# Impact of the Mean Daily Air Temperature Calculation on the Rainfall-Runoff Modelling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Data and Study Area

_{i}, T

_{MAX}, and T

_{MIN}are hourly air temperature values, and maximum and minimum daily air temperature values, respectively. Moreover, T

_{7}, T

_{13}, T

_{14}, T

_{19}, and T

_{21}are hourly air temperature values measured at 7:00, 13:00, 14:00, 19:00, and 21:00, respectively. According to Conner and Foster (2008), the T

_{0}(Equation (1)) was selected as the “true” mean daily air temperature in this study and deviations between the Equations (1) and (2), Equations (1)–(3), and Equations (1)–(4) were evaluated. Mean Absolute Error (MAE) was applied for this purpose. It should be noted that many other equations for the estimation of the mean daily air temperature can also be found in the literature [12]. Additional information about the origin of different equations and regions, where these are mostly used can be found in the existing literature [12,13,16,18,23,24]. It should be noted that Equation (2) is the one that is the most frequently used in Slovenia for calculating the mean daily air temperature. In this case, manual readings at 7:00, 14:00, and 21:00 are used as an input and not continuous hourly measurements. Some historical background of various equations that can be used to calculate the mean daily air temperature is provided by Conner and Foster [23]. Most of the equations used today originated more than 100 years ago and are based on the assumptions and knowledge of that time [23].

#### 2.2. Hydrological Modelling

#### 2.3. Measurement Error of the Model Input Data

_{i}is the discharge measurement, Q

_{c}is the corresponding discharge taken from the rating curve, N is the number of discharge measurements, and t is Student’s t correction at the 95% confidence level, which is around 2 for 20 or more measurements [37].

## 3. Results and Discussion

#### 3.1. Mean Daily Air Temperature Calculation Using Multiple Equations

#### 3.2. Impact of the Mean Daily Air Temperature on the Rainfall-Runoff Modelling Results

^{3}/s and 1.53 m

^{3}/s for the Equation (4) and (1), respectively). Furthermore, somewhat larger differences were obtained for maximum simulated values in case results using Equation (1) are compared with the rainfall-runoff models results using Equations (2)–(4). More specifically, in some cases, the differences were within the range of 5% but mostly in the range of 1%–2%. The differences in the mean daily air temperature values using Equations (1)–(4) were within the range of 5% (i.e., for the 2014–2018 period) (Figure 6). Moreover, the differences in the maximum daily air temperature values derived using different equations (i.e., Equations (1)–(4)) for stations that are used for the rainfall-runoff modelling (i.e., Rateče, Postojna, Murska Sobota) were within the range of 8% (Figure 6). Therefore, it is clear that rainfall-runoff process minimizes the differences in the mean daily air temperature data since precipitation is the main input variable (i.e., compare the scatter shown in Figure 5 and Figure 6). Thus, these differences in the rainfall-runoff model results can be regarded as relatively small, especially if one considers uncertainty of rainfall-runoff modelling results related to the parameters’ estimation or model structural uncertainty [38,39]. Additionally, the next section discusses the impact of other factors on the rainfall-runoff modelling results.

#### 3.3. Impact of the Measurements Error on the Rainfall-Runoff Modelling Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location of the considered meteorological and climatological stations (white dots) and location of four selected catchments (red areas, green dots show water level gauging stations) with a river network in Slovenia (blue lines).

**Figure 2.**Mean Absolute Error (MAE) values between Equations (1) and (2) (blue bar), Equations (1)–(3) (green bar), and Equations (1)–(4) (red bar) using mean daily air temperature data.

**Figure 3.**Evaluation of the CemaNeige GR6J model for the calibration (

**left**) and validation (

**right**) period. Top two panels show input precipitation and air temperature data. Middle panel shows modelled snow pack values. Lower two panels show a comparison between simulated and observed discharge data using daily discharge values, 30-days rolling mean, flow duration curve, and scatter plot.

**Figure 4.**Comparison of different model results using different estimates of mean daily air temperature values for the Kranjska Gora station on the Sava Dolinka river.

**Figure 5.**Comparison among the CemaNeige GR6J results for the Kranjska Gora station on the Sava Dolinka river using different input values for the mean air temperature estimates. X-axis in all three cases show results obtained using Equation (1) while the y-axis show results using Equation (2), Equation (3), and Equation (4) (from left to right).

**Figure 6.**Comparison of mean daily air temperature values for the Rateče station on the Sava Dolinka river using different input values for the mean air temperature estimates. In all three cases, the x-axis shows the mean daily air temperature estimates using Equation (1) while the y-axis show values obtained using Equation (2), Equation (3), and Equation (4) (from left to right).

**Figure 7.**Impact of the efficiency criteria used for the CemaNeige GR6J model calibration (

**left figure**) and impact on the length of the model warm-up period (

**right figure**) on the model results in the validation period for the Blejski most station on the Sava Dolinka river. In both cases, the x-axis shows discharge values obtained using Equation (1), RMSE as a criterion for the calibration, and one year as a warm-up period. In the left figure, the y-axis shows the results using the KGE2 for model calibration. In the right figure, the y-axis shows the results using half-year data for the model warm-up period instead of one year.

**Figure 8.**Rating curve (yellow line) for the Blejski most station on the Sava Dolinka river with periodical discharge and water level measurements (points).

**Table 1.**Main characteristics of catchments used in this study for hydrological modelling. Mean discharge in the period of 2010–2018 as well as mean annual precipitation, number of snow days, and mean min., mean, and mean max. air temperature for the 2010–2019 period are presented.

River | Sava Dolinka | Sava Dolinka | Ledava | Nanoščica |
---|---|---|---|---|

Discharge gauging station | Kranjska Gora | Blejski most | Polana | Mali Otok |

Catchment area (km^{2}) | 40 | 509 | 209 | 51 |

Mean discharge (m^{3}/s) | 1.6 | 25.3 | 1.2 | 1.6 |

Precipitation and air temperature station | Rateče | Rateče | Murska Sobota | Postojna |

Station elevation | 864 m a.s.l. | 864 m a.s.l. | 187 m a.s.l. | 533 m a.s.l. |

Mean annual precipitation (mm) | 1655 | 1655 | 841 | 1532 |

Number of days with snow cover | 105 | 105 | 30 | 35 |

Mean min., mean and max. air temperature (°C) | 2.5, 7.5, 13.7 | 2.5, 7.5, 13.7 | 6.3, 11.2, 16.8 | 5.5, 10.4, 15.7 |

**Table 2.**Root Mean Square Error (RMSE) criteria values for discharge results (mm) in case of four investigated catchments for the calibration and validation periods using different input values for the mean air temperature estimates.

Station | Period | GR4J | GR6J | CemaNeige GR6J | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Equation (1) | Equation (2) | Equation (3) | Equation (4) | Equation (1) | Equation (2) | Equation (3) | Equation (4) | Equation (1) | Equation (2) | Equation (3) | Equation (4) | ||

Kranjska Gora | Calibration | 1.76 | 1.76 | 1.76 | 1.76 | 1.76 | 1.76 | 1.76 | 1.77 | 1.52 | 1.52 | 1.48 | 1.49 |

Validation | 1.67 | 1.66 | 1.64 | 1.67 | 1.66 | 1.64 | 1.66 | 1.67 | 1.08 | 1.08 | 1.14 | 1.18 | |

Polana | Calibration | 0.69 | 0.69 | 0.69 | 0.68 | 0.64 | 0.64 | 0.64 | 0.64 | 0.46 | 0.46 | 0.48 | 0.48 |

Validation | 1.09 | 1.09 | 1.09 | 1.07 | 1.35 | 1.32 | 1.34 | 1.29 | 1.26 | 1.22 | 1.24 | 1.20 | |

Mali otok | Calibration | 2.52 | 2.52 | 2.52 | 2.52 | 2.51 | 2.51 | 2.50 | 2.50 | 2.35 | 2.41 | 2.32 | 2.39 |

Validation | 2.55 | 2.55 | 2.55 | 2.56 | 2.54 | 2.54 | 2.55 | 2.55 | 2.58 | 2.58 | 2.60 | 2.59 | |

Blejski most | Calibration | 2.08 | 2.08 | 2.08 | 2.08 | 2.00 | 2.00 | 2.00 | 2.00 | 1.76 | 1.80 | 1.71 | 1.77 |

Validation | 2.60 | 2.60 | 2.60 | 2.60 | 2.46 | 2.45 | 2.46 | 2.49 | 1.76 | 1.80 | 1.73 | 1.76 |

**Table 3.**Overview of the measurements and rating curve errors on the rainfall-runoff modelling results using GR4J and CemaNeige GR6J models. The difference between the minimum and maximum simulated discharge values of the 100 model runs, calculated for all mean and maximum simulated discharge values, is shown.

Model | Statistics | T | T, P | T, P, Q | T, P, Q, RC |
---|---|---|---|---|---|

GR4J | Mean | ~0% | <1% | <1% | 3% |

Maximum | ~0% | 1% | 1% | 14% | |

CemaNeige GR6J | Mean | ~0% | <1% | <1% | 2% |

Maximum | ~0% | 1% | 1% | 13% |

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**MDPI and ACS Style**

Bezak, N.; Cerović, L.; Šraj, M.
Impact of the Mean Daily Air Temperature Calculation on the Rainfall-Runoff Modelling. *Water* **2020**, *12*, 3175.
https://doi.org/10.3390/w12113175

**AMA Style**

Bezak N, Cerović L, Šraj M.
Impact of the Mean Daily Air Temperature Calculation on the Rainfall-Runoff Modelling. *Water*. 2020; 12(11):3175.
https://doi.org/10.3390/w12113175

**Chicago/Turabian Style**

Bezak, Nejc, Lazar Cerović, and Mojca Šraj.
2020. "Impact of the Mean Daily Air Temperature Calculation on the Rainfall-Runoff Modelling" *Water* 12, no. 11: 3175.
https://doi.org/10.3390/w12113175