# Predictions of Geological Interface Using Relevant Vector Machine with Borehole Data

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## Abstract

**:**

## 1. Introduction

## 2. Relevant Vector Machine

**α**and ${\sigma}^{2}$ values. The noise variance ${\sigma}^{2}$ can be updated by,

## 3. The Width Parameter Optimized by PSO

## 4. Results and Discussion

#### 4.1. Two-Dimensional Analysis

#### 4.2. Three-Dimensional Analysis

## 5. Conclusions and Discussions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Source Codes of the Key Program in MATLAB

## References

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**Figure 3.**Stratigraphic stratification predictions for a two-dimensional section (four training boreholes).

**Figure 5.**Stratigraphic stratification prediction of two-dimensional section (eight training holes).

Kernel Function | $\mathbf{\Omega}$ for Training Data | $\mathbf{\Omega}$ for Validation Data | $\mathbf{\Delta}$ (m) for V2 |
---|---|---|---|

Spline | 1.0 | 0.8544 | 8.4 |

Cauchy | 1.0 | 0.8576 | 12.4 |

Gauss | 1.0 | 0.8479 | 9.4 |

Kernel Function | V1 | V2 | V3 | ALL |
---|---|---|---|---|

Spline | 0.9712 | 0.9053 | 0.9733 | 0.9515 |

Cauchy | 0.9712 | 0.7684 | 1.0 | 0.9159 |

Gauss | 0.9640 | 0.5684 | 0.9867 | 0.8544 |

Kernel Function | V1 | V2 | V3 | ALL |
---|---|---|---|---|

Spline | 0.8 | 1.8 | 0.4 | 1.0 |

Cauchy | 0.8 | 4.4 | 0.2 | 1.8 |

Gauss | 1.0 | 8.0 | 0.2 | 3.1 |

Kernel Function | V1 | V2 | V3 | ALL | Execution Time [sec] |
---|---|---|---|---|---|

Spline | 0.9856 | 0.9053 | 1.0 | 0.9644 | 6910 |

Cauchy | 0.9928 | 0.8842 | 0.9733 | 0.9547 | 7220 |

Gauss | 0.9856 | 0.9368 | 0.9867 | 0.9709 | 7015 |

Kernel Function | V1 | V2 | V3 | ALL |
---|---|---|---|---|

Spline | 0.4 | 1.7 | 0 | 0.7 |

Cauchy | 0.2 | 2.0 | 0.4 | 0.9 |

Gauss | 0.4 | 1.2 | 0.2 | 0.6 |

Kernel Function | Spline | Cauchy | Gauss |
---|---|---|---|

V1 | 0.9784 | 0.9640 | 0.9712 |

V2 | 0.9504 | 0.8582 | 0.9078 |

V3 | 0.9589 | 0.8973 | 0.9110 |

V4 | 0.9291 | 0.9764 | 0.9606 |

V5 | 0.9927 | 0.9927 | 1.0000 |

ALL | 0.9623 | 0.9362 | 0.9493 |

Execution Time (sec) | 8012 | 9040 | 8520 |

Kernel Function | Spline | Cauchy | Gauss |
---|---|---|---|

V1(m) | 0.4 | 1.0. | 0.8 |

V2(m) | 1.4 | 4.0 | 2.6 |

V3(m) | 1.0 | 3.8 | 2.4 |

V4(m) | 1.8 | 0.6 | 1.0 |

V5(m) | 0.4 | 0.0 | 0.0 |

Mean(m) | 1.0 | 1.8 | 1.4 |

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**MDPI and ACS Style**

Ji, X.; Lu, X.; Guo, C.; Pei, W.; Xu, H.
Predictions of Geological Interface Using Relevant Vector Machine with Borehole Data. *Sustainability* **2022**, *14*, 10122.
https://doi.org/10.3390/su141610122

**AMA Style**

Ji X, Lu X, Guo C, Pei W, Xu H.
Predictions of Geological Interface Using Relevant Vector Machine with Borehole Data. *Sustainability*. 2022; 14(16):10122.
https://doi.org/10.3390/su141610122

**Chicago/Turabian Style**

Ji, Xiaojia, Xuanyi Lu, Chunhong Guo, Weiwei Pei, and Hui Xu.
2022. "Predictions of Geological Interface Using Relevant Vector Machine with Borehole Data" *Sustainability* 14, no. 16: 10122.
https://doi.org/10.3390/su141610122