Question 11 of 13: Ordinary, Simple and Indicator Kriging

Question 5: Ordinary, Simple and Indicator Kriging (15 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.

5.1 Ordinary versus simple kriging. Ordinary Kriging (OK) estimates a block grade as a weighted linear combination of nearby samples, $$Z^*=\sum_i \lambda_i Z_i,$$ with the weights constrained to sum to exactly 1. This constraint makes OK locally unbiased WITHOUT requiring the population mean to be known – the estimator effectively re-estimates its own local mean from the neighbourhood samples, which is why OK is the most widely used kriging variant in practice and is robust to gentle local trends. Simple Kriging (SK) instead assumes the stationary (global) mean m is known with confidence and uses it explicitly: $$Z^*=m+\sum_i\lambda_i(Z_i-m)$$ Here the weights λ_i are NOT constrained to sum to 1 – they can (and typically do) sum to less than 1. SK is theoretically more efficient (lower kriging variance) than OK when the assumed mean is genuinely correct, but is sensitive to bias in areas whose local grade deviates from that global mean, since it always pulls the estimate back toward m. In practice OK dominates general resource estimation because a trustworthy global mean is rarely available; SK is reserved for special contexts such as indicator/multiple-indicator kriging within a class, or as a component of conditional simulation.

5.2 Why the incomplete SK weight sum is not a problem. Because SK's weights sum to less than 1, the "missing" weight, (1−Σλ_i), is deliberately applied to the known stationary mean m rather than left unassigned – that IS the mechanism built into the SK estimator above. This anchoring is precisely what prevents systematic bias: in a sparsely-sampled area (where the sample weights alone sum to a small fraction), the estimate is pulled strongly toward the known population mean rather than being an unweighted (and therefore biased) function of only the few nearby samples. Applying the sample weights alone, without the mean term, would systematically under-estimate high-grade zones and over-estimate low-grade zones wherever data is sparse; the (1−Σλ)·m term is the correction that keeps SK conditionally unbiased.

5.3 Indicator kriging for ore/waste tonnage and grade. Indicator kriging (IK) transforms each sample into a binary indicator at a chosen cut-off grade z_c: I(x; z_c)=1 if the sample grade ≥ z_c, else 0. Kriging this indicator variable (using its own, separately-modelled indicator variogram) at a block location gives the estimated probability that the block's grade exceeds z_c – directly, this is the estimated ore proportion (volume fraction) of the block, with the complementary proportion being waste; multiplying each proportion by the block volume and density gives ore and waste tonnage directly. Repeating IK at a SERIES of cut-offs across the grade range builds an estimated conditional cumulative distribution function (ccdf) of grade within the block, without assuming any parametric (e.g. lognormal) shape for that distribution; the mean grade of the ore portion is then obtained by integrating z·dF(z) over z>z_c (and similarly the mean grade of the waste portion below z_c). This is particularly valuable for highly skewed, non-Gaussian grade populations (common in precious-metal and some base-metal deposits) and for accurately quantifying the ore/waste split at the selective mining unit (SMU) scale – something ordinary or simple kriging, which estimate only the MEAN grade and not the full within-block distribution, cannot directly provide.