# Influence of the Sampling Density in the Coestimation Error of a Regionalized Locally Stationary Variable

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

## 3. Methodology

#### 3.1. Synthetic Case Study Creation

^{2}is defined as spatial domain with coordinates ranging from the origin to 1 km in the East (X) and in the North (Y). A $200\times 200$ squared geometry is considered, thus, 40,000 cells of 25 m

^{2}are considered to be target population.

#### 3.2. Semivariographic Modeling of Variables

#### 3.3. Sample Extraction at Different Densities

#### 3.4. Estimation of Different Scenarios

#### 3.5. Comparison of Estimations and Simulated Reality

#### 3.6. Analysis of the Sampling Density and Errors

## 4. Application

- Semivariographic fitting model for the primary variable (${Z}_{1}$).
- Semivariographic fitting model for the exhaustive auxiliary variable (${Z}_{2}$).
- Linear correlation coefficient between ${Z}_{1}$ and ${Z}_{2}$.

^{2}is considered for ${Z}_{2}$. A maximum sample size of 5% of the population has been considered, associated with high costs when retrieving information of metallic elements from the earth’s crust. It is typical to have case studies with smaller sample size than the 1% of the target population [17,32]. The limit scenarios (lowest and highest sampling densities) are presented in Figure 6.

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PERC | Pan European Reserves and Resources Reporting Committee |

SGS | Sequential Gaussian Simulation |

GSLIB | Geostatistical Software Library |

MSE | Mean Squared Error |

SGSIM | Sequential Gaussian Simulation Program |

SD | Sampling Density |

SDP | Sampling Density Percentage |

MGE | Mean Global Error |

## References

- Tulcanaza, E. Evaluación de Recursos y Negocios Mineros: Incertidumbres, Riesgos y Modelos Numéricos; Instituto de Ingenieros de Minas de Chile: Santiago de Chile, Chile, 1999. [Google Scholar]
- Battalgazy, N.; Madani, N. Stochastic Modeling of Chemical Compounds in a Limestone Deposit by Unlocking the Complexity in Bivariate Relationships. Minerals
**2019**, 9, 683. [Google Scholar] [CrossRef][Green Version] - 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. [Google Scholar] [CrossRef][Green Version] - Madani, N.; Yagiz, S.; Coffi Adoko, A. Spatial Mapping of the Rock Quality Designation Using Multi-Gaussian Kriging Method. Minerals
**2018**, 8, 530. [Google Scholar] [CrossRef][Green Version] - Nevskaya, M.A.; Seleznev, S.G.; Masloboev, V.A.; Klyuchnikova, E.M.; Makarov, D.V. Environmental and Business Challenges Presented by Mining and Mineral Processing Waste in the Russian Federation. Minerals
**2019**, 9, 445. [Google Scholar] [CrossRef][Green Version] - PERC Reporting Standard. Pan-European Standard for Reporting of Exploration Results, Mineral Resources and Reserves; The Pan-European Reserves and Resources Reporting Committee: Brussels, Belgium, 2017.
- Matheron, G. The theory of regionalized variables and its applications. In Les Cahiers du Centre de Morphologie Mathematique de Fontainebleu; École Nationale Supérieure des Mines de Paris: Paris, France, 1971. [Google Scholar]
- Isaaks, E.H.; Srivastava, M.R. Applied Geostatistics; Oxford University Press: New York, NY, USA, 1989. [Google Scholar]
- Journel, A.G.; Huijbregts, C.J. Mining Geostatistics; Academic Press: New York, NY, USA, 1978. [Google Scholar]
- Sinclair, A.J.; Blackwell, G.H. Applied Mineral Inventory Estimation; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Matheron, G. Principles of geostatistics. Econ. Geol.
**1963**, 58, 1246–1266. [Google Scholar] [CrossRef] - Oliver, M.A.; Webster, R. Basic Steps in Geostatistics: The Variogram and Kriging; Springer: New York, NY, USA, 2015. [Google Scholar]
- Pan, G.; Gaard, D.; Moss, K.; Heiner, T. A comparison between cokriging and ordinary kriging: Case study with a polymetallic deposit. Math. Geol.
**1993**, 25, 377–398. [Google Scholar] [CrossRef] - Wackernagel, H. Multivariate Geostatistics. An Introduction with Applications; Springer: Berlin, Germany, 1995. [Google Scholar]
- Almeida, A.S.; Journel, A.G. Joint simulation of multiple variables with a Markov-type coregionalization model. Math. Geol.
**1994**, 26, 565–588. [Google Scholar] [CrossRef] - Chen, F.; Chen, S.; Peng, G. Using Sequential Gaussian Simulation to assess geochemical anomaly areas of lead element, computer and computing technologies in agriculture VI IFIP advances. Inf. Commun. Technol.
**2013**, 393, 69–76. [Google Scholar] - Soltani, F.; Afzal, P.; Asghari, O. Delineation of alteration zones based on sequential gaussian simulation and concentration-volume fractal modeling in the hypogene zone of Sungun copper deposit, NW Iran. J. Geochem. Explor.
**2014**, 140, 64–76. [Google Scholar] [CrossRef] - Paravarzar, S.; Maarefvand, P.; Maghsoudi, A.; Afzal, P. Correlation between geological units and mineralized zones using fractal modeling in Zarshuran gold deposit (NW Iran). Arab. J. Geosci.
**2015**, 8, 3845–3854. [Google Scholar] [CrossRef] - Brus, D.; De Gruijter, J. Random sampling or geostatistical modelling? Choosing between design-based and model-based sampling strategies for soil (with discussion). Geoderma
**1997**, 80, 1–44. [Google Scholar] [CrossRef] - Brus, D.J.; Heuvelink, G. Optimization of sample patterns for universal kriging of environmental variables. Geoderma
**2007**, 138, 86–95. [Google Scholar] [CrossRef] - Oliver, M.; Webster, R. A tutorial guide to geostatistics: Computing and modelling variograms and kriging. Catena
**2014**, 113, 56–69. [Google Scholar] [CrossRef] - Alperin, M. Introducción al AnáLisis EstadíStico de Datos GeolóGicos; Editorial de la Universidad de La Plata: La Plata, Argentina, 2013. [Google Scholar]
- Rossi, M.E.; Deutsch, C.V. Mineral Resource Estimation; Springer: Dordrecht, The Netherlands, 2014. [Google Scholar]
- Hohn, M.E. Geostatistics and Petroleum Geology; Springer: Boston, MA, USA, 1988. [Google Scholar]
- Pyrcz, M.J.; Deutsch, C.V. Geostatistical Reservoir Modeling; Oxford University Press: New York, NY, USA, 2014. [Google Scholar]
- Goovaerts, P. Geostatistics for Natural Resources Evaluation; Oxford University Press: Nueva York, NY, USA, 1997. [Google Scholar]
- Rocha, M.M.; Yamamoto, J.K.; Watabane, J.; Fonseca, P.P. Studying the influence of a secondary variable in collocated cokriging estimates. An. Acad. Bras. Cienc.
**2012**, 84, 335–346. [Google Scholar] [CrossRef][Green Version] - Xu, W.; Tran, T.T.; Srivastava, R.M.; Journel, A.G. Integrating seismic data in reservoir modeling: The collocated cokriging alternative. In Proceedings of the 67th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Washington, DC, USA, 4–7 October 1992; Society of Petroleum Engineers Inc.: London, UK, 1992; pp. 833–842. [Google Scholar]
- Journel, A.G. Markov models for cross-covariances. Math. Geol.
**1999**, 31, 955–964. [Google Scholar] [CrossRef] - Remy, N.; Boucher, A.; Jianbing, W. Applied Geoestatistics with SGeMS; A User’s Guide; Cambridge University Press: New York, NY, USA, 2009. [Google Scholar]
- Armstrong, M. Basic Linear Geostatistics; Springer: Berlin, Germany, 1998. [Google Scholar]
- Ortiz, J.M.; Emery, X. Geostatistical estimation of mineral resources with soft geological boundaries: A comparative study. J. S. Afr. Inst. Min. Metall.
**2006**, 106, 577–584. [Google Scholar] - Rivoirard, J. Two key parameters when choosing the kriging neighborhood. Math. Geol.
**1987**, 19, 851–856. [Google Scholar] [CrossRef]

**Figure 3.**Maps of ${Z}_{1}$ and ${Z}_{2}$ with Gaussian sequential simulation and corrective affine transformation.

**Figure 12.**Relationship between the mean global error and the collocated ordinary cokriging deviation.

Parameter | Value | Unit of Measurement |
---|---|---|

X coordinate in the origin | 0 | meters (m) |

Y coordinate in the origin | 0 | meters (m) |

Last coordinate in X | 1000 | meters (m) |

Last coordinate in Y | 1000 | meters (m) |

Cell width in X | 5 | meters (m) |

Cell width in Y | 5 | meters (m) |

Number of cells in X | 200 | unities |

Number of cells in Y | 200 | unities |

Total number of cells | 40,000 | unities |

Parameter | Value | Units of Measurement |
---|---|---|

Anisotropy azimuth ${Z}_{1}$ | 45 | degrees |

Anisotropy azimuth ${Z}_{2}$ | 45 | degrees |

Maximum value of ${Z}_{1}$ | 500 | meters (m) |

Minimum value of ${Z}_{1}$ | 200 | meters (m) |

Maximum value of ${Z}_{2}$ | 300 | meters (m) |

Minimum value of ${Z}_{2}$ | 80 | meters (m) |

Parameter | Value |
---|---|

Mean of ${Z}_{1}$ | 10 |

Mean of ${Z}_{2}$ | 2 |

Standard deviation of ${Z}_{1}$ | 2.5 |

Standard deviation of ${Z}_{2}$ | 1.5 |

Scenario | Number of Samples | Sampling Density ${\mathit{Z}}_{1}$ (m ^{2}) | Sampling Size ${\mathit{Z}}_{1}$ (%) | Mean | Variance | Deviation |
---|---|---|---|---|---|---|

1 | 25 | 625 | 0.06 | 10.16 | 2.97 | 1.72 |

2 | 40 | 1000 | 0.10 | 10.16 | 2.43 | 1.56 |

3 | 60 | 1500 | 0.15 | 9.86 | 2.55 | 1.60 |

4 | 70 | 1750 | 0.18 | 9.87 | 2.88 | 1.70 |

5 | 80 | 2000 | 0.20 | 10.11 | 3.32 | 1.82 |

6 | 120 | 3000 | 0.30 | 10.10 | 3.32 | 1.82 |

7 | 160 | 4000 | 0.40 | 10.15 | 3.60 | 1.90 |

8 | 200 | 5000 | 0.50 | 9.91 | 4.67 | 2.16 |

9 | 240 | 6000 | 0.60 | 9.98 | 4.51 | 2.12 |

10 | 280 | 7000 | 0.70 | 9.95 | 4.77 | 2.18 |

11 | 320 | 8000 | 0.80 | 9.95 | 4.95 | 2.23 |

12 | 360 | 9000 | 0.90 | 10.01 | 5.14 | 2.27 |

13 | 400 | 10,000 | 1.00 | 10.02 | 5.25 | 2.29 |

14 | 1600 | 40,000 | 4.00 | 10.02 | 5.41 | 2.33 |

15 | 2000 | 50,000 | 5.00 | 10.01 | 5.55 | 2.36 |

Scenario | Sampling Density | Linear Correlation Coefficient | Ellipse | Minimum Number of Samples ${\mathit{Z}}_{1}$ | Maximum Number of Samples ${\mathit{Z}}_{1}$ |
---|---|---|---|---|---|

1 | 625 | 0.91 | 300/150 | 1 | 4 |

2 | 1000 | 0.96 | 300/150 | 1 | 12 |

3 | 1500 | 0.92 | 300/150 | 1 | 20 |

4 | 1750 | 0.87 | 300/150 | 1 | 20 |

5 | 2000 | 0.88 | 300/150 | 1 | 20 |

6 | 3000 | 0.88 | 300/150 | 1 | 20 |

7 | 4000 | 0.84 | 300/80 | 1 | 20 |

8 | 5000 | 0.88 | 300/80 | 1 | 20 |

9 | 6000 | 0.86 | 300/80 | 1 | 20 |

10 | 7000 | 0.85 | 300/80 | 1 | 20 |

11 | 8000 | 0.85 | 200/80 | 1 | 20 |

12 | 9000 | 0.91 | 150/80 | 1 | 20 |

13 | 10,000 | 0.89 | 150/80 | 1 | 20 |

14 | 40,000 | 0.90 | 150/40 | 1 | 20 |

15 | 50,000 | 0.90 | 150/40 | 1 | 30 |

Scenario | Sampling Density | SDP | MSE | MGE | ${\mathit{\mu}}_{\mathit{C}\mathit{O}\mathit{C}\mathit{K}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{O}\mathit{C}\mathit{K}}^{2}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{O}\mathit{C}\mathit{K}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{O}\mathit{C}\mathit{K}}/{\mathit{\mu}}_{\mathit{C}\mathit{O}\mathit{C}\mathit{K}}$ | Inferior Limit | Superior Limit |
---|---|---|---|---|---|---|---|---|---|---|

1 | 25 | 0.06 | 4.77 | 2.18 | 10.16 | 6.20 | 2.49 | 0.25 | 7.67 | 12.65 |

2 | 40 | 0.10 | 4.43 | 2.11 | 10.16 | 5.11 | 2.26 | 0.22 | 7.90 | 12.42 |

3 | 60 | 0.15 | 3.78 | 1.94 | 9.86 | 4.32 | 2.08 | 0.21 | 7.79 | 11.94 |

4 | 72 | 0.18 | 3.43 | 1.85 | 9.87 | 4.16 | 2.04 | 0.21 | 7.83 | 11.91 |

5 | 80 | 0.20 | 3.06 | 1.75 | 10.11 | 3.67 | 1.92 | 0.19 | 8.19 | 12.03 |

6 | 120 | 0.30 | 2.78 | 1.67 | 10.10 | 3.20 | 1.79 | 0.18 | 8.31 | 11.89 |

7 | 160 | 0.40 | 2.60 | 1.61 | 10.15 | 2.93 | 1.71 | 0.17 | 8.43 | 11.86 |

8 | 200 | 0.50 | 2.19 | 1.48 | 9.91 | 2.54 | 1.59 | 0.16 | 8.31 | 11.50 |

9 | 240 | 0.60 | 1.89 | 1.37 | 9.98 | 2.34 | 1.53 | 0.15 | 8.45 | 11.51 |

10 | 280 | 0.70 | 2.03 | 1.43 | 9.95 | 2.23 | 1.49 | 0.15 | 8.46 | 11.45 |

11 | 320 | 0.80 | 1.44 | 1.20 | 9.95 | 1.68 | 1.30 | 0.13 | 8.65 | 11.24 |

12 | 360 | 0.90 | 1.43 | 1.20 | 10.01 | 1.58 | 1.26 | 0.13 | 8.75 | 11.27 |

13 | 400 | 1.00 | 1.40 | 1.18 | 10.02 | 1.53 | 1.24 | 0.12 | 8.78 | 11.26 |

14 | 1600 | 4.00 | 0.85 | 0.92 | 10.02 | 0.80 | 0.89 | 0.09 | 9.13 | 10.91 |

15 | 2000 | 5.00 | 0.76 | 0.87 | 10.01 | 0.68 | 0.82 | 0.08 | 9.19 | 10.84 |

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**MDPI and ACS Style**

Hernandez Guerra, H.; Alberdi, E.; Goti, A. Influence of the Sampling Density in the Coestimation Error of a Regionalized Locally Stationary Variable. *Minerals* **2020**, *10*, 90.
https://doi.org/10.3390/min10020090

**AMA Style**

Hernandez Guerra H, Alberdi E, Goti A. Influence of the Sampling Density in the Coestimation Error of a Regionalized Locally Stationary Variable. *Minerals*. 2020; 10(2):90.
https://doi.org/10.3390/min10020090

**Chicago/Turabian Style**

Hernandez Guerra, Heber, Elisabete Alberdi, and Aitor Goti. 2020. "Influence of the Sampling Density in the Coestimation Error of a Regionalized Locally Stationary Variable" *Minerals* 10, no. 2: 90.
https://doi.org/10.3390/min10020090