a ne ear | Gee ka.ce, CETA “BI 3 WHO! DOCUMENT COLLECTION A Model for the Distribution Function for Significant Wave Height by Edward F. Thompson COASTAL ENGINEERING TECHNICAL AID NO. 81-3 JANUARY 1981 ¢ \ : po Ce / Ne 97 ee eee “tains Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS =z COASTAL ENGINEERING 330 RESEARCH CENTER Us Kingman Building no, 31-3 Fort Belvoir, Va. 22060 Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: Nattonal Technical Information Service ATTN: Operations Diviston 5285 Port Royal Road Springfield, Virginia 22161 The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. =< 5 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO] 3. RECIPIENT'S CATALOG NUMBER CREAN SI=3) 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Coastal Engineering Technical Aid 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(s) A MODEL FOR THE DISTRIBUTION FUNCTION FOR SIGNIFICANT WAVE HEIGHT 7. AUTHOR(S) Edward F. Thompson 10. PROGRAM ELEMENT, PROJECT, TASK 9. PERFORMING ORGANIZATION NAME AND ADDRESS AREA & WORK UNIT NUMBERS Department of the Army Coastal Engineering Research Center (CERRE-CO) Kingman Building, Fort Belvoir, Virginia 22060 A31463 12. REPORT DATE January 1981 13. NUMBER OF PAGES 16 15. SECURITY CLASS. (of this report) . CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center Kingman Building, Fort Belvoir, Virginia 22060 MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 14. UNCLASSIFIED DECL ASSIFICATION/ DOWNGRADING SCHEDULE 15a, DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Nags Head, North Carolina Wave heights Shallow-water gage data Wave model ABSTRACT (Continue on reverse side if necesaary and identify by block number) A model based on a three-parameter Weibull distribution function is given for the long- term distribution of significant wave height. The model, formulated in dimensionless terms, is believed to provide a more general representation than the corresponding models given in the Shore Protection Manual (SPM). A procedure for using available data from a site to estimate model parameters is described. The procedure extends the use of available data and leads to a model which more closely follows the data than the procedures in the SPM. The procedure is applied to shallow-water gage data from Nags Head, North Carolina. Exam- ple problems are given to illustrate the use of the model at the Nags Head site. FORM DD , jan 73 1473. EDITION OF 1 Nov 65 1S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) i" a ” iii se a d 4 , at ony Fest “dah a clenathand é j J 7 phe le 1 ped agama - 4 Selleeky } ee KA nee! 6 Pi Ti rf Gar ALL ATT HK eee Lie) gas ma 4 h + gare Seed Aertib'Nee Ait te 4 ay, . aonetiehya OPE SAR es nescartilt ates (ny este on SAN i, fount eee hin bit SOR” e447 Cy hela Hb eevee sob 3 oe | ner Soe wait ie fran ia ‘ areas te oe fh a eS not cnr eh” P Vent geibe ba Baie aed ane vee poem it Dba ae Mma T Bk wel Sie pt a Be Piaemes vail \ I q Setar. * ee ton + t 4 Aikite Sita 09 qr : t Aone €tah oT F Leet Deere Dek weey ATM ero he -earebendin’ Wi nha’ GoBe, Wale, psig) bolt is cise rire ely go | hes a wly Bah LH aaW % ie TES RE Ged PD ‘iN co emma err eah mT rl A> mea 1% PREFACE Two empirical models for the distribution of significant wave height are given in Section 4.332 of the Shore Protection Manual. A similar model is presented in this report. The model is based on a three-parameter Weibull distribution function. Parameters in the model are evaluated from a large sample of shallow-water gage data at Nags Head, North Carolina. The model, which more closely represents available data than either of the previous models, is particularly useful for statistical prediction of extreme signifi- cant wave heights in shallow water at Nags Head. The technique is applicable for other gage sites. This work was carried out under the waves and coastal flooding research program of the U.S. Army Coastal Engineering Research Center (CERC). This report was prepared by Edward F. Thompson, Hydraulic Engineer, under the supervision of Dr. C.L. Vincent, Chief, Coastal Oceanography Branch. The author gratefully acknowledges Dr. D.L. Harris, formerly CERC Senior Scien- tist, who provided valuable comments on this study, and J. Peworchik of CERC who processed the Nags Head data for the study. Comments on this publication are invited. Approved for publication in accordance with Public Law 1966, 79th Congress, approved 31 July 1945, as supplemented by Public Law 1972, 88th Congress, approved 7 November 1963. TED E. BISHOP Colonel, Corps of Engineers Commander and Director CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)... . SNAMIOVS ANID) IDIDRIENIEIMIIONS 5 56 6 6 6 o 0 6 0 o 1 ENTRODUCTIONGHA S75, SoS ROU! Ve ei Ra ERM stein deter mee ere fale WEIBULL DISTRIBUTION FUNCTION ......... JEILIL COMPILATION (OF EMPIRECAL, DESTREBULION® 2 5 29922 2°22 IV APPLICATION TO SHALLOW-WATER GAGE SITE AT NAGS HEAD, MOISE CAROIGION 6 6 6160 0 06,6 6 6-6 0 oF 0 6 Bho © V GNUIEL IINOILITUIS, 5 5 6 0 666005060 0 6 VI SUMMARY) 82. dene 2 5, arc che ese came eee ee ee APPENDIX METHOD FOR ESTIMATING PARAMETERS IN THE WEIBULL DISTRIBUTION POM (CAEILON Giseereien haem RO TGCwG. 0c ec tar 70. auvor TOL eGIROn cD! 6..'0!-.0 TABLE Monthly and annual significant wave heights statistics, Nags Head, NorehiCaroildimaios, We. oa yet vay sadoheey iat eet liey hears aeons fimyeuint ana Mees c - FIGURE Distribution of significant wave height, Nags Head, North Carolina. CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: To obtain Multiply inches square inches cubic inches feet square feet cubic feet yards square yards cubic yards miles square miles knots acres foot-pounds millibars ounces pounds ton, long ton, short degrees (angel) Fahrenheit degrees by 2524 2.54 62452 16.39 30.48 0.3048 0.0929 0.0283: 0.9144 0.836 0.7646 1.6093 259.0 1.852 0.4047 1.3558 1.0197 28-35 453.6 0.4536 1.0160 0.9072 0.01745 5/9 x 1073 millimeters centimeters Square centimeters cubic centimeters centimeters meters Square meters cubic meters meters square meters cubic meters kilometers hectares kilometers per hour hectares newton meters kilograms per square centimeter grams grams kilograms metric tons metric tons radians Celsius degrees or Kelvins To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, HS eMeLormuUl'a) Ce —n5)/9)) CHa 32). To obtain Kelvin (K) readings, use formula: K = (5/9) (2 32) se 27/SoilSc SYMBOLS AND DEFINITIONS water depth probability that the ratio H,/H, = Hj, is greater than or equal to a specified ratio He: acceleration due to gravity significant wave height mean significant wave height significant wave height divided by mean significant height parameter in Weibull distribution function specified value of Blog minimum expected value of He» parameter in Weibull distri- bution function wave period parameter in Weibull distribution function standard deviation of significant wave height A MODEL FOR THE DISTRIBUTION FUNCTION FOR SIGNIFICANT WAVE HEIGHT by Edward F. Thompson I. INTRODUCTION The long-term distribution of significant wave height at a site can be estimated from empirical data or by either of two empirical models in the Shore Protection Manual (SPM), Section 4.332 (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977)!. Where sufficient data are avail- able, direct use of empirical data is usually preferable to either SPM model. However, proper estimation of the long-term distribution of significant height requires the use of an integral number (preferably 3 or more) of reasonably complete years of data. This requirement is often difficult to meet because of intermittent failure to obtain observations and limited number of years of data collection at a site. Tt is often convenient to model an observed distribution of significant wave height. A model provides a simple parameterization of the observed dis- tribution as well as a systematic method for extrapolating to probabilities beyond the data (although extrapolations are always much more uncertain than the part of the distribution well supported by data because of long-term variability in storms producing extreme wave conditions). Since there is no compelling physical basis for favoring any particular model, models are chosen to fit observed distributions of significant height. The models in the SPM were proposed as a tool for representing the distribution of the highest 50 to 80 percent of observed significant heights. The model presented in equation 4-6 of the SPM is a two-parameter modified exponential distribution which is further simplified in equation 4-9 of the SPM to a one-parameter distribution. The model presented in this report is based on a three-parameter Weibull distribution function. The three-parameter model can better fit observations than either the one- or two-parameter models in the SPM. Parameters are eval- uated to optimize the fit to empirical data from a gage at Nags Head, North Carolina. The model is formulated in dimensionless terms so that the effect of mean significant wave height level is removed. The advantages of using dimensionless terms are that more complete use is made of available data and that general characteristics of the distribution of significant height in addition to the mean can be readily examined. The dimensionless distribution function may be relatively invariant compared to mean significant height vari- ations along a short section of coast. Hence, the model presented in this report is believed to provide a more general representation than the models in the SPM. ly.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protection Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. II. WEIBULL DISTRIBUTION FUNCTION A general model which can be used to approximate the empirical distribution of significant wave height is a o " st sc Heo nin] > = 5 FB = He S Hse } ie) where Heya = significant wave height divided by mean significant height Hee SespeetitedsvallucvorssHor lige > Hla = probability that H,. is greater than or equal to a speci- fied ratio eS floes ain = minimum expected value of H,. Heat, a = other parameters in distribution function. Equation (1) is a form of the Weibull distribution function with three param- eters (H H eG ©) sc min® “sc? IIL. COMPILATION OF EMPIRICAL DISTRIBUTION The parameters in equation (1) must be evaluated for each site by opti- mizing the agreement between equation (1) and the empirical, dimensionless distribution of significant height at the site using the following procedure: (a) Assemble all significant heights obtained by reliable, consist- ent analysis methods at a particular site; (b) delete significant heights from months in which more than 50 percent of the possible observations are missing; (c) compute mean significant height for each remaining month; (d) divide each significant height by the appropriate monthly mean Significant height; and (e) combine the dimensionless significant heights in step (d) from all months into one distribution. The implicit assumption in step (e) is that monthly variations in wave condi- tions can be completely represented by variations in monthly mean significant height but that variations relative to the mean are consistent from month to month. The assumption may not be valid at sites strongly affected by hurri- cane waves unless hurricane waves are treated separately. IV. APPLICATION TO SHALLOW-WATER GAGE SITE AT NAGS HEAD, NORTH CAROLINA Wave data used to test the model were obtained from a pier-mounted staff gage in a 16-foot water depth at Nags Head, North Carolina (see Thompson, 1977)2. Digital records were collected and analyzed at 6-hour intervals, with numerous interruptions, from December 1968 to March 1978. Significant wave heights from each of 54 relatively complete months were processed, as discussed previously, to form one dimensionless distribution containing 5,220 observa- tions (see Fig.). The empirical distribution extends to an exceedance percent-— age of 0.02. Many cases at low exceedance percentages may be affected by the limited water depth at the Nags Head site, as indicated by the hatched area in the Figure. A ant re Z tj ff GY Z \ Y A CAUTION: Hg m Gyn 4-6, SPM) y J | be limited By wat YE y yy v, depth at site ae Yj wn _ = pf = ° 2 238° VU ez ZONA YY 2s tj. (3,220 Obsn = 35 ify), =, © LYf, pp A= 2 a on So =5 o @ = ° | 0 0.0! 0.1 | 10 100 Percent Greater than Indicated, F X |OOpct Figure. Distribution of significant wave height, Nags Head, North Carolina. 2THOMPSON, E.F., "Wave Climate at Selected Locations Along U.S. Coasts," TR 77-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Jan. 1977. Values for the parameters in equation (1) were estimated from the empirical distribution of significant heights at Nags Head by the method described in the Appendix to give Z 1.65 “(le =" On 198) e [eee He. | = 3 0.885 (2) Equation (2) is shown in the Figure. This distribution function fits the empirical distribution at Nags Head better than the comparable models in the SPM. Equation (2) can be rearranged to give 0.606 an( ra |B & ise] Oot22| ae a Hee =e The assumption that variations relative to monthly mean significant height are consistent from month to month was tested at Nags Head by comparing empir- ical distributions of significant height by season. Distributions for fall (September to November), winter (December to February), and spring (March to May) are comparable to the empirical distribution in the Figure. However, the distribution for summer (June to August) indicates higher values of H,. than the empirical distribution in the Figure at probabilities below about 10 per- cent. This discrepancy is due to exceptionally low monthly mean significant heights in summer and, in many cases, to nearshore hurricanes. Equation (2) seems to be satisfactory for estimating the annual or nonsummer distribution of significant height at Nags Head, but the summer distribution must include special consideration of hurricane-generated waves at this site. V. EXAMPLE PROBLEMS kk ek KK KK KK A KOK ® & & EXAMPLE PROBLEM 1 * * * ¥ *¥ ¥ ¥ KKK KK KK XK GIVEN: Mean annual significant height of approximately [El = 3.0 feet (0.91 meter) at Nags Head, North Carolina (see Table). FIND: (a) The significant height which is equaled or exceeded during 6 hours every year. (b) The significant height which is equaled or exceeded during 1 hour every year. SOLUTION: (a) The exceedance percentage Cis > Heel x 100 percent) is “ts x 100 = 0.0685 percent 6 Table. Monthly and annual significant wave height statistics, Nags Head, North Carolina ; No. of = No. of i= Month Year Obsns. Hs fo} Month Year Obsns. Hs (Gee) (Cele) Cee) Cee) 12 1968 86.) 2:62 1594 || O2 gra 7S DSO 1.29 Ol 1969 112 3,04 1.59 09 1974 75 255 O98 02 1969 103 4ot2 2oOil 2 1974 118 3009) 1559 03 1969 106 3.66 1.85 Ol 1975 93 3.38 1.58 04 1969 90 Dos) s27 ||| O3 IO) 7/5 63 Dol edz 05 1969 85 2.32 1.08 |] 08 1975 63 1.86 0.86 07 1969 112 2.01 1.02 09 1975 86 2o9Y) LolO 08 1969 94 DoD) iNoO7 10 1975 Q2 3535 1,40 09 1969 105 3o35 lo 74 11 1975 90 3.02 Io4e Annual Dec. 1968-Oct. 1969 1,006 3500 Lo72 02 1976 87 2.30 1.09 09 IG)7/ iL 117 3A Io 7s ||) O3 1976 99 Dofit ite i3 10 1971 117 3534 2.02 ||| @4 1976 72 ADS 1533 1l IG) 7/ it 78 Bo ilo 71 06 1976 65 206 1 @il 12 1971 120 3539 1.94 10 1976 113 Zo lola 01 1972 82 Zod WoYO i il 1976 113 BoA 1a Ue 02 1972 116 3.81 2.03 12 1976 109 2503 1.26 03 1972 123 293 Uo Ol 1977 89 2590 lols 04 UO) 72 110 3.10 1.67 02 1977 98 2ZoAl 118 05 1972 120 Jol U653) |} O3 1977 118 234 O.93 06 1972 101 2.15 0.98 || 04 1977 93 1.97 O.78 08 1972 88 Do Ah 1,33 |\| OS 1977 82 1560 Oo7/2 Annual Sept. 1971-Aug. 1972 1,173 3olQ 17 06 1977 91 1.68 0.82 09 1972 106 3.06 2.03 || 08 1977 88 Lo 40) 0)559) 10 IQYZ 109 3007 Wo 12 1977 93 3620 1,40 ib 1972 96 53 40) {I} Ol 1978 100 3526 Wo2il 12 1972 97 29 Wal 03 1978 82 3.48 1523 O01 UG) 7/3} 97 3539 ool 02 1973 92 4.43 2.09 03 UTS) 114 443333 2ZoBS 05 1973 89 2o33 Io OO Annual Sept. 1972-May 1973 854 3555 1,93 From the Figure or equation (3), H ae, UWA paste Age = G = 3-13, Ss H. = 9.4 feet (2.9 meters). Ss (b) The exceedance percentage is eo x 100 = 0.0114 percent. From the Figure or equation (3), i Hs Hg Sy, > 85: H, = 10.7 feet (3.3 meters). Check to see if H, exceeds depth-limited height. ides H, 10.7 From Figure 2-66 in the SPM, depth-limited breaking may be possible if H, IG 0.0172 eT This condition corresponds to janis (a ‘ SN ORO a NOsOlI2 2 22.2 7 4.4 seconds. Since a period of 4.4 seconds or less is unreasonably short for a 10./7-foot- high wave at this site (see Thompson, 1977)3, depth-limited breaking is not expected to be a consideration in this example. kok kk & KK KK RK & K X & EXAMPLE PROBLEM 2 * * * *¥ & XK RX KR KX KK KK GIVEN: Mean significant height of approximately Hg = 3.4 feet (1.0 meter) in February at Nags Head, North Carolina (see Table). FIND: The significant height which is equaled or exceeded during 6 hours every February. SOLUTION: The exceedance percentage is ears x 100 = 0.89 percent 3THOMPSON, E.F., op. cit., p. 9. From the Figure or equation (3), jae) i =: Ss Boe ae 2oAll 5 8.4 feet (2.6 meters). 2") 7) Ml VI. SUMMARY A three-parameter model for the distribution of significant wave height is given. A procedure for using available data from a site to compile a dimension- less distribution of significant height and to estimate parameters in the model is presented. The procedure extends the use of available data and leads to a model which more closely follows the data than procedures given in the SPM. The procedure is applied to shallow-water gage data from Nags Head, North Carolina. “cobain AD athens. o nit & tesa deh aloe te geten tod, ‘7 + Gn Ne ane rind J a | ‘- i? revat tas tay, oe ‘ y 7 th WG ay, ae ae . oo — 7 | I Be, |.) ” ln an S:." sieigrem é ey des? 8, “at a i i” . eae we at ie ial bint ye ‘oo o 7 at MAgtsil avnw 1) mrs Ve ‘pate “fi ions tie abe oft ot e322 sino) e Bre Tie Snkadtinyts oo. 2 & 69 shail hae note oda tar he me aii te wtrw sis ry aA? sMeZ ott ok nevis: mL ete f° | rhe) sgt ete” Wits eee reels bibiaaed sis vol bask a; oh oss ® ate oGhR ween sig “no karina oa bi: ‘ 1 ee Uae aa 7 A ‘4 "i i iN > on a’ i } oo - } y = , f i i ry H si te i ’ 3 , ] ah 4 ‘ i F ’ hi i 1 i i i; Y \ 1 1 i is ease we. has be Se Gee Be eneny nes € ei Ap. Ut eaten al i re a fon ai " x ‘ : 4 \ , Ph) Bre Wy i a Kikh weve At 5 eek. tae ee OLIVER eee RP ah a yale -3 eh } j " 4 - if if ‘ , f ie i ‘y I f ci ql j ae 7: 4 aya 4 * tee Are § ar ~ ’ f * ey An pel. rh ne t e MH bd fi iva 8 b, : i THOMPS ow * iy ? APPENDIX METHOD FOR ESTIMATING PARAMETERS IN THE WEIBULL DISTRIBUTION FUNCTION The Weibull distribution function, equation (1), can be transformed into a form suitable for linear regression analysis. First, the natural logarithm of equation (1) is taken H.. - H __\% On Pe _ (is i sc si (A-1) Both sides of equation (A-1) are multiplied by -1, and again natural logarithms are taken on (- ’n F) = on (isc = Bsc nin) (A-2) Hc Equation (A-2) can be rewritten as fim (oe > Hye atin) = 8 Hse $5 Qn (- £n F) (A-3) Equation (A-3) is in the form Yeatp (A-4) where i ie in(Hs. = lee ‘ita a = &n Hee ie! X = Mn (© Qn F). An initial value of the parameter Hgce min was obtained from Table 1 of Thompson and Harris (1972) as the "minimum significant height" divided by the observed mean significant height. The value for Nags Head was 0.31. Alterna- tively, Hsc min could be estimated initially as 0.38 from equation 4-8 in the SPM. The estimated value of Hge min and empirical tabulation of F asa function of Hg, are used to compute a table of X and Y values. Linear regression analysis is then used to estimate optimum values of a and b in 4THOMPSON, E.F., and HARRIS, D.L., "A Wave Climatology for U.S. Coastal Waters," Proceedings of Offshore Technology Conference, May 1972, pp. 675-688 (also Reprint 1-72, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., NTIS AD 746 365). 15 equation (A-4). Values for the parameters Hs, and a in equation (1) are easily calculated from a and b by using the relationships defined for equation (A-4). The distribution function given by equation (1) with the estimated para- meters Hoo min» Hsc; and a is compared with the empirical distribution. A new value of Hgc min is estimated to attempt a better fit to the empirical distribution. The new Hgce min is used to compute a new table of X and Y values which is then used to estimate new values of Hc and a as before. The process is continued until an Hgce min has been found which leads to a satisfactory model. The parameters for Nags Head in equation (2) are con- sidered satisfactory because they specify a distribution function which fits the lower 99 percent of the empirical distribution reasonably well but is con- servatively high in comparison to the highest 1 percent of the empirical dis- tribution. 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