# Automatic Parameter Tuning of Multiple-Point Statistical Simulations for Lateritic Bauxite Deposits

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## Abstract

**:**

## 1. Introduction

## 2. Overview of Underlying Methods

#### 2.1. Direct Sampling Algorithm

#### 2.2. Comparing Patterns with Smooth Histograms

#### 2.3. Generalised Simulated Annealing

- Choose a high initial temperature ${T}_{0}$ value, an initial solution ${x}^{0}$ and evaluate the objective function, ${E}^{0}=f({x}^{0})$.
- Propose a new solution ${x}^{i+1}$:
- Generate a candidate solution ${x}^{i+1}$ from the current one (${x}^{i}$) using a predefined visiting distribution.
- Evaluate the change in the objective function for the candidate solution, $\Delta =f({x}^{i+1})-f({x}^{i})$.
- Accept the iteration if it reduces the objective function, $\Delta <0$.
- Otherwise, accept or reject it based on a probability of acceptance criterion.

- Repeat step 2 for L number of iteration times keeping T constant.
- Reduce the temperature to ${T}_{n+1}$ using a cooling function.
- Repeat steps 2–4 until the convergence criteria is satisfied.

## 3. Problem Setup and Methodology

#### Methodology

## 4. Results and Discussion

#### 4.1. Implementation and Analysis of the Tuned Parameters on the Simulations

#### 4.2. Effect of the Optimisation Method and the Initial Parameters on the Final Results

#### 4.3. Effect of the Chosen Parameters on the Pattern Reproduction

#### 4.4. Analysis of the Automatically Tuned Parameters

## 5. Summary and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SG | Simulation Grid |

MPS | Multiple-Point Statistics |

TI | Training Image |

DS | Direct Sampling |

SA | Simulated Annealing |

JS | Jensen-Shannon Divergence |

L-BFGS-B | Limited Memory Broyden-Fletcher-Goldfarb-Shanno Optimisation with Box Constraints |

BOBYQA | Bound Optimisation by Quadratic Approximation |

Glossary of Notations | |

${Z}^{*}(x)$ | Simulated value at location x |

${d}_{n}(x,L)$ | Data event constructed at location x of the simulation grid |

${d}_{n}(y,L)$ | Data event constructed at location y of the training image grid |

$Z(y)$ | Value of the attribute at y location of the training image |

t | Acceptance threshold to accept a pattern compatible |

f | Fraction of the training image to be scanned |

$d\{{d}_{n}(x,L),{d}_{n}(y,L)\}$ | Distance between the conditioning data and training image patterns |

${\alpha}_{i}$ | Weighting factor applied to the ith data event node |

$\delta $ | Power for computing the ${\alpha}_{i}$ weighting factor |

w | Special weight attached to the conditioning data events |

T | Search template used to construct the pseudo-histograms |

$pa{t}_{j}^{TI,un}$ | jth unique pattern collected from the training image |

${H}^{d,m}$ | Histogram class of the model image |

${H}^{d,TI}$ | Histogram class of the training image |

$pa{t}_{i}^{m}$ | Pattern i observed in the model image |

${N}^{TI,un}$ | Number of unique pattern categories in the training image |

${d}_{JS}(p,q)$ | Jensen-shannon divergence of p and q density distributions |

$f(x)$ | Objective function |

$\Delta $ | Change in the objective function due to a candidate solution ${x}^{i+1}$ |

$pa{t}_{i}^{C}$ | ith conditioning pattern collected from the simulation grid |

$\overline{pa{t}_{i}^{C}}$ | Mean of the values at the nodes of the pattern $pa{t}_{i}^{C}$ |

${N}^{C}$ | Number of patterns captured from the SG. |

$pa{t}_{i}^{SIM}$ | Pattern i collected from the simulation grid |

$\overline{pa{t}_{i}^{SIM}}$ | Mean value of the nodes in pattern $pa{t}_{i}^{SIM}$ |

${H}_{j}^{{}^{d,SIM}}$ | Value of the jth category of the realisation image pseudo-histogram |

${H}^{d,SIM}$ | Pseudo-histogram of a realisation |

${N}^{SIM}$ | Number of patterns captured in the realisation image generated |

${{H}_{j}}^{{{}^{d}}_{Nor}}$ | Normalised weight in the jth histogram category |

${H}_{j}^{{}^{d,{C}_{Nor}}}$ | Normalised weight of the jth category in the conditioning data pseudo-histogram |

${H}_{j}^{{}^{d,SI{M}_{{}_{Nor}}}}$ | Normalised weight of the jth category in the simulation image pseudo-histogram |

${w}_{hd}$ | Weighting factor attached to the hard data |

${w}_{GPR}$ | Weighting factor attached to the GPR data |

${n}_{hd}$ | Maximum number hard data nodes to be used in the MPS simulations |

${n}_{GPR}$ | Maximum number GPR nodes to be used in the MPS simulations |

${t}_{hd}$ | Acceptance threshold for the hard data |

${t}_{GPR}$ | Acceptance threshold for the GPR data |

$L(\alpha ,\psi )$ | Likelihood as a function of model parameters ($\alpha ,\psi $) |

${L}_{p}(\alpha )$ | Profile of the likelihood as a function of the parameter $\alpha $ |

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**Figure 1.**Simulation and the TI variables: (

**a**) GPR variable used as the secondary simulation variable; (

**b**) borehole data used to condition the simulations (primary variable-black dots overlying the GPR survey); (

**c**) floor survey of a previously extracted mine area (primary TI variable) and (

**d**) GPR data of the previously mined area collected prior to the extraction (Secondary TI variable).

**Figure 5.**(

**a**) Initial average of the realisations; (

**b**) final average of the realisations. Interquartile range maps: (

**c**) before the parameter optimisation and (

**d**) after the parameter optimisation.

**Figure 6.**Evolution of the distributions of the IQR values calculated at each grid node. Mean of the IQR values dropped from 0.60 to 0.47 in automatically found parameters.

**Figure 7.**Variograms of the realisations and the experimental variogram of the borehole data: (

**a**) Variograms before the optimisation; (

**b**) variograms after the optimisation and (

**c**) 99.9% confidence interval of the variograms before and after the optimisation.

**Figure 8.**Convergence comparisons of three different optimisers with three different set of initial DS parameters.

Parameters | ${\mathit{w}}_{\mathit{hd}}$ | ${\mathit{n}}_{\mathit{hd}}$ | ${\mathit{n}}_{\mathit{GPR}}$ | ${\mathit{t}}_{\mathit{hd}}$ | ${\mathit{t}}_{\mathit{GPR}}$ | ${\mathit{d}}_{\mathit{JS}}$ |
---|---|---|---|---|---|---|

Lower Boundaries | 1 | 1 | 1 | 0.0001 | 0.0001 | 0.1520 |

Initial Parameters | 10 | 5 | 20 | 0.010 | 0.10 | 0.1020 |

Automatic Tuning | 13.07 | 14 | 45 | 0.009 | 0.305 | 0.0667 |

Upper Boundaries | 15 | 50 | 50 | 0.5 | 0.5 | 0.1793 |

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**MDPI and ACS Style**

Dagasan, Y.; Renard, P.; Straubhaar, J.; Erten, O.; Topal, E.
Automatic Parameter Tuning of Multiple-Point Statistical Simulations for Lateritic Bauxite Deposits. *Minerals* **2018**, *8*, 220.
https://doi.org/10.3390/min8050220

**AMA Style**

Dagasan Y, Renard P, Straubhaar J, Erten O, Topal E.
Automatic Parameter Tuning of Multiple-Point Statistical Simulations for Lateritic Bauxite Deposits. *Minerals*. 2018; 8(5):220.
https://doi.org/10.3390/min8050220

**Chicago/Turabian Style**

Dagasan, Yasin, Philippe Renard, Julien Straubhaar, Oktay Erten, and Erkan Topal.
2018. "Automatic Parameter Tuning of Multiple-Point Statistical Simulations for Lateritic Bauxite Deposits" *Minerals* 8, no. 5: 220.
https://doi.org/10.3390/min8050220