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The variability of the curve number (^{2} to the gauging station) in central Poland has been assessed using the probabilistic approach: distribution fitting and confidence intervals (CIs). Empirical

For the estimation of design floods and volumes, often, event-based hydrological models are applied. As inputs, they often use the components of rainfall, infiltration, soil properties, land use,

The rainfall-runoff curve number method was introduced in the 1950s by the United States Department of Agriculture-Soil Conservation Service (USDA-SCS, now NRCS (Natural Resources Conservation Service)) to estimate the runoff depth from small catchments. It is included also in many complex models, e.g., for the estimation of the rainstorm-generated sediment yield [

The method allows the relationship among the rainfall depth, _{a}, the initial abstraction ratio _{a}/S_{tab}

The primary source reference describing the _{a}/S

In practice, a variability of

The asymptotic approach recognizes the drift of the

The Zagożdżonka watershed, presented in this paper, is, since 1962, an experimental watershed of the Department of Water Engineering of the Warsaw University of Life Sciences (Szkoła Główna Gospodarstwa Wiejskiego), Poland. The studies of

The aim of the paper was to assess the variability of _{tab}

The paper is structured as follows: in the second part, the watershed is described; in the third section, the data and methods used are depicted; in the fourth section, the results are presented and discussed, and the last part contains the conclusions of the study.

The Zagożdżonka watershed is located in central Poland, with an area of 23.4 km^{2} to the Czarna gauging station (_{tab}_{tab}^{3}/s) and July (0.166 m^{3}/s), respectively, and the mean discharge is 0.277 m^{3}/s. The SCS-CN method could be adapted here, due to the small area and agriculture-dominated land use [

The Zagożdżonka watershed with the Czarna gauging station, Poland.

The study was carried out using empirical

The calibration of the initial abstraction coefficient, _{tab}_{theor}

Eventually, the main

This data set of 34 elements selected for further analysis had a large range of rainfall events, _{50%} = 37 mm. The rainfall events of a depth not larger than 37 mm were considered as moderate rainfalls and larger than 37 mm as heavy rainfalls. We used another set of data from the period 1963–2011, namely the annual maxima of daily rainfall depth for the estimation of _{50%}. Eventually, three cases, A, B, C, of

The identification of the theoretical distribution function was executed with statistical tests, to estimate the quantiles of _{S}_{CN} , the densities of

Empirical

Subsequently, the families _{1}_{n}

The maximum of

The candidates were the distributions, which passed the Kolmogorov–Smirnov (KS) [

The above three tests compare _{n}_{i=1,…,n}|_{i},θ_{n}_{i}

The AD test gives more weight to the tails than the KS and CM tests do, which was essential in further interval estimation. Therefore, the AD test was joined to the study, although it was not used in the further selection criteria. If the distribution was supposed to be normal, then the sample was additionally required to pass the Shapiro–Wilk (SW) test [

The identification of the distribution closest to the one supposed to have generated the data was performed using two criteria. The criterion ^{2})

This value is recommended for flood discharge estimation [_{i}_{i}

We faced the problem that the above tests allowed for accepting various distributions. Furthermore, measures

The confidence interval (

On the other hand, three moisture conditions in the SCS-CN model are represented by the antecedent runoff conditions (ARCs): _{tab}

These formulas preserve the negative skewness of _{Haw} − CN_{tab} < CN_{tab} − CN_{Haw}

The Hjelmfelt proposition [

In the following study, the assumption of lognormality of _{Hjel}_{10%}_{Hjel}_{90%}
_{10%} and _{90%} are the 10% and 90% theoretical quantiles of

The results of the analysis of Samples A, B and C were collected in _{tab}_{tab}_{tab}

Basic characteristics of Samples A, B and C.

Sample | Mean Value | Median | Coeff. of Var. (%) | Range | Skewnes | |||||
---|---|---|---|---|---|---|---|---|---|---|

A | 81.1 | 75.8 | 83.6 | 75.2 | 59.1 | 14.1 | 233.5 | 44 | 1 | |

B | 62.3 | 81.1 | 63.6 | 80 | 51.8 | 10 | 124 | 29.9 | 0.4 | |

C | 130.2 | 66.1 | 127.2 | 66.6 | 36.1 | 11.8 | 175.8 | 27.3 | 1 |

Division into two subsamples revealed a discrepancy: the mean and median were larger in B and smaller in C when compared to _{tab}

The families, _{S}, F_{100−CN} , of competing distributions of the variables, _{S}_{100−CN} = {

The values of

GEV | N | GLO | WE | LN | G | GEV | N | GLO | WE | LN | G | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A | 33.2 | 33.1 | 33.1 | × | 365.1 | ||||||||

B | × | 34.9 | × | × | 219.6 | × | × | ||||||

C | 29.8 | 29.6 | 30.7 | × | 130.7 | 130.9 | 130.8 | ||||||

A | 19.1 | × | × | × | 263.8 | × | × | × | |||||

B | × | × | × | 160.7 | × | × | × | ||||||

C | 25.5 | 37.4 | 33.1 | 26.5 | 88.6 | 88.7 | 88.2 | 88.1 |

Note:

Criterion

Criterion

The differences in measures were small between distributions; therefore, the first three, GEV, GLO, LN for

The GEV distribution is featured by a thick, moderate or thin tail. It relates to the shape parameter greater than zero (Frechet distribution), when the pdf decreases slower than exponential, equal to zero (Gumbel distribution) or lower than zero when the pdf has an upper limit (reversed Weibull distribution). In Cases A, B and C, the results for 100-

A similar analysis, based on events with the largest discharges, was performed by McCuen [

Subsequently, the 95% and 99% confidence limits of _{Hjel}_{Hjel}

Equations (12) and (13) for the ARC of Hawkins led to:
_{Haw}_{Haw}

The above values were located approximately symmetrically around _{tab}

Case A (_{tab}_{tab}_{Haw}_{Haw}_{Haw}_{Hjel}_{Haw}_{Hjel}

Case B (17 _{tab}_{tab}_{Haw}_{Haw}_{Haw}_{Hjel}_{tab}_{Haw}_{Haw}

Case C (_{tab}_{tab}_{tab}_{Hjel}_{tab}_{Haw}_{Haw}_{Hjel}_{Haw}_{Hjel}_{tab}_{Haw}_{Haw}

Characteristics of the distribution of

A | B | C | |||||||
---|---|---|---|---|---|---|---|---|---|

Characteristic | GEV | N | GLO | GEV | N | GLO | GEV | N | GLO |

median | 76.2 | 75.8 | 76.6 | 81.1 | 81.1 | 81.8 | 66.5 | 66.1 | 67.3 |

_{Hjel} |
62.0 | 62.1 | 61.3 | 70.9 | 70.7 | 69.8 | 56.3 | 56.1 | 55.0 |

_{Hjel} |
89.1 | 89.5 | 88.8 | 91.3 | 91.5 | 91.0 | 75.4 | 76.1 | 75.3 |

_{95%}^{(∗)} |
57.9 | 58.2 | 55.7 | 68.3 | 67.8 | 65.4 | 53.3 | 53.3 | 50.3 |

_{95%}^{(∗)} |
92.4 | 93.4 | 92.2 | 94.0 | 94.4 | 93.5 | 77.6 | 78.9 | 77.1 |

_{99%} |
50.9 | 50.9 | 43.3 | 64.0 | 62.2 | 55.7 | 47.9 | 48.0 | 39.7 |

_{99%} |
98.3 | 99.0 | 99.0 | 99.0 | 99.0 | 98.3 | 81.5 | 84.3 | 80.1 |

Note: ^{(∗)}

The tabulated _{Hjel}_{Hjel}_{Haw}_{Haw}_{tab}_{Haw}_{Haw}

A general observation is that in Case A, both of the ARCs of Hawkins were covered by all three CIs, while in Case B, the ARC for dry conditions and in C the ARC for wet conditions were out of the CIs. Large differences between the ARCs of Hawkins and Hjelmfelt indicate the inapplicability of _{Haw}_{Haw}_{tab}

The study completed by McCuen [

To support our conclusions, an analogous study was conducted for the Stupavský catchment, located in the Inner Western Carpathians in Slovakia. The stream length is 16.7 km and the catchment area 33 km^{2} for the gauging station. The catchment is forested, with a forest cover ratio equal to 0.9. The tabulated _{tab}_{Haw}_{Haw}_{Haw}_{tab}_{Haw}_{Haw}_{Haw}

A comparison of the ARCs of Hawkins and Hjelmfelt is the motivation for the formulation of the statement that even the perfect calibration of the SCS-CN model, based on soil and land use and on empirical

The conclusions, presented below, are catchment specific. To observe the

The main conclusion follows from the comparisons of _{tab}_{tab}_{Haw}

Generally, ARCs should be applied with caution. Even if they agree with the 10% and 90% quantiles for all collected rainfall-runoff data, which is what suggests the justification for their use, a serious inconsistency is observed in subsamples. It is worth pointing out that, although the inconsistency of

instead of _{Haw}_{Hjel}

instead of _{Haw}_{Hjel}

Moreover, the Hjelmfelt method may be simplified. The direct calculation of quantiles of

The authors are thankful to all the reviewers for the insightful comments that highly improved the paper. The investigation described in the contribution has been partially performed by the second author’s research visit to the Technical University of Bratislava, Slovakia, in 2012 within an STSM (Short Term Scientific Mission) of the COST (European Cooperation in Science and Technology) Action ES0901. The data have been provided by research project No. N N305 396238, founded by the Ministry of Science and Higher Education in Poland, as well as by the research project, KORANET-EURRO-KPS (the Korean scientific cooperation network with the European Research Area - Estimation of Uncertainty in Rainfall RunOff modeling - Korea, Poland and Slovakia), founded by PL-National Center of Research and Development (Narodowe Centrum Badan´ i Rozwoju, Poland). The research was also supported by the University of Agriculture, Cracow, Poland. The compilation of the Slovakian data set was partially supported by the Agency for Research and Development under contract No. 0496-10. The support provided by these organizations is gratefully acknowledged.

The first author has been responsible for field data collecting and preparing them to analysis, including calibration of

The authors declare no conflict of interest.