# Possibilities of Using Neuro-Fuzzy Models for Post-Processing of Hydrological Forecasts

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}and the profiles varied in their character, both in terms of elevation as well as land cover. After finding the suitable model architecture and introducing supporting algorithms, there was an improvement in the results for the individual profiles for selected criteria by on average 5–60% (relative culmination error, mean square error) compared to the results of re-simulation of the hydrological model. The results of the application show that the method was able to improve the accuracy of hydrological forecasts and thus could contribute to better management of flood situations.

## 1. Introduction

## 2. Materials and Methods

## 3. Application

_{a}. Furthermore, for the individual profiles, values of flows are given, at which the individual degrees of flood activity are announced, which are defined by the Water Act of the Czech Republic 254/2001 of the Collection of Laws. The values of the flood level danger (FLD) levels themselves represent the individual limit values for the declaration of a flood danger (FLD 1—low level of danger—yellow color; FLD 2—high level of danger-orange color; FLD 3—extreme level of danger—red color). The particular level of danger usually corresponds with this color set in a global warning system. The individual values of flows in the FLD columns were determined according to the measurement curve valid at the time of validation of the post-processing model. Column N1 shows the one-year flood flow rate value and column N100 shows the 100-year flood flow rate value. The last column of the table shows the percentage of forest area with respect to the total catchment area FIP. The FIP parameter is used for better understanding of the basin.

^{2}. The required inputs are data on river discharge, precipitation and air temperature in 1-h step. The precipitation data, which were used for resimulation, were merged with rainfall data (combination of adjusted radar observation and values measured by automatic weather station). Aqualog re-simulation data were calculated on the observed data in continuous intervention-free operation. Deviations of the calculated flow from the observed values therefore represent the error of the input observed data and hydrological modeling. The re-simulation period 2005–2016 includes a number of significant flood episodes. The data were grouped into individual months and the tendency of the simulation model compared to the reality in the individual months was determined. Data from a particular month were selected for the training of the model for the that particular month, and if the neighboring months had the same tendency (underestimated, overestimated), the training matrix was extended by these data. Data of distant months were not considered, even if they had the same tendency. The reason for this selection was the time variability of the influence of the model sensitivity on the individual input parameters of the simulation model and the variable boundary conditions of the simulation model. Data from neighboring months that met the condition of the same tendency were included only due to the lack of training data during the individual months.

^{2}) from the last value of the precipitation Hs, which was higher than 0. From the events defined above, those events were selected in the target behavior matrix, for which the peak flow exceeded the half the flow value of the FLD 1 of the given profile basin.

_{sim}and the actual flow Q

_{real}for the observed period. The results showed that the NFM models showed a lower error of criterion E by 20 to 50% compared to the results of unadjusted (without post-processing) outputs of the hydrological model. These results set the limits that can be achieved in this study when NFM models are used for post-processing. It is possible to achieve very different values of criterion E for selected episodes.

_{sim}as input the gradient between the current and previous value ΔQmodp and the average of hourly precipitation total Hspi for the selected time period Γ [h]. The architecture described above will hereafter be referred to as the NFM 1 model. Figure 5 shows a diagram of the NFM 1 model and Figure 6 shows one of the local model (N-F).

## 4. Results

_{p}

_{,i}and the predicted value Q

_{s}

_{,i}for the entire validation period, where the individual members are averaged in absolute values.

## 5. Discussion

## 6. Conclusions

- The method was successfully applied on 12 profiles (base on chosen criteria).
- Improved forecast can lead to better estimation of risk.
- The method itself is transferable with certain limitation.
- The method has short calculation time and can be used for chosen length of forecast.
- The method is applied directly on hydrological model results (hydrological model do not have to be recalculated).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Results for the selected episode in the Bechyně profile. Figure 7 shows the course of the selected episode, where the dashed blue line shows the flow value N1 (period of return = 1) and the green value N100 (period of return = 100). The measured data are plotted in orange. The re-simulation of the episode using the Aqualog model is shown in gray. The results are shown in pink with the use of post-processing without the use of the EEA algorithm and the selection of episodes was not used during the training. The results of post-processing using the EEA algorithm during training and without selection of individual episodes are displayed in blue. The yellow color shows the results using post-processing using the EEA algorithm during training and using the selection of episodes.

**Figure 8.**Calibration results for the selected episode in the Bechyně profile. Figure 8 plots the training data (blue) and the result of the trained post-processing model (red). An artificial training episode provided by the EEA algorithm is also shown (black circles).

Value | Marking | Unit |
---|---|---|

Instantaneous flow value | Q_{sim} | m^{3}/s |

Total precipitation amount (1 h) | Hs | Mm |

Previous flow value | Qpm | m^{3}/s |

Next flow value | Qfm | m^{3}/s |

Topsoil saturation indicator | UTWZ | Mm |

Average of the hourly sum of precipitation totals over a period of time | Hspi | Mm |

Corrected flow (post-procesing flow) | Qpost | m^{3}/s |

Difference between current and previous flow value | ΔQmodp | m^{3}/s |

Difference between current and next flow value | ΔQmodf | m^{3}/s |

Average of the flow values over time | Qpmodi | m^{3}/s |

Num. | Profile | River | Catchment Area [km ^{2} ] | Qa [m ^{3}/s] | 1. FLD [m ^{3}/s] | 2. FLD [m ^{3}/s] | 3. FLD [m ^{3}/s] | N1 [m ^{3}/s] | N100 [m ^{3}/s] | FIP [%] |
---|---|---|---|---|---|---|---|---|---|---|

1 | České Budejovice | Vltava | 2850 | 106 | 244 | 361 | 489 | 172 | 908 | 35 |

2 | Lenora | Tepla Vltava | 177 | 3.06 | 29 | 53.5 | 70.8 | 26 | 113 | 78 |

3 | Ličov | Cerna | 127 | 1.29 | 12.7 | 21.4 | 29.7 | 21 | 188 | 53 |

4 | Bechyně | Luznice | 4057 | 22.2 | 87.9 | 140 | 187 | 111 | 577 | 33 |

5 | Rodvínov | Nezarka | 297 | 2.2 | 18.7 | 26.9 | 43.7 | 20 | 91 | 10 |

6 | Písek | Otava | 2914 | 23.4 | 135 | 214 | 297 | 146 | 837 | 27 |

7 | Katovice | Otava | 1134 | 13.8 | 118 | 169 | 255 | 133 | 510 | 42 |

8 | Sušice | Otava | 533 | 10.5 | 64.2 | 94.3 | 127 | 101 | 369 | 77 |

9 | Modrava | Vydra | 90 | 3.01 | 30.5 | 41.2 | 54.6 | 29 | 120 | 96 |

10 | Stodůlky | Kremelna | 135 | 3.24 | 24 | 37 | 52.4 | 40 | 153 | 89 |

11 | Bohumilice | Spulka | 105 | 0.97 | 24 | 35.1 | 47.7 | 11 | 84 | 48 |

12 | Podedvory | Blanice | 204 | 2.04 | 15.3 | 28.2 | 38.6 | 25 | 165 | 60 |

Num. Basin | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Γ [h] | 6 | 2 | 2 | 4 | 3 | 6 | 6 | 5 | 3 | 2 | 2 | 3 |

Area [km^{2}] | 2848 | 177 | 127 | 4057 | 297 | 2914 | 1134 | 534 | 90 | 135 | 105 | 203 |

Name: České Budějovice (Area = 2850 km^{2}, N1 = 172 m^{3}/s) | ||||||
---|---|---|---|---|---|---|

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.33 | - | 0.33 | 0.63 | - | 0.63 |

E | 665 | - | 665 | 1539 | - | 1539 |

RMSE | 2.44 | - | 2.44 | 3.44 | - | 3.44 |

NSE | 0.77 | - | 0.77 | 0.70 | 0.70 | |

Num. events | 24 | 0 | 24 | 24 | 0 | 24 |

Name: Lenora (Area 177 = km^{2}, N1 = 26 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.23 | 0.24 | 0.23 | 0.38 | 0.22 | 0.40 |

E | 559 | 3092 | 228 | 1074 | 5081 | 551 |

RMSE | 2.16 | 4.10 | 1.79 | 3.03 | 5.02 | 2.93 |

NSE | 0.81 | 0.86 | 0.80 | 0.77 | 0.81 | 0.77 |

Num. events | 26 | 3 | 23 | 26 | 3 | 23 |

Name: Ličov (Area = 127 km^{2}, N1 = 21 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.35 | 0.33 | 0.36 | 0.37 | 0.38 | 0.37 |

E | 1138 | 3972 | 429 | 1249 | 4788 | 364 |

RMSE | 2.31 | 2.41 | 2.67 | 2.54 | 2.73 | 2.45 |

NSE | 0.65 | 0.71 | 0.63 | 0.62 | 0.69 | 0.60 |

Num. events | 20 | 4 | 16 | 20 | 4 | 16 |

Name: Bechyně (Area = 4057 km^{2}, N1 = 111 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.09 | 0.07 | 0.02 | 0.14 | 0.14 | 0.03 |

E | 596,330 | 715,530 | 331 | 952,960 | 1,143,461 | 453 |

RMSE | 15.23 | 24.55 | 1.55 | 17.23 | 26.93 | 1.79 |

NSE | 0.94 | 0.93 | 0.99 | 0.91 | 0.89 | 0.98 |

Num. events | 6 | 5 | 1 | 6 | 5 | 1 |

Name: Rodvínov (Area = 297 km^{2}, N1 = 20 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.31 | 0.32 | 0.26 | 0.39 | 0.32 | 0.41 |

E | 2711 | 5476 | 2020 | 3231.02 | 6476 | 2420 |

RMSE | 5.21 | 5.65 | 4.72 | 6.07 | 11.53 | 5.71 |

NSE | 0.6 | 0.66 | 0.56 | 0.58 | 0.66 | 0.52 |

Num. events | 5 | 1 | 4 | 5 | 1 | 4 |

Name: Písek (Area = 2914 km^{2}, N1 = 146 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.10 | 0.07 | 0.17 | 0.12 | 0.07 | 0.22 |

E | 18,062 | 22,748 | 16,890 | 21,276 | 27,573 | 19,402 |

RMSE | 12.62 | 11.75 | 14.13 | 15.11 | 14.16 | 16.78 |

NSE | 0.81 | 0.94 | 0.77 | 0.77 | 0.87 | 0.75 |

Num. events | 20 | 4 | 16 | 20 | 4 | 16 |

Name: Katovice (Area = 1134 km^{2}, N1 = 137 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.09 | 0.12 | 0.08 | 0.15 | 0.18 | 0.15 |

E | 3806 | 12,554 | 1965 | 6846 | 14,946 | 5141 |

RMSE | 6.36 | 10.54 | 5.21 | 9.17 | 11.94 | 8.17 |

NSE | 0.83 | 0.94 | 0.81 | 0.81 | 0.91 | 0.79 |

Num. events | 23 | 4 | 19 | 23 | 4 | 19 |

Name: Sušice (Area = 533 km^{2}, N1 = 101 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.11 | 0.12 | 0.09 | 0.09 | 0.14 | 0.08 |

E | 3806 | 9726 | 1500 | 3843 | 7294 | 3046 |

RMSE | 7.76 | 14.39 | 4.28 | 7.78 | 11.58 | 6.81 |

NSE | 0.80 | 0.86 | 0.78 | 0.61 | 0.83 | 0.56 |

Num. events | 16 | 3 | 13 | 16 | 3 | 13 |

Name: Modrava (Area = 90 km^{2}, N1 = 29 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.28 | 0.27 | 0.28 | 0.43 | 0.30 | 0.45 |

E | 1096 | 3839 | 613 | 1311 | 3621 | 903 |

RMSE | 3.73 | 8.11 | 2.95 | 4.50 | 7.99 | 3.88 |

NSE | 0.83 | 0.71 | 0.86 | 0.64 | 0.7 | 0.62 |

Num. events | 20 | 3 | 17 | 20 | 3 | 17 |

Name: Stodůlky (Area = 135 km^{2}, N1 = 40 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.31 | 0.31 | 0.31 | 0.36 | 0.34 | 0.36 |

E | 732 | 2889 | 463 | 1093 | 4035 | 726 |

RMSE | 2.86 | 6.31 | 2.43 | 3.73 | 7.25 | 3.29 |

NSE | 0.68 | 0.74 | 0.67 | 0.53 | 0.63 | 0.51 |

Num. events | 27 | 3 | 24 | 27 | 3 | 24 |

Name: Bohumilice (Area = 105 km^{2}, N1 = 11 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.57 | 0.45 | 0.59 | 0.96 | 0.46 | 1.05 |

E | 177 | 678 | 89 | 176 | 674 | 88 |

RMSE | 1.54 | 3.44 | 1.20 | 1.52 | 3.42 | 1.19 |

NSE | 0.75 | 0.63 | 0.78 | 0.61 | 0.61 | 0.61 |

Num. events | 20 | 3 | 17 | 20 | 3 | 17 |

Name: Podedvory (Area = 204 km^{2}, N1 = 25 m^{3}/s) | ||||||

Method | Post-procesing | Clean data | ||||

Data set | All | N > 1 | N < 1 | All | N > 1 | N < 1 |

Ek | 0.17 | 0.14 | 0.18 | 0.18 | 0.21 | 0.17 |

E | 1681 | 2902 | 1070 | 2598 | 6141 | 827 |

RMSE | 3.46 | 3.69 | 3.78 | 3.75 | 4.50 | 3.27 |

NSE | 0.75 | 0.76 | 0.74 | 0.62 | 0.63 | 0.61 |

Num. events | 15 | 5 | 10 | 15 | 5 | 10 |

Criteria | Post-Procesing | Clean Data | ||||
---|---|---|---|---|---|---|

All | N > 1 | N < 1 | All | N > 1 | N < 1 | |

Ek | 0.25 | 0.22 | 0.21 | 0.34 | 0.26 | 0.31 |

E | 52,500 | 71,218 | 2133 | 83,099 | 111,281 | 2851 |

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**MDPI and ACS Style**

Kozel, T.; Vlasak, T.; Janal, P. Possibilities of Using Neuro-Fuzzy Models for Post-Processing of Hydrological Forecasts. *Water* **2021**, *13*, 1894.
https://doi.org/10.3390/w13141894

**AMA Style**

Kozel T, Vlasak T, Janal P. Possibilities of Using Neuro-Fuzzy Models for Post-Processing of Hydrological Forecasts. *Water*. 2021; 13(14):1894.
https://doi.org/10.3390/w13141894

**Chicago/Turabian Style**

Kozel, Tomas, Tomas Vlasak, and Petr Janal. 2021. "Possibilities of Using Neuro-Fuzzy Models for Post-Processing of Hydrological Forecasts" *Water* 13, no. 14: 1894.
https://doi.org/10.3390/w13141894