The Spatial Distribution of Low Flows in Poland Not Exceeded at an Assumed Probability

GEOGRAPHIA POLONICA 2010, 83, 1, 39–50 THE SPATIAL DISTRIBUTION OF LOW FLOWS IN POLAND NOT EXCEEDED AT AN ASSUMED PROBABILITY ADAM BARTNIK Department of Hydrology and Water Management University of Łódź Narutowicza 88, 90-139 Łódź, Poland E-mail: [email protected] Abstract: Since most hydrological quantities are regarded as random variables, it is important that their distributions of probability be identifi ed. This paper therefore presents the results of a fi tting of most frequently-applied probability distributions to a series of minimum yearly fl ows. Specifi cally, the series of minimal annual discharges derived from 119 stream gauges located throughout Poland and relating to the period 1971-1990. The main task was to adjust one of the probability distribution functions applied most frequently (Fig. 1). Maps generated provide for the identifi cation of several regularities. Key words: hydrology, Poland, fl ow, rivers, frequency distributions, low-fl ow INTRODUCTION associated with any known probability of ap- pearance. For this reason, fl ows at an assumed Annual minimal fl ows play an important probability are used to calculate constructional role in hydrology, since knowledge of them, profi les (for example as water intakes on rivers their temporal variability, periodicity and spa- are planned). tial diversity is crucial if the use of water re- A Low-fl ow Frequency Curve (LFFC) sources on the local and national scales is to shows the proportion of years in which be rational. Annual minimal fl ows can be used a fl ow is exceeded (or equivalently the average to generate a series of parameters and charac- interval in years (‘return period’ or ‘recurrence teristics of economic and management value interval’) after which a river once again falls (navigation parameters of rivers, inviolable below a given discharge) (Smakhtin 2001). fl ows, available fl ows, etc.). Many authors use The LFFC is constructed on the basis of a se- low discharges to estimate the boundary val- ries of annual fl ow minima (daily or monthly ues used in setting low fl ows, and it is equally minimum discharges or fl ow volumes), which important that there be knowledge and an ap- are extracted from available original continu- propriate interpretation of low discharges in ous fl ow series (one value from every year the estimation of inviolable fl ows, i.e. the min- recorded). imum fl ow rate needing to be maintained in a particular discharge section line of a river in line with biological and social considerations. METHOD The conventional characteristic fl ows applied in Poland (e.g. Average Low Flow, As most hydrological characteristics Smallest Low Flow), are usually values not are regarded as random variables, it is im- http://rcin.org.pl 40 Adam Bartnik portant that distributions of probability for bution and to quantify the parameters thereof. them be identifi ed. The procedure involved The procedure here entails fi tting several theo- usually includes two stages: estimation of retical distribution functions to observed low- the parameters to the accepted probability fl ow data and deciding which distribution fi ts distribution and verifi cation of the hypoth- the data best, on the basis of statistically- and esis that a real probability distribution of graphically-based tests. Among the distribution a given value is involved. functions, most of those referred to frequently As the ‘true’ probability distributions of in the literature in connection with low-fl ows low fl ows are unknown, the practical problem are different forms of the Weibull, Gumbel, is to identify a reasonable ‘functional’ distri- Pearson Type III and log-normal distributions. Figure 1. Examples of fi tted types of distribution of analysed annual low fl ow series from twenty-year period (1971-1990) (fl ow duration and density curves). A – Fisher-Tippet III distribution (Biała Lądecka-Żelazno), B – log-normal distribution (Gowienica- Widzieńsko), C – gamma distribution (Łososina-Jakubkowice). http://rcin.org.pl The spatial distribution of low fl ows in Poland not exceeded at an assumed probability 41 Many studies have been devoted to the prob- 1995; Ozga-Zielińska, Brzeziński 1997). ability distributions most suitable for fi tting the However, on the other hand, its computa- sequences of annual minimum fl ows in differ- tional simplicity makes it popular and rec- ent regions and for minima of different aver- ommended offi cially for application (Za- aging intervals, as well as evaluating methods sady obliczania … 1973). by which to estimate distribution parameters The basic information relating to the fi t (Matalas 1963; Jozeph 1970; Prakash 1981; Be- types of the probability distribution is as ran and Rodier 1985; Loganathan et al. 1985; presented below. Further details on proper- McMahon and Mein 1986; Singh 1987; Waylen ties of these functions and questions con- and Woo 1987; Khan and Mawdsley 1988; Sefe cerning estimation of parameters for them 1988; Leppajarvi 1989; Polarski 1989; FREND are to be found in many papers on the sub- 1989; Nathan and McMahon 1990b; Russell ject (Kaczmarek 1970; Byczkowski 1972; 1992; Loaiciga et al. 1992; Durrans 1996; Law- Kite 1988; Węglarczyk 1998). Examples al and Watt 1996a; Lawal and Watt 1996b; Bulu of three fi t distributions of probability are and Onoz 1997, Jakubowski 2005). shown in Fig. 1. The author’s task was not to investigate the real distribution of examined series but only The Fisher-Tippett type III distribu- to adjust one of the functions applied most tion (Weibull) – Fig. 1 A frequently to ensure fulfi lment of the Kołmo- Examined by Fisher and Tippett in 1928 gorov λ test condition of conformability. The (Byczkowski 1996), this asymptotic distri- scheme applied to achieve this entailed: bution is well conditioned when sample size – an attempt to adjust a Fisher-Tippett III proceeds to infi nity. Theoretically, then, it distribution, frequently recommended in should only be applied when sample size is reference to minimum fl ows, by means of suffi ciently large. On the other hand, there the maximum likelihood method; is no general criterion allowing it to be eval- – adjustment of the function of the gamma uated if a given size is indeed suffi ciently distribution to the series for which the large (Kaczmarek 1960). The probability Kołmogorov λ test indicated lack of com- density function is: patibility with the aforementioned Fisher- β β −1 §·x−ε βε− −¨¸ Tippett III distribution; §·x ©¹αε− fx(;αβε , , )=⋅¨¸ ⋅ e , (1.) – further verifi cation followed by adjust- αε−−©¹ αε ment of the log-normal distribution to α >0, (β xx ) > 0, ≥ε the remaining series by means of the mo- ments method, as well as verifi cation of where α is a scale parameter equal to the compatibility with the empirical distribu- order of the lowest derivative of the prob- tion (test above). ability function that is not zero at x=ε, β the The distribution functions for all inves- characteristic drought (a location or central tigated basins were recognized in this way. value parameter) and ε the lower limit to x. Matalas (1963) recommended the types of Equations are a solution employing the distributions listed above for the analysis of method of maximum likelihood: low fl ows in the United States. n The use of the Kołmogorow λ test in de- βεˆ − ˆ βˆ−1 n nx¦()i ()βεˆ −⋅ −ˆ −1 −i=1 = ciding on the suffi cient level of adjustment 1(¦ xi )n 0 (2.) i=1 −εβˆ ˆ of a theoretical distribution to an empirical ¦()xi one can be controversial. It is sometimes i=1 considered to be too gentle (Brzeziński http://rcin.org.pl 42 Adam Bartnik n with respect to α, β and ε thus equating to βˆ βεˆ ⋅−⋅−()()ˆˆ ε n nx¦ iiln x zero give the equations: +⋅βεˆ () − −i=1 = nxlni ˆ n 0 ¦ βˆ i=1 ()−εˆ ∂ n β ¦ xi lnLn 1 = =⋅()x −−=ε 0 (6.) i 1 ∂αα2 ¦ i α (3.) i=1 No further simplifi cation is possible and ∂ ln L n⋅Γ′()β n =− +ln()xn −εα − ⋅ ln = 0 (7.) ∂Γββ¦ i equations 2 and 3 must be solved as simulta- () i=1 neous equations. The procedure follows the method of Condie and Nix (1975). Know- ∂ ln Ln n §·1 =−()β −⋅10 = ing α and ε, the β parameter can be solved ∂−εα¦¨¸ ε (8.) i=1 ©¹xi using the formula: 1 Further transformations are solved nu- n ªºβˆ βˆ ()−εˆ merically. Maximum likelihood methods «»¦ xi αεˆˆ=+«»i=1 (4.) may not always be applicable, and in gen- «»n eral it is not possible for any automatic pro- ¬¼«» cedure to guarantee determination of the minimum variance solution each time. Each Gamma distribution (Pearson type III) dataset requires careful investigation of the – Fig. 1 C shape of the function. This is one of the distributions applied most frequently in hydrology. It was worked The three-parameter lognormal dis- out at the beginning of the 20th century by tribution (Galton) – Fig. 1 B British statistician Karl Pearson, as one The three-parameter lognormal repre- of the biggest systems of the function. The sents the normal distribution of the loga- gamma distribution has gained widespread rithms of the reduced variable (x-ε), where ε use in the smoothing of hydrological data is a lower boundary. The density probability series. The density probability distribution distribution is given by: of the Pearson type III distribution is: ªº−−εμ2 − [ln(x ) ] «»2 1 ¬¼«»2δ β − −ε fx(;μδε , , )=⋅e , −ε 1 −§·x (9.) 1 §·x ¨¸α ()2x −εδ π fx(;αβε , , )=⋅¨¸ ⋅ e©¹ , (5.) αβ⋅Γ()©¹ α where μ and σ2 are form and scale param- x ≥εα,0, > β >0 eters, shown later to the mean and variance of the logarithms of (x-ε).

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