# 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 |

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**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