Question 9 of 13: Kriging Techniques and the Volume–Variance Relationship

Question 3: Kriging Techniques and the Volume–Variance Relationship (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.

Simple Kriging (SK). Assumes the deposit's mean grade m is known and constant (stationary) across the whole domain; the estimator is a weighted average of the surrounding samples PLUS a term that pulls the estimate toward the known mean where data are sparse, with weights unconstrained to sum to one. SK is the theoretically minimum-variance linear unbiased estimator when m truly is known, but real deposits rarely justify assuming a single known global mean, so SK is used mainly as a building block for other estimators (e.g. indicator kriging) rather than for primary grade estimation.

Ordinary Kriging (OK). Does not require the mean to be known; instead the estimator's weights are constrained to sum to exactly one, which makes the estimate locally unbiased for whatever the LOCAL mean happens to be, without ever explicitly estimating that mean. OK is the workhorse resource-estimation technique in industry – it needs only a fitted variogram and produces both a grade estimate and its kriging variance for every block.

Indicator Kriging (IK). Recodes each sample's continuous grade into a 0/1 indicator relative to a chosen cutoff (or a family of cutoffs), then krige each indicator variable separately (each cutoff gets its own indicator variogram). The result at each cutoff is an estimate of the PROBABILITY that the true grade exceeds that cutoff, and assembling estimates across many cutoffs reconstructs a full local grade distribution (an "ogive"). IK is favoured for highly skewed, erratic deposits (e.g. gold) where a single Gaussian-style linear estimator handles high-grade outliers poorly, and it is the natural tool when the real decision is "is this block ore or waste at cutoff Z" rather than "what is this block's mean grade".

Interaction of volume–variance, mineable reserves, SMUs, David's rule and block estimates. The volume–variance relationship (Krige's relationship) states that the variance of grades measured over a larger support (volume) is always less than the variance measured over a smaller support – averaging smooths out extremes. This directly governs what a selective mining unit (SMU) can be: the SMU is the smallest volume of ground that can realistically be selectively mined and hauled separately as ore or waste given the fleet and grade-control practice, and its grade distribution (via volume–variance) is necessarily SMOOTHER, with a narrower spread, than the distribution of the individual drill-hole samples that informed it – a resource model that estimates SMU-sized blocks directly (rather than smoothing point-support kriging results down after the fact) avoids double-counting this smoothing. M. David's one-quarter drill-spacing rule is a practical rule of thumb linking drilling density to reliable block estimation: to krige a block reliably at the SMU scale, the drill spacing should be roughly one-quarter (or less) of the SMU's own dimension, ensuring enough informing samples fall within the block's own footprint and its immediate search neighbourhood rather than relying on extrapolation from distant holes. All of this converges on mineable reserves: a resource only becomes a mineable reserve once it has been estimated on SMU-sized blocks (respecting volume–variance smoothing), with drilling dense enough (per David's rule) to support that block size at an acceptable confidence, and only the blocks whose kriged (or simulated) grade clears the operating cut-off – net of the dilution and ore-loss that selective mining at SMU scale actually implies – are counted as reserve tonnage.