# Impact of the Grid Resolution and Deterministic Interpolation of Precipitation on Rainfall-Runoff Modeling in a Sparsely Gauged Mountainous Catchment

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of Study Area

^{2}and is entirely located in the Outer Carpathians. The catchment can be divided into two parts, upper- and lower-one [22], where the upper part is more exposed to the risk of flooding. Within the river basin, the elevation varies from 435 to 1038 m a.s.l. To the southwest of the catchment is the Babia Góra massif—the highest peak of the Polish part of the Carpathian Mountains. The highest rainfall is observed in the Babia Góra region and the lowest in the lower part of the catchment. Most of the catchment area is dominated by a warm temperate (up to approx. 700 m a.s.l.) or a cold temperate cold (at altitudes 700–1100 m a.s.l.). There are four rain gauges in the catchment area, of which one is directly located in the investigated study area. The rain gauges are not well distributed over the entire study area, which makes the areal estimation of precipitation based on them more challenging.

^{2}), which is particularly at risk of flooding [23]. This part of the catchment consists of 6 sub-catchments. The upper Skawa catchment area is characterized by a dense water network with dominant short streams with large slopes, resulting from the mountainous nature of the Skawa river [22]. The area of the catchment is dominated by low permeable soils, which is one of the most major factors contributing to the formation of flash floods caused by excessive rainfall [24,25]. Discharge data for the catchment are available at the gauging station in Osielec, which is located downstream.

#### 2.2. Data Collection and Processing

#### 2.3. Hydrological Modeling and Assesment

#### 2.3.1. Selection of the Model

- precipitation fields interpolated using the IDW interpolation method (IDP = 2.0) for 6 different grid sizes (250, 500, 750, 1000, 2500, and 5000 m);
- precipitation fields interpolated using the IDW interpolation method and different IDP values (0.5, 2.0 and 5.0) for 2 grid sizes (250 and 2500 m);
- precipitation fields interpolated using the first-degree polynomial interpolation method for 6 different grid sizes (250, 500, 750, 1000, 2500, and 5000 m).

#### 2.3.2. Model Assessment

- Nash-Sutcliffe efficiency (NSE)—frequently used metric to determine the relative magnitude of the residual variance in relation to the measured data variance [35]. The NSE values range from −1 to 1. The closer to 1, the more accurate the model is. If NSE value = 0, it means that the model predictions are as accurate as the mean value from the observed data. Values < 0 indicate that the mean value from observed data is a better predictor than the model results. The NSE is defined as:$$\mathrm{NSE}=1-\frac{{\sum}_{i=1}^{n}{\left({Q}_{obs}-{Q}_{sim}\right)}^{2}}{{\sum}_{i=1}^{n}{\left({Q}_{obs}-\overline{{Q}_{obs}}\right)}^{2}}$$
_{sim}and Q_{obs}are consecutively simulated and observed river discharge, ${\overline{Q}}_{obs}$ represents the mean of observed values, and n stands for the number of observations. - Kling-Gupta efficiency (KGE)—developed by Gupta et al. [36] is one of the alternatives to the NSE criterion [37], which is based on its decomposition (correlation, variability, and mean bias). Similarly, like NSE, the KGE value equal to 1 indicates a perfect agreement between model results and observation data, and values <0 means that the mean of observation data serves as a better predictor than the model outputs. The KGE is expressed as follows:$$\mathrm{KGE}=1-\sqrt{{\left(r-1\right)}^{2}+{\left(\alpha -1\right)}^{2}+{\left(\beta -1\right)}^{2}}$$
- Percent bias (PBIAS)—this metric is used to assess the model performance regarding the tendency of the simulated flow to be over- or underestimated [26]. If the value of PBIAS is greater than 20%, then it is considered to be unacceptable [38]. The formula for PBIAS is expressed as follows:$$\mathrm{PBIAS}=\frac{{\sum}_{i=1}^{n}\left({Q}_{sim}-{Q}_{obs}\right)}{{\sum}_{i=1}^{n}{Q}_{obs}}$$
_{sim}and Q_{obs}are consecutively simulated and observed river discharge.

#### 2.4. Spatial Interpolation of Precipitation

#### 2.4.1. Interpolation Grid Resolutions

#### 2.4.2. Inverse Distance Weighting

_{i}—rainfall amount measured at the rain gauge, D

_{0i}—distance between the location of the estimated part of the precipitation field and the rain gauge i, n—number of rain gauges used to estimate the precipitation amount at the location 0, p—power exponent responsible for assigning significance weights to individual rain gauges.

#### 2.4.3. Polynomial Interpolation

_{1}–a

_{6}—regression function coefficients.

## 3. Results and Discussion

#### 3.1. Impact of the Grid Resolution on the IDW Method

#### 3.2. Impact of the IDP Value on the IDW Method

#### 3.3. Impact of the Grid Resolution on the Polynomial Interpolation

## 4. Summary and Conclusions

- When analyzing Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, it must be noticed that the curves for various sizes of the grid and different IDP values for the IDW method are very correlated when compared to the curve of the observed flow. Therefore, the choice of a different grid size (or IDP for the IDW method) does not change much the picture with respect to the observed discharge. However, when looking at the data more precisely with statistical analysis, some differences can be detected.
- The impact of the grid resolution is more visible for the IDW method than for the first-degree polynomial interpolation. As for the IDW method, the maximum difference for the NSE criterion is 0.26 for both, calibration, and validation phases. For the first-degree polynomial method, the maximum differences for the NSE are 0.12 and 0.16, respectively. As the IDW method is frequently used in hydrological applications, the appropriate choice of the interpolation grid is of particular importance.
- Among the analyzed grid resolutions, the best results for the IDW method were obtained for the grids of 250 m and 2500 m (average values of the NSE were 0.62 and 0.65 for the calibration and 0.74 and 0.76 for the validation respectively). For the first-degree polynomial method, higher grid resolutions (smaller or equal to 750 m) outperformed the lower ones (greater or equal to 1000 m). The mean value of the NSE for the calibration phase for grids up to 750 m was 0.63 and 0.67 for validation. As for the lower resolution grids, the results were 0.60 and 0.65 consecutively.
- The applied value of the IDP in the IDW method has a significant impact on the outputs of hydrological modeling. In most of the cases, more accurate results were obtained using different values of IDP than traditionally applied value equal to 2.0. Therefore, the choice of the appropriate IDP value when using a semi-distributed hydrological model cannot be neglected and should be taken into account.
- The IDP value in the IDW interpolation method has more impact on the simulation results than the grid size. That can be clearly seen when comparing the results presented in Table 6.
- Within the analyzed deterministic interpolation methods, slightly better results were obtained for the first-degree interpolation method than for the IDW interpolation considering the results of the evaluation criteria presented in Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8. Tobin et al. [39] reported that the IDW method tends to significantly underestimate rainfall volume, but this study shows that when using the right grid size and appropriate IDP value, this method can also be effective. It should also be noted (Figure A3) that the first-degree polynomial method can lead to significant underestimation of precipitation over relatively large areas (horizontal), especially when using low-resolution grids.
- For small mountainous catchments, the best data source on the precipitation field would be rain gauge data interpolated using the first-degree interpolation method and grid size smaller or equal to 750 m. This method, unlike the IDW, is more straightforward in application, and does not require subjective investigation of the method’s parameters (the IDP value in the IDW interpolation method).
- Kling-Gupta efficiency (KGE), which is considered as one of the alternatives to the Nash-Sutcliffe efficiency (NSE), generally tends to provide higher and less varied values, which makes it less useful for the evaluation of the results.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Comparison of precipitation fields obtained using the Inverse Distance Weighting interpolation method (IDP = 2.0) for different grid sizes: (

**a**) 250 m, (

**b**) 500 m, (

**c**) 750 m, (

**d**) 1000 m, (

**e**) 2500 m, (

**f**) 5000 m.

**Figure A2.**Comparison of precipitation fields obtained using the Inverse Distance Weighting interpolation method and a grid size of 250 m: (

**a**,

**c**,

**e**) and 2500 m: (

**b**,

**d**,

**f**) for the IDP values of 0.5, 2.0, and 5.0 consecutively.

**Figure A3.**Comparison of precipitation fields obtained using the first-degree polynomial as an interpolation method for different grid sizes: (

**a**) 250 m, (

**b**) 500 m, (

**c**) 750 m, (

**d**) 1000 m, (

**e**) 2500 m, (

**f**) 5000 m.

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**Figure 1.**Characteristics of the Upper Skawa River catchment in reference to digital elevation model (DEM).

**Figure 2.**Comparison of the interpolation grids investigated in the study (

**a**) 250 m, (

**b**) 500 m, (

**c**) 750, (

**d**) 1000 m, (

**e**) 2500 m. (

**f**) 5000 m.

**Figure 3.**Comparison of observed and simulated hydrographs using different grid sizes and inverse distance weighting (IDW) (inverse distance power (IDP) = 2.0) as an interpolation method for the calibration events (

**a**) Event 1: May 2014, (

**b**) Event 2: May 2015, (

**c**) Event 3: July 2016, (

**d**) Event 4: October 2016.

**Figure 4.**Comparison of observed and simulated hydrographs using different grid sizes and IDW (IDP = 2.0) as an interpolation method for the validation events (

**a**) Event 1: April 2017, (

**b**) Event 2: July 2018, (

**c**) Event 3a: May 2019, (

**d**) Event 3b: May 2019.

**Figure 5.**Comparison of observed and simulated hydrographs using two grid sizes (250 m and 2500 m) and three IDW (IDP = 0.5, 2.0, 5.0) as an interpolation method for the calibration events (

**a**) Event 1: May 2014, (

**b**) Event 2: May 2015, (

**c**) Event 3: July 2016, (

**d**) Event 4: October 2016.

**Figure 6.**Comparison of observed and simulated hydrographs using two grid sizes (250 m and 2500 m) and three IDW (IDP = 0.5, 2.0, 5.0) as an interpolation method for the validation events (

**a**) Event 1: April 2017, (

**b**) Event 2: July 2018, (

**c**) Event 3a: May 2019, (

**d**) Event 3b: May 2019.

**Figure 7.**Comparison of observed and simulated hydrographs using different grid sizes and first-degree polynomial as an interpolation method for the calibration events (

**a**) Event 1: May 2014, (

**b**) Event 2: May 2015, (

**c**) Event 3: July 2016, (

**d**) Event 4: October 2016.

**Figure 8.**Comparison of observed and simulated hydrographs using different grid sizes and first-degree polynomial as an interpolation method for the validation events (

**a**) Event 1: April 2017, (

**b**) Event 2: July 2018, (

**c**) Event 3a: May 2019, (

**d**) Event 3b: May 2019.

Rain Gauge Station | Station Code | Acronym | Longitude | Latitude | Altitude [m a.s.l] |
---|---|---|---|---|---|

Maków Podhalański | 249190190 | RG-1 | 19°40′36.59″49° | 49°43′51.29″ | 367 |

Markowe Szczawiny | 249190390 | RG-2 | 19°30′58.55″ | 49°35′17.05″ | 1184 |

Spytkowice Górne | 249190460 | RG-3 | 19°50′0.57″ | 49°34′38.78″ | 525 |

Zawoja | 249190350 | RG-4 | 19°34′1″ | 49°40′1″ | 604 |

**Table 2.**Methods applied in the Hydrologic Engineering Center-Hydrologic Modelling System (HEC-HMS) hydrological model.

Catchment Model | Meteorological Model | ||
---|---|---|---|

Parameter | Method | Parameter | Method |

Rainfall losses | SCS Curve Number | Precipitation | Specified hyetographs for each of the data sources of precipitation |

Transformation of effective precipitation | Snyder Unit Hydrograph | ||

Baseflow | Recession baseflow | ||

Routing | Muskingum-Cunge |

**Table 3.**The results of the evaluation criteria for the calibration events using different grids sizes and IDW (IDP = 2.0) as an interpolation method.

Event | Nash-Sutcliffe Efficiency (NSE) | Kling-Gupta Efficiency (KGE) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Grid Resolution [m] | Grid Resolution [m] | |||||||||||

250 | 500 | 750 | 1000 | 2500 | 5000 | 250 | 500 | 750 | 1000 | 2500 | 5000 | |

Event 1 | 0.61 | 0.56 | 0.58 | 0.56 | 0.60 | 0.56 | 0.79 | 0.64 | 0.73 | 0.74 | 0.80 | 0.76 |

Event 2 | 0.65 | 0.65 | 0.64 | 0.65 | 0.60 | 0.59 | 0.69 | 0.64 | 0.62 | 0.69 | 0.60 | 0.67 |

Event 3 | 0.52 | 0.50 | 0.47 | 0.47 | 0.57 | 0.59 | 0.55 | 0.54 | 0.53 | 0.53 | 0.56 | 0.61 |

Event 4 | 0.75 | 0.80 | 0.73 | 0.72 | 0.77 | 0.8 | 0.85 | 0.88 | 0.86 | 0.85 | 0.88 | 0.89 |

Percent bias (PBIAS) | ||||||||||||

Grid resolution [m] | ||||||||||||

Event 1 | 2.6 | −14.8 | −5 | −3.5 | 2.6 | 1.3 | ||||||

Event 2 | 3.3 | 12.4 | 5.8 | 1.5 | 14.6 | 12.2 | ||||||

Event 3 | 33.4 | 26 | 23.2 | 31.9 | 27.5 | 28.3 | ||||||

Event 4 | −3.9 | −3.3 | −3.2 | −5.9 | −2.7 | −4.2 |

**Table 4.**The results of the evaluation criteria for the validation events using different grids sizes and IDW (IDP = 2.0) as an interpolation method.

Event | NSE | KGE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Grid Resolution [m] | Grid Resolution [m] | |||||||||||

250 | 500 | 750 | 1000 | 2500 | 5000 | 250 | 500 | 750 | 1000 | 2500 | 5000 | |

Event 1 | 0.56 | 0.6 | 0.65 | 0.72 | 0.67 | 0.46 | 0.67 | 0.67 | 0.67 | 0.73 | 0.7 | 0.62 |

Event 2 | 0.64 | 0.65 | 0.66 | 0.64 | 0.64 | 0.65 | 0.8 | 0.81 | 0.81 | 0.8 | 0.8 | 0.81 |

Event 3a | 0.84 | 0.72 | 0.69 | 0.78 | 0.79 | 0.83 | 0.87 | 0.77 | 0.77 | 0.82 | 0.79 | 0.82 |

Event 3b | 0.93 | 0.93 | 0.93 | 0.93 | 0.93 | 0.92 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 |

PBIAS | ||||||||||||

Grid Resolution [m] | ||||||||||||

Event 1 | −30.4 | −32.3 | −32.6 | −25.6 | −29.7 | −35.2 | ||||||

Event 2 | 6.5 | 5.9 | 6.3 | 6 | 10.6 | 8.8 | ||||||

Event 3a | 4.2 | 0.5 | −0.2 | −2 | 0.2 | 0.4 | ||||||

Event 3b | 1 | −0.6 | 0.1 | 0.1 | 1.6 | −1.7 |

**Table 5.**The results of the evaluation criteria for the calibration events using two grid sizes (250 m and 2500 m) and three IDW (IDP = 0.5, 2.0, 5.0) as an interpolation method.

Event | NSE | KGE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Grid Resolution [m] | Grid Resolution [m] | |||||||||||

250 | 2500 | 250 | 2500 | |||||||||

IDP | 0.5 | 2.0 | 5.0 | 0.5 | 2.0 | 5.0 | 0.5 | 2.0 | 5.0 | 0.5 | 2.0 | 5.0 |

Event 1 | 0.52 | 0.61 | 0.65 | 0.54 | 0.60 | 0.63 | 0.56 | 0.79 | 0.81 | 0.60 | 0.80 | 0.80 |

Event 2 | 0.64 | 0.65 | 0.63 | 0.55 | 0.60 | 0.60 | 0.66 | 0.69 | 0.70 | 0.61 | 0.60 | 0.65 |

Event 3 | 0.38 | 0.52 | 0.53 | 0.38 | 0.57 | 0.54 | 0.30 | 0.55 | 0.58 | 0.37 | 0.56 | 0.57 |

Event 4 | 0.69 | 0.75 | 0.79 | 0.72 | 0.77 | 0.79 | 0.84 | 0.85 | 0.89 | 0.82 | 0.88 | 0.89 |

Event | PBIAS | |||||||||||

Grid Resolution [m] | ||||||||||||

250 | 250 | |||||||||||

IDP | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | ||||||

Event 1 | −23.0 | −23.0 | −23.0 | −23.0 | −23.0 | −23.0 | ||||||

Event 2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | ||||||

Event 3 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | ||||||

Event 4 | − 6.9 | − 6.9 | − 6.9 | − 6.9 | − 6.9 | − 6.9 |

**Table 6.**The results of the evaluation criteria for the validation events using two grid sizes (250 m and 2500 m) and three IDW (IDP = 0.5, 2.0, 5.0) as an interpolation method.

Event | NSE | KGE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Grid Resolution [m] | Grid Resolution [m] | |||||||||||

250 | 2500 | 250 | 2500 | |||||||||

IDP | 0.5 | 2.0 | 5.0 | 0.5 | 2.0 | 5.0 | 0.5 | 2.0 | 5.0 | 0.5 | 2.0 | 5.0 |

Event 1 | 0.75 | 0.56 | 0.62 | 0.60 | 0.67 | 0.63 | 0.73 | 0.67 | 0.65 | 0.66 | 0.70 | 0.69 |

Event 2 | 0.68 | 0.64 | 0.62 | 0.68 | 0.64 | 0.61 | 0.83 | 0.80 | 0.79 | 0.83 | 0.80 | 0.77 |

Event 3a | 0.81 | 0.84 | 0.86 | 0.83 | 0.79 | 0.85 | 0.77 | 0.87 | 0.84 | 0.81 | 0.79 | 0.84 |

Event 3b | 0.92 | 0.93 | 0.90 | 0.92 | 0.93 | 0.93 | 0.96 | 0.95 | 0.92 | 0.96 | 0.95 | 0.96 |

Event | PBIAS | |||||||||||

Grid Resolution [m] | ||||||||||||

250 | 250 | |||||||||||

IDP | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | ||||||

Event 1 | −26.2 | −26.2 | −26.2 | −26.2 | −26.2 | −26.2 | ||||||

Event 2 | 3.7 | 3.7 | 3.7 | 3.7 | 3.7 | 3.7 | ||||||

Event 3a | −2.1 | −2.1 | −2.1 | −2.1 | −2.1 | −2.1 | ||||||

Event 3b | −1.1 | −1.1 | −1.1 | −1.1 | −1.1 | −1.1 |

**Table 7.**The results of the evaluation criteria for the validation events using different grids sizes and first-degree polynomial as an interpolation method.

Event | NSE | KGE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Grid Resolution [m] | Grid Resolution [m] | |||||||||||

250 | 500 | 750 | 1000 | 2500 | 5000 | 250 | 500 | 750 | 1000 | 2500 | 5000 | |

Event 1 | 0.47 | 0.52 | 0.48 | 0.50 | 0.48 | 0.48 | 0.72 | 0.73 | 0.72 | 0.74 | 0.72 | 0.72 |

Event 2 | 0.74 | 0.75 | 0.75 | 0.71 | 0.63 | 0.63 | 0.70 | 0.81 | 0.75 | 0.75 | 0.73 | 0.73 |

Event 3 | 0.48 | 0.56 | 0.53 | 0.52 | 0.50 | 0.50 | 0.56 | 0.56 | 0.58 | 0.55 | 0.57 | 0.57 |

Event 4 | 0.78 | 0.78 | 0.73 | 0.78 | 0.75 | 0.75 | 0.88 | 0.89 | 0.85 | 0.87 | 0.87 | 0.87 |

Event | PBIAS | |||||||||||

Grid Resolution [m] | ||||||||||||

Event 1 | 11.06 | 4.0 | 12.3 | 9.2 | 12.0 | 12.0 | ||||||

Event 2 | −0.2 | 2.9 | 9.9 | 7.8 | 3.3 | 3.3 | ||||||

Event 3 | 21.7 | 20.3 | 29.0 | 23.1 | 29.2 | 29.2 | ||||||

Event 4 | −6.3 | −0.3 | −8.7 | −0.8 | −3.4 | −3.4 |

**Table 8.**The results of the evaluation criteria for the validation events using different grids sizes and first-degree polynomial as an interpolation method.

Event | NSE | KGE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Grid Resolution [m] | Grid Resolution [m] | |||||||||||

250 | 500 | 750 | 1000 | 2000 | 5000 | 250 | 500 | 750 | 1000 | 2500 | 5000 | |

Event 1 | 0.61 | 0.54 | 0.68 | 0.52 | 0.57 | 0.57 | 0.67 | 0.66 | 0.70 | 0.64 | 0.65 | 0.65 |

Event 2 | 0.72 | 0.79 | 0.79 | 0.74 | 0.74 | 0.74 | 0.78 | 0.85 | 0.85 | 0.83 | 0.82 | 0.82 |

Event 3a | 0.38 | 0.41 | 0.37 | 0.38 | 0.40 | 0.40 | 0.65 | 0.66 | 0.64 | 0.66 | 0.65 | 0.65 |

Event 3b | 0.92 | 0.89 | 0.89 | 0.91 | 0.90 | 0.90 | 0.91 | 0.91 | 0.91 | 0.91 | 0.88 | 0.88 |

Event | PBIAS | |||||||||||

Grid Resolution [m] | ||||||||||||

Event 1 | −32.7 | −32.9 | −30.0 | −34.4 | −34.0 | −34.0 | ||||||

Event 2 | −18.3 | −11.0 | −11.6 | −12.5 | −14.4 | −14.4 | ||||||

Event 3a | 19.1 | 19.6 | 20.0 | 17.6 | 19.4 | 19.4 | ||||||

Event 3b | 1.7 | 0.5 | 25.3 | 0.9 | −2.8 | −2.8 |

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**MDPI and ACS Style**

Gilewski, P.
Impact of the Grid Resolution and Deterministic Interpolation of Precipitation on Rainfall-Runoff Modeling in a Sparsely Gauged Mountainous Catchment. *Water* **2021**, *13*, 230.
https://doi.org/10.3390/w13020230

**AMA Style**

Gilewski P.
Impact of the Grid Resolution and Deterministic Interpolation of Precipitation on Rainfall-Runoff Modeling in a Sparsely Gauged Mountainous Catchment. *Water*. 2021; 13(2):230.
https://doi.org/10.3390/w13020230

**Chicago/Turabian Style**

Gilewski, Paweł.
2021. "Impact of the Grid Resolution and Deterministic Interpolation of Precipitation on Rainfall-Runoff Modeling in a Sparsely Gauged Mountainous Catchment" *Water* 13, no. 2: 230.
https://doi.org/10.3390/w13020230