Calibration of the kriging variance

§6.1 introduced LOO-CV and §6.2 the accuracy plot. Together they DIAGNOSE kriging-variance miscalibration. §6.3 develops the iterative refinement process: given diagnostic feedback, how do you adjust the variogram to achieve well-calibrated uncertainty?

Why calibration matters

The kriging variance $\sigma_K^2(x_0)$ is used downstream in:

• Confidence intervals: $\hat{z}(x_0) \pm 1.96 \sqrt{\sigma_K^2(x_0)}$ for prediction intervals.
• Conditional simulation: SGS (§7.3) draws from N($\hat{z}(x_0)$, $\sigma_K^2(x_0)$) at each node.
• Risk assessment: P(z > cutoff) for resource estimation, contamination analysis, etc.
• Sampling design: identifying high-uncertainty zones needing additional sampling.

If kriging variance is MISCALIBRATED, every downstream uncertainty quantification is biased. Over-stated → too-wide intervals, over-conservative decisions, wasted resources. Under-stated → too-narrow intervals, false confidence, costly mistakes.

Calibration via iterative refinement

Standard workflow:

• Fit an INITIAL variogram visually (experimental variogram + manual model fit).
• Compute LOO-CV variance ratio.
• Compute accuracy plot.
• If ratio > 1.2 or accuracy curve below diagonal: variogram parameters too restrictive. Increase range, sill, or nugget.
• If ratio < 0.8 or accuracy curve above diagonal: too generous. Decrease parameters.
• Iterate until BOTH ratio ≈ 1 AND accuracy curve hugs diagonal.
• Document final variogram + calibration in the deployment report.

Parameter sensitivities

• Increasing RANGE generally INCREASES kriging variance (data farther away contribute less); helps if model is over-confident.
• Increasing SILL increases the kriging variance everywhere (limit at infinity); same direction.
• Increasing NUGGET adds a fixed component to kriging variance at unsampled locations; also tends to SMOOTH predictions.

The interaction matters: increasing nugget DECREASES the apparent spatial correlation (since the nugget represents un-correlated variation). This may simultaneously calibrate the kriging variance UPWARD while reducing prediction accuracy. There's no universal best refinement direction — depends on diagnostic feedback.

Joint refinement of variogram model AND parameters

Sometimes the variogram MODEL is wrong (e.g., assumed spherical when truth is exponential). If parameter refinement plateaus without good calibration, try a different parametric family:

• Spherical: typical for hard-boundary phenomena (mineral grades, soil properties).
• Exponential: smooth decline; common in environmental data.
• Gaussian: highly smooth field; tends to oversmooth predictions.
• Nested combinations: multiple structures captured by combining models. Often necessary for real-world data.

Modern software (R's gstat fit.variogram with multiple model candidates) automates the model selection step within an iterative variogram-fitting workflow.

Documentation requirements

A complete kriging deployment report includes:

• Final variogram model + parameters.
• LOO-CV variance ratio with associated CI.
• Accuracy plot with predicted vs actual coverage at multiple levels.
• Statement of model SELECTION process (which models considered, why this one chosen).
• Sensitivity analysis: how do results change if variogram parameters move within their uncertainty?

Modern resource-estimation and reservoir-modeling reports include this entire suite.

Try it

• Default: assumed = true variogram (5, 1.0, 0.1). Variance ratio ≈ 1, tail coverage matches predicted (~95% at 95% level).
• Crank range to 15 (too long). Kriging is now OVER-confident in extending influence. Variance ratio drops below 1; 95% tail coverage may exceed 95%. Over-conservative.
• Drop range to 1.5 (too short). Kriging believes points are nearly independent. Variance ratio explodes (under-stated kvar); 95% tail coverage drops below 95%. Over-confident.
• Increase nugget to 0.5. Now kriging variance is artificially inflated; ratio drops below 1; tail coverage approaches 100%. Smoothing also affects predictions.
• Find the sweet spot manually by tweaking all three parameters. The "Reset to truth" button returns to optimal calibration as a benchmark.

A kriging deployment shows LOO-CV variance ratio = 1.0 but tail coverage at 95% is only 88%. What might be going wrong, and what variogram refinement could help?

What you now know

Variogram calibration is iterative: tune range, sill, nugget guided by LOO-CV variance ratio + accuracy plot. Modern workflow combines visual fit + parameter tuning + model selection. Document the final variogram + diagnostics in the deployment report. §6.4 next: jackknife and split-sample validation, complementary to LOO-CV.

Mark section complete →

References

• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford.
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford.
• Cressie, N. (1993). Statistics for Spatial Data. Wiley. (Theoretical foundations of kriging variance.)
• Wackernagel, H. (2003). Multivariate Geostatistics, 3rd ed. Springer.
• Pebesma, E.J. (2004). "Multivariable geostatistics in S: the gstat package." Computers & Geosciences 30, 683–691. (R gstat: variogram fitting + LOO-CV.)