Post-processing realisations into decision metrics

Realisations are MEANS, not ends. The final deliverables are DECISION METRICS extracted from the ensemble: recoverable tonnage, average grade, total metal, expected value, risk-adjusted reserves. §7.6 covers the standard post-processing methods that transform an ensemble of grids into the numbers decision-makers actually use.

Grade-tonnage curve

For each candidate cutoff $c$ and each realisation $i$:

• Tonnage above cutoff $T_i(c) = \sum_{x} \mathbb{1}[z^{(i)}(x) > c]$ (fraction of grid nodes above cutoff).
• Average grade above cutoff $\bar{g}$i(c) = \sum{x} z^{(i)}(x) \mathbb{1}[z^{(i)}(x) > c] / T_i(c).

The grade-tonnage curve plots $(c, T(c), \bar{g}(c))$ as cutoff varies. Each realisation gives one curve; the ensemble gives a BAND. P10/P50/P90 of tonnage at each cutoff define risk-stratified reserves.

Total metal

For each realisation: $M_i(c) = T_i(c) \times \bar{g}_i(c)$, total recoverable metal above cutoff. This is the economic-decision metric: how much VALUE is in the deposit above the given cutoff?

P10/P50/P90 of total metal give risk-stratified value estimates. Modern mining: discount each by cost (haulage, processing, refining) to get expected net present value.

Non-linear post-processing

Beyond simple aggregates, realisations support NON-LINEAR post-processing:

• Reservoir flow simulation: each permeability realisation is input to a flow simulator; production schedule comes out; ensemble of production schedules characterises flow uncertainty.
• Hydraulic head / groundwater: each permeability realisation drives a flow model; ensemble characterises water-availability uncertainty.
• Contaminant transport: each permeability realisation drives a transport model; ensemble characterises plume-extent uncertainty.
• Mining cut-off optimisation: each grade realisation is input to a mine-planning model; ensemble of optimal cutoffs characterises value uncertainty.

The key insight: kriging map alone CANNOT support non-linear post-processing without bias. Each non-linear function applied to the smoothed kriging map gives the wrong answer (Jensen's inequality). Apply the non-linear function to each realisation, then take the ensemble, that's the unbiased way.

Local accuracy vs decision accuracy

A subtle point: PER-LOCATION quantiles (P10 map, P90 map) are LOCAL estimates. They are NOT the same as the GLOBAL P10 reserves. Computing the global P10 reserve (the optimistic high case, exceeded only 10% of the time):

• For each realisation, compute the total tonnage above cutoff.
• Collect 100 totals.
• P10 reserve = empirical 90th percentile of the collection (90% of realisations fall below it); the conservative P90 reserve is the 10th percentile.

NOT: P10 map then sum. The two procedures give DIFFERENT answers because high values aren't spatially independent, they tend to cluster. Modern reporting: BOTH local and global metrics with clear labeling.

Recoverable resource standards

Mining-resource estimation standards (JORC, NI 43-101, SEC S-K) increasingly require:

• Both kriging-based AND simulation-based estimates.
• P10/P50/P90 reserves at the chosen cutoff.
• Sensitivity analysis: how do reserves change as the cutoff varies?
• Variogram and LOO-CV diagnostics.
• Discussion of model limitations and uncertainties.

The full report is comprehensive and supports peer/regulatory review.

Try it

• Defaults: mean grade = 1.5, SD = 0.8, cutoff = 1.2. 100 realisations × 500 nodes each. Recoverable tonnage (P50) is about 65%; avg grade above cutoff is about 2.0 (notably higher than the population mean, selecting above cutoff biases upward).
• Increase cutoff to 2.5. Tonnage drops dramatically; avg grade above cutoff is much higher; total metal may decrease (less ore overall). The grade-tonnage curves visualise this trade-off across all cutoffs.
• Crank SD to 1.5 (wide grade distribution). High-grade tail is enhanced; tonnage above moderate cutoffs increases.
• Drop SD to 0.3 (tight grade distribution). Very binary: at cutoffs below the mean, recover most material; above the mean, very little.
• P10/P50/P90 of tonnage tell the risk story: tight P90-P10 spread = high-confidence estimate; wide P90-P10 spread = high uncertainty, more data needed.

A copper mine has 100 realisations of grade. At cutoff = 0.5%, P50 recoverable tonnage = 50 million tonnes. Why might the engineer report this together with the conservative P90 (40 Mt) and optimistic P10 (60 Mt) rather than just the P50?

What you now know

Post-processing extracts decision metrics from the ensemble: recoverable tonnage, average grade, total metal, grade-tonnage curves, non-linear functional outputs (flow simulation). Local vs global metrics distinct. Modern mining-resource standards require both kriging-based and simulation-based estimates with comprehensive documentation. PART 7 COMPLETE.

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References

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