When one or several zero values are present in a long-term series of observations of river runoff it is practically impossible to plot a KritskiiMenkel' cumulative probability distribution curve, which is legally assigned in the existing building codes (SNIP) [7] as the main probability distribution function.For the present the procedure of using bootstrap [8] is the only variant of solving such a problem. A simpler method of determining the calculated value of a certain hydrologic characteristic of the required exceedance probability is proposed. We will take another cumulative probability distribution function (CPDF) solved with proper substantiation --a Pearson type III.Using the property of invariance introduced into hydrology in [4], we can prove that the known parameters of the Pearson type III distribution curve --standard deviation and skewness coefficient --do not change as a result of adding a certain positive constant to each value of the series.Only one parameter of the Pearson type IH distribution curve --the arithmetic mean value --increases by the amount of a positive constant.Then traditional calculation of the ordinates of the "new" (with the added constant) Pearson type IH probability distribution curve is carried out by the known algorithm [ 1, 5, 6,9].The next and last step of calculating the CPDF is subtraction of the constant added at the first step from the ordinates obtained. Then graphic plotting of the CPDF is carried out by means of computer or manually.The region of allowable values of the positive constant for various numerical structures of the investigated series will be different. Its selection should be made by an expert statistician.Such series of geophysical observations are found rather often [5]. If river runoff is examined, then they can be both in arid regions and in the permafrost region.The permafrost region of Russia was selected for an illustrative calculation. As a result of a special screening analysis of more than 50 rivers of Eastern Siberia, rivers freezing up in the winter whose runoff in individual years is equal to zero were selected. Furthermore, the criteria of the analysis were: the presence of outliers in the observation series; a large variability of water discharges.As a result a unique series of minimum annual discharges, a probability calculation of which can be considered exclusive, was found (Table 1). This is the Yana River near the region of DThangky. During 20 years of observations the river froze up complete seven times, and in two more years the minimum annual discharge was 0.01 m3/sec. Thus, 45% of all members of the observation series are almost equal to zero. The drainage area is 216,000 km 2, the distance from the river mouth is 381 km (the gauging station is located near the Arctic Circle).The maximum outlier (the maximum values standing out from the series) is more than 10 times greater than the arithmetic mean value of the observation series and 3.15 times greater than the value of the second discharge in the series of observed minimum ann...

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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."

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