# Possibility of Using Selected Rainfall-Runoff Models for Determining the Design Hydrograph in Mountainous Catchments: A Case Study in Poland

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Research Area Characteristic

^{2}. The length of the main watercourse to the outlet cross-section is 15.0 km. The average catchment slope is 8%. The density of the river network is 0.70 km

^{−1}. This information was derived using a Digital Elevation Model (DEM) provided by USGS—United States Geological Survey, with a grid resolution of about 25 m. Based on the Hydrographic Division Map of Poland 2010 and on the Corine Land Cover 2018 database, the following forms of land uses were identified together with the percentage of the catchment area respectively occupied. Loose urban buildings (4.5%), arable lands without irrigation (5.1%), meadows and pastures (14.4%), areas occupied mainly by agriculture with a high proportion of natural vegetation (2.1%), coniferous forests (29.8%), mixed forests (43.4%), and deciduous forests (0.2%). The case study is dominated by poorly permeable and impermeable soils. The average annual rainfall in the catchment area exceeds 765 mm. The average annual temperature is 6.8 °C. Figure 1 shows the land use and the topography of the catchment.

#### 2.2. Hydrometeorological Data Verification

_{max}) and the maximum annual flow (Q

_{max}), using the Mann-Kendall test (MK). The H

_{0}null hypothesis of the test assumes no monotonic data trend, while the H

_{1}alternative states that such a trend exists. The calculations were done for the significance level α = 0.05. The S Mann-Kendall statistics were determined based on the equation [20,21]:

- n—number of elements in the time series.

_{max}and Q

_{max}analysis, such relationships may occur, which in consequence leads to an underestimation of the Var(S) variance. Therefore, a correction for correction of variance has been included, calculated only for data with significant partial autocorrelation:

- Var*(S)—corrected variance;
- n—the real number of observation;
- ${n}_{s}^{*}$—effective number of observations calculated as:$$\frac{n}{{n}_{s}^{*}}=1+\frac{2}{n(n-1)(n-2)}\xb7{\Sigma}_{k=1}^{n-1}(n-k)(n-k-1)(n-k-2){\rho}_{k}$$

- k—next group with repeating elements;
- ρk—value of the next significant autocorrelation coefficient.

#### 2.3. Calculation of Maximum Annual Rainfall and Flows with Specific Occurrence Frequency

_{T}) is caused by the occurrence of rainfall with the same frequency [22]. The Q

_{T}flows were determined in order to assess the quality of the hydrological rainfall-runoff models. The tests were performed for the frequencies related to the return periods of 500, 100, and 10 years. The calculations were performed applying the log-normal distribution, which is described as [23]:

- x
_{p}—quantile of the theoretical log-normal distribution; - ε—lower string limit;
- erf(2(1 − p) − 1)—Gauss error function.

#### 2.4. Determination of the Design Hydrograph

- t
_{o}—wave fall time (h); - t
_{s}—wave rise time (h);

- P
_{e}—excess rainfall (mm); - P—total rainfall (mm);
- S—maximum potential catchment retention (mm).

_{L}= C

_{t}·(L·L

_{c})

^{0.3}

- T
_{L}—delay time (h); - C
_{t}—factor related to catchment retention (-); - L—maximum distance along the watercourse from the outlet cross-section to the drainage divide (km);
- L
_{c}—distance along the main watercourse from the outlet cross-section to the centroid of the catchment (km).

- Q
_{p}—peak flow of the unit hydrograph (m^{3}·s^{−1}·mm); - C
_{p}—empirical coefficient resulting from the simplification of the hydrograph to triangular shape (-); - A—catchment area (km
^{2}).

- q
_{p}—peak flow of the unit hydrograph (m^{3}·s^{−1}·mm); - c—conversion factor (c = 0.208) (-);
- T
_{p}—flood rise time, (h), calculated as:$${T}_{P}=\frac{D}{2}+{T}_{LAG}$$

- D—duration of excess rainfall (h);
- T
_{LAG}—lag time in the SCS-UH method, (h), calculated as:$${T}_{LAG}=\frac{{(3.28\xb7L\xb71000)}^{0.8}\xb7{(\frac{1000}{CN}-9)}^{0.7}}{1900\xb7\sqrt{I}}$$

- L—maximum length of the runoff path (km);
- CN—Curve Number value (-);
- I—average catchment slope (%).

- q
_{0}—infiltration indicator; - t
_{p}—ponding time; - K
_{s}—saturated hydraulic conductivity; - I—cumulative infiltration;
- Δθ—change in soil-water content between the initial value and the field saturated soil-water content;
- ΔH—difference between the pressure head at the soil surface and the matric pressure head at the moving wetting front.

_{s}parameter. At the beginning, the value of this parameter is assumed based on the case study soil group. Then the cumulative infiltration is calculated, and its value is compared to that obtained from the NRCS-CN method. The value of the K

_{s}parameter changes until the cumulative infiltration from Equation (17) is equal to that calculated using the NRCS-CN method. After determining the amount of excess rainfall, the instantaneous unit hydrograph is determined based on the width function described by the relationship [38]:

- L
_{c}, L_{h}—hillslope and channel flow paths, functions of DEM cell x, respectively; - V
_{c}, V_{h}—runoff velocity for hillslope cells and flow channel cells.

- A—catchment area (km
^{2}); - T—duration of rainfall (h);
- P
_{n}(t)—excess rainfall determined by the CN4GA method (mm/h).

#### 2.5. Assessment of Quality of Analysed Hydrological Models

- Q
_{m,max}—maximum flow with a certain frequency of occurrence, calculated using rainfall-runoff models (m^{3}·s^{−1}); - Q
_{s,max}—maximum flow with a specified frequency of occurrence, calculated using the log-normal distribution on observed data (m^{3}·s^{−1}).

## 3. Results

#### 3.1. Hydrometeorological Data Verification

_{c}Mann-Kendall statistics, which assume smaller quantities than the critical value, for the significance level α = 0.05 at 1.96. In the case of the results for the P

_{max}observation series, the effective number of observations (n/n *) takes values other than 0, which indicates autocorrelation between individual variables. However, it does not affect the conclusions made using the analyzed test. Similar results for rainfall in southern Poland were obtained by Niedźwiedz et al. [40] and Młyński et al. [41]. These studies showed that the rainfall in this region is characterized by growing but not statistically significant trends, and all changes in the amount of rainfall are caused by irregular fluctuations. Slight trends in changes in the amount of rainfall are reflected in the flood flows in the upper Vistula catchment. Research conducted by Kundzewicz et al. [42], Wałęga et al. [43] and Młyński et al. [44] have shown a steady state of flood flows in southern Poland over the last several years. Due to their geological structure, morphometric characteristics, and land use, mountainous basins of the upper Vistula river catchment are sensitive to the occurrence of intense rainfall. Hence, the rhythm of their course is repeated by the rhythm of rainfall [45]. Therefore, it can be pointed out that there is a relationship between the maximum annual daily rainfall and the maximum flow.

#### 3.2. Determination of Rainfall and Peak Flows at a Specific Occurrence Frequency

^{3}∙s

^{−1}for return periods of 500, 100, and 10 years, respectively. The use of log-normal distribution in the analysis for calculating the maximum daily annual rainfall with a specific frequency of occurrence was supported by research conducted by Młyński et al. [46] It has been shown that this function is best suited to the empirical rainfall distribution P

_{max}, hence it can be the basis for determining the course of design rainfall. In the work Młyński et al. [47] it has been shown also that the same function describes at best the empirical distributions of Q

_{max}flows in the upper Vistula catchment. The log-normal distribution belongs to the family of right-handed heavy-tail functions, hence it is widely used to describe extreme hydrometeorological phenomena, as supported by Kuczera [48] and Strupczewski et al. [49]

#### 3.3. Determination of Design Hydrographs Employing the Selected Rainfall-Runoff Models

^{3}, 1 million m

^{3}and ca. 0.4 million m

^{3}for return periods of 500, 100, and 10 years, respectively. Similar volumes result from the fact that the basic information on excess rainfall in the CN4GA model are values determined using NRCS-CN. The CN4GA method first calculates the cumulative excess rainfall using the NRCS-CN method. The CN4GA model is then used to determine the distribution of the excess rainfall over time by taking into account the variable infiltration capacity during soil in the catchment. The shorter peak flow time is the result of the distribution of excess rainfall over time, determined by the CN4GA method (Figure 5). The duration of this fallout is 0.75 h (500 and 100 years return period) and 0.5 h (10 years return period) where for the NRCS-CN method it is 2.8 h and 2.5 h, respectively. Therefore, the effective rainfall intensity for CN4GA is concentrated over a shorter period of time, which is reflected by the shape of the design hydrograph. The slenderness factor, whose values in each case are greater than 1.0, indicates that the water level increased rapidly until the peak and then it fell slowly. Hence, it can be concluded that there is an imbalance between volumes for the rising and falling parts, where the falling volume was in each case larger. Due to its simplicity, the EBA4SUB model can be alternatively used to determine the shape of design hydrographs in uncontrolled catchments. Its use consists basically of three stages: determining the distribution of the design rainfall, determining the excess rainfall, and determining the design hydrograph from the catchment. The simplicity of application of the model in hydrological analyses means that it can be successfully used by practitioners to determine flood hazard zones or to determine the size of reliable flows for the design of hydrotechnical constructions. All analysed models have some limitations. They are sensitive to changes in the CN parameter, which determines the amount of excess rainfall, which translates into the values of the peak flow. Research carried out by Maidment and Hoogerwerf [53] showed that an increase in the CN parameter by 1% increases the peak flow determined from the model also by approximately 1%. In the case of the Snyder model, the coefficient values depending on the retention capacity of the catchment must be estimated for its identification. Currently, there are no specific guidelines indicating which values of these parameters should be assumed in relation to particular characteristics of the catchment, which is why designers do it in a subjective way. The first attempts to make the parameters of the Snyder model dependent on the catchment characteristics in Polish conditions were carried out by Wałęga [54], but the obtained results require verification in the Carpathian catchments. Changes in the values of individual parameters may, as a consequence, affect the sizes of floodplains or planned water management facilities. The conducted research allowed to state that the EBA4SUB model does not have such limitations. Model parameters are determined directly on the basis of physiographic characteristics of the catchment, which precludes the subjectivity of their determination.

#### 3.4. Evaluation of the Quality of Analysed Hydrological Models

_{T}values determined using the Snyder and NRCS-UH models are lower, as respect to the corresponding values determined using the log-normal distribution. For the EBA4SUB model, the calculated Q

_{500}and Q

_{100}are higher than the statistical method. Based on the values listed in Table 4, it was found that for Q

_{500}, the NRCS-UH (PRF 600) model was characterized by the smallest relative error. It should be emphasized, however, that the average value of this error, for all return periods, is lower for the EBA4SUB and NRCS-UH (PRF 600) models and amounted to 26% for both. For Snyder it was 33%, while for NRCS-UH (PRF 484) it was 38%. The calculations carried out confirm that the EBA4SUB model can be an alternative to the Snyder and NRCS-UH models. In the case of lower return periods, the EBA4SUB model gives peak flows more similar to the observed ones. This provides some security for uncontrolled catchments, where there is no hydrometric information. This could reduce the risk of designating too-narrow flood hazard zones or undersizing hydrotechnical constructions in the light of weather phenomena, the course of which is becoming increasingly extreme.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Excess rainfall hyetographs against the background of the total rainfall determined by the following methods: (

**a**) NRCS-CN for return period 500 years; (

**b**) NRCS-CN for return period 100 years; (

**c**) NRCS-CN for return period 10 years; (

**d**) CN4GA for return period 500 years; (

**e**) CN4GA for return period 100 years; (

**f**) CN4GA for return period 10 years.

**Figure 8.**Q

_{T}values obtained with the analysed hydrological models against the background of log-normal distribution.

**Table 1.**The results of the significance analysis of the trend for the observation series P

_{max}and Q

_{max}in the Grajcarek catchment.

Characteristic | Z_{c} | p_{c} | Var_{c} | n/n * | Z | p | Var |
---|---|---|---|---|---|---|---|

P_{max} | 1.582 | 0.114 | 7721.554 | 0.846 | 1.455 | 0.146 | 9129.333 |

Q_{max} | 1.143 | 0.253 | 9766.667 | 1.000 | 1.143 | 0.253 | 9766.667 |

_{c}—modified value of normalized MK statistics; p

_{c}—modified value of test probability, Var

_{c}—modified value of variance, n/n *—effective number of observations, Z—value of normalized statistics MK, p—test probability, Var—variance.

Return Period | T_{c} (h) | CN | P (mm) | P_{net} (mm) |
---|---|---|---|---|

500 | 4 | 68.1 | 95.8 | 24.7 |

100 | 76.6 | 14.6 | ||

10 | 50.2 | 4.1 |

Characteristic | Snyder | NRCS-UH | EBA4SUB | ||||||
---|---|---|---|---|---|---|---|---|---|

500 | 100 | 10 | 500 | 100 | 10 | 500 | 100 | 10 | |

Q_{max} [m^{3}·s^{−1}] | 122.909 | 74.191 | 21.975 | 113.235 * | 68.424 * | 20.329 * | 212.531 | 125.602 | 35.779 |

135.483 ** | 82.059 ** | 24.529 ** | |||||||

V [mln m^{3}] | 2.291 | 1.377 | 0.403 | 2.291 | 1.377 | 0.403 | 2.078 | 1.226 | 0.346 |

t [h] | 15.250 | 15.250 | 15.000 | 21.750 * | 21.750 * | 21.500 * | 6.800 | 6.800 | 6.500 |

16.000 ** | 15.750 ** | 15.500 ** | |||||||

α [–] | 2.100 | 1.952 | 1.905 | 3.150 * | 3.000 * | 2.950 * | 1.545 | 1.545 | 1.500 |

2.095 ** | 2.095 ** | 2.150 ** |

Model | MAPE [%] | ||
---|---|---|---|

500 | 100 | 10 | |

Snyder | 22.0 | 26.9 | 50.5 |

NRCS-UH (PRF 484) | 28.2 | 32.6 | 54.2 |

NRCS-UH (PRF 600) | 14.1 | 19.1 | 44.7 |

EBA4SUB | −34.8 | −23.8 | 19.4 |

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## Share and Cite

**MDPI and ACS Style**

Młyński, D.; Wałęga, A.; Książek, L.; Florek, J.; Petroselli, A.
Possibility of Using Selected Rainfall-Runoff Models for Determining the Design Hydrograph in Mountainous Catchments: A Case Study in Poland. *Water* **2020**, *12*, 1450.
https://doi.org/10.3390/w12051450

**AMA Style**

Młyński D, Wałęga A, Książek L, Florek J, Petroselli A.
Possibility of Using Selected Rainfall-Runoff Models for Determining the Design Hydrograph in Mountainous Catchments: A Case Study in Poland. *Water*. 2020; 12(5):1450.
https://doi.org/10.3390/w12051450

**Chicago/Turabian Style**

Młyński, Dariusz, Andrzej Wałęga, Leszek Książek, Jacek Florek, and Andrea Petroselli.
2020. "Possibility of Using Selected Rainfall-Runoff Models for Determining the Design Hydrograph in Mountainous Catchments: A Case Study in Poland" *Water* 12, no. 5: 1450.
https://doi.org/10.3390/w12051450