Probabilistic properties of the Curve Number Agnieszka Rutkowska 1) , Kazimierz Banasik 2) , Silvia Kohnova 3) , Beata Karabova 3) European Geosciences Union General Assembly, Vienna, 07-12 April 2013 1) Department of Applied Mathematics, University of Agriculture in Krakow, Poland 2) Department of Water Engineering, Warsaw University of Life Sciences - SGGW, Poland 3) Department of Land and Water Resources Management, Slovak University of Technology in Bratislava, Slovakia 1 The main problem What is the variability of the Curve Number in the SCS-CN method? 2 Abstract The SCS-CN method allows to predict the runoff volume in small, ungauged catchments. Due to observed variability of the CN, its probabilistic properties in Slovakian and Polish catchments were analyzed. Results contain: the determination of the theoretical distribution function, confidence intervals, comparison to ARC I and ARC III and the asymptotic fitting. 3 Catchments A. The Zago ˙ z d˙ z onka River, centre of Poland B. Carpathian Slovakian catchments: Stupavsk´ y, Raˇ ciansk´ y, Gidra, Viˇ stuk, Petrinovec, C. Carpathian Polish catchments: Poniczanka, Mszanka, Kasinianka, Lubie´ nka, Skawica. 4 The SCS-CN method The SCS-CN equations: H =                                (P −λS ) 2 P +(1−λ)S if P ≥ λS , 0 otherwise and S = 254( 100 CN − 1) H − the direct runoff, P − the rainfall depth, S − the watershed storage parameter λ− the initial abstraction ratio. CN depends on land use and soil type, in the model is assumed to be constant and is tabulated ⇒ CN theor In practiceCN varies and is a random variable 5 Sample analysis The solution of the SCS-CN equations if λ =0.2 is CN = 25400 S +254 where S = 5(P +2H − √ 4H 2 +5PH ) For events (P i , H i ) we get a sample CN i . The empirical density function of the CN is negatively skewed The variable 100 − CN is positively skewed. It was investigated to fit a typical theoretical distribution function [3]. 6 Distribution fitting • Statistical tests of goodness of fit: Kolmogorov-Smirnov (KS), Cramer-von Mises (CM), Anderson-Darling, Shapiro-Wilk. • Quantile-Quantile plots and correlation coeff. r . Criteria: (a) W = KS + CM +1 − r 2 achieves minimum (b) Akaike Information Criterion (AIC) Theoretical distribution for 100 − CN is Generalized Extreme Value. 7 Confidence intervals and ARC I, III Confidence intervals indicate the most probable CN s. The comparison to the CN (I ) and CN (III ) conditions (Hawkins formula) implies the SCS-CN model may • overestimate the direct runoff (Stupavsk´ y, Raˇ ciansk´ y) • underestimate the direct runoff (Petrinovec, Poniczanka, Skawica). 8 The asymptotic fitting The drift of CN s with increasing rainfall depth [2] is recognized using lim P ′ →∞ CN (P ′ , H ′ ) where P ′ , H ′ are independently sorted P and H . Stupavsk´ y: CN (P ) = 41.23 + 58, 77e −0.02P Petrinovec: CN (P ) = 63.53 + 36.47e −0.03P Zago ˙ z d˙ z onka (result of Banasik, Woodward [1]): CN (P ) = 74.03 + 25.97e −0.06P 9 Acknowledgments The investigation described in the contribution has been initiated by first Author research visit to Technical University of Bratislava in 2012 within a STSM of the COST Action ES0901. Polish data used here have been provided by research project no. N N305 396238 founded by PL-Ministry of Science and Higher Education. This work was also supported by the Slovak Research and Development Agency under the contract No. APVV-0015-10 and APVV-0496-10. The support provided by the organizations is gratefully acknowledged. 10 References [1] Banasik K., Woodward D. (2010) Empirical determination of runoff Curve Number for a small agricultural watershed in Poland 2nd Joint Federal Interagency Conference, Las Vegas, NV, June 27 - July 1. [2] Hawkins RH. (1993) Asymptotic determination of curve numbers from data Journal of Irrigation and Drainage Division, American Society of Civil Engineers, 119(2), p. 334-345. [3] McCuen R. (2002) Approach to Confidence Interval Estimation for Curve Numbers Journal of Hydrologic Engineering, 7, p. 43-48, DOI: 10.1061/(ASCE)1084-0699(2002)7:1(43).