Coastal Engneerng Techncal Note CETN V-10 Even-Odd Functon Analyss of Shorelne Poston and Volume Change by Jule Dean RoSllt and Nchollls C. Kraus Purpose: To present the background and methodology for separatng shorelne poston and volume change data nto symmetrc (even) and ant-symmetrc (odd) functons. Applcablty and nterpretaton of the even-odd functon analyss for engneerng applcaton s also dscussed. As appled to nlet data sets, the rates of shorelne and volume change whch occur symmetrcally about an nlet (e.g., storm eroson, eroson and accreton due to relatve sealevel change) can be dstngushed from those changes whch are ant-symmetrc (e.g., updrft mpoundment, downdrft eroson). Background: Even-odd functon analyss s a drect and easly appled method for examnng shorelne (or volume change) data. The power behnd the method s ts capablty to unambguously separate shorelne poston and volume changes whch are symmetrc (even) about a chosen pont along the shorelne from those that are ant-symmetrc (odd). An even functon.fe (x) does not change sgn f ts argument changes sgn,.e.,.fe (-x) =.fe ( +x); an odd functon/, (x) does change sgn.fc, (-x) = -.fo (+x). The sum of the even and odd functons reproduces the orgnal data. Because of ts ease n applcaton, the method s becomng popular for determnng the alongshore extent of nlet mpacts, such as s requred for Secton 111 studes. However, the method requres engneerng judgement n ts nterpretaton, as dscussed heren. As an llustratve and dealzed example, assume that a par of mpermeable jettes are constructed along a sandy beach wth a sgnfcant net longshore transport. From tme t=o to t= 1, gnorng possble nlet-nduced adjacent beach losses, the left jetty mpounds materal updrft, wth a commensurate eroson pattern downdrft of the rght jetty (Fgure 1a). The even-odd functon method appled to ths data set wth the center pont at the centerlne of the nlet yelds an odd functon sgnature that returns to a neglgble value at the alongshore lmt of the jetty system's mpact (Fgure lc). Ths example llustrates the most common nterpretaton of the even and odd functon analyss, that the odd functon s an ndcator of the alongshore extent of a jetty system's mpact. However, suppose that the same jettes were US Army Engneer Waterways Experment Staton. Coastal and Hydraulcs Laboratory 3909 Halls Ferry Road. Vcksburg, Msssspp 39180-6199
Report Documentaton Page Form Approved OMB No. 0704-0188 Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for nformaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 1204, Arlngton VA 22202-4302. Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE DEC 1997 2. REPORT TYPE N/A 3. DATES COVERED - 4. TTLE AND SUBTTLE Even-Odd Functon Analyss of Shorelne Poston and Volume Change (CETN V-10) 6. AUTHOR(S) Rosat, Jule Dean, and Kraus, Ncholas 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNT NUMBER 7. PERFORMNG ORGANZATON NAME(S) AND ADDRESS(ES) US Army Engneer Waterways Experment Staton, Coastal and Hydraulc Laboratory, Vcksburg, MS 8. PERFORMNG ORGANZATON REPORT NUMBER 9. SPONSORNG/MONTORNG AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONTOR S ACRONYM(S) 12. DSTRBUTON/AVALABLTY STATEMENT Approved for publc release, dstrbuton unlmted 13. SUPPLEMENTARY NOTES 11. SPONSOR/MONTOR S REPORT NUMBER(S) 14. ABSTRACT Ths Coastal Engneerng Techncal Note (CETN) presents the background and methodology for separatng shorelne poston and volume change data nto symmetrc (even) and ant-symmetrc (odd) functons. Applcablty and nterpretaton of the even-odd functon analyss for engneerng applcaton are also dscussed. As appled to nlet data sets, the rates of shorelne and volume change whch occur symmetrcally about an nlet (e.g., storm eroson, eroson and accreton due to relatve sealevel change) can be dstngushed from those changes whch are ant-symmetrc (e.g., updrft mpoundment, downdrft eroson). 15. SUBJECT TERMS 16. SECURTY CLASSFCATON OF: 17. LMTATON OF ABSTRACT UU a. REPORT unclassfed b. ABSTRACT unclassfed c. THS PAGE unclassfed 18. NUMBER OF PAGES 8 19a. NAME OF RESPONSBLE PERSON Standard Form 298 (Rev. 8-98) Prescrbed by ANS Std Z39-18
. ~ CETN V-10 / / e\ r:"" ' ;!! a:::: :c~: ws 2:5:. 0 extent 0~ nfluence (a) (b) even (c) odd Fgure 1. Example of jetted nlet on coast wth sgnfcant net longshore sedment transport. constructed along a: sandy shore wth a near-zero net longshore sedment transport, but a sgnfcant gross longshore sedment transport. The jettes would mpound materal on both the left and rght, wth eroson at some dstance away outsde ther shadow zone (Fgure 2a). The even-odd functon method appled to ths example yelds a neglgble odd functon, because all shorelne changes were symmetrc. However, the jettes obvously mpacted the shorelne. For ths case, determnaton of the extent of mpact of the jettes s reflected by the zone at whch the even functon approaches a neglgble value (Fgure 2b). Thus, accurate nterpretaton of the analyss requres knowledge of ste (e.g., drecton and relatve magntude of net and gross longshore sedment transport rates) and structure (e.g., permeablty) condtons. t s reterated that, although the even-odd functon analyss s unambguous, ~. c eve" ode 0 0 E). :e:-- c (a) (b) even (c) odd Fgure 2. Example of jetted nlet on coast wth zero net longshore sedment transport. 2
CETN V-10 consderable coastal engneerng judgement may be requred n nterpretaton of the results. A lstng of references dscussng even-odd functon applcatons s provded at the end of ths Techncal Note. Procedure: Shorelne (or volume) change, denoted as f (x), between two tme perods at some alongshore poston x, can be represented by even (symmetrc)/. (x) and odd (ant-symmetrc).fo (x) functons, f(x) = fe (x) + fo (x) (1) The even functon s determned as f. (x) = f(x) + /( -x) e 2 (2) and the odd functon s gven by f. (x) = f(x) - /(-x) 0 2 (3) \,.._... t s readly verfed that substtuton of Equatons 2 and 3 nto Equaton 1 produces an dentty. Note that the even-odd functon method can only be appled for the length of shorelne for whch both/ (x) andf (-x) are avalable. Equatons 2 and 3 are appled n the followng example problem. Example Problem: Table 1 presents shorelne change rate data from July 1929 and May 1933 (pre-nlet) to May 1996 (post-nlet and jetty constructon) for the shorelnes adjacent to Ocean Cty nlet, Maryland, whch s located on the east coast of the Unted States. The nlet was created when a hurrcane broke through the exstng barrer sland on August 23, 1933, and jettes were constructed from September 1933 through May 1935. Ths ste has a sgnfcant net longshore sedment transport to the south (from top to bottom n Table 1), wth mpoundment at the updrft jetty and sgnfcant eroson of the downdrft beach occurrng durng ths tme perod. Shorelne change data are gven n 500-m ncrements, extendng from 15 km updrft (north) of the nlet to 14 km downdrft (south) of the nlet; however, the evenodd functon analyss was conducted for the regon for whch both/ (x) andf (-x) were avalable ( + 14 km). The July 1929 and May 1933 shorelnes were dgtzed from topographc maps (T-sheets) created through a cooperatve Natonal Ocean Servce (NOS)- Coastal Engneerng Research Center study. The May 1996 shorelne was derved from a Global Postonng System-controlled feld survey of the foreshore. Shorelne data sources and ther relatve accuracy are dscussed n CETN-11-39 (Kraus and Rosat 1997). 3
Tablet Shorelne Change Rate Data for Shorelnes Acjacent to Ocean Cty nlet, Maryland: Nov 1929 and May 1933 to May 1996 Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 Column 7 ColumnS Column 9 Column 10 (x} S(x} De B Ad f(x} (-x} f(-x} (f(x)+f( -x)v2 (f(x}-f(-x})/2 D1stance Ocean Ocean Ocean Ocean Ocean nverted nverted from Center Shorelne Depth of Actve Actve Vol Chg D1stance Vol Chg Even Odd of nlet Change Closure Benn Elev Depth Rate from nlet Rate Functon Functon (km} Rate (m/yr} (mngvd} (mngvd} (m} (m3/m/yr} (km} (m3/m/yr} (m3/m/yr} (m3/m/yr} -15-0.47 6.1 3 9.1-4.3-14.5-0.32 6.1 3 9.1-2.9-14 0.11 6.1 3 9.1 1.0 14-2.0-0.5 1.5-13.5 0.17 6.1 3 9.1 1.5 13.5-2.6-0.5 2.1-13 0.27 6.1 3 9.1 2.5 13-2.2 0.2 2.3-12.5 0.24 6.1 3 9.1 2.2 12.5-4.0-0.9 3.1-12 0.07 6.1 3 9.1 0.6 12-4 -29 3.5-11.5 0.05 6.1 3 9.1 0.5 11.5-84 -4.0 4.4-11 -0 46 6.1 3 9.1-4.2 11 -.8-5.5 1.3-10.5-0 18 6.1 3 9.1-1 6 10 5 -.2-3.9 23-10 -0.02 6.1 3 9.1-0.2 10 -.5-3.4 3.2-9.5 0.57 6.1 3 9.1 5.2 95-7.3-1.1 6.2-9 0.07 6.1 3 9.1 0.6 9-8.9-4.1 4.7-8.5-0.49 6.1 3 9.1-4.5 8.5-14 3-94 4.9-8 -0.4 6.1 3 91-16 8-16 3-10.0 6.4-7.5 0.41 6.1 3 9.1 3.7 7.5-20.1-8.2 11.9-7 0.5 6.1 3 9.1 4.6 7-21.2-8.3 12.9 -.5 0.37 6.1 3 9.1 3.4 65-19 4-8.0 11.4-0.55 6 1 3 9.1 5.0 6-24.1-9.5 14.5-55 0.63 6 1 3 9.1 5.7 5.5-35.7-15.0 20.7-5 0.37 61 3 9.1 3.4 5-38.6-17.6 21.0-4.5 0.44 6.1 3 9.1 4.0 4.5-401 -18.1 22.1-4 0.74 6 1 3 91 6.7 4-46.1-19.7 26.4-3.5 0.88 61 3 9.1 8.0 35-49.0-20.5 28.5-3 0.58 6 1 3 9 1 5.3 3-51.9-23.3 28.6-2.5 0.92 61 3 9.1 8.4 2.5-56.3-239 32.3-2 0.97 6.1 3 9.1 8.8 2-56.7-23.9 32.8 1 12 6 1 3 91 10.2 1.5-51.6-20.7 30.9 _, 5 _, 2 61 3 91 18.2 1-41.6-11.7 29 9-0.5 3.42 61 3 91 31.1 0.5-33.4-1.1 32.3 0 0 51 3 9 1 00 0 00 00 00 05-3.93 61 24 85-33 4-0 5 31 1 _, 1-32 3 18 2 _, 1.7-29.9 1 5-6.07 51 2 4 8.5-51 6 1 5 10 2-20.7-30.9 ~ -4 89 "1 2 4 8.5-41.6 2-6.67 61 24 8.5-56 7-2 88-23.9-32 8 25-6 62 6.1 24 8.5-56.3-2.5 8.4-23.9-32.3 3-6.11 61 24 8.5-51.9-3 53-233 -28.6 35-5.76 61 24 8.5-49.0-3.5 8.0-20.5-28.5 4-5.42 61 2.4 8.5-46.1-4 6.7-19.7-26.4 4.5-4.72 6.1 24 8.5-40.1-4.5 4.0-18.1-22.1 5-4.54 6.1 2.4 8.5-38.6-5 3.4-17.6-21.0 5.5-4.15 6.1 25 8.6-35.7-5.5 5.7-15.0-20.7 6-2.8 6.1 2.5 8.6-24.1-50 -9.5-14.5 65-2.25 6.1 2.5 8.6-19.4 -.5 3.4-8.0-11.4 7-2.47 61 25 8.6-21.2-7 4.6-8.3-1-2.9 7.5-2.34 6.1 2.5 8.6-20.1-7.5 3.7-8.2-11.9 8-1.9 6.1 2.5 8.6-16.3-8 -3.6-10.0 -.4 85-1.66 6.1 2.5 8.6-14.3-8.5-4.5-9.4-4.9 9-1.03 61 2.5 8.6-8.9-9 0.6-4.1-4.7 95-0.85 61 2.5 8.6-7.3-9.5 5.2-1.1 -.2 10-0.76 6.1 2.5 8.6 -.5-10 -0.2-3.4-3.2 10 5-0.72 6.1 2.5 8.6 -.2-10.5-1 6-3.9-2.3 11-0.79 6.1 2.5 8.6 -.8-11 -4.2-5.5-1.3 11.5-0.98 6.1 2.5 8.6-84 -11.5 0.5-4.0-4.4 12-0.74 6.1 2.5 8.6 -.4-12 0.6-2.9-3.5 12.5-0.47 6.1 2.5 8.6-4.0-12.5 2.2-0.9-3.1 13-0.25 6.1 2.5 8.6-2.2-13 2.5 0.2-2.3 13 5-0.3 6.1 2.5 8.6-2.6-13.5 1.5-0.5-2.1 14-0.23 6.1 2.5 8.6-2 0-14 1.0-0.5-14.5-2.9-15 -4.3 _, CETN V-10 _, 5 4
CETN V-10 The alongshore coordnate x = 0 was chosen as the center of the nlet, wth negatve values ndcatng dstances updrft and postve values ndcatng dstances downdrft from the nlet, respectvely (column 1). The shorelne change rates have not been adjusted to remove the volume of beach fll placed durng ths tme perod. The actve depth (Ad n m, column 5) s gven as the sum of the depth of closure (De n m, column 3) and the actve berm crest (B n m, column 4): (4) The shorelne change rate (S (x) n m/year, column 2) has been converted to a volume change rate per unt length of beach (/ (x) n m 3 /m/year, column 6) by multplyng by Ad, whch ranges from 8.5 to 9.1 m (see column 5): f(x) = S (x) Ad (5) For the Ocean Cty data set, volumetrc change rate data are preferred over shorelne change rate data to account for A 11, whch vares alongshore, and to account for beach flls (dscussed n the next secton). Note that the volume change rate per unt length of beach,/ (x) (n unts of m 3 /m/year), s readly converted nto a volume change rate at each cell (n unts of m 3 /year) by multplyng by the cell wdth, W = 500 m. The frst step n the even-odd functon analyss procedure s to nvert x and/ (x) about x = 0, producng values for -x (column 7) and/ (-x) (column 8). The even (column 9) and odd (column 10) functons can then be calculated usng Equatons 2 and 3, respectvely. Fgure 3 shows/ (x) (noted as "total" n the legend),/e (x), andfo (x). -... <- 40- (;) -+ : ~,r Al E 1-- E 20 --~. r! -l.-----t--t--1--r---- - /:, '! Q) J-"-.,11 ~ o~ ;~ ;, ~-vg_2o --~J--+:--7-'~~h~+~~/-*4\~~1/~/~~/--r-,_- ro! :! '>.../ '--V..c J ( o o! ~ --. lj 4 --~~--~~-4--+\~r-~-+--r-,_- Q) :! \! J E V ~ -60-'- l > -15-10 -5 0 5 10 15 Dstance from Center of nlet (km) ----Total-- Even - Odd Fgure 3. Example problem: Ocean Cty even-odd functon analyss, 1929/1933 to 1966 5 For ths ste wth a sgnfcant. net longshore sedment transport rate (approxmately 115,000 m 3 /year at present), the alongshore pont at whch the odd functon returns to a zero value s an ndcator of the alongshore dstance nfluenced by ant-symmetrc processes, such as mpoundment, whch s a projectnduced mpact. The alongshore pont at whch the even functon approaches a constant value s an ndcaton of the alongshore dstance nfluenced by symmetrc processes, such as storm mpacts and nlet-nduced mpacts such as shorelne retreat due to feedng
CETN V-10 the ebb and flood tdal shoals. For the regon of shorelne for whch the even functon approaches a constant value, the even functon can be nterpreted to represent the "background" shorelne (or volumetrc) change rate (e.g., sea level rse, nfluence of storms, etc.). From a cursory observaton, because the odd functon does not return to a zero value, t appears that the nlet's extent of nfluence n terms of mpoundment extends beyond the longshore extent of the data lmts. Qualtatvely, the even functon appears to approach a constant value at x = 9 to 12 km from the center of the nlet. However, two other factors must be consdered. Frst, as mentoned prevously, the data presented n Table 1 and Fgure 3 were not corrected to adjust the 1996 shorelne poston for beach fll. For ths partcular ste, ths adjustment s mportant due to the volume of dredged materal placed on the downdrft beach (a total volume of approxmately 2.9 mllon m 3 ), and the June 1988 through Aprl 1995 beach fll project on the updrft beach (a total volume of approxmately 7.4 mllon m 3 placed from the north jetty at -0.2 km to -14 km). To account for the volume of beach fll placed, a three-step process was used. Frst, the 1929/1933 to 1996 volume change rate (n unts of m 3 /year) was converted to a volume change (unts of m 3 ) by multplyng by the number of years (63 years). Second, the volume of beach fll estmated to be remanng n 1996 at each alongshore cell was subtracted from the 1929/33 to 1996 volume change. Ths estmaton was based on analyss of seven profles from June 1988 (mmedately pre-fll) and May 1996 (present-day condton) for the updrft beach. From ths analyss, t was estmated that the updrft beach fll lost materal at a rate of approxmately 2 percent per year. Ths rate was used for all beach fll placed, resultng n estmates -,_ - E 60 >. -E 4o..._..... Q) 20 ro 0:: 0 Q) 0>-20 c ro.c -40 (..) Q) -60 E ::l -80 0 '! :!! _/- : '"'- /,.,' r1! ( ~ A,.,) ' ""- ~~.A.: f""' r "\ '... " --,.,...,..! > -15! v~ J ~.J.r-1-../ '1-- / G... L--,... ' \ : \ r' 'odd: -8.3 to 10 km ' \ / even: - 11.4 km 1--- -- -- -1 0-5 0 5 1 0 15 Dstance from Center of nlet (km) - - Total - Even - Odd Fgure 4. Ocean Cty even-odd functon analyss, 1929/1933 to 1996, adjusted for beach fll of 0. 72 mllon m 3 remanng on the downdrft beach, and 5. 8 mllon m 3 remanng on the updrft beach. Fnally, the shorelne change rate accountng for beach fll was calculated by subtractng the fll remanng from the volume change at each cell, whch was then converted back nto a volume change rate (unts of m 3 /year). Fgure 4 shows the total, even, and odd functons for the 1929/33 to 1996 volumetrc change rate as 6
CETN V-10 adjusted to remove the volume of beach fll placed (note that these data are presented for the full data set wth cell wdths W=50 m). The second consderaton concerns confdence n the accuracy of the shorelne data set (see CETN-11-39). The shorelne postons are estmated to have an uncertanty or tolerance of + 10 m, based on consderaton of four potental error sources: accuracy of shorelnes plotted on NOS T-sheets, photo-nterpretaton of the hgh-water lne, lne thckness of shorelnes on the maps, and equpment/ operator error ncurred durng the dgtzaton process. Placng the uncertanty estmate nto Equaton 5 results n a "data-capture estmate" of+ 1.4 m 3 /m/year. The mnmum dstance at whch the odd functon equals the negatve value of the data-capture estmate ( -1.4 m 3 /m/year) s approxmately 8. 3 km from the center lne of the nlet, and the dstance represented by the postve value ( + 1. 4 m 3 /m/yr) s approxmately 10 km from the center. lne of the nlet. The even functon approaches a constant value at the dstance for whch the standard devaton of ths functon at greater dstances equals the data-capture error estmate. For the 1929/33-1996 data set, the even functon approaches a constant value approxmately 11.4 km from the center lne of the nlet (.e., the standard devaton ofthe even functon from 11.4 km to the data lmt equals± 1.4 m 3 /m/year). Thus, analyss of shorelne response data from 1929/33 to 1996 ndcates that the nfluence of the Ocean Cty nlet has extended from 8. 3 to 11.4 km from the center of the nlet. Addtonal nformaton: A comprehensve bblography s provded for the reader to consult for addtonal dscusson of nterpretaton of the even-odd functon method. Questons about ths CETN can be addressed to Ms. Jule Dean Rosat (601-634-3005, Fax 601-634-4314 '~ emal: jd.rosat@cerc.wes.army.ml) or Dr. Ncholas C. Kraus (601-634-2016, Fax 601-634-2055, emal: n.kraus@cerc.wes.army.ml). A spreadsheet verson of the example problem s avalable from the authors. Bblography Berek, E. P., and Dean, R. G. (1982). "Feld nvestgaton of longshore transport dstrbuton." Proc. ll" Coastal Engneerng Conference, ASCE, NY, 1620-1638. Dean, J. L., and Pope, J. (1987). "The Redngton Shores breakwater project: ntal response." Proceedngs Coastal Sedments '87. ASCE, NY, 1369-1384. Dean, R. G., and Work, P. A. (1993). nteracton of navgatonal entrances wth adjacent shorelnes," Journal of Coastal Research 18: 91-110. Douglas, B. D., and Walther, M.P. (1994). "Odd/even analyss modfed to account for sedment losses to ebb and flood shoals." Proceedngs 1994 Natonal Conference on Beach Preservaton Technology. FSBPA, Tallahassee, FL, 354-373. Kraus, N. C. "Dstngushng cross-shore and longshore processes n shorelne change evaluaton." Proceedngs 1997 Natonal Conference on Beach Preservaton Technology. n press, FSBPA, Tallahassee, FL. ~. 7
CETN V-10 Kraus, N. C., and Rosat, J. D. 1997. "nterpretaton of Shorelne-Poston Data for Coastal Engneerng Analyss," Coastal Engneerng Techncal Note CETN-11-39, U.S. Army Engneer Waterways Experment Staton, Vcksburg, MS. Rosat, J. D., and Ebersole, B. A. "Lttoral mpacts of Ocean Cty nlet, Maryland, USA." Proceedngs 25'h Coastal Engneerng Conference. n press, ASCE, NY. Work, P. A., and Dean, R. G. (1990). Even/odd analyss of shorelne changes adjacent to Florda's tdal nlets." Proceedngs 2(Jh Coastal Engneerng Conference. ASCE, NY, 2522-2535. 8