Bölgesel Taşkın Frekans Analizi

Abstract

Birçok hidrolik yapının planlanmasında ve taşkın yatağındaki riskin belirlenmesinde taşkın frekans analizi önemli bilgiler sağlamaktadır. Taşkın frekans analizinin amacı belli bir dönüş aralığına karşı gelen taşkın debisinin tahminidir. Bu tahminin yapılması, kabul edilen taş kın frekans dağılımının fiziksel bir temele dayanmaması ve gözlenmiş kayıtlardan daha uzun süreli dönüş aralıklarındaki taşkın riskinin belirlenmesinde karşılaşılan zorluklar nedeniyle oldukça karışıktır. Taş kın debisinin dönüş aralığı ile olan ilişkisinin gerçekçi tahminini, bir istasyondaki yüksek örnekleme hatalarına sahip küçük bir örnekle elde etmek mümkün değildir. Bölgesel bilgilerin kullanılmasının gerekliliği, istasyonlardaki tahminlerin iyileştirilmesi veya ölçüm olmayan havzalarda taşkın tahmin ihtiyacından doğmaktadır. Bu çalışmada, bölgesel taşkın frekans analizi (BTFA) incelenmiş ve güçlü (robust, duyarsız) taşkın tahminlerine imkan veren modeller araştırılmıştır. Bölüm l'de BTFA'nin amaçları açıklanmış ve Bölüm 2'de taşkın frekans analizi için gerekli teorik bilgiler verilmiştir. Bölüm 3 'de BTFA'nde kullanılan çeşitli yöntemler ortak bir notasyonla açıklanmış ve bu konuda yapılan çalışmalar özetlenmiştir. Bölüm 4'de BTFA'de homojen bölgelerin belirlenmesi için önerilen bir yöntem ile kullanılan dağılımların yapısı hakkında fikir veren değişim katsayısı ve çarpıklık katsayısının bölgesel ortalama değerlerinin el de edilmesi için gerekli ifadeler verilmiştir. Bölüm 5'de çeşitli yöntemlerin bir uygulaması Yeşilırmak Havzası'nda belirlenen iki homojen bölgede yapılmış ve bölgesel taşkın frekans eğrileri çeşitli modeller kullanılarak elde edilmiştir. Bölüm 6 'da güçlü bir modelin belirlenebilmesi için çalışmalar yapılmış tır. Taşkın debilerinin toplumları için çeşitli olasılık dağılımları kabul edilerek homojen bölgelerde taşkın serileri türetilmiş ve yeni serilerin BTFA çeşitli modeller ile yapılmıştır. Seçilen dönüş aralıklarında yapılan taşkın tahminleri, toplum eğrileriyle taraflılık (bias) ve değişkenlik (karelerin ortalama karekökü, KOK, RMS) açısından karşılaştırılmıştır. Ayrıca modelleme çalışmalarında örnek sayısının, taşkın tahminlerinde kullanılan modellerin performanslarına etkisi araştırılmıştır. Çalışmanın sonunda elde edilen neticeler çeşitli yönlerden karşılaştırılmış ve bölgesel yöntemlerin istasyon yöntemlerinden, homojen bölgelerdeki taşkın frekans analizinin ise heterojen bölgelerdekinden daha az taraflı ve değişken sonuçlar verdiği görülmüştür. Homojen bölgeler de beş ayrı taşkın toplumu kabulüyle yapılan BTFA'de lognormal (MOM) modelin en yüksek performansa sahip olduğu sonucuna varılmıştır.

Flood frequency analysis provides vital information for the planning of many hydraulic structures and for determining the risk of flooding in flood plain. The objective of flood frequency analysis is to estimate the flood magnitude corresponding to a certain return period. Estimation of this magnitude is quite complex due to lack of physical basis in choosing the flood frequency distribution that will be accepted and due to difficulties in determining flood peaks at longer return periods than that of the observed record. It is not possible to determine a realistic estimation of flood return period relation by means of a small sample at a site which would be associated with high sampling error. The requirement for using regional information arises from the possibility of improvement of at-site estimation or the need of flood estimation for ungauged basins. In this study, Regional Flood Frequency Analysis (RFFA) has been investigated and robust models which would allow flood estimation for basins especially with short records, have been analysed. In Part 1, methods used in flood frequency analysis and then basic approaches are considered, the aims of RFFA are explained and the subject of the study is introduced. In Part 2, statistical methods have been widely investigated since the study regards flood peaks statistically and some parameter estimation methods such as method of moments (MOM), method of maximum likelihood (ML), method of probability weighted moments (PWM) have been explained. Some commonly used distributions in hydrology, especially in flood frequency analysis, namely lognormal, Pearson Type III, log-Pearson Type III, extreme value, Pareto, Wakeby and loglogistic distributions have been analysed and estimations of parameters according to various methods have been given. In Part 3, the following methods in RFFA have been discussed: - Station-year method - Dalrymple's method - Methods based on regional average coefficients - Methods based on order statistics - Method based on standardised probability weighted moments (PWMs) - Threshold and censored sample methods xiv In RFFA methods, where series of annual maximums are used, the following common notation is adopted : Qijf {1-L2,....Nj ; j=l,2,....M } being the ith year's maximum value at the j station, at M stations L=Ef\al N, observed values are present in total. { Q1j} i=l,2 N, } values at any j**1 station constitute a random sample taken from the flood population. In many methods the flood series are standardized. Methods in which this standardization is made by the median or mean of the flood series are named as index flood methods. In case the standardization is made by the mean, the statistics of the new variable in X=Q/Q form are: E(X)-1, ax=Cv(Q), CsX=CsQ In this study, the flood series is standardized by its mean and the assumption that the distribution of variable X at all stations in the region remains same, is defined as " Regional Homogeneity ". When homogeneity assumption is made in many of the RFFA methods, the more homogeneous a region is the greater is the gain in using regional instead of at-site estimation. In Part 4, a method used in determination of homogeneous regions given by Wiltshire (1986) is explained. In this method the variance of Cv's at sites in a region is minimized and the variance of regional mean Cv's for P regions is maximized. For this reason Fisher F test, which is a variance ratio test, is used. In this method the F statistic is expressed as follows: (Variance of regional mean Cv's at P regions) F= (Variance of Cv's at sites with in a region) For an efficient subdivision of basins into significantly different groups the F statistic should be large. This means that the numerator is made maximum and the denominator is made minimum. The homogeneity test of each subregion within itself can be done by means of Sk. S. which is distributed as *2 statistic with Mk-1 degrees of freedom and can be used as a test of region homogeneity under the null hypothesis that there is no difference between site Cv's within a region. In this part, regional averages of dimensionless coefficients of variation and of coefficients of skewness are defined which would give an idea about the structure of the selected distribution to be used in regional average coefficients method. Also the necessary expressions for regional estimation of probability weighted moments are given. In part 5, an application of RFFA is made. For this purpose the 11- year annual maximums in Yeşil ırmak Basin have been chosen since there are longer records at relatively more sites. The sites having records over 20 years have been used, the ones having shorter records have not been included in the study with the thought that they would give biased results. Firstly, the basin is tested for homogeneity by the method given in xv Part 4 and it is concluded that the basin cannot be accepted homogeneous as a whole. By analysing the position of sites and coefficients of variation, the basin is divided into two parts in north south direction. By repeating the tests it is concluded that the two regions are homogeneous within themselves and they are significantly different from each other. The following expressions of regional coefficient of variation and coefficient of skewness have been adopted in order to obtain flood frequency curves: BCv=((^=i (Nj-UV)/^., 1/2 BCS=C3/BCV3 C3=lfj=1 ((NJ/(NJ-2)).S3J)/I?J.1 (Nrl) To obtain Q/Q - T relation from observed data, the annual maximums at sites have been arranged from low to high values forming an ordered sample, after being standardized by the site means. The distribution of these values have been expressed as a function of y (Gumbel variable). The value of y depends on sample length and the form of the distribution. The frequency of the random variable being equal or less than the i value is assessed by Gringorten formula: Fij=(i-0.44)/(Nj+0.12) Thus by using these F,, values, in case the EVİ distribution is assumed the y,j values can be found by the following expression: y^-lnt-ln^)] The y^ values calculeted at sites are brought together at homogeneous region and the y axis being divided into k=l,2,... intervals of width 0.5, average Xk and y. pairs are defined falling into each interval. In this way it is provided that the graph of X. versus yk has a smooth form. This relation between X and y values is at the same time an estimation of the relation between X-T. Among the probability distributions, extreme value distributions (GEV), log-Pearson Type III, lognormal, Wakeby, Pearson Type III, loglogistic and Pareto distributions have_been selected to present the observed values and by calculation of Q/Q values corresponding to return periods of T=2, 5, 10, 25, 50, 100, 200 regional flood frequency curves have been obtained. When the curves are compared to observed values; following conclusions are drawn: - Regional flood frequency curves obtained in both regions fit well to observed data generally. - As the return period increases, differences between the curves increase also. xvi - Differences between flood frequency curves for longer return periods are greater for Region 1 whose coefficient of variation is larger. By using different methods varying estimates for RFFA of observed data have been obtained at various return periods. In general interpretation of these differences is difficult, and it is not possible to decide which one of the methods is better since real values are unknown. In flood frequency analysis the recent approaches in the selection of the distribution to which the maximum series fits, depend on behaviour and robustness analyses. While behaviour analysis involves testing of the selection of the probability distribution fitting best to the observed series, the objective of the robustness analysis is the testing of the estimation abilities. While in behaviour analysis observed values are used, in robustness analysis generated flood series are used. This generation is made by selecting a probability distribution which presents the flood population. In Part 6, robustness analysis is explained aiming to test estimation abilities of different methods. In these analyses, a distribution function together with a parameter estimation method is taken into consideration as a " model ". Investigation of robust modelling carries importance for being able to make reliable estimates especially for basins with short records. Studies can be categorized in 3 groups: 1) Modelling for regional flood frequency analysis. 2) Modelling for at-site flood frequency analysis. 3) Modelling in heterogeneous region. 1) MODELLING FOR REGIONAL FLOOD FREQUENCY ANALYSIS: In this section flood population is thought respectively as extreme value, log-Pearson Type III, lognormal, Pearson Type III and Wakeby population. For each population: a) For 2 regions determined as homogeneous, 18 flood series each consisting of 25-years at 5 sites for Region 1, 30-years at 6 sites for Region 2 have been generated, meanwhile preserving the present number of sites and average record periods at the 2 regions. b) Regional flood frequency curves have been obtained by determining regional Q/Q-T relation of each series using pre selected methods. c) The assessed regional curves have been compared to the regional curve assumed for the population with regard to bias and variability (RMS) at different return periods. The following expressions have been used for bias and variability: Bias-d/t).^ (XTpr-XTr) xvii RMS-Ed/t).!*^ (XTpr-XTr)2]1/2 In the expressions p Js the simulation number, t is the total number of simulation runs, XT is the T-year dimensionless flood estimate obtained by regional analysis of generated series and XTr is the T-year dimensionless flood (population value) calculated according to the distribution accepted for the flood population. 2) MODELLING FOR AT-SITE FLOOD FREQUENCY ANALYSIS; Lognormal and GEV(PWM) methods have been used for at-site flood frequency analysis of the series generated by using Wakeby distribution selected as the flood population, which is needed for being able to compare estimates obtained by RFFA with at-site estimates. The means of bias and RMS values of estimates have been compared with regional values, the variation intervals of at-site estimates have been analysed. 3) MODELLING IN HETEROGENEOUS REGION: Lognormal and GEV(PWM) methods have been used in regional flood frequency analysis in heterogeneous region by bringing together the series generated under the assumption that they fit Wakeby distribution in homogeneous regions. The bias and RMS values of estimates with both regions' population curves have been evaluated and these have been compared to those found in homogeneous regions. Separately, in modelling, the effect of sample number on the performance of the methods used has been investigated. CONCLUSIONS; 1) In homogeneous regions, among 5 different flood populations used in the modelling, the performance of lognormal (MOM) distribution is the highest. 2) Pareto (ML), Wakeby (PWM), Pearson Type III (MOM), GEV (PWM), are alternative distributions which give reliable estimates. 3) The loglogistic distribution gives always the greatest positive bias and variability. 4) The performances of regional methods are much higher than average performances at sites. 5) Application of RFFA by dividing the basin into homogeneous regions results in decrease of variabilities of estimates. 6) In modelling studies the performance of methods are only slightly effected by an increase of sample number.