# A Hybrid Approach for Joint Simulation of Geometallurgical Variables with Inequality Constraint

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Inequality Constraint

#### 2.2. Minimum/Maximum Autocorrelation Factor (MAF)

- Transform the original variables to normal score values with a mean of zero and variance one $N\left(0,1\right):$ this can be implemented by normal score transformation methodologies such as Gaussian anamorphosis [32] or quantile-based approach [33].$$Z\left(u\right)={G}^{-1}\left(F\left(Y\left(u\right)\right)\right)$$
- Compute the experimental variance-covariance matrix $\left(V\right)$: since we are dealing with normal score values, this matrix is identical to the sample correlation matrix. In the case of two variables, this matrix $\left(V\right)$ is given by:$$V=Corr\left\{Z\left(u\right),Z\left(u\right)\right\}=\left[\begin{array}{cc}{\rho}_{11}\left(0\right)& {\rho}_{12}\left(0\right)\\ {\rho}_{21}\left(0\right)& {\rho}_{22}\left(0\right)\end{array}\right]$$
- Perform the spectral decomposition of above matrix $\left(V\right)$ to derive the orthonormal eigenvectors matrix $\left({M}_{1}\right)$, associated with the underlying diagonal eigenvalues matrix $\left({E}_{1}\right)$, such that:$$V={M}_{1}{E}_{1}{M}_{1}^{T}$$It is necessary to check that the entries of $E$ are in decreasing order.
- Calculate the PCA transformations at locations $u$ by:$$PCA\left(u\right)={E}_{1}^{-1/2}{M}_{1}Z\left(u\right)$$
- Choose a proper nonzero lag distance $h$ and calculate the sample covariance and cross-covariance matrices ${\widehat{\mathsf{\Lambda}}}_{PCA}\left(h\right)$ over the $PCA$ scores, so its related spectral decomposition with diagonal eigenvalues matrix $\left({E}_{2}\right)$ and orthonormal eigenvectors matrix $\left({M}_{1}\right)$ is:$${\widehat{\mathsf{\Lambda}}}_{PCA}\left(h\right)={M}_{2}{E}_{2}{M}_{2}^{T}\text{}$$It is worth mentioning that since the $PCA$ scores are standard values, the variance-covariance matrix is identical to correlogram matrix.
- Finally, the MAF factors at location $u$ can be derived:$$\tau \left(u\right)={M}_{2}PCA\left(u\right)$$$$Z\left(u\right)={M}_{2}^{-1}\tau \left(u\right)$$

- Convert the original cross-correlated variables to the new variables free of inequality constraint
- Transform the declustered converted variables into normal score data (Gaussian random field with mean 0 and variance 1) (Equation (1))
- Transform the normal score data into orthogonal MAF factors (Equation (6)).
- Calculate the experimental variograms for each MAF factor
- Independent Gaussian simulation of MAF factors
- Back-transformation of the simulation results (realizations) into normal score space (Equation (7))
- Back-transformation of the normal score realizations into the original space in order to restitute the intrinsic cross-correlation

## 3. Application to an Actual Case study

#### 3.1. Conventional Co-Simulation

#### 3.2. Joint Simulation with MAF Transformation

#### 3.3. Validation of Results

#### 3.3.1. Validation of Global Statistical Parameter

#### 3.3.2. Validation of Local Statistical Parameter

#### 3.3.3. Sensitivity Analysis of the Number of Simulations

#### 3.3.4. Post Processing the Realizations: Probabilistic Domaining of Geometallurgical Domains

#### 3.3.5. Validation against Actual Data

## 4. Discussion on Application and Limitations of the Proposed Approach

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Macfarlane, A.S.; Williams, T.P. Optimizing value on a copper mine by adopting a geometallurgical solution. J. South. Afr. Inst. Min. Metall.
**2014**, 114, 929–935. [Google Scholar] - Almeida, J.A. Modelling of cement raw material compositional indices with direct sequential cosimulation. Eng. Geol.
**2010**, 114, 26–33. [Google Scholar] [CrossRef] - Tercan, A.E.; Sohrabian, B. Multivariate geostatistical simulation of coal quality by independent components. Int. J. Coal Geol.
**2013**, 112, 53–66. [Google Scholar] [CrossRef] - Brissette, M.; Mihajlovic, V.; Sanuri, S. Geometallurgy: New accurate test work to meet required accuracies of mining project development. In Proceedings of the XXVII International Mineral Processing Congress, Santiago, Chile, 20–24 October 2014. [Google Scholar]
- Deutsch, J.L.; Szymanski, J.; Etsell, T.H. Metallurgical Variable Re-expression for Geostatistics. In Geostatistical and Geospatial Approaches for the Characterization of Natural Resources in the Environment; Springer: Cham, Switzerland, 2016; pp. 83–88. [Google Scholar]
- Tolosana-Delgado, R.; Mueller, U.; van den Boogaart, K.G.; Ward, C.; Gutzmer, J. Improving processing by adaption to conditional geostatistical simulation of block compositions. J. South. Afr. Inst. Min. Metall.
**2015**, 115, 13–26. [Google Scholar] [CrossRef] [Green Version] - Emery, X. Co-simulating total and soluble copper grades in an oxide ore deposit. Math. Geosci.
**2012**, 44, 27–46. [Google Scholar] [CrossRef] - Hosseini, S.A.; Asghari, O. Simulation of geometallurgical variables through stepwise conditional transformation in Sungun copper deposit, Iran. Arab. J. Geosci.
**2015**, 8, 3821–3831. [Google Scholar] [CrossRef] - Bai, S.; Fu, X.; Li, C.; Wen, S. Process improvement and kinetic study on copper leaching from low-grade cuprite ores. Physicochem. Probl. Miner. Process
**2018**, 54, 300–310. [Google Scholar] - Pizarro, S.; Emery, X. Geostatistical joint modelling of total and soluble copper grades. In Proceedings of the 2nd International Seminar on Geology for the Mining Industry GEOMIN 2011, Antofagasta, Chile, 8–10 June 2011; Beniscelli, J., Kuyvenhoven, R., Hoal, K.O., Eds.; Gecamin Ltda: Santiago, Chile, 2011. [Google Scholar]
- Cáceres, A.; Riquelme, R.; Emery, X.; Díaz, J.; Fuster, G. Total and soluble copper grade estimation using minimum/maximum autocorrelation factors and multigaussian kriging. In Proceedings of the 2nd International Seminar on Geology for the Mining Industry GEOMIN 2011, Antofagasta, Chile, 8–10 June 2011; Beniscelli, J., Kuyvenhoven, R., Hoal, K.O., Eds.; Gecamin Ltda: Santiago, Chile, 2011. [Google Scholar]
- Lange, W.; Emery, X. Joint simulation of total and soluble copper grades in an oxide copper deposit. In Proceedings of the 5th International Conference on Innovation in Mine Operations, Santiago, Chile, 20–22 June 2012; Kuyvenhoven, R., Morales, J.E., Vega, C., Eds.; Gecamin Ltda: Santiago, Chile, 2012; pp. 62–63. [Google Scholar]
- Dubrule, O.; Kostov, C. An interpolation method taking into account inequality constraints: I. Methodology. Math. Geol.
**1986**, 18, 33–51. [Google Scholar] [CrossRef] - Leuangthong, O.; Deutsch, C.V. Stepwise conditional transformation for simulation of multiple variables. Math. Geol.
**2003**, 35, 155–173. [Google Scholar] [CrossRef] - Abildin, Y.; Madani, N.; Topal, E. Geostatistical Modelling of Geometallurgical Variables through Turning Bands Approach. In Proceedings of the 25th World Mining Congress, Astana, Kazakhstan (WMC 2018), Astana, Kazakhstan, 19–22 June 2018. [Google Scholar]
- Emery, X. Testing the correctness of the sequential algorithm for simulating Gaussian random fields. Stoch. Environ. Res. Risk Assess.
**2004**, 18, 401–413. [Google Scholar] [CrossRef] - Journel, A.G.; Huijbregts, C.J. Mining Geostatistics; Academic Press: Cambridge, MA, USA, 1978. [Google Scholar]
- Goovaerts, P. Spatial orthogonality of the principal components computed from coregionalized variables. Math. Geol.
**1993**, 25, 281–302. [Google Scholar] [CrossRef] - Bandarian, E.M.; Bloom, L.M.; Mueller, U.A. Direct minimum/maximum autocorrelation factors within the framework of a two structure linear model of coregionalisation. Comput. Geosci.
**2008**, 34, 190–200. [Google Scholar] [CrossRef] - Switzer, P.; Green, A.A. Min/max autocorrelation factors for multivariate spatial imagery. Comput. Sci. Stat.
**1985**, 16, 13–16. [Google Scholar] - Maleki, M.; Madani, N. Multivariate Geostatistical Analysis: An application to ore body evaluation. Iran. J. Earth Sci.
**2017**, 8, 173–184. [Google Scholar] - Kim, H.J.; Song, Y.; Lee, K.H. Inequality constraint in least-squares inversion of geophysical data. Earth Planets Space
**1999**, 51, 255–259. [Google Scholar] [CrossRef] [Green Version] - Abrahamsen, P.; Benth, F.E. Kriging with Inequality Constraints. Math. Geol.
**2001**, 33, 719–744. [Google Scholar] [CrossRef] - Journel, A.G. Constrained interpolation and qualitative information—The soft kriging approach. Math. Geol.
**1986**, 18, 269–286. [Google Scholar] [CrossRef] - Vargas-Guzmán, J.A.; Dimitrakopoulos, R. Successive nonparametric estimation of conditional distributions. Math. Geol.
**2003**, 35, 39–52. [Google Scholar] [CrossRef] - Tran, T.T.; Murphy, M.; Glacken, I. Semivariogram structures used in multivariate conditional simulation via minimum/maximum autocorrelation factors. In Proceedings of the XI International Congress, IAMG, Liège, Belgium, 3–8 September 2006. [Google Scholar]
- Davis, B.M.; Greenes, K.A. Estimation using spatially distributed multivariate data: An example with coal quality. J. Int. Assoc. Math. Geol.
**1983**, 15, 287–300. [Google Scholar] [CrossRef] - Suro-Perez, V.; Journel, A.G. Indicator principal component kriging. Math. Geol.
**1991**, 23, 759–788. [Google Scholar] [CrossRef] - Rondon, O. Teaching aid: Minimum/maximum autocorrelation factors for joint simulation of attributes. Math. Geosci.
**2012**, 44, 469–504. [Google Scholar] [CrossRef] - Desbarats, A.J. Geostatistical modeling of regionalized grain-size distributions using min/max autocorrelation factors. In geoENV III—Geostatistics for Environmental Applications; Springer: Dordrecht, The Netherlands, 2001; pp. 441–452. [Google Scholar]
- Desbarats, A.J.; Dimitrakopoulos, R. Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Math. Geol.
**2000**, 32, 919–942. [Google Scholar] [CrossRef] - Rivoirard, J. Introduction to Disjunctive Kriging and Non-Linear Geostatistics; No. 551.021 R626i; Oxford University Press: Oxford, UK, 1994. [Google Scholar]
- Deutsch, C.V.; Journel, A.G. GSLIB: Geostatistical Software Library and User’s Guide, 2nd ed.; Oxford University Press: New York, NY, USA, 1998. [Google Scholar]
- Emery, X.; Lantuéjoul, C. Tbsim: A computer program for conditional simulation of three-dimensional gaussian random fields via the turning bands method. Comput. Geosci.
**2006**, 32, 1615–1628. [Google Scholar] [CrossRef] - Paravarzar, S.; Emery, X.; Madani, N. Comparing sequential Gaussian and turning bands algorithms for cosimulating grades in multi-element deposits. Comptes Rendus Geosci.
**2015**, 347, 84–93. [Google Scholar] [CrossRef] - Madani, N.; Ortiz, J. Geostatistical Simulation of Cross-Correlated Variables: A Case Study through Cerro Matoso Nickel-Laterite Deposit. In Proceedings of the 26th International Symposium on Mine Planning and Equipment Selection, Luleå, Sweden, 29–31 August 2017. [Google Scholar]
- Eze, P.N.; Madani, N.; Adoko, A.C. Multivariate Mapping of Heavy Metals Spatial Contamination in a Cu–Ni Exploration Field (Botswana) Using Turning Bands Co-simulation Algorithm. Nat. Resour. Res.
**2018**, 1–16. [Google Scholar] [CrossRef] [Green Version] - Wackernagel, H. Multivariate Geostatistics: An Introduction with Applications; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Goovaerts, P. Geostatistics for Natural Resources Evaluation; Oxford University Press: New York, NY, USA, 1997. [Google Scholar]
- Rossi, M.; Deutsch, C.V. Mineral Resource Estimation; Springer: New York, NY, USA, 2014. [Google Scholar]
- Emery, X. A turning bands program for conditional co-simulation of cross-correlated Gaussian random fields. Comput. Geosci.
**2008**, 34, 1850–1862. [Google Scholar] [CrossRef] - Chilès, J.P.; Delfiner, P. Geostatistics: Modeling Spatial Uncertainty; Wiley: New York, NY, USA, 2012. [Google Scholar]
- Goulard, M.; Voltz, M. Linear corregionalization model: Tools for estimation and choice of cross variogram matrix. Math. Geol.
**1992**, 30, 589–615. [Google Scholar] - Emery, X. Iterative algorithms for fitting a linear model of coregionalization. Comput. Geosci.
**2010**, 36, 1150–1160. [Google Scholar] [CrossRef] - Madani, N.; Emery, X. A comparison of search strategies to design the cokriging neighborhood for predicting coregionalized variables. Stoch. Environ. Res. Risk Assess.
**2018**, 1–17. [Google Scholar] [CrossRef] - Lantuejoul, C. Geostatistical Simulation, Models and Algorithms; Springer: Berlin/Heidelberg, Germany, 2002; p. 256. [Google Scholar]
- Emery, X. Conditioning simulations of Gaussian random fields by ordinary kriging. Math. Geol.
**2007**, 39, 607–623. [Google Scholar] [CrossRef] - Deutsch, C.V. All Realizations All the Time. In Handbook of Mathematical Geosciences; Daya Sagar, B., Cheng, Q., Agterberg, F., Eds.; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Jakab, N. Stochastic modeling in geology: Determining the sufficient number of models. Central Eur. Geol.
**2017**, 60, 135–151. [Google Scholar] [CrossRef] [Green Version] - Pyrcz, M.; Deutsch, C.V. Geostatistical Reservoir Modeling; Oxford University Press: New York, NY, USA, 2014. [Google Scholar]
- Emery, X.; Ortiz, J.M. Enhanced coregionalization analysis for simulating vector Gaussian random fields. Comput. Geosci.
**2012**, 42, 126–135. [Google Scholar] [CrossRef]

**Figure 1.**Different settings of inequality constraint in a dataset with their marginal distributions; ${Z}_{1}$ and ${Z}_{2}$: the underlying variables, (

**a**): unequal variables with the same distribution and linear relationship, (

**b**): unequal variables with a different distribution linear relationship, (

**c**): unequal variables with different distributions and non-linear relationship.

**Figure 3.**Base map for the location of samples in the plane (the coordinates are local). (

**a**) Total copper grade (tCu); (

**b**) Soluble copper grade (sCu).

**Figure 5.**Normal score transformation (left: declustered histogram of the original data and right: histogram of the normal score values.

**Figure 7.**Direct and cross-variograms of normal score values of total copper grade and solubility ratio. (

**a**) Direct variogram of SR (

**b**) Direct variogram of tCu (

**c**) Cross-variogram between SR and tCu.

**Figure 13.**The scatter plot of tCu(%) and sCu(%) obtained by MAF (left-up); TBCOSIM (right-up); original dataset (center-down) for realization No. 1.

**Figure 14.**Direct and cross-variograms of simulated realizations with TBCOSIM and MAF. (solid blue line: fitted model, dashed red line: average of the realizations; dashed green line: individual realizations). (

**a**) Direct variogram of tCu(%) for TBcosim values; (

**b**) Direct variogram of tCu(%) for MAF-TBsim; (

**c**) Direct variogram of sCu(%) for TBCOSIM; (

**d**) Direct variogram of sCu(%) for MAF-TBSIM; (

**e**) Cross-variogram of tCu(%) vs sCu(%) for TBCOSIM; (

**f**) Cross-variogram of tCu(%) and sCu(%) for MAF-TBSIM.

**Figure 15.**Codispersion functions showing the same trend for both approaches (TBCOSIM and MAF) compared to original data.

**Figure 16.**QQ-plot of distributions of mean values of total and soluble copper grades for three different numbers of realizations (20, 50 and 100) obtained from TBCOSIM and MAF simulation. (

**a**) Total copper grade; (

**b**) Soluble copper grade.

**Figure 17.**QQ-plot of distributions of variances of total and soluble copper grades for three different numbers of realizations (20, 50 and 100) obtained from TBCOSIM and MAF simulation. (

**a**) Total copper grade; (

**b**) Soluble copper grade.

**Figure 18.**Sensitivity analysis of mean values versus the number of realizations; blue line: MAF, black line: TBCOSIM and red line: original declustered line.

**Figure 19.**Probabilistic domaining of geometallurgical domains in the oxide copper mine obtained from post processing of 100 realizations produced by MAF approach.

**Figure 20.**Division of sample points into test and analysis for jackknife analysis; red points: analysis and black points: test.

**Figure 21.**Scatter plot for jackknife validation (prediction versus true values); left: MAF and right: turning bands co-simulation; black line: diagonal and red line: linear regression line.

Variable | Mean | Variance | Maximum | Minimum | COV | |
---|---|---|---|---|---|---|

Original | tCu(%) | 0.723 | 0.233 | 3.150 | 0.100 | 0.667 |

sCu(%) | 0.587 | 0.232 | 2.950 | 0.000 | 0.820 | |

Declustered | tCu(%) | 0.632 | 0.209 | 3.150 | 0.100 | 0.723 |

sCu(%) | 0.493 | 0.203 | 2.950 | 0.000 | 0.913 |

**Table 2.**Reproduction of global mean, variances and correlations obtained from initial data, TBCOSIM and MAF over 100 realizations.

Mean | Variance | Correlation | |||
---|---|---|---|---|---|

Parameters | tCu(%) | sCu(%) | tCu(%) | sCu(%) | tCu(%) vs sCu(%) |

Original | 0.723 | 0.587 | 0.233 | 0.232 | 0.991 |

Declustred | 0.632 | 0.493 | 0.209 | 0.203 | 0.989 |

Average TBcosim | 0.616 | 0.487 | 0.228 | 0.219 | 0.990 |

Average MAF | 0.611 | 0.483 | 0.225 | 0.216 | 0.990 |

tCu | sCu | |
---|---|---|

TBCOSIM | 0.284 | 0.678 |

MAF | 0.268 | 0.667 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abildin, Y.; Madani, N.; Topal, E.
A Hybrid Approach for Joint Simulation of Geometallurgical Variables with Inequality Constraint. *Minerals* **2019**, *9*, 24.
https://doi.org/10.3390/min9010024

**AMA Style**

Abildin Y, Madani N, Topal E.
A Hybrid Approach for Joint Simulation of Geometallurgical Variables with Inequality Constraint. *Minerals*. 2019; 9(1):24.
https://doi.org/10.3390/min9010024

**Chicago/Turabian Style**

Abildin, Yerniyaz, Nasser Madani, and Erkan Topal.
2019. "A Hybrid Approach for Joint Simulation of Geometallurgical Variables with Inequality Constraint" *Minerals* 9, no. 1: 24.
https://doi.org/10.3390/min9010024