog Wrz cares ST Ogme 76°6 002 786 T 2) 002 ‘ Bile eee ise I SrZ/e Oa Teeeac iva BOGE og Gea 02°S 00z 6£9°E a Zoiy OSs Gen GIO 092 Ve? O0SE €16‘FI Vv 0fS6 AG o/T Yc pesn Vv 0 OOT* syutod NZ eoiy teddy 19Mo'Ty 72 % 4 OAS) [eon Diet Om Pp Orhent 7 (aru) ON G spunoqg SLOJDeCF DOUeITFIUSIS souUeTIeA aouepryuoy aoueptyuoy 3801 "Z 32S eyeq UT seery qng jo stshyeuy “2°11 P1deL The variance of area A was 0.50[(mm)*% x 100] less than the variance of the total area. The variance of area C was over twice as large as that of area A. The degrees of freedom actually used in the F test were less than the computed degrees of freedom, and yet at the 5 percent significance level the hypothesis that the sample of points from area C is from the same popula- tion as the sample of points from area A must be rejected. The grid of points for the numerical analysis must be rectangular. Areas B and ©, D and C; and B, C, and D were combined, and their combined variances were tested against area A. In all combinations, the areas could be rejected at the 5 percent level. Moreover the lower confidence bounds on the variances of area C, areas B+C, C+D, and B+ C+D were all greater than the upper confidence bound on area A. Figure 11.12 shows a comparison of the probability histograms (number of points in class interval divided by total number of points) from areas B, C, and D, with the probability histogram from area A. Area C is quite a bit different from area A. Note also that the histogram for area A appears to be normal. The spot heights for Data Set 3 were analyzed ina similar way. A study of the contours suggested that a tendency toward vertical instead of diagonal crest orientation existed on both edges of the area of analysis and the points in Data Set 3 were broken up into five areas as indicated below. 176 AREA B 700 POIWTS “95 9.8 - 85 -@0-78 -70 -65 "G0 -6.5 -80 -45 -40-35 -20-25 - TH -70 -G8 “GO “65 “BO -40 -40-35 -30-25-20-1.0-10-B 9 05 LO 16 20 28 303540 45 60 55 €065 70 75 80 85 90 88 100 AREA C 200 POINTS a -95 -90 -8.5 -6.0 -75 -7.0 -65-6.0 -85-5.0-4.5 40 -3.5-430 -25-2.0-15-LO -5 O 8S 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 9.0 95 100 50 10 AREA D 1,000 POINTS 70 3500 POINTS ae mae 795 -90-85 -€0-7.5 -70 -65 -6.0-05 -50-45-40-35 -3.0-25-20-1.5 “10 -5 0 6 LO 15 2.0 2.5 3.0 3.6 4.0 45 5068 606.5 7.0 7.5 820 85 20 96 10.0 EMPERICAL PROBABILITY HISTOGRAMS FOR AREAS B,C, ANDD FOR DATA SET 2 COMPARED Fig. 11.12 WITH AREA A,(TO CONVERT TO FEET DIVIDE HORIZONTAL SCALE BY 1.0/6. ) 177 a cl —_- FO rr Ir a a a a a ee ee 89,0 1 39,5. .. 89,9 | 89,10 .... 89,49} 89,50... 89,54! 89,55... 89,59 | | | | | | | | : : : | AREA FATS | AREA | AREA | AREA A | B Cc | D | E 450 | 450 | 3600 | 450 | 450 points | points points | points | points | | | | | | | | | | | | | | | | GNOMNOI 04 10,5 5.0202 019.) | OO Olt 9N OLE 0,54 | 0,55..0, 59 ST a aos us | al aap | net 09 aaliyehienaahiialaR meee agra |e Seem yah ee niu lank Aaa The results of the analysis are shown in Table 11.3. Area C inthis set of data had the smallest variance (a reduction of about 0.24l(eenn)- (x1G0)]) over the various points. However there is no significant discrepancy with any combination of areas at the 5 percent level. The above results show something definitely wrong with Data Set 2 and suggest something wrong in Data Set 3, especially since some spectral esti- mates are negative in Set 3. it was therefore decided to do the computations over again on a reduced portion of the data. The computations were performed on area A (with 3500 points) in Data Set 2, and on area C (with 3600 points) in Data Set 3. Some badly needed degrees of freedom were sacrificed by this procedure, but the results were quite encouraging. For example, the covari- ances actually became negative on the vertical axis of the covariance surface of Data Set 2, and there were no negative values in the smoothed spectral estimates for either data set. A discussion of ‘the corrected computations will follow. 178 EOS wets Be eeu 97 T en G?7T cel efi 26°F 758'8 0081 H+G+d+V ce e o e e e 002 ° * € Ce 9 “Ss ee t GLO v9 1 evil scon WS etl Lee OG a 006 aA+d cy ° ° . cy aca 002 3 s é EY Oe es Wl Gb O 3) Il evi 7con VY eae OG WSIS 006 a+V 9L6°E€2Z 00FS TeIOL © . xe e e ) 002 ° 2 c VY iB? GH gO c0°2 v9'T Te €e 60°T Oy ZW ¢ OST a oC ° Se ° cue me . e 002 . e € Silk, FOE SW ta 230) 20°? VII we ce LIT HAD = AECL TE OSV a et ON t3 SGI LEO 892 O2*> P2I‘st 009¢€ e] e e e = e e 6 002 . e « € Gi OES Siz eo) 20°? vA Ik ase ce Z2'l GSE°*S = BOF Z OST da 08°9 F6°2 SH ZNO) 20°? ¥9‘T eos €€ Zi LL? 8FIZ OST Vv G6 GC if g pesn 5) Oil syutod teddy irtemo7T T2eA2T J} JO p fF FO p oney NZ ON BOY) spunoq SIOJOET SOUCITFIUSTS aouelsreA, e eoduepljuoy sdueptTzuOD 4893 4 "€ 30g eJeq UT seery qng jo stsk{Teuy “E'TT PTIeL 4 = ) Before the analysis of the corrected results is made, a discussion of welt what went wrong with the original results is needed. The basic source of the difficulty can be traced back to a statement made in Part 6. The photographs were taken with reconnaissance type film instead of the more dimensionally stable topographic base film. The film magazines used in the cameras were labeled to contain the correct film but they had actually been loaded with the wrong film. Such a mistake would not be detectable until after the film had been developed. This dimensionally unstable film then underwent differential changes in areas (that is, small areas of the film shrank by greater amounts than others) which introduced a complicated error pattern in the spot height data. Fortunately most of the error (but possibly not all) appears to have been concentrated on the edges of the areas analyzed. The question might be asked as to why the errors in the original leveled spot heights were not detected prior to making the laborious computations of the covariances and spectra given above. A close comparison of figures ll. 1 and 11.3 suggests, since hindsight is always better than foresight, that the error in the spot heights might have been detectable simply on a comparison basis. To be really sure, however, computations similar to the ones given above would have had to have been made, and they could not have been made without a knowledge of the effective number of degrees of freedom of the sub- samples. This effective number of degrees of freedom was estimated from the incorrect spectra. The use of theories valid for correct data on incor- rect data to show that the data are incorrect is quite similar to pulling 180 oneself up by one's own bootstraps (with perhaps the bootstraps being broken in this case). Thus all of the above analyses and comments serve only to sug- gest the nature and source of the error and a possible way to remove it, What was done did remove the error, so in this sense the analysis of the error was correct, All of the numerical results obtained in the original analyses of the full sets of stereo spot heights were kept in the tables along with the preceding figures in order that this report would be complete, They represent a wealth of data which can be used for additional analysis and study, This report is unique in that it is a study of a random process ina plane, and the complete set of original data and computations should be of value to geophysicists, statisticians and physicists. Re-analysis of reduced areas As stated above both spectral computations were carried out over again for reduced sets of spot height data, For Data Set 2 the area was area A as defined before as bounded on the four corners by the points 20,0; 20,49; 89,0; and 89,49. In what follows these 3500 numbers will be called Data Set 2A, Similarly for Data Set 3, area C (consisting of 3600 points) bounded by 0, 10; 0,49; 89,10; and 89,49 will be called Data Set 3C. The covariance surfaces for Data Sets.2A and 3C and the average of the values for the two data sets are shown in figures 11,13, 11.14, and 11.15. The patterns are better defined than they were for the surfaces given previously 181 O.1016mm, » | FOOT LENGTH OF BIDE OF SQUARE® 1200 FEET COVARIANCE SURFACE SET 2A FIGURE 11,13 IN UNITS OF TENTHS OF mm. SQUARED 2 , a ; ; ; H phn te \ fz OO i hw joa eels Be 4 aa a : [P| PAA ile eee | as |p ea Ss yO i = ty \ al v 2 V/ SSS Fa = >» SF —— | 0.1016 mm. = | FOOT LENGTH OF SIDE OF SQUARE» 1200 FEET COVARIANCE SURFACE SET 3C FIGURE 11.14 183 TN UNITS OF TENTHS OF mm, SQUARED ee é We le er NIM } Ney | I» fs Lys \ ( is ia || |, HARK w 2 a : < 8 : g a! =3 | 5 ) 12 33 oe i |S OF mm. SQUARED IN UNITS OF TENTH: FIGURE 11.15 184 in figures 11.6, 11.7, and 11.8. The negative areas are better defined. The ridge along the vertical axis is weakened although there is still a trace of column noise. For Data Set 2A, the covariance surface actually becomes slightly negative on the vertical axis. The covariance surfaces still need some minor corrections, but they will not be too difficult to make. The spectra for the reduced data sets The spectra for Data Sets 2A and 3C (in terms of variance) and the sum of the values for the two (in terms of E value) are shown in figures 11.16, 11.17, and 11.18. There are no negative values! The numbers at the grid intersections should be divided by 10* to put them in units of (ft). The con- tours are labeled in units of (£t)2, and as mentioned before they should be interpreted as the integral over the spectrum on a square of the same size as the grid of the plotted numbers. These spectra definitely show the effects of column noise. There is a strong ridge along the horizontal axis of the spectral coordinate system. The spectrum for Data Set 3C shows a decrease in the effect of curvature in producing high values at the origin. In general the above two spectra appear consistent with each other. The 0.0100 and 0.0050 contours are in roughly the same positions on the two spectra. The peak in the spectrum for Data Set 2A has a value of 0.2052 (ft)2 whereas the corresponding value in the spectrum for Data Set 3C is 0.0797. The ratio of 0.2052 to 0.0797 is equal to 2.57. For Data Set 2A, the number of degrees of freedom is given by equation (11.8). 185 ~~ _.0088 ~ FIGURE 11.16 VARIANCE SPECTRUM FOR DATA SET 2A 186 AOS (oo nt \ 67 ¢ Figure 11.17 VARIANCE SPECTRUM FOR DATA SET 3C 187 Fig. 11.18 + 7 5 Fr ia es ao ter Ge aS WOON 120 BT isk 160. Uealr,s) * Usclr,s) 188 ef ie 3 169 ++— 03 pe A, zi tar / / fi ON + fics \70 ' N a4) + a 7 aN + ry (11. 8) rh {I eLG EC ga 158 [58 | E 4| 1.58(6) = 9.48 if For Data Set 3C, the number of degrees of freedom is given by equation (11.9). 40 J Bye (11.9) F = 5649-4) apy = 9,48 Thus each individual spectral estimate is distributed according to a Chi- square distribution with slightly more than 9 degrees of freedom. 1200 LAO) WeAoNe) ‘ on both the horizontal and vertical axes (1200 = 30 x 2x 20). A simpler system of notation will be used by assigning the values 0, 1, ...., etc. tothe spectral coordinates just as the values of k were used in studying the wave pole spectrum. Then R is given by R* times 1200/21, and it becomes 1200(k -4)¢ Ml RS (96) 2 (5.12) Ut 0.025429(k - 4)2 = C(k - a8, The values of R for k -4 and k + 4 determine two concentric circles. The total contribution to the value of E of all estimates within these two circles should correspond to the value determined from the wave pole spectrum except for residual errors of white noise and curvature. To esti- mate this, the value of one half of the area between the inner circle and the outer circle is needed. The area of the inner circle is nf C(k - 4)*] and the area of the outer circle is nf Ctk +1)7]* . One half of the difference is the areaofone half the circular ring as given by 194 Dey ae 14 1)4 (11.14) A= 3 CU( K+ 5)" - (k--5)"] 20.315: 1074(2k> +k u The values of A are tabulated in Table 11.4 for future reference, A value of k equal to 27 corresponds to the largest circle that can be drawn in the plane of the directional spectrum. The values of A increase slightly more rapidly than the cube of the values of k. Also tabulated in Table 11.4 are the values of A divided by 36 for future reference, A Cartesian coordinate grid was constructed by drawing heavy lines at the yalues of r and s corresponding to 0.5, 1.5, 2.5, ...., 19.5. This divided the plane of the spectrum into 741 squares assigned unit area, 116 half squares, and 4 quarter squares for a total of 800 full squares. The radii given by setting k equalto 1, 2, 3, .... and 28 in equation (11.3) were then computed and semicircles with these radii were superim- posed on the grid. The semi-circles divided the squares into pieces. The number and size of the pieces depended on the geometry of the system. The areas of the pieces were then computed from geometrical consider- ations which depended essentially on differences between areas of sectors of circles and triangles. The squares along the r axis, the squares at 45° to the r axis and those in between out to the largest radius were the ones that were analyzed because all others could be obtained by reflection in either 195 the r axis or the 45° line. Table 11.4, Half the areas of the circular rings associated with the different values of k. x pole A i 0.0051 2 0,0345 3 0.1127 4 0. 2641 5 0. 5130 6 0. 8837 7 1, 4007 8 2.0681 "} 2.9711 10 4,0732 Il 5.4190 l 7.0331 is 8.9396 14 11.1631 15 13. 7279 16 16. 6583 17 19.9788 18 23. 7137 19 27. 8874 20 32. 5243 21 37. 6488 22 43. 2852 23 49.4579 24 56. 1913 25 63. 5098 26 71.4377 27 ho 9995 A/36 .0142 0244 0389 OSS 0825 1131 - 1506 5 UIE) 2483 . 3100 3814 . 4628 5550 . 6586 . 7747 = 9035 . 0458 . 2024 . 3738 » 5609 - 7642 - 9844 2. 2ace eel oll eel eel oe el © 2 2 Se OO Oe Ol OE OE SO OO Oe) The final result was that each piece of each square as cut up by the circles was assigned a percentage between zero and 100. The calculations depended on the difference between large numbers, and the results on summing around circles did not check with the results of Table ] 1.4. to the various areas so that the sum around circles would check with Table 11,4. Small adjustments of the order of one or two percent were made 196 The pattern employed, the values of the radii, and the numbers finally obtained are shown in figure 11.20 for a quarter sector of the full area of the directional spectrum. All other points can be obtained by symmetry. Note that half the values on the horizontal axis should be used on the vertical axis. The values of Uza(r,s) + Uzecl(r, s) corrected for column ngise were then entered in the corresponding squares. To determine U(k), the percentages of the squares falling between circles with radii corresponding to k -4 and k rs were multiplied by the AE values for the appropriate squares and all contri- butions for that particular semicircular ring were summed. | The results are shown in Figure 11.21, The values of AE in (ft) 4 obtained upon summation are plotted as a function of k in the upper curve. | The spec- trum obtained from the wave pole data is also shown. | An additional correction is needed before the two curves can be compared. The effect of the white noise variance of 0.54 (£t) 2 must be removed. Since this error variance is spread evenly over the entire plane of the directional spectrum each square in this analysis has an expected value of 1,08/800 (£t) assigned to it in terms of E value. When the entries in Table 11,4 are multiplied by 1.08/800 and subtracted from the values shown on the top curve in figure 11.21 the result is the middle curve which shows the frequengy ¢ spectrum corrected for the white noise estimate given previously. The effect of assuming that the white noise error variance is twice as great is shown by a third curve in the figure. Such a correction would be much too big, 197 94 36 83 17 |" | “Ih FIG. 11.20 TRANSFORMATION FROM RECTANGULAR TO POLAR GOORDINATES. 198 fl 100 73 p “S3190YNID INAS GNNOYV GSANWNS SV WNYLD3dS TIWNOILD 3Q4NIG HLIM WNYlo3adS 310d 3AVM 4O NOSIYWNOD ov sf 9¢ “WNYLIADS 310d JAVM “ALVWILSS 3SION JLIHM JOIML YO4 JOVW NOILD3SYNYOO “SLVWILSS 3SION SUHM 40s SO0VW NOILOSHYOS “S319NID INSS GNNOYV GSWWNS WNYLI3dS IVNOILI3AYIC ot “Old ce OF B82 S +e 22 O2 BI Cy val eal Ol 8 Seay, 2 I99 The curve to use for further analysis then is the middle curve of the three curves for the directional spectrum. The agreement at first sight is not too striking since the only points that are close are k= li, 23, 24, 25, 2eq and 27. A further study of these results will be made later. Table 11.5 shows the values obtained for different k by summing around the semicircular rings and the effect of applying the corrections due to white noise. Table 11.5. AE values as a function of k summed around semicircular rings in the directional spectrum. k AE AE -white noise AkE-2 white noise O21, 2, 3,4 -0S57 0351 .0346 5 0338 20331 0324 6 .0536 -0524 0512 7 .0850 0831 0812 8 1412 1384 21356 9 2432 s2o92 22352 10 .3771 23716 23661 11 .5898 25825 sD SZ Ae Cae 7016 6921 13.202 7081 -6961 14 .6345 6194 6044 15 .5500 25315 5129 16 .4544 -4319 4094 17 + .3823. 203553 3284 18 3423 3103 22183 19 .3086 2710 22333 AQ) AZ 22194 21743 21 .2402 1894 01385 PA oBe'a5) 1771 1186 23 .2100 1432 0765 24 .1791 21032 0274 25 .1836 0979 0121 26 1760 .0796 - .0169 PAY llishiliry 0737 = 0.343 z200 The angula rt variation In order to study the angular variation of the spectrum, the results shown in figure 11.20 were employed. Radii at 5 degree intervals were superimposed on the figure by overlays. The plane of the directional spectrum was thus di- vided into small areas bounded by arcs of two circles and two adjacent radii, Two adjacent circles bounded 36 such small areas, and the areas are tabulated in Table 11.4. Since the effect of the circles in breaking the squares up into suh- areas had already been computed, it was not too difficult to compute the effect of the radii since each small area had to have a known value. The percentages which resulted were then multiplied by the appropriate U(r, s) values and sum- med for each small area. The number thus obtained is an estimate of the con- tribution to the total E value of the short crested sea for spectral components with frequencies between 2m(k - £196 and 2n(k + 4) /96 and with directions be- tween 9 and 6+ 5° as k varies from 11 through 27 and as @ varies from -90° to +85°, The results of this computation are shown in figure 11.22. The white noise estimate has been removed by subtracting (A/36)(1/800)(1.08) from each value. The curves are erratic, mainly due to sampling variation, but there is a rather definite indication of the presence of a swell for curves corresponding to k = 11 through 17. The swell was removed by estimating the shape thatthe curve would have had, had the swell not been there. This estimate is shown by the dashed lines in figure 11.22, Tables 11.6 and 11.7 show the resolution of the data into frequency and direction intervals. 201 O -G0 -70 -60 -30 -40 -30 -2 40 0 1 20 30 40 SO G0 70 & 90 “90 -€0 -70 -€0 -50 -40 -30-20-10 0 © 20 30 40 50 60 70 80 90 FG. 11.22 ANGULAR VARIATION FOR A CONSTANT FREQUENCY IN THE DIRECTIONAL SPECTRUM. 202 ou rH MN NMNN tf Www MnXN nn Vn as Table 11.6. Resolution of directional spectrum into frequency.and direction intervals, swell, white noise and column noise removed, Se Sa I a Dc abl 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 +0025 20035 20034 .0047 .0039 .0035 .0040 .0031 .0034 .0039 .0027 .0032 .0032 .0029 .0035 .0019 .0025 +0017 0028 .0025 .0033 .0030 .0026 .0034 .0030 .0022 .0048 .0022 .0028 .0026 .0028 .0023 .0015 .0022 +0011 .0018 .0016 .0022 .0020 .0018 .0026 .0026 .0020 .0018 .0020 .0023 .0023 .0035 .0022 .0006 .0012 20009 .0011 .0010 .0014 .0011 .0012 .0022 .0022 .0020 .0019 .0016 .0015 .0017 .0020 .0007 .0005 .0010 +0004 .0005 .0005 .0007 .0006 .0008 .0015 .0019 .0018 .0013 .0008 .0006 .0007 .0007 +0010 .0009 20006 20002 .0001 .0002 .0001 .0002 .0008 .0008 .0017 .0016 .0009 .0006 .0006 .0006 .0012 .0022 .0017 .0018 0001 _O _0O _0 .0003 .0010 .0004 .0014 .0010 .0010 .0012 .0017 .0014 .0018 .0019 .0012 .0O11 _0 .0002 O .0003 .0006 .0014 .0004 .0025 .0014 .0012 .0018 .0028 .0016 .0014 .0027 .0019 ~.0008 +0001 .0011 .0003 .0008 .0012 .0019 .0006 .0040 .0024 .0019 .0019 .0026 .0022 .007.9 .0032 .0018 .0041 e0010 .0017 .0008 .0014 .0019 .0026 .0013 .0039 .0028 .0023 .0019 .0025 .0028 .n19 .0022 .0019 .0022 +0019 .0028 .0029 .0027 .0027 .0035 .0024 .0039 .0040 .0034 .0023 .0027 .0031 .0025 .0026 .0024 .0029 +0026 .0038 .0028 .0037 .0036 .0042 .0027 .0052 .0050 .0046 .0037 .0037 .0041 .0033 .0018 20015 .0017 20038 .0049 .0038 .0051 .0045 .0062 .0071 .0066 .0064 .0055 .0046 .0052 .0045 .0035 .0028 .0033 .0027 -0049 .0083 .0058 .0065 .0081 .0097 .0085 .0086 .0076 .0071 .0071 .0065 .0040 .0035 .0022 .0019 .0033 20059 .0087 .0103 .0095 .0091 .0122 .0115 .0102 .0092 .0081 .0085 .0074 .0054 .0056 .0034 .0023 .0031 20059 .0084 .0107 .0122 .0155 .0158 .0131 .0131 .0128 .0071 .0076 .0127 .0046 .0082 .0055 .0045 .0035 20236 0211 .0181 .0134 .0160 .0175 .0175 .0199 .0156 .0097 .0095 .0158 .0100 .0062 .0073 .0054 .0044 20265 0290 .0333 .0246 .0257 .0254 .0236 .0211 .0192 .0147 .0137 .0140 .0081 .0030 .0079 .0026 .0036 20265 0290 0333 .0246 .0260 .0269 .0252 .0211 .0196 .0169 .0154 .0113 .0096 .0051 .0044 .0047 .0060 20287 .0342 .0384 .0306 .0299 .0288 .0246 .0199 .0156 .0142 .0121 .0085 .0084 .0075 .0035 .0043 .0052 20421 .0425 .0455 .0312 .0298 .0255 .0183 .0147 .0126 .0098 .0074 .0060 .0067 .0035 .0022 .0019 .0026 20421 0457 0443 .0306 .0262 .0192 .0159 .0141 .0110 .0080 .0064 .0057 .0062 .0033 .0025 .0025 .0021 20421 .0479 ..0321 .0300 .0273 .0160 .0120 .0135 .0130 .0054 .0088 .0055 .0047 .0036 .0038 .0036 .0015 20413 0413 .0317 0315 .0234 .0136 .0119 .0123 .0116 .0103 .0094 .0058 .0040 .0036 .0044 .0027 .0013 20319 0415 .0353 20303 .0224 .0150 .0104 .0103 .0084 .0079 .0073 .0067 .0046 .0037 .0032 .0023 .0016 -0314 .0415 .0347 .0300 .0232 .0138 .0113 .0097 .0064 .0047 .0059 .0064 .0044 .0037 .0029 .0020 .0011 20308 .0351 .0354 .0321 20245 .0167 .0151 .0102 .0060 .0046 .0039 .0035 .0033 .0022 .0020 .0017 .0010 20306 .031B .0355 .0298 .0250 .0192 .0150 .0101 .0074 .0050 .0038 .0032 .0033 .0021 .0017 .0015 .0011 20285 .0341 .0333 .0255 .0220 .0131 .0115 .0108 .0082 .0052 .0043 .0037 .0032 .0023 .0022 .0017 .0011 20262 .0341 .0352 .0250 .0167 .0138 .0101 .0065 .0094 .0062 .0043 .0030 .0027 .0018 .0014 .0016 .0015 20165 .0324 .0235 .0209 .0146 .0100 .0089 .0091 .0084 .0058 .0039 .0019 .0016 .0016 .0013 .0015 .0015 -0135 .0169 .0222 .0167 .0121 .0094 .0082 .0083 .0080 .0068 .0039 .0018 .0014 .0019 .0025 .0012 .0014 20135 .0150 .0142 .0109 .0112 .0096 .0079 .0071 .0068 .0062 .0048 .0029 .0024 .0010 .0011 .0013 .0018 20135 .0139 .0134 .0077 .0075 .0073 .0070 .0060 .0042 .0050 .0054 .0045 .0027 .0021 .0017 .0013 .0011 20044 .0091 .0095 .0075 .0073 .0066 .0050 .0036 .0036 .0057 .0055 .0044 .0021 .0013 .0010 .0016 .0010 20029 .0040 .0053 .0060 .0054 .0052 .0043 .0034 .0040 .0055 .0042 .0032 .0031 .0024 .0013 .0017 .0024 Table 11, 7. Swell contribution to directional spectrum, white noise and column noise removed. 20004 .0005 .0019 -.0013 -.0015 -.0014 20012 .0011 .0050 -.0067 -.0041 -.0025 20018 .0036 .0072 -.0082 -.0055 -.0031 20030 .0047 .0082 -.0099 -.0093 -.0042 0 20025 .0060 .0122 -.0123 -.0110 -.0048 -.0012 20035 .0075 .0126 -.0134 -.0103 -.0055 -.0030 -0053 .0077 -.0108 =-.0132 =30097 -.0065 -.0050 20048 .0075 =.0103 -.0130 -.0130 -.0068 -.0050 20036 .0068 -.0110 -.0115 -.0076 -.0057 -.0047 20027 .0054 -.0105 -.0075 -.0065 -.0030 -.0038 -0023 .0032 =.0055 -.0042 +.0034 -.0010 -.0015 +0020 .0022 -.0035 -.0030 -.0008 -.0005 -.0005 . .0020 .0010 =-.0022 -.0015 -.0003 -0010 -.0005 ooo 203 Table 11.6 shows the local sea and Table 11.7 shows the disturbance from a distance. Corresponding values of Tables 11.6 and 11.7 add up the values graphed by the solid lines in figure 11.22. Table 11.6 corresponds to the solid lines continued by the dashed lines where appropriate. The maximum value for a given k and the minimum value are underlined in Table 11.6. Note that the minimum continues quite smoothly into that region where the swell is not present. Confidence bounds on the sums around circles Let the sums of the corresponding entries in Tables 11.6 and 11.7 be de- signated by D,g. Each value of D,g can be thought to have 19(A/36) degrees an freedom if the variation between nearby values of U(r, s) is not too rapid. For example, if a square in the U(r, s) plane were cut in half, then each half would be assigned the yalue U(r, s)/2 and 19/2 degrees of freedom. The degrees of freedom of the sum of the two values would then be ig|L(U(r, s)/2 + U(r, s) /2)¢ (U(r, s)/2)° + (U(r, s)/2)2 or 19, and the sum of the two E values would again be U(r, s). For the sum around a circular ring, this can be generalized to give the degrees of freedom for a AE value corresponding to those frequencies be- tween 21(k - 4) /96 and 2r(k + 4)/96. The equation for the degrees of free- dome is then given by equation (11.15). 2 (11.15) pu LUN [=P igl The factor of 4 enters in the denominator of equation (11.15) because only every fourth value of the spectral estimates is independent. 204 For a more strict analysis, the degrees of freedom should be computed from the data with the white noise still present since at each value it is also distributed according to Chi square with 19 degrees of freedom. The error made in the above computations is small for low values of k, but for large k the variation in white noise may be falsely reflected into variation in spectral estimates because the white noise is a rather large proportion of the total contribution. The degrees of freedom for low values of k are quite low. For k equal to 11, there are only 22 degrees of freedom. Had all these stereo pairs been satisfactory for analysis and had there been no distortion at the edges of the stereo data, the 19 degrees of freedom for each value of U(r,s) actually ob- tained would have been raised to 50 degrees of freedom, and the 22 degrees of freedom for k = 11 would have been close to 50 degrees of freedom. The number of degrees of freedom given by equation (11.15) can be com- bined with the entries in Table 11.5 to give the 90 percent confidence bounds on the estimates of AE as corrected for white noise. The results are shown in Table 11.8. The values given at the 90 percent confidence bounds will enclose the true value of AE nine times out of ten in repeated tests of this same type under the same conditions. Of course, for a given set of data, the true value either does or does not fall within the confidence bounds and one can never know whether it did or did not, The entries shown in Table 11.8 can be combined with the entries in Table 10.1 to compare the wave pole data and the stereo data. The resultis 206 Table 11.8. Confidence bands on data from stereo spectrum i Lower 5 percent Upper 95 percent k confidence band AE confidence band 11 . 3778 . 5825 1, 0387 12 - 4510 . 7016 Zr WZ 13 ~4853 , 081 1, 1487 14 .4478 . 6194 - 9402 15 . 3950 “BSS 7715 16 - 3279 4319 - 6065 17 - 2144 . _ B55 - 4862 18 » 2435 . 3103 . 4089 19 5 AOE) . 2710 . 3384 | 20 - 1784 - 2194 - 1792 2) . 1555 - 1894 » 2319 22 . 1466 L771 , 2201 23 , 1210 . 1432 _ 1732 24 - 0884 - 1032 . 1228 25) 0845 0979 21153 26 -0695 .0796 -0925 27 . 06046 »O0737 - 0852 shown in figure 11.23. For k equal to Tl, 12, 13, 22, 23, 24, 25,, 26, and. ay the agreement is satisfactory, but for k equal to 14, 15, 16, and 21 the con- fidence bounds do not even overlap. Thetworesults are therefore inconsistent. 206 1.25 1.20 115 UPPER 95% CONFIDENCE BAND. 1.00 ——STEREO SPECTRUM. LOWER 5% CONFIDENCE BAND. .90 UPPER 95% CONFIDENCE BAND. WAVE POLE SPECTRUM. LOWER 5% CONFIDENCE BAND. 80 10 -50 -40 +30 20 FIGealines CONFIDENCE BANDS ON WAVE POLE AND STEREO SPECTRUM. 207 Another way to show the lack of agreement in the data is by means of the F test. The ratio of the stereo values to the wave pole values are tabulated in Table 11.9 along with the appropriate degrees of freedom for each esti- mate. At values of k equal to 15 and 21 there is less thay one chance in 100 that the two values of AE could have come from the samme population. Table 11.9. F test applied to wave pole and stereo spectra. F test significance level k ddf Stereo Wave pole Ratio 5% 1% _ Conclusion 11 ae 5825 5804 1.0036 1. 82 2. 38 Accept at 5% 12 Zl Oe 6201 1.1314 1.85 2.43 Accept at 5% 13 30.~=s iw. 7081 4682 1.5124 1. 56 2.08 Accept at 5% Reject at 5% Accept at 1% Reject at 5% Reject at 1% Rejlectra teas Accept at 1% 14 41 .6194 50 1.7547 1736 Wg the) 15 BO BSS .2838 1.8728 1. 49 1.77 16 Be) Bash) 2611 1.6542 1.45 1. 69 17 Oe oAaas 2562 1.3858 1.41 1. 63 Accept at 5% Reject at 5% Accept at 1% Reject at 5% Accept at 1% Reject at 5% Accept at 1% 18 82 .3103 2358 S59 1, 31 1.58 19 128 ,2710 pee 1.4085 1. 32 1. 48 20 LOT ang 4 .1509 1.4539 1. 34 1. 51 Bie 2A) ae PG BIA ee 1, AD Reject at 1% 22 183 JI77l 51320 Re a7 fe an 4 un miccre Wisiecom Moric Mlj,28° sal Accept at 5% 24 203 1032 -1137 9077 1. 26 1. 39 Accept at 5% 25 227% .0979 OT) 1.0000 Ilo AS 1. 39 Accept at 5% 26 267 .0796 .0815 roe munleyaa mee 39 Accept at 5% 27 284 0737 OSS1 ude. ce mina 2 ou males Accept at 5% 208 These results are made even more interesting by considering the values obtained by summing the columns in Table 11.6 where the effect of a disturb- ance from a distance has been removed. These values can be plotted against a Neumann spectrum for 18.7 knots and against the wave pole spectrum as shown in figure 11.24. One could not ask for much better agreement between theory and observation than is shown between the theoretical Neumann spec- trum and the frequency spectrum obtained from the directional spectrum. The agreement between the wave pole spectrum and the theoretical spectrum is actually a little (but not much) better than shown because the contribution of the swell for k equal to 11 and 12 will reduce the sharp peak. One dis- advantage of wave pole data is evidently that there is no way to see the swell if ithas the same frequencies in it as the local sea. Another question to be asked before entering into a discussion of the above results is what would the wave pole calibration have had to have been in order to provide agreement with it and both the theoretical Neumann spec- trum and the directional spectrum. This result can be obtained by dividing the values for the directional spectrum, including swell, by the values for the wave pole spectrum before multiplication by the calibration curve. The result is shown in figure 11.25. There is the possibility of some sort of amplified response in the wave pole, undetected by still water damping and resonance tests, as » equal to 2n(15)/96. The agreement between the two spectra would be fairly good if something like one of the dashed curves were paed for calibration instead of the original theoretical curve. 209 Soe ig_Os| oF Gb lv VIVG 310d SAVM SHI GNV VIVG O3YS1S AHL WOUS GSNIVIGO VuloadS SHI HLIM GSyVdWOD GNIM LON 281 NV YOS WNYLOAdS NNVWAAN IVOILSHOSHL dll SuNdIS | i i +) ve 72E 8b 96 Sze ose ai ea pe £222 2 of 6i_sl_4 9! si_+i_si “WNYLO3dS 310d 3AVM ~~~ ~~~ “WNULOIdS NNVWNEN WWOILSYOSHL “WNYLOSdS O3NS1S oro 020 o¢0 0390 010 Itz ‘NOILVYSITVD 310d SJAVM TVIIYIdNWA ONY IWOILSYOSHL Gell Sls 9€ wE ce OF BS 92 we 22 O2 BI cy Ov BE 91 vl ‘NOLLVEGINVI 310d JAVM TVIILIYOSHL ‘NOILVUGINVD 210d 3AVM 3181 SSOd “ANULIADS 310d JAVM 3V OL WNYLI3dS OFYRLS IV Olive él oo"! ler oSs"2 Discussion of wave pole, stereo and theoretical spectra If a physicist were to measure the acceleration of gravity at the same place by two different methods and obtain 980 cm/sec? by one method and 1400 cm/sec* by another method, he would be positive that there lo some- thing wrong with the second method. In this study one is not in so fortunate a position. There is no background of previous experience, and sampling varia- tion must always be recognized as a source of any disagreement. The results obtained so far are that: (1) A frequency spectrum obtained from stereo wave data agrees witha theoretical curve derived by Neumann after correction for the presence of swell and the effects of white noise and column noise in the original data. (2) A frequency spectrum obtained from a wave pole observation does not agree with either the one derived theoretically or obtained from the stereo data at two points at the one percent significance level. However, the wave pole spectrum does agree with the theoretical spectra given a one knot variation in the winds as pointed out in Part 10. The following hypotheses are among those that could be advanced to ex- plain the results: (1) The agreement between the stereo spectrum and the theory is fictitious. It has been obtained by choosing just the right weighted average of winds reported quite a few hours before the actual observations of the waves and by rather prejudiced choices of just the right amounts of noise and swell to get agreement. Also the reduced stereo data may still be distorted. 212 (2) Variability in the winds and background distortion in the stereo data is sufficient to explain the difference in the two different sets of observed values. (3) Sampling variations at the 1 percent significance level have actually occurred. (4) The wave pole calibration is incorrect. (5) Weighted combinations of modifications of the above four hypotheses taken 2,3, or 4 at atime suchas, for example (2-3-4). The wave pole cali- bration is wrong by 30 percent at k = 15, sampling variation was at the 20 foe cent level and the variability in the winds explains the rest of the differences. The first hypothesis can be checked by study of the original data as tabu- lated. The fact that the histogram shown in figure 11.12 shows no effect of distortion in area A at least suggests that most of this effect has been re- moved. Also the uncorrected spectra come closer to agreeing with the theo- retical Neumann spectrum than to agreeing with the theories of Roll and Fischer [1956] and Darbyshire [1955]. The analysis has only served to re- fine the results by what are in total rather small corrections, and the cor- rections appear to be logically justifiable in all cases. If agreement with the theoretical Neumann spectrum is not obtained, then the result would be that there is no adequate theoretical wave spectrum in existence, In the light of these new results, the hypothesis of wind variability is much less attractive than it was in Part 10. The wave pole and stereo obser- vations were simultaneous in the sampling sense. The variation in the three 213 | different wave pole spectra as originally tabulated shows no effect of wind variability at the 5 percent level, and the low value at k= 15 occurs in all three cases, The third hypothesis is one that cannot be tested except by doing the same experiment over again using the same wave pole under similar meteorological conditions, Jt will be rejected as a working hypothesis solely because it can- not be tested, However, due to the possibility of this hypothesis combining with some of the others in part, the possibility of incorrectly rejecting it with a chance of more than 0,01 (say 0.15) must be borne in mind. The fourth hypothesis is a very attractive one, If the wave pole cali- bration curye were more like the one shown by the dashed curves in figure 11.25 than the theoretical one, there would be agreement between both obser- ved curves and the theory. It will therefore be assumed that this hypothesis is the dominant explanation for the discrepancies which have occurred. This hypothesis can be tested by modeling the wave pole in a scaled down long crested Gaussian sea with the correct model frequencies present, and comparing the record it makes with a record made by a wave pole held fixed in position, An irregular sea is suggested for the tests because a non-linear effect of considerable magnitude may be present. In Part 8 it was shown that the wave pole moves upward when the crest of a long period wave passes, The submerged tanks are therefore closer to the mean level in the crest of a long period wave, Ifthe crest of a shorter period wave is present at the same 214 time by superposition, the calibration constants would be considerably modi- fied due to the fact that the depth of the submerged tanks is less. This would cause the wave pole to move up in the crest of the shorter period wave even more than the theory would predict. This effect is not compensated for by an equal and opposite effect when the trough of a short period wave is present on the crest of a long period wave due to the exponential behavior of these factors. Thus the response may be non-linear and the heights of the shorter period waves may be underestimated. For very short period waves such effects would again be negligible, If the calibration of the wave pole fails to explain the discrepancy ae tween the two sets of observations then the other possible explanations will have to be investigated. On the basis of the above considerations, a predic~ tion is ventured that the wave pole calibration will explain the discrepancy. If the above hypothesis is a correct one, then the study of ocean waves is ina very odd position. The wave pole data were to have heen a primary calibration for the stereo data. The stereo data appear to have detected, to the contrary, a faulty theoretical calibration see the wave pole, The shipborne wave recorder developed by Tucker [1956a] has been compared with the WHOI wave pole, and agreement was not obtained in this comparison either (Tucker [1956b]). This does not necessarily lead to the conclusion that the shipborne instrument is correctly calibrated. Im fact, its response at high frequencies is known to be poor (Tucker [1956 a}). Therefore at present, there is no primary instrument capable of measuring waves as a function of time ata fixed point in deep water. ALS H. G. Farmer in conversations with the author has described how he would raodify the WHOI wave pole by putting the tanks at greater depths so as to improve the response of the instrument. This should certainly bea subject for further investigation and study both theoretically and by means of model studies. Comaposite frequency spectrum. The results of the frequency analysis of the stereo data and the wave pole data, as given in Tables 11.5, 11.6 and 10.1, can now be combined to yield a composite frequency spectrum over a full range of frequencies. The) spectrum for the stereo data is assumed to be correct for low frequencies, - and the wave pole spectrum is surely quite reliable at high frequencies ex- — cept perhaps for a small amount of white noise. As k varies from 0 to 10) the entries in the second column of Table 11.5 from the stereo data will be used, As k varies from 11 to 22 the sumas of the columns in Table 11.6 : will be used in order to remove swell from the spectrum, One can note small differences between the entries in Tables 11.6 and 11.5 due to round- off errors at high frequencies. The errors are small compared to the vari ability in the sample. For k from 23 to 27 the stereo values and the wave pole values agree and an average weighted according to the computed num- : ber ef degrees of freedom is used. For k greater than 27, the wave pole values are used. This composite spectrum is given in Table 11.10. 216 Table 11.10. Composite frequency spectrum for the local sea determined from the wave pole and the stereo data. AE AE Com- Degrees AE AE Com- Degrees k stereo wave bined of stereo wave hined of a pole freedom| k pole freedom 7 0.0831 34 0.0212 174 8 0.1384 35 0.0193 174 9 0.2392 36 0.0200 174 10 0.3716 37 0.0180 174 11 0.5496 Ze 38 0.0160 174 12 0.6498 21 39 0.0150 174 13 0.6208 30 40 0.0140 174 14 0.5135 Al Al 0.0110 174 15 0.4545 50 42 0.0100 174 16 0.3818 59 «| 43 0.0100 174 17 0.3272 68 44 0.0090 174 18 0.3056 82 45 0.0090 174 19 0.2656 128 46 0.0080 174 20 0.2144 110 A7 0.0080 174 21 0.1894 WZ, 48 0.0080 174 22 0.1766 133 49 0.0070 174 Z2OestS O1336 0.1354 345 50 0.0060 174 24 0.1086 0.1137 0.1110 377 51 0.0060 174 25 0.0984 0.0979 0.0982 401 52 0.0050 174 26 0.0769 0.0815 0.0787 441 53 0.0050 174 27 0.0785 0.0591 0.0711 458 54 0.0050 174 28 0.0491 174 55 0.0050 174 29 0.0443 174 56 0.0050 174 30 0.0395 174 57 0.0050 174 31 0.0392 174 58 0.0050 174 Be 0.0420 174 59 0.0040 174 33 0.0327 174 60 0.0040 174 The values for the composite spectrum are plotted against the family of theoretical Neumann spectra for various wind speeds in figure 11.26. The agreement is good for an 18.7 knot wind. Note that the family of theoretical spectra grows up to and then through the composite spectrum as the wind speed is varied. (all7 Comparison with other families of theoretical spectra This composite spectrum can be compared with the families of theo- retical spectra derived by Darbyshire [1955] and Roll and Fischer [1956]. In both cases the agreement between the computed spectrum and the theo- retical spectrum is poor. There is no value for the wind speed which will give agreement between the theoretical curves and the numbers given in Table 11.10. These comparisons are discussed in greater detail by Neu- mann and Pierson [1957a] and Neumann and Pierson [1957b]. Removal of white noise from the directional spectrum Upon summation around semicircles, the predicted effect of the white noise was verified and the original estimate of the error in the spot height readings as made by the Photogrammetry Division ofthe Hydrographic Office was verified. The total contribution of the white noise to the E value for the waves under study is thus about 1.08 Tae). and 1.08/800 (£t)* must be sub- tracted from each value of the energy spectrum obtained from summing the values of Up alr.s) and Uzcl(r, s), after correction for column noise. This amounts to 0.00135 (£t) 2 per unit square in the spectral plane. Since only four significant figures were tabulated either 0.0013 or 0.0014 was sub- tracted from each particular square. Each of the above values was subtracted an equal number of times so as to even out the total effect to 1.08 (ft). A few very small negative values occurred due to extremes of sampling vari- ation in the white noise where it was a large part of the total contribution. The negative values were removed by "'borrowing'' from nearby points. 218 zt (°S}]0UY UT PUTM 9dPTINS ayy ST 9BAIND YoOrS sAOGe Iequinu syy) “uimnajoeds ojtsodwioo snsiz3A eijyoeds uueuInsN [TeotZe10Eey4 Jo ATIuUIe |, Sp eee ‘OZ IT eansty Sp eee oro Oro oso 030 ogo zH 219 Orientation of the sea and swell in the directional spectrum The heading of the airplanes taking the stereo data for Data Set 2 was 330°. Since the planes flew one behind the other, correctly directed arrows with shafts parallel to the short sides of figure 6.2 will point toward 330°. Since the buoy shown in figure 6.2 drifted generally downwind and since the wind was from 330°, or so, as reported in Part 7, an arrow parallel to the short sides of figure 6.2 and pointing to the left will point toward 330°. Due to the 180° indeterminancy in direction in the directional spec- trum, this is equivalent to letting the positive r axis in the directional spectrum point toward 1509. The peak in the directional spectrum indi- cates waves traveling toward 180° approximately. With the direction fixed, the secondary peak in the spectrum indicates that the swell is traveling either toward 90° or toward 270°. It is im- probable that the swell is traveling toward 90° because there is no area where it could have been generated between the point of observation and the east coast of the United States. The assumption that the spectral components are traveling within + 90° of the direction toward which the wind is blowing is not correct for the swell and thus the final directional spectrum may have to have a range of more than 180° in direction. The secondary maximum shown in figure 11.18 should be considered to be composed ot two parts. One part is the continuation of the local sea by means of the dashed lines of the energy as a function of direction as shown 220 in figure 11.22 and tabulated in Table 11.6. Temporarily let all of this energy be assigned to the first quadrant. The contribution from the swell as given in Table 11.7 then belongs in the third quadrant. Tables 11.6 and 11.7 were then recombined separately to provide estimates of each of these contributions to a square area in the U(r, s) plane. The values due to the swell were mapped by reflection through the origin into the third quadrant. The minima indicated in Table 11.6 were then assumed to be one ex- treme in the angular range of the sea. Aline forming an angle of about 30° with the positive vertical axis could then be determined. Those values of U(r, s) between this line and the vertical axis were then transferred to the third quadrant. The final spectral estimates in the U(r,s) plane are shown in figure 11.27. The values should be divided by 104 to put them in units of (ft)*. The range of directions toward which the spectral components are travel- ing varies from 80° to 320°. The sea has components traveling toward directions ranging from 80° to 260°. The swell is traveling toward di- rections ranging from 240° to 320°. The quantities shown in this figure have been obtained by applying corrections for the effects of column noise and white noise to the original data and by expanding the spectrum to a range of more than 180° from considerations of the local wind direction and the geography of the area where the data were obtained. The effects of curvature do not seem to be very great. The values at the origin must be excluded, and perhaps the Bak 6 — Ber i) “20 1 we 88 280° 270° 5 5 405 6 6 3 203 5 6 5S 8 Ry oy id Cee oi / mm te Be 260° Fig. Il. 59 138122 143 390 378 194 396 522 126 235 369 OS 4 127 2 34-43 23 28 39 14-232 2 21 20 J bm 2 18 20 4 15-20 3B 24 8 2 23 17 6 9 10 1) 4029) ey m wm an 1 16 1 1 1 hm & i. 27 225 299 762 2041 °2i27 1253 ~ 196 lO 897 1749 1427 +966 33 252 9 98 823 6m 325 121 7 2 39 45 45 1 173 27 182 188 67 6S 107 «143-139-145 4853 57 992 IS lll 49 S54 84 7 a 69 SS 6 6 3B 4M FT 2 27 2 2B OS MO 2 23 BT B49 6 2 9 9 +S 6 22 20 8 6 4 9 4 45 9 6 10 8 ‘8 -8 -8 -6 Ty AD Dy Sey a ts) 2 eG 4G 7 0 7 10 4 6 ' 1 1 t 1 1 B° £2EE & \ 240° 230° BR a i} 8 4 ® 8 f 8 z 62 39) 32) -39) +26) 5 25 30 a C C) 7) ny C) 220° 49 “52 BR —3h a & & 8 ° [+ Se + Ses Melt att Ce Oo lo ° a) BP A tt © 1 2 0 0 2) 40) 0) <0 Gy a g Oe: 20S ee O 7 oF ©) 18 5 3 B 1S 0 9 4 8 5 8 22 B 4 5 20 5 a 7 3 ar 2B 7 6 Ey) a a ] 6 26 36 4 4 44 96 2 28 63 8 20 6 27 7 52 37 24 63 37 3B 52 BB 47 B OB 27 6 OB 27 20 7 6 2830 7 2 20 2 2 9 B 2 5 B 10 9 6 Oo” OD 6 5 7 10 2 @ & 7 6 10 2 9 4 7 2 vo O FW Gow a Ga 2 f 2 9 8 © 2 1 0 0 ' 1 U ! me & \ 210° aR RB $ 8 te] 26 FINAL NUMERICAL VALUES FOR THE DIRECTIONAL SPECTRUM 222 ° ° 1 fe} 1 ge pk sk 8 o 1 ty a o o i aly 7 “10 sil) =z 0 6 " 2 —% a 6 5 wy as ye 5 5 6 —- 8-9 9 -— 105 a) i 22 -= 200 © 24 24 - +4 00 26 25 «25 —35 18 20 24 - 9 10 7 —2e 20 8 12 18 =a5) 2 W -20)—=s= ma eo -o 4 (i) 0 4 3:4 -d, 2 4 4 ah 190° 200° forward face of the spectrum should be somewhat steeper. If the plotted numbers in figure 11.27, for the third quadrant, are transferred to the first quadrant, and if the column noise and white noise are added to the values obtained, the result would be essentially the numbers shown in figure 11.18. The sum of the numbers in figure 11.27 will equal the total E value of the sea plus the swell excluding a small circle near the origin. Strictly speaking, the values at the borders of the rectangular area formed by the data in the first and second quadrant before any reflections through the origin should be halved before summing. However, the values on the s axis of the U(r,s) plots are used only once in the direction of 240°. The values at the outer edge are so small that only a minor error is made in not halving these values. Contours drawn as precisely as possible for the numbers shown in figure 11.27 are shown in figure 11.28. The contours are not very smooth due to sampling variation. The contour analysis can be considerably smoothed when this sampling variation is taken into account. Each of the original spectral estimates had 19 degrees of freedom. Due to the corrections made so far, the smaller values of the spectral estimates and the values for the transferred swell do not have 19 degrees of freedom, but values near the peak of the spectrum of the sea still have essentially 19 degrees of freedom. If a spectral estimate has 19 degrees of freedom, it can be multi- plied by 1.88 and 0.63. Then 9 times out of 10 the true spectral value, as might be obtained by taking a sample with many more degrees of freedom, will lie between these bounds. Similarly, if the spectral estimate is multiplied by 229 70° 80° 90° 100° eA, 75: es a 7, 9 oil Ce € | lo} 20 I is 2 280° a / g 270°, ts | = 3 a ay l eh a 8 if 1 ig BR BR at SR Bly aiealaget 2 2 of 3s 6075S 708 FIGURE 11.28 PRECISE CONTOUR ANALYSIS OF THE STEREO SPECTRUM (CORRECTED FOR WHITE NOISE, COLUMN NOISE, AND WITH THE SWELL TRANSPOSED) 224 0.875 and 1.24, the true value will lie between these bounds four times in ten, The contours in figure 11,28 can be smoothed by taking these facts into consideration and by assuming that the true spectrum is basically a smoothly varying function, The resulting smoothed spectrum is shown in figure 11.29. An attempt to indicate the ‘very steep forward face has been made. In order to obtain this smoothed version it was only necessary to go outside the 40 percent bounds about 10 times in the area where the estimates were greater than 0,0050. Analytic OE of the directional spectrum The curves shown in figure 11,22 and the data tabulated in Table 11.6 pro- vide a way to find an analytic representation for the directional spectrum of the sea. The results of the frequency analysis show that the theoretical Neumann spectrum as a function of frequency fits the data as summed around semicircles quite well. The spectrum as a function of frequency and direction can therefore be written as equation (11.16). -2g7 [yuav? (11,16) [A(us @)]* = a0 ad a [£(4, 9)] where c= 3,05x 10* and all values are inc.g.s. units, The function, f(y, 6) should have the property that it is zero over half the plane, that w/2 (11.17) J f(y, 8) d@ = 1 -a/2 and that f(y, 6) >0. 225 1 \ Hd rl® H8 ne 2a ug HE ish te elec = He wg Tr 150° FINAL SMOOTHED SPECTRUM 226 Such a function is given by N (11, 18) f(p, 0) s2[1 + = a_(y) cos 2n0 + b,(p) sin 2n@] T n=] © for 04 (H) -a7O and if 6, (¥) is the angle in the first quadrant where {(p, 8) is a minimum as a function OH [lc If the values of the entries in Table 11.6 are divided by the sum for each column, k, and called F{k, 6), then (11,19) ZF(k. @) = 1 and (11.20) ¢ | 2nk (m+a)"] (m +9) le 0 0 AG 36=F\* 36 If the Fourier series given by equation (11.18) is truncated at sine parti- cular N as indicated, the effect is to smooth out some of the sampling vari- ation in the data under the assumption that the spectrum is not too complex a function. Since there are only 36 points to fit for a given k, for N large enough a perfect fit within the resolution of the data could be obtained. The coefficients, a (e) and bi (H), in equation (11.18) can be computed for a given }, = amk/96, by equations (11.21) and (11.22). +17 (m. +5)" 2n(m +3)n| (11,21) a(k) = een F[k, aa * cos 36 +17 (m +4)n 2n(ma + 4)n) (et 2)2;) b,(k) =2 05 | Bikes bh inl m=-18 36 a6 227 A measure of the variation in F(k,6) is given by ~ Je (11.23) Il = 2 ae 8) ] and since 7 N 2 (11.24) My -f [(u, @)]* a0 = + +4 = [la ly)” + (oC) )7] T i n 1/2 n= the closeness of the fit for a given N is given by i (11.25) Ry = M)/M-+ It Ry is one. the fit is perfect for the available data, i Equation (11.18) can also be put in the form of equation (11.26), & 1B oe Ne (11,26) f(y, 8) = 2 1 + ea cf) cos(2n(6 - Y,)) for a () ~ <6 <6, (iH) ; where . 2 Ze pe lee (11.27) c ty) = fa, (4) +b, (H)] and mis -] b, (4) (11.28) Ynit) = 2, tan an (y) | The values of c and — for n equal to 1, 2, 3, 4, and 5 3were ns No computed by means of the IBM 650 for each k in Table 11.6. The results are given in Table 11.11. The values of c, and Y, 2re plotted as a function of inp hioune pil 30: The values of c, and y, show a fairly smooth variation with kas do also the values of co and y2. The values of C3, C4; and Cp are low and somewhat erratic, and the values of Y3: Y4: Ys are highly variable 228 +10° -10° -20° -30° it (25S SIS SES Si IB ISs 20n zien 2s 24) 25) 26) 27, Figure 11.30 Cp AND yn AS A FUNCTION OF k 229 Table 11.11. for the analysis of the angular variation, 12 13 14 15 16 1,24 1.21 1,16 1,10 0.97 22g z° 228,29 =ze7” 25475 09.69 1516" -969 963 .983 .975 .934 0.24 G16 “Gul TNO? 0429 =27,2° “=21 5) =26,6°) =1s00" 40.5" -985 -970 991 2255 2700 230 17 0.93 911 0.37 £1,.8° -954 18 0.85 -14,2 934 0.31 +2.1° 967 0.23 =1,6° (Cont, ) ie) Fourier coefficients, phases and goodness of fit 20 0.66 Hide? 118° 974 0.45 +3.8° 949 0,15 55° -956 Table 11,11 (Cont. ) al 0.76 885 0.45 2,1° 22 0,78 ~3,3° « 847 23 0.70 0,23 °980 0,08 44,59 °982 (ait: ey ea’ 231 especially when one notes the way in which y,, is defined. The values of R, range from 0.983 to 0.847 and the average value is 0.918. Thus over 90 percent of the angular variation on the average is ex- plained by the values of Cy and Vy The values of R, range from 0,991 to 0.882 and the average value is 0.955, Over 95 percent of the angular variation ~ is explained, on the average, by the values ofc), y,, cz and yz. The erratic behavior of the other coefficients is explained as an attempt to fit the sampling © variation of the data. The graphs of Cy Yas Cops and y> do not vary as a function of k very rapidly. It would not be difficult to express them as somewhat smoothed func- : tions of k (and hence p.) over the range of k from 1] to 27. The result would then be given by equations (11.29), (11.30), (11.31) and (11.32). (11.29) ce, = y(n) (11.30) Yi v1 (11.31) eg = cz (4) (11.32) Y> = ¥> () The directional spectrum could then be defined analytically as a function of frequency anddirection by equation (11.33) in which precautions would have to be takento insure that the square bracket on the right was always positive. A Re ce ZB IH = 2 6 ee A au + cy *(H) cos(2(8-y, *(H))) # c2"() cos(4(@ - y,"(H))) ] Also f(y, 8) would have a minimum in the first quadrant as a function of 8 232 for a fixed p. Let this minimum be Q~ (n). Then (11.33) would be defined as above for “ < ote 6 n(H) - ™< O< Oy (H) and by zero otherwise, The analytic expression determined as outlined above could then be trans- formed to Cartesian coordinates in the a, plane as described in Part 8. If the function [A(a, 8) ]° so obtained were integrated over a square of ie area of one of the squares in the U(r,s) plane the resulting number would then be quite close to the computed values of U(r.s) and it would certainly agree within possible sampling variations with the computed number. However, such an analytic expression would still reflect certain features of the observed data and the wind field which generated the waves which would be difficult to generalize to other cases. In what follows this point will be discussed in more detail and a simpler analytical expression derived for wave forecasting purposes. Properties of the directional spectrum By means of the data tabulated and graphed so far, in particular by means of Tables 11.6, 11,10; and 11.11 and by means of figures 11,22, 11,26, 11.27, and 11.29, certain properties of the sea generated by the local ae in the area where the data mec obtained can be summarized. | These properties are (1) that the integral over Wi ucetos of the Girecy tional spectrum agrees remarkably well as a function of frequency with the f theoretical spectrum derived by Neumann for an 18.7 knot wind, (2) that the angular spectrum is concentrated over narrower angular range for long waves 233 (low frequencies) and spread out over a wider range for short waves (high frequencies) and (3) that the integrated spectrum continues as predicted into higher frequencies as determined by the wave pole data, The properties should be expected to be the same for other spectra obtained for other con- ditions at other times. There are other properties of the particular spectrum studied which are in part probably due to sampling variation and in part due to the particular local wind field which generated the waves. The values of if show that the peak in the angular variation of the spectrum shifts from what corresponds to 180° in figure 11.29 to 140° as the frequency increases from 2n(11)/9%6 to 2mi27)/96. Also a secondary peak at frequencies corresponding to 2m(i6)/96, 2m(i7)/96, 2m(18)/96. and perhaps even for higher frequencies, is indicated in figure 1i.22, and by the high values of c> and the values of ‘f in figure 11.30. This secondary peak causes the graphs in figure 11.22 to have the property that they are not even functions about some central value of the direction. The change in % can be explained partly by sampl- ing variation and partly by the fact that the local wind direction was reported to be from 330° and the winds further to the north were from 360°: Pos- sibly the winds to the north were the ones which generated the longer waves. The skewness of the curves for the angular variation may or may not be real in the sense that it would still show up in a spectrum with a larger number of degrees of freedom. However, it should also be noted that there is a wind shear present over the area of wave generation with the property that the 234 wind speed increases from east to west across the area under study. A possible effect of the shear would be to produce the skewness in the angular variation as indicated, At some future time it may be possible to extend the concepts of wave theory to permit a representation of the local wave spectrum as a function of wind velocity and wind direction locally and as a function of the change of wind direction up wind and the shear in wind velo~ city cross wind. Todo this would require a greater number of degrees of freedom than this study has obtained, several different spectra for different wind conditions, and a very detailed study of the wind fields. An idealized directional spectrum For the present purpose, however, it is desirable to attempt to ideal- ize the results obtained so as to reflect the three results pointed out above and so as to eliminate sampling variation and the effects of changing wind direction and wind shear. It will therefore be assumed that [A(p, 9) 17 is an even func- tion about the local wind direction and that its peak value falls at 6=0. The values of cj(y) as tabulated above thus determine the amplitude of the cos 20 term and 01 (4) is assumed to be zero. (This implies a rotation of -30° for the axes in the figures given above if it is desired to approximate the peak of the spectrum. ) After considerable subjective curve fitting and trying a number of pos- sible functions which did not do as well, it was found that cj could be approxi- mated by the following function of frequency and wind speed where the values are inc. g, s, units and v is (18.7 x 51.5) cm/sec. 235 VA Se) ey = 0,50)-4 0.82 o(ev/e)"/2 The function f(p, 6) can then be given by ~(uv/g)*/ 2 (11.35) — f(w,@) = 4 [1 +(0.50+0.82e ) cos 26 +c, cos 46] 7 2 for -7/2 < @< T/2. Since the values of c, are greater than one for small yp, f(y,8) becomes negative for © near + 1/2, and this is not permissible, To avoid this, cp must be chosen so as to make f(y, 6) everywhere positive. Since (11.36) cos 20 = 2(cos 6)“ - 1 and since (11.37) cos 40 = 8(cos 9)* - 8(cos 9) +1 equation (11.35) can be rewritten as equation (11,38). 4 (11.38) f(u, 8) = [1 - 0.50 -0,82e Hv/e) /4 4 <1] 4 + [1.00+1.64 e (hv /g) [2 8 ca](cos 9) 74 8c,(cos 9)* In order to keep the term independent of @ always positive, the small- est possible value of C5 is given by 4 (11.39) Sp = 0.326 v/s) /2 The function, fi, 6) can then be written in two alternative forms as equations (11.40) and (11.41), 236 ‘a ihe 4 4 (11.40) £(p, 6) =] +(0.50+0.82e HV/8) -/4) 55264 (0.326 bv/8) /4 cos 40] ie airoo sorseien >.) (ceciens (11.41) £(y,0) = 4[0.50(1 - e 2 (2,86 e(ev/e)*/2, (cos @)4] A value of cz greater than 0.33 would make the coefficient of (cos @)2 in (11.41) negative for small » with the accompanying possibility of negative values for f(y, 6). The curves for cj and cz are graphed against the observed values of c) and cp in figure 11.31. The fit is fairly good for c); and for cz for frequencies corresponding to k equal to 11 through 15, the fit is not too bad. The extension of the curves outside of the region where data are available is quite arbitrary. For the longer waves the value of (11.38) has little total effect on the spectrum because the energy is very low there. For the shorter waves if c) became less than 0.50, the effect would be even greater angular spreading. Note that in figure 11.30 Y, and y2 are close together for k equal to 11 through 14, and that in a sense the value of cz used above is only the in-phase part of cos 40 with respect to the original data when k is larger. A possible functional form for the directional spectrum of a wind generated sea is finally given by equation (11.42) if the wind is uniform in direction and speed over the area of wave generation and if the sea is fully developed. ze oo 2le/Hv) We 4 (lue42) . [A(e, 8) 17 -1fi+ (0.5040.82e7#V/8) /2) cos 26 a 2 4 + (0.32 e (Hv/g) ye, cos 40 | for -1t/2< @< 1/2, and zero otherwise. 237 SEZ 7D GNV 'D ‘SINSINI4S5O9 QSL ATSAILOSPENS = tert “Bra Ov SE 9E VE 2 OF B2 92 we 22 O2 BI 9! vi 2 Ol 8 9 » 2 oO (sar) H-2 280 +0G0 ='9 v This idealized directional spectrum still comes fairly close to agreeing with the curves in 11.22, After proper angular rotation, the (cos 9)4 term will give good agreement with the curves for low frequencies. Agreement with the higher frequencies is also good. The secondary peak and the skew- ness at intermediate frequencies is missed. Caution is recommended in the use of equation (11.42). Within the limita- tions mentioned above it comes close to describing the sea observed for a wind near 18.7 knots. For higher or lower values of the wind speed, however, it may not work although as a working hypothesis it may lead to useful results. Since only one spectrum was observed the variation in v of f(y, 6) as fitted cannot be tested. One could on the basis of the available data put v = 18.7 knots inside the square brackets of equation (11.42) and say that variation in [A(p, 9) ]7 as a function of v is caused solely by the occurrence of v in the first term. However, there are two additional points that can be made in favor of equation (11.42) as written. They are that it would appear to give more real- istic swell forecasts than previously used formulas, and that the mean square slope of the sea surface still varies linearly with wind speed as observed by Cox and Munk [1954]. A previously given equation for the directional spectrum of a wind generated sea [Pierson, 1955] is shown in equation (11.43). 2 -2(g/pv) (11.43) [Als )]* = 2&7 p for -7/2< @< 1/2, and zero otherwise. (cos 9)° 239 Equation (11.42) can also be written as equation (11.44). 2 -2 (11.44) [A(y,6)]7 vie Saleh u [o.25(1- elev /2)*/2) 4 (0,50 -0.46 e(iv/e)"/2 (cos @) + (1.28 e(hv/8)" 12) (cog @)4] for -17/2< 0< 1/2, and zero otherwise. If is small, the angular term in (11.44) becomes 0.04(cos @)* + 1.28(cos@)* which shows that the spectrum is more peaked at low frequencies than had been assumed in (11.43). Conversely if pis larger, the angular term in (11.44) be- comes 0.25+40.50(cos 9) 2 which shows that the spectrum is more evenly spread out at high frequencies than had been assumed previously. The angular spreading factor used in Pierson, Neumann and James [1955] can be derived from equation (11.43) and it is given by equation (11.45). 8, sin 26 (11.45) F(o) = 4494 51028 for -n/2< 0< 0/2. The angular spreading factor from equation (11.42) can be writtenas equation (11.46), e 4 i 4 (11.46) F(u,6) = 240+ 10.50+0.826 Evie) fe sin20, 0.32 (uv/g)"/2) in 40 5 Zee en 4t for -t/2< @< w/2. The curves for f(6) as given by (11.45) and for F(p, 6) with » = 0 and Ww = © as given by (11.46) are given in figure 11.32. Equation (11.43) is seen to be a compromise between the two extremes indicated by equation (11.42). The new results, if correct, indicate that long period swell will be higher on a line through the center of the generating area parallel to the wind direction than it would be using the methods of Pierson, Neumann and James[1955] and that the short period waves which follow later would be lower. Stated another way, the long period components of the spectrum are more concentrated in 240 90 70. 50 40. 30 20: PERCENT — Fig. 11.32 DEGREE S—> ANGULAR SPREADING FACTOR FOR F(o,e),F(e) ond F(s.e) 241 the direction of the wind and hence in general they should be observed at a greater distance than the short period components which spread angularly over a wide area outside of the generating area. These results are thus an- other reason, apart from possible effects of viscosity, why swell has a higher period than the waves in the area of generation and why shart period swell is seldom observed. It should be noted that pv/g is just another wavy to write v/e where c is the phase velocity of the speciral component, and it will not be too difficult to write a brief modification of Chapter 3 of H. O. Pub. 603 which will employ a family of angular spreading diagrams as a function of v/c and permit better swell forecasts. Cox and Munk: [1954] have found that the variance of the slope of the sea surface increases linearly with the wind velocity and that the theovetical sper- trum of Neumann [1954] correctly predicts the total slope variance of the gravity wave part of the spectrum. The upwind slope variance is given by equation (11.47) and the crosswind slope variance is giver by equation (11.48}. (See Pierson, [1955]. ) co ,7/2 pn (11.47) cae -| { [A(p, 9)]° - (cos 0)“ dOcu G@ =n/2 Co 0/2 ; (11.48) a = [A(u, 6)]? ie (sin 0)“ d@du 0 -n/2 When equation (11.42) is substituted into equation (11.47) the result can be simplified to the form of equation (11.49) where v is in meters/sec. 24 2, 0O 3 AP -a*/2-8/a4 (11.49) of ey xy LO > yfo.soso.12s+ 2205 f aunt, a da | A yar 0 The contribution of the integral to equation (11.49) is quite small and the 2 value of g, can be given by equation (11.50) where v is in meters per second. (11.50) ox” = 0.99: 1079 v Similarly of can be found to be equal to (11.51) ae Sioleowmon. These values of the upwind and crosswind slope contributions are in better agreement with the observations than those which result from equation (11.43) although Cox and Munk found of and oy. to be nearly equal. Perhaps the dis- crepancy can be explained by the nature of the site at which they obtained their observations. * If the ratio, jv/g, used in deriving equation (11.42) had been of the form RV, /g where Vv, is @ constant equal to the wind observed at the time of the ob- servation, then the integral over a@ would be a function of v such that the ex- ponent would be (-a" /2 - 8/{va)*). For a surface wind of 15 m/sec there would be a tendency toward a greater contribution to o,, than observed by Cox and Munk, and similarly a smaller contribution to oy Barber [1954] has studied the angular variation of waves with a period near two seconds in Waitemata Harbour, Auckland, He found an angular vari- ation somewhat like [cos el*. However, his results cannot be compared with these results as he writes that "the wind was about 15 knots and 2 sec waves * See also the end of this chapter. Dae were dominant; but because the fetch in the wind direction was much greater than elsewhere, it is not expected that the [results] will apply to open water." Aliasing in the directional spe ctrum As shown in figure 11.26 the eos wave pole spectrum at high fre- quencies is a little high compared to the theoretical Neumann spectrum. The computed energy at frequencies greater than a value corresponding to a k of 275 ici O57 (£t)@. Some of this energy is aliased in the directional spectrum back into longer wavelengths. The amount aliased is certainly less than 0.57 because part of the above value is probably white noise and part is correctly located in the corners of the rectangular area of the directional spectrum analy- sis. Only about 0.37 (4t)2 lies above k equal to 31 and hence some part of 0.20 (ft)? is correctly located. Thus as a very crude estimate something of the order of 0.25 (£t)@ is actually aliased over the directional spectrum. When spread out over a wide frequency and angular range, this aliased energy is un- detectable because of the cevaniellinn variation in the higher unaliased values. Correction to the covariance surface The covariance surface given in figure 11,15 still has errors due to white noise and column noise a it. The correction for white noise is to sub- tract 0.56 (mam* x 100) (that is, 0.54 x 1.032) from (0,0) and 0.191 from the central column (0.186 x 1.032). The result is the estimated covariance surface of the sea plus the swell as shown in eee 11,33, The major effects are to re- duce the peak at the center, and hence increase the correlation of the edges with 244 +l + + + * + + ORR ~.104 -,310 +240 130 + + .220 076 + 7 O31 =.097 -200 202 -314 _-2i5 _=.020 Sea Rae + I+ =.285 -.283 -.177_-.033 = .018 nea ae heats 4 055-155-208 -.242 -192 =060 042, = rs ch, Le RE EO CT Ee 226.090 080 ~012 + + + SS 1210 046.024 -,006 Tate. > > + + 23 {096 _=230_-.282 -.200_-J75 + + + + + ea . ry + + Nig 72026 098 136 _=138_=050_ 4 =106 _=24 +258 ae om 244 a © + =247_=223 -140 c c 1040-025 -078 -063 042 136 oe Ars + + 061 © \ =.090 073-0185 r a + 2277 _=200_-.092' 4 080 ~033__-.1I7 FIGURE 11.33 CORRECTED COVARIANCE SURFACE 245 the center, and to push the zero contour more realistically to the left along the -q axis. It would not be too difficult to remove the effect of the swell from the co- variance surface and to obtain an estimated covariance surface for the sea. How- ever, as Tukey and Hamming [1949] have pointed out the covariance estimates are subject to even more erratic sampling variation than the smoothed spec- tral estimates and this sampling variation is not well understood. For example, Tukey [1951] has shown covariance functions computed from portions of the same time series. They were markedly different and yet the spectra computed from the different covariance functions were very similar. For many types of problems in which knowledge of the covariance function is needed, it has been found that reinverting the smoothed spectral estimates will yield a more reliable covariance surface. Also for simpler problems the simplified spectrum given above which is symmetrical about 9 = 0 would give a more tractable covariance surface. Alternate procedures for determining directional spectra A number of alternate procedures for determining directional properties of waves have been proposed and attempted. The methods used by Barber [1954], essentially directional antenna arrays, are by far the simplest and most economical if fixed positions for the wave poles can be maintained. The effects of refraction and perhaps bottom friction and percolation, however, make it difficult to generalize to open sea conditions and study the full range of components in the spectra. 246 Another way is to take wave records from a moving ship by means of the shipborne wave recorder as described by Cartwright[1956]. The ship is run on courses corresponding to an n sided polygon and the shift in frequency of the spectral components is studied. With enough degrees of freedom per spectral estimate, it should be possible (in principle) to resolve the spectral estimates into a directional spectrum by the inversion of some simultaneous linear equations in an appropriate number of unknowns somewhat along the lines of the method described by Pierson[1952]. However, if the response of the instrument to the wavesis different for different headings due to the pres- ence of the ship and if the records are too short so that sampling variation from record to record is pronounced, then the difficulties tobe encountered will be even greater than those encountered in this report. Although some of the data reduction might be eliminated by analogue methods, the procedure would have essentially the same degree of complexity asthe one used in this report. The latest proposed procedure for determining directional spectra is given by Longuet-Higgins [1957]. The records from an airborne altimeter capable of measuring 1(x,y) and 01(x,y)/dt are assumed at the starting point, and then by computing various moments fromthe data as determined by such quantities asthe average distance between successive zeros at various head- ings and the velocity distribution of zeros, the moments of the spectrum are obtained. Then by an inversion technique the spectrum is deduced. Pierson[1952] proposed the use of anairborne altimeter to determine the directional spectrum. The method of analysis involved the study of the 247 spectra obtained at different headings and the solution of a set of simul- taneously linear equations. From the results of this present study, it can be stated that the method proposed by Longuet-Higgins[1957]is not likely to be successful, especially with respect to a sea. Since the data are taken at different times at different headings, each record has a different sampling variation for each spectral estimate. The various moments thus have wide sampling variation. Moreover, eighth moments are required to give any sort of definition to the spectrum. For the true sea surface the eighth moment is entirely determined by the capillary waves on the water. Some sort of filtering action would be needed in the recording instruments to maintain pure gravity wave conditions otherwise a problem in resolution would arise due to the extreme range of wavelengths covered. The effect of such filters would have to be incorporated in the theory. Even with the capillary waves filtered out there would be high frequency error noise of some sort or another present in the data. In computing an eighth moment, this noise would blow up beyond all recognition and com- pletely obviate the value of the estimated moment. In contrast the methods used in this study effectively suppress high fre- quencies whether real or due to errors in the data.- Also various sources of error which will undoubtedly be present in any method of recording waves were isolated and removed. The very valuable results of Longuet-Higgins [1957] on the statistical 248 properties of a random moving surface can most efficiently be applied by using the moments computed from the corrected spectrum obtained in this study and allowing for the effects of the white noise and column errors. The winds as observed by the R. V. Atlantis were measured by means of a three cup anemometer and a wind vane. The three cup anemometer and the vane were mounted at the end ofthe main boom above theupper laboratory of the Atlantis at a height of 15 to 18 ft above sea level. The dial of the ane- mometer was read visually to get the wind speeds. The winds as observed might have been a little high compared to undisturbed measurements over open water due to the presence of the ship. The theory of the Neumann spectrum is based on observations of the wind at a height of about 25 feet above the sea level. If a logarithmic wind profile is used with a roughness coefficient of 0.75 cm (Neumann [1948]), and if 15 feet is used for the anemometer height of the Atlantis, the 18.7 knot wind becomes a 20 knot wind at 25 feet. It becomes a 19.5 knot wind if 18 feet is used. The theoretical spectra for 19 and 20 knots are also shown in figure 11.26. On consideration of the confidence bands of the composite spectrum, especially near the peak where there are only 22 degrees of freedom, the variability of the winds during the time when the 18.7 knot average was ob- tained, and the compensating effects of the presence of the ship and the cor- 249 rection to a greater height, it is only possible to conclude that the agreement is satisfactory within the range of possible variation of wind speed and true spectral values, and that there is certainly no justification for changing the constant in the Neumann spectrum. Added notes on the results of Farmer Farmer [1956] has made further measurements of wave slopes on the windward side of Bermuda. He therefore had an unlimited fetch of open water over which the sea was generated in contrast to the results of Cox and Munk [1954] in which some islands may have interfered with the fetch as pointed out by Darbyshire [1956]. Farmer [1956] found essentially the same total slope variance as Cox and Munk [1954]. The ratios of upwind downwind to total slope variance found by Farmer were 0.57, 0.60, and 0.77, and these compare quite favorably to the theoretical value of 0.625 given by equation (11.49). References to Parts 10 and 11 Barber, N.F., [1954]: Finding the direction of travel of sea waves. Nature, v. 174, p. 1048. Cartwright, D. E., [1956]: On determining the directions of waves from a ship at sea. Proc. Royal Soc., A, 234, 382-387. Cox, C., and W.H. Munk [1954]: Statistics of the sea surface derived from sun glitter. Jour. of Marine Research, 13(2), pp. 198-227. Darbyshire, J., [1955]: An investigation of storm waves in the North At- lantic Ocean. Proc. Royal Soc., A, v. 230, pp. 560-569. Darbyshire, J., [1956]: An investigation into the generation of waves when the fetch of the wind is less than 100 miles. Q.J.R.M.S., v. 82, Oct. 1956, pp. 461-468. 250 Farmer, H.G. [1956]: Some recent observations of sea surface elevation and slope. Woods Hole Oceanographic Institution, Ref. No. 56-37, (unpublished manuscript). Longuet-Higgins, M.S. [1957]; The statistical analysis of a random moving surface. Phil. Trans. Roy. Soc., Ser. A, no. 966, v. 249, pp. 321-387. Neumann, G., [1948]: Uber den Tangentialdruck des Windes und die Rauhigkeit der Meeresoberflache. Zeit. Meteorol., H7/8, 193-203. Neumann, G., [1954]: Zur Charakteristik des Seeganges. Archiv. f. Meteorol. Geophysik und Bioklimat., ser. A, v. 7, pp. 352-377. Neumann, G., and W. J. Pierson, [1957a]: A comparison of various theo- retical wave spectra. To appear in ''Proceedings of a Symposium on Ship Motions in Irregular Seas.'' Ned. Scheepsbouwkundig Proef- station, Haagsteeg 2, Wageningen, Netherlands. Neumann, G. and W.J. Pierson, [1957b]: A detailed comparison of various theoretical wave spectra and wave forecasting methods. (In preparation, to be submitted to Deut. Hydrogr. Zeits.) Pierson, W.J., [1952]: A unified mathematical theory for the analysis, propagation and refraction of storm generated ocean surface waves. Parts land Il. Research Division, College of Engineering, New York University, Department of Meteorology and Oceanography. Prepared for the Beach Erosion Board, Department of the Army, and the Office of Naval Research. Pierson, W.J , [1954]: An electronic wave spectrum analyzer and its use in engineering problems. Tech. Memo. No. 56, Beach Erosion Board, Washington, D.C. Pierson, W.J., [1955]: Wind Generated Gravity Waves. In Advances in Geophysics, v. 2, Academic Press, Inc., Publishers, New York, N.Y. Pierson, W.J., G. Neumann, and R. W. James, [1955]: Practical methods of observing and forecasting ocean waves by means of wave spectra and statistics. H.O. Pub. 603. Press, H., and J. W. Tukey, [1956]: Power spectral methods of analysis and application in airplane dynamics. Flight Test Manual, Vol. IV, Instrumentation NATO Advisory Group for Aeronautical Research and Development, edited by E.J. Durbin. Part IVc, pp. ivc-l to ivc-41. 251 Roll, H.U., and G. Fischer, [1956]: Eine kritische Bemerkung zum Neumann-Spektrum des Seeganges. Deut. Hydrogr. Zeits., Band 9, Ines tke Tucker, M.J., [1956a]: Comparison of wave spectra as measured by the N.I.O. shipborne wave recorder installed in the R. V. Atlantis and the Woods Hole Oceanographic Institution wave pole. N.1I.O. Internal Report No. A6. Tucker, M.J., [1956b]: A shipborne wave recorder. Trans. of the Institution of Naval Architecture, v. 98, pp. 236-250. Tukey, J.W., [1949]: The sampling theory of power spectrum estimates. Symposium on Applications of Autocorrelation Analysis to Physical Problems. Woods Hole, Mass., June 13-14 (Office of Naval Research, Washington, D.C.). Tukey, J. W., [1951]: Measuring noise color. Bell Telephone Laboratories in Murray Hill, N.J. Prepared for distribution at a meeting of the Metropolitan Section Institute of Radio Engineers, 7 Nov. 1951. Acknowledgments The help of Mr. Raymond Stevens, Mr. Rudolph Hollman, and Mr. Roy E. Peterson in the preparation of this part of the work is greatly appreciated. 252 BARTZ RECOMMENDATIONS AND CONCLUSIONS Conclusions The directional spectrum of a wind generated sea has been determined from stereo data after correcting the data for differential shrinkage, column noise, white noise and the presence of a swell. This spectrum shows a single peak and the contributions from different wavelengths cover a wide range of wavelengths and directions. When transformed to a frequency spectrum, with directional effects eliminated, the results are remarkably close to the theoretical spectrum derived by Neumann. The longer waves in the spec- trum are concentrated over a narrower range of angles about the wind di- rection than the shorter waves. The actual spectrum reflects some effects of sampling variation, wind shear, and changing wind direction upwind which are difficult to isolate be- cause of the nature of the wind data and the sampling variation. When these are removed by simplifying assumptions, it is possible to obtain an analytic representation for the spectrum which appears consistent with known pro- perties of swell and sea surface slopes. The analytic representation which has been obtained rests upon some- what shaky foundations as far as angular effects are concerned. However, for forecasting it would appear advisable to incorporate these results into the forecasting method without awaiting further verification. Certainly the re- sults on which such a revision would be based are on firmer theoretical ground 253 than the results on which the original material was based. The spectrum computed from the wave pole data does not agree with the spectrum computed from the stereo data nor with the corresponding theoretical Neumann spectrum. Variability in the spectrum dueto wind variation of the order of one knot would explain the discrepancy. However, a more likely reason for the discrepancy appears to be in the calibration of the wave pole. The numerical results which have been obtained provide valuable data on a wind generated sea for a fairly low wind speed. It will be particularly useful in studying the topography of the sea surface and in problems con- nected with seaplanes and small vessels. Recommendations The use of stereo photographs to determine the directional spectrum of a sea has proved feasible. Due to attrition, an originally desired 50 degrees of freedom per spectral estimate was reduced to only 19. The com- putations were lengthy and difficult, but nevertheless results of consider- able value were obtained. It is difficult to generalize the results obtained to higher wind speeds, and one determination of a directional spectrum is not enough to provide comprehensive details on fully generated seas for a range of wind speeds. It is therefore recommended that an experiment similar to the one de- scribed in this report be repeated for a fully developed sea at at least one higher wind speed. A wind of 24 knots, a fetch of 130 NM and a duration of 254 14 hours should not be too difficult to find. For these conditions the signifi- cant wave height would be nearly double and the E value would be nearly four times those observed for the 18.7 knot wind according to the results of Neumann. If such distorting effects as differential shrinkage and column noise could be eliminated by proper choice of film and preliminary studies of their causes it would then be possible to allow a four-fold increase inthe white noise vari- ance without seriously affecting the results. This would permit the planes to fly higher, thus covering a larger area in one stereopair and providing better resolution and more degrees of freedom. An appendix written by Simeon Braunstein, the Research Division photographer at New York University, follows these recommendations and conclusions. In it is given a discussion of the stability of different types of film bases and of film processing methods which should eliminate the effects of differential shrinkage. In such an experiment more careful attention should be paid to the wind field, and winds should be recorded at least every hour for as long a time as possible prior to the observations. A decrease of wind speed in the wind field should be avoided. The winds, if possible, should be measured at several heights. Moreover, now that a fairly good method of analysis for the results has been developed, it should be possible to program additional operations on the spectrum to carry out in just a few minutes all the computations made in Part 11. 255 As mentioned in Part 11, a calibration study of the wave pole is recom- mended, but for a new stereo study, it is recommended that the design of the wave pole be altered along the lines suggested by H.G. Farmer. At or near the same time that the stereo and wave pole data are taken, it might be advisable to take records with anairborne altimeter as developed at the U.S. Navy Hydrographic Office and with shipborne wave recorders in- stalled on several different types of vessels, The airborne altimeter could be flown at a number of different headings and the ships could be operated both hove to in head seas and on polygonal patterns as described by Cart- wright. This would require an advance forecast of a stationary state for at least four hours, but this should not be too difficult to achieve. The wave pole data at the present time appear to be the only data capable of reproducing the higher frequencies correctly, and such data would still be needed. With stereo data, wave pole data, airborne altimeter data, and shipborne recorder data it will be possible to make exhaustive cross checks of the calibrations and responses of all the instruments and to study the relative utility of each. With such exhaustive measurements of the sea state, additional data of interest to naval architects and electrical engineers could also be ob- tained at the same time. This would permit their theories and calculations to be based on a firm foundation consisting of adequate knowledge of the state of the sea at the time of their observations. 256 APPENDIX The Dimensional Stability of Photographic Films Abstract The dimensional stability of several photographic film bases to relative humidity, temperature, processing, handling, and storage is discussed. Permanent and temporary size changes are outlined, Recommendations for the choice, processing, and storage of film intended for photogrammetric use are made. Sources of errors in stereo-photogrammetry In addition to optical factors, platform stability, camera tilt, etc., the inherent dimensional instability of flexible photographic film bases may contribute to error in measurements made from aerial stereo photo- graphs. In order to minimize error due to the last cause, care must be taken in the choice of film, storage before and after exposure, processing, and handling. Dimensional changes in photographic films may be classified under two headings: temporary and permanent. There are two factors involved in temporary changes: temperature and relative humidity. Temperature effects The thermal coefficient of expansion of most common film bases (1944) is approximately 5 x107° inches per inch per degree F, or about 0.05 per- cent per 10°F. Table I shows the effect of temperature, as well as relative humidity and processing, on several film bases (Fordyce, Calhoun, and Moyer, 1955). The expansion is generally 10 to 40 percent greater in the widthwise than in the lengthwise direction. This is the result of the partial orientation of the molecules in the base in the machine direction. It is evi- dently easier, under these conditions, to increase the distance between them, either by thermal agitation, or by the introduction of moisture, in a direction perpendicular to this alignment. Humidity effects The humidity coefficient of linear expansion of common films varies from a low of 1.0 x 107° for DuPont "Cronar" to about 10 x 107° inches per inch per 1 percent relative humidity change, for standard cellulose acetate. This effect is essentially linear between 20 and 70 percent relative humidity, and somewhat greater below 20 percent and above 70 percent. Photographic films exchange moisure with the air continually. The mois- ture content of a film is determined almost solely by the relative humidity of the air with which it is in equilibrium. 257 TableI. Average Processing Shrinkage, Humidity Expansion and Thermal Expansion of Current Eastman Motion Picture Films. Humidity expane Processing sion per 10% Thermal expan- shrinkage, % R.H., % sion per 10 F, Film Base (Tray (Range: (Range development) 20%-70% R.H.) 0-100 F) Length Width Length Width Length Width Black-and-White Negative and Eastman Color Triacetate -06 .07 -07 -08 -03 .035 Negative Black-and-White Positive and Triacetate -05 -05 -05 .06 -03 2035 Sound Recording Eastman Color Triacetate -07 -08 -06 .07 03 -035 Print Kodachrome Films Acetate (16mm) propionate -09 -10 -08 .10 .035 .04 Cronar A new polyester film support, 'Cronar", is now being produced by the Photo Products Division of E. I. Du Pont de Nemours and Company. At pre- sent, it is being coated only with the slow, high contrast ''photolith'’ emulsion. Reference to Tables IJ and III, and Figures 1 and 2, will indicate that "Cronar'' shows considerable improvement in dimensional stability over other flexible film supports, in regard to temperature, relative humidity, and processing. There is reason to hope that when ''Cronar" is available in larger quantities, it will be coated with an aerial emulsion, in addition to the litho emulsion now available. Permanent changes There are three principal causes of permanent dimensional changes in film supports. The first, and most important, is the gradual loss of volatile chemicals (plasticizer and solvents). Film base is cured for about five hours, which eliminates about 96 percent of the volatile chemicals. The subsequent loss of the remaining 4 percent causes shrinkage and re- lated troubles. Shrinkage from this cause is accelerated by heat and moisture, and reduced by preventing free access to air. As with tempor- ary changes in dimension, shrinkage is greater in the widthwise direction. The compressive force of the emulsion upon the base results ina certain amount of plastic flow or permanent shrinkage. Dimensional changes from this cause are increased by heat, because of increased film plasticity at high temperatures. Moisture also increases base plasticity but inhibits the contraction of the base, and the latter has the greater effect. Thus, an increase in relative humidity, at constant temperature greatly decreases this type of shrinkage. Plastic flow of the base may also be the result of stretching in handling and processing -- resulting in extension lengthwise. Such changes are increased by heat, moisture, amount of tension applied, and the duration of the tension. 258 Base Type TABLE 11 Humidity Coefficient (length change in inches/inch of length/1% RH change) Size change example of a 30” litho nega- tive with a 20% in- crease in RH, using the midpoint of the Humidity-Coefficient Range .0042’” *‘Cronar’’ based PHOTOLITH 1.0 to 2.0x10-° .009” (1.5x10-5) .0058” Standard cellu- lose acetate based PHOTOLITH 8.0 to 10.0x10-° .054” (9.0x10-5) .0058” Litho sensitized high acetyl cellu- lose acetate base 5.0 to 7.0x10°° + .036” (6.0x10°*) Litho sensitized polystyrene base 1.0 to 2.0x10-° + .009” (1.5x10-*) .0120” Litho sensitized vinyl base 1.0 to 2.0x10-° .009” (1.5x10-*) (Coefficient x film length in inches x % RH change = film size change in inches.) TABLE III Size change example Temperature of a 30” litho Coefficient negative with a 20° Effect of Temperature Changes Unsensitized Base (in./in./1 °F.) rise inT on Film Size “Cronar’’ base Standard cellulose acetate base High acetyl cellulose acetate base Vinyl base Polystyrene base Glass Aluminum These average temperature coefficients indicate the rela- tive temperature stability of various photographic supports and can be used to calculate negative size change with varying temperatures at constant humidity. (Coefficient x film length in inches x temperature in °F = film size change in inches.) JONVHO ALIGIWNH JAILW13Y Boy + Boe + Boz + Bor + 0 *SUOI}IPUOD jo @Bups epiM D 418A eoUDULIOJJed ;DUO!sUeWIP eBDIeAD Bulyow! -4S8 U! @pinB yua!UBAUOD D sD BAJ@S ||IM 4IDYD SIUy “J@AQMOHY *9UO0|JOUNS IDAUI| JOU BID syUBID14je0> AyiprwiNy 41944 OoUIS “payopoiddd eip Ayiprwny yBiy pup/Mo] jo sewesyxo eu} sD A\qoioeiddp Aipa Adu syonpoid espq esojnj|a> 4103 sen|DA “UOIJDWOJU! 40] S14} WOIy payDjnd|D> AjxD1nb eq uDD seAlyD -Beu 1061p] yo eBuny> ezIg * E00" U!AYs PjNOM swj1y pasog wtDUOID,, PUD 4.8/0" 4ULIYsS PjMOM WI} 94DJ9DD Bsojn|]9> Pupp -UDJS D “47/9” 4NOgD yULJYS PjNOM WII} @sDq 94DJ99D 9sojN|]e9 [44990 yBiy 0] D “ejdwioxe siyy uy *eA!yDBeu oO] D yo eGupyo @zIs eyy Aj49011p seyDD|puy YO!YM e/D9s (@GuDYD ezis) puDy 448] 944 Of UOIJOasIO4U! sIyy Woy A} |DyUOZIJOY poel Uayy puD asn ul addy wij!y ays Of Spucdse.toOD yDYy yaxDDIq Ay1]1qGDys OUy JO 1ejue> ayy OF Aj]DD1448A poss MON] *(%0Z-) @U!| esDq aus uo jurod Buypuodsesi0o9 ayy @4D90; pup (doip Hy %Oz D Ads) Pesejunosue eq {]3M yOu, BBuDYS Ay!piuiny @AIyO] a4 ayy Ys!| -qojse ‘seBupys ezis wi! Bulyouiyse ul y4DYD si4yy @sN OF SNOILSNYLSNI W1ld OHLIT 01 V 40 S3HONI NI SJONVHO 3ZIS 260 (LIGHN3YHV) JUNLWaddW3L NI JONVHO o0b + o0E + 002 + ol + all il Sse|9) wnuiwnyy TTT. TLL LES YWNOYD,, ayejady asojnj|ad |. * [acy 43iH z10 + ayejaoy asojniia9 piepueys QUaIA}SA|Od plo + IAUIA Pil? 290s e2aceaGD ono UNG DD GoODONOEOUGODOUN (VEY rid(or PP OPDoODAACAtDODBODoROUSROSNGOONG PEN Key p yo aBunyo ez] yj seyDo1pu! Aj49041p YDIYM e[Dos (e6upy> uZlO° CCTs aybyaoD BsojN]{@5 PADpUu_IS @ZIs) PudY 449] SY4 Of UO!JOBSIBjuU! PSiIsep ayy WOl} Aj |Dyuoz w800° ‘CCT 84b499D BsojN]/a5 |Ape0p yBiH -!40y pded puD “asn ul asDq Buy OF spuodsesioo 4DYy SUl| B44 OF uZ00° OOOODCOGODODDDOODONFODORDOGOOOO00DS ose, Aj |P91448A poas MON| "(,G€- ) eur asbq 944 uo yulod Bulpuod tt GOO Fie so sc al ed eee rae eed “ss wun -SO1105 SYf 84D50] pud (doip 4 oS D Abs) sleyusiyD4 seeiBep ul uZ00° SOTO DIO 0 ODIO CODON US OE OMCHESIOLS T=) aBupyo ainypieduiay peyoadxe aut ysi}qo4se *y1DYD siut esn Oo] *sjUsWAaINSDOW yyBua| [DI1411D QO] © 10J payDoipul st abou 14Yys *eBupyo ozis aAlypBeu pup seBupyo einyosadwiey useMjeq diysuoty aypw1xoiddp Buimoj [oy tl “a|dupxe siuy ul * @AlypBau PX0}| -D[91 844 BZI[DNSIA JO ByDWI}sa Of PasN 8q UDS MO/sq 4{4DYS oy] SNOILSNULSNI W1l4d OHLIT 01 V 40 SHON! NI SONVHO 3ZIS 261 The third cause of permanent dimensional change is release of strain, or recovery from deformation. If film base is stretched during manufacture under conditions which do not permit reorientation of the molecules, deformation, or creep, occurs, resulting in lengthwise extension and widthwise contraction. Rapid cooling retards recovery of the deformation (primary creep) due to "freezing in of strain'! This strain may be released at some time during the life of the film, with con- sequent lengthwise shrinkage, and widthwise expansion. Where sucha strain exists, the rate of recovery is increased by both heat and moisture, Table IV shows the effect of temperature on the rate of shrinkage of an earlier film base, EK16 mm safety reversal. Shrinkage was mea- sured in the lengthwise direction, on processed film strips exposed freely to air at the indicated temperatures, and 20 percent relative humidity. TABLE IV % Shrinkage TIME (months) 70° F 262 Processing shrinkage Films swell during development, and shrink again during drying. Most films undergo a small permanent shrinkage during processing. How- ever, if the film is not brought to equilibrium with air at the same rela- tive humidity after development as it was before, the permanent process- ing may be completely masked by the temporary expansion or contraction due to change in relative humidity. Table V shows the effect of processing on several film bases. Values are given for materials conditioned 4 hours before and after processing at 20 percent relative humidity, 50 percent relative humidity, and 70 per- cent relative humidity, all at 70°F. Effect of Processing on Litho Flim Size Representative sensitized films were measured before and after for materials conditioned 4 hours before and after processing processing to determine processing stability. Values are given at 20% RH, 50% RH and 70% RH, all at 70°F. TABLE V AVERAGE SIZE CHANGES IN % Relative Humidities Before and After Processing 20% RH 50% RH ““CRONAR” base Standard cellulose acetate base High acetyl cellulose acetate base Vinyl base Polystyrene base (All films were developed 212 min. in Du Pont 7-D Developer, rinsed 20 sec. in clear water, fixed 3 min. in Du Pont 20-F Fixer and washed 10 min., dried below 100°F. and reconditioned at the indicated RH at 70°F.). As indicated in Table IV, photographic film shrinks during storage. This shrinkage is accelerated by high temperatures and by free contact with air. Table Vl illustrates shrinkage of EK nitrate MP film (no longer used) in the lengthwise direction for various periods, under three storage con- _ ditions, all at 70°F and 50-65 percent relative humidity. Fig. 3 shows the shrinkage rate of the newer triacetate base. Du Pont literature states that ''Cronar'' polyester photographic film base is chemically inert, and contains no plasticizer or solvents to be lost gradually as it ages. Normal storage studies, it is added, have given no indication of base change or deterioration, and forty day accelerated stor- age tests at 100°C have caused no significant change in processed film properties. Hence, it is expected that this base will remain substantially unchanged over long periods of time. 263 TABLE VI. Shrinkage of EK Nitrate MP Film. | % Shrinkage % Shrinkage % Shrinkage % Shrinkage | 10 weeks 20 weeks 30 weeks 40 weeks | Rolls in taped cans Rolls in untaped cans Strips open to air ° a % ° wn fo} rs ° w& LENGTHWISE SHRINKAGE, % LENGTHWISE SHRINKAGE, % 0 ° 1 2 3 fo) 1 2 3 5 c) TIME, YEARS AGE, YEARS Fig. 3. Shrinkage vs. age for triacetate and nitrate prints scrapped afte: Fig. 4. Average rate of shrinkage of processed triacetate normal theater use. Measurements made at 70 F and 50% R.H. 35mm motion-picture positive film at 90 F and 90% R.H. Con- trolled tests on strips freely exposed to circulating air; all meas- urements made after reconditioning at 70 F and 50% R.H. Kodak aerial films are now coated on two bases. Kodak Aerographic films, (Type 1A) are made on low shrink topographic base, suitable for use in accurate mapping work. Regular Safety Aero base is used for Kodak Ektachrome Aero film, and Recon film. The latter has somewhat higher shrinkage characteristics than the Type 1A. A comparison of dimensional changes in the two bases is shown in Table VIL. Since 1941, Type 1A (topographic) film base has been made from cellulose acetate butyrate. Between 1938 and 1941, it was made from cellu- lose acetate propiomate. Both these bases have substantially lower humid- ity expansion coefficients than cellulose acetate, used prior to 1938. 264 NA dav s0U8IaIIT¢ = q UPTM = M UZsueT = TH °9S6T em 2 peyep *(aVSN) OL09Z-4-TIW » eqTUM-pus=yOBTg “Tetzey SoTtyderZ0q0ug ‘wrty, suoFyeoTsToedg AxezIT TW Aq peatnber yey se oes au} SE 4Sey 3upzy PpezeteTeooy eu, °sTetieyeu of ydesdo oud Jo ageo UE adeyULIys Zutsseooid sepntour asexUupiyg Juypzy euy (aie %09 - 49d 38 ceok T ) ¥ ‘SadexHuTaYs 3updy eupl-suo7 (°H °H 0% = JO2T 38 skep )) % ‘adexup.yg Suysy pe,eseTeooy y% ‘edeyupayg Jupsseooig OT X q 9eudeq sed udtsuedicy Ieeufq jo PUSTOTIJe0) [euLreyy T X *H°Y ¥T sed uofsusdxy Jeeufy jo queTotsjeop £47 pFumy #891 Jo wotq09I7q at edfy vt eddy A£IY. 5 LIW ATEYITIW sdyzjg 4eTy go (2Fudesz0doz-uoy) Why Oley (2 Fudesdodoz) WIOg OU UP WT Ty sworyoeyAW Aepoy pus wlTy Wa oFydeas uo epREW S789] Tly SQUPSSEBUUODeY [eTley yepoy -oJey yepoy WIF-d ®9s8eq-e0uBssteuu0oey (esegq AZa58S9) AHAVUOOLOHd TWIUAV NI Gusn STVIUALVN MVGOM 4O SOLLSIYMLOVYVHO AOVUNLIYHS ALWWIxXOuddV 265 Shrinkage of photographic film is extremely complex. Several different processes are going on at once, and each is affected in a different manner by heat and moisture, and other factors. It is not always easy to predict howa given film will react when subjected to unknown conditions of storage and handling. Recommendations Film choice. The film which shows the least amount of processing and storage change should be used. This, at present, is the Type A ,cellulose acetate butyrate base. When it becomes available for aerial film, Du Pont ''Cronar" should show some improvement over other bases. The two rolls intended for stereophotography should be chosen from the same emulsion lot. Making the photographs Dimensional errors in aerial negatives caused by humidity or thermal expansion may be reduced by printing (or measuring the negatives) in an air conditioned laboratory, preferably at about 70°F, and 50 percent relative humidity, and by thermostating the cameras at the same tempera- ture. Ideally, the negative should be in equilibrium with air of the same tem- perature and relative humidity at the time of printing or measurement as at the instant of exposure. Film is in equilibrium with air at approximately 55-60 percent relative humidity when packed in air-tight (taped) cans, and will change very little in the camera if exposures are made in rapid succession; however, temperature changes inside the camera cannot be prevented except by some method of automatically controlled heating. Completely air-con- ditioned cameras, which provide both temperature and relative humidity con- trol, have been used quite successfully, in the recent past. Dimensional errors have been reduced considerably by this method, as well as markings by static electricity. Processing Film should be processed at normal temperatures -- 68-70°F. It should be subjected to as little tension as possible, especially while wet, and should be dried at a relative humidity of about 50 percent, and temperature not in excess of 85°F. Handling Film should be handled gently, and, insofar as possible, both rolls intended for stereo photography should receive identical treatment. 266 Storage Since access to air increases shrinkage -- '"'Cronar" pos sibly excepted-- film should be stored in sealed cans before and after use and processing. Heat also speeds shrinkage; therefore film should be stored in a cool place. Both rolls of a stereo pair must be stored under identical conditions, both before and after processing. REFERENCES 1. Calhoun, J.M., 1948: The physical properties and Dimensional behavior of motion picture films. Journal of the Society of Motion Picture Engineers, October. 2. Fordyce, Calhoun, and Moyer, 1955: Shrinkage behavior of motion- picture film. Journal of the Society of Motion Picture and Television Engineers, February. 3. Eastman Kodak Company, 1956: Kodak Materials for Aerial Photography. 4, E.I. Du Pont de Nemours and Co.: ''Technical Data on Experimental "Cronar'' Polyester Film Base Sensitized with Du Pont Photolith Emulsion. " 267 | 1 \ ii ‘ i} ‘at Ge \ it 7 ! p i 4 « ‘| 1 ' , “ vali ,) ae , ay be | + { i n- 1 % f i tt oa Distribution List for Project SWOP. cee. + * ayers + 1 Assistant Secretary of Defense Navy Research and Development Attn: Committee on General Sciences Chief, Bureau of Ships Pentagon Building Navy Department Washington 25, D.C. Washington 25, D.C. 1 Attn: Codo gul2 Navy ; a 320 . S : 1 531 5 Office of Naval Research 1 B74 Geophysics Branch {Code 416) 7 845 Washington 25, D.C. 1 Air Branch, Attn: Filodil Us. Navy Hydrographic Off. 1 Amphibious Branch, Hashing vous); Dele Attn: Maj. Crownover 10 Division of Oceanography 5 Photogrammetry Division Chief of Naval Research Washington 25, D.C. 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