Question 28 of 29: Indicator Kriging for Epithermal Gold

Question 6.2: Indicator Kriging for Epithermal Gold (8 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.

6.2.1 Why simple/ordinary kriging struggle with epithermal gold. Epithermal gold grade populations are extremely positively skewed and "nuggety" – bonanza-grade veinlets sit within otherwise low-grade to barren host rock (Question 6.1.1's narrow, boiling-level ore shoots) – so the underlying distribution is far from the roughly-Gaussian, single-population behaviour that linear (simple/ordinary) kriging implicitly handles best. Linear kriging of raw grades in this setting over-smooths: it systematically under-estimates true high-grade zones (a few extreme assays get averaged down by many surrounding low values) and cannot directly answer the practically essential question – what is the PROBABILITY this block exceeds cutoff – since it produces only a single expected-value estimate with a Gaussian-shaped error model that poorly represents a skewed reality.

6.2.2 Indicator variograms. Data are first transformed to a binary indicator at a chosen cutoff z_c: I(x;z_c)=1 if Z(x)≥z_c, else 0. A separate experimental variogram is then computed on this 0/1 indicator series for EACH of several cutoffs spanning the grade distribution. Indicator variograms typically show "DESTRUCTURATION" with increasing cutoff – at low cutoffs (most of the deposit is "1") continuity is high (long range, low relative nugget); at high cutoffs (only rare bonanza samples are "1") continuity degrades sharply (short range, high relative nugget), directly reflecting the increasingly sporadic, structurally-controlled nature of the highest-grade material.

6.2.3 Basic results of indicator kriging. At each cutoff, kriging the indicator variable produces, for every block/point, an estimate of the LOCAL PROBABILITY (or proportion) that true grade exceeds that cutoff – one point on that location's local conditional cumulative distribution function (ccdf). Repeating across a full suite of cutoffs builds the complete local ccdf of grade at every block, from which the expected (mean) grade, and a full local uncertainty distribution (not just a single point estimate and variance), can both be derived – this multi-cutoff procedure is Multiple Indicator Kriging (MIK).

6.2.4 Ore/waste volume and average grade at a cutoff. For a given cutoff z_c and a given block, the IK-derived probability p(z_c) that grade exceeds z_c gives the estimated ORE proportion of that block directly: ore volume = block volume × p(z_c); waste volume = block volume × (1−p(z_c)). The average grade of the ORE portion is obtained by integrating the block's estimated local ccdf ABOVE z_c (the conditional expectation E[Z|Z≥z_c]); the average grade of the WASTE portion is the corresponding integral BELOW z_c. Summing/integrating these block-by-block results across the full model at a given cutoff produces the deposit's overall tonnage-grade curve at that cutoff – the standard grade-tonnage output every MIK resource model is built to deliver.