Block kriging and change-of-support

§§5.1–5.4 estimated point values: $Z^*(\mathbf{s}_0)$. But real decisions are usually at the block support. A mine extracts ore by SMU (selective mining unit, typically 10×10×5 m). A reservoir simulator solves flow on grid cells (50–200 m horizontal). A remediation crew treats lots, not points. §5.5 is how to estimate block averages directly — and why doing so changes the answer.

The block-kriging problem

Instead of estimating $Z(\mathbf{s}_0)$ at a single point, estimate

— the average value of $Z$ over a block $V$ of volume $|V|$. The block-kriging estimator is again a linear combination of sample values: $\hat{\bar{Z}}$V = \sum{i=1}^N w_i Z(\mathbf{s}_i).

The block-kriging system

The kriging system has the SAME structure as §5.2's ordinary kriging system — only the right-hand side changes. Where point kriging uses $C(\mathbf{s}_i, \mathbf{s}_0)$, block kriging uses the average covariance from sample $\mathbf{s}_i$ to every point inside $V$:

For ordinary block kriging the augmented system is:

where $\mathbf{K}_{ij} = C(\mathbf{s}_i, \mathbf{s}$j)Kij​=C(si​,sj​) (unchanged) and $\bar{k}${V,i} = \bar{C}(\mathbf{s}_i, V).

Computing the block-to-point covariance

The integral $\bar{C}(\mathbf{s}_i, V)$ is approximated numerically. The standard method is to discretise $V$ into $M$ points $\mathbf{u}_1, \ldots, \mathbf{u}_M$ (Gauss-Legendre grid, Monte Carlo, or a regular $n \times n$ subgrid) and average:

Typical practice: 4×4 or 5×5 grid in 2D, 4×4×2 in 3D. Increase if blocks are large compared to the variogram range.

The block-kriging variance

The block-kriging variance includes the WITHIN-BLOCK variability:

where $\bar{C}(V, V) = \frac{1}{|V|^2} \int_V \int_V C(\mathbf{s}, \mathbf{s}') , d\mathbf{s} , d\mathbf{s}'$ is the average within-block covariance. This term is SMALLER than $C(0) = \sigma^2$ — averaging within $V$ removes the high-frequency variability. Consequently, block-kriging variance is typically SMALLER than point-kriging variance at the same location.

Krige's additivity revisited

The relationship from §1.5 in compact form:

As block size $|V|$ grows: between-block variance shrinks (block averages converge toward the global mean); within-block variance grows (more variability is "averaged out" inside each block). Total variance is conserved.

The support effect on grade-tonnage curves

This is where block kriging matters in mining decisions. Apply a cutoff grade $g_c$ to point samples: the grade-tonnage curve is computed from point values. Apply the same cutoff to block averages: the curve is DIFFERENT.

• POINT support: more extreme values (both high and low). At a high cutoff $g_c$, a fraction of points are above — some by a lot.
• BLOCK support: smoothed values (extremes averaged out). At the same $g_c$, fewer blocks are above, and the mean grade above the cutoff is closer to $g_c$.

For a mining reserve estimate: tonnes above cutoff × mean grade above cutoff × recovery factor. Using POINT support typically OVERESTIMATES tonnage (selection bias for high values); using BLOCK support is more realistic because mining actually happens at SMU support, not point support. The "support effect on tonnage" is a multi-million-dollar issue in real mining valuations.

Direct vs indirect block kriging

Two ways to get block estimates:

• Direct: solve one block-kriging system using $\bar{\mathbf{k}}_V$. Best for small numbers of blocks.
• Indirect (point-then-average): krige each discretisation point $\mathbf{u}_m$ inside $V$ separately, then average. Equivalent to direct block kriging when the discretisation is the same and the kriging neighbourhood is the same.

Direct is computationally cheaper for many blocks (one system per block vs M systems). Indirect is conceptually clearer for understanding what's happening.

Try it

• In the block-kriging widget, slide block size from "very small" (point-like) to "very large". Watch the kriged map: smaller blocks resemble point kriging (more extreme values); larger blocks smooth out the variability. Verify that block variance shrinks with size.
• For a fixed sample configuration, increase block size and watch the kriging variance map change. Does it become more uniform across the field? Why? (Larger blocks average over more variability; the within-block term grows; the between-location differences shrink.)
• In the grade-tonnage widget, pick a cutoff grade. Read the tonnes-above-cutoff for point support vs block support. Estimate the dollar-impact of using the wrong support (10% deposit value swing is realistic).
• Increase the block size in the grade-tonnage widget. The cutoff at which 50% of blocks are above shifts toward the mean. Why? (As $V$ grows, block averages concentrate toward the mean; very few blocks reach extreme values.)
• Compare block kriging variance to point kriging variance at the SAME centroid location. Which is smaller? Why? (Block variance is smaller because $\bar{C}(V, V) < C(0)$ — within-block averaging removes high-frequency variance.)

If point variance is 1.0 and a block of size V has internal point-point covariance of 0.6 on average, what is the within-block variance D²(point|V)? Hint: D² = average variance - average covariance.

What you now know

Block kriging estimates the average over a finite-volume support directly, using block-to-point average covariances. The kriging system is structurally identical to point kriging; only the RHS uses average-over-V covariances computed by discretisation. Block kriging variance is smaller than point variance because within-block averaging removes high-frequency variability — quantified by Krige's additivity. Crucially, grade-tonnage curves at block support differ MATERIALLY from point support, with real dollar implications for mining reserves. §5.6 turns to the kriging NEIGHBOURHOOD — how to pick which samples to include in the kriging system — and §5.7 introduces cokriging with secondary data.

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, 119–139. (The historical origin of "kriging" and the support-effect framework.)
• Journel, A.G., Huijbregts, C.J. (1978). Mining Geostatistics. London: Academic Press. (Chapter 5 — the comprehensive treatment of dispersion variance and change of support. Still the canonical reference 45 years later.)
• Matheron, G. (1971). The Theory of Regionalized Variables and Its Applications. Fontainebleau: Les Cahiers du Centre de Morphologie Mathématique. (Foundational text on regionalised variables; block estimation is a worked example.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. New York: Oxford University Press. (§5.5 develops block kriging with worked numerical examples and the connection to indicator-based methods.)
• Rossi, M.E., Deutsch, C.V. (2014). Mineral Resource Estimation. Berlin: Springer. (Modern industry treatment of block estimation for resource and reserve reporting under JORC / NI 43-101.)