ROLE OF SYNTHETIC STORMS ON PEAK FLOW ESTIMATION

Transcription

1 1 ROLE OF SYNTHETIC STORMS ON PEAK FLOW ESTIMATION Abstract BETÜL SAF Departent of Hydrology&Hydraulic&Water Resources Paukkale University Kınıklı Capüs, Denizli TURKEY In this study, synthetic annual axiu stors distributed as Gubel with rando effective stor durations and specific tie distribution of precipitation for a given population ean and variance are generated by Monte Carlo ethod. Population eans and variances are defined by a three paraeter nonlinear regression odel and by a cubic logarithic odel, respectively. Effective durations of the synthetic annual axiu stors are related to basin characteristics, length and haronic slope of the ain course through Kirpich s tie of concentration relationship. Annual peak flow series of the hypothetical basins with physiographic characteristics are generated through a deterinistic rainfall-runoff odel, naely the Soil Conservation Service synthetic unit hydrograph ethod and discrete convolution technique. Saple statistics and frequency distributions of the generated annual axiu stors of rando effective durations and of the resulting peak flow series are investigated. Eight well-known probability distribution odels with oent and axiu likelihood paraeters are tested by chi-square and probability plot correlation goodness of fit tests. Key-words: synthetic rainfall, onte carlo, annual peak flow, SCS abstraction, frequency distribution, goodness-of-fit. 1. Introduction The design of water resources structures for use in ungaged basins requires soe estiate of flood flows and their frequency of occurence. If there is no historical streaflow data for these watershed areas, regional flood frequency analysis, or paraetric rainfall-runoff event siulation can be used to estiate floods. Available precipitation data with axiu value are used as input variables in paraetric rainfall-runoff odels. If both rainfall and runoff data are not available, rando input variables are generated by the Monte Carlo ethod and frequency analysis techniques are used to analyze output variables. Soeties rainfall data recorded at eteorological stations ay be uch longer than corresponding streaflow observations. Hence, frequency odels fitted to extree rainfall data are ore reliable than the frequency odels of peak flows. This advantage can be transferred into iproving or perforing decisions on the appropriate frequency distribution odel for the peak flows. Of course, occurrences and frequencies of peak flows are also dependent on the tie distribution or pattern of the rainfall, basin physiography, vegetation, land anageent, and antecedent oisture conditions of the watershed. Methods of generating design stors are available and in wide use, but they are general in nature and assue stors occur with the sae teporal distribution over a wide area. Because of extree cliatic differences between the areas, design curves are not likely to be representative of the actual tie distribution of stors in sei-arid regions. For exaple, [12] decided to develop a new design stor generation procedure applicable to the State of Wyoing based on observed stor rainfall in Wyoing. Synthetic stor series, with siilar statistical properties to observed series, were developed by [17]. Data of the Alcorta raingauge (with a 5-year series of data) were analyzed by eans of five variables: duration of the rain, tie between events, average and axiu intensity of the rain, and stor advance coefficient. The variables were classified as independent (the first three variables) and dependent (the last two variables). Probability distribution functions were fitted to the independent variables. Multiplicative relationships were proposed for dependent variables and their coefficients were adjusted. The statistical characteristics of the synthetic series were calculated and copared with the observed data series. A good agreeent between calculated and observed series was obtained [17].

2 2 2. Scope of the Study The ajor objectives of this study were to find satisfactory answers in the following questions and to give an insight into the proble of rainfall-runoff transforation on a probabilistic basis. (1) Provided that the stor events of different durations have the sae type of probability distribution (say, Gubel), does the type of the probability distribution of ixed stor events reain unchanged? (2) Does, or how does, a rainfall-runoff transforation process transfer the distributional characteristics of rainfall events into the output (peak flows)? (3) Are there significant and physically eaningfull relations between the statistics of input and output? (4) The stor events which are responsible for the annual peak flows ay be drawn fro different populations. Therefore, return periods of input and output ay or ay not be the sae. In order to find answers to the above given questions, a Monte Carlo siulation is followed in the study. Twenty-five sets of synthetic stors of different durations each N=100 size are generated. Kirpich s epirical relationship between the tie of concentration of the sall watershed and the length and slope of the ain course is used for deciding the critical rainfall durations over the hypothetical watersheds. Synthetic stor inputs are transfored into peak flows by following Soil Conservation Service (SCS) triangular unit graph and discrete convolution procedures. Statistical descriptors of both input (rainfall) and output (peak flow) series and paraeters of various probability distribution odels by the ethods of oents and axiu likelihood are estiated. Tests for the goodness of fit of those distribution odels by the chi-square test, as well as the probability plot correlation test, are perfored and frequency of acceptance of each odel is found. Relations between the types of probability distributions of the input and the output are investigated on the basis of relative acceptance frequencies of the postulated probability distribution odels. 3. Methodology 3.1 Probability Models, Paraeter Estiations and Goodness of Fit In this study; the noral (NOR), lognoral with two- and three paraeter (LN2 and LN3), gubel (GUM), loggubel (LGUM), gaa with two-and three paraeters (G2 and G3), and the log-pearson III distributions (LP3) are used for synthetically generated stors and peak flows which are generated fro these stors by using a rainfallrunoff odels. Nuerous ethods of estiation are available and two ost coon approaches that ethod of oents (MOM) and axiu likelihood (ML) ethods are used in this study. Detailed inforation about these probability odels and paraeter estiation ethods are given by [3] and [8]. The selection of a distribution type is crucial and propounds a fundaental challenge to hydrologists of the flood frequency approach. For any cases, custo and convenience often play a significant role in the selection of a flood distribution. In this study, chi-square (χ 2 ) and probability plot correlation (PPC) test were used for goodness of fit. 3.2 Synthetic Unit Hydrograph Deterination of the unit hydrograph of a basin is very iportant for the design of water structures, because it gives inforation about peak discharge, tie of the peak discharge, and duration of excess runoff. When it is necessary to deterine a unit hydrograph for a ungaged basin, therefore, one of the synthetic unit hydrograph deterination ethods is used. Synthetic unit hydrographs can be estiated for ungauged drainage basins by eans of relationships with paraeters of characteristics of the drainage basin. The ost coonly used ethods are the Snyder (1938), the Mockus (1957), and U.S. Soil Conservation Service (SCS) (1972) ethods. In this study the Mockus synthetic unit hydrograph ethod was used.

3 3 4. Coputational Algorith for Generation Synthetic Stors and Peak Flows 4.1 Generate Synthetic Stors Tie of concentration, effective stor duration and unit duration A very siple event-based rainfall-runoff odel is applied to a 100 k 2 hypothetical watershed area without baseflow. Durations of synthetic stors conforing basin lag are generated by considering that the length (L) and haronic slope (S) of the ain channel are doinating factors on the tie of concentration. The effective stor duration, D e, is given as [2] and [10]; D e =2(t c ) 0.5 (1) t c = (L) 0.77 /S (2) where L is the length of the ain course in eters, S is the haronic slope, t c is the tie of concentration of the watershed and D e is the effective stor duration, both in hours. Tie of concentration (t c ) according to Kirpich s forula Eq.2 for various lengths of ain course (L) and haronic slopes (S) are given in Table 1. For t c 4 hour, effective stor durations are assued to be equal to tie of concentration (D e =t c ). Table 1 Tie of Concentration (t c ; hour ) According to Kirpich s Forula for Various Lengths of Main Course (L) and Haronic Slopes (S) S (Haronic Slope) L() A typical rainfall hyetograph is a coposition of M nuber of discrete pulses of t tie increents. Therefore the critical rainfall duration, D, generated in a rando anner fro a unifor distribution in the range D e D 2D e is rounded off as ultiples of t: D = M. t (3) In order to siplify calculations of coposite hydrograph M is assued as 5, 6, or 7. This assuption is in accordance with unit graph durations used in hydrologic practice, 0.15t c t 0.20t c Type-I Extreal Distribution of Annual Maxiu Stors with Asyptotic Moent Paraeters In this study, the ean of rainfall depth-rainfall duration and standard deviation of rainfall depth-rainfall duration relationships of Uşak Meteorological Station developed by Benzeden (2001) are used as population statistics of the synthetic stors. The ean of rainfall depth-rainfall duration and the standart deviation of rainfall depth-rainfall duration relationships are as follows: [ ( ) ] M D = ln D (4)

4 2 [ ln( D) ln ( D) ln ( D) ] 3 S D exp + = (5) where D is in inutes and M D and S D both are in. According to these relationships, synthetic rainfall depths, Y D, distributed as the Extree Value Type-I (Gubel) with asyptotic paraeters are generated through Eq.6. Y D,T = M D +S D.K T (6) where K T = - { ln[-ln(1-1/t)]} (7) is the frequency factor corresponding to the return period T or probability of nonexceedance P T which can be calculated fro generated unifor rando nubers 0 <P T <1, that is T=1/(1-P T ) (Kite, 1977) Tie Distribution of the Generated Stors The SCS diensionless cuulative rainfall curves were developed for various stor types, stor durations and regions in the United States [10]. In this study, SCS 6-hour tie distributions are used in order to calculate rainfall hyetographs of given synthetic stors. A design stor, Y D, is divided into increents Y using an appropriate tie distribution curve for the project site. A tie distribution curve represents the cuulative percentage of the precipitated rainfall f = Y YD, during the percentage tie X = t D, where D=M. t is the rainfall duration and t = t (=1,2,...,M). Having f values, the cuulative rainfall aount (Y ) precipitated during period 0 to t, and increental rainfall aount ( Y ) can be coputed as: Y = fd, =1,2,...,M (8) Y = Y Y, =1,2,...,M (9) 1 where y 0 =0 for = Genaration Synthetic Peak Flows According to Synthetic Annual Maxiu Stors The SCS-curve nuber ethod with CN=90 is applied in order to account for the initial abstractions and its role on the excess rainfall hyetograph and coposite hydrographs. For the sake of siplicity, it is assued that the Mockus unit hydrograph is a sufficient tool to siulate the rainfall-runoff transforation process. The triangular unit graph [10] of a unit duration t includes the size of the basin drainage area, A, the length of the ain channel L, and slope, S because of the relation between tie to peak, t p, and tie of concentration, t c. t p q p = t tc = f1( L, S, t) (10) = KA t p = KA f1 ( s) = f2( A, L, S, t) (11) Ordinates of t-hour triangular unit hydrograph at ties t, 2 t, 3 t,... are used in calculating superposed output resulting fro given ERH of M nuber of increental excess rainfall pulses, R 1, R 2,... R M. 4 Q n = n M = 1 R U n + 1 (12)

5 5 5. Results 5.1 Probability Distribution of Mixed-Stor Durations When the results of χ 2 tests for 25 synthetic stor series are evaluated the ost suitable odel for the ixed-duration stor series cannot be deterined clearly (Table 3). Though LN2 distribution for the case with MOM paraeters has the highest acceptance frequency, GUM, G2, LN3 and G3 distributions are also copatible distributions. For the axiu likelihood paraeters, LN2, G2 and GUM distributions are the ost suitable distributions. On the other hand the PPC test results reveal that the ost suitable odels are LN2, G2, GUM, LN3 and G3 either with MOM or ML paraeters. 5.2 Relations Between the Types of Probability Distribution Functions In order to copare the probability distribution odels ost frequently accepted (that is, the goodness of fit test is passed) for the synthetic stor saples (input) and for the peak flows (output), relative acceptance frequencies of each odel, f I and f O, coputed fro Eq.13 and 14, at a significance level α=5% for the χ 2 test are presented coparatively with and axiu likelihood paraeters. Relative acceptance frequency of a specific odel (f I ) is defined as: f I = 100 (TNCH)/25 (13) 2 where TNCH is total nuber of series that passed the χ test for a specific distribution. Siilarly, using paraeters estiated by ethod of oents, relative acceptance frequencies of each odel at a significance level α=5% are presented coparatively. The results of the PPC tests are evaluated in the sae way except that the relative acceptance frequencies of the output series are calculated now fro Eq.14. f O = 100 (TNPP)/25 (14) where TNPP is total nuber of stor series that have a PPC coefficient greater than r c =0.95. The PPC results are shown in Fig.5 and 6. Relative acceptance frequencies according to the chi- test results with MOM paraeters LN2 is the ost appropriate odel. Siilarly, with ML paraeters; LN3 is the ost appropriate odel. For PPC test, it LN2 and LN3 odels are ost suitable both of paraeter estiation ethods. Table 3 Results of godness-of fit tests of synthetic stors (input) and peak flow series (output) with MOM and ML estiators (GOF:Goodness-of-fit, PEM:Paraeter estiation ethod) GOF ethods PEM Data NOR LN2 G2 GUM LGUM LN3 G3 LP3 Chi-square PPC MOM ML MOM ML Input Output Input Output Input Output Input Output

6 6 5.3 Relations Between Distributional Characteristics of the Input and Output Relation between the types of the probability distribution of input and output according to the chi-square tests are investigated on the basis of correlations (r) between the relative acceptance frequencies and are calculated with axiu likelihood paraeters according. The correlations between the acceptance frequencies of input and output are calculated. The input-output relationships are and for chi-square goodnes-of fit test with MOM and ML, respectively. When PPCC odel is used, input-output relationships are and for MOM and ML, respectively. As it can be seen, result of ML paraeter estiation ethod is better than the MOM. But, the relationship between input and output distribution is not good in both of case. 6. Conclusions and Recoendations As can be seen fro Table 3, when the twenty-five synthetic stor series were evaluated with chi-square goodness of fit test, the ost suitable distribution with MOM paraeters is LN2; and LN2, G2 and GUM with ML paraeters. This eans that when the generated synthetic stors distributed as Gubel are put into a ixed duration series the type of the appropriate distribution ay change. Siilarly, according to the results of probability plot correlation test given in Table 3, LN2, G2, GUM, LN3 and G3 distributions are the ost suitable distributions either with MOM or ML paraeters. These results of the study reveals that the probability distribution of the rainfall input ay even diverge fro their parent (Type-I Extreal) distributions because of the sapling, and since the generated input series is a ixture of rainfall events of variable durations. Provided that the stor events of different durations have the sae type of probability distribution, the probability distribution types of ixed stor events have not been seen to change drastically. The answer of the second question ay be given as, a rainfall-runoff transforation process doesn t transfer the distributional characteristics of rainfall events into the peak flows. The answer of third question is there aren t significant and physically eaningfull relations between the probability distribution of stor and peak flow. The last answer is the stor events which are responsible for the annual peak flows ay be drawn fro different populations. Therefore, return periods of input and output ay not be the sae. This study is based on various assuptions and siplifications and therefore the conclusions should be evaluated carefully. A rather coplicated and coprehensive research should be conducted by taking into account for the coplicated echanis of rainfall-runoff transforation process which ust include the storage effects, the nonlinearities in the watershed syste, and alternative probability distribution odels for the input. It is hoped that these conclusions of this study will be beneficial in sall basins where significant inforation on precipitations is available in case of liited inforation on peak flows. References [1] Benzeden E., Standart Süreli Maksiu Yağışların Frekans Analizinde Karşılaşılan Sorunlar, DSI Teknoloji D. Bşk. Bası Şb. Md., III. Ulusal Hidroloji Kongresi Bildirileri, Izir, p.11-18, [2] Chow, V. T., Handbook of Applied Hydrology, McGraw-Hill, [3] Kite, G. W., Frequency and Risk Analysis in Hydrology, Water Resources Publications, Fort Collins, Colorado, USA, [4] Lettenaier, D. P., & Burges, S. J., Gubel s Extree Value I Distribution: A New Look, Journal of Hydraulic Division, , [5] Linsley, R. K., Kohler, M. A., & Paulhus, J. L. H., Applied Hydrology, McGraw-Hill, New York, 1949.

7 7 [6] Marien, J. L., & Vandewiele, G. L., A Point Rainfall Generator with Internal Stor Structure, Water Resources Research, 22, , [7] Mockus, V., Use of Stor and Watershed Characteristics in Synthetic Hydrograph Analysis and Application, Aer. Geophys., Union, Southwest Region Meeting, Sacraento, Calif., [8] Rao, A. R. and Haed K. H., Flood Frequency Analysis, CRC Pres LLC, Boca Raton, Florida, [9] Saf, B.,Role of Rainfall Pattern and Basin Physiography on the Probability Distribution of Floods, Ph. D. Dissertation, Dokuz Eylül University, Turkey, [10] SCS, National Engineering Handbook, Section 4, Hydrology Soil Conservation Services, U.S., Departent of Agriculture, Washington, D. C., [11] Snyder, F. F., Synthetic Unit-Graphs, Transactions of the Aerican Geophysical Union,19, , [12] Tyrell, Patrick T., Developent of Design Rainfall Distribution for the State of Wyoing, Msc. Thesis, University of Wyoing, 71 pp., [13] Vogel, R. M., The Probability Plot Correlation Coefficient Test for the Noral, Lognoral and Gubel Distributional Hypthesis, Water Resources Research, 22, , [14] Vogel, R. M., & McMartin, D. E., Probability Plot Goodness-of-fit and Skewness Estiation Procedures for the Pearson Type 3 Distribution, Water Resources Research, 27, , [15] Wanieliesta, M., Kersten, R., & Eaglin, R., Hydrology: Water Quantity & Quality Control, John Wiley & Sons, [16] Wei, Tseng C., and C. L. Larson, Effects of Areal and Tie Distribution of Rainfall on Sall Watershed Runoff Hydrographs, Minnesota Univ., Minneapolis, Water Resources Research Center, Bulletin No. 30, 130 pp., [17] Yen, Ben Chie, and Ven Te Chow, Design Hyetographs for Sall Drainage Structures, Journal of the Hydraulics Division, ASCE, Vol. 106.,No. HY6., [18] Zierann, Erik D., Synthetic Stor Generatıon in a Flatland Region, Santa Fe, Argentina, Journal of the International Association for Environental Hydrology, 6, 2, 1998.