# Application of Artificial Neural Networks in Forecasting a Standardized Precipitation Evapotranspiration Index for the Upper Blue Nile Basin

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data Used

#### 2.1. Study Area

#### 2.2. Data Sources

## 3. Methodology

#### 3.1. SPEI

_{i}values were then aggregated at different time scales. Following the Vicente-Serrano et al. [14] approach, the log-logistic distribution F(x) was applied to transform the original D series into standardized units at different time scales. Finally, the F(x) distribution was utilized to calculate the SPEI following the inverse normal function discussed in Reference [51]. The complete derivation and theory of the index can be found in [52].

#### 3.2. Artificial Neural Networks (ANNs)

#### 3.3. ANN Model Development

#### 3.4. Statistical Performance Measures

^{2}), root-mean-square error (RMSE), Willmott’s index of agreement (d), and the Nash–Sutcliffe coefficient of efficiency (E). The following mathematical equations define these metrics.

_{i}is the observed SPEI value, and P

_{i}is the ANN-predicted SPEI value. RMSE was used to measure the average ANN model prediction error to indicate how close the predicted values were to the observed. Lower values of the index indicate high prediction accuracy. The coefficient of determination R

^{2}was determined from the scattered plot of the observed and predicted SPEI values from the fitted regression line. The best model should have an R

^{2}value close to unity. R

^{2}= 1 denotes an exact linear relationship between the observed and predicted values. However, correlation-based measures are oversensitive to extreme values. As a result, a model might appear to be a good predictor when it is not [60]. The Nash–Sutcliffe coefficient of efficiency [61] is 1 minus the absolute difference between the sum of the squared differences between the predicted and observed values, standardized by the variance of the observed values during the study period. The Nash–Sutcliffe coefficient of efficiency (E) statistical measure ranges from −Infinity to 1, and a value closer to unity indicates a better relationship of the observed to the predicted data. However, it has drawbacks, as the errors are calculated as square terms and hence larger values in a time-series are overestimated, whereas lower values are neglected [62]. To overcome the insensitivity of Nash–Sutcliffe coefficient of efficiency (E) and the coefficient of determination (R

^{2}), Willmott’s index of agreement (d) was proposed [63]. Willmott’s index of agreement (d) gives a value between 0 and 1, and a value close to unity indicates the realization of the best model, whereas a value close to 0 indicates no agreement at all. The index improves upon those mentioned above, yet remains sensitive to extreme events. For sound scientific model calibration and evaluation, a combination of different performance measures is recommended. The performance of the ANN in terms of the score metrics between the observed SPEI and predicted ANN outputs were examined and the results are presented in the next section. The best ANN architecture was trained with the resilient back-propagation learning algorithm with the tangent sigmoid hidden transfer function. Furthermore, the output transfer function was chosen to be linear.

## 4. Results and Discussions

#### 4.1. Model Performance

^{2}), RMSE, Willmott’s index of agreement (d), and the Nash–Sutcliffe coefficient of efficiency (E).

#### 4.2. Comparison of Different Models

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The geographical location of the Upper Blue Nile (UBN) river basin and metrological stations used in this study.

**Figure 2.**Land cover map of the UBN river basin in 2016 at 20 m spatial resolution, extracted from the European Space Agency.

**Figure 3.**Monthly climatology of rainfall and climatic water balance, CWB

_{i}= P

_{i}− PET

_{i}(precipitation − potential evapotranspiration) for the 1986–2015 period.

**Figure 8.**Density plots and histograms of the prediction error (PE) values calculated for the test period.

**Figure 11.**(

**a**) A scatterplot of the observed versus predicted plots for the two-layer ANN (ANN_2), one-layer ANN (ANN_1), and linear models (LM) with a 1:1 reference line plot. (

**b**) The 10-fold cross-validation root-mean-square errors for the two-layer ANN and one-layer ANN models.

Station Name | Geographical Locations | Elevation ASL | Annual Mean Rainfall (mm) | Mean Annual Temperature (°C) |
---|---|---|---|---|

Alemketema | 10.03° N, 39.03° E | 2280 m | 1049.16 | 19.74 |

Asossa | 10.02° N, 34.52° E | 1590 m | 1198.57 | 24.61 |

Bahir Dar | 11.59° N, 37.38° E | 1770 m | 1387.37 | 20.43 |

Bedele | 8.45° N, 36.33° E | 2030 m | 1809.18 | 17.92 |

Chagni | 10.97° N, 36.5° E | 1620 m | 1699.58 | 20.34 |

Debremarkos | 10.33° N, 37.74° E | 2515 m | 1334.15 | 16.27 |

Gondar | 12.61° N, 37.45° E | 1967 m | 1145.87 | 19.89 |

**Table 2.**Drought characterization based on standardized precipitation evapotransporation index (SPEI) values.

SPEI Values | Drought Category |
---|---|

SPEI ≥ 2 | Extremely wet |

1.5 ≤ SPEI < 1 | Severely wet |

1 ≤ SPEI < 1.5 | Moderately wet |

−1 ≤ SPEI < 1 | Near normal |

−1.5 ≤ SPEI < −1 | Moderately dry |

−2 ≤ SPEI < −1.5 | Severely dry |

SPEI < −2 | Extremely dry |

Model | No. of Input Variables | Year | Month | Rainfall | Max T | Min T | PET | SOI | IOD | PDO | N3 SST | N3.4 SST | N4 SST |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

M1 | 12 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

M2 | 11 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

M3 | 10 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||

M4 | 8 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||

M5 | 6 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||

M6 | 6 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||

M7 | 4 | ✓ | ✓ | ✓ | ✓ |

**Table 4.**A measure of ANN model performance based on all statistical measures of the observed SPEI and predicted SPEI.

Station Name | R^{2} | RMSE | d | E |
---|---|---|---|---|

Alemketema | 0.870 | 0.335 | 0.965 | 0.863 |

Asossa | 0.892 | 0.349 | 0.966 | 0.884 |

Bahir Dar | 0.820 | 0.428 | 0.946 | 0.818 |

Bedele | 0.856 | 0.338 | 0.959 | 0.854 |

Chagni | 0.908 | 0.290 | 0.975 | 0.905 |

Debremarkos | 0.865 | 0.363 | 0.964 | 0.862 |

Gondar | 0.949 | 0.263 | 0.987 | 0.949 |

Overall station average | 0.880 | 0.338 | 0.966 | 0.876 |

Station Name | Maximum PE | Minimum PE | Standard Deviation |
---|---|---|---|

Alemketema | 1.674 | −0.631 | 0.337 |

Asossa | 0.877 | −1.561 | 0.346 |

Bahir Dar | 1.614 | −1.189 | 0.430 |

Bedele | 0.956 | −0.588 | 0.338 |

Chagni | 1.075 | −0.518 | 0.288 |

Debremarkos | 0.799 | −0.964 | 0.364 |

Gondar | 0.690 | −0.653 | 0.265 |

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**MDPI and ACS Style**

Mulualem, G.M.; Liou, Y.-A. Application of Artificial Neural Networks in Forecasting a Standardized Precipitation Evapotranspiration Index for the Upper Blue Nile Basin. *Water* **2020**, *12*, 643.
https://doi.org/10.3390/w12030643

**AMA Style**

Mulualem GM, Liou Y-A. Application of Artificial Neural Networks in Forecasting a Standardized Precipitation Evapotranspiration Index for the Upper Blue Nile Basin. *Water*. 2020; 12(3):643.
https://doi.org/10.3390/w12030643

**Chicago/Turabian Style**

Mulualem, Getachew Mehabie, and Yuei-An Liou. 2020. "Application of Artificial Neural Networks in Forecasting a Standardized Precipitation Evapotranspiration Index for the Upper Blue Nile Basin" *Water* 12, no. 3: 643.
https://doi.org/10.3390/w12030643