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 global P10:

• For each realisation, compute the total tonnage above cutoff.
• Collect 100 totals.
• P10 = empirical 10th percentile of the collection.

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 P10-P90 = high-confidence estimate; wide P10-P90 = 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 P10 (40 Mt) and P90 (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.