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saeeotey athe _ a SECURITY CLASSIFICATION DF THIS PAGE (Whan Data Entered) REPORT DOCUMENTATION PAGE 1. REPORT NUMSER 2. GOVT ACCESSION NO.| 3. RECIPIENT’S CATALOG NUMBER TYPE OF REPORT 4 PERIOO COVERED Master's Thesis December 1984 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT CR GRANT NUMBER/(a3) Sn . TITLE (and Subtitle) A Comparison of Methods of Least Squares Adjustment of Traverses » AUTHOR(s) Saman Aumchantr + PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS . PERFORMING ORGANIZATION NAME ANO AOORESS Naval Postgraduate School Monterey, California 93943 12. REPORT DATE December 1984 13. NUMBER OF PAGES - CONTROLLING OFFICE NAME AND AOORESS Naval Postgraduate School Monterey, California 93943 151 - MONITORING AGENCY NAME & AODORESS (If different from Controlling Office) 18. SECURITY CLASS. (of this report) UNCLASSIFIED 1Sa. DECLASSIFICATION/ DOWNGRADING SCHEOULE | ae ki Ve ps att 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited 17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Adjustment, Approximate Method, Comparison, Condition Equation Method, Least Squares Method, Traverse, UTM grid coordinates. 20. ABSTRACT (Continue on reverse side If neceseary and identity by block number) Traverse is a method of surveying in which a sequence of lenqths and directions of lines between points on the Earth are measured and used in determining positions of the points. This method is one of several used to find the accurate geodetic positions which various agencies use. Traversing is a convenient, rapid method for establishing horizontal control. The theoretical background is provided here to explain the method of traverse station position computations and adjustments in the Universal] FORM DO , 1473. EDITION OF 1 NOV 65 1S OBSOLETE JAN 73 SN 0102- LF- 014-6601 1 Ee I a ee SECURITY CLASSIFICATION OF THIS PAGE (Whon Deta Entered) ——————————E——E—E— EE SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) Block 20 (continued) Transverse Mercator grid coordinates. Closed traverse station positions were computed and adjusted using the Approximate Method and by the Least Squares Method. The adjusted coordinates of both methods were transformed from the Universal Transverse Mercator grid coordinates to geographic coordinates and compared with the coordinates which were adjusted by the U.S. National Ocean Service. S/N 0102- LF- 014-6601 a eS ee 2 SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) aes es ee ae y Approved for public release; distribution unlimited. A Comes esau toe Methods fo) Least Squares Adjustment of Traverses by Saman Aumchantr Lieutenant, "Royal Thai Navy Boose, KOyar thas Naval Acadeny, (976 SUM tted an partial tulriliment of the requirements for the degree of MASTER OF SCIENCE IN HYDROGRAPHIC SCIENCES from the NAVAL POSTGRADUATE SCHOOL December 1984 DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA 93943 ABSTRACT Traverse is a method of surveying in which a sequence of lengths and directions of lines ketween points on the Earth are measured and used in determining positions of the points. This method is one of several used to find the accurate geodetic positions which various agencies use. Traversing iS a convenient, rapid method for establishing hocizontal control: The theoretical background is provided here to explain the method of traverse station position computations and adjustments in the Universal Transverse Mercator grid coom- dinates. Closed traverse station positions were computed and adjusted using the Approximate Method and by the Least Squares Method. The adjusted coordinates of both methods were transformed from the Universal Transverse Mercator grid coordinates to geographic coordinates and compared with the i coordinates which were adjusted by the U.S. National Ocean Service. DUDLEY KNOX LIBRAR NAVAL POSTGRADUATE SCHOOL MONTEREY, C/ (FORT 98543 TABLE? OFF CONTENES ales INTRODUCTION CMe eRe arHancie =o 6) Patate: cet eiiteu ieee, van wie. ve. 1 mi, TRAVERSE Sonoita se Mme Gils ss (Seek eae sv se we V2 eee GE Er Seana mee ol ee epee) ied Skah We wn) s. eos je ye Be PONG ESANDRDERECTMONINEARSUREMENT) (Sic « « «© « V3 Cae DS LANCEe MEASURE MENS yams takes ets we Sse 6S DEN COUR ACY meme wne a el os 6 of = «© So a « = « 16 Pe DAL Ap eC OUs SON e lyase 6 ws = = i «6 « « « 16 cet. TRAVERS recOMeOn kt LOS a AND, ADUUSTMENTS < 2 « 6 or 21 eee Ne samen ee Rio COM PM mon, LavONS: —s: e ste iasne. ie “ee 2a Dep COME U Ae sONwOr “Di SC REDPANC TES: on sii. @ Jeli 06 ie, 23 Gan AD US CEN OF A LRAVERSE BY AN, APPROXIAATE [ieee OO Ree Mn ch ict ta etmauiiee Jo, @ 0:6 rem rte OMe NO PA PON Moura: lvewn tor! (ary len i vor.iae ner fell tet et) wy! oe, be, a> OF Bee Gra mr ROGRA Me We (siie: “s)! Sle, fer Mer oh 6) Su oy ae. US ea ee ws 68 Per est OX ese NOG RIAU 2 ea ven inc) ee epi s- oc) 5) viene) ss Sts) poh ws J's. «- BY APPENDIX Cs “PROGRAM 37 - LiST “OF RERERBNCESe < = « BEB EEOGRAP AY) 5). «amen ENEDEAL DESTREBUTION EES se 121 143 149 150 Ix. x. ), oe rst) vOF ABE ES Classrtmie@eLony,, Standards of) Accuracy ,,, and General (Specriitecatt ons, fOr Horrzontal Control” - Coordinates Of Known (Stations and \AZimutos,” 9; 7": Grid Distances and Standard Deviations |. . s. \< The Observed Angles and Standard Deviations . . Pata conn twee Traverse Conuputatlons: (. ks) 4 Rites Azmi \Caeecu at MOMs Nis. se) acs) ol lets | @! Peal era yet oe OOMPUtat TONS <2, /<) sn “. je ea jie Initial Traverse Computations (Approximate Merhod wart nsAdwusted Azo mabh) 2 6) 2 se ee Adjusted (Coordinates (Approximate Method) . 6. . The Comparisons Between the Computer Storag= Maca and COU Taine Of ,PrOgnams (WN, 2) amd 3s Geograr uc COOEGIMaALOS! ie Ys) Si vin ce) Veer a sh hsp Ow he The Comparisons Between the Standard Deviations Grom ea SiteroigitaiaeSu isin ofa ys)) epi ev Ue) pie fia es ek ey eee LES OF Zen Closed ravers ci. =e. Closed-Voep iraverso i. PAIN | Horizontal) cAng ise. wan. ms Shel A. Closed: “Tl rawenrrsci ome. FIGURES 12 1a 15 22 ACKNOWLEDGEMENTS Raexpress sincere gratitude tomy Thiesas) Advisor, Dr. poselrand (i. “Hard yy candrsiccond, Reader, Cdr. \Gleni sR. Schaefer, LOnReneme Suggestions and wassitstance > Finally,.:I thank Ms. fataray Ms Haylang "ht. eNicholas Ei (Perugingd, cand) Messrs. Mark L. Faye, Peter J. Rakowsky, and James R. CcCherry who made this thesis possible. I. INTRODUCTION Hydrographic surveying includes many branches of science for the purpose o£ the production of nawtreal hares specially designed for use by the mariner. The determina- tion of position in hydrographic surveying is aS impontame as the measurement of depth. Before determining an accurate Lydrographic position, accurate geodetic positrvons* sor Shore control must be established. There are many methods available to establish geodetic control in the survey area. These methods are: 1. Triangulation 2. Trilateration 3. Traverse 4. Intersection 5. Resection The process of making a proper nauticai chart consisSeey first of all, in setting up: a framework of marks on the ground. Before 1950 the main framework of a geodetic survey almost always consisted of triangulation, which was replaced by traverse if the topography made triangulation impracti- cable. During the last decade, the introduction of elecs trotiic distance measuring (EDM) equipment has made both trilateration and traverse economical, and an acceptable substitute for triangulation. In fact, it appears proObabme that these new methods will replace triangulation as the main framework for new geodetic surveys [Ref. 1]. It is evident that triangulation and traverse are the main methods used for establishing control. There seems to be no 1A position of a point on the surface of the wiamea Se0h es ce in, berms Oc geode tac tate geodetic longitude and geodetic height. geodetic position implies an adopte geodetic datun. 10 agreement among the various’ agencies as to which of these BuOMmetiOds is MOStuywwseds. ithe lt. S. NationalkjOcear Service Mos tiges the Majority Of LES hori Zontal control surveys LOm MydrOgraphy with itraverse j(about °90%). Pref. 12]. The Main factor for the selection of one or the other method depends on the geographical configuration of the survey project area and the availability of good EDM equipment. Traversing is a convenient, rapid method for establishing HecHiZOn tal) controls ites \ipartiicularlywuseful in idensely built areas, when the coastline tends to be even, along a Tailroad track, and in heavily forested areas where lengths Guisaght areishortssogthdt sneither triangulatwvon inor trilateration is suitable. The objectives of this thesis are to show nethods of traverse station position computation and a comparison of methods of least squares adjustment. Closed traverse station positions were computed and adjusted using the Approximate Method and by the Least Squares Method in the Universal Transverse Mercator (UTM) grid coordinates. The computer programs were written to calculate the adjusted traverse station positions. Test data included those data obtained during the Geodetic Survey Field Experience course seamen NavalsPostgraduate:Schood:(NPS)—ken~ OGtober 19-83. The results of comparative computations are shown to iomemsrGniticant Elgures, in this thesis, than are, normally considered desirable in production work. The same observed Gata are used with each of three computational methods. It is important to recognize that what is being compared is not G>servational precision but computational precision. » Hence, mt 2S comsidered necessary for a rigorous comparison of Gemputational precision, including round off error, to show EBeswilts to several more decimal places than is justifiable based on observational precision alone. a A. GENERAL Traverse is a method of surveying in which a series Of straight lines connect successive established points along the route of a survey. An angular measurement is taken using a theodolite at each point where the traverse changes direction. Distances along the iine between successive traverse points are determined by EDM equipment. The points defining the ends of the traverse lines are called turning points, traverse points, or traverse stations. §efach straight section of/a traverse is*called a»*legvor a teaver line. A closed traverse originates at a point of known posi- tion and is closed:or another point of |krown horizontal position. Traverse -I-1-2-3-F originates at portnteie yee backsight along line IA of known azimuth and closes on? pomme F, with a foresight aleng line FB also of known azinuth (Figure 2.1). This type ‘of traverse is*preferable toeams | | | | | | | | | | | | ee | = es ie oo Figure 2a Closed Traverse. ae ethers since computational checks“are: possible which allow Gevection Of Systematic eCErorse in Doth drstanse and direction. nh eLosed—lgep (closed polygon) traverse is a special ease Of a Closed traverse in which the originating and terminating points are the same point with a known hori- Mba DOsSvevOn. \ Traverse. —-I—-2-3-4-1, originates ard BeEinmdtes On Ppolne © (rigure 2.2). “This type o£ traverse permats an internal check on the angles, but there is no check on the linear measurements. Therefore, there is a Pessupairty thal an, GrEOr proportional, to distance may, occur and not ke detected. oki AEN al= ees SP Closed-loop Traverse. B. ANGLE AND DIRECTION MEASUREMENT Angles and directions may be defined by means of bear-.- mgs, azimuths, deflection. angles,.angles, to the’ right,)or interior angles. These quantities are said to be observed 2Systematic errors may be caused ea instruments or factors such as temperature or humidity changes which affect the performance of measuring instruments: 13 ee | when obtained directiy in the field and calculated when obtained indirectly by computation. A theodolite is an instrument desijgnec to observe hori- zontal directions and measure vertical angles. [It consioee of a telescope mounted to rotate vertically on a horizoneam axis supported by a pair of vertical standards’ 2€=tached@ecme revolvable circular plate containing a graduated cirelewees observing horizontal directions. Another graduated anew attached to one standard so that vertical angles can be measured. Repeating and direction theodolites have features that are common to poth types of instruments. Repeating theodol- ites are read directly to 20" or 01" and by estimatvonmmua one-tenth the corresponding direct reading. Direction theo- dolites are usually read directly to 01 and can be coe matec to tenths of seconds [Ref. 3, p. 215]. _in genecuam direction theodolites are more precise than are repeating theodolites. The direction theodolite observes directions only and angles are computed by subtracting one direction from another. Assume that the horizontal angle AIB (Figure 233) is to be measured with a direction theodolite. The theo- dolite is set over point I, leveled and centered andea sight is taken on point A. The Eorizontal circle as eemen viewed through the optical-viewing system and the circle reading is observed and recorded. Assume the reading is 459 02' 4O". The telescope is then sighted on point Bo etme horizontal circle is then viewed through the optical-viewing system and the circle reading is observed and recorded. Assume the reading is 1249 11' 59". These two observations constitute directions which have a common reference direc-— tion that is completely arbitrary. The clockwise horrzomeass angle is i = (126° 171" 59) =. (95° 02° (20) 7 9°oe Go ao 14 (anna eee ee kr Nn i se a a a a el | | I | | | I | | | | | | | | | | | Elgure™ 2.3 Horizontal Angle. CG. HDISTANCE SEASUREMENTS There are several methods of determining distance, the Choice of which depends on the accuracy required and the cost. For example, tacheometry, taping, and EDM equipment Cam be used. The general availability of EDM eguipment has practically eliminated the use of taping for the measurement Or traverse lengths: Accuracies are comparable, or superior, to those obtained with invar tapes. Distances measured using EDM ezguipment are subject to SEEOES arising from the instrumental components, calibration of the ecuipment, inaccuracies in the meteorological data, Elevation discrepancies, and centering of the instruments or reflectors. The reduction of measured distances involves converting the slope distance to a horizontal distance, eonverting the chord distance to an eguivalent arc distance, euapleducing the arc distance to ,the ellipsoid....The reduc- evony Or (slope distance to horizontal distance is mecessary to compensate for the difference in elevation of the end pornts of the measured line.... The horizontal distance is 15 reduced to the geodetic distance by applyinigia seawbever corrector ard a chord-are cobmector to the) Hormzomeas distance [Reis 0) "pps 125-NiZei- D. ACCURACY In general, the accuracy of a traverse is judged on the basis of the resultant error of elosure of ther teaveru=em This resultant closing error is a function of the accvEedewea in the measurement of directions and distances. The classi- fication, standards of accuracy, and general specificatvoms for horizontal control have been prepared by the Federal Geodetic Control Committee (FGCC) ; Ref. 5] and have been reviewed by the American Society of Civil Engineers, The American Congress on Surveying and Mapping, and the American Geophysical Union (Table I). The third-order traverse class I and class II are of particular interest because thesevame the orders of accuracy the hydrographer is usuaily required to accomplish. Es DATA ACQUISITION The data for this thesis were acquired at Moss: Landang: California, by NPS Hydrographic students during the, Geodetre Survey Field Experience course in Qctober. 198325 All fosegene knowr station positions and azimuths (Table II) were adjusted by the Coast and Geodetic Survey by third-order methods [Ref. 6]. The distances were observed by using a Kern DM102 anda Tellurometer MRA5S. ‘The Kern DAS1T02 15 an“electro-opezeae distance meter. 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Funo2j09 Ag anjRa painseaw 24) PUP INTRA FrUy FYI UPaMjaq JoUaJAYIP ay) St IP Apioexa peuirigo 2q 1OUULD Jeqi Anjuenb e& st sousa yenjor so ans) aqy ‘aunieu ul Wopurs Apiouis qe S101 We deqy Yondwnsse ay) J2puN prindwoos si 224 pasn Jou? parpuris,, WII WyY HUswWINsTIwW JO JIQuINU Jy) SIU pur ‘(aur & Jo sysua] painseaw yje jo uraw ay) pur yiduay PINnscaw eB VIIMIIq BUI YIP IYI ‘SH Fey) Feapisoys (J — U) uU eS! A ‘URaW YP JO JOUa psrpuris ayr st "oO JayM AZ Aq pairtunsa aq OF St JoIs2 Parpuris ayy (1) FLON (panutzuog) I daTave a a Sg ee aT 18 TABLE, Pi Coordinates Of Known Stations and AZinmuths a i a i oe | Rie C8 ae coordinates ; | Station ee ola TAG Gard Easting | | me | | Moss /2 Te aA 608,279.04404 | | Holn MONS p20. 31 1a 4 (Si) PEN PAR els Wists) | | Grid azimuth clockwise ! | from North AZinuth | | eae | BEOM MOSS 2 to Pipher. Stations TOO 9 V6.) S23 S72 | From Foln to Moran Stations 136 S33) 2623384 | | | : | determination. The distances were observed in the field, cocrected by temperature and pressure for propagation error. Unfortunately, meteorological data are generally accuired G@ulyy at the end points of a measured line. Using the mean of these meteorological values only approximates the actual conditions of the entire measured line and does not feompletely correct for errors in the propagation velocity of electromagnetic radiation. By applying the elevation, sea Havel and scale factor corrections’ they were reduced jto.U?M mardadtstances [ff Rer. 47) pp." 124-1254." The OTM grid distances and standard deviations of the distances were determined in meters (Table III). The angles were observed by using a Wild T-2 theodolite. POowensune the Correctness of the beginning, and ending azimuths, a check azimuth to a second station of known posi- tion was observed. The angles were observed at stations Boos) 2, MOssback, Dune Temp,,and Holm, (Table IV). All observations were made by NPS students and conform to epecitication for a third-order class I traverse. ive) . Moss. 2 Mosskback Dune Temp JRacksight station Pipher Moss 2 |Mossback Dune Temp Statauons and and and | | Grid distances between | | | | TABLE, LEE Mossback Dune Temp Holn TABLE Grid Distances and Standard Deviations f Standard Distances deviatior ( ae) C “a. 1,424.004 O. O07 BR Sq 7/ Ell 0.001 674 7 Gaece a 0.008 IV Observed Angles and Standard Deviations Center Sstatron Koss. 2 Mossback Dune Temp Holn Foresight Sita on Mossback Dune Holn Temp Moran Observed Standard angles deviation fe) ' "” " 246.05. 43.200 _0O1 5968 222, 51. .08s600. _ Of 2a 190,.15, 02.5600, — Oclee2Zaee 277, 905. ATs000.) Cia | | | | | | | | | | | Pa uweoroAL, TRAVERSE, (COMPUTATIONS Ae travesse (Fagume 3.1) originates at station 1 "(known position) and terminates at station 4 (also known position) Rape Vie To Compute the forward azimuth (Table VI) of an Unknown leg, the angle is added to the back azimuth of tne meev ous, leg (cquation” 3.1). | Az = BN ec =a 1)g1180e (3i41) Where: i = the number of legs, Az, = the forward azimuth of an unknown leg, and AZ. = the faxed anitval azimuth. EGhnthenconputataon of st TM grid coondinates, «let ANE, and ON, be designated as the departure and latitude for leg mae igure) 3. l) so that the general formulas’ to Compute the departure and latitude are AE, ON. d; Sin Az; (322) d; cos Az, (323) Nene San ey tS taie. foods Mo, Dos hn = the number of the observed argles. where 3: The algebraic signs of the departure and latitude for a meaverse leg depend. on,the signs,of the sine;and cosine of Peemazimuth of that (leg. The algebraic sign of the depar- ture and iatitude is determined by the following rules: i ULM igaitd sazimnuthsvare, refered. to, grid north. 2. For azimuths, between 0° and 180°, the departure is pius 2 )for lab yother azimuths, ;thesdeparture -is, minus. ae Or azimuths: between -90° and 270%, the latitude.is NERWS Ee Om dull other azimuths, «tbe. Latitude.is, plus. Ae | a a ee | | | | Figure 354 A Closed Traverse. The calculators and computers with internal routines for trigonometric functions yield the proper sign automatically given the azimuth. The major portion of traverse computa— tions consists of calculating departures and Tatutudesmeon successive legs and cumulating the values to determine the coordinates for consecutive traverse stations: 22 an a ee es Os ee a TO a ce a OS as ee ae NS SO ES OS ca OS cS ON es OS a A ce a pa SS a es cere Hoel . TABLE V Data for Initial Traverse Computations a | | Leg Heer ane a Angle | Me fe) t] " | A 2 d, h=) 424.004 Co) =) 2UGem Ot & tS) 12.010 | 2h = OS doet= 365.744 Qa. = 22:2) 4) 51) 410'8%/6.0:0 | Ban 4 dopo) Ow er6 27 1 Oe = 190 a 02000 | | | | The initial UTM grid azimuth Az,- = 100 16 PEM UTARS) | | ULM gradsNorthing N, = LOM 2 Doo ocO6 Mm. i Stacrron” | : | DiM grad Easting E, = S08, 27900804 me il io . i Ly | TreKcoondiund Hesror Station (i ij = air Me ane iE and Ny Hhe-coordinates~of station j are ae ON. (355) Wee her lea we 2 eS bes ee Ie The values of departures, latitudes, and coordinates are Shown in Table VII and were computed from the data in Tables Paand Vi by application on-Equations (322) through (3.'5). Bee COBPUTATION OF _DISCREPANCIES AV ezosedPtraverse: at Moss Landing” (Figure 3.1) “origi- maeTned at Station “l'( Moss’ 2)" arid iclosed, at station 4° (Holm). The closing errors for a traverse are caused by observation errors in the observed angles and the measured distances. Bye~ Closing “errors may be computed by applying Equations ee) through: (3'.'6)*. 23 TABLE VI The Azimuth Calculation ——_——_—-_---—_—_-—-x-_--“—-“—“-_---—-_ 1 | Forward Back Observed | | Azimuth Azimuth Angle ¥ ; ® | AZ 100 6 2eacaae | a, 206 057% aa.2we | | Az. 346 “2268 06.998 | ="4809° =00n! Olon cae a 166, ~22) eodienncme | a, 222 54 08.600 | AZ» 599° "43 “gsmouie | - 180 00 00.000 | | hee 209 13, some | | a, 190 15 02.600 | | Bee 399. .29eeuAea tas | | - 180 00. saOouoOs | | M2) a 219 28 aisoage | i i Laas.) The angular error of closure may be computed by applying Equation (3.1) becomes n AZ tO. = (Bot 0 ae ee ae (3-6) i 1 where Az, is the fixed closing azimuth and W, is the angular error of, ,chosune- AZ) is the grid-azinwtheteon stations Holm to Moran (AZ, = 136° 33". 26.334"), “(tables The computed closing azimuth is egual to AZ4_3 plus, the observed angle at station, Holm = ,(2.199. 26 "te fo ae (277° O5S* 17") = 4969 33)" 35.178" = 1369 338 35S eee Eguation (3.6), the angular error (of closmmesis (136° 33° 35.778") = (136° 33° 26.3990) (oor eeee 24 TABLE, Were Initial Traverse Computations — SC — 7 | Values in meters Values in meters | N, UPA0T 92), too. 6 ZUG E, 608,279.04404 On, Teo. COZ T OS ES =n 5) SerOUNr oro N, LOS, FS 9 7457 3 E. 607,943.44238 ia 2 92200164 AE, i654 8 74 N, BOT, 25 8.94534 E3 6.08% 121299 109 1%; 4,999.28284 QE, UP lo. 96755 | Ny Uy OT LIL LOS Ey Giese. 93664 | ne a a Re Tohconpure. the errors ian closure in. position, for).a “closed EeeMeCESes G-ngGvalrons)/(S.2) to (3'.5) are applied to obtain: n-1 Say: 28, Ti weer ee (Boy) Is n-1 wee ei) -(N,- WN, ) (3. 8) or i Ween oo EB (Seva) W, =p Nip No (3.8a) Where; nh = the number of observed angles, Ww « We = pEecalculated discrepancies, Si i ete EIxed, anitial Coordinates, E, . N; = the fixed closing coordinates, and Eoin ene. compuced ‘closing, coordinates. Hiemaesilt Of the errors in closure in position ofa traverse is Ze CO ee | Qa 0 Waa eae (3.9) The computed closing coordinates of this traverse are Lee 612,238.93864 m and N, = 4,079,258.22818 m. The fixed closing coordinates are E, = 612,238.85256 m and N= 4&,079,258.31754 m. The precalculated, discrepancrtes ean computed by using Equations (3.7a)* and °(3. 6a)e § The secsumae are W, = 0.08608 m and W, = -0.08936 m. By applying Equation (3.9), Qd is 0.12408 m.. From Table Tit, the vee measured distance is 8,266.019 m. The ratio of the distance error of closure to the total distance is 0.12408/8,266 705 er 1 part in 66,618. The ratio isan indication longer goodness of this traverse. C. ADJUSTMENT OF A TRAVERSE BY AN APPROXIMATE METHOD The angular error of closure of traverse may be distrape utel equally among the observed anyles. The angular error of closure of this traverse is +8.844", which Corres ponds a correction for each angle of -2.211". The observed angtes were correctei by this value. The adjusted azimuth was then computed. The departures, latitudes, and coordinates were recomputed Fy using the adjusted azimuth (Table VIII). The closure corrections in E and N coordinates are +0709635am and -0.04326 m, respectively. The resultant closures 0.10562 m. The ratio of the distance error of Closuneire the total distance is 0.10562/8,266.019 or 1 part rn 7 e72eee The adjustments of a closed traverse by the Approximate Method are completed by the compass rule which proportions the errors in E and N coordinates according to the distanec of the course [Ref. 4, p. 354]. The corrections arevappeeee to the departures and latitudes prior to computationsor coordinates. 26 \ TABLE VILLI Initial Traverse COopuese st ts aE peor taate Method with Adjusted Azimuth) Se aA bigeunesscs a ms | | Observed angle Correction Adjusted azimuth | Be | o2 1OOe) 169,235 778 | | Cre ROe OSV se ZI 2.27 1." 296) 015). 406 9169 | | AZ, SGN 22a 4 0 7.67 ="1808. COTS OCS000 | Aza TEC? 2) FOO 2767 | | Hi 2o20lstuvossenes= 229TH PADI Al Sed NS NESIS | | AZo SO Seed hate | | Setsoe roo "00.000 Wz 5 > 209F Pi sya Ae a6 | | OO wAseIsts! FO2SG eS, 22TH" PITS ee CS 89 | | AZ. BOOP, 22 re Iso 45 | i138 0'' 200) MOOsC00 | | Moy Pie) lA) ee i peiko es | | el TOPOS INITIO 2.23274" 2ETOLMO SON S789 | Az, Was60, 338. 262334 | Values in meters Values in meters | | Nat Or 29955-85206 B, 608,279.04404 | | ON, IPeestesI07u CEL = 335.61649 | | N, 4 ,073,939.74113 Boe RoOr oS 42155 | | AN, 319.20444 QE, Ie S5 8187 | N, 4,074, 258.94557 Enim GOelm2 1396942 | | ON, bi 9 SORUGS.2:3 as 4,116. 78679 | | 4,079, 258. 36080 By a Onhae 238-7 9621 | | He 4,079,258. 31754 Ea edz 2a 8e8 5256 | | dn 0.04326 dE =) 0.09635 | | | ea ee ed OB. = ( db 7D yaad aig} ONS 9= (aNe7 = payee d. (3% Th Where: bE. = COLEec tion (Eo AVE, ON, = Correction ito #CUN, 7; dE = total closure, correction in thenSicoordivaecs dN =- total closure correction in the N coordm@idaece and D =total dastances The, adjusted E_and N coordinates (Table IX) were different from-the fixed .E and N coordinates (Zable Gree 0.00001-m due to-round off error. Thescaleulatvonseana adjustments were illustrated in sections A, B, and e€uay using the hand,calculator (TI-59) and rounded of eager decimal places... The computer program was written to caleu= late the adjusted traverse station positions by the Approximate Method by using 16 decimal places (Appendix A). The approximate traverse adjustment is based on an assumed condition that the angular precision equals the precision es linear, distance. D. ADJUSTMENT OF A TRAVERSE BY LEAST SQUARES MEDAODS The method of least squares provides a rigorous adjustment and best estimates for positions of all traverse stations. The Least Squares Method is used to Simultaneously eliminate closing errors in azimuths and coordinates of traverses. 1. The Principle of Least Squares The fundamental condition of the least squares tech- hnigue in surveying requires that the sum of the squares of the residuals be minimized. A residual is defined as the difference between the true and observed values. In making 28 TABLE Ix Adjusted Coordinates (Approximate Method) a --—- - — > —— _ —_—— Departure Correction Grid Values in meters AE, .-335.616499 +.0.01659 ES 178.54187 + 0.00426 QE, a loas Soyo (+ O20 7549 Ey 608,279.04404 = SSSA SSSI 607,943.44414 WLS So kiG aS ede, IZ ls 99027 O21 36a SC O2]5 Latitude Correction Grid Values in meters N ON, 319204888 5—20200 191 | | | | | | | | | OMe s83288907 Te0400745 | 4,999.41523 - 0.03389 aes | | | | | | | | | | | | | | U,116.86228 | | | | | 4 ,972,555.85206 | 38S 28S 16200 | a0 73. 9392.7 336 | 319520253 | 4,074,258.93621 pragoaso rsa Ue ROEM ee Yes ig Sd | | | | | | J! physical measurements, the true values can never be deter- Mined. The least squares principle establishes a criterion for obtaining the best estimates of the true values. If the best estimates of the by x, and observed values by xr the as v, Hix b=, x The fundamental condition of uncorrelated observations with equal as ZS true values are stated residuals are expressed Leas tyscuanes,: or precision are expressed n > ( ¥, )2 = (¥,)2 + (v,)2 # (Way? + . . 5 #) (wee (eet Or an Natrix form viy = Minimum (3. 12) In general, the observed values are of unequal precision. The observed value of high precision has a small vactiance. Conversely, a low precision of the observed value has a large variance. Since the value of the variance goes in opposite direction to that of the precision, the) obsSeega] tion is assigned a value called weight corresponding Coma quality that is inversely proportional to the observations variance. For uncorrelated measurements X,4 Xor £34 7 © ume i < 2 2 xX,» WIth variances C,, O.. O,, zoe coe Ol respectively, the n? weights or these uncorrelated measurements are, 2 2 2 2 2 2 = 6. (6) = ©. = 6, o) e e ) e Ei we aE, 1 One P53 oe 2 2 p= G7 On (3. 13) 2 where 6, is the proportionally constant of an observetren of unit weight [Ref. 7, p. 67]. These wertghts! mayeee collected into a corresponding diagonal P matrix, callecuime WOLGht Matrix; Ot Oe ann) 144] 0 OS Bee 0 2 0 ee ou nn 30 The weight matrix consists of weighted observations Gea itraverse with Mixed) kinds of measurements. They are Gistances and angles. The variance of a measured distance is expressed in meters and an angle in radians. For uncorrelated observations*of wnegual precision, the funda- Hegtal condstionof*least (squares 1S expressed as fe = Paine 2 + ToeRe Site et ied ey Sea v2 os Ein Pi, % Poo %5 P53 3 P Mininmuno Ske in Natrix efiorn v'pv = Mininun (3.14) In the problems involving the adjustment of observed values, all observed quantities are expressed by functions ep ythe quantities ‘tobe determined. In simphe cases these Beolattons arevlinear, butewhen this is not the case, the relations must be converted into the linear form by expanding them into Taylor's series. The terms of higher ‘order are neglected, so as to obtain linear relations, solving the resulting linear equations, then iterating until the effect of the neglected higher order terms are Minimized. 2- Least Squares The observed values are related to the desired unknown values through formulas or functions which are called observation equations. One observation equation is written for each measurement. To solve for the best value of each unknown parameter, at least one redundant observa- tion equation must be written. That is, the number obsServa- tions must be greater than the number of unknowns. The linear observation equations can be written in the general form as follows: Sut 1@) Wt 4 1222 Wan Sian 1 1 + + So eamiey pou es = ss Poqul cot ot A o0%> 2om%m= So * Y% + w=, 2 + = + eh ani% PnoX > bbl tam rH ce vy where n is the number of observations; m is the numberman the unknowns; A190" Anp4e + + + Ang are cConstaness Ai, 4 Aye - + » 4pm abe coefficients of “the unknowns sce, Bad 3 and Vi ¢ Yor + + « Vy are the residuals. Because the observations G; (2 = \lg7.Zads 420. See not free from random errors, each G-, must be correcteds by ae residual value, V; let b. = (Gat aio: Thus @4qX, F Qo kg F = eo * 847 oe Q5,%, + 4n5%5 + 2 - © * A5,,%,,7- Dg = Vy a,,%, a qo oS) ee SA Gan Di , Li or in the matrix forn VY =.AxX- 8B (3.15) This equation is called the observation equation or the observation equation matrix. For uncorrelated observations of unequal preecrstome substitute the value for the V matrix from the observation Equation (3.15). into Equatzonm «(a2 04) 32 vVipvy = (AX - By’ P(AX - 8B) =) (ax) ASB) Pp (Ax .=.8B) = (x’a’ - B')P(AX - B) = (x’a’p - Blpy (ax - B) ViAlMipA x = Al PR. BD PAK #, B! PB Y alone Sy? a) pp xT pps Bl Ds x'a’ pax -— 2x’alpp + Bi pB (3. 16) {from matrix algebra, (ax)! =aha0 and “BS Pax = x'al pp The minimum of this function can be found by taking sie partial derivatives of the’ function with respect to each unknown variable (i.e., with respect to the x,, X5, -- - aan) must “equal” zero." Hence, Ovipy) = 2a’ pax - 2aTpp = 0 (Ein) x Dividing Equation (3..17)0 by Z, the following result ms obtained: A’PAX = A! PB This represents the normal equations, and by multi- plying this equation by (A’PA)-1, the, solutzon ws obtained: x = (a! pay-1al pp (3.18) For uncorrelated observations with equal precision, Pee WweLlght matrix 1S ane-identirty. matrix, “and*equation (3.18) becomes x = (aA! ajy-ta's (3.19) This equation can be derived similarly to the Pmedudie precision. case. jEGguations (3.18) and (3.19) are the basic least squares matrix equations. 33 If the relations are nonlinear, the relations must be converted into the linear form by using Taylor's series expansion [Ref. 3, p.- 919]. Let F = £ (4.2 soy = eee be the general observation equation that is a nonlinear function.° The Taylor's series exrpansvoneas Ba fat ee - « x0.) OF Ox, + Oe 6x56 2 eee 2G a ° ° ° Ox, Xo axe mM + Higher order terms ° ° where > ae X50 + + + Xp are the approximate values of the variables at which the, function is, evaluated.» Rometre traverse problems, an approximate value can be the precalculated value, and Om. Oe, eked Ms ome can be the corrections. The higher order terms in the series are neglected and only the zero and first order terms are Maintained, the approximate value must be improved by successive iterations until the erfiiect of the. neghleered higher order terms is minimized. After linearization, me observation equations become: \ £, (x‘, Xe ones =) +O£,6x,+ Of Ox, + siais + Of G, Oe ax‘, Ox, a £, (x, x oe a a ee) + 9£,dx,+ O£,bx5+ ad % + 2£,0x_- G, Ox; oe Xm Vy = fylke Xpep-.+ Ey + tid eye OE Ouse, 25 pe Bee eee Ox", OX, Ox", OF inwthe matrix orn yore Max a | (3.20) 34 where OLS nner A MeN arr A) 6 INOFE v, Ox‘, Ox, ex Obe QUEM eile. or eigen, OE v 2 2 2 2 Ox, Ox*, ox iN 5S E 4 Y= . Cee ROrod 46 Ae tH eONE A ve Ox; 0x, Ox? G, EGA « Se ee. can eu ee § > & Gps EONX, 5 oat EO | koxe) om B= - 4 b ‘. c xX = : es many @ 1) eee Wie) lee sande: Or in the Eunctional £orn Thus, the linearized form of the distance observation eguation is vi =f; (E> N;, Bi, N; ) + OF ORR © som ome J J OE J a zl i zi Nj +, 0.6 58+ jt) OifyOmye aud = J J } OE* ON’ “For the adjustment of the traverse which was conducted at Moss Landing with the technique of adjustment of indirect observations, four angle and three distance 36 observation equations need to be written. The seven equa- EPOnS would GneLudesas unknown!parameters the four” coordi- Rates O2 Statmonsiz (Mossbaick) fand 3! (Dune Temp) .. Four normal equations are formed and solved for corrections to the approximate values (precalculated values) of the parane- tecs. The corrections are added tothe’ approxinations to update their’ values; and the Solution is repeated until the taste set, of Corrections, is insignificantly small. The linearized eguations of this problems are wa a FOE, + au ON, + a,3 Oa: 4) ay) ON, = b, = 1) = V5 a,, OE, + ae (ON. + a,, OE. + a, , ON, Db, v= = aOR. + a,,0N, + a, ae + as, ON, - b_ The zero order terms of the angle and distance Sunctions are eS Ee = 2 Let i = takc stain NS—N] AZ_ ee ee a ° £. ssarcytan j=2 == -“arcytan =! = 2 2 Na N Ny—N% —E,-€&, E5—E- f, = arc tan X ae — anc tain —— 4 aie 2 3 £ A ae = AZ - are tan -—3 4 L NB aa Seay gy Boeyaet (8 Noes.) = 1272 £. = ( (E, SOE 54); + (W, rr) aa Kh ll PREG aa Epa e ty) (N, A My yeagytge By using the data of the known coordinates (Table II) and the approximate values (Table VIT) Zing See = Noe = 0.68246 x 10-3 CES | KS Cla aa 2 1 2 1 Bi = 2£, an E> - 2; ib 0.16550 x 10-3 ON? 9 2 9 2 2 (MeN ae ee =f, — = (0) OE, = Of, =, 0) ONS =n. =" N3 - N> + N, - N5 =) (N3-N3)* + (E%- 5 2 (Nj- NGS 4+ (E,-£5)* =-3 06863 -x 11053 =e, = Nz - Ns | = 2538621 x Wed 963 NS- N50? + (63-63)? =?f, = “| By - Eo. |= = 1,33947602 Weg N3 (N3-N3)* + eS ee == = N> - N i 2238621 x916—2 OES (N5 NGJ° (ES ie Si =e = E> - £3 = - 1.33476 x 10-3 ONS (NB=N2,)2 + (E5— €5)2 =e = “f Ny - N3 + N> - N53 Oey (Ng N3)? Fey Boe (Ny = N',)? i es-€ 2 = - 2.5054 1 "x "0-2 =a 2 Eq = E> - E> - E3 Ne Lene mt One Oe ome 0 -9 2 3 (N,-N3)? +E - €3) (NEN Aelia = 1 43291 “x 10-3 Si2) Oo 0 | N5 - Na i 0.11920 x 10-3 Se 2 ou 2 GaSe (5 =) E3 - E4 ig - 0.09816 x 10-3 ° 2 2 - E; - Eo i - 235.67466 x 10-3 8 ve om Ae [(e,— E,)? + (N, — NY)? ] “| N, - No J" 971263201 x 10-3 0) We 0,2 we ) 0 E> - £3 |" - 488.17948 x 10-3 ° 2 ° V2 (E5-E3)° + (NZ—NG ) | N5 - NG iy = 972. 74326 x 10-3 ° Ce) ° oF NA HI 72 eg-€2) + (NZ —N3) | - E> > ES J" 488.17948 x 10-3 (E5-E3)° +(N3—N3)3]' 72 - N> - N3 i. BF 25743216, x01 O=3 ° One ° Ce Welt 42 (SES +(N5— N%) 0 89 ~- 635.68254 x 10h : E3 - Ey S (een en a74 = Sf, i i ~ ~_Ny No ° ° A MNS 3 (E3- E+ (NS —Ny) ] - T71.95059"x AG OF In the mateix form 0.68246 OS 16550 0.00000 0. 00000 -3. 06 868 1S 16326 2.38621 =17. 33476 22386 21 = Tan3 S476 =2e00 541 V. 432390 A. = BOSS 0.00000 0.00000 OSt1920 —0.09816 =-235.67466 ~S7A,83200 0.00000 0.00000 -448.17948 -872.74326 488.17948 872.743 26 0.00000 0.00000 -635.68254 -7715950e2 The computation’ of B matrix, by * wsang the daca from Tables AI, (TEL, sv; and Verr 0.00000 radians 0.00000 radians 0.00002 radians 0.00002 radians 0.00000 meters 0.00000 meters -0.01426 meters —s 4 A =F io a W = A, - 1 i Fh Fh FP) fh Fh 6h SO) wo & W d = d= d 3 Or ir the matrix. torn Bl =[{ 0 0 0.00002 0.00002 0 0) =0. 014265) The weight matrix can be computed by using the waene ance of the observed angles and distances from Tables III and IV by applying to Equation (3.173). “Fou puncomechaued observations of unequal precision, the weight matrix is defined as the diagonal P matrix. 40 P, = 1 / sin2(1.984") = 10,808,537,426.79957 P, = 1 7 sin2(1.405") “= 21,552,498,219.02474 P,, = 1/7 sin2(1.203") = 29,398,062,997.43487 Po, = 17 sin2(1,619") = 16,332,144,194. 08730 Pha pve 7, (0200112 = 1,000,000.00000 Pheva s¢, (3001- = 1,000,000.00000 Peepze ded, (099022 = 1A, Wythe 101.011 For uncorrelated observations of unegual precision, Biewvalues Of A, B,land PP matrices ane applied to squation mee). ~The correction’ vector vs found to: be SE, = 0.01284 eS Sn, = 10. 010336 SEs = 7001079 SN, = 0.00495 The approximation values were improved by adding the GOrrections to the first approximation values. The inproved approximation values after the first iteration are £5 = 607,943.44238 + 0.01284 = 607,943.45522 nm. N, = 4,073,939.74473 + 0.00336 = 4,073,939.74809 Be 1g COSI 21299110 OL 0TO79° = 6,087,122 00189" m. Na S074 2 56 oo 4534" OO 04 IS =) 4,074,258. 95029 mn. Using these values, the solution is iterated. After the Eesond Lteration, the correction. vector is, zero to..six decimal places, so the improved approximation values are the final estimates of the coordinates. For uncorrelated observations of ecual precision, PiewcOLEection vector is, found by solving Equation (3.19). mren COLrectaion vector atter) the) first iteration. is SE, = 0.01756 or ON, = 0.00426 SE, = 0.01660 ON, = 0.00479 4] The impreved approximation values after the first iteration are E, = 607,943.44238 + 0.01756 = 607,943.45994 nm. el 4,073,939 74873 + 0.00426 = 4,073, 939° 74859" n- E3 = 6087 12099110 + 0.2.0 1660 «— 608, 122. 00770"m- N= 4,074,258.94538 + 0.00479 = 47074, 2538-95013" a- The solution is iterated Fy using these adjusted values. The correction vector is zero to six decimal” places after the second iteration. These values are the final estimates of the coordinates. The standard deviation of an observation which has unit weight can be found by the following equations, for uncorrelated observations of unequal precision, Gn = [eee | 1/2 (3.22) asp 0) for uncorrelated observations of equal “precision, 0, = Bak: eke (3.235 n—-m™ where n is the number of observation eyuations and m is the humber of unknowns [({ Ref. 7, p. 249]. After rcalculating eae best estimate values of the unknowns, the V matrix can be computed from Equation (3.20). The standard deviations of the best estimate values for the unknowns are then given by the following equations: 6.= 6 En iZ2 (3. 24) | ° where oF is’ the standard deviation of the 1th adjuseece quantity. The quantity in the 1th row of the X maemo S:: is an element of the (AT PA) -1 matrix for uncorrelated observations of unequal precision. For uncorrelated observations of equal precision, Ss is an element of the 1 (ALA) <29 matrixsf Refo.g, po. 250m. 42 11 S15 e e e eo e Sy et 2X) S55 e e oe e e 2m (A’PA)-1 or (A’A)-1 = f ’ Mies «ae ie m1 Sm 2 e e e e e nS) The standard deviations of adjusted angles and distances can be computed by the following eguation: 6. = 6,10] 1/2 (3.25) where 67us the standard deviations of the Lth adjusted fiieriy. The (quantity im the ith row of the V matrix, 9: is an element of aca’ pay-1a" Natnx for unicorrelated observations of unequal precision. For uncorrelated observations of egual precision, Q. is an element of the Menta" matrix PRef. 3, p. 912}. oA ewan Pechy Q>, Qo ay Re Se Ts Con A(A’ PA) -1a7 . : Pe ay eee or = ° e ey let el ile) 1%) ue a(a’ a) -1aT : : a a ne ae Qn Qn2 la Sih Nach hb oo MA clLosed traverse was) conducted at Moss Landing. After the second iteration, the Vi matrix can be found by epoivyong the ivalues) of tAy Bywand! Ky matrices *to Equation (20). 4 3 For uncorrelated observations of unegual precision, the Vo matrix is found? tor be le 0.93149 x 10-5 radians or > 1.9212. Wa = -1.63120 x 10-S radians of —323605 5" qe = -1. 28363 x 10-5 radians o©F —-2,6476e" Vie ha -2.30436 x 10-S radians oF —4.975e0eu Vent ai 0.00024 meters Ve olka 0.00039 meters v 5oG 0.00358 meters The. standard deviation of unit weight rs: found by Solving Equation? (3.22) ,- thes result as 6. = + 2.69685 The values of (aA’ PA)—-2 are 1.25178 0. 22009 1.225575, 02 TO ee 10-5 | O.22089 0. 138d9'" O21 21810 Oeiiial 2a 1.25575 O02321810 T.46808_ 0. 0S8ii5 0.10788 0.701127 0.03845) s0e225i5 The standard deviations of the best estimate values for positions can be, found by solving Equation (2225))) seue results are O-, = + 0.00954 meters Ono = + 0.00317 meters Oc3 = + 0.01032 meters O13 = + 0.00405 meters 44 The C064 =O 2052 0202, Oleu2 25 =O 2a) Olney. lOia2.0 0041/02 1.0. 002 = 165) ©3945 =-6.963 4.690 =635.88 841559 values of A(aA'’PA)-1a' are Ole: Oa ei lenlate Oe Zaid =O 094 ZrO tae lero So SOF hee. 30 1 OO AO eH Oo) aoe JOS) 03-06 O2002) S415 too 630" 471.59 SOO Mi ee OO SME SetoM de Sis 7 9 ORO OZ ere Sas ele SiG SS0u1 VOU ee ly sO Ge laliclec tee SIS) SO dice op) Sita, Oe 950 Ve Oo ae soz 5 T6323 The standard deviations of adjusted angles and distances are found by solving Equation + I+ Reine lac tne: = tne (3225)'; 1.40346 seconds 2-64103 seconds 2.55260~ “seconds 326137 s.seconds 0.00269 meters 0.00267 meters 0.00745 meters the results are For uncorrelated observations of equal precision, mehiey VW. matrix is found ‘tobe Vv d1 Vv d2 43 Ve268964x 10-5 radians =". 568935: (x 910-5 “cadiars —TS 75414 ox 610-5 ‘cadians =2ecosoO,xe pops CadLans 0.14037 x 10-7 meters 0.23844 x 10-7 meters 0.24365 x 10-7 meters Oc ocr ONE or 2261742" SICA SH lest =—S.61d 17" =45 607 108 The standard deviation of unit weight is found by Soeving Equation (3. 23), 6. the result is E20 VT eix Oss 45 The values of (al a) —1 are 0.28193 0.06827 -—0. 8544S" 0270240 10=4¢ 0.06827 0.501664 =—0. 2073s roca Ome =—0. 85449-02073) 2. SISO 0 eee 0270380 ~O217075'-2. 1IS7 Zea eee The standard deviations of the best estimate values for positions can be found by solving Equation (3224) ) the results are O-5 = + 0.00107 meters Oyo = + 0.00026 meters O-3 = + 0.00324 meters Ons = + 0.00267 meters The values of Aca’ ay—tal are 1.472 =27.197 26.960 -1.235 -0.516 —-07 943s Ose -27.197 503.1 -498.7 22.855 J2388 OSes Orme 26.960 -498.7 494.4 =-22.66 0.370 02523) 0eeee 10-3 -1.235 22.855 -22.66 1.2038 -02:209) —02 1590 0eeiees =O. 578 0.348 05370 -0.:2015 10000) —020Gne—02 Cm -0.. 843 0.480 0.523 -0..159 -02007) =100008—020c8 -0.856 0-473 0.578 -—0.,.134 —0 2001" 00h icioies The standard deviations of adjusted angles and distances are found -by, solving Equation 0(325)f7the-resulesiare O63; = +4 0.15917 seconds 6,5 = + 2.94270 seconds O33 = + 2.91732 seconds or = + 0.13370 seconds Ou = + 0.00002 meters Oyo = + 0.00002 meters 643 = + 0.00002 meters 46 The computations in’ this subsection were perforned DyneMewWPSeLBe 3033 .conputer using 6 "decimal "places .and Were rounded "OLE tom5 decinal places \prior ito output. es a —— = = _——— ee Se The general principle of least squares adjustment by the condition equation method in surveying is EOPiINin Ze a 2unertom /CONSPSHiIng Sf the sum of the Ssquanes Of the CObrecEVonS £oO the, observations, plus the necessary mathematical conditions involving some or all Bue COERGeECtELZOnS: eCach Condition by Ltseli is made equal POMZeLO bY acgdaimg the coprections ts, Ehe discrepancy determined from a preadjusted calculation (i.e. calculated PaOMlicne ONServed values, | Tather than Erom adjuste . Values thus, the magnitude of the sum of the Squares is not changed by adding conditions [Ref. 8]. For uncorrelated observations of unequal precision, Ene principle condition of least syguares function may be expressed intMmeatrrex Lorn as follows: U = Vipy - 2K( BY + Wj = Vipy - 2[BV + W]'K = v'ipv - 2¢(pvy’ + w'iK = yi py 2acpv'Bl+ w#! JK =) py — 2) B'kK € QW K mininun miere U is the least squares function matrix, K is a matrix SeelLagrange multipliers, P is the weight matrix, Vis the S9Eerect ton Vector, 'B Js Behe constant icoefficients of the corrections, and W is the precalculated discrepancy Peet. 9]. It is numerically more convenient for later development to multiply by 2. Taking the partial Geet vatives Of the U matrix with respect to each of the corrections and equating to zero leads to 47 09'= 2Pv —s2nK =. 0 OV Or Vv = p-1B/K (3.26) This equation is called a correlate equation Or compre ace equation matrix. The solution of the Lagrange nultipliers matrix can be derived by multiplying Equation §(3%.26)) by gee B matrix then adding the Vimaitrax BV + W = BP-1B/K + ® (BY + W) is the condition equation matrix which mnustecgquat zero. The solution of the Lagrange multiplier vector maa =" BP-1B! K K (BP-1B!l) -1 (-wW) (3. 27) The correction vector is derived by substituting Eguation (3.27) ino, Equation "(S.425) v = P-15B! (BP-1B7)-1 (-§) (3.28) For uncorrelated observations of equal precision, the correction vector can be derived similarly to tie unequal precision case or by using the inverse of the weight matrix which is the identity matrix. Equation (6423) becomes v = B! (BB! )-1 (-W) (3.29) Closed traverse station positions may be adjusted bv using the technique of least squares adjustment by the Condition Equation Method. There are two condition equa- tions. They are the azimuth and coordinate condition equa- tions. The coordinate condition eguations are divideduemes two parts, which are E and N coordinate condition equatromean 48 The azamuth condition equation is the sum of the SOELectZONnSaco thevanglesvinta traverse plus a precalculated discrepancy W, (Equation 3.6) which must equal zero. Where imedudlsuthe: CObrections to they angles, the general equ'a- at mon can be written: or n Dien tae Oat: ow = 40 (3.30) where Lier TS bhe OrEreECtLOn to Ehe dustances. However, this term equals zero and does not effect the eguation, but it does reserve space in matrix notation for distance Gortections. A closed traverse which was conducted at Moss iiaG@ing) GCantbe>written with’an azinuth condition of es + LJ + se + e + es + oe = Uae 1 ees 1-V_, US ee 0.%4, 0.V,, + O.v,, amere the precalculated discrepancy (W,) is +8.844" or +0.00004288 radians. The simple linearized form of an E coordinate condi- tion equation is the sum of the adjusted departures and must egual i Zeros: Byvapplying Equations) (3.2) amd (3.7), the General formula for E coordinate condition equation is —1 eli oe Ostmu AZ = (Eo EG yer) 0 (2222) where Aes = AZ # Diet tev pS (P= 21) 1800 3.53 3) al J lea Dee peonenepadjusted azimuth. By using Equations (34,32) and (8233) with the data conducted at Moss Landing, these equations become ct Vg1)-Sin Az, + (d. + Ygo)-sin AZ,» + (d, + Vy3)-Ssin Az. =~ ¥(5 E =r aE pia 0 (3.34) 49 where a alte A, + vy =" Az, P 4 ¥, AZ.» AZ— . oO, * Nag $ oo V3 soho va AZ. Vat ¥ ME ee Tea ao) ca ae OV 2 VB GS =4R2g SVL eae te Az, is the precalculated azimuth or unadjusted azimuth wW(Equataonjgsad)< From trigonometry sin (A +A ) = Sin AwcosQa& + cos) Asin in which QA is a very small angle in radians, cosa = 1. and “sana —2x therefore, sin ( A +A.) = sin 2 +QAccossa from the first term of Equation. 43. 3:4) (da st-V¥j, sin Az, = (d, + vj, )sin (A240 * 354) (d, + Vu,) (sin AZ, FON 1 cos Az, J 1 d, sin AZ, + Vv = d, cos AZo) Fav sin AZ, + V d1 d1 OP a's Te) is very small, by Letting thers. term Equation (3.35) becomes (d, + Vy,) sin AZ. = ] ob d,sin Az, v,, 4,cos AZ, + V 44 sin Az, 50 let Vv. 51 Cos AZ, (3.35) equal zero, (3.36) Drensiecoma cand rthiinas tens) Or Equation) (3. 34). can be, derived Sam nars, co. ehGuia Clon O(S.19'6)i,.. (So (1d, + Vy5)Sin AZz., = d,sin Az, + (VL, ae V,2)d,cos Az, + ¥,, Sin AZ, C35 3m) (d, + V43)Sin Az., = d,sin Az, + (vV,, +V,5+V,3;) d,c0s AZ, + Vy3 Sin Az, (6-3/8) Subst icmen Nop Equa talons; (3296) >, (3.37), and (3.38) into Equation (3.34) and rearranging terms Va, (dycos Az, + d,cos Az, + d,cos Az.) + V,> (d,cos Azi5, We d,cos AZ3) + V,34,¢00S Az, vy sin AZa #8 SanvAzZ. + v d1 d2 2 q3 S10 AZ, Pedy sam AZ, t9:d5sin AZp) tad {sinyAzse> (E+) =o EA) =) 0 This equation can be written in the general formula as =) n-1 n=" nai > (va; Didvcos AZo 2,4; sin Az. + > 4d, sin AZ. i =1 k=i ea Ts Fi a E,) = 0) (639) T Moaseimi tng EqGwataons (3.2) ,7 (3.3) ,"and, (3.7) into Equation (3.39) n-1 n-1 n-1 2(%5 2OM) eyes MZ. EY WE, OOS 0) (3.40) = =f] = For example, if the number of the observed angles ({n) at Moss Landing is four, this equation can be expanded to Wag (OO; + OND * On) + v4 (ON, +ON,) + ¥,, 4%, row Sin Az, 4 & San AZ, ees iam) waz: d1 d2 d3 3 Sh Substituting the numerical values from the precalculated of this traverse into this equation, the resumemis (6702.37612)v,, + (5318-48345) v,>5 + (4999 28200) + Onvae -(0.23567)v,, +(0.48818)¥,, +(0-63570)¥,, + W, = 0 | (aaa where the precal culated drscrepancy (a is 0.08608 m and >) O0.v,,1s equal to zero, which does not effect this equation, but space is reserved in matrix notation for the last angle COETreCction. The N coordinate condition equation is the sum of the adjusted latitudes and must equal zero. By applying Equations (3.3) and (3.7), the general condition equatvenmae nis} (d. Hiv “COs Az.. uta iz1 This equation can be derived similarly to the E coordinate condition equation case, resulting in nisl n-1 Misat - = Diya AAS) + Ds 4c0s Az. to Wap = 0 (3.42) {sal Keo teat From the data acquired at Moss Laniing, this equation can ke expanded to =W.4 (C8, +E. + OE,) > Win (AE. + OE,) # 7 Gee + V cos AZ, + Vv cos AZ» Sf + W = a a 143 COS AZ. 3 0 Substituting the numerical values from the calculationpes this traverse into this equation, the result’ is ~ (3959. 89460) v_, ~ (4295. 49626) v_, 7 (4116. 94755) v_5 + 0.VL, #(0.197 VS3)iy & (0.872749 )0 + (0.77 194)iv a oe Wee (3.43) where the precalculated discrepancy (W,) is -0.08936 nm. a2 Equations" (3.30) (3. 40)> and’ (3-42) can be written PME hetatrt x Horn as (Biv +. WN) where: BY 1s “the constant Eaeenicients OL thie GOEnecttOns,.WeasS the correction vector Welch 1s GOmposed of the angle and.distance corrections, W tseche precaleulated discrepancy.,,Equations, (3.31), (3-41), amide s.43) are written in matrix, Eorm.as 1 | 6702. 3176 12°9= 395 9.89460 ee 1 5318. 9$8345"9=4905.'96 26 wy, 1 4999.28284 -4116.94755 wae BY = 1 0.00000 0.00000 eS YA 0 -0. 23567 0. 971183 ee 0 0.48818 0.87274 Vis 0 0.63570 0.77194 vee -W, = -0.00004 See tictw, = 0.08608 -W¥, = 0.08936 For uncorrelated observations of unequal precision, the inverse of the weight matrix is the diagonal matrix, which is equal to the observation's variance. The diagonal elements of the inverse of the weight matrix is Pi tee= Sine (129848 }0— 092599 x 10-10 Po Be sin2(1.405") = 0.46398 x 10-10 Pat ee= Sineq152039) (= 0234076 x 10-10 Bete a sine (le tayg = h026t229 x 10-10 Jee eee Ua ues = 0.10000 x 10-5 Ba = (0-001) 2 = 0.10000 x 10-s Dose an (0.003) 2 =O. 90000 x 10-5 he) For uncorrelated observations of unequal precision, the correction vector is. found by solving Equa trons (oes) Voy Re ee Oe Eaddians of 1.920 Y5.. = =1.63174 x 10-5 radians oc —-3. 36046" Yio = =1..28358 x 10-5 radians of —Z.08dome v= Vo4 > eoes0tso x 0s Tadians of —afodmee a is 0.00024 meters Van = 0200039 meters Vu3 0.00357 meters The corrected angles are CO, = 2469 05" 45.12117" Xs = 2220.51" 05.23554" = 1909 14" 59.95242" CL. = 2779 05' 12.24688" The corrected distances are d, = A424.0042% -n- d, =) 365274435. a. d. (= 6ua6s27457 ame Using the corrected data to recompute the coordinates es similar to the initial traverse computations. The adjwsreg coordinates are E5les0 =60 77903295520 =m: N, = 4,073,939.74809 an. BE, = 608,122.00789° 9 me N, = 4,074,258.95029 ao. £, = 612,238 285256507 WH, = 4,079, 258.3054 9 me The last adjusted coordinates are the same values as the fixed coordinates. Hence; it is Mot mecessany toweeraeee For uncorrelated observations with equal precision, the correction vector is found by solving Equatwone(s—- 54 Weak Wi IZSESsee WORKS), radvans. of 2561717" a PO yeas =—Wesoocoux 10-5) | radians) or) —3. 23601" Ve > —q 75908) x2 10—> . tadians or —3.619808" v= Va ae 2300 x 10R> Eadians of '—4.60713" eran 0.14034 x 10-7 meters Veo = O223892 x 105" meters wa. 2 QE2tso2 "xy 10-" i meters The corrected angles are CA, = 2469 05" 45.81717" OX, = 2229 51* 05.36399" CA, = 1909 14° 58.98196" O22 770 0581 12.3920 71" The corrected distances are d, do a By uSing the corrected 1424.00400 ao. 365.74400 mo. 6476.27 100 om. data to recompute the adjusted coordinates are 607, 943.45994 4,073,939.74899 608, 122..00770 4,074, 258.95013 612,238.85256 4,079, 258.31754 the coordinates, The last adjusted coordinates are the same values as the mesed COOordinates. Hence, Teas not nece Sa) Ssary to mwterate. The standard deviation of an observation which has unit weight can be found by the following equations. * For uncorrelated observations of unegual precision, 6, = apy] +72 (3.44) r and, for uncorrelated observations of egual precision, 0, = [ata] 2 (3.45) = where r is the number of condition equations. There are three condition equations for a closed traverse (r=3). The standard deviations of adjusted angles and distances can be found by the following equation: 6. = 6.| 0,,| 172 (3.46) | where 6, is the standard deviation of the ith adjusted quantity. The quantity in the ith row of the V mates is an element of the P-! - p-137(pp-1B1)-1pp-1 latrisx em uncorrelated observations of unequal precision {Ref. 3, p. 918]. For uncorrelated observatrons of equa precision, Q;} is the element of the I - B’ (BBT)-1B matrix. The matrix P-1 - p-1B!(BP-1B7)—1BP-1 or the matrix t - 3! (BBT)-1B is equal to aa) 5 “Dita set aoe ood Dost oO sai hat asi et cree ate 56 For uncorrelated observations of unegual precision, Ehewstanddatd deviation Of Unit weight is tound by solving Equation (3.44), the result is 6 = + 2.69679 The results of P-1! - p-13' (BP-137)-158p-1 are OnViG RO n057 0 20/211) PAO U0) 4 S65) 76.1916 3-63.58 SOS. 22OenOpr oe, OSI02 SS. 415). 45639 E1159 tO FOIA Malan Mot A POaea tie O S051 Heme 2 paiers) ts Se 4 Shak. 739 HO =2.0 ORO MGS BO COZ 0! SONAR | FO O)02 i si 5S) 3) FEIS.01i 99) 3500 ello Solos LerOS! he SNS ASSAY BS = m0G7 ee SOS Hao Siva mire sO he) Line. e320 sh= 1525 SOSIHOC Me de Se NS Teoh SoC VO iat5.25 TOSZ 3 ene Standard deviations .of, adjusted angles and distances are Bound by Solving Equatzon «(3.4%6)fs\ithe results are sues es. 1.40340 (yseconds on = + 2.64096 seconds O33 = + 2.55253 seconds On, =40'-:2'6 1316 “seconds O41 = + 0.00269 meters Og> = + 0.00267 meters Gis = + 0.00745 meters For uncorrelated observations of egual precision, the Seamadrd deviation of unit weight is found by solving Equaezon (3.45), the result is O, = + 2.01139 x 10-5 Sp thes resus of k= Bl (BBT)—18 are 1.472 -27.199 26.962 -1.235 -02518 —O=6ese Oe —27.199' 503. 19=-898.7 22.855 O25348 “02a c0 germs 26.962 -4968.7 894.4 —22.66 06370 803523 Creo 1O>3 -1.235 22.855 -22.66 1.038 -0). 209) —@315o Gee =07;9:18 0.348 0.370 =O0.220T 310000 °-G20 01. ea tae =0'..843 0.480) 0.523 -0.159 -0.001 T0000) [Gees =0. 856 0.473 0.518 -0.134 -0.-001 —=02001) sce The standard deviations of adjusted angles and distances are found by solving Equation (3.46); the results are O.; = + 0.15918 seconds O,5 = + 2.94262 seconds 6.9 = 222691722. ‘seconds 6,4 = + 0.13369 seconds O,, = + 0.00002 meters Oyo = + 0.00002 meters 653 = + 0.00002 meters The computations in this subsection were also performed on the NPS IBM 3033 computer; they used 16 decimal places, and were rounded off to 5 decimal places in the final solutions. These are more decimal places than are normally used in practice because such precisions are not generally attainable Fy corresponding observations. However, in comparing two computational methods that are theoretically equal, one method may be more sensitive to roung off error than the other. This will be commented van in the next chapter. 58 IV. DISCUSSIONS AND ANALYSIS OF RESULTS Three programs were used to compute the initial traverse andmadjust: the, closedctraverse ‘station ipositions. \Prograns 1, 2, and 3 were used to’ compute and adjust the traverse Etat ton rpositions) iby the ‘Approximate Method, ithe Indirect Observations Method, and the Condition Equation Method, Eespectively. ) The programs were written in WASFIV language for implementation on the NPS IBM 3033 computer. The EConputer output and listings ‘of (Programs 1,).2, and'i3 are Provided in Appendices Ay B, and Cy; (respectively. \The laximum number of intermediate traverse station positions that can be computed and adjusted by these programs is 30. The programs were tested using several fictitious data sets to ensure their performance in handling the various intermediate traverse station positions. the, CcOuputer, storage, areawsanGd CPU \time tof Programs 1, °2, and 3 has been compared (Tabie X). The adjustment of a closed traverse by the Approximate Method did not use a weight matrix in the matrix computations. The computer program for this method was written by using the variables in only one dimension for storage of both data and results. This method did not require any iterations. The least Squares adjustment of a closed traverse characteristically is used to simultaneously eliminate closing errors in azimuths and distances, and Programs 2 and 3 both utilized Even tan adjustment.) The, computation by either Least Squares Method used matrices in two dimensions to compute the correction vector. Likewise the computer programs were written uSing variables in two dimensions. Included are all Subroutines from Program 1 and extra subroutines for each individual program. Therefore, both Programs 2 and 3 used mote Computer storage area and CPU time than Program 1. 59 "99S sowhkq eek Wee: E Gaal sl Gre-'b Oi Lew Sra vel e Gel L°Gst Grek kt 8ZL“6ZL} 958 °H6Z L el omg Gc Uy Lace | 6 Lest 8h9 HL8 € c sweiboig JO 9UTL Ndd pue = G@- co SULIDOIg Forslta mds FOXct pes Get Lo @- leswetboig JO suLaw ds JO «raed Coe SUOT}EFS OMG poysn(pe ATOZ autq nd Z:¢€ SutTbOIg 30 - raze ahei04s ATaynduod Fo ct}eI ayy Es, pue cs “Sswepborq jo eal traae abeatoys Jaynduod fo ctzeI JEG! Cz Peete afei04s aaynduoo Te}OoO4 = sueibCig Jo szuauazeYS Fo ctzeI €-| pue oe | sweaifoig Jo szuawazeyYS Fo ctzeI 0Sh S}uaUazyeYS FO Jaqunu €_pue “2 “4 sweaborg aud aus Z3€ aL cth oOuUL aul eoiy a96bt31034S AazNndUOD 9Yy UaaKNaG SUOSTIedUOD ay X ATAVS OTe eee 60 The Indirect Observations Method requires that the number of observation equations be equal to the sum of observation equations of both observed angles and distances. The mMunber cof observation equationstisinot constants)? It changes depending on the number of the traverse stations-- this means that the row dimension of each matrix will change depending on the number of observation equations. Therefore, the subroutines affected by tne number of trav- erse stations are difficult to write in the general progran. The coefficients in the A matrix are computed by taking. the Partial derivatives of the functions with respect to each unknown variabie, where the number of unknown variables is not, CONnStant.»©s Theo numbers of unknownivariables changes depending on the number of the traverse stations. Therefore, the subroutine for the computed A matrix is also gar it) toy write! in: thetgeneral i progran:) The Condition Equations Method has only three equations which makes it easier to derive the general form and write the computer program. The adjustment using the Indirect Observations Method does not apply the corrections to the observed values directly; therefore, it requires the approximate values of the unknown coordinates be computed for the correction vector of the unknown coordinates. This method requires at least two iterations to check the insignificance of the cOEEection vector when it is compared with an arbitrarily selected small number. The adjusted coordinates from the Condition Equations Method can be checked at the first iter- aevon., Thus, Program 3911S more ;|economicaloas )it uses less computer StObagerareanand GPUytimeythansProgramiZ2ud{Tablerxy. The closed traverse at Moss Landing originated and terminated at control points with known positions which were determined by third-order methods. The azimuth closure was eae |) pernstatitondandgthe position¢closureé was .1:66,617. Biepelassi fication, pstandards .of saccuracy,, and general 61 specifications: for a third-order: Class Ij)traverser( taoreu indicate the azimuth closure is not to exceed 3" per station and: position-closure must be Letter) thang 12mi07 0002 ines traverse met the specifications and standards o£ aceuEaGcy for av third-ordergclasspl traverse. Using Programs 1, 2, and 3, the traverse was computed and adjusted in UTM grid coordinates. The fanal adjusted coordinates were transformed from UTM grid coordinates to geographic coordinates [Refs 4, pps 319-3217]. {Pheri aces ence between the coordinates for each method and NOS {Ref. 10] have been computed (Table XI). The techniquer oe Least Squares for both methods yields identical computa- tionalsresuits' to iat leastfive decimal places Mtanmcmam Methods 3.1 and 4.1;°3.2 and 4:2).. Least) Squares {prow the kest estimates for positions of all traverse stations Program 2 performs a statistical test, yieldingo the Stamiaed deviations of both adjusted positions and observed values. Conversely, Program 3 only yields the standard deviation of adjusted observed vaiues.- Comparisons of the standard devi- ations of adjusted observed angles and distances were made between these computed by the Indirect Observations Method and the Condition Equations Method (fable XII) 2 §The computed standard deviations of the adjusted angles differed significantly in the fourth decimal place’ in several cases, which is a slight indication that the Indirect Jbservations Metnod is more sensitive to round off error. The standard deviation estimates are larger for the Indirect Observations Method - in every case oftavsignificant ditferencetsn gene fourth decimal place. However, a significant difference in the fourth decimal place is insigniitveant in teens omeere observational precision. The standard deviations of ere adjusted distances for both methods yield identical resuits (Table XII). Both Methods differ from the NOS positions in the fourth decimal place in seconds of are for both Jatitude 62 and Longutude. the borazontal)' control for third-order stan- dards EecquvuEes! accuracy to three decimal places’ in. seconds of arc for both latitude and longitude--thus, these methods eam, oe Used -Or \Computing. thind-order. positions’. 6 3 (a eens i i ee ee eS SLOOO"O B8E000"O+ GSLOO90°O+ 8E000°0+ GEO00 0- “JFta fl 0O0O0°O+ TE COCO 0+ hl CCO°O+ hEO00O0°O+ LEOQ00*O- “JJFtd CGloe oF Le Tze 6croe OL Lf ct CSGLGE Sl Lt Ut Galoe OL Li cl OCcne ol Ln wc BOLGe 91 Lact “ 4 Le) (°H) epnythuoT 688SL°EZ Lh LZL GORGES Et ecl GESGE- Ge Lt WC 698GE “Geo ER CL WE6GL EC Li cboel B0GSGH EC Lt Let 6 4 oO (°H) epnytbuoT 60000 °0- B8000C “0- 6000C °0- 80000 °0- 3t-0.0'C 0+ “jjTd uotze4s 600C0C°0- EL OOC =0- 6000C°O0- GEOG: O= LE OUC-20 + GCLBE SE ViGy Gree Se CGL Ble «SIE UCESE SE LLOBE “SE EELS & + Sie 8h 8 t) 8h Ze Se eis JE ie] CN splat yey dway oung iV 6S260' "Sc CIES TSC SSI OOrSiC CPLEEO SC ELLGO SC OSL60 “Sc 8h 8h 8h 8t 8h 8t IE GE tS JE GE Oe Oo (°K) opngtqed 3S YORQSSOWK 4V SozeuTpIoO) DTYydeahoayg IX dATEvs uotst9eid [Tenboaugn Z°h uotstoeid yenbs | °h uotT}enby uotzyTpuog uotstoeid Tenhaun 7Z°¢e uotstoeid [Tenby 1 °¢ SUOT}JBATASAQ }OSITpUL a zeutxoiddy SONG Se Saul Spoy zen uotstoaid Tenboun Z*h uotstoeid Tenby | *h uotzenby uotztpuog uotstoeid Tenboun 7°¢E uotStoaid jyenbg | °¢ SUOTJEATISAQ }OSIITpUL azeutxoiddy SON *%S 30 Spoy ew 64 = = = LOOO0R =~ 20 T0002 = 80000, = 0000 + souUsTeF FTC Ie000'020 ZO000S0 COO00Z0 OLEEL “0 CELLG"? OLTHE"°C LIEGL “0 WOT ZeATSSqO YOOTTpUT °z CUO0OF 0 CZO000 20 -cO00'070 6 VEEL 0 -CCLLG G, CIC -c CLOSE 0 uotTzenbg uotytpuog*|] “ul ul "Ul “ TT " un SpoyzoW § C L t S Cc L saoueqystd paysnl(py soTbuy peysn(py JO suot}etasd prPppurys SUOT}PATOSGO UOTSTIaIg Tenhy” eee ee ee = = = E0000, —.L0000; = LO00 00> =% 9:00:00 “= souetTaFFTd GSHLOO*O L9COC"O 69200°0 LELIT°O O0972SS°Z ECLHO°S AQHEOhH°L WOTZeATISKCO YPOOITPUT °zZ GhL00°O L9ZOC°O 69Z00°0 YEL9Z°O ESZSS°Z 960H9°Z OHEOH'L uotzenby uotzytpuon’y “Ul “Wl “ul u u " u SPpOyyoHW € c L ti iS 6 L seaueystqd paysnl py saTbuy peysnl(py JC SUOT}ETARG prepuerys SUOTJEATOSGQ UOTSTI3IIg Tenhsaug Serenbs 4seeT jo suotyetasg prepueys ayy waesaqyog suostiaedwoy) ayy IITX JTGVL o5 V. CONCLUSIONS AND RECOMMENDATION A. CONCLUSIONS The computer programs developed by the author ard contained in this -thesis have a variety of useful and prac. tical hydroyraphic applications. In hydrography, geedemees rield work plays a key role. dorizontal control is negess Sary in determining positions and, even in areas which appear to initially have adequate control, aadrevona control stations often need to be established aGter aprameee in the field. These field-gqenerated control points muse adjusted-to Zit withinm-the existing survey net ani )t ome Ml1zé errors in the 2i€leé neasurssents. The Grpoes) canmuea Minimized by these computer programs. Direct application of these programs would be enormcusly beneficial, since no equivaient software exist at NPS to perform such adjustments at the present time. In the Approximate Method, a traverse is adjusted by computations uSing a hand calculator. This method /|is @sweee able for field computation and the adjusted coordinates meet the specifications and standards of accuracy f05 a) Gimmes order class I traverse. Although«the accuracy 50m seis method is less than the Least Squares Metnod, it does not require a computer and it furnishes-coordinates which ange checked in the £ield by” the £frelideoarey. The Least Squares Method provides the best estimates for positions of all traverse stations. Least Squares does reguire more of a computational effort than the Approximate Method. However, the accuracy of Least Squares is better than the Approximate Method, and the Least Sguares Method should be used for the final office computations. Jeorn 66 Leastisquares: Methods yield adentical results, but the Gordition Eeuations Method, is more econonical and easier to aqderive thar the Indirect Observations “4ethod. Therefore, ene Condition ,bGuationsuMethod should .be applied at NPS. B. RECOSMENDATION Studies shouldbe ‘continued at_NPS to compare the economy of adjustment methods used in this thesis with other methods of least squares adjustment. 67 OOP ee -9C el JET OCB -ec GT OO'T 00€CO0 "0 OO0TOO*O COTOO*O "Ww 7Gus NV 4dOW OOF COGIC CLT SS Lid W 10H WOpLE 9L79 OOEOCP 1 (KONG? 2 GT O6T dwWw3l sWNNG OOvVFL° SIE 00SO0%°T 00009°8 UG eece AIVESSOW 00400 °4c47T O0OF86°T .O000C°EV G 942 c SSOW YSHdId SYd3leiw “5 Sa Wk > @ 3INVISIG ~GHs 3 IONV SNOILVLS dO AWVN Oiiedic G3AX=SS EC yO ORK ao tok ok tok ok VIVG G2AY42S980 NVYOW Vi KO AS 9GC2GB°BECCIIO YSGLTE°8S26L0¥ W 10H U o ND e (&) O oa oe ae uu aqtVYe Oo Uy 5 at Wr = CG ae) (S) So ce fale >U os) ONC2 oO >) a ZnN N= Fw <— ere O mem SB) oS > a Ww OZ °2=4) mu Ay He & D>DrrRe Fe O (Sy a Co >~>RO ON NY WwW NNW AXoOe Sa aa) GS oO Ohi a Ww FeZzaNwW Y~erowW Vex SS ened) ae oO e OMe a O> fen x 4 ys an wtt at rFyitstoezeaz « (Oye Oo = GC. 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OratOD AODOKRAaID OCUIFr-ZZOOCZOeW2 el? 4p) el Ha zore ro m od = z Ook oes, Eo OOM OO oO ro) fo) co) + wn We) Lom! a to] Ww = © _ = <= Ke w Cs OS - = t oO ay = oss = a (oes a, i th eas eo) os fe Pat ree ww) x a -! ie .02 7 1 ' <— FSF e =) aR 0 a —_ A za — fC & oy fay Kb Cael) (OC wm <= ce OVUOVUU0OUVUUO om fa aN} COs see a oo On @ meaner OO —_J @ CO009009 sQwo ODAINM FOO BWEF- QWO00000 oo SAS aS Se Se | aaa) en nrnrne rem PSO 00000 00ers eh COLL LULU LL LL FER ERERE Ln heat oe Loe heen oe | oe) CYarerarcO Se oe eae) 76 NGRID(T) »EGRID(1),STA D(l)sANGM(1),4NGS(1),9S _— - Wy) uw) mO Cas) UVLLI Le Pel eo aes ——4 Ww <—c-_ - ae nn ey e~ N YO us 4 ~O =u qu) Se a | Sl LL ~ ke -~— EWE = 2m ZRrOnrZeH uw (ees) OW OereZzecor _ Oa Or OsSsatszu=z + + © (s} io) (=) oO jo) = N Lom N N N Ww — < = ms Ww x WN ore 4 Oe Lu Ol Olr) 105) © a = a a as O < ! oat eS = > eee — — crea (ab sgn) (ea) arm wu a al uw < F OVUVOUOOOU Tat ) WY — fo) = + =) N Q 7~7~ Oo 2y ke _ a CZ =ae io) (ake) = ve) Ovo (ajax faa) =~ oO ee oe] z= a QBS: <= a << —~DD « a =O4 (o>) ~O 1 => > as Se) QA~~— => oO OMe F.« 38) (95) 225 DRERREMNO e N aqaq=- SlD77H (o) (je) eS Sees Re ee) e+ ON NROOD > &- 2 ret aNrRHOW~ ac e D@ea << e O ame OY til i ui le (oy (ey) wi OW ad w~ Se) ENN BNee - eee “O- _ af¢>5D ” (2 ae ow == RS ete te >, act = SNe i} z= 4 OaN NN2-ai~~~ _ e iT] ee ene O — wil fa an ai nl? 4 a=ng rag AOU = EN OUNNN- [og oF MWZ Il Or Z2aioiagaan ON awe a ONL OO = pul] I SA ay OUOUUUO Ot ar 240 N > ~~ = = va) =) ns (osn' = - =) et fan) > rs = O a N = - an ke On > >= = eD a fam on) NJ O e-_-— *OO aC ZOOr'-K NE aa —a > pa Pa ke _——_ =a] > wilt = «AO _ <> | ORANtIanNaond~—~ eS eo oe QOf==I4ItSZTADFEFrwH OFSSwwTst>+ozre ~DDAADD= emn= *OQ OO=VA0D Qa SNR O Ou ruaa <> Sap ace UO>> Se De RHItH NZS nny rE oe | elo = Oe Wow Ke~~—~—> 1! CO Ox> COVUO ba Ga Pol nl oe] O=Me jj I ar rOOCKHIIINNNES Jao dE TOOIIaatetu oo fwWDDwWSzataetONOONnNvoo OQANNVOYOO UO Lx > ++ woudl aN >< O > x< tu > a Co - = oO O 250 PRINT DETAILS. 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T1V9 ( 0601'9 ) 3iTUM ( GOLSENSENS ENS ILEG*EN* + I—Ye ) WIMSN_ T1V9 ( O80T'9 ) 3LTUM ( OOOT'S ) SLUM ( GOLSENSENS ENS LEGSENS .9—Ye ) WAMSN T1V9 ( OLOT'S ) 3LTuM ( GOLSENSLWANS EWAN SLOId*EN Ss 9—Ye ) WAMSN_ 11V9 ( 0901'S ) ALTIUM ( GOI SENS LWANSLWANS LWOSENS oI—de ) W4MSN_ 11V9 ( 0S0T'9 ) 311MM ( OOOT'S ) 31TUM ( GSOT*LWANSENSENSWOSENS eO—Ue ) WAMSN 179 ( OF0T'9 ) SLIUM ( GOISLWANS LWANS WAN! IWd* EN! 0O—Ye ) WIMST_ 11¥9 ( O€OT'S } JLIuM 3NNTLNOI OvT ( I) WM ¢ OzOT‘9 ) SLIUM e*t = T O4T OG ( OTOT'9 ) 3LIUM ( OOOT'9 ) 3LiuM OST OL 09 ( O °0S° GINIdd ) JI NOTSViNdWOd 3HL 40 TWIVL130 LNIdd ( YSIS IWAN SWOSENS EWAN! LWANS ENS AWANSWId9S0418Id ) SATNWA 11V9 O OVO 146 OO VOU SQRT(DABS(DM(T,J)))) A GO TO 200 ) ) R-C*sN390M~NUMT,NUMT »NUMT,IO5 ) ) ) QORMWR ae OF-Decrat ae One) Or 6,1000 ) ul) ~— em pa LL! OC e-eD> zaer- Caw osa MATRIX! y ~ o- se S< — fad e at A -~ at * ey (28) ~ - + ~ ere = « <) oS ° ee OB | = me OO mae wn fo (eg ONT *#O. ke REN e@r mit au —q <= 0 -ao > = = exa m~e NZ ~ eo rit Ou = ARK OXYORKN HN! ~— ~ CL eee Hr Oe XH ye UNNLYtUSN Ne olUpit OnNCRrTEONNXKO>o& 8 - e <¢ eS wt Se OO <— Be wWrteZe ee COR — OX. ed © e ® e@ &@ & @® ewe eTeee ao MLL KO OK KKK OK BR RRB MH BLN OM ASL << OK OK OX SIX MO FANN SS LALA LALA LA eV Fe Fe Fe F wwe wre =e ~SNN SSN NS HNN NNN AN FRR RNAS SRN NSN -SNSSNR NAN RNR NS NNN et et ee es ee ee et ee ee et ee ee ee ee ee ee RREREREERREREEEEEFE dididtqcqadqdaqdaqcqaaac SiS s25s55555>= OODOCCOOOOCO0O0O0OO OR Ce oy Ce ee ee eee * lelelelelelelelelelelelelelelel@) OANMHFINOEH DHOANMSN eleleleleleleleleleol. [alanis lenlen! SSS SAS SS SS ea 147 (=) Zz TE OVO OL 10. LIST OF REFERENCES Bomford, Brigadier G., Geodesy, Second Edition, pease Oxford Gniversity PEeSSipmiooz Palikaris, Athanasios E., Methods of Hydrographic Surveying used by Different Countries, H.S. Thesis, Naver ees o> ae e School, Monterey, Californiay Gee v e pp. 6-7, Ta. - Department of Commerce, Environmental Science U.S Services Administration, Coast and Geodetic - Survey,Horizontal Control Data guae 361214, Stations 1037, 1058, 059m, andwlUec; ces - Mikhail, Edward M., and Gracie, Gordon, Analysis and Adjustment of SN Measurements, Van Nostrand Reinhold Company, TY8T. Hardy, Rolland L., A Brief Outline and Demonstration of the Theory and»Practice.of Teast Squapes ipa a Unpublished Note. Hardy, Rolland I., Least Squares Adjustment of Traverse, p. 4, Unpublished Note. Pacific Marine Center, NOS, Horizontal Control Report for Moss Landing = Naval Postgraduate School Survey, 148 BIBLIOGRAPHY MiGente Casperona amd sutchitie,) Walter D.,\ Manual of _ First-order Traverse, Coast and seodetic Survey Special PiibicadteTOn NOs Sv, U.S. Department of Commerce; 1927. Peg gnc lark Ejgand) Matehell,, NMachael 4.7 Introduction to seodesy, Elsevier Nortk Holland Publishing Company, 19569. Graham and Neil, Introduction to Computer Science A Structure Approach, West Publishing Company, 1982. Hodgson, C.V., Manual of Second and Third Qrder Triangulation and Traverse, Coast and Geodetic Survey 28S Sot aoe deka Wo. 75, U.S. Department of Commerce, iM. 12. INITIAL DISTRIBUTION LIST No. Defense Technical Information Center Cameron Station Alexandria, VA 22314 Labrar Code 0142 Naval Postgraduate School Monterey, CA 93943 Chairman, Code 68Mr Naval Postgraduate School Monterey, CA 93943 Department of Oceanography NOAA. Liaison. O£ ficer Post Office Box .8688 Monterey, CA 93943 OFF ice Ofnthe: Darectorsa = Naval Oceanogr euhy DAVESLOM (O\P=—952) Department ot the Nav Washington, D.C. 203 Ccmmander Naval See en sor. Command NSIL6 NS1o3; S529 Commanding Officer ; Naval Oceanographic Office Bay St. Louis NSTL, MS. 39522 Ccmmanding Officer ae Naval Ocean Research and Development Activity Bay St. Louis NSTL,. NS") 39522 Chief of Naval Research 800 North Quancy Street Arlangton, VA AOA I | Chairman, Oceanography Department U. S. Naval Academy Annapolis;..MDo» 21402 Chief, Boe OgE ape Programs Division Defense us ppend Agency {Code PPdH) Building 5 U.S. Naval Observator Washington, D. CC. (20305-3000 Lt. Saman Aumchantr THAILAND 150 Copies 2 13. 14. 15. 16. VW. 18. Se 20. Personnel Department Royal Thai Navy Bangkok 10600 THAILAND Library 4 Hydrographic Department Royal Thai Navy Bangkok 10600 THAILAND Chief Bes) eae te Surveys Branch N/CG24, -oom 404, WSC-1 National Oceanic and Atmospheric Administration Rockville, MD’ “20852 Program ee eee itarson, and Training Division NC2, Room 105, Rockwall Building oe ; National Oceanic and Atmospheric Administration Rockyilie, MD "20852 Director, Pacific Marine Center N/MOP National Ocean Service, NOAA 1801 Fairview Avenue, East Seattle, WA 98102 preegr ore Atlantic Marine Center L National Ocean Service, NOAA 439 West York Street Norfolk, VR. 23510 Dee Rolland LL. hardy Department of Oceanogra Naval Postgraduate Scho Monterey, CA 93943 phy, Code 68 Xz ol Associate Deputy Director for Hydrography Defense epee ad Agency (Code DH} Building 5 U.S. Naval Observator Washangton, DOC. 20305—30100 jon Y 8RDO ia >. 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