From kriging to simulation: why one map is never enough

Part 7 — Sequential Gaussian simulation

Learning objectives

Recognise the LIMITS of a single kriging map for uncertainty quantification
Distinguish the BEST-ESTIMATE map (kriging) from the FULL POSTERIOR distribution of maps
State the conditional-simulation principle: realisations honour data + variogram
Recognise that simulation REVERSES SMOOTHING by restoring data variance
Apply the kriging-plus-simulation workflow as modern best practice

A kriging map gives ONE prediction at every unsampled location. It minimises mean squared error — useful for visualisation and area delineation. But it answers only "what is the best estimate?". Many downstream decisions need to answer DIFFERENT questions: "What is the probability that the grade exceeds 1.0 g/t?", "What are the P10 and P90 reserves?", "How risky is this development decision?". These require knowing the FULL POSTERIOR DISTRIBUTION over possible maps — not just one best estimate.

The kriging map is one sample, not the truth

The kriging map $\hat{z}_K(x)$ is the conditional mean $E[z(x) \mid \text{data}]$ . The data also constrain the FULL CONDITIONAL DISTRIBUTION $p(z(x) \mid \text{data})$ . Different draws from this distribution give different maps — all CONSISTENT with the data, but DIFFERENT between data points where uncertainty exists.

Conditional simulation generates these alternative maps. Each realisation:

EXACTLY honours the data — same value at each known point.
HONOURS the variogram — its empirical spatial-correlation structure matches the assumed model.
Has the FULL DATA VARIANCE — unlike the smoothed kriging map.
Represents ONE plausible realisation of the spatial-correlated field.

The collection of realisations (typically 50-500) characterises the spatial uncertainty in a way the kriging map alone cannot.

Why simulation matters for decisions

Resource estimation: P10/P50/P90 reserves come from the ensemble of realisations. Kriging map alone gives only the P50.
Risk assessment: probability that grade exceeds cutoff = fraction of realisations exceeding cutoff at each location.
Optimal sampling design: variance across realisations reveals high-uncertainty zones for additional sampling.
Downstream simulation: e.g., reservoir flow simulation requires permeability realisations; one smoothed map gives systematically biased flow.

The modern workflow

Build a variogram + validate via LOO-CV (Part 6).
Run kriging to get the SINGLE BEST-ESTIMATE map (for visualisation).
Run CONDITIONAL SIMULATION (SGS, §7.3) to get ENSEMBLE of realisations (for uncertainty quantification).
Post-process realisations for decision metrics: P10/P50/P90, conditional probabilities, expected costs.
Document both kriging map and simulation summary in the deployment report.

This is best practice in modern mining-resource estimation, reservoir characterisation, environmental modelling, and any geostat application where uncertainty matters.

Why simulation reverses smoothing

Kriging map = conditional mean (smoothed). Simulation = draw from conditional distribution. The conditional MEAN smooths; individual DRAWS preserve the full variance. Mathematically: Var(draw) = Var(data); Var(kriging map) < Var(data) by an amount equal to the kriging variance. Simulation realisations look "noisy" individually but their ensemble correctly characterises uncertainty.

Try it

Default range = 2.0. The kriging map (red) interpolates smoothly between the 6 data points. The 95% CI band widens between data and narrows at data points.
The 4 simulation realisations (different colours below) all PASS THROUGH the 6 data points but DIFFER between them. Each is a plausible realisation of the spatial field consistent with the data.
Crank range to 5.0 (long correlation). Kriging map smooths more aggressively; simulations also smooth more (longer correlation = more similar nearby).
Drop range to 0.5 (short correlation). Kriging map shows sharper transitions between data points; simulations show much more variability between points — short-correlation means more rapid variation.
Re-simulate (different seed). Same data, same variogram, but new realisations. The data are honoured identically; the unsampled regions vary across realisations — the essence of uncertainty quantification.

A mining-resource estimation report contains only the kriging map (single best estimate). The report quotes P10 = 500 oz/t, P50 = 750 oz/t, P90 = 1000 oz/t. What is wrong with this report?

What you now know

One kriging map answers one question. Multiple simulation realisations answer many uncertainty questions. Modern geostatistical practice: combine BOTH — kriging for visualisation, simulation for uncertainty quantification. §7.2 next: the LU decomposition approach for unconditional simulation, then §7.3 sequential Gaussian simulation (SGS) for conditional simulation.

References

Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation, Chapters 7-8. Oxford.
Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed., Chapter 5. Oxford.
Journel, A.G. (1989). "Fundamentals of geostatistics in five lessons." AGU Short Course, Vol. 8.
Deutsch, C.V., Journel, A.G. (1998). GSLIB, 2nd ed. Oxford. (SGS implementation.)
Caers, J. (2011). Modeling Uncertainty in the Earth Sciences. Wiley-Blackwell.