# Geostatistical Estimation of Multi-Domain Deposits with Transitional Boundaries: A Sensitivity Study for the Sechahun Iron Mine

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- The selection of input data, and consequently, the choice between a global model (considering all available data from various geologic domains) or a local model (selecting only one class of data for each particular geologic domain and defining the neighborhood used for the estimation);
- The uncertainty of the selected model (if the local model is chosen) and the natural continuity of variables across boundaries [23];
- The qualitative data, such as geologic information, that should be coded in quantitative variables (indicators) and considered in the estimation procedures;
- The evaluation of the advantages and disadvantages of estimation results while adding qualitative geologic information to the model.

- Statistical studies and data spatial analysis (borehole and blast hole data if available) for a general understanding of the deposit and geologic domain spatial variability;
- Spatial variability studies of the target variable and continuity with nearby boundaries (with comparison of local and global models);
- Transforming geologic information into indicators so as to conduct spatial analysis and evaluate the correlation with the target variable;
- Contact analysis to identify the type of geologic domains (hard or soft boundaries), using tools such as contact plots, preferential relationship schemes, variogram ratio, etc.;
- Determining possible appropriate geostatistical methods:
- V-a.
- OK with the local model;
- V-b.
- OK with the global model;

Adding geologic information for grade estimation:- V-c.
- Indicator co-kriging (ICK e.g., using geological domains as indicators) to identify the probability of each geologic domain in the ore body;
- V-d.
- Indicator co-kriging (e.g., using geological domains as indicators) and a spatial variable (e.g., grade), in the case of having indicators in all points of the ore deposits. This method can be used when geological information (indicators as auxiliary variables) are known at the target points (collocated CK);
- V-e.
- Indicator co-kriging (e.g., using geological domains as indicators) and a spatial variable (e.g., grade), in the case of having indicators only in borehole samples but not in all points of the ore deposits. In this method, indicators (as auxiliary variables) are unknown at the target points;
- V-f.
- Border effect study the between geologic domains and the possibility of performing co-kriging of PGs [21];

- Cross-validation of models, validating estimation results and interpretation of appropriate methods with particular regard to transitional boundary-domains.

#### 2.1. Statistical Studies

#### 2.2. Local and Global Model

#### 2.3. Geologic Information

#### 2.4. Type of Boundaries between the Geologic Domains: Hard or Soft

#### 2.5. Geostatistical Estimation Methods in the Case of Transitional Boundaries

_{i}(x) (different geologic domains) and grade Z(x).

_{i}(x) is called partial grade (PG). In fact, PG is an isotopic CK system based on the indicators of the geologic domains and their products with the target variable:

#### 2.6. Cross Validation of Models and Validation of Estimation Results

_{α}from the set of variables Z(x) and then estimating them by kriging from neighboring data Z(x

_{β}), α ≠ β. Accordingly, at every sample point x

_{α}the Kriging estimate Z

_{α}

^{*}and the associated Kriging variance σ

^{2}

_{Kα}are calculated. Since the measured value Z

_{α}= Z(x

_{α}) is known, the empirical Kriging error (E

_{α}) and standardized error (e

_{α}) can be computed:

#### 2.7. Application: The Sechahun Iron Mine

- High-grade magnetite, or rich iron ore (w(Fe) > 45%);
- Low-grade magnetite, or poor iron ore (w(Fe) < 45%);
- Oxidized high-grade magnetite (hematitized).

#### 2.7.1. Borehole Samples

#### 2.7.2. Blasthole Samples

- Waste zone: Fe < 20%;
- Poor Zone: 20% ≤ Fe < 45%;
- Rich Zone: Fe ≥ 45%.

## 3. Results

#### 3.1. Local or Global Model

#### 3.2. Adding Geologic Information

#### Waste, Poor, Rich, Crush Zones and Metasomatite

#### 3.3. Hard or Soft Boundaries

#### 3.4. Estimation Results

1. Waste: Fe (%) < 20 (cut-off) | yellow |

2. Poor: 20 < Fe (%) < 45 | orange |

3. Rich: 45 < Fe (%) | red |

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Variogram ratio showing at point (1) the probability of entering the j domain when leaving the i domain and at point (2) the preferentiality value [21].

**Table A1.**Preferentiality values. Each cell contains one value per direction from top to bottom: N–S, W–E and vertical directions.

From/To | Waste | Poor | Rich | Crush Zone | Metasomatite |
---|---|---|---|---|---|

Waste | Dir-1 | −0.25 | 0.05 | 0.30 | 0.50 |

Dir-2 | −0.24 | −0.20 | 0.00 | 0.00 | |

Dir-3 | −0.34 | 0.00 | 0.00 | −0.18 | |

Poor | −0.30 | Dir-1 | 0.40 | −0.10 | 0.10 |

−0.14 | Dir-2 | 0.26 | −0.10 | 0.12 | |

−0.12 | Dir-3 | −0.22 | −0.08 | 0.20 | |

Rich | −0.15 | 0.30 | Dir-1 | 0.00 | −0.05 |

−0.28 | 0.18 | Dir-2 | −0.16 | −0.14 | |

−0.28 | −0.34 | Dir-3 | 0.00 | −0.18 | |

Crush Zone | 0.45 | −0.25 | −0.15 | Dir-1 | 0.25 |

0.20 | 0.28 | −0.16 | Dir-2 | 0.00 | |

−0.24 | 0.12 | −0.16 | Dir-3 | −0.02 | |

Metasomatite | 0.00 | 0.05 | 0.25 | 0.00 | Dir-1 |

−0.12 | 0.24 | 0.08 | 0.00 | Dir-2 | |

−0.18 | −0.14 | −0.10 | 0.00 | Dir-3 |

## Appendix B

_{i}(x) is defined as:

## References

- Matheron, G. The Theory of Regionalized Variables and its Application; École Nationale Supérieure des Mines de Paris: Paris, France, 1971; ASIN:B0007ALEG2. [Google Scholar]
- Krige, D.G. A Statistical Approach to Some Mine Valuations and Allied Problems at the Witwatersrand. Master’s Thesis, University of Witwatersrand, Johannesburg, South Africa, 1951. [Google Scholar]
- Armstrong, M. Basic Linear Geostatistics; Springer: Berlin/Heidelberg, Germany, 1998; pp. 15–115. [Google Scholar] [CrossRef]
- Journel, A.G.; Huijbregts, C.J. Mining Geostatistics; The Blackburn Press: New York, NY, USA, 1991; pp. 77–218. ISBN 978-1930665910. [Google Scholar]
- Wackernagel, H. Multivariate Geostatistics an Introduction with Applications; Springer: Berlin/Heidelberg, Germany, 2003; pp. 121–208. [Google Scholar] [CrossRef]
- Chiles, J.P.; Delfiner, P. Geostatistics Modeling Spatial Uncertainty, 2nd ed.; WILEY: Hoboken, NJ, USA, 2012; pp. 118–177. ISBN 978-0-470-18315-1. [Google Scholar]
- Dowd, P.A.; Pardo-Igúzquiza, E. Estimating the boundary surface between geologic formations from 3D seismic data using neural networks and Geostatistics. Geophysics
**2005**, 70, 1–11. [Google Scholar] [CrossRef] - Larrondo, P.F.; Deutsch, C.V. Methodology for Geostatistical Model of Gradational Geological Boundaries: Local Non-stationary LMC. Cent. Comput. Geostat.
**2004**, 6, 1–17. [Google Scholar] - Emery, X.; Ortiz, J.M. Estimation of mineral resources using grade domains: Critical analysis and a suggested methodology. J. South. Afr. Inst. Min. Metall.
**2005**, 106, 247–256. [Google Scholar] - Ortiz, J.M.; Emery, X. Geostatistical estimation of mineral resources with soft geological boundaries: A comparative study. J. South. Afr. Inst. Min. Metall.
**2006**, 106, 577–584. [Google Scholar] - Emery, X.; Ortiz, J.M. Cáceres A.M. Geostatistical modeling of rock type domains with spatially varying proportions: application to a porphyry copper deposit. J. South. Afr. Inst. Min. Metall.
**2008**, 108, 284–292. [Google Scholar] - Madani, N.; Emery, X. Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields. Stoch. Environ Res. Risk Assess.
**2017**, 31, 893–913. [Google Scholar] [CrossRef] - Adeli, A.; Emery, X.; Dowd, P. Geological Modelling and Validation of Geological Interpretations via Simulation and Classification of Quantitative Covariates. Minerals
**2017**, 8, 7. [Google Scholar] [CrossRef] - Soares, A. Geostatistical estimation of multiphase structures. Math. Geol.
**1992**, 24, 149–160. [Google Scholar] [CrossRef] - Dimitrakopoulos, R.; Dagbert, M. Sequential modeling of relative indicator variables: Dealing with multiple lithology types. In Geostatistics Troia; Soares, A., Ed.; Springer: Dordrecht, The Netherlands, 1993; pp. 413–422. [Google Scholar] [CrossRef]
- Gossage, B. The Application of Indicator Kriging in the Modeling of Geological Data, Symposium on Beyond Ordinary Kriging. 1998. Available online: http://www.gaa.org.au/pdf/bok%20gossage.pdf (accessed on 14 January 2019).
- Marinoni, O. Improving Geological Models Using a Combined Ordinary–Indicator Kriging Approach. Eng. Geol.
**2003**, 69, 37–45. [Google Scholar] [CrossRef] - Gholamnejad, J.; Ansari, A.H.; Yarahmadi Bafghi, A.R.; Taqizadeh, M. Determination of ore/waste contacts by using indicator kriging, case study: Choghart Iron Mine of Iran. Int. J. Eng.
**2010**, 23, 269–276. [Google Scholar] - Kameshwara Rao, V.; Narayana, A.C. Application of nonlinear geostatistical indicator kriging in lithological categorization of an iron ore deposit. Curr. Sci.
**2015**, 108, 413–421. [Google Scholar] - Séguret, S.A. Block model in a multi facies context-Application to a porphyry copper deposit. In Proceedings of the 2nd International Seminar on Geology for the Mining Industry, Antofagasta, Chile, 8–10 June 2011. [Google Scholar]
- Séguret, S.A. Analysis and estimation of multi-unit deposits: application to a porphyry copper deposit. J. Math. Geosci.
**2013**, 45, 927–947. [Google Scholar] [CrossRef] - Rivoirard, J. Introduction to Disjunctive Kriging and Non-Linear Geostatisics; Oxford University Press: Oxford, UK, 1994; ISBN 978-0198741800. [Google Scholar]
- Kasmaee, S.; Torab, F. Risk reduction in Sechahun iron ore deposit by geological boundary modification using multiple indicator Kriging. J. Cent. South Univ.
**2014**, 21, 2011–2017. [Google Scholar] [CrossRef] - Afzal, P.; Fadakar Alghalandis, Y.; Khakzad, A.; Moarefvand, P.; Rashidnejad Omran, N. Delineation of mineralization zones in porphyry Cu deposits by fractal concentration–volume modeling. J. Geochem. Explor.
**2011**, 108, 220–232. [Google Scholar] [CrossRef] - Rossi, M.E.; Deutsch, C.V. Mineral Resource Estimation; Springer: London, UK, 2014; pp. 29–65. [Google Scholar] [CrossRef]
- Matheron, G. La Déstructuration des Hautes Teneurs et le Krigeage des Indicatrices; Technical Report N.761; Centre de Géostatistique: Fontainebleau, France, 1982. [Google Scholar]
- Glacken, I.M.; Snowden, D.V. Mineral Resource Estimation. 2001, pp. 189–198. Available online: https://pdfs.semanticscholar.org/b76f/b443827f8392bfac434d694b9f19ec53be0f.pdf (accessed on 14 January 2019).
- Matheron, G. The internal consistency of models in geostatistics. In Geostatistics; Quantitative Geology and Geostatistics Book Series; Armstrong, M., Ed.; Springer: Dordrecht, The Netherlands, 1989; Volume 4. [Google Scholar] [CrossRef]
- Wilde, B.J.; Deutsch, C.V. Kriging and simulation in presence of stationary domains: Developments in boundary modeling. In Geostatistics Oslo 2012; Quantitative Geology and Geostatistics Book Series; Abrahamsen, P., Hauge, R., Kolbjørnsen, O., Eds.; Springer: Dordrecht, The Netherlands, 2012; Volume 17. [Google Scholar] [CrossRef]
- Séguret, S.A. Geostatistical comparison between blast and drill holes in a porphyry copper deposit. In Proceedings of the 7th World Conference on Sampling and Blending, Bordeaux, France, 10–12 June 2015. [Google Scholar] [CrossRef]
- Bonyadi, Z.; Davidson, G.J.; Mehrabi, B.; Meffre, S.; Ghazban, F. Significance of apatite REE depletion and monazite inclusions in the brecciated Se–Chahun iron oxide–apatite deposit, Bafq district. Iran: Insights from paragenesis and geochemistry. J. Chem. Geol.
**2011**, 281, 253–269. [Google Scholar] [CrossRef] - Kasmaee, S.; Raspa, G.; de Fouquet, C.; Bonduà, S.; Tinti, F.; Bruno, R. How different data supports affect geostatistical modelling: the new aggregation method and comparison with the classical regularisation and the theoretical punctual model. Int. J. Min. Reclam. Environ.
**2018**. [Google Scholar] [CrossRef] - ISATIS. Isatis Software Manual. Available online: https://www.geovariances.com (accessed on 14 January 2019).

**Figure 1.**Schematic figures showing hard boundaries with sharp changes between geological domains (

**a**) and soft boundaries with transitional changes between geological domains (

**b**).

**Figure 2.**Location of Sechahun iron mine (

**a**) and localized vertical boreholes (Z above the sea level) (

**b**).

**Figure 3.**Histograms of borehole samples obtained from Fe (%) (

**a**), P (%) (

**b**), S (%) (

**c**) and the distribution of borehole samples in each geologic domain (

**d**).

**Figure 4.**The proportions of the grades data in different geologic domains. (

**a**) shows the overlapping of poor, rich and other geological zones, (

**b**) shows the overlapping for waste, metasomatite and other geological zones.

**Figure 5.**Elevation Z = 1585 m of blastholes classified by cutoff grade and threshold (horizontal section).

**Figure 6.**Vertical sample variogram (black points) and variogram model (red line) for regularized 2.0 m samples for the global model.

**Figure 7.**Example of an estimated geologic vertical section using local models (

**a**), and zoom of one specific area (

**b**).

**Figure 8.**Sample and modeled variograms of indicators and Fe (%) (direct and cross variograms): Regularized 2.0 m samples in vertical direction.

**Figure 10.**Contact plot showing the mean Fe (%) in a rich domain (left), and poor domain (right) (

**a**); contact plot showing the mean Fe (%) in a poor domain (left), and metasomatite domain (right) (

**b**).

**Figure 11.**Preferential relationship schemes in North-South direction (45˚ tolerance). The top section shows domains above cut-off (Fe > 20%), while the lower section indicates domains with Fe < 20%.

**Figure 12.**Variogram ratio between indicator and partial grade divided by indicator variogram in three directions (vertical direction: blue, horizontal directions (0, 90): red and green).

**Figure 13.**Sample and modeled variograms of indicators and partial grades (direct and cross variograms): Regularized 2.0 m samples in vertical direction.

**Figure 14.**Histogram of true block values obtained from the mean of blastholes (

**a**) and histogram of number of blastholes used for averaging the block values (

**b**).

**Figure 15.**Maps of real block values obtained from mean of blastholes with optimum estimation method (e.g., three exploited levels). (

**a**) is the horizontal section z = 1560 m, (

**b**) is the horizontal section z = 1570 and (

**c**) is the horizontal section z = 1580 m.

Variable | Number of Samples | Minimum | Maximum | Mean (m) | Standard Deviation (σ) | Coefficient of Variation (σ/m) |
---|---|---|---|---|---|---|

Fe (%) | 1537 | 2.38 | 66.37 | 31.88 | 15.74 | 0.49 |

P (%) | 1214 | 0.01 | 0.42 | 0.05 | 0.06 | 1.2 |

S (%) | 1114 | 0.01 | 0.27 | 0.05 | 0.04 | 0.8 |

Geological Zones of the Iron Deposit | Number of Data | Grade Fe (%) | ||
---|---|---|---|---|

Mean | Minimum | Maximum | ||

Waste | 43 | 12.54 | 2.38 | 26.10 |

Poor | 482 | 30.50 | 11.90 | 53.23 |

Rich | 278 | 57.29 | 26.16 | 67.79 |

Crush Zone | 51 | 15.18 | 4.84 | 30.98 |

Dike | 62 | 15.62 | 3.05 | 46.42 |

Metasomatite | 136 | 15.06 | 4.11 | 28.37 |

**Table 3.**Histogram, sample variogram (black points) and the variogram model (red line) for regularized 2.0 m samples for local models (waste, poor and rich domains).

Exploited Domains | Histogram | Variogram |
---|---|---|

Poor zone | ||

Rich zone | ||

Waste zone |

Method | OK-Global Model | OK-Local Poor | OK-Local Rich | OK-Local Waste |
---|---|---|---|---|

Mean error (%) | −0.14 | 0.47 | 1.55 | −2.25 |

Variance error (%)^{2} | 26.45 | 41.64 | 35.72 | 24.85 |

Variance standardized error | 0.89 | 1.03 | 0.70 | 1.37 |

Method | OK-Global Model | OK-Local Poor | OK-Local Rich | OK-Local Waste |
---|---|---|---|---|

Mean error (%) | −0.14 | 0.14 | −0.15 | −0.13 |

Variance error (%)^{2} | 26.45 | 27.88 | 27.20 | 27.68 |

Variance standardized error | 0.89 | 0.80 | 0.82 | 0.63 |

**Table 6.**Cross-validation using the co-kriging (CK), partial grade (PG) and ordinary kriging (OK)-global models.

Methods | Mean-Error (%) | Variance-Error (%)^{2} | Variance of Standardized Error (%)^{2} |
---|---|---|---|

Partial Grade | −0.12 | 27.73 | 0.83 |

CK-with indicators at target points | −0.02 | 17.90 | 0.82 |

CK- without indicators at target points | 0.12 | 26.08 | 0.99 |

OK global model | −0.09 | 28.22 | 1.04 |

Levels (m) | Total Number of Blocks | Number of Blocks for Zone and Geological Methods | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Rich Zone | Poor Zone | Waste Zone | ||||||||

OK | CK | PG | OK | CK | PG | OK | CK | PG | ||

Z = 1540 | 31 | 5 | 0 | 2 | 5 | 8 | 7 | 1 | 0 | 3 |

Z = 1550 | 100 | 18 | 14 | 7 | 17 | 17 | 20 | 4 | 0 | 3 |

Z = 1560 | 145 | 19 | 18 | 13 | 34 | 22 | 35 | 0 | 1 | 3 |

Z = 1570 | 183 | 24 | 17 | 17 | 35 | 38 | 41 | 4 | 1 | 6 |

Z = 1580 | 203 | 46 | 8 | 10 | 40 | 50 | 45 | 3 | 1 | 0 |

Z = 1590 | 203 | 52 | 12 | 4 | 39 | 43 | 52 | 1 | 0 | 0 |

Z = 1600 | 203 | 58 | 9 | 2 | 41 | 40 | 48 | 1 | 3 | 1 |

Z = 1610 | 189 | 64 | 6 | 0 | 50 | 40 | 22 | 2 | 4 | 1 |

Total | 1257 | 286 | 84 | 55 | 261 | 258 | 270 | 16 | 10 | 17 |

Total (%) | 1257 | 22.7% | 6.68% | 4.4% | 20.8% | 20.5% | 21.5% | 1.3% | 0.8% | 1.3% |

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## Share and Cite

**MDPI and ACS Style**

Kasmaee, S.; Raspa, G.; de Fouquet, C.; Tinti, F.; Bonduà, S.; Bruno, R.
Geostatistical Estimation of Multi-Domain Deposits with Transitional Boundaries: A Sensitivity Study for the Sechahun Iron Mine. *Minerals* **2019**, *9*, 115.
https://doi.org/10.3390/min9020115

**AMA Style**

Kasmaee S, Raspa G, de Fouquet C, Tinti F, Bonduà S, Bruno R.
Geostatistical Estimation of Multi-Domain Deposits with Transitional Boundaries: A Sensitivity Study for the Sechahun Iron Mine. *Minerals*. 2019; 9(2):115.
https://doi.org/10.3390/min9020115

**Chicago/Turabian Style**

Kasmaee, Sara, Giuseppe Raspa, Chantal de Fouquet, Francesco Tinti, Stefano Bonduà, and Roberto Bruno.
2019. "Geostatistical Estimation of Multi-Domain Deposits with Transitional Boundaries: A Sensitivity Study for the Sechahun Iron Mine" *Minerals* 9, no. 2: 115.
https://doi.org/10.3390/min9020115