From global statistics to local: blocks and panels

§§1.1–1.4 built the toolkit for the GLOBAL univariate distribution of a geo-dataset: histograms and CDFs (§1.1), the normal-score transform that gives a clean Gaussian working variable (§1.2), declustering to fix the histogram's SHAPE under spatially-biased sampling (§1.3), and robust descriptive statistics for the OUTLIERS that wreck classical mean and variance (§1.4). Every one of those tools operated on the WHOLE study area at once. The output was a single histogram, a single mean, a single variance, characterising the spatial domain as a featureless statistical population.

But mining reserves are extracted block-by-block. Reservoir fluid flow is simulated cell-by-cell on a finite-difference grid. Environmental remediation decisions are made one parcel at a time. The estimate a geostatistical workflow has to deliver is almost never "the global mean grade of the deposit"; it is "the grade of THIS 10×10×5 m block, given everything we know about it." That move — from GLOBAL to LOCAL — is the conceptual pivot of §1.5, and it brings with it a precise statistical consequence: BLOCK statistics are not the same as POINT statistics, even when the spatial population is the same.

This section makes the move explicit. We define LOCAL statistics over finite-volume blocks, derive the SUPPORT EFFECT (the volume-variance relationship) as the central change-of-support phenomenon, write Krige's additivity relation that conserves total variance as the partition refines, and run two widgets that make the support effect visible. Part 3 then puts the spatial-correlation machinery (the variogram) under the hood; Part 5 turns it into block kriging. §1.5 is the conceptual scaffold both rest on.

Three industries, three reasons local matters

• Mining: reserves are block-grade-driven. A polymetallic deposit may have a point-sample histogram spanning 0.1–50 g/t Au, with a global mean of 4 g/t. But ore is extracted in SELECTIVE MINING UNITS (SMUs) — typically 10×10×5 m blocks ranging from a few hundred to a few thousand tonnes each. The cutoff grade for the mining plan (say 1.5 g/t Au) is applied to the BLOCK grade, not to the point sample. A 1.5 g/t block can sit on top of point samples ranging from 0.2 g/t to 30 g/t — what matters is the AVERAGE of all the point grades the block contains. The grade-tonnage curve (tonnes of ore above any cutoff and the mean grade of that ore) computed from POINT samples is dramatically different from the curve computed from BLOCKS: the block curve is SMOOTHER (no extreme high or low values), has lower mean grade at any high cutoff, and reports more tonnage in the medium-grade range. Getting this right is worth hundreds of millions of dollars on a real deposit.
• Petroleum: reservoir simulation grids are finite cells. A reservoir simulation grid block is 50–200 m horizontal and 1–10 m vertical — a single cell may average 10⁵–10⁶ m³ of rock. The porosity, permeability, and saturation values fed to the flow simulator at each cell are BLOCK averages of properties that vary at the scale of individual depositional layers (sub-metre vertical) and pore-throat networks (sub-millimetre). A point porosity measurement from a core plug is 1 cm³; the cell it lives in is 10¹¹ times larger. Reporting "the porosity of cell (i,j,k)" without acknowledging the block-vs-point support gap is one of the most common errors in reservoir characterisation. Block kriging (Part 5) is the workflow that turns point measurements into block estimates honestly.
• Environmental: remediation is parcel-by-parcel. A contaminated-soil survey samples discrete points (drill cuttings, surface scrapes) on a roughly regular grid. The regulator's cleanup threshold applies to LOTS or PARCELS — the spatial unit at which property ownership and remediation contracts operate. A lot may have point samples ranging from clean to ten times the threshold. What matters is the LOT-AVERAGE concentration (or, for some contaminants, the LOT-MAX) — not the histogram of point values. Decisions made on the point histogram alone are systematically wrong about which lots need cleanup.

The common pattern: the decision is made at a SPATIAL UNIT of finite volume, the data are at a different (almost always much smaller) volume, and the move from point statistics to block statistics is a statistical operation in its own right. Globally-correct statistics need to be transformed into locally-correct statistics before they enter the decision.

The support effect: same mean, smaller variance

The central phenomenon goes by two names. The mining literature calls it the volume effect; the geostatistics literature calls it the support effect. They are the same thing: as you average a spatial variable over a larger volume, the resulting block averages have the same MEAN but a SMALLER VARIANCE than the underlying point values.

The mean part is easy. Partition the domain into N blocks of equal area, each containing the same number k of point cells (assume an exact partition for the moment). The block-average Z̄_v at block v is

Averaging the block-averages over all N blocks gives

i.e. the average of block averages equals the global point average. The mean is INVARIANT under block-averaging of a complete partition.

The variance part is where the support effect lives. If the k point samples inside each block were INDEPENDENT draws from the same distribution with variance σ², the block average would have variance σ²/k. The block-distribution variance — the variance computed across the N block averages — would also be σ²/k (for large N, ignoring finite-sample subtleties). The ratio

is the independent-samples bound. It is the variance reduction you would get from averaging k independent measurements of a random quantity — and it is exactly the formula behind "the standard error of the mean shrinks as 1/√n" in elementary statistics.

For SPATIALLY CORRELATED data — the realistic case — adjacent point cells inside a block are NOT independent. They carry overlapping information: knowing the value at one cell narrows down the value at the next. As a result, averaging k correlated values shrinks the variance by LESS than 1/k. Equivalently, the empirical variance ratio σ²_block / σ²_point lies ABOVE the 1/k bound, sometimes well above. The size of the gap is a direct measure of how strongly spatially correlated the field is at the block scale.

The widget instantiates a 96 × 96 synthetic field (the same field shape used by the §1.3 declustering widgets) and lets the reader choose a block edge length from 1 to 32 point cells. At edge length 1, "block" = "point cell" and the block-support histogram coincides with the point-support histogram; the variance ratio is exactly 1. As the edge grows, the block-support histogram NARROWS — its variance shrinks — while its mean stays locked to the point-support mean. The empirical ratio is displayed alongside the 1/k independent-samples bound, and the verdict line tells the reader whether the field at that block size is behaving close to the independent regime (small blocks compared to the correlation length) or far from it (block edge comparable to the correlation length, where the support effect is dramatic).

What to read from the experiment: at small block edge (2–4 cells), variance reduction is small and the block histogram is barely distinguishable from the point histogram. At medium block edge (8–16 cells, comparable to the field's correlation length), the block histogram is visibly narrower; the variance ratio drops to 0.2–0.4. At very large block edge (24+ cells), the block histogram is a tight spike near the global mean — every block now spans multiple correlation lengths and the support effect dominates.

Krige's additivity relation: variance is conserved across change of support

The relationship between point variance, block-distribution variance, and within-block variance is one of the foundational identities of geostatistics, first proved by Daniel Krige in his 1951 South African gold-mine paper. Define

• D²(point | domain): the variance of point values across the whole domain. This is the global σ² from §1.1, computed on the point support.
• D²(block | domain): the variance of BLOCK AVERAGES across the domain. The block-distribution variance — what the §1.5 widget computes.
• D²(point | block): the variance of point values WITHIN a block, averaged across blocks. The mean within-block point variance.

Krige's additivity relation states

Total point variance equals between-block variance plus mean within-block variance. The relation is EXACT for an exhaustively-sampled domain with an exact partition (it is the law of total variance specialised to a spatial partition). With a finite sample or a ragged partition, the equality is approximate up to boundary effects.

Reading the relation as a CONSERVATION law clarifies the support effect. The total point variance is fixed by the population. When we change the support — make blocks larger — we redistribute that fixed variance budget between two pots: BETWEEN-block (which shrinks because larger blocks all look more like the global mean) and WITHIN-block (which grows because larger blocks now span more of the population's variability). At the limit b = 1 cell, between-block holds everything and within-block is zero. At the limit b = whole domain, between-block is zero (one block = global mean, no spread) and within-block holds everything.

The second widget makes the partition visible. The sweep chart on the left plots BETWEEN-block variance (blue) and WITHIN-block variance (orange) as functions of block edge length from 1 to 32 point cells; the total variance is drawn as a flat dashed reference line and the sum (between + within) is plotted as a dashed green line that overlays the reference — confirming additivity. The stacked bar on the right shows the same partition at the currently-selected block size as a single column: a small orange band of within-block at the bottom and a large blue stack of between-block above, both summing to the dashed total line. As the slider moves right, blue shrinks and orange grows; the total stays flat.

The transition between regimes is precise. At block edge ≈ 1, the bar is essentially all blue (between-block ≈ total, within-block ≈ 0). At block edge ≈ correlation length (around 8–10 cells for this field), between and within are comparable. At very large block edge, the bar is essentially all orange (within-block dominates, between-block is small).

This is the operational meaning of "change of support is variance-preserving in total but redistributes between components." Part 3's variogram will turn this same partition into an ANALYTIC formula: the variogram γ(h) gives the half-squared-difference of point values at lag h, and integrating γ over a block size gives D²(point | block) directly, without resorting to the explicit grid-tile computation done here. For now, the experimental partition is the lesson.

Block, panel, cell: the geo-domain vocabulary

The three industries that this textbook serves use distinct terms for "spatial unit of finite volume at which decisions are made." Worth committing them to memory.

• Mining: SMU and panel. The SELECTIVE MINING UNIT (SMU) is the smallest block size at which an ore-vs-waste decision is operationally made — usually constrained by the mining method (a 10×10×5 m block for open-pit, smaller for selective underground mining, larger for bulk-mining methods). The PANEL is a larger volume (typically 25×25×10 m or bigger) used for geological domaining and resource categorisation. SMU statistics drive the grade-tonnage curve and reserves; panel statistics drive geological zone classification and mine planning at the level of stope geometry. Both are computed from the same underlying drill-hole samples but operate at different supports.
• Petroleum: simulation cell. The reservoir model is built on a 3D grid of CELLS — typically 50–200 m horizontal, 1–10 m vertical for a high-resolution model; coarser (200–500 m horizontal) for a basin-scale model. Each cell carries a single porosity, permeability tensor, and saturation per phase. The cell is the support at which the flow simulator operates, and the inputs need to be honest block-averages of the underlying lithology and petrophysics.
• Environmental: lot, parcel, or grid cell. The remediation unit is whatever spatial unit ownership, regulatory monitoring, or remediation contracts operate on. For an industrial site this is typically a 10–100 m lot; for a regional groundwater survey it may be a 1 km grid cell.

What unites them is the support-effect formalism: the point sample is one thing, the decision unit is another, and the geostatistical workflow must move from one to the other honestly.

How §1.5 ties back to §§1.1–1.4

Every Part 1 tool acted on a particular support. Now we can say which.

• §1.1 (histograms, CDFs, Q-Q plots). Computed on the SAMPLE — which for typical drill-hole / core / well data is the POINT support (or close to it: a core plug is ~1 cm³, much smaller than any decision unit). The histograms the §1.1 widgets show are POINT histograms. The block histogram, computed from the same field, is a different and NARROWER distribution.
• §1.2 (normal-score transform). Applied to the SAMPLE — point support. The N-scored values are Gaussian-margin at the POINT support. Back-transforming kriged or simulated values back to original units is also a point-support operation; if a downstream task wants BLOCK averages of the back-transformed variable, the right workflow is to back-transform the SIMULATIONS at point support, average them to block support, and report the block-mean statistics. Doing block-kriged-then-back-transformed is statistically wrong (the back-transform is non-linear, so averaging then back-transforming is not the same as back-transforming then averaging). This subtlety is properly developed in Part 7.
• §1.3 (declustering). Corrects the sample HISTOGRAM's shape — at whatever support the samples sit on. For typical drill-hole data this is point support, and the declustered histogram is the best estimate of the POINT-SUPPORT histogram of the population. Converting that to a BLOCK-SUPPORT histogram is exactly the §1.5 problem.
• §1.4 (robust statistics). Apply to any support — but they are most useful at POINT support, where ultra-rich drillholes, fracture-corridor spikes, and contamination hot-spots arrive raw. The very act of block-averaging is itself a (heavy-handed) robustness operation: block averages over many points dilute outliers automatically. A block at the SMU size that contains one 100×-bulk drillhole and 99 normal samples reports a block average roughly 2× the bulk (the outlier is averaged in but does not dominate). That observation is also the warning: block-averaging HIDES outliers but does not REMOVE them — the high-grade material may not extend volumetrically, and treating the block average as if every point inside it were near the average can substantially overestimate the recoverable resource.

Preview: grade-tonnage curves at different supports

One concrete consequence of the support effect, deferred to Part 5 / Part 10 but worth previewing because it is the single biggest commercial driver in mining geostatistics: the GRADE-TONNAGE CURVE depends critically on support.

The grade-tonnage curve is two related curves plotted against cutoff grade g_c: TONNAGE above cutoff (the tonnes of material with grade ≥ g_c) and MEAN GRADE above cutoff (the average grade of that material). Both are needed for every mining reserve calculation; both depend on the histogram of grades at the support at which the cutoff is applied.

For ANY cutoff above the global mean, the BLOCK-support grade-tonnage curve reports LESS TONNAGE and LOWER MEAN GRADE above cutoff than the POINT-support curve. The reason is the support-effect smoothing: at point support, isolated high-grade samples push their bin's tail far above the cutoff; at block support, those same samples are averaged with their neighbours, and the resulting block sits closer to the mean. The block histogram's upper tail is SHORTER than the point histogram's upper tail.

The COMMERCIAL implication: reporting reserves on POINT samples systematically over-estimates the recoverable ore at any high cutoff. The reserves that actually come out of the ground are at BLOCK support — SMU support, specifically. Reporting at the wrong support is the single most common cause of a mine producing less ore (and less grade) than its feasibility study promised. The full grade-tonnage / block-kriging workflow is developed in Part 5; the support-effect machinery established here is what makes it work.

What §1.5 deliberately leaves to later sections

This section sets up the conceptual machinery for change of support. Three natural extensions belong to later sections and we flag them honestly so the reader knows what comes next.

• §3.x — the variogram is the spatial-correlation framework. §1.5's within-block variance D²(point | block) was computed by gridding the field and averaging. The variogram γ(h) gives the same quantity ANALYTICALLY: D²(point | block) = (1 / V²) ∫∫_v γ(s − s') ds ds'. The classical formulae for block dispersion variance (Journel-Huijbregts 1978 §5) all reduce to integrals of γ over block geometries. Building those integrals is Part 3 / Part 4 work.
• §5.x — block kriging. §1.5 has shown that block statistics differ from point statistics; it has NOT shown how to ESTIMATE a single block's value from sparse point samples. That is block kriging's job. Block kriging takes a set of point samples plus a variogram model and returns the best linear unbiased estimate of the block average — and crucially, it accounts for the support effect by design.
• Part 5 / Part 10 — change-of-support transforms. Sometimes you do not want a single block estimate but a HISTOGRAM of block grades over the domain (for grade-tonnage). The classical change-of-support methods — affine correction, indirect lognormal correction, discrete Gaussian model (Hermitian, Anamorphosed) — provide ways to derive the block-grade histogram analytically from the point-grade histogram plus the variogram. These are developed properly in Part 5 and applied in the mining capstone in Part 10. The §1.5 widgets compute the same quantities numerically from a known field; the analytical machinery comes later.

Two cautions worth flagging early.

• The mean is preserved only in expectation, not always sample-by-sample. For a complete partition with equal-sized blocks of a stationary field, the mean of block averages equals the mean of point samples. For irregular blocks, partial coverage, or non-stationary fields, the equality is approximate. Always check your specific case.
• Block-averaging is non-linear under non-linear transforms. If you N-score the data (§1.2) and then block-average the N-scored values, you do NOT get the same answer as block-averaging the raw values and then N-scoring the block averages. The N-score is non-linear; commuting it with averaging gives different distributions. The §7 simulation workflow handles this by simulating at POINT support and averaging to block support at the end — never the other way around.

Try it

• In the block-stats-explorer widget, set block edge = 1. Confirm that the green block-support histogram exactly coincides with the gray point-support histogram and the variance ratio reads 1.000. Now slide to edge = 4. By how much (as a percentage of the point variance) does the block variance shrink? Compare this number to the independent-samples bound 1/k = 1/16 ≈ 0.063. Which is larger? Why?
• Continue sliding to edge = 8 (k = 64), 16 (k = 256), 24 (k = 576). Plot or tabulate the empirical ratio against the independent-samples ratio 1/k. At which block edge does the empirical ratio cross 0.5? At which does it cross 0.1? The crossings give you a numerical handle on the field's effective correlation length.
• In the dispersion-variance widget at block edge = 1, what is the within-block variance? Confirm it is zero (a 1×1 block has no internal variation). Now slide to edge = 8. Read between-block and within-block. Verify that their sum equals the total (within a small residual from ragged boundary blocks). At what block edge are the two components equal? That edge is approximately the field's effective correlation length.
• Without coding: a mining geologist has 10,000 drill-hole composites with a global mean Au grade of 4.0 g/t and a point-support variance of 36.0 (g/t)². The mining plan uses 10×10×5 m SMUs, and a variogram model predicts that the average within-block point variance for that SMU size is 24.0 (g/t)². What is the BETWEEN-SMU variance? What does that imply for the standard deviation of SMU grades — i.e. roughly what range of SMU grades will the mine actually encounter?
• Without coding: a regulator sets the contamination cleanup threshold at 10 mg/L for a remediation lot, applied to the LOT-AVERAGE concentration. The survey reports point measurements at concentrations ranging from 0.1 to 35 mg/L, with a global mean of 8 mg/L. If the lot is large (containing many point samples), is the lot-average likely to be ABOVE or BELOW any single point measurement that the regulator sees? What error would a regulator make by applying the 10 mg/L threshold to the point measurements directly?

Pause and reflect: the support effect is sometimes described as "averaging smooths everything out." That phrasing is misleading. Block averages do NOT bring every block toward the global mean uniformly — high-mean blocks stay high, low-mean blocks stay low. What averaging removes is the EXTREME values from the BLOCK histogram. The high tail of the point histogram (the rare ultra-rich drillholes) gets smeared into nearby blocks; those blocks end up moderately above the mean, not at the point-sample extreme. In a mining context this is good news for stable production but bad news for high-grade recovery. Why? Because the high-grade BLOCKS the mine wants to find are necessarily fewer (and lower-grade) than the high-grade POINT samples in the data. The grade-tonnage curve at SMU support is the right tool for this question.

What you now know — and what Part 1 has built

You have the LOCAL-vs-GLOBAL distinction and three industries' worth of vocabulary for the spatial decision units involved (SMU, panel, cell, lot). You have the SUPPORT EFFECT (block averages have the same mean as point samples but smaller variance) and its central operational consequence: BLOCK histograms are NARROWER than POINT histograms, and grade-tonnage curves at block support differ from grade-tonnage curves at point support for any cutoff above the mean. You have the INDEPENDENT-SAMPLES bound σ²_block / σ²_point ≥ 1/k and you have seen, in the §1.5 widget, that real spatially correlated fields exceed that bound by a factor that scales with the block size relative to the correlation length.

You have Krige's ADDITIVITY RELATION D²(point | domain) = D²(block | domain) + D²(point | block) as the conservation law that organises change of support: total variance is fixed, it just redistributes between between-block and within-block components as the partition refines. You have seen the partition swing from "all between" at point support to "all within" at full-domain support, with the cross-over at the field's effective correlation length.

Part 1 is now complete. Across five sections you have built the full toolkit for univariate description of a geo-dataset: the histogram and CDF (§1.1), the Gaussian working variable (§1.2), the declustered histogram (§1.3), robust summaries that survive contamination (§1.4), and the block-support change-of-support framework (§1.5). What every Part 1 tool LACKED was the spatial-correlation framework — the variogram — that converts point statistics to block statistics analytically rather than by explicit gridding. Part 3 supplies it; this section is the conceptual bridge.

The next big move is from one variable seen as a marginal distribution to one variable seen as a SPATIAL PAIR — the variogram's view. Part 2 takes a detour into the practical machinery of declustering (the methods §1.3 motivated). Part 3 then introduces the variogram. Everything Part 3 and beyond does ultimately answers the question: given a point-support variogram, what is the block-support story? §1.5 is the framework that asks that question; the rest of the textbook is the answer.

Mark section complete →

References

• Krige, D.G. (1951). A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa, 52, 119–139. (The foundational paper of mining geostatistics; the additivity relation D² = D²_between + D²_within is implicit in Krige's analysis of the variance reduction with increasing block size on Witwatersrand gold-mine data — the first quantitative demonstration of the volume effect.)
• Matheron, G. (1971). The Theory of Regionalized Variables and Its Applications. Les Cahiers du Centre de Morphologie Mathématique 5, Fontainebleau. (Matheron's formal theory. Defines the support, the dispersion variance D²(v | V), and proves the additivity relation as a consequence of stationarity. This is the technical foundation that Part 3 builds on.)
• Journel, A.G., Huijbregts, C.J. (1978). Mining Geostatistics. Academic Press. (Chapter 5 develops dispersion variance and change of support in full mining-applied form; the formulae for D²(v | V) in terms of the variogram, and the practical change-of-support transforms — affine correction, indirect lognormal — are tabulated here. The canonical reference for §1.5's material.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press. (§5.5 on block estimation and §5.6 on change of support; develops the volume-variance relationship with worked examples on environmental and mining data. The most readable modern textbook treatment of this material.)
• Isaaks, E.H., Srivastava, R.M. (1989). An Introduction to Applied Geostatistics. Oxford University Press. (§19 on block kriging makes the practical move from point to block support; the support effect is illustrated with the Walker Lake dataset and grade-tonnage examples.)
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty, 2nd ed. Wiley. (Chapter 2 on dispersion variance and Chapter 6 on change of support; the modern reference for the discrete Gaussian model and its anamorphic generalisations.)
• Rossi, M.E., Deutsch, C.V. (2014). Mineral Resource Estimation. Springer. (Practical mining-applied treatment; Chapter 7 on grade-tonnage curves and Chapter 9 on change-of-support transforms. The applied mining counterpart to Journel-Huijbregts 1978.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford University Press. (Petroleum-applied counterpart; §4 on simulation cell support, §10 on upscaling of petrophysical properties to flow-simulation cells.)