# Spatial Interpolation of Soil Temperature and Water Content in the Land-Water Interface Using Artificial Intelligence

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Dataset

^{2}) were downloaded from the freely accessible website of Environment and Climate Change Canada, which is the department of the Government of Canada responsible for coordinating environmental policies and programs (https://weather.gc.ca/grib/grib2_reg_10km_e.html, accessed on 15 May 2021). Both soil temperature and soil moisture variables were collected 0–10 cm below ground level.

#### 2.2. Description of Applied Methods

#### 2.2.1. Deterministic Interpolation

#### 2.2.2. Radial Basis Function Neural Networks

#### 2.2.3. Deep Learning

#### 2.3. Methodological Overview

## 3. Results

## 4. Discussions

#### 4.1. Interpolation of the Water Content of the Soil

#### 4.2. Evaluation of Methods’ Performance along the Railroad

## 5. Conclusions

- The spline interpolation method, which belongs to the deterministic category, showed weaknesses in calculating interpolated values in areas with sudden variations due to its inherent property of fitting a minimum curvature surface. This limitation did not improve relatively by increasing the nonlinearity of the fitted function.
- AI methods used in this study were able to demonstrate a confident and stable performance in zones with sudden changes and can provide an alternative for deterministic interpolation methods.
- Although both RBF and deep neural networks showed successful performance in interpolating data even over sharp change areas, deep learning demonstrated overall better accuracy in validation. Therefore, interpolated temperatures estimated along the railroad, calculated with a deep neural network model, were more reliable.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Graphical demonstration of gridded values of soil temperature in the study area, reported by Environment and Climate Change Canada (dashed circle shows the area of interest).

**Figure 3.**Location of 6640 reference points (orange dots), 1300 interpolation points (blue dots) on the railroad, and 12 evaluation points (red dots).

**Figure 5.**Interpolated soil temperature calculated by (

**a**) linear method (

**b**) cubic spline method (

**c**) quintic spline method (

**d**) 3D graphs for spline results.

**Figure 6.**Scatter plots of interpolated and actual soil temperature, identity, and regression lines of (

**a**) RBFN (

**b**) Deep learning; confidence and prediction bands of (

**c**) RBFN (

**d**) Deep learning.

**Figure 8.**Interpolation soil temperature results along the railroad by (

**a**) RBFN (

**b**) Deep learning (

**c**) 3D graph of RBFN (

**d**) 3D graph of Deep learning.

**Figure 9.**Interpolation soil water content results along the railroad by (

**a**) RBFN (

**b**) Deep learning (

**c**) 3D graph of RBFN (

**d**) 3D graph of Deep learning.

Maximum iteration(RBFN) | 100 | 500 | 700 | 1000 | ||||

R-squared | 0.54655 | 0.54378 | 0.54957 | 0.54641 | ||||

Neurons in hidden layer(Deep learning) | 300 | 500 | 300, 30 | 300, 100 | 500, 30 | 500, 100 | ||

R-squared | 0.83668 | 0.84645 | 0.85651 | 0.87846 | 0.88696 | 0.89011 |

Error index | MaxE (K) | MAE (K) | MSE (K ^{2}) | RMSE (K) | NRMSE (-) | RRMSE (-) |

Method | ||||||

RBFN | 14.89 | 2.58 | 16.50 | 4.06 | 16.25% | 1.41% |

Deep Learning | 8.13 | 1.63 | 5.12 | 2.26 | 9.05% | 0.78% |

Error index | MAPE (-) | Bias (K) | R^{2}(-) | NSE (-) | VAF (-) | AIC |

Method | ||||||

RBFN | 0.90% | 0.08 | 53.81% | 53.78% | 53.80% | 23100 |

Deep Learning | 0.57% | 1.13 | 89.24% | 85.65% | 89.22% | 20800 |

Error index | MaxE (kg/m ^{2}) | MAE (kg/m ^{2}) | MSE (kg ^{2}/m^{4}) | RMSE (kg/m ^{2}) | NRMSE (-) | RRMSE (-) |

Method | ||||||

RBFN | 0.76 | 0.10 | 0.03 | 0.17 | 17.54% | 69.55% |

Deep Learning | 0.66 | 0.04 | 0.01 | 0.08 | 7.92% | 31.39% |

Error index | MAPE (-) | Bias (kg/m ^{2}) | R^{2}(-) | NSE (-) | VAF (-) | AIC |

Method | ||||||

RBFN | 48.00% | 0.00 | 56.91% | 56.88% | 56.91% | 8600 |

Deep Learning | 20.69% | 0.01 | 91.32% | 91.21% | 91.32% | 5900 |

**Table 4.**Error analysis of soil temperature and water content values interpolated along the railroad using different AI models.

Variable | Soil Temperature | Water Content | ||
---|---|---|---|---|

Interpolation Method | RBFN | Deep Learning | RBFN | Deep Learning |

RMSE | 2.26 | 1.30 | 0.09 | 0.06 |

R^{2} | 26% | 67% | 39% | 34% |

Bias | −1.34 | 0.32 | 0.05 | 0.03 |

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**MDPI and ACS Style**

Imanian, H.; Shirkhani, H.; Mohammadian, A.; Hiedra Cobo, J.; Payeur, P. Spatial Interpolation of Soil Temperature and Water Content in the Land-Water Interface Using Artificial Intelligence. *Water* **2023**, *15*, 473.
https://doi.org/10.3390/w15030473

**AMA Style**

Imanian H, Shirkhani H, Mohammadian A, Hiedra Cobo J, Payeur P. Spatial Interpolation of Soil Temperature and Water Content in the Land-Water Interface Using Artificial Intelligence. *Water*. 2023; 15(3):473.
https://doi.org/10.3390/w15030473

**Chicago/Turabian Style**

Imanian, Hanifeh, Hamidreza Shirkhani, Abdolmajid Mohammadian, Juan Hiedra Cobo, and Pierre Payeur. 2023. "Spatial Interpolation of Soil Temperature and Water Content in the Land-Water Interface Using Artificial Intelligence" *Water* 15, no. 3: 473.
https://doi.org/10.3390/w15030473