Debiasing checks and conditional bias

Kriging is a MINIMUM-VARIANCE estimator. By construction, this means predictions are SMOOTHER than the data: high values get under-predicted, low values get over-predicted, and the dynamic range is compressed. This smoothing is INTRINSIC to kriging, not a bug. But excessive smoothing can mask real spatial structure and bias downstream uncertainty quantification.

Conditional bias defined

From LOO-CV, regress the actual values on the predictions: $\text{actual} = \alpha + \beta \cdot \text{predicted} + \varepsilon$. Under perfectly unbiased estimation, $\beta = 1$ — predicted and actual move 1:1. Under SMOOTHING, $\beta < 1$: a unit change in predicted corresponds to a less-than-unit change in actual. Predictions are compressed toward the mean.

The complementary metric: SMOOTHING RATIO = SD(predicted) / SD(actual). Equals 1 under unbiased; less than 1 under smoothing.

Why kriging smooths

The kriging objective minimises mean squared error: $\sigma_K^2(x_0) = E[(\hat{z}(x_0) - z(x_0))^2]$. The minimum-MSE estimator is, by definition, the CONDITIONAL EXPECTATION $E[z(x_0) \mid \text{data}]$. Conditional expectations are SMOOTHER than the underlying variable: they average out random variation.

Mathematically: $\text{Var}(\hat{z}) + \sigma_K^2 = \text{Var}(z)$. The total data variance decomposes into the predictor's variance + the kriging variance. Higher kriging variance → smaller predictor variance → more smoothing.

Mitigating smoothing

Three approaches:

• Reduce variogram nugget: smaller nugget = less assumed uncorrelated variability = less smoothing. But CHECK with LOO-CV — too-low nugget can also lead to over-confidence.
• Use conditional simulation (§7.3): SGS realisations restore the full variance of the data — predicted realisations match the empirical SD of observations. The smoothing problem is fundamentally bypassed.
• Use ordinary kriging (OK) with appropriate variogram: simple kriging assumes a known global mean, which can introduce smoothing toward that mean. OK estimates the mean locally and tends to be less smoothed.

When is smoothing acceptable?

For SPATIAL PREDICTION at unsampled locations (the kriging map), some smoothing is unavoidable. Modest slopes (0.85–1.00) are acceptable. Slopes below 0.6–0.7 indicate excessive smoothing that should be addressed via nugget reduction or conditional simulation.

For RESOURCE ESTIMATION applications (mining grades, oil-and-gas), conditional bias has DIRECT financial implications: under-predicted high-grade ore → missed value; over-predicted high-grade ore → development decisions based on wrong reserves. Modern mining-resource reports REQUIRE smoothing diagnostics in regulatory submissions (JORC, NI 43-101, SEC S-K).

The complement: conditional simulation

Kriging produces a SINGLE best estimate. Conditional simulation (§7.3) produces MULTIPLE realisations, each respecting the data AND the variogram. The simulations DON'T SMOOTH — each realisation has the same SD as the observed data. Trade-off: each simulation is "noisy" individually; insight requires averaging over multiple realisations. Modern practice: use kriging for the single-best-estimate map; use simulation for uncertainty quantification.

Try it

• Default: nugget = 0.1, N = 40. The regression slope is around 0.85 — moderate smoothing. The smoothing ratio is around 0.92.
• Increase nugget to 0.6. The slope drops to around 0.5 — strong smoothing. The regression line is much flatter than the diagonal. Predicted range is heavily compressed.
• Reduce nugget to 0. Slope approaches 1 — minimal smoothing. But also check LOO-CV variance ratio: too-low nugget can lead to over-confident kriging variance.
• Increase N to 100. Smoothing pattern persists; conditional bias is a model property, not a sample-size issue.
• Note: even with perfect variogram (nugget = true value), there's still some smoothing — minimum-variance estimators always smooth. Mitigation, not elimination, is the goal.

A mining-resource estimation report shows LOO-CV slope = 0.45 for high-grade ore predictions. Why does this matter for the resource estimate, and what action would you recommend?

What you now know

Kriging smooths by construction. Diagnose via regression slope of actual on predicted from LOO-CV (target = 1) and smoothing ratio (target = 1). Mitigate via nugget reduction or conditional simulation. Modern mining and oil-and-gas resource estimation reports include conditional-bias diagnostics. PART 6 COMPLETE.

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References

• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford. (Comprehensive treatment of smoothing and conditional bias.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford.
• Journel, A.G. (1989). "Fundamentals of geostatistics in five lessons." AGU Short Course in Geology, Vol. 8. (Classical treatment of conditional bias.)
• Rivoirard, J. (1994). Introduction to Disjunctive Kriging and Non-linear Geostatistics. Oxford. (Alternative non-linear methods to bypass smoothing.)
• Rossi, M.E., Deutsch, C.V. (2014). Mineral Resource Estimation. Springer. (Mining-specific treatment of smoothing in resource estimation.)