Multiple realisations and uncertainty maps

An ensemble of B simulation realisations on N grid nodes provides B × N values — a rich uncertainty resource. §7.5 develops the standard post-processing: per-location quantile maps, exceedance probability maps, ensemble summary statistics.

Per-location quantile maps

For each grid node $x$, the B values $z^{(1)}(x), \ldots, z^{(B)}(x)$ form an empirical distribution. Compute quantiles:

• P10 map: 10th percentile at each location. "What value is the grade likely to exceed 90% of the time at this location?"
• P50 map: median ≈ ensemble mean ≈ kriging map.
• P90 map: 90th percentile. "What value is the grade likely to fall below 90% of the time?"
• P10-P90 band: uncertainty width at each location.

Resource estimation: P10 reserves = recoverable tonnage above cutoff using P10 grade maps; P90 reserves = same with P90 grade map. Industry uses these for risk-stratified reserve reporting.

Exceedance probability maps

For each grid node, count the fraction of realisations whose value exceeds a chosen cutoff:

The result is a P-map: at each location, the probability of exceeding the cutoff. Used to identify high-confidence high-grade zones (P > 80%), moderate-confidence zones (P 30-80%), low-confidence zones (P < 30%).

Modern mining: combine P-maps for multiple cutoffs to produce "grade-tonnage" curves with associated uncertainty.

Local vs global uncertainty

• Local uncertainty: per-node variability across realisations. Captured by P10-P90 band per location.
• Global uncertainty: variability of full-grid statistics (mean grade across the deposit, total tonnage, etc.). Captured by aggregating across the grid then taking ensemble distribution.

A common error: confuse local uncertainty with global uncertainty. The variability of MEAN grade across the deposit (small, with averaging) is much less than the per-location grade variability (full). Modern reporting distinguishes carefully.

Sample sizing

For B = 100 realisations, Monte Carlo SE on a quantile estimate is approximately:

P10 has Monte Carlo precision ~3%; P90 the same. Modern practice: 100 realisations for routine work, 500 for high-stakes decisions, 1000+ for definitive resource reports. Diminishing returns beyond 500.

Try it

• Defaults: 50 realisations, range = 2, cutoff = 5.5. Top panel shows ensemble mean (red), P10-P90 band (orange shaded), and the 6 hard data points (green). Bottom shows exceedance probability across the transect.
• The P10-P90 band collapses to a point at hard data locations (zero local uncertainty) and widens between (high local uncertainty). Spatial uncertainty visualised.
• The exceedance curve shows where the model is confident in exceeding the cutoff. Near data above the cutoff, P > 80%. Far from data, P approaches 50% (uninformed). Below-cutoff data give P → 0%.
• Drag cutoff to 4.0 (low). Most locations exceed it; exceedance curve approaches 1.0 everywhere.
• Drag cutoff to 7.0 (high). Few locations exceed; curve approaches 0 except near the data peaks.

A copper-mining resource report uses 100 SGS realisations and produces P10/P50/P90 reserves. The P90 reserves are 1.5× the P10 reserves. What does this ratio tell the reader, and how does it compare to a similar deposit with P90/P10 ratio of 4?

What you now know

Multiple realisations give per-location quantile maps (P10/P50/P90) and exceedance probability maps. These are the natural outputs for decision support beyond the kriging map. Local vs global uncertainty are distinct concepts. Sample sizing: 100 realisations for routine, 500+ for high-stakes. §7.6 closes Part 7 with post-processing realisations into decision metrics — recoverable tonnage, expected costs, etc.

Mark section complete →

References

• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation, Chapter 8. Oxford.
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford.
• Caers, J. (2011). Modeling Uncertainty in the Earth Sciences. Wiley-Blackwell.
• Journel, A.G., Huijbregts, C.J. (1978). Mining Geostatistics. Academic Press.
• Rossi, M.E., Deutsch, C.V. (2014). Mineral Resource Estimation. Springer.