# Saturated Hydraulic Conductivity Estimation Using Artificial Neural Networks

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Soil Database

#### 2.3. The ANNs’ Setup

#### 2.4. Cross-Validation

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation for an ANN structure. Each circle represents a neuron, where the ${I}_{j}$ means the neurons in the input layer, and the ${O}_{j}$ are the neurons in the output layer. All ${H}_{j}$ circles are the neurons in the hidden layers. The arrows represent the weighted connections.

**Figure 2.**Location map of the sampling points in Irrigation District 023 San Juan del Río, Querétaro.

**Figure 3.**ANN test for choosing the ideal number of hidden neurons varying the input data. From top to bottom and left to right the number of input data goes from 7 to 3. The x axis represents the number of neurons in the first hidden layer and the y axis is the RMSE value. Each color is a different number of neurons in the second hidden layer.

**Figure 4.**RMSE matrix for different number of neurons in the hidden layers. From top to bottom and left to right, the number of input data goes from 7 to 3. The x axis represents the number of neurons in the first hidden layer and the y axis is the number of neurons in the second hidden layer.

**Figure 5.**Density plots for (from left to right) RMSE, R${}^{2}$, and MAE resulting from the 25 bootstrap replications. From top to bottom, the number of input data goes from 7 to 3.

**Figure 6.**Importance plots referring to the weight of each variable in the calculations. From top to bottom and from right to left, the number of input data goes from 7 to 3.

**Figure 7.**Plots for the comparison between K${}_{\mathrm{S}}$ measurement in the field with the K${}_{\mathrm{S}}$ value obtained applying the different ANN configurations. The dotted line is a 1:1 desirable relation. The histograms represent the residual distribution ($\Delta $K${}_{\mathrm{S}}$).

**Figure 8.**Boxplot for each run of the train package. The names means two hidden layers and the number of input data used in each run.

Variable | Min | Max | Median | Mean | SD | Q1 | Q3 |
---|---|---|---|---|---|---|---|

Sand (%) | 0.07 | 77.83 | 28.35 | 31.14 | 20.22 | 13.75 | 52.00 |

Clay (%) | 2.12 | 59.46 | 21.74 | 21.95 | 12.06 | 13.44 | 30.00 |

Silt (%) | 0.80 | 92.00 | 45.27 | 46.91 | 23.48 | 27.30 | 59.79 |

${\rho}_{a}$ (g/cm${}^{3}$) | 1.18 | 1.70 | 1.40 | 1.41 | 0.11 | 1.32 | 1.47 |

PWP (cm${}^{3}$/cm${}^{3}$) | 0.07 | 0.35 | 0.13 | 0.15 | 0.05 | 0.10 | 0.17 |

${\theta}_{S}$ (cm${}^{3}$/cm${}^{3}$) | 0.35 | 0.56 | 0.47 | 0.47 | 0.04 | 0.45 | 0.50 |

FC (cm${}^{3}$/cm${}^{3}$) | 0.17 | 0.47 | 0.29 | 0.30 | 0.06 | 0.25 | 0.32 |

K${}_{\mathrm{S}}$ (cm/h) | 0.05 | 5.15 | 0.78 | 1.42 | 1.42 | 0.40 | 1.80 |

**Table 2.**The top three configurations for ANN structure and their statistical measurements. (1) # Input data (2) contains the ANN neurons’ structure (Input-Hidden1-Hidden2-Ouput), where each number represents the quantity of neurons used in each layer, (3) the RMSE measurements, (4) the MAE measurements, and (5) the R${}^{2}$ measurements.

# Input Data | ANN Structure | RMSE | MAE | R${}^{2}$ |
---|---|---|---|---|

(cm/h) | (cm/h) | |||

7-9-3-1 | 0.0459 | 0.0159 | 0.9725 | |

7 | 7-7-6-1 | 0.0460 | 0.0164 | 0.9720 |

7-10-4-1 | 0.0465 | 0.0162 | 0.9715 | |

6-5-7-1 | 0.0445 | 0.0171 | 0.9740 | |

6 | 6-6-3-1 | 0.0455 | 0.0171 | 0.9742 |

6-8-4-1 | 0.0447 | 0.0163 | 0.9739 | |

5-8-3-1 | 0.0413 | 0.0152 | 0.9780 | |

5 | 5-4-9-1 | 0.0417 | 0.0156 | 0.9774 |

5-8-8-1 | 0.0418 | 0.0152 | 0.9777 | |

4-9-10-1 | 0.0449 | 0.0152 | 0.9736 | |

4 | 4-8-8-1 | 0.0450 | 0.0156 | 0.9735 |

4-9-9-1 | 0.0452 | 0.0155 | 0.9734 | |

3-9-6-1 | 0.0433 | 0.0155 | 0.9757 | |

3 | 3-8-9-1 | 0.0434 | 0.0154 | 0.9757 |

3-8-6-1 | 0.0436 | 0.0160 | 0.9755 |

Model | RMSE | ${\mathbf{R}}^{2}$ | Type |
---|---|---|---|

This work | 0.0413 | 0.9780 | ANN |

Tamari et al. [31] | 0.0707 | NA | ANN |

Brakensiek et al. [9] | 0.1370 | 0.9953 | PTF |

Erzin et al. [12] | 0.1700 | 0.9970 | ANN |

Saxton et al. [32] | 0.1895 | 0.9915 | PTF |

Parasuraman et al. [33] | 0.1900 | NA | ANN |

Trejo-Alonso et al. [23] | 0.1983 | 0.9901 | PTF |

Cosby et al. [34] | 0.4325 | 0.9546 | PTF |

Ahuja et al. [35] | 0.6498 | 0.8910 | PTF |

Schaap & Leij [36] | 0.7130 | NA | ANN |

Vereecken et al. [37] | 0.7143 | 0.9307 | PTF |

Minasny et al. [38] | 0.7330 | NA | ANN |

Ferrer-Julià et al. [39] | 1.3018 | 0.4083 | PTF |

Merdun et al. [40] | 3.5110 | 0.5240 | ANN |

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**MDPI and ACS Style**

Trejo-Alonso, J.; Fuentes, C.; Chávez, C.; Quevedo, A.; Gutierrez-Lopez, A.; González-Correa, B.
Saturated Hydraulic Conductivity Estimation Using Artificial Neural Networks. *Water* **2021**, *13*, 705.
https://doi.org/10.3390/w13050705

**AMA Style**

Trejo-Alonso J, Fuentes C, Chávez C, Quevedo A, Gutierrez-Lopez A, González-Correa B.
Saturated Hydraulic Conductivity Estimation Using Artificial Neural Networks. *Water*. 2021; 13(5):705.
https://doi.org/10.3390/w13050705

**Chicago/Turabian Style**

Trejo-Alonso, Josué, Carlos Fuentes, Carlos Chávez, Antonio Quevedo, Alfonso Gutierrez-Lopez, and Brandon González-Correa.
2021. "Saturated Hydraulic Conductivity Estimation Using Artificial Neural Networks" *Water* 13, no. 5: 705.
https://doi.org/10.3390/w13050705