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 moments 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). where α, β and ε are parameters to be de- The lognormal distribution curve is ﬁ ned, and Г(β) the gamma function. asymmetrical (displays positive skewness), The Pearson type III function is asymmet- limited at the point ε=0 (the range of variabil- rical in relation to the straight line crossing the ity is unlimited), and with a function of densi- mode point (i.e. is positively skewed), limited ty that has one maximum. Using the formula at the point ε=0, and asymptotic, approach- presented below (Kite 1988), estimators of ing the vertical axis in the upper part. There is the distribution were calculated by means of also one maximum in the mode point. the method of moments. The likelihood function is established as the logarithm of the product of the density 2 δ =+ln() 1 ()z (10.) probability distributions. Differentiation

http://rcin.org.pl The spatial distribution of low ﬂ ows in Poland not exceeded at an assumed probability 43

sents a very signifi cant feature, leading to δ ln(z + 1) its being described as a “multi-year period μ =−ln( ) (11.) z 2 digest”. Examining a shorter time-interval, we obtain information on the scale of vari- δ εμ=− (12.) ability to a phenomenon being addressed z over a longer period. 2 There is now a common view among hy- § ()cc 2 ++− 4 · 3 1− ¨ 55 ¸ drologists that gauging records need to meet ¨ 2 ¸ = © ¹ where: z 1 (13.) conditions of homogeneity so that they can § ()cc 2 ++− 4 · 3 ¨ 55 be statistically tested and analysed (Ozga- ¨ ¸ © 2 ¹ Zielińska, Brzeziński 1994; Węglarczyk 1998). With a view to potential non-uni- and c5 is the skewness coeffi cient. formity in series of NQ fl ows being identifi ed, several statistical tests are recommended. To detect elements whose values stray MATERIALS more markedly from remaining values, the Grubbs-Beck test was applied. The follow- Studies were based on the series of an- ing step of a uniformity analysis procedure nual minimal discharges (NQ) from 119 examined all the records series for the inde- stream gauges closing off autochthonous pendent occurrence of elements. The Wald- basins. As the catchments referred to cover Wolfowitz test was applied to that end. The 19% of Poland, from the point of view of fi nal stage of the analysis entailed checks for the size criterion of the sample, it is possible series stationarity of selected basin samples to affi rm with 95% confi dence that this is using a rank Kruskal-Wallis sums test. enough for derived analyses to be regard- The fact that all observed cuttings were ed as credible (Boczarow 1976). Flow se- detected with a Grubbs-Beck test is of a ries (Q) [m3s-1] were transformed into their great importance, as this statistic only in equivalent series (q) [dm3s-1km-2] of specifi c fact allows for detection of elements with runoffs more suitable for comparative anal- values differing markedly from others. Next yses and for the illustration of the spatial a Wald-Wolfowitz series test and a Kruskal- variability to a phenomenon. Areas of ba- Wallis test qualifi ed the researched series as sins are between 104 and 1534 km2 (small uniform at the α = 1% level of signifi cance. and medium-sized in Polish conditions). At the same time, in accordance with Ney- Data used for the analysis come from man’s and Scott’s statement (1971), some the period 1971-1990. It was a phase of pe- probability distributions including the log- culiar “hydrological anxiety” with frequent normal, Pearson and Weibull were entirely high-water events of wide amplitude (1979, resistant to outstanding elements in a records 1982), as well as prolonged droughts extend- series. This means that the occurrence of ing over large areas of Poland (1983, 1989). elements outstanding in the sequence de- At that time, the anthropogenic infl uence on scribed by means of those distributions of hydrological processes increased. Equally random variables should be treated as to- important was the fact that relatively few tally natural, and the elements in question analyses treated low fl ows during this pe- should not be rejected. As is demonstrated riod. The twenty-year period of 1971-1990 in the following part of this paper, all exam- was in many respects atypical, however, and ined series of records present distributions from the research point of view this repre- from this particular group.

http://rcin.org.pl 44 Adam Bartnik

ample, that the rivers with minimum fl ow series ANALYSIS have a similar distribution to the Fisher-Tippett function, being grouped in areas of high reten- The results of fi tting probability distribu- tiveness. This is especially true of the rivers in tions to the studied series of annual minima young glacial areas, as well as upland rivers rel- for discharges are presented in Fig. 2. Only atively abundant in water. Lake District basins about 15% of the empirical distributions of are examples here, like those of the Glomia, studied series (17 basins) were recognised as Fisher-Tippett distributions. Likewise, only about 15% of the studied cases fi tted the gamma distribution well. Overall, in more than 70% of the examined series the log-normal distribution curve approximated the empirical distribution suffi ciently. This type of distribution may be assumed to pre- dominate in Poland. The spatial distribution of the types of probability distribution adjusted to the series of minimum yearly fl ows does not indicate any considerable order (Fig. 3). However, several Figure 2. Frequency of distribution types of ana- regularities may be observed. It seems, for ex- lysed set of annual low fl ow series (NQ).

Figure 3. Types of frequency distributions of annual low ﬂ ow series. 1 – log-normal distribution, 2 – Fisher-Tippet III distribution, 3 – gamma distribution, 4 – areas in which basins with yearly minima coming under Fisher-Tippet III distribution are grouped, 5 – areas in which basins with yearly minima coming under gamma distribution are grouped.

http://rcin.org.pl The spatial distribution of low ﬂ ows in Poland not exceeded at an assumed probability 45

Rega and Parseta, as well as some rivers fl ow- The estimators of maximum specifi c ing off the Lublin Upland, Roztocze Hills and low runoff - Nq(90%), change within the sam- Kraków-Czestochowa Upland (e.g. the Tanew, ple of basins, over a comparatively wide Warta and Liswarta). Most of these series are range (from 0.6 to 8.7 dm3s-1km-2) - Fig. 4. characterized by high negative coeffi cients of The minimum was obtained for the Ochnia skewness. basin, while the maximum appeared in ba- Many regions of Poland are found to sins of the Pomeranian Lake District (the have pairs of adjacent rivers whose minima Studnica - 8.73 dm3s-1km-2, the Wieprza - 3 -1 -2 for low fl ows assume a gamma distribu- 8.61 dm dm s km ). Nq(90%) ranged from tion (like the Myśla and Mała Ina; the Orla 1.8 to 3.4 dm3s-1km-2 in half of the studied and Czarna Struga; and the Osobłoga and cases, and the median obtained for the whole Biała). Such a situation occurs often applies sample of basins is equal to 2.57 dm3s-1km-2. in Lower Silesia and on the Wielkopolska The distribution for these runoffs shows Lowland. Beyond these areas, series of clear, if not very strong, positive skewness minimum annual fl ows to which the gamma (coeffi cient of skewness = 1.37). distribution fi ts best occur singly and rela- Some mountain areas are also repre- tively seldom. sented in the group of river basins in which

The great majority of the studied basins Nq(90%) values differ signifi cantly from the are distinguished by a lognormal distribu- average, alongside the aforementioned tion, so we can assume that this type of Lake District basins (the Biała Lądecka and distribution dominates in Poland where av- Łomnica in the Sudety Mountains and the erage-sized basins are concerned. Although Biały Dunajec in the Tatras) - Fig. 4. the regularities observed are not marked, High absolute minima are connected the author is of the opinion that they offer with the occurrence in these basins of a strong hint as regards further research and waste-mantles of great thickness and con- analysis in this area. siderable water-absorptiveness. According to Jokiel (1994), the resources of the zone of active exchange are considerable; much EXTREMES IN LOW-FLOW SERIES greater than on average in Poland. In remaining mountain areas, foothills, The spatial and temporal nature of ex- uplands and the greater part of the lake dis- tremes are very important in analyses of low tricts, runoff Nq(90%) estimators are much fl ow. Conclusions are to be drawn with care, lower, and do not exceed 4 dm3s-1km-2. Some- on account of the temporal inhomogeneity what greater maxima are only noted for the of the extremes. Thus, to maintain compara- basins in the Roztocze and Podhale regions, bility of results and eliminate this problem, as well as in the Beskid Slaski and Beskid quantiles were accepted estimators of the Żywiecki ranges. Low Nq(90%)’s also occur in minimum and maximum specifi c low run- the eastern part of the Malopolska Upland off, calculated on the basis of probability and among the Great Mazurian Lakes. distributions presented above. Specifi c low On the basis of the spatial distribution runoff of a probability not exceeding 10% for the estimator of minimum specifi c low was used as the estimator of minimum low runoff (Nq(10%)) there is found to be a quite fl ow, while the estimator of the maximum distinctive, compact area of low runoff ac- was probable runoff, measured in respect counting for a considerable part of the Polish of specifi c low runoff of probability not ex- Lowland (Wielkopolska and Mazowsze). ceeding 90%. The greatest runoffs appear consistently in

http://rcin.org.pl 46 Adam Bartnik

Figure 4. Spatial arrangement and diversity of maximal 90% specifi c low fl ow estimator – Nq(90%) [dm3·s-1·km-2] the lake districts of Pomerania and Suwałki, might also be related to low runoffs because as well as in the Tatra Mountains. Account of the identical origin of the two phenomena. should be taken of the way this relates to the The main characteristic of the Carpathian specifi c hydrogeological conditions in this area is the more major role played by falls area. The greatest minimum runoffs occur in and evaporation processes in the formation regions richest in near-surface underground of the runoff, as compared with other parts waters. For example, there are basins in ar- of Poland. A lack of prolonged dry periods eas with large aquifers, such as those in the is also an important feature of mountainous deposits of end moraine (of the Pomeranian terrain. This property, and the fact that riv- and Suwałki Lake Districts), or in creviced ers are supplied by crevice waters (springs karst rocks (of the Uplands: of Lublin and with a quite stable yield) during low-fl ow the Małopolska region). periods ensures that in mountain rivers The Carpathians (especially the West- a certain small amount of groundwater is ern Beskids) also feature elevated Nq(10%) always present. values. In this case, the estimator is diversi- The average estimator of minimum low fi ed in relation to the affl uence of aquifers runoff for the sample of basins in the period and the speed with which they become de- 1971-1990 was 0.99 dm3s-1km-2. Half of all watered. The differentiation to the environ- estimated runoffs are in the range 0.56 to mental conditioning of low fl ows was the 1.46 dm3s-1km-2. Similarly, as in the earlier reason for the regional division of this area cases, a group of values signifi cantly differ- by Tlałka (1982). The results of her work ent from average is also apparent here. The

http://rcin.org.pl The spatial distribution of low ﬂ ows in Poland not exceeded at an assumed probability 47

absolute maximum Nq(10%) was observed racy of low-fl ow estimation. The previously an- in the Studnica basin (6.36 dm3s-1km-2). alyzed estimators of extremes of low-fl ow were Large runoffs also occur in different ba- between the limits of empirical distribution sins of the Pomeranian Lake District, the (e.g. Nq(10%) - 1 per 10 years in the case of Wieprza (5.88 dm3s-1km-2) and the Brda 20-year series). The low probability, for short (4.95 dm3s-1km-2). Wielkopolska Lake Dis- sequences, can be obtained through the extrap- trict basins are characterized by the absolute olation of the theoretical function. This means lowest Nq(10%) runoffs Nq(10%). The distribu- that, for probabilities of 1, 2 and 5% (once per tion of the basins in the studied sample is 100, 50 or 20 years), we are not able to obtain positively asymmetric (skewness coeffi - quantiles on the basis of the empirical distribu- cient = 2.17) - Fig. 5. tion. These were counted by using the distribution functions of the probability, which had been fi tted previously. LOW-FLOWS NOT EXCEEDED Frequencies of specifi c low runoffs of the AT THE PROBABILITY OF 1% assumed probability not exceeding 1% were counted. More than 70% of all counted values As available observed fl ow records are nor- (85 basins) are below 1 dm3s-1km-2, and only mally insuffi cient for reliable frequency quan- 7% are runoffs of more than 2 dm3s-1km-2. tifi cation of extreme low-fl ow events, different Among the counted runoffs of probability 1%, types of theoretical distribution functions are there are also values below zero. Theoreti- used to extrapolate beyond the limits of “ob- cally, there is a 1% probability (8 basins) that served” probabilities and to improve the accu- these rivers can periodically be dry. However,

Figure 5. Spatial arrangement and diversity of minimal 10% speciﬁ c low ﬂ ow estimator – Nq(10%) [dm3s-1km-2].

http://rcin.org.pl 48 Adam Bartnik it should be remembered that this fi nding can also be found in the Atlas Hydrologiczny Polski only be the result of the mathematical smooth- [Hydrological Atlas of Poland] (1986). Despite ing points. Moreover, the lowest fl ows usually the fact that this map relates to another time pe- have the largest errors, because of rating curve riod (1951-1975), it is practically identical to inaccuracy. Nevertheless, it is visible that values the map presented here. The smallest runoffs below or close to zero are grouped in the ma- (below 0.05 dm3s-1km-2) occur in Wielkopolska jority of cases in the Wielkopolska region and and in the eastern part of the Mazowsze Low- in Mazowsze. These are areas poorest in water land as well. The largest (as in this paper) – are (e.g. the Rivers Ochnia, Liwiec, Gąsawka and in the Pomeranian Lake District, in the Tatra Noteć). and Sudety Mountains, and in the centre of the The highest specifi c low runoffs, counted Silesian Upland. This shows that the distribu- for low-fl ow of probability 1% Nq(1%) are lo- tion of low runoffs does not depend on climatic cally greater than 4 dm3s-1km-2, and they gener- conditions, but refl ects overlapping hydrogeo- ally occur in basins abundant in water situated logical and climatic conditions, and outfl ow due in the middle of the Pomeranian Lake District. to anthropogenic activity. These are areas of end moraine of particularly The maps presented here can also be great thickness, as well as waterlogging (along used as a background to more detailed inter- the Studnica, Wieprza and Brda, for example) regional analysis. Observed regularities can - Fig. 6. also be helpful in various designs and tasks A similar map of minimum specifi c low relating to opinions on water resources for runoffs at a 1% exceedance probability can the national economy.

Figure 6. Spatial arrangement and diversity of speciﬁ c low ﬂ ows not exceeded at 1% probability – Nq(1%) [dm3s-1km-2].

http://rcin.org.pl The spatial distribution of low ﬂ ows in Poland not exceeded at an assumed probability 49

Durrans, S.R. (1996), Low-fl ow analysis with REFERENCES a conditional Webull tail model, Water Resources Research 32 (6): 1749–1760. Beran, M.A., Rodier, J.A. (1985), Hydrologi- FREND: Flow Regimes From Experimental cal aspects of drought, UNESCO-WMO And Network Data (1989), I: Hydrologi- Studies and Reports in Hydrology, vol. cal Studies; II: Hydrological Data, Wall- 39. ingford, UK. Boczarow, M.K. (1976), Metody statystyki Atlas Hydrologiczny Polski [Hydrological matematycznej w geografi i [Methods of Atlas of Poland] (1986), Stachý J. (ed.), mathematical statistics in geography], Wydawnictwo Instytutu Meteorologii Państwowe Wydawnictwa Naukowe i Gospodarki Wodnej (IMGW), Warsza- (PWN), Warszawa. wa. Brzeziński, J. (1995), Obliczanie przepły- Jakubowski, W. (2005) Rozkłady prawdopo- wów maksymalnych rocznych o okre- dobieństwa niżówek maksymalnych [Di- ślonym prawdopodobieństwie przewyż- stributions of low fl ow maxima], Prze- szenia dla długich ciągów pomiarowych gląd Geofi zyczny 3-4, Warszawa. [Counting the maximal yearly fl ows with Jozeph, E.S. (1970), Probability distribution a defi ned probability of exceedance for of annual droughts, Journal of the Irri- the long measurment sequences], Gospo- gation and Drainage Division, ASCE 96 darka Wodna, 8, Warszawa: 201-205. (4): 461–474. Bulu, A., Onoz, B. (1997), Frequency analy- Kaczmarek, Z. (1960), Przedział ufności sis of low fl ows by the PPCC tests in Tur- jako miara dokładności oszacowania key. FRIEND’97- Regional Hydrology: prawdopodobnych przepływów pow- Concepts and Models for Sustainable Wa- odziowych [Bracket of confi dence as ter Resources Management, International a measure of accuracy in estimating the Associations of Hydrological Sciences probable fl ood fl ows], Wiadomości Służ- (IAHS) Publications, 246: 133–140. by Hydrologicznej i Meteorologicznej, 7, Byczkowski, A. (1972), Hydrologiczne 4., Warszawa: 133-185. podstawy projektowania budowli wodno- Kaczmarek, Z. (1970), Metody statystycz- melioracyjnych. Przepływy ekstremalne ne w hydrologii i meteorologii [Statistical [Hydrological basis for designing hy- methods in hydrology and meteorology], dromelioration facilities. Extreme fl ows], Wydawnictwo Komunikacji i Łączności Państwowe Wydawnctwo Rolnicze i Le- (WKiŁ), Warszawa. śne (PWRiL), Warszawa. Khan, L.R., Mawdsley, J.A. (1988), Reliable Byczkowski, A. (1996), Hydrologia [Hy- yield of unconfi rmed aquifers, Hydrolog- drology], Wydawnictwo Szkoły Głów- ical Sciences Journal 33 (2): 151–171. nej Gospodarstwa Wiejskiego (SGGW), Kite, G.W. (1988), Frequency And Risk Warszawa. Analyses In Hydrology, Water Resources Condie, R., Nix, G. (1975), Modeling of Publications, Littleton. Low Flow Frequency Distributions and Lawal, S.A., Watt, W.E. (1996a), Frequency Parameters Estimation, International analysis of low-fl ows using the Akaike Water Resources Association, Proceed- information criterion, Canadian Journal ings of the Symposioum “On Water for of Civil Engineering 23: 1180–1189. Arid Lands”, 8-9 Dec. 1975, Teheran. Lawal, S.A., Watt, W.E. (1996b), Non-zero lower limit in low fl ow frequency analysis, Water Resour. Bull. 32 (6): 1159–1166.

http://rcin.org.pl 50 Adam Bartnik

Leppajarvi, R. (1989), Frequency analysis fl ows of Malawian rivers, Hydrological of fl ood and low fl ow. FRIENDS in Hy- Processes 2: 75–91. drology, International Association of Hy- Singh, V.P. (1987), Derivation of extreme drological Sciences (IAHS) Publications value (EV) type III distribution for low No. 187: 435–442. fl ows using entropy, Hydrological Sci- Loaiciga, H.A., Michaelsen, J., Hudak, P.F. ences Journal 32 (4): 521–533. (1992), Truncated distributions in hydro- Smakhtin, V.U. (2001), Low fl ow hydrol- logic analysis, Water Resources Bulletin ogy: a review, Journal of Hydrology 240: 28 (5): 853–863. 147–186. Loganathan, G.V., Kuo, C.Y., McCormic, Tlałka, A. (1982), Przestrzenne zróżni- T.C. (1985), Frequency analysis of low cowanie niżówek letnich w dorzeczu fl ows, Nordic Hydrology 16: 105–128. górnej Wisły [Spatial diversifi cation of Matalas, N.C. (1963), Probability distribu- low summer fl ows in the Upper Vistula tion of low fl ows, US Geological Survey River catchment], Rozprawy habilita- Professional Paper 434-A, Washington, cyjne Uniwersytetu Jagiellońskiego, 63, DC. Wydawnictwo UJ, Kraków. McMahon, T.A., Mein, R.G. (1986), River Waylen, P.R., Woo, M.-K. (1987), Annual and reservoir yield, Water Resources low fl ows generated by mixed processes, Publications, Littleton, Colorado. Hydrological Sciences Journal 32 (3): Nathan, R.J., McMahon, T.A. (1990), Practi- 371–383. cal aspects of low-fl ow frequency analy- Węglarczyk, S. (1998), Wybrane problemy sis, Water Resources Research 26 (9): hydrologii stochastycznej [Selected prob- 2135–2141. lems of stochastic hydrology], Series: Neyman J., Scott E. (1971), Outlier hydro-sanitary engineering, Monografi a proneness of phenomena and of related 235, Wydawnictwo Politechniki Kra- distributions, in Rustagi J. S. (ed.), Op- kowskiej Kraków. timizing Methods in Statistics, Academic Zasady obliczania największych przepływów Press, New York: 413-430. rocznych o określonym prawdopodobień- Ozga-Zielińska, M., Brzeziński, J. stwie pojawiania się przy projektowaniu (1997), Hydrologia stosowana [Applied obiektów inżynierskich i urządzeń tech- hydrology], Państwowe Wydawnictwa nicznych gospodarki wodnej w zakresie Naukowe (PWN), Warszawa. budownictwa hydrotechnicznego [The Polarski, M. (1989), Fitting distributions to principles for counting the largest annual annual minimum fl ows of different dura- fl ows with a defi ned probability of occur- tions, International Association of Hy- rence while designing the ingineer facili- drological Sciences (IAHS) Publications ties and technical installations for water No. 187: 97–104. management as regards the hydrotechnic Prakash, A. (1981), Statistical determination construction] (1973), Załącznik nr 26, of design low fl ows, Journal of Hydrol- Zarządzenie Prezesa CUGW z dn. 9 lipca ogy 51: 109–118. 1968 [Annex no. 26, Regulation of the Russell, D.S.O. (1992), Estimating fl ows President of the Central Offi ce for Water from limited data, Canadian Journal of Management], Wydawnictwo Katalogów Civil Engineering 19 (1): 51–58. i Cenników, Warszawa. Sefe, F.T.K. (1988), Some aspects of the hydrology of one-day annual minimum low

http://rcin.org.pl