What declustering changes downstream

§2.1 built cell declustering. §2.2 built the polygonal alternative. §2.3 put them head-to-head with a four-scenario comparison and a five-question decision rule. Each section answered the question "how do I compute the weights?" — the algorithmic and decision-rule layers. §2.4 answers a different question, and one that determines whether anyone outside the geostatistics team will ever care about your declustering choice: "what happens to the rest of the workflow if I get this wrong?"

The short answer is that EVERY downstream product depends on the declustering correction. The §1.2 normal-score transform uses the empirical CDF; if that CDF is biased, every Y_i is miscalibrated. The §3 experimental variogram counts pairs; if pairs from the over-sampled high zone dominate, γ̂(h) at short lags is inflated. The §5 simple-kriging mean uses the global mean; if that mean is biased, every kriged estimate is offset. The §7 SGS reference distribution is the declustered F_n; realisations back-transformed against the wrong reference inherit the bias. The §10 reserves calculation integrates the declustered histogram above a cutoff; raw histograms over-state recoverable ore in any high-grade-preferential project. §2.4 walks the cascade explicitly and puts numbers on each stage.

The §2.4 thesis can be stated in one sentence: on a realistic preferential-sampling severity, the cumulative downstream impact of skipping declustering is large enough to fail a qualified-person sign-off under NI 43-101 (Canadian mining), JORC (Australasian), SAMREC (South African) or SPE-PRMS (petroleum) — and the corresponding dollar value typically exceeds the entire data-acquisition cost of the project. The two §2.4 widgets make this concrete; the prose develops the cascade stage by stage; the references trace the regulatory framework that drives the production protocol.

The downstream cascade — stage by stage

The cascade is consistent and traceable. At each stage, the operation that the rest of the workflow expects is a SAMPLE-AVERAGED quantity — a mean, a histogram, a covariance, a quantile — and the appropriate "average" is the DECLUSTERED average rather than the unweighted one. Concretely:

• §1.2 N-score transform. The normal-score transform maps each sample value $Z_i$ to a standard-Gaussian counterpart $Y_i = \Phi^{-1}(\hat{F}_n(Z_i))$ via the empirical CDF. Under preferential sampling, the unweighted $\hat{F}_n$ over-represents the high zone — the rank of a high-grade sample is artificially LOW (the high tail of the unweighted CDF is "fat"), so $\Phi^{-1}$ assigns it a Gaussian Y-value that is systematically too SMALL relative to its true population rank. The declustered CDF $\hat{F}_n^{d}(z) = \sum_i w_i \mathbf{1}{Z_i \le z}$ corrects this: the high-zone samples are downweighted, the high tail of $\hat{F}_n^d$ is no longer fat, and the resulting Y-values are properly Gaussian on the population. Every SGS realisation that is back-transformed through $\Phi^{-1}$ then maps the simulated Gaussian values back to original units via the SAME $\hat{F}_n^d$; using the unweighted $\hat{F}_n$ instead would propagate the rank-miscalibration into every realisation's marginal distribution.
• §3 Experimental variogram. The classical Matheron estimator $\hat{\gamma}(h) = (1 / 2N(h)) \sum_{(i,j) \in N(h)} (Z_i - Z_j)^2$ weights every pair equally. Under preferential sampling, pairs within the over-sampled high zone form a disproportionate fraction of every lag bin, and the resulting variogram is biased toward the within-cluster covariance — often producing an artificial nugget effect at short lags and an inflated overall sill. The declustering-weighted variant $\hat{\gamma}$d(h) = (1 / 2 W(h)) \sum{(i,j)} w_i w_j (Z_i - Z_j)^2 uses the pair weight $w_i w_j$ — pairs from over-sampled regions contribute proportionally less. The Cressie–Hawkins robust estimator (§3.6) extends this further with absolute-difference moments. Part 3.6 gives the full machinery; for §2.4 the point is that the SAME declustering weights from §2.1 feed both the marginal CDF and the spatial-pair count.
• §5 Simple kriging. Simple kriging at a query point $\mathbf{s}_0$ writes $Z^$(\mathbf{s}_0) = \mu + \sum_i \lambda_i (Z_i - \mu)Z∗(s0​)=μ+∑i​λi​(Zi​−μ). The kriging weights $\lambda_i$ come from solving $\mathbf{K} \boldsymbol{\lambda} = \mathbf{k}$ where $\mathbf{K}, \mathbf{k}$ are covariance matrices derived from the variogram. The constant $\mu$ is the (assumed-known) global mean of the population. Under preferential sampling, using the UNWEIGHTED sample mean for $\mu$ is wrong by the same percentage as the marginal bias — typically 15-25% for the canonical case. Every kriged estimate inherits that offset, since $Z^$(\mathbf{s}_0) - \mu = \sum_i \lambda_i (Z_i - \mu) is on the order of $Z_i - \mu$ but the assumed $\mu$ is biased. Ordinary kriging (§5.2) replaces $\mu$ with an unbiasedness-constrained estimate, but THAT estimate itself uses the data as inputs — and under preferential sampling the OK estimate of the local mean is also biased toward the cluster mean. The fix in both cases is the declustered mean from §2.1.
• §5.5 Block kriging and change-of-support. Block-kriged estimates $Z^$(V)Z∗(V) average over a finite block $V$. The estimator is the same form $Z^$(V) = \mu + \sum_i \lambda_i^V (Z_i - \mu) with block-averaged covariance vectors $\mathbf{k}^V$. The same global-mean bias propagates into block estimates — and into the change-of-support distribution that gives block-grade variances (§1.5's dispersion variance). A biased $\mu$ shifts the entire dispersion-variance computation, which then propagates into selectivity curves (Part 10).
• §6 Cross-validation. Leave-one-out cross-validation removes each sample in turn and re-krigs the remaining set. Every cross-validation metric — mean error, mean squared error, accuracy plot, calibration of the kriging variance — is a SAMPLE-AVERAGED diagnostic. Under preferential sampling, the cross-validation metrics inherit the bias: the mean error is computed against the (biased) sample distribution, and the accuracy-plot calibration line drifts away from 1:1. Declustering-weighted versions of these metrics give the right answer. Part 6 develops this in detail.
• §7 Sequential Gaussian simulation. SGS produces conditional realisations of the spatial field whose marginal distribution matches a REFERENCE histogram. The reference is the declustered $\hat{F}_n^d$ on the data — every realisation's marginal Q-Q-matches that reference by construction. Using the unweighted $\hat{F}_n$ instead gives realisations whose marginal is the (biased) sample histogram, which then propagates into every post-processing step (Part 7.6 — grade-tonnage curves, ranked-realisation envelopes, P10/P50/P90 reserves quantiles). The declustering decision made at §2.1 propagates into every one of (say) 200 realisations.
• §10 Reserves and grade-tonnage curves. The grade-tonnage curve plots tonnes-above-cutoff $T(c)$ and mean grade $G(c)$ for every cutoff $c$. Contained metal $M(c) = T(c) \cdot G(c)$; dollar value $$(c) = M(c) \cdot \text{price}$. All three depend on the histogram of grades. The declustered histogram is the right histogram; using the raw histogram overstates $T$ and $G$ at high cutoffs (the high tail is over-represented) and therefore overstates $M$. The dollar magnitude of the overstatement depends on the deposit volume, the metal price, and the severity of preferential sampling — but on a 10 Mt deposit at role="main" aria-label="Lesson content" tabindex="-1"900/oz with canonical sampling severity, the §2.4 widget below puts realistic numbers in the multi-tens-of-millions range.

The cascade is causal — each stage feeds the next. Errors do not always compound multiplicatively; in some cases the kriging operator partially absorbs the global-mean bias (because the $\sum \lambda_i (Z_i - \mu)$ deviation term is partially self-correcting when $\lambda_i$ and $Z_i - \mu$ have correlated structure). In other cases the bias amplifies — particularly in reserves, where the cutoff operation $\mathbf{1}{Z > c}$ is non-linear and a small shift in the high tail produces a large shift in contained metal. The first §2.4 widget shows this directly, stage by stage.

The cascade made interactive

The first widget steps through the four key downstream operations on the same preferentially-sampled dataset. Pick a severity level (mild / canonical / severe), then step through the operations: N-score transform, variogram preview, simple-kriging at a query point, reserves at a cutoff. Each operation reports its result TWO WAYS — RAW (unweighted sample, no declustering) and DECLUSTERED (cell weights from the §2.1 GSLIB DECLUS sweep). The four-stage strip at the top of the widget tracks the per-stage relative bias, so you can see the cascade build as you switch between operations.

What to look for. On the CANONICAL setting (the §2.1 / §2.2 default), the global-mean raw bias is +15-25%. Step through the four operations and read the per-stage bars: N-score bias on the top sample is typically 5-15% of the top-Y value (visible in the bottom strip); variogram inflation at $h \approx 0.15$ is typically 10-25%; kriging shift at the centre query point is 2-8% (kriging weights partially absorb the bias because the centre point has neighbours from both high and low zones); reserves bias on contained metal at cutoff 3.5 is typically 8-18%. Switch to SEVERE: every stage's bias roughly doubles. Switch to MILD: every stage drops by a factor of ~2-3. The cascade is NOT uniform across stages — reserves is consistently the largest because of the non-linearity of the cutoff operation. This is why the dollar-value story (next widget) is dominated by the reserves end of the chain.

A worked numerical example

Concrete numbers help the cascade feel real. Consider a synthetic gold-style deposit of 10 Mt, with a true population mean grade of 2.7 g/t, sampled preferentially in the high zones (cubic acceptance, the §1.3 / §2.1 canonical case). 200 samples are drawn; the raw sample mean is 3.3 g/t — a +22% bias. Run the cell-declustering sweep: the declustered mean is 2.75 g/t — within 2% of truth. So far we have a global-mean correction; the cascade adds the rest.

• The §1.2 N-score transform on the raw sample assigns the highest-grade sample (say $Z = 8.4$ g/t) the Y-value $\Phi^{-1}((200 - 0.5)/200) = 2.50$. The declustered transform assigns the same sample a smaller cumulative-probability mass (because the high-zone samples are downweighted), say $p_d = 0.985$, giving $Y_d = \Phi^{-1}(0.985) = 2.17$. A 0.33 standard-deviation difference — material on the high tail.
• The experimental variogram at lag $h = 0.15$ (typical short-range): the raw $\hat{\gamma}(0.15) \approx 5.4$; the declustering-weighted estimator $\hat{\gamma}_d(0.15) \approx 4.2$. A 22% inflation in the raw estimate — the same percentage as the global-mean bias, because short-lag pairs are dominated by the over-sampled high zone (where the variance is also higher).
• Simple kriging at the centre $\mathbf{s}$0 = (0.5, 0.5)s0​=(0.5,0.5) with neighbours drawn from the same sample. With $\mu${\text{raw}} = 3.3: the SK estimate is 3.0 g/t. With $\mu_{\text{decl}} = 2.75$: the SK estimate is 2.4 g/t. A 0.6 g/t shift at this single query point — about 25% of the local value. Compounded across a 100 × 100 grid this is the average kriging-map bias.
• Reserves at cutoff $c = 3.5$ g/t. Raw: 32% of tonnes above cutoff, mean grade above cutoff 4.6 g/t. Declustered: 18% of tonnes above cutoff, mean grade above cutoff 4.3 g/t. Raw contained metal: 10 Mt × 0.32 × 4.6 g/t = 14.7 Mg gold = 474 koz. Declustered contained metal: 10 Mt × 0.18 × 4.3 g/t = 7.7 Mg = 248 koz. Difference: 226 koz × $1900/oz =$429 M of "phantom ounces" in the raw reserves estimate.

The exact numbers vary with the sample realisation, the preferential-sampling exponent, and the cutoff — but the order of magnitude is robust. The §2.4 reserves widget below lets you adjust deposit volume, severity, cutoff, and metal price and read the dollar value directly. For canonical settings the avoidable error (raw minus declustered) is typically 5-15% of deposit value; for severe settings it can reach 20-30%. These are not academic numbers — they routinely show up in NI 43-101 technical-report peer review.

What declustering CAN and CANNOT fix

Declustering is a powerful correction for ONE specific source of bias: the over-representation of certain regions of the domain due to preferential sample LOCATION. Within that scope, declustering works reliably across a wide range of realistic designs and is the standard production protocol. Outside that scope, declustering does nothing — and importantly, it is silent about its own limitations.

• CAN fix: histogram bias from preferential location. Samples are clustered in some regions of the domain and sparse in others; the unweighted histogram over-represents the cluster regions and biases the marginal distribution. Cell and polygonal declustering correct this by downweighting the cluster regions. This is the canonical use case from Journel (1983) onward; the §2.1 / §2.2 / §2.3 widgets demonstrate the correction; the §2.4 cascade widget shows the corresponding fix at each downstream stage.
• CAN fix: spatial covariance bias. Pair counts in the experimental variogram are dominated by the over-sampled regions; declustering-weighted estimators (§3.6) correct this. The fix is essentially automatic once per-sample declustering weights are available — the same weights from §2.1 feed both the marginal CDF and the spatial-pair count.
• CANNOT fix: missing sub-populations. If a sub-population of the deposit was never sampled — the reduced zone of an orebody, the deep over-mature zone of a reservoir, the south basin of a hydrology study — no choice of cell size and no number of grid origins can recover the unobserved values. Declustering corrects the WEIGHTING of each sample; it does not extrapolate to unsampled regions of feature space. The fix here is geological judgement, additional samples, and possibly explicit boundary modelling (Part 8 indicator methods) — not declustering parameters.
• CANNOT fix: measurement bias unrelated to location. If high-grade samples are assayed by a different laboratory than low-grade samples, with different calibration, declustering cannot correct the cross-lab bias. The fix is QA/QC sample analysis, blank / standard / duplicate verification, and possibly a lab-correction model — not declustering parameters.
• CANNOT fix: errors in the deposit-model DOMAIN. If the domain mask itself is wrong — e.g. the lease boundary excludes mineralised ground that should be inside — no declustering choice within the wrong mask recovers the right answer. The fix is geological re-interpretation, not statistics. Polygonal declustering with explicit domain clipping (§2.2) helps when the domain is correct but irregular; it does not help when the domain is wrong.
• CANNOT fix: unmeasured covariates. If grade depends on a covariate (e.g. lithology, depth, hydrothermal alteration) that is not measured at every sample location, declustering does not impute it. Cokriging (§5.7) and kriging-with-external-drift (§5.3) can — but only if the covariate IS measured on a denser support than the primary variable. Declustering operates on the primary variable alone.

The honest framing for any reserves-estimation report: declustering corrects ONE source of bias, the one that is most amenable to a per-sample-weight fix. The other potential bias sources require different machinery — additional sampling, lab QA/QC, geological re-interpretation, or higher-level modelling. The §2.4 reserves widget below is a "do you decluster or not" comparison; it is NOT a "declustering solves all problems" demonstration. The dollar magnitudes are real, but they bound the value of the SPECIFIC fix that declustering offers, not the value of comprehensive bias-correction practice.

The regulatory framework — NI 43-101, JORC, SAMREC, SPE-PRMS

Reserves and resources reporting in mining and petroleum is governed by codes that require defensible methodology. The four leading codes and their declustering treatment:

• NI 43-101 (Canadian Securities Administrators, 2011, revised). Requires a "Qualified Person" sign-off on resource and reserve estimates. Companion CIM Best Practice Guidelines (CIM 2003, 2019) specifically address spatial-sampling bias and require either declustering or an equivalent correction whenever sampling is not approximately uniform. A technical report submitted to a Canadian securities regulator that uses unweighted sample statistics in the face of visible clustering is at material risk of regulatory deficiency. The §2.4 protocol — always run, compare, document, sensitivity-sweep — is consistent with NI 43-101 CIM best-practice.
• JORC Code (Joint Ore Reserves Committee, Australasia, 2012 ed.). The JORC Table 1 disclosure requires sampling-method documentation and explicit treatment of clustering. The 2012 edition strengthened the Table 1 requirements after several high-profile reserves restatements (e.g. the Bre-X scandal and subsequent industry reviews). Declustering or an equivalent bias-correction step is explicitly named in the JORC competent-person checklist.
• SAMREC Code (South African Mineral Resource Committee, 2016 ed.). The South African equivalent of JORC. Same Table 1 structure, same declustering requirement. The 2016 edition aligned closely with JORC 2012.
• SPE-PRMS (Society of Petroleum Engineers — Petroleum Resources Management System, 2018 ed.). Petroleum-industry analogue. Requires defensible statistical estimation of in-place hydrocarbon volumes and recoverable reserves. The Pyrcz & Deutsch (2014) reservoir-engineering treatment of declustering is the standard reference for SPE-PRMS-compliant geostatistical workflows.

Across all four codes the pattern is consistent: explicit treatment of preferential sampling is required; declustering is the canonical implementation; alternative methods (e.g. polygon-clipping by geological domain) are acceptable if documented; no treatment is NOT acceptable for any non-uniform sampling design. A §2.4-compliant reserves report names the method (cell vs polygonal), the cell-size selection rule (MIN / MAX / a-priori / Bourgault), the number of grid origins, the resulting declustered mean alongside the unweighted mean, and ideally a sensitivity sweep showing how the answer moves under different declustering parameter choices.

Putting a dollar value on the choice

The first §2.4 widget showed the per-stage relative biases as the cascade builds. The second widget commits to a concrete reserves calculation under user-configurable inputs and reports a DOLLAR value for the declustering choice.

The widget exposes four inputs: deposit volume (default 10 Mt, range 0.5-500 Mt), clustering severity (a 0-1 slider that maps to a cubic-acceptance exponent), cutoff grade (default 3.5 g/t, range 0-10 g/t), and metal price (default $1900/oz, range 800-5000$/oz). For each setting the widget computes the reserves THREE ways: from the full deposit (TRUE), from the raw unweighted sample (RAW), and from the cell-declustered sample (DECLUSTERED). The headline numbers are the three errors — Raw minus Truth, Declustered minus Truth, and Raw minus Declustered (the AVOIDABLE error, i.e. the dollar value of running declustering).

The sensitivity sweep at the bottom plots both the raw and the declustered errors as functions of the severity slider, 0 to 1. On a 10 Mt / $1900/oz deposit at cutoff 3.5 g/t the typical pattern is: at severity 0 (uniform sampling) both errors are <2$1900/oz on a 10 Mt deposit the severity-0.7 case translates into an avoidable error of $50-150 M — the §2.4 thesis made concrete.

The widget also reveals an important second-order pattern: the declustered error is not zero. Declustering corrects most of the bias but not all of it, and at very high severity the declustered error grows to 3-5%. This is the §2.1 / §2.2 / §2.3 honest framing: declustering is a HEURISTIC. The residual error at severe settings reflects (a) the heuristic nature of the $1/(n_c \cdot N_{\text{occ}})$ approximation to the inclusion probability and (b) the finite-sample noise from a 200-sample dataset. Even the best declustering choice does not recover the full population statistics from a biased sample; what it does is REDUCE the avoidable error from "fails-peer-review" to "well within Code tolerance".

The §2.4 production protocol

Putting the cascade story, the regulatory framework, and the dollar magnitudes together, here is the explicit production protocol that §2.4 recommends — and that satisfies the standard codes:

• ALWAYS run declustering when clustering is visible. "Visible" can be tested algorithmically (compute the polygon-area distribution from §2.2 — a max-to-min ratio above ~5:1 signals clustering) or visually (a scatter plot of sample locations). The cost of running declustering is trivial — milliseconds of compute on thousands of samples — so the threshold for triggering it should be near zero. If sampling is genuinely uniform, declustering gives back essentially-uniform weights and changes nothing; if not, it corrects what needs correcting.
• COMPARE raw and declustered global statistics. A large gap (>3-5% on the mean, or visibly different histograms) signals a sensitive downstream workflow. Small gaps (<2%) signal a benign sampling design where the declustering correction is essentially cosmetic. Either outcome is informative: a large gap motivates explicit sensitivity analysis, a small gap motivates a "methods agree, declustering corroborates" disclosure in the report.
• DOCUMENT the declustering choice. Method (cell vs polygonal), cell-size selection (MIN / MAX / a-priori / Bourgault average), number of grid origins, polygonal clipping mask (if applicable), and per-sample weights as a sortable table appended to the technical report. NI 43-101 / JORC / SAMREC Table 1 disclosure requires this level of detail.
• PROPAGATE declustering through every downstream product. The N-score transform uses $\hat{F}_n^d$; the experimental variogram uses pair-weighted estimators (§3.6); the kriging mean uses $\mu_d$; the SGS reference distribution uses $\hat{F}_n^d$; the reserves calculation uses the declustered histogram. Consistency across the chain matters: a declustered mean fed into an unweighted variogram is a logical inconsistency that peer review WILL catch.
• SENSITIVITY-sweep the declustering parameters. Run the calculation under several reasonable declustering choices — different cell sizes, MIN vs MAX criterion, cell vs polygonal — and report the SPREAD in the headline numbers. A small spread (~1-2% on contained metal) is a strong statement; a large spread (~5-10%) indicates a project where the declustering choice is consequential and additional samples or a stronger geological prior would materially reduce uncertainty.
• If the spread is large, REPORT THE RANGE. Do not pick the answer that flatters the project. The §2.3 "committee" approach extended to the §2.4 dollar value: present the raw-versus-declustered envelope and let the regulator / qualified person assess. Modern practice is to attach a P10 / P50 / P90 envelope on reserves from the spread of defensible declustering choices.

This protocol is consistent with the regulatory codes and with the multi-method-critique argument from Pyrcz, Strebelle & Deutsch (2005) developed in §2.3's prose. It is also consistent with the §1 / §2.1-§2.3 framing of declustering as a heuristic family: running multiple methods and reporting the spread is the responsible move precisely because no single method is provably optimal across all designs.

Try it

• In the downstream-cascade widget, set the scenario to "Canonical" and step through the four operations in order: N-score, Variogram, Kriging, Reserves. Read each per-stage bias percentage from the bar in the strip. Are they all the same sign? Are they all the same magnitude? Which stage shows the largest relative impact, and why? (Hint: reserves is non-linear in the cutoff operation, so it amplifies any tail bias.) Now switch to "Severe": how much does each stage's bias change?
• Resample (advance seed) 5-10 times at the "Canonical" setting. The per-stage biases jitter with each draw, but the pattern should be stable: most resamples show the cascade in the same direction with similar magnitudes. The seed-by-seed jitter is the realisation-level noise; the pattern across resamples is the structural impact. Compare to "Mild": at mild severity the seed-to-seed jitter is comparable to the signal, so a single resample is unreliable — typical of the small-bias regime.
• In the downstream-cascade widget, set Operation to "Reserves" and slide the cutoff slider from 1.5 to 6.0 g/t. The per-stage biases change with cutoff — high cutoffs typically show larger relative biases (the high tail dominates the calculation). What is the cutoff at which the raw and declustered reserves are most divergent? Is it the cutoff your operation would actually use? (Most realistic mining cutoffs are in the 2-4 g/t range for gold; petroleum-style net-pay cutoffs are 8-12% porosity. The §2.4 bias is sensitive to the operational cutoff.)
• In the reserves-impact-calculator, set Volume = 10 Mt, Severity = 0.7, Cutoff = 3.5 g/t, Price = 1900 $/oz. Read the three headline cards. What is the avoidable error in dollars? How does it compare to the entire data-acquisition cost of a 200-sample drillout (~$2-5 M for an exploration-stage gold project)? (Answer: the avoidable error is typically 20-50× the data-acquisition cost. Declustering is one of the highest-leverage tools in the workflow.)
• In the reserves-impact-calculator, hold all other inputs fixed and slide the Severity from 0 to 1. Watch the sensitivity-sweep plot and the headline cards. At what severity does the raw error first exceed 5% of deposit value? At what severity does it exceed 10%? Does the declustered error EVER exceed 5%? Is the declustered curve monotone in severity, or does it have structure?
• In the reserves-impact-calculator, set Severity = 1.0 (severe) and increase the Volume slider from 10 Mt to 500 Mt. The avoidable error in dollars scales linearly with volume; at 500 Mt the avoidable error reaches the multi-billion-dollar range. Is this realistic? (Yes — major-mining-province deposits do reach 200+ Mt and reserves restatements have been in the multi-billion range historically. The 2010s nickel-laterite reserves restatements and the late-1990s Bre-X gold scandal are textbook examples of bias-propagation failures, though Bre-X was outright fraud rather than statistical error.)
• Without coding: a mining company has 1500 drillhole samples in a clean rectangular lease with a 100 Mt deposit at 2 g/t gold. The raw mean is 2.6 g/t and the cell-declustered mean is 2.0 g/t — a 30% bias correction. The cutoff is 1.5 g/t, the price is 2000 $/oz, and the recoverable fraction is assumed 0.7. Estimate the avoidable error in dollars. (Hint: 100 Mt × 0.6 g/t shift × 0.7 recoverable × 0.0322 oz/g ×$2000/oz ≈ $2.7 B. The dollar impact of a 30% declustering correction on a 100 Mt deposit is in the billions.)
• Without coding: an environmental scientist has 400 contaminant-concentration measurements in a watershed. Sampling is clustered around three suspected sources. The raw mean concentration is 28 μg/L; the declustered mean is 16 μg/L. The regulatory action threshold is 20 μg/L. Does the declustering choice change the regulatory decision? What is the §2.4-compliant way to present this finding? (Hint: the raw mean exceeds the threshold; the declustered mean is below it. The §2.4-compliant presentation discloses BOTH numbers, names the declustering method and parameters, and discusses the sampling-design assumption that the clustering reflects measurement convenience rather than a hot-spot that the regulator should know about.)
• Without coding: a petroleum reservoir engineer has 200 well-log porosity samples in a clean rectangular reservoir block. Raw mean porosity is 0.18; declustered is 0.16. The reservoir volume is 5 km × 3 km × 50 m = 0.75 km³ and the oil saturation is 0.6. What is the avoidable error in in-place oil volume? Why does even a 2-percentage-point declustering correction matter at field scale? (Hint: 0.75 km³ × 0.02 × 0.6 ≈ 9 million m³ of oil ≈ 56 MMbbl. At $80/bbl this is$4.5 B in in-place value. The proportional correction is small but the field is large; declustering matters at every scale.)
• Without coding: a project geologist runs declustering and reports the declustered mean. A peer reviewer asks "what would the dollar value of the reserves be without declustering?" Is this a reasonable question to ask, or is the un-declustered number not appropriate to even compute? (Answer: it IS reasonable, but ONLY as a sensitivity-analysis number to demonstrate that the declustering correction is material. The un-declustered number should never be the primary reported value when sampling is clustered. The §2.4 protocol's "sensitivity-sweep" step is exactly this — show the range and disclose the choice.)

Pause and reflect: the §2.4 cascade shows declustering propagating through five downstream stages, each of which is a SAMPLE-AVERAGED quantity. Are there downstream operations that are NOT sample-averaged and therefore not affected by the declustering choice? (Hint: deterministic geological constraints — domain boundaries, hard cutoffs on lithology, fault throws — are not affected by sample weighting; they are inputs to the model rather than outputs of sample statistics. Visualisation choices — colour scales, axis ranges, kriging-grid resolution — are also unaffected. But every statistic of the data ITSELF is affected. The §2.4 chain is therefore exhaustive within the data-driven part of the workflow.)

What you now know — and what closes Part 2

You have the explicit downstream cascade of the declustering correction: N-score transform, experimental variogram, simple and ordinary kriging, block kriging and change-of-support, cross-validation, sequential Gaussian simulation, reserves and grade-tonnage. At each stage you know the SAME per-sample weights from §2.1 (or §2.2) feed in, and you know what changes if those weights are wrong. You have the worked numerical example on a canonical 10 Mt / 200-sample gold-style deposit, showing how a +22% global-mean bias translates into 5-15% biases at each downstream stage and a $200-400 M reserves error at$1900/oz. You have the regulatory framework — NI 43-101, JORC, SAMREC, SPE-PRMS — that REQUIRES declustering or an equivalent correction whenever sampling is preferential, and you have the §2.4 production protocol (always run, compare, document, propagate, sensitivity-sweep, report the range) that satisfies all four codes.

You have the honest scope: declustering corrects ONE specific bias source — preferential location within a properly-defined population — and is silent about the others (missing sub-populations, measurement bias unrelated to location, wrong domain, unmeasured covariates). The §2.4 widgets are "do you decluster or not" comparisons, not "declustering fixes everything" demonstrations. The dollar magnitudes are real but they bound the value of the SPECIFIC fix that declustering offers.

This closes Part 2. The four sections built declustering from motivation (§1.3 / Part 2 intro) through the cell algorithm (§2.1), the polygonal alternative (§2.2), the comparison and decision rule (§2.3), and the downstream cascade with dollar magnitudes (§2.4). The reader has the full declustering toolbox plus the honest framing of what it can and cannot do. The same per-sample weights computed in §2.1 / §2.2 will reappear throughout Part 3 (variograms — pair counts will be weighted by $w_i w_j$ under the robust estimators), Part 4 (variogram modelling — weighted-least-squares fits use the same weights), Part 5 (kriging — $\mu_d$ replaces the unweighted mean), Part 6 (cross-validation — accuracy plots use the declustered metrics), and Part 7 (SGS — back-transform uses $\hat{F}_n^d$). The declustering decision is not just Part 2's deliverable; it is an input to every product that follows.

Part 3 takes up the experimental variogram — the central spatial-pair tool that finally moves beyond marginal distributions. The declustering weights from Part 2 will weight the pair counts in §3.6's robust estimators. The same Part 2 honesty — heuristic methods, run multiple, report the spread — extends naturally to Part 3's variogram-estimator choices. The bridge between the marginal world (Part 1 / Part 2) and the spatial world (Part 3 onward) is finished; the rest of the textbook builds on it.

Mark section complete →

References

• Journel, A.G. (1983). Nonparametric estimation of spatial distributions. Mathematical Geology, 15(3), 445–468. (The foundational paper. The downstream-cascade argument — declustered F_n^d propagates to every product — is implicit in Journel's use of the declustered CDF as the universal input to univariate workflow steps.)
• Deutsch, C.V. (1989). DECLUS: a Fortran 77 program for determining optimum spatial declustering weights. Computers & Geosciences, 15(3), 325–332. (The GSLIB DECLUS program. The "feed every downstream tool" framing is the operational consequence of producing per-sample weights as the universal output.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press. (§4.1 on cell and polygonal declustering, then §5.1-§5.6 on kriging that uses the declustered mean. The §2.4 cascade story is implicit in Goovaerts' integrated treatment of the workflow.)
• Isaaks, E.H., Srivastava, R.M. (1989). An Introduction to Applied Geostatistics. Oxford University Press. (Chapter 10 on declustering; Chapter 11 on kriging that uses the declustered statistics; Chapter 16 on simulation. The textbook-canonical illustration of the cascade.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford University Press. (Reservoir-engineering treatment. Chapter 4 on declustering; subsequent chapters develop the downstream propagation explicitly in the petroleum context. §2.4's widget framework — N-score, variogram, kriging, reserves — mirrors Pyrcz & Deutsch's chapter ordering.)
• Rossi, M.E., Deutsch, C.V. (2014). Mineral Resource Estimation. Springer. (Mining-focused treatment with explicit dollar-value examples on real deposits. The §2.4 reserves widget's economic model — V × P(Z > c) × E[Z | Z > c] × price — follows Rossi & Deutsch's formulation.)
• Sinclair, A.J., Blackwell, G.H. (2002). Applied Mineral Inventory Estimation. Cambridge University Press. (NI 43-101 / JORC perspective on mineral inventory. Chapter 4 on sample compositing and declustering as preliminary statistical treatments before any spatial estimation step. §2.4's protocol-and-disclosure framing follows Sinclair & Blackwell's qualified-person checklist.)
• Deutsch, C.V. (2002). Geostatistical Reservoir Modeling. Oxford University Press. (Mining and petroleum hybrid treatment. Earlier edition of the Pyrcz & Deutsch 2014 text. The same cascade argument is developed in Chapter 4 with worked numerical examples that informed §2.4's worked example.)
• Bourgault, G. (1997). Spatial declustering weights. Mathematical Geology, 29(2), 277–290. (The multi-cell-size averaging refinement. §2.4 references Bourgault as one of the defensible declustering choices in the sensitivity-sweep protocol — running the calculation under several reasonable declustering methods.)
• Pyrcz, M.J., Strebelle, S., Deutsch, C.V. (2005). Multiple-point geostatistics for stochastic modeling of clastic reservoirs. Mathematical Geology, 37(1), 1–34. (The multi-method-critique argument that §2.4's "report the range" protocol extends from MPS down to declustering. Run multiple defensible methods and disclose the spread as model uncertainty.)