Global Estimation-Cell Declustering

Introduction

The subject of cluster mitigation has been thoroughly discussed and well-documented, supported by sources provided by Deutsch (2001, pp. 53-62), Isaaks and Srivastava (1997, pp. 237-248), and Goovaerts (1997, pp. 77-82).

In the cell de-clustering approach, the entire area is divided into rectangular regions called cells, with each sample receiving a weight inversely proportional to the number of samples in the same cell. This often results in clustered samples receiving lower weights due to the presence of multiple samples within the same cell.

Cell De-clustering

Cell declustering involves several steps: First, a regular grid is overlaid onto the data. Then, each cell is assigned an intermediate weight, calculated as the reciprocal of the number of data points in that cell. These weights are then standardized by dividing them by the total number of cells with data. Next, the grid is shifted, and this process is repeated for each origin offset. Finally, the resulting weights are averaged over each origin offset.

In the provided arrangement, there is a set of clustered samples placed over cells of size 115x115 m². The middle cell in the top row contains only one sample, hence it receives a weight of 1. The middle cell in the bottom row, however, contains six samples, with each sample receiving a weight of 1/6.

Since all cells collectively carry a weight of 1, all samples within a specific cell receive equal weights. Therefore, the cell clustering method can be viewed as a two-step procedure. Firstly, we use our samples to compute the mean value within moving windows; then, we take these moving window averages and use them to compute the mean of the global area. The estimation obtained from this cell clustering method will depend on the size of the cells chosen. If the cells are very small, each sample will fall into its own cell, thus all samples will receive equal weights. Conversely, if all samples fall into the same cell across the entire global area, they will again receive equal weights. We need to find an appropriate balance between these two extremes.

The graph shown above will assist in selecting the optimal cell size. The result for a zero-cell size is the equal weighted value. A too-small minimum cell size should not be chosen because the number of cells would increase significantly. The correct cell size should be consistent with the range of data in sparsely sampled areas. The maximum cell size should not be set too large. It should be less than half of the domain size. The most challenging decision is how to choose the optimal cell size. The cell size that gives the minimum average on this graph will provide the optimal cell size.

Our dataset may not typically produce curves similar to the graph above and may not provide a clear idea about the optimal cell size. We might encounter a scenario with cell sizes that yield multiple minimum average values, as shown in the graph below.

Utilizing a high-resolution data spacing model aids in selecting an appropriate cell size for spatial analysis. The computed data spacing on a fine grid reveals clustered samples in the lower tail, nearly constant spacing in steep regions, and increasing spacing near study area borders in the upper tail (Deutsch, 2015).

In this case, it might be possible to map the range between samples and make an inference from the cumulative histogram graph of the data range.

The steep region in this graph is where the data range remains nearly constant. The upper tail is where the data range increases close to the boundaries of the study area. This approach could provide insights when selecting the optimal cell size, and comparing the results with polygon declustering could serve as another verification method.

The data is often not evenly spaced; some areas may have more densely sampled points than others, and there could be various reasons for this preference. The crucial point is to ensure that the method used to address clustering is applied correctly and that the histogram of the sample accurately reflects the histogram of the entire population. However, when the data is subjected to spatial clustering, either in cells or polygons, the histogram obtained through the preferred sampling method may not fully reflect the population's histogram.

If we use the areas of impact polygons as clustering weights, the estimated value is found to be 889.8. However, when the clustering method with a cell size of 115 x 115 m² is used, the estimated value calculated with weights is 933.0.

The cell decluster method, based on a grid structure, performs well when clustered samplings occur only in areas with high or low values. The arithmetic average of 324 samples in our dataset is 1179.9. The estimated value obtained with the cell decluster method (933.0) is closer to the value obtained with the polygon decluster method (890.0).

Global Estimation

Using drill data, global estimations were made based on chemical analyses and horizon thickness values by assigning samples with the nearest neighbor algorithm. Cell-declustering method was employed to determine cell sizes for this estimation study. The plan and section views below belong to this study. A result table regarding the global estimations is provided. In the previously published article, global estimation results obtained through the Voronoi polygon method were compared with these cell-based global estimation results.

Conclusion

The study delves into the intricacies of spatial analysis, particularly focusing on the cell declustering method and its implications for estimating values within a given area. Through meticulous examination, it becomes evident that the choice of cell size significantly impacts the resulting estimates, as illustrated by the weight assignments and considerations of clustered samples. The provided graphs serve as invaluable tools for determining the optimal cell size, with the cumulative histogram graph offering additional insights, particularly in scenarios where multiple minimum average values are observed. The comparison between the cell and polygon declustering methods further underscores the importance of method selection and verification in ensuring accurate estimations. The findings suggest that while both methods yield estimations close to the arithmetic average, the cell decluster method demonstrates greater proximity to the values obtained through the polygon decluster method. Moreover, the global estimation exercise, utilizing drill data and employing the nearest neighbor algorithm, reaffirms the utility of the cell-declustering method in spatial estimation studies. The juxtaposition of results obtained through different methodologies enhances our understanding of the intricacies involved in spatial analysis and underscores the importance of methodological choices in obtaining reliable estimations.

The polygonal method, unlike the cell de-clustering method, provides a distinct estimate. In scenarios where sampling doesn't support selecting the minimum of various cell de-clustered estimates, determining an appropriate cell size becomes challenging (Isaaks & Srivastava, 1989)

References

1- Deutsch, C. (2015, October 5). University of Alberta. "Cell Declustering Parameter Selection." Retrieved from: https://geostatisticslessons.com/ lessons/celldeclustering

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9- Journel, A. G. (1983). Non-parametric estimation of spatial distributions. Math Geology, 15(3), 445–68.

10- Pyrcz, M. J. ve Deutsch, C. V., "Declustering and Debiasing," University of Alberta, Edmonton, Alberta, CANADA, URL: https://www.ccgalberta.com/ccgresources/report04/2002-124-declusterdebias-ccg.pdf.