Cell declustering

§1.3 closed with a promise: the cure for preferentially-sampled histograms is to replace the equal-weight 1/n in the empirical CDF with a non-uniform set of weights $w_i$ that downweight clustered samples and upweight isolated ones. The §1.3 widget previewed cell declustering in action and showed the central phenomenon: as you sweep the cell-size slider, the corrected histogram is barely different from the biased histogram at extreme cell sizes, but tracks the truth closely at intermediate ones — a U-shape that says "in the middle, this works". §2.1 takes that preview apart and makes it operational. It is the FIRST section of Part 2, and Part 2 has a clean four-section arc: cell declustering here (§2.1), the polygonal alternative (§2.2), a head-to-head comparison on the same dataset (§2.3), and the downstream-impact story — what declustering changes in the variogram, the kriged map, and the simulation reference distribution (§2.4).

By the end of this section you will know the cell-declustering algorithm in detail: how to compute the weights from a sample, how to choose the cell size by an explicit optimisation criterion, and how to average over multiple grid origins to make the answer robust. You will run the GSLIB DECLUS workflow in a widget. And you will hold the honest scope warnings — what cell declustering can fix, what it cannot, when it loses statistical power.

The cell-declustering algorithm, step by step

Cell declustering was introduced in its modern form by Journel (1983) and turned into the standard GSLIB program DECLUS by Deutsch (1989). The recipe is short enough to write in five lines.

• Discretise the domain. Tile the study area with a regular grid of rectangular cells of size $L \times W$ (in 2D) or $L \times W \times H$ (in 3D). For an isotropic problem, take $L = W$ (and $= H$); for an anisotropic problem, set the cell aspect ratio to match the field's anisotropy ellipse (Part 3.4 develops that machinery; for now assume isotropic).
• Count samples per cell. For each cell $c$, count $n_c$, the number of samples that fall inside $c$. Let $N_{\text{occ}}$ be the number of cells with $n_c \geq 1$.
• Assign weights. Each sample $i$ in a cell $c$ receives weight

• Verify normalisation. Summing over all samples gives

The weights sum to 1 by construction — they are a proper probability mass on the sample.

• Compute declustered statistics. The declustered mean is

and the declustered empirical CDF is

Higher moments and quantiles follow analogously: every place in §1.1's univariate toolkit where you would write "(1/n) Σ" you now write "Σ w_i". The §1.2 normal-score transform also uses the declustered Fₙ in place of the unweighted Fₙ — the recipe $Y_i = \Phi^{-1}(\hat{F}_n^{d}(Z_i))$ delivers Gaussian margins under the declustering correction.

Reading the weight formula geometrically. The weight $w_i = 1/(n_c \cdot N_{\text{occ}})$ has two pieces. The factor $1/N_{\text{occ}}$ is the same for every sample — it normalises the total weight to 1 across the $N_{\text{occ}}$ occupied cells. The factor $1/n_c$ is the local adjustment: in a crowded cell with $n_c = 10$ samples, each sample gets 1/10 of the cell's share; in a singleton cell with $n_c = 1$, that single sample gets the entire cell's share. Equivalently, isolated samples are amplified relative to crowded ones, by a factor up to $n_c$. This is exactly the qualitative behaviour §1.3 described — and now we have the precise formula.

Watching the algorithm run

The first widget makes the algorithm visible. It samples 80 points preferentially from the §1.3 synthetic field (the same field shape, so your intuition from earlier widgets carries over) and lets you adjust two knobs:

• The cell size, from 5 cells per side (large cells) to 30 cells per side (small cells). The cell-side count is the standard GSLIB DECLUS parameterisation — smaller "cells per side" means larger physical cells.
• The grid origin, in two dimensions, with offsets in [0, cell_size]. Sliding the origin shifts the entire grid laterally, exposing the cell-origin sensitivity you will read about formally a few paragraphs below.

What the widget shows on each redraw: (LEFT) the field heatmap with the 80 sample points overlaid as filled dots, the regular cell grid drawn as faint lines on top, and each sample coloured by its weight (red = low weight, green = high weight). The cell-count overlay shows $n_c$ for each occupied cell so you can read how the crowded cells are sharing their weight. (RIGHT) the histogram with three overlays: the true population histogram (grey, fixed reference), the biased sample histogram (red, unweighted), and the declustered sample histogram (green, weighted by $w_i$) on a shared axis. Below the panels, four numbers report: total samples (always 80), occupied cells $N_{\text{occ}}$, true population mean, biased sample mean, declustered mean.

Slide the cell-size first. At "30 cells per side" (small cells, ~0.033 unit-square wide), almost every cell has at most one sample, $N_{\text{occ}}$ approaches $n = 80$, every weight collapses to ~1/80, and the declustered histogram coincides with the biased histogram — the green and red bars match. Sliding the cell-size down to "5 cells per side" (large cells, 0.20 unit-square wide) goes to the opposite extreme: most samples sit in just a handful of cells, the cells in the high zone have $n_c$ as large as 30–40, and the weights inside those crowded cells get aggressively divided. The declustered histogram now shifts visibly LEFT toward the population mean — declustering has done its job.

Now slide the grid-origin while holding the cell size at, say, "12 cells per side". The cells shift sideways across the field, and the $n_c$ values on the high zone re-shuffle. Watch the declustered-mean readout. It is NOT constant: a small origin shift can move it by 0.05–0.10 in this field. This is the cell-origin sensitivity that §2.1's §3 paragraph below addresses — and that the second widget tackles head-on.

The cell-size dilemma — the U-shape made formal

Cell declustering has exactly two free parameters: the cell size and the grid origin. The cell size carries the heavy lifting — change it by a factor of 5 and the declustered mean can swing by 20% — so most of the literature focuses on how to choose it. The grid origin matters less per unit shift but matters in aggregate (multiple origins are averaged, as in the next subsection).

What is the right cell size? The §1.3 widget showed the answer is U-shaped: extremes do nothing, the middle works. Here is the formal version.

• Cell size much smaller than typical inter-sample spacing. If the cells are so small that almost every cell contains at most one sample, then $n_c \in {0, 1}$ everywhere, $N_{\text{occ}}$ approaches the sample size $n$, and the weight formula reduces to $w_i = 1/(1 \cdot n) = 1/n$ — equal weights, no correction. The declustered histogram is the biased histogram. The bias remains.
• Cell size much larger than the field extent. If the cells are so large that all samples fall into a single cell, then $N_{\text{occ}} = 1$, $n_c = n$, and $w_i = 1/(n \cdot 1) = 1/n$ — equal weights again. Same as the previous case. The declustered histogram is once again the biased histogram.
• Cell size matched to the clustering scale. In the sweet spot, the cells are large enough that a crowded cluster of samples in the high zone gets bundled into one cell (so each one is divided down by $n_c$), but small enough that an isolated sample in a barren region sits alone in its cell (and gets the full cell weight). The contrast in weights between "crowded" and "isolated" samples is maximal — and that contrast is exactly what corrects the bias.

What "matched to the clustering scale" means in practice: the cell should hold roughly 5–50 samples in the crowded regions and 1–5 samples in the sparser regions. As a rough heuristic, set the cell side length to roughly the typical inter-cluster spacing — the distance between distinct clusters of samples on a scatter plot of locations. There is no analytical formula for the optimum cell size as a function of the underlying population. It depends on the sampling pattern (which we know) and the spatial correlation structure of the field (which we typically do not know in detail at the declustering stage). So the cell size is chosen by inspection or via an explicit optimisation criterion.

Cell-size optimisation criteria

The standard practice is to sweep a range of cell sizes, compute the declustered mean at each, and pick the size that satisfies the chosen criterion. Three criteria are in common use:

• Minimum declustered mean (high-grade preferential sampling). If you know a priori that samples are over-sampled in the high zone — the canonical mining-drillout case, where the drill programme targets visible mineralisation — then the unweighted mean over-estimates the population mean, and any declustering correction should pull the mean DOWN. The optimum cell size is the one that produces the SMALLEST declustered mean over the sweep. This is the criterion the GSLIB DECLUS manual recommends as the default for mining data, and it is what produces the U-shape you saw in the §1.3 widget.
• Maximum declustered mean (low-grade preferential sampling). Less common, but it arises in environmental work where stations cluster around low-contamination "control" sites and the sample histogram under-represents hot spots; or in mining where the cheap-to-drill zones happen to be low-grade waste rock. The correction pulls UP, and the optimum cell size produces the LARGEST declustered mean over the sweep. The choice is symmetric to the high-grade case.
• A-priori target (independent justification). If an INDEPENDENT line of evidence — a geological model, an analogue deposit, a previous mining campaign in the same district — gives you a believable value for the population mean, then you can pick the cell size whose declustered mean is CLOSEST to that target. This is the most defensible criterion when it is available, because the target is not derived from the same sample, so it is not vulnerable to the bias in question. The two practical drawbacks are that the target may not exist for new prospects, and that anchoring on a wrong target can introduce a worse bias than the one you are trying to fix.

Choosing between criteria 1 and 2 requires a prior judgement about the SAMPLING DESIGN, not the data. For a mining drillout chasing visible ore, criterion 1 (minimum). For an environmental survey of contamination hot spots near suspected sources, criterion 2 (maximum). For a petroleum well programme targeting structural highs, criterion 1 again (high-porosity zones over-sampled). Get this judgement wrong and you optimise in the wrong direction; cell declustering then makes the bias WORSE, not better. The literature is consistent on this point: cell declustering is a HEURISTIC tool, and the user is responsible for setting it up correctly.

Bourgault's 1997 multi-cell-size criterion. The minimum/maximum criteria are sensitive to noise in the declustered-mean curve — at finite sample size, the curve can have multiple local minima or a flat region, and picking "the" optimum is fragile. Bourgault (1997) proposed a robustified alternative: rather than picking a single optimum cell size, AVERAGE the declustering weights over a range of cell sizes. The resulting weights are still a proper probability mass on the sample (linear combinations of valid weights are valid), and the resulting declustered mean is more stable. The Bourgault approach is the modern default when the cell-size curve is ambiguous, when sample size is modest, or when there is no clear a-priori target.

Cell-origin sensitivity and the multi-origin averaging trick

The cell-size determines the GEOMETRY of the partition, but it does not pin it down completely. A grid of $L \times L$ cells can be ANCHORED at any offset $(\delta_x, \delta_y) \in [0, L) \times [0, L)$ — sliding the whole grid by less than one cell width yields a different partition, with different cells, different $n_c$ counts, and different declustering weights.

How much does the origin matter? Take a simple example. Consider two samples 0.05 apart along the x-axis, in a cell-size grid of 0.10. If the grid origin places them in the SAME cell, each gets weight 1/(2 · N_occ). If the grid is shifted so the cell boundary falls BETWEEN them, each ends up in a singleton cell and gets weight 1/(1 · (N_occ + 1)) — roughly double, because each is now in its own cell, and there is one more occupied cell. A 0.05-unit shift of the origin can swing individual weights by a factor of two. Over an 80-sample dataset, these per-sample swings rarely cancel exactly, and the declustered MEAN can move by 1–5% depending on the sampling pattern.

The first widget exposes this directly: hold the cell size fixed, slide the grid-origin, and watch the readout. The declustered mean is not a deterministic function of cell size alone.

The standard fix, baked into the GSLIB DECLUS program from its inception, is to AVERAGE OVER MULTIPLE GRID ORIGINS at each cell size. Take 3–5 origin shifts (usually drawn pseudo-randomly within $[0, L) \times [0, L)$), compute the declustering weights for each, and report the AVERAGE of the per-sample weights. The averaged weights are still a valid probability mass on the sample (each origin-specific set sums to 1, so their average does too), and the resulting declustered mean is much smoother as a function of cell size — exactly the smoothness you want when you intend to pick the cell size by an optimisation criterion.

The second widget visualises this trick.

The widget sweeps cell sizes from 4 cells per side (very large cells) to 30 cells per side (very small cells), with 5 random grid origins per cell size. The left panel plots the declustered mean as a function of cell size: thin coloured lines show the 5 single-origin curves (each one noisy with bumps and dips), and a thick black line shows the average-over-origins curve (smooth, monotone-bumped, and visibly the natural function to optimise over). The right panel highlights the chosen optimum (default: minimum declustered mean, configurable to maximum) with a vertical marker, and the readout reports the optimum cell size and the recommended declustered mean alongside the true population mean (because this is a synthetic field where we know the truth) and the biased (no-declustering) mean.

Three readings to take away. (1) The single-origin curves cross each other — at some cell sizes, one origin gives the lowest declustered mean, at another cell size a different origin gives the lowest. There is no "best origin" per se; the answer is the average. (2) The average curve is U-shaped (for high-grade preferential sampling): biased at both ends, minimum in the middle. The minimum is shallow — the curve is FLAT over a fairly wide range of cell sizes around the optimum — which is actually good news, because it means the choice of cell size is not knife-edge. The classical recommendation "anywhere within a factor of 2 of the optimum gives nearly the same answer" is visible directly. (3) The optimum declustered mean for this field is within 1–3% of the true population mean, while the biased mean overshoots by 15–25%. The cell-declustering algorithm — properly tuned with multi-origin averaging — recovers most of the lost accuracy.

The complete GSLIB DECLUS workflow

Put the pieces together and you have the standard workflow that the GSLIB DECLUS program implements (Deutsch 1989; Deutsch & Journel 1998, pages 213–218):

• Choose a cell-size range. Typically 5% to 50% of the domain extent (in units of the cell side length), with 10–20 cell sizes spaced log-linearly or linearly across the range. Smaller than 5% gives the degenerate "every cell has one sample" case; larger than 50% gives the degenerate "all samples in one cell" case.
• Choose a number of grid origins per cell size. Typically 3–5. Each origin is a random offset in $[0, L) \times [0, L)$. The exact number is not critical; the diminishing-returns sweet spot is around 5.
• For each (cell size, origin) combination, compute the declustering weights. Iterate the $w_i = 1/(n_c \cdot N_{\text{occ}})$ formula over all samples.
• Average the per-sample weights across origins. The averaged weights at each cell size define a smoothed weight vector.
• Compute the declustered mean at each cell size.

• Plot the declustered mean as a function of cell size. Visual inspection should show a clear U-shape (high-grade case) or inverted-U (low-grade case), with a stable minimum (or maximum) around the optimum.
• Pick the optimum cell size by the chosen criterion. Minimum (high-grade preferential), maximum (low-grade), or closest-to-target (a-priori value). Bourgault's multi-cell-size average is a robust fallback when the curve is ambiguous.
• Report the per-sample declustering weights at the optimum cell size. These feed every downstream geostatistical operation: declustered histogram (§1.3, §1.1), declustered normal-score transform (§1.2), declustered variogram pair counts (§3), declustered kriging mean (Part 5), declustered SGS reference distribution (Part 7).

The whole workflow is computationally trivial — GSLIB DECLUS runs in well under a second on a dataset of thousands of samples, even with 20 cell sizes × 5 origins = 100 weight-vector computations. It is one of the cheapest tools in the geostatistical workflow, and one of the most consequential.

What cell declustering can and cannot do

Cell declustering is a HEURISTIC, not a population-model-derived correction. The Horvitz–Thompson estimator $w_i \propto 1/\pi(\mathbf{s}_i)$ from §1.3 is the mathematically right thing — but the inclusion probability $\pi$ is unknown in practice. Cell declustering approximates $\pi$ by local sample density (a crowded cell signals high inclusion probability there, hence small weight per sample). The approximation is sensible, and it works well across a wide range of realistic sampling designs. But it is an approximation, and it has well-documented limitations.

• It cannot fix MISSING populations. If a sub-population of the deposit was never sampled — the reduced zone of an orebody, the dim seismic reflectors of a reservoir, the unmonitored land-use class of an environmental site — no choice of cell size and no number of grid origins can recover the missing data. Cell declustering corrects the WEIGHT of each sample, not the COMPOSITION of the sample set. If the geological model says there is a sub-population you have not sampled, you need geological judgement (and probably additional samples), not a declustering parameter.
• It is a HEURISTIC, not a derived estimator. The weight formula $w_i = 1/(n_c \cdot N_{\text{occ}})$ is not derived from a population model. It is a pragmatic local-density adjustment that happens to work. The Horvitz–Thompson framework gives a theoretical justification for "weight inversely proportional to inclusion probability", and cell declustering implements that via local-density estimation; but the local-density estimate itself is heuristic and depends on the cell-size and origin choices. Validate against secondary information whenever possible (different declustering method, geological constraints, independent estimates).
• It loses statistical power at small sample sizes. For fewer than about 50 samples, the cell counts $n_c$ are too sparse to support reliable weight assignment — most cells hold 0 or 1 sample regardless of cell size. The declustering correction collapses to ≈ 1/n weights even in the sweet spot. The honest move at small $n$ is to acknowledge that any histogram (biased or declustered) is unreliable, and to support summaries with as much auxiliary information as possible (geological domain boundaries, analogue datasets, prior distributions).
• For very irregular sampling patterns, polygonal declustering is more principled. Cell declustering assumes a roughly regular grid that approximates the sample density well. For samples that are themselves placed on a roughly regular grid with localised clustering (the common case), cell declustering is exactly the right tool. For samples that are scattered very irregularly — say, a few large clusters separated by big empty gaps — polygonal declustering (§2.2) handles the geometry better because the Voronoi cells naturally adapt to the local sample geometry without a free cell-size parameter. §2.3 compares the methods side by side on the same dataset.
• 2D vs 3D. The recipe extends directly to 3D: replace $L \times W$ with $L \times W \times H$, count samples per box, weight by $1/(n_c \cdot N_{\text{occ}})$. The GSLIB DECLUS program does both. In 3D the cell-size choice becomes a 3-dimensional sweep (typically isotropic, but anisotropic if the field anisotropy is known), and the origin shift becomes a 3D offset.

What declustering changes downstream (preview of §2.4)

The per-sample declustering weights are not just an academic correction — they propagate into every subsequent step of the geostatistical workflow. §2.4 takes this story up in detail; here is the preview.

• Declustered histogram → unbiased N-score transform (§1.2). The §1.2 recipe $Y_i = \Phi^{-1}(\hat{F}_n^{d}(Z_i))$ uses the DECLUSTERED empirical CDF, not the unweighted one. Without declustering, the ranks assigned to high-zone samples are too high (high tail under-counted in the declustered Fₙ) and the resulting Gaussian-margin values are systematically wrong.
• Declustered mean → kriging mean. Simple kriging (Part 5.1) uses the population mean as a known parameter; ordinary kriging (5.2) estimates it on-the-fly via the unbiasedness constraint. Either way, the appropriate "mean" is the declustered mean, not the unweighted sample mean. Using the wrong mean shifts every kriged estimate by the bias.
• Declustered samples → variogram pair counts (Part 3). The classical experimental variogram weights every pair equally; under preferential sampling, pairs from the high zone dominate and the variogram's short-range behaviour is over-fit to the high-zone covariance. Declustering-weighted variogram estimators (Part 3.6 covers these as one robustness option) give pairs from sparser regions more relative weight.
• Declustered samples → SGS reference distribution (Part 7). Sequential Gaussian simulation conditions on the sample data and produces realisations whose marginal distribution matches a reference distribution. That reference is the declustered Fₙ, not the unweighted Fₙ; otherwise the simulations inherit the preferential-sampling bias as their target.
• Declustered grade-tonnage (Part 5, Part 10). The grade-tonnage curve — tonnes above any cutoff and mean grade of that ore — depends on the histogram of grades. The declustered histogram is the right histogram. The biased histogram over-states recoverable ore at high cutoffs by 10–20% in typical mining datasets, with corresponding millions of dollars in feasibility-study error.

The chain is consistent: every place §1.1's univariate toolkit, §1.2's N-score, or §1.5's local-statistics framework writes "(1/n) Σ", the production workflow writes "Σ w_i" with the declustering weights from §2.1. Cell declustering is the input. The downstream tools — variogram, kriging, simulation — are the consumers. §2.4 walks through each consumer in detail; §2.2 and §2.3 develop the alternative method (polygonal) and compare; this section ships the workhorse.

Try it

• In the cell-declustering-step widget, set the cell size to "12 cells per side". Read the declustered mean. Now slide the cell size to "30 cells per side" — observe the declustered mean approach the biased mean (and the declustered histogram collapse onto the biased histogram). Slide back to "5 cells per side" — observe the same collapse from the other extreme. Confirm the U-shape: extremes give no correction; the middle does.
• Hold the cell size at "10 cells per side" and slide the X-origin from 0 to its maximum value. How much does the declustered mean swing? How much does the occupied-cell count $N_{\text{occ}}$ swing? Now also slide the Y-origin. The two-dimensional swing is the cell-origin sensitivity that the multi-origin averaging trick addresses.
• Click "Resample" to draw a fresh 80-sample realisation. The biased mean is now different (different specific samples, same sampling rule). At your previously-favourite cell size, the declustered mean is also different. Is it CLOSER to the true mean? The point of the next widget — multi-origin averaging — is to smooth out this realisation-level noise as much as possible at fixed cell size.
• In the cell-size-optimizer widget, read the optimum cell size flagged by the vertical marker. By how much does the average-over-origins declustered mean at the optimum differ from the true mean? By how much does it differ from the BIASED mean? Compare the magnitudes. Now look at the 5 individual-origin curves — are they all monotone? At what cell sizes do they cross each other most prominently? The crossing pattern justifies the multi-origin averaging: no single origin is "best" across all cell sizes.
• Without coding: a mining engineer reports a raw-sample mean grade of 4.3 g/t and a declustered mean grade of 3.4 g/t (using cell declustering with the GSLIB DECLUS default workflow). The deposit is 60 Mt at $50 / (g · t). What is the difference in nominal in-ground value between the two estimates? Which one would NI 43-101 / JORC accept? Why does the answer not depend on the cell-size choice as long as the chosen size is "near the optimum" — i.e. anywhere on the flat part of the average-over-origins curve?
• Without coding: a petroleum engineer has 30 well log samples and tries to apply cell declustering with the GSLIB DECLUS default sweep. The declustered-mean-vs-cell-size curve is noisy and has no clean minimum. What is the likely problem, and what would you advise the engineer to do? (Hint: see "It loses statistical power at small sample sizes" in the caveats above. Two options exist: (a) trust the raw mean and accept the bias risk, with a clear caveat in the report; (b) use polygonal declustering — §2.2 — which has no free cell-size parameter and is less affected by small $n$.)

Pause and reflect: the cell-declustering weight formula $w_i = 1/(n_c \cdot N_{\text{occ}})$ says every occupied CELL contributes 1/N_occ of the total weight, regardless of $n_c$. Is that fair? In a cell with 30 samples, the algorithm treats them collectively as ONE cell's worth of evidence — even though there is more information about that cell's local mean than there is about a cell with 1 sample. An alternative would be to weight cells by SAMPLE COUNT $n_c$ rather than equally, which would partially restore the bias. Why does GSLIB DECLUS go the equal-cells way instead? (Hint: the goal is to estimate the POPULATION mean, not the SAMPLED mean. Each cell's evidence about the local sub-population is what matters; sampling 30 times within a small cluster gives you 30 measurements of essentially the same place, not 30 measurements of 30 different places.)

What you now know — and where Part 2 goes next

You have the cell-declustering algorithm in full: a five-step recipe that turns a preferentially-located sample into a set of per-sample weights summing to 1, replacing the equal weights of the unweighted Fₙ with the proper declustering weights for use in every downstream geostatistical product. You have the cell-size dilemma formalised — the U-shape from §1.3 explained by the two degenerate limits — and you have THREE optimisation criteria (minimum, maximum, a-priori target) plus Bourgault's 1997 multi-cell-size average for robustness. You have the cell-origin sensitivity story and the GSLIB DECLUS standard practice of averaging over 3–5 random origins to smooth the declustered-mean-vs-cell-size curve. You have the complete DECLUS workflow as an 8-step procedure that the §2.1 widgets implement faithfully.

You have the honest scope: cell declustering is a heuristic, not a derived estimator. It corrects sampling bias but cannot fix missing populations. It loses power below ~50 samples. For very irregular sampling patterns polygonal declustering is better; for the somewhat-irregular common case cell declustering is the GSLIB default. And you have the downstream-impact preview: declustering weights feed the N-score transform, the kriging mean, the variogram (under robust estimators), the SGS reference distribution, and the grade-tonnage curve — the entire workflow downstream.

§2.2 develops the polygonal alternative — Voronoi-cell weighting, no free parameter, more principled but more sensitive to boundary effects. §2.3 puts cell and polygonal head-to-head on the same dataset, with a clear set of criteria for choosing between them. §2.4 walks the downstream chain quantitatively: how much does declustering change the variogram, the kriged map, the SGS reference distribution? By the end of Part 2 the reader has the full declustering toolbox, with explicit pre-/post-declustering comparisons on every product. Then Part 3 introduces the experimental variogram — the spatial-pair tool that finally moves beyond marginal distributions — and the declustering weights you compute here will reappear, weighting the variogram pair counts.

Mark section complete →

References

• Journel, A.G. (1983). Nonparametric estimation of spatial distributions. Mathematical Geology, 15(3), 445–468. (The foundational paper. Introduces the cell-declustering weight formula in its modern form and proves that the resulting weighted empirical CDF is a consistent estimator of the population CDF when the cell size is chosen well.)
• Deutsch, C.V. (1989). DECLUS: a Fortran 77 program for determining optimum spatial declustering weights. Computers & Geosciences, 15(3), 325–332. (The original DECLUS program. Specifies the cell-size sweep, the random-origin averaging trick, and the minimum/maximum/a-priori criterion for picking the optimum cell size. Every subsequent implementation traces back to this paper.)
• Bourgault, G. (1997). Spatial declustering weights. Mathematical Geology, 29(2), 277–290. (The multi-cell-size averaging refinement. Argues that picking a single optimum cell size is fragile under small-sample noise, and shows that averaging weights across a range of cell sizes produces more robust corrections. The Bourgault average is the modern default for ambiguous datasets.)
• Deutsch, C.V., Journel, A.G. (1998). GSLIB: Geostatistical Software Library and User's Guide, 2nd ed. Oxford University Press. (The canonical reference. The DECLUS program is documented on pages 213–218 with full input-file format, parameter defaults, and worked examples on the Walker Lake dataset. The 2nd edition adds discussion of Bourgault's multi-cell-size criterion.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press. (§4.1 on cell declustering with the Walker Lake worked example; the cleanest modern textbook treatment of the algorithm with both the high-grade and low-grade preferential cases worked out.)
• Isaaks, E.H., Srivastava, R.M. (1989). An Introduction to Applied Geostatistics. Oxford University Press. (Chapter 10 introduces cell and polygonal declustering side-by-side; the §2.1 / §2.2 / §2.3 arc of Part 2 mirrors Isaaks & Srivastava's chapter structure.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford University Press. (Reservoir-engineering treatment; emphasises the downstream chain — declustered N-score, declustered variogram, declustered SGS reference distribution — that §2.4 develops in detail.)