# Geological Modelling and Validation of Geological Interpretations via Simulation and Classification of Quantitative Covariates

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study Presentation

_{2}), phosphorus (P), alumina (Al

_{2}O

_{3}), manganese (Mn), loss on ignition (LOI), and the granulometric fraction of fragments with size above 6.3 mm (G). In addition, for each sample, the dominant rock type is available from geological logging, which is coded into ten categories: friable hematite (code 1), compact hematite (code 2), alumina-rich hematite (code 3), alumina-rich itabirite (code 4), manganese-rich itabirite (code 5), compact itabirite (code 6), friable iron-poor itabirite (code 7), friable iron-rich itabirite (code 8), amphibolitic itabirite (code 9), and canga (code 10). There are dependent relationships between the quantitative variables and the rock codes [23], as summarised in Table 1. This implies that information about the former may help to detect inconsistencies in the interpretation of the latter, which is the basis of the proposed geostatistical methodology.

#### 2.2. Modelling and Simulation of Quantitative Variables

_{2}, P, Al

_{2}O

_{3}, Mn, LOI, G) and the spatial correlation structure of these variables depends on the prevailing rock type domain. In the following, we will reverse this point of view and assume that the rock type is subordinate to the quantitative variables. In other words, the quantitative variables will be modelled and simulated throughout the deposit without any previous geological domaining. Because of the relationship between the grades, granulometry, and rock types (Table 1), the rock type will then be allocated on the basis of the simulated values of these quantitative variables by means of a classification algorithm. Unlike the aforementioned hierarchical model, this approach does not produce discontinuities in the values of the quantitative variables near the rock type boundaries, which conforms with the concept of a disseminated ore deposit [14,15]. In this deposit, the quantitative variables are spatially correlated across the rock type boundaries, as shown in [10,24].

#### 2.2.1. Change of Variables Based on Stoichiometric Closure

_{2}+ 2.2913 P + Al

_{2}O

_{3}+ 1.2912 Mn + LOI = 100

_{2}O

_{3}), phosphorus pentoxide (P

_{2}O

_{5}), and manganese monoxide (MnO), respectively. A convenient way of reproducing the stoichiometric closure in the simulated grade values is to conduct a change of variables. Some alternatives for such a change of variables are the additive logratio (alr), centred logratio (clr), or isometric logratio (ilr) transformations that are often used in compositional data analysis [25], but these transformations are not suitable for variables that can take zero values, as is the case in the present case study. We therefore opt for a ratio transformation that does not use logarithms, as proposed in [10], where the quantitative variables are successively normalised by the residual of the closure:

_{2}) in order to minimise the distortion induced by the ratio transformation (the correlation coefficients between Z

_{1}and P, Z

_{2}and Mn, Z

_{3}and Al

_{2}O

_{3}, Z

_{4}and LOI, and Z

_{5}and SiO

_{2}are all greater than 0.995 [10]). The transformed variables have no stoichiometric constraint and take their values in the interval [0–1). Note that there are only five unconstrained transformed variables (Z

_{1}–Z

_{5}) instead of six constrained grade variables (Fe, SiO

_{2}, P, Al

_{2}O

_{3}, Mn, LOI). The back-transformation is obtained from Equations (1) and (2):

#### 2.2.2. Projection Pursuit Multivariate Transformation

_{1}–Z

_{5}) and granulometry (G) are transformed into multivariate Gaussian data, hereafter called “normal scores”. Because of the heteroscedastic dependence relationships between the variables prior to transformation (Figure 2), the normal scores transformation of each variable separately [26] does not provide truly multivariate Gaussian data. For instance, the scatter diagram between any two transformed variables does not have an elliptical shape, which indicates that these transformed variables do not correspond to jointly Gaussian random fields. To avoid this inconvenience, a joint normal scores transformation can be used, such as stepwise conditional transformation (SCT) [27], flow transformation (FT) [28], or projection pursuit multivariate transformation (PPMT) [29,30,31]. All these methods require all the variables to be known at all the data locations (isotopic sampling), which is the case in the present case study; otherwise, the data set should be completed by multivariate imputation techniques [32] before joint normal scores transformation. In practice, the first two approaches are still limited to few variables (SCT) or to small data sets (FT), and for this reason we chose the third approach (PPMT) here. The PPMT transformation is based on an iterative algorithm and allows the complex dependence relationships (such as nonlinearities and heteroscedasticities) between cross-correlated variables to be removed, providing a set of new variables that are normally distributed and uncorrelated at collocated locations [29,30,31]. The transformation uses declustering weights to account for the uneven positions of the drill hole data in space. For each rock type, the weights are obtained by considering the ratio of the rock type proportion in the interpreted geological model and the rock type proportion in the drill hole data. It is assumed here that the interpreted model, which is constructed from the drill hole information and geological knowledge of the deposit, is globally accurate, i.e., it provides a reliable estimate of the true rock type proportions, although it may be locally inaccurate as some blocks may be misinterpreted.

_{1}, Z

_{2}, Z

_{3}, Z

_{4}, Z

_{5}, G) into a multi-Gaussian one. The marginal distributions (histograms) are bell-shaped, while the bivariate distributions (scatter plots) exhibit the typical circular shape of uncorrelated Gaussian variables.

#### 2.2.3. Spatial Continuity Modelling

#### 2.2.4. Conditional Simulation

#### 2.2.5. Checking the Realisations

#### 2.3. Construction of Simulated Geological Scenarios by Classification

#### 2.4. Determining the Prior Probability of Occurrence of Each Rock Type

_{1}, …, p

_{9}) of the different rock types by counting the numbers of occurrences of each rock type across the 1000 realisations (Table 4). This only requires one block to be simulated because the prior probabilities are the same for all the blocks in the deposit.

#### 2.5. Comparing the Prior and Posterior Probabilities of Rock Type Occurrences to Identify Potentially Misinterpreted Blocks

## 3. Results and Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Isometric view of the interpreted rock type model, showing the plan view and vertical cross-sections passing through the origin (local coordinate system). Waste and air are shown in dark blue and grey, respectively.

**Figure 2.**Histograms and scatterplots of Z

_{1}vs. Z

_{2}, Z

_{3}vs. Z

_{4}, and Z

_{5}vs. G before (

**left**) and after (

**right**) PPMT transformation.

**Figure 3.**An example of experimental (crosses) and fitted (solid lines) direct and cross-variograms for the PPMT-transformed variables of Z

_{1}, Z

_{3}, and G, along the horizontal (black) and vertical (blue) directions.

**Figure 4.**Isometric view of two realisations of the grades and granulometry (

**left**: realisation 1 and

**right**: realisation 2), showing the plan view and vertical cross-sections passing through the origin (local coordinate system). Waste and air are shown in grey. From top to bottom: iron grade, silica grade, phosphorus grade, alumina grade, manganese grade, loss on ignition, granulometry.

**Figure 5.**Scatter diagrams of Fe vs. SiO

_{2}, Fe vs. LOI, and LOI vs. P for drill hole data (red) and simulated values (blue).

**Left**: realisation 1,

**right**: realisation 2.

**Figure 6.**Isometric view of two realisations of the classified rock type (

**left**: realisation 1 and

**right**: realisation 2), showing the plan view and vertical cross-sections passing through the origin (local coordinate system). Waste and air are shown in dark blue and grey, respectively.

**Figure 7.**Isometric view of block classification according to criteria in Table 5, showing the plan view and vertical cross-sections passing through the origin (local coordinate system). Waste and air are shown in dark blue and grey, respectively.

**Table 1.**Associations between rock types and quantitative variables (for each variable, “poor” and “fine” refer to the rock types with the lowest values, and “rich” and “coarse” to the rock types with the highest values) [23].

G | Fe | SiO_{2} | Al_{2}O_{3} | Mn | LOI | P | Rock Code |
---|---|---|---|---|---|---|---|

Coarse | Rich | Poor | 2 | ||||

Coarse | Poor | Rich | 6 | ||||

Fine | Rich | Poor | Rich | 3 | |||

Fine | Rich | Poor | Poor | 1 | |||

Fine | Intermediate | Intermediate | Rich | Rich | 5 | ||

Fine | Intermediate | Intermediate | Rich | Poor | Rich | Rich | 9 |

Fine | Intermediate | Intermediate | Rich | Poor | Poor | Poor | 4 |

Fine | Intermediate | Intermediate | Poor | 8 | |||

Fine | Poor | Rich | 7 | ||||

Intermediate | Rich | Poor | Rich | Poor | Rich | 10 |

**Table 2.**Correlation coefficients of drill hole data and simulated outcomes of grades and granulometry (correlation observed on drill hole data: bold entries above main diagonal; average correlation over 20 outcomes: regular entries above main diagonal; minimum correlation over 20 outcomes: bold entries under main diagonal; maximum correlation over 20 outcomes: regular entries under main diagonal).

Variable | Fe | Si | P | Al | Mn | LOI | G |
---|---|---|---|---|---|---|---|

Fe | 1 | −0.98/−0.99 | 0.21/0.18 | 0.27/0.30 | −0.04/−0.01 | 0.23/0.23 | −0.11/−0.07 |

Si | −0.99/−0.98 | 1 | −0.30/−0.29 | −0.39/−0.42 | −0.08/−0.08 | −0.36/−0.37 | 0.14/0.11 |

P | 0.14/0.22 | −0.32/−0.25 | 1 | 0.33/0.43 | 0.11/0.16 | 0.72/0.73 | −0.07/−0.10 |

Al | 0.27/0.32 | −0.44/−0.39 | 0.41/0.45 | 1 | 0.19/0.17 | 0.59/0.62 | −0.38/−0.32 |

Mn | −0.03/0.02 | −0.10/−0.06 | 0.13/0.20 | 0.14/0.21 | 1 | 0.19/0.15 | −0.06/−0.08 |

LOI | 0.20/0.27 | −0.40/−0.35 | 0.71/0.75 | 0.60/0.64 | 0.12/0.19 | 1 | −0.18/−0.13 |

G | −0.12/−0.04 | 0.07/0.16 | −0.14/−0.08 | −0.34/−0.30 | −0.10/−0.05 | −0.17/−0.10 | 1 |

**Table 3.**Classifiers tested for the case study with their rates of correct classification on drill hole data.

Classification Algorithm | Algorithm Type | Correct Classification Rate (Cross-Validation) |
---|---|---|

Simple Cart | Decision tree | 82.6 |

BF Tree | Decision tree | 81.7 |

Classification via Regression | Meta-learning algorithm | 81.7 |

REP Tree | Decision tree | 81.6 |

Random Forest | Decision tree | 81.1 |

Multilayer Perceptron (Neural Network) | Function | 80.6 |

Bayes Network | Bayesian | 77.5 |

RBF Network | Function | 75.1 |

Naive Bayes | Bayesian | 73.3 |

Random Tree | Decision tree | 70.6 |

Symbol | Rock Type Code | Prior Probability |
---|---|---|

p_{1} | 1 | 0.023 |

p_{2} | 2 | 0.051 |

p_{3} | 3 | 0.003 |

p_{4} | 4 | 0.048 |

p_{5} | 5 | 0.025 |

p_{6} | 6 | 0.345 |

p_{7} | 7 | 0.384 |

p_{8} | 8 | 0.090 |

p_{9} | 9 | 0.031 |

Symbol | Condition |
---|---|

+1 | Block under consideration is not affected significantly by the drill hole data |

+2 | Prior probability of the rock type interpreted by the geologists is lower than its posterior probability (the interpreted rock type “agrees” with the posterior distribution) |

+3 | Block under consideration does not meet condition +2, but there is not any evidence for a misinterpretation (neither +4 nor +5) |

+4 | Prior probability of the rock type interpreted by the geologists is greater than its posterior probability, posterior probability of the interpreted rock type is less than 0.15 (unlikely), and another rock type has a higher posterior probability |

+5 | In addition to the criteria of condition +4, a rock type with higher posterior probability has been logged at some drill hole sample less than 60 m from the block |

**Table 6.**Numbers of blocks with no evidence of misinterpretation (conditions +1 to +4) (diagonal line) and numbers of potentially misinterpreted blocks (condition +5) (off-diagonal) for each interpreted rock type (row) and each suggested rock type based on classification of 20 realisations (column).

Rock Type | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1 | 12,058 | 35 | 51 | 22 | 35 | 21 | 76 | 61 | 29 |

2 | 198 | 7712 | 45 | 14 | 8 | 41 | 239 | 223 | 27 |

3 | 70 | 61 | 5289 | 26 | 22 | 5 | 41 | 55 | 14 |

4 | 93 | 39 | 99 | 15,917 | 22 | 6 | 99 | 78 | 70 |

5 | 96 | 15 | 38 | 55 | 5788 | 3 | 85 | 89 | 4 |

6 | 535 | 216 | 73 | 410 | 165 | 349,723 | 2607 | 415 | 619 |

7 | 2292 | 990 | 1169 | 1458 | 776 | 1161 | 172,299 | 1623 | 2072 |

8 | 369 | 172 | 336 | 149 | 110 | 72 | 403 | 18,259 | 233 |

9 | 50 | 48 | 32 | 36 | 15 | 8 | 46 | 29 | 11,875 |

**Table 7.**Simulated grades and granulometry, and associated rock type, for 20 realisations of block n°1, interpreted as rock type 6 (compact itabirite) by mining geologists.

Realisation | Fe | SiO_{2} | P | Al_{2}O_{3} | Mn | LOI | G | Classified Rock Type |
---|---|---|---|---|---|---|---|---|

1 | 47.22 | 30.21 | 0.019 | 1.411 | 0.026 | 0.798 | 55.92 | 6 |

2 | 40.56 | 40.97 | 0.017 | 0.418 | 0.011 | 0.566 | 20.90 | 7 |

3 | 35.79 | 47.56 | 0.017 | 0.678 | 0.010 | 0.533 | 16.69 | 7 |

4 | 39.74 | 41.97 | 0.022 | 0.540 | 0.010 | 0.613 | 40.73 | 7 |

5 | 36.71 | 46.70 | 0.010 | 0.416 | 0.010 | 0.368 | 23.99 | 7 |

6 | 37.02 | 46.01 | 0.016 | 0.451 | 0.010 | 0.573 | 11.24 | 7 |

7 | 41.50 | 38.96 | 0.017 | 0.536 | 0.020 | 1.113 | 32.39 | 7 |

8 | 59.11 | 14.72 | 0.010 | 0.384 | 0.010 | 0.343 | 10.60 | 8 |

9 | 52.29 | 20.23 | 0.072 | 1.703 | 0.029 | 3.113 | 31.50 | 8 |

10 | 38.68 | 44.03 | 0.012 | 0.330 | 0.010 | 0.301 | 49.71 | 7 |

11 | 37.93 | 43.16 | 0.036 | 0.570 | 0.010 | 1.950 | 43.22 | 7 |

12 | 37.42 | 45.59 | 0.010 | 0.311 | 0.010 | 0.570 | 37.23 | 7 |

13 | 30.67 | 55.29 | 0.014 | 0.320 | 0.010 | 0.502 | 14.52 | 7 |

14 | 32.97 | 52.01 | 0.011 | 0.254 | 0.010 | 0.563 | 42.12 | 7 |

15 | 30.54 | 54.77 | 0.012 | 0.702 | 0.010 | 0.814 | 34.56 | 7 |

16 | 44.76 | 33.78 | 0.017 | 1.146 | 0.027 | 1.013 | 16.65 | 7 |

17 | 45.96 | 32.21 | 0.017 | 0.988 | 0.014 | 1.032 | 33.37 | 7 |

18 | 40.43 | 41.54 | 0.013 | 0.303 | 0.010 | 0.314 | 45.39 | 7 |

19 | 47.19 | 30.01 | 0.032 | 1.164 | 0.018 | 1.263 | 41.08 | 7 |

20 | 42.17 | 37.34 | 0.048 | 0.901 | 0.015 | 1.343 | 44.33 | 7 |

**Table 8.**Simulated grades and granulometry, and associated rock type, for 20 realisations of block n°2, interpreted as rock type 7 (friable iron-poor itabirite) by mining geologists.

Realisation | Fe | SiO_{2} | P | Al_{2}O_{3} | Mn | LOI | G | Classified Rock Type |
---|---|---|---|---|---|---|---|---|

1 | 67.59 | 1.07 | 0.013 | 1.193 | 0.010 | 1.054 | 26.67 | 1 |

2 | 66.88 | 2.50 | 0.016 | 1.240 | 0.010 | 0.584 | 46.37 | 1 |

3 | 66.17 | 3.46 | 0.015 | 1.275 | 0.010 | 0.613 | 48.08 | 1 |

4 | 64.99 | 5.14 | 0.014 | 1.317 | 0.010 | 0.576 | 3.89 | 1 |

5 | 67.46 | 1.68 | 0.018 | 1.020 | 0.010 | 0.805 | 19.45 | 1 |

6 | 68.66 | 1.24 | 0.010 | 0.346 | 0.010 | 0.216 | 57.84 | 2 |

7 | 62.30 | 7.40 | 0.015 | 2.319 | 0.010 | 1.161 | 10.35 | 1 |

8 | 66.55 | 2.07 | 0.020 | 1.663 | 0.010 | 1.061 | 25.53 | 1 |

9 | 67.89 | 2.27 | 0.010 | 0.361 | 0.010 | 0.272 | 2.92 | 1 |

10 | 68.74 | 0.83 | 0.015 | 0.388 | 0.010 | 0.448 | 31.09 | 1 |

11 | 62.57 | 9.15 | 0.012 | 0.851 | 0.010 | 0.503 | 10.12 | 1 |

12 | 61.69 | 9.59 | 0.024 | 0.970 | 0.010 | 1.168 | 18.37 | 8 |

13 | 65.41 | 3.59 | 0.016 | 1.816 | 0.010 | 1.021 | 35.01 | 1 |

14 | 67.70 | 2.10 | 0.012 | 0.679 | 0.010 | 0.389 | 10.40 | 1 |

15 | 67.63 | 0.65 | 0.028 | 1.240 | 0.011 | 1.339 | 37.19 | 1 |

16 | 67.66 | 1.84 | 0.021 | 0.704 | 0.010 | 0.659 | 14.63 | 1 |

17 | 59.19 | 14.67 | 0.010 | 0.397 | 0.010 | 0.272 | 1.93 | 8 |

18 | 62.59 | 8.48 | 0.012 | 1.255 | 0.010 | 0.743 | 22.11 | 1 |

19 | 66.79 | 1.55 | 0.032 | 1.533 | 0.016 | 1.338 | 31.18 | 1 |

20 | 68.48 | 0.71 | 0.016 | 0.581 | 0.010 | 0.748 | 32.40 | 1 |

**Table 9.**Simulated grades and granulometry, and associated rock type, for 20 realisations of block n°3, interpreted as rock type 8 (friable iron-rich itabirite) by mining geologists.

Realisation | Fe | SiO_{2} | P | Al_{2}O_{3} | Mn | LOI | G | Classified Rock Type |
---|---|---|---|---|---|---|---|---|

1 | 63.608 | 0.80 | 0.052 | 0.873 | 0.410 | 6.736 | 9.90 | 3 |

2 | 62.235 | 2.79 | 0.076 | 3.258 | 0.021 | 4.770 | 14.71 | 3 |

3 | 62.653 | 1.17 | 0.042 | 1.497 | 0.053 | 7.595 | 4.63 | 3 |

4 | 64.02 | 2.96 | 0.037 | 2.453 | 0.064 | 2.884 | 1.80 | 1 |

5 | 64.817 | 0.80 | 0.095 | 1.886 | 0.095 | 4.310 | 20.67 | 3 |

6 | 62.642 | 2.64 | 0.105 | 3.058 | 0.083 | 4.397 | 14.68 | 3 |

7 | 63.305 | 1.23 | 0.047 | 2.737 | 0.073 | 5.326 | 34.15 | 3 |

8 | 62.068 | 1.11 | 0.126 | 3.221 | 0.034 | 6.600 | 21.30 | 3 |

9 | 65.543 | 0.39 | 0.069 | 1.635 | 0.031 | 4.065 | 23.33 | 3 |

10 | 62.09 | 0.97 | 0.070 | 2.925 | 0.095 | 7.048 | 44.50 | 3 |

11 | 63.943 | 1.32 | 0.037 | 1.387 | 0.024 | 5.758 | 32.72 | 3 |

12 | 63.108 | 1.14 | 0.039 | 2.357 | 0.039 | 6.141 | 7.71 | 3 |

13 | 59.538 | 2.04 | 0.073 | 6.104 | 0.080 | 6.466 | 8.94 | 9 |

14 | 62.418 | 0.52 | 0.099 | 3.669 | 0.075 | 6.244 | 5.13 | 3 |

15 | 62.12 | 0.41 | 0.096 | 3.155 | 0.033 | 7.361 | 6.21 | 3 |

16 | 63.739 | 0.52 | 0.040 | 1.420 | 0.148 | 6.644 | 23.91 | 3 |

17 | 57.725 | 0.49 | 0.095 | 5.816 | 3.110 | 6.932 | 11.19 | 5 |

18 | 64.591 | 1.03 | 0.067 | 1.721 | 0.033 | 4.707 | 17.59 | 3 |

19 | 62.706 | 2.49 | 0.079 | 2.092 | 0.018 | 5.559 | 21.10 | 3 |

20 | 64.873 | 0.57 | 0.043 | 1.131 | 0.025 | 5.423 | 40.42 | 3 |

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**MDPI and ACS Style**

Adeli, A.; Emery, X.; Dowd, P.
Geological Modelling and Validation of Geological Interpretations via Simulation and Classification of Quantitative Covariates. *Minerals* **2018**, *8*, 7.
https://doi.org/10.3390/min8010007

**AMA Style**

Adeli A, Emery X, Dowd P.
Geological Modelling and Validation of Geological Interpretations via Simulation and Classification of Quantitative Covariates. *Minerals*. 2018; 8(1):7.
https://doi.org/10.3390/min8010007

**Chicago/Turabian Style**

Adeli, Amir, Xavier Emery, and Peter Dowd.
2018. "Geological Modelling and Validation of Geological Interpretations via Simulation and Classification of Quantitative Covariates" *Minerals* 8, no. 1: 7.
https://doi.org/10.3390/min8010007