Concept of robust and invariant stochastic hydrology for hydrotechnical construction

Abstract

During the past 50 years numerous theoretical and practical works have been devoted to a study of the degree of correspondence of the probability distribution of S. N. Kritskii and M. F. Menkel' to empirical probability distributions by means of various statistical tests and adaptation of these tests for skew distributions; to improvement of the probability distribution --to supplementing the tables of ordinates and expanding the limits of constructing the distribution with a different Cs/C v and span of variation of this ratio; to an original formulation and development of the maximum likelihood method for determining the parameters of the Kritskii--Menkel' probability distribution, etc.The Kritskii--Menkel' probability distribution widely used in research and development actually became a universal one-dimensional probability model.In regional investigations C v is also widely used for determining it with respect to groups of rivers during hydrotechnical designing.Thus, the conviction of the universality of the Kritskii--Menkel' probability distribution and also of its parameters -the coefficients of variation C v and skewness C s --has developed in hydrology and its various engineering applications.In this connection the problem is being solved: is the Kritskii--Menkel' probability distribution actually universal and are its parameters universal? To solve such a nontrivial problem, the author thoroughly analyzed modern methods of the mathematical theory of random phenomena [3, 8, 9].As a result the most characteristic mathematical concepts were revealed --indicators attesting to the universality of a particular probability distribution and its parameters: robustness (statistical stability) and invariance. There are grounds to assume that for hydrology these concepts are cardinal.As is known, the Kritskii--Menkel' probability distribution curves are constructed on the basis of the Pearson type III probability distribution for C s = 2C v. The further account is given according to Kritskii and Menkel' (1946).The functional dependence is established in the formwhere a, b are coefficients. Coefficient a is selected so as to reduce the curve being sought to the given value of the center of the distribution 0 = 1 (analogously as with the center of the initial curve R = 1).The initial curve for R = 1 is exhaustively characterized by one parameter Cv(x), since Cs(x ) = 2Cv(x ). The sought curve is related by Kritsldi and Menkel' to two independent parameters Cv(U) and Cs(U ). The transforming equation contains two parameters a and b. Consequently, the problem is to establish the relation between Cv(U) and Cs(U) as arguments and Cv(x), a, b as functions, i.e., Kritskii and Menkel' solve the inverse problem.The exponent b in Eq.(1) can take on not only positive but also negative values by definition. In the latter case the limits of variation of U are equal to a/O = oo > U > O = a/oo ,i.e., "the conditions which the transformed curves should satisfy are fulfilled."