The Kriging Oxymoron

The following example illustrates one of the most poorly understood topics in mining geostatistics which is Conditional Bias. Most of the geostatistical literature written on conditional bias as it pertains to the estimation of mineral resource block models is not correct and often very misleading. For example, consider a block model of a mineral deposit that is minable by open pit methods. In the early stages of study, it is quite common to prepare a block model of the deposit for the purposes of a feasibility study. In this case, the model is generally used to prepare a mine plan and a production schedule. For example, the mine plan may divide the deposit (block model) into annual production blocks over the life of the mine. The accumulation of blocks mined within each annual period block are summarized to provide an estimate of the recoverable tons and grade of ore material for each annual period. Thus, all that is required is a good estimate of the histogram of true block grades within each mining period -- nothing else. Whether or not the block estimates are conditionally biased is irrelevant to the problem. In fact, if the histogram of estimated block grades within each mining period is equal to the unknown but true histogram of block grades, then:

1. The grade tonnage curves provided by the estimated blocks will provide an accurate estimate of the in situ tons and grade above cutoff for each mining period.
2. The estimated block grades are NECESSARILY conditionally biased.

Conversely,

1. If the estimated block grades are conditionally unbiased, then necessarily, the histogram of block grades is smoothed (the estimated block variance is less than the true block variance).
2. Thus, the histogram of smoothed estimated block grades CANNOT accurately predict the in situ tons and grade above cutoff grade for each mining period.

The following figures provide an example where 1000 estimated block grades provide an excellent (accurate) prediction of the in situ tons and grade above cutoff for one annual mining period. However, the example also shows that these block estimates are severly conditionally biased.

Figure 1: the histogram and true block grades for 1000 blocks from an annual block.

Figure 2: The histogram and statistics of the corresponding 1000 estimated block grades.

Figure 3: A scatterplot of the 1000 true block grades versus their estimates. The regression line shows an obvious conditional bias. Note the slope of the regression line is 0.717. Conditional unbias requires a slope exactly equal to 1.0

Figure 4: "Grade Above Cutoff" curves for both the true and estimated block grades. Thus, the estimated block grades provide excellent predictions of the in situ grade above cutoff for all cutoff grades inspite of the conditional bias.

Figure 5: "Tons Above Cutoff" curves for both the true and estimated block grades. Thus, the estimated block grades provide excellent predictions of the in situ tons above cutoff for all cutoff grades inspite of the conditional bias.