8 Parameter estimation of series of maximum flood flows

Part I1

Application of estimation theory to hydrology and water engineering 8 Parameter estimation of series of maximum flood flows

8.1 Fundamental problems of processing N-year flows The problems of processing culminating flood flows were given considerable importance as early as at the time when probability theory had just begun to develop and when the methods of the theory of estimation had not yet been satisfactorily elaborated. The variable properties of hydrological regimes, a large number of factors affecting flood runoffs, the limited length of observation, and the related problems of estimating the law of probability distribution, as well as the extrapolation in the region of low probabilities of transgression - all these circumstances are the reason why finding the desired hydrological design quantities has always been one of the most difficult tasks in the processing of hydrological information. With the development of knowledge in this field further complex problems gradually started to be investigated, such as for instance the probability properties of the flood flows and their genesis in the different seasons of the year (rain-induced and snow-induced floods), extreme runoffs from smaller river basins not subjected to measurement, the effect of historical floods on the parameters of a series of culminating flows, regional relationships of extreme runoffs. At present, use is currently made of statistical and genetic methods for determining the design parameters of the flood waves, and empirical formulae have also been derived. These methods, as well as the conditions of their 149

Parameter estimation of series of maximumJloodflowss

application, have been described in the already quite voluminous hydrological literature. One of the most comprehensive and important works published in Czechoslovakia concerning the fundamental characteristics of hydrological phenomena TABLE18. Characteristics of maximum annual flows in a set of 250 stations in Czechoslovakia (according to [38])

Length of the periods examined

C"

OC"

(%I

the shortest period of 25 years considered

0.4 to 1.0

I8 to 29

the most frequent period of 30 years

0.3 to 1.2

15 to 30

the second most frequent period of 55 years

0.4 to 0.9

12 to 18

the period of 80 to 85 years

0.5 to 0.8

10 to 14

TABLE 19. Characteristics of maximum annual flows in a set of 250

cs

C"

ors (%)

minimum observation of 25 years

0.9 to 2.6

0.4 1.0

21 to 19 65 to 189

the most frequent 55-year observation

0.6 to 2.8

0.4

17 to 80 36 to 166

the longest observation series of 85 years

1.5 to 2.5

Type of observation

0.9 0.5

o.8

18 to 30 15 to 25

are the Hydrologicke pomEry CSSR (Hydrological Regimes of the Czechoslovak Socialist Republic*)[51]). In this work, properties of the flood flows of Czechoslovak streams are dealt with in Part I11 Chapter 7, which presents the statistical characteristics of culminating flows in a set of 250 stations, their hydrological and geographic characteristics, as well as an analysis of the basic factors affecting runoffs. *)

In the year of publication the official name of the state was the Czechoslovak Socialist Republic (CSSR).

150

Fundamental problems oiprocessing N-year JIows

From the point of view of the theory of estimation, and the justification of the application of that theory, the statistical characteristics of the culminating flows must be regarded as the most valuable. They were computed with the help of the moments method, or the quantiles method,') using the shortest, 25-year, series and the longest, 115-term, series (at DEin on the Elbe) of the set. The most significant results are given in Tables 18 and 19. The lowest C, value, 0.23, was recorded at Komarno, and the highest value, 1.30, at Husinec. The most frequent C, value ranged between 0.50 and 0.69 (with the median equal to 0.6). For the coefficient of asymmetry the lowest value, 0.03, was recorded at Michalovce, and the highest, 3.2, at Spalov. From the results quoted it is evident that the real series of maximum annual flows exhibit considerable fluctuation and skewness, and that the length of observation was in several cases rather limited. These properties of the culminating flow series substantiate the necessity to estimate for their unbiassed parameters. The greatest attention should be given to the distribution of probability and to the ascertainment of the systematic errors with its asymmetrical types. Only in this way can dependable values of the N-year flows be approximated. The application of the theory of estimation can be traced back to 1977, when basic material was being processed for the research project entitled The Complex Solution of the Water Engineering Problems of the North-Bohemian Lignite Basin and the Related Problems of the Protection of the Environment. It was then that studies of the laws of the flood regimes of the smaller streams started appearing, based upon the theory of estimation and the application of simulation models of hydrological processes [19]. The research led to new knowledge concerning the behaviour of the culminating flows and their sample characteristics. It particularly turned out that mechanical application of the hitherto current methodological procedures based upon the assumption of the representativeness of a single short sample could on the average lead to systematic underestimation of N-year flows. The research also confirmed the fact that for smaller streams the estimation of the parameters of their culminating flows is of particular importance, because their regimes are characterized by high variability and skewness of distribution (the differences in the N-year maximum flows amounting to as much as several hundred percent). Figure 36 presents a characteristic example of a period of chronologically ordered culminating flows in a modelled 10 000-years series with the parameters estimated. The example shows that under the extreme conditions of the smaller streams, catastrophic flows can occur by sheer chance after a calm period of *I As in the preceding works, no corrections were considered of the characteristics calculated as far as the systematic errors are concerned;at that time the required relationships between characteristics and parameters had not yet been formulated.

151

Parameter estimation of series of maximum jZoodjows

several decades. A proposal for concrete antiflood measures to be adopted based upon short-term observation and underestimating outcomes of the laws of statistics can be extraordinarily risky, and could lead to serious economic consequences. 20

9:

a my

= 20.64

P,in

= 0.867

-

years

Fig. 36. Characteristic section in a 10 000-year random series of maximum annual flows with Pearson’s IIIrd type distribution Inputs: = 1, C, = 0.8, C, = 12.

The processing of the N-year maximum flows has received much attention at the Czech Hydrometeorological Institute in Prague. Its report [45] contains a summary assessment of the latest achievements in the application of the estimation theory to bulk processing of the culminating flows. The recommendation of the most suitable types of theoretical distributions as well as theemethods of estimating their parameters is most important. As far as practical application is concerned, it is suggested that use should primarily be made of Pearson’s IIIrd type, triparametric log-normal, and logarithmic Pearson’s IIIrd type distributions. Gamma distribution is not recommended in view of the fact that in the region of the lower probabilities of transgression it will lead to results analogous to those of the computationally simpler triparametric log-normal distribution. In agreement with the results of the research conducted by the Department of Hydrotechnology of the Czech Technical University in Prague, the moments method, with the systematic bias involved in the estimation of the coefficients of variation and asymmetry corrected, is recommended for bulk processing of the N-year flows for all the types of theoretical distribution quoted above. For reasons mentioned in Part I of this book, neither the maximum likelihood method nor the quantiles method is reccommended for bulk data processing. The automatic bulk processing of the series of culminating flows and the determination of the N-year maximum flows revealed the necessity to upgrade 152

Fundamental problems of processing N-year j7ows

the efficiency of the moments method and to convert the graphical relationships concerning the estimation of the unbiassed parameters to analytic form, which is of course more computer-friendly.The conclusions formulated in the foreign literature available to us [12,98] were of course taken into account, as well as the results of our own comprehensive research [81,82]. In view of the general importance of the solution, i. e. also for parameter estimation of other types of series and their mathematical modelling, this subject is treated separately in k t i o n 11.1. The Czech Hydrometeorological Institute has completed a draft for complex automatic processing of the N-year maximum flows [45,46] corresponding to the contemporary level of knowledge supplied by the theory of estimation achieved both in Czechoslovakia and abroad. The programme is a valuable outcome of the long research conducted by the Department of Hydrotechnology of the Czech Technical University in Prague in close cooperation with the Czech Hydrometeorological Institute in Prague. Apart from the programme itself, aids have been prepared to facilitate the computation of the design variables in cases where a computer is not available. As far as the probability properties of the culminating flow series are concerned, a most important problem is the determination of the design N-year flows with due account taken of historical floods. In the literature on this subject [30,45]we find expressions for the estimation of the mean values of the coefficients of variation and asymmetry of the culmination flow series, with the occurrence of historical floods duly considered. These problems have recently been dealt with by KaSparek, who in his study [42] gave an evaluation of the significance of the floods on the Litavka in the years 1872 to 1981 for the estimation of the N-year flows. In a number of variants their computations revealed that the effect of the historical floods on the determination of the magnitudes of the N-year flows could be most significant. A full-scale investigation and adequate processing of the data on extraordinary floods, whether they have occured only recently or in the past, could, in a number of cases, reduce the risk of estimating wrongly the design flows. Whenever antiflood precautions are to be adopted, it is thus essential that these circumstances should be taken fully into account. Despite the results achieved so far in the field of application of the theory of estimation, research must be continued and attention should be given to the problems that have so far remained unsolved, such as the problem of theoretical distributions of the flow series with historical floods; smoothing of the results achieved with the help of numerical procedures so as to make them applicable to the whole river-basin, with due account taken of its hydrological regularities; and determination of the N-year flows in the smaller river-basins where the required observations may be lacking. Apart from the culminating flows, more attention will have to be given to the shape and the volume of the flood waves,which should be regarded as basic information, besides the culminating flows, upon which the design the protective effect of storage reservoirs is based. 153

Parameter estimation of series of maximum Poodjows

8.2 Probability properties of intervals between culminating flows By “interval between culminating flows” we mean the interval at which the culminating flow selected repeats itself. In practice use is invariably made of its mean value. In our research we conceived of it as a random variable describable with the help of the respective statistical characteristics. It was the aim of our research to investigate the fundamental probability

i=

I

0 5467 P = a0042 T = 237

-

I0 Yo

Fig. 37. Lines of transgression of all Ti times between selected maximum annual flows of a 10 000year random series with Pearson’s IIIrd type probability distribution Inputs: 0 = 1, C, = 0.8, C, = 12; Outputs: p = 0.990, C, = 0.682, C, = 10.800.

154

Probability properties of intervals between culminating flow

properties of these intervals and to explain more profoundly the relationships between the sample observation and the population. Use was made of the modelled 10 000-year random series of culminating flows generated as absolutely random sequences with specified parameters. The printed output of the model were the lines of transgression of all the intervals of repetition covering the whole scope of the culminating flows, Figure 37 presents examples of the lines of transgression of the intervals of repetition obtained from a 10 000-year random series with Pearson’s type 111 distribution under the extreme conditions of a small (unwooded) river-basin, where the culminating flows could exhibit a high degree of fluctuation and skewness. The theoretical values of the average intervals of repetition, T, were calculated by using the following formula:

where p is the probability of transgression. Formula (8.1) can be derived from Poisson’s law of distribution. The values of T were in all the cases compared with the expected values of the empirical lines of transgression. From the examples presented in Fig. 37 it can be seen that it is fully justified to consider the intervals of repetition as random variables exhibiting relatively high variance. Thus, for instance, 1 13-year maximum flows (Q = 4.467) occured (i. e. were reached or exceeded) in one case within a 4-year period, another extreme being the period of 637 years for which that flow did not reappear. Analogous properties are manifested by the curves of transgression of the intervals of repetition (Fig. 38), which correspond approximately to the regimes of small, partly wooded and sloping river-basins. For instance, in one case a 101-year flow was repeated in the next year but one, the contrary extreme being the period of 385 years, during which that climax did not reappear. It is also characteristic of the lines of transgression that with the N-year flow rising, the potential variance of the intervals of repetition grows quite rapidly. A 5.826 climax which repeats (i. e. it is reached or exceeded)in 196years on average, may not occur for as long as 892 years; and a 9.326 climax repeated in 1000 years on average, may reappear in 115 to 3 460 years (in Fig. 38 these extremes have of course not been plotted). The results achieved fully support the previous results of the research into the behaviour of the sample characteristics and their relationship to parameters. From the lines of transgression of the intervals between the climaxes it can be seen that in the shorter periods of observation, such as periods of several tens of years, no significant extreme flow may occur at all, leading to more dependable extrapolation of the lines of transgression into the region of the lower probabilities. The contrary case can however not be excluded either, viz. the

Parameter estimation of series of maximum joodponw

occurrence of several extremes in a shorter period, which may lead to overestimation of the probability properties of the given phenomenon. Whereas the systematic bias can relatively easily be eliminated using contemporary methods, the estimation of random errors is more difficult, because in view of the nature of the given hydrological phenomenon it may not be so easy to ascertain whether the series observed is representative or not. The checks on the representativeness of the culminating flow series will henceforward have to be based primarily upon the genetic and comparative methods.

Fig. 38. Lines of transgression of all Ti times between selected maximum annual flows ol' a 10 OOOyear random series with Pearson's IIIrd type probability distribution Inputs: 0 = I , C, = 0.7, C, = 8; Outputs: 0 = 1.002, C, = 0.717, C, = 7.953.

156