Support and scale: the volume problem

Every geostatistical measurement has a SUPPORT: the volume, area, or footprint of the measurement device or aggregation unit. Core plugs (~10 cm³), well logs (vertical 1 ft segments), seismic samples (CMP gathers ~50 m horizontal), 3D grid cells (50 × 50 × 5 m) — all different supports. The same property has DIFFERENT STATISTICS at different supports, and ignoring this is one of the great sources of error in applied geostatistics.

Why support matters: variance reduces with averaging

For an arithmetic average over a block $V$ containing $k^2$ points:

Krige's relation: $\sigma^2(V) = \sigma^2(v) - \bar\gamma(V, V)$. The block variance is the point variance minus the AVERAGE within-block semivariance $\bar\gamma(V, V)$. For uncorrelated point data, the reduction is exactly $1/k^2$. For correlated data, the reduction is LESS because neighbouring points are partly redundant.

Why MEANS preserve but quantiles don't

E[Z(V)] = E[Z(s)] — mean is preserved under block averaging. The HISTOGRAM at block support, however, is COMPRESSED (lower variance). High percentiles drop; low percentiles rise. Block P10 > point P10; block P90 < point P90.

This matters enormously in mining: a deposit's economically-mineable fraction is determined by what percentage of grade exceeds a cutoff. The point-support histogram (cores) and the block-support histogram (mining selective-mining-unit) give different fractions. Using point-support histogram to plan block-support mining → SYSTEMATIC OVER-ESTIMATION of recoverable reserves.

The dispersion variance and Krige's formula

For nested supports v ⊂ V, Krige's additivity:

(simplified for the equal-weight case). Practical: dispersion variance of point support within a block depends on the variogram structure within that block.

Practical implications

• If you measure on cores (10 cm support) and want to model 50 × 50 × 5 m blocks, you must REGULARISE the variogram from core to block support before kriging. Tools: average-variogram γ̄(V, V).
• If you decision-relevant statistic is P90 at the decision support (e.g., per-well drainage area), you cannot estimate it from point-support histograms — you need block-support realisations (SGS at the block scale).
• If you mix supports (cores + well logs + seismic-derived attributes), each has different support; cosimulation or change-of-support corrections are needed.

Try it

• Default range = 4. The 1×1 (point) panel shows the full-resolution field with variance ≈ 2.25 (σ = 1.5). The 8×8 panel averages 64 cells per block. Variance is reduced — note the ratio in the readout.
• Set range = 1 (nearly-uncorrelated). The point-to-8×8 variance ratio should approach 64 (the uncorrelated limit). Each 8×8 block averages 64 nearly-independent samples → variance drops by ~64×.
• Set range = 16 (highly correlated). The variance ratio is much smaller, perhaps 5-10. Heavily-correlated neighbours don't add independent information; block averaging only modestly reduces variance.
• Look at the histogram shapes. The point-support histogram is widest; each coarser support gives a tighter histogram centred at the same mean. The tails compress markedly — high and low percentiles drift toward the mean.
• Consider: if a regulator wants the "P10 grade" for a mining-grade decision, do you want the point P10 or the block P10? The widget shows they're different. Match the support to the decision.

A mining company has core data with mean grade 1.8 g/t and grade SD 0.6. Their selective-mining unit is 5 m × 5 m × 2 m. Will their SMU block-support distribution have higher or lower variance than the core distribution? And what about the FRACTION of SMU blocks exceeding the 2.0 g/t cutoff — higher or lower than the core fraction?

What you now know

Support is the volume / footprint of the measurement and the decision unit. Block-support variance is less than point-support variance; tails compress. Krige's relation quantifies the reduction. Mean is preserved; quantiles aren't. Decision relevance: match the estimation support to the decision support. Section §0.4 turns to coordinate systems and spatial reference, the technical infrastructure underlying support analysis.

Mark section complete →

References

• Krige, D.G. (1951). "A statistical approach to some basic mine valuation problems on the Witwatersrand." J. Chem. Metall. Min. Soc. South Africa 52(6), 119-139. (The eponymous block-vs-point insight, mining context.)
• Journel, A.G., Huijbregts, C.J. (1978). Mining Geostatistics. Academic Press. (Chapter 1 — support discussion.)
• Cressie, N. (1993). Statistics for Spatial Data. Wiley.
• Gotway, C.A., Young, L.J. (2002). "Combining incompatible spatial data." JASA 97(458), 632-648. (Change-of-support methods.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford. (Chapter 8 — block kriging and change of support.)