Question 9 of 18: Kriging Methods – Ordinary, Simple and Indicator

Question 2.3: Kriging Methods – Ordinary, Simple and Indicator (12 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Ordinary kriging (OK). Estimates a block/point grade as Σλ_iZ_i with the constraint Σλ_i=1, so it does NOT require the population mean to be known – it re-estimates a local mean implicitly within each search neighbourhood, making it robust to a gently drifting global mean (local stationarity only).

(b) Simple kriging (SK). Estimates a RESIDUAL from a known, stationary global mean m: Z₀−m=Σλ_i(Z_i−m), with NO sum-to-one constraint on the weights (they may sum to less than one, with the shortfall implicitly assigned to the known mean m). SK requires the analyst to supply a reliable, deposit-wide stationary mean up front – a much stronger assumption than OK's local-mean re-estimation.

(c) Indicator kriging (IK). Transforms the continuous grade variable into a binary indicator I(x;z_c)=1 if Z(x)≥z_c, else 0, for one or more chosen cutoff grades z_c, then krige EACH indicator variable (its own indicator variogram) separately using ordinary kriging. The output at each cutoff is a direct, distribution-free estimate of the PROBABILITY (or local proportion) that the block grade equals or exceeds that cutoff, avoiding the assumption of any particular grade distribution (e.g. lognormal) that OK/SK implicitly carry.

(d) Resulting output values. OK and SK each directly produce a single estimated GRADE Z₀ at each block, plus its kriging (estimation) variance σ²_OK or σ²_SK. IK instead produces, for EACH chosen cutoff z_c, an estimated PROBABILITY (proportion) that the block is at or above that cutoff – i.e. IK's raw output is a set of points on the block's local cumulative distribution function, not a single grade.

(e) Post-kriging calculations for (b) and (c). For SIMPLE kriging, the block grade must be recovered by adding the known stationary mean back to the estimated residual: Z₀=m+Σλ_i(Z_i−m) – the SK weights alone give only the mean-corrected residual. For INDICATOR kriging, the set of estimated probabilities at multiple cutoffs (F(z_c)≡1−probability(≥z_c)) must be assembled into a full local cumulative distribution function, order-corrected for any cutoff-to-cutoff non-monotonicity, and then the EXPECTED grade of the block is recovered by numerically integrating that CDF (E[Z]=∫(1−F(z))dz) – a materially heavier post-processing step than OK/SK's direct grade output.

(f) Recovering tonnage and grade above cut-off from IK. Because each indicator-kriged probability directly estimates the LOCAL PROPORTION of the block at or above that cutoff, the tonnage above cut-off is obtained simply as (block tonnage)×(estimated probability at that cutoff), summed over all blocks in the model. The GRADE above cut-off requires one further step: from the assembled local CDF (part e), compute the conditional mean of the distribution restricted to the interval above the cutoff – i.e. integrate z·(the local probability density implied by the CDF) over z≥z_c, then divide by the probability mass above z_c to get the mean grade of just that above-cutoff fraction. Because this is derived from the FULL local distribution rather than a single point estimate, IK-derived tonnage/grade-above-cutoff curves are markedly more reliable at high cutoffs (where OK/SK single-point grade estimates are known to be conditionally biased/over-smoothed) than an equivalent calculation built from an OK or SK block model alone.