Question 3 of 18: The Ordinary Kriging Matrix

Question 1.3: The Ordinary Kriging Matrix (7 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.

Ordinary kriging estimates an unknown block grade Z₀ as a weighted linear combination of n surrounding sample grades, Σλ_iZ_i, choosing the weights λ_i that minimize the estimation variance subject to the unbiasedness constraint Σλ_i=1 (which is what makes it "ordinary" – the local mean is not assumed known, unlike simple kriging). Introducing a Lagrange multiplier μ to enforce that constraint produces the (n+1)×(n+1) linear system:

$$\begin{bmatrix}\gamma_{11}&\cdots&\gamma_{1n}&1\\ \vdots&\ddots&\vdots&\vdots\\ \gamma_{n1}&\cdots&\gamma_{nn}&1\\ 1&\cdots&1&0\end{bmatrix}\begin{bmatrix}\lambda_1\\ \vdots\\ \lambda_n\\ \mu\end{bmatrix}=\begin{bmatrix}\gamma_{10}\\ \vdots\\ \gamma_{n0}\\ 1\end{bmatrix}$$

Inputs. The left-hand (n×n) block γ_ij holds the modelled semi-variogram value between every pair of surrounding SAMPLE points i and j – read directly off the fitted variogram model (Question 1.2) at each pair's separation distance. The right-hand vector γ_i0 holds the modelled variogram value between each sample i and the BLOCK (or point) being estimated, 0 – for a block (rather than a point) estimate this is itself an average of the point-to-point variogram over the block's discretized volume, i.e. a γ(sample, block) rather than γ(sample, point) term. Both come entirely from the already-fitted variogram model and the sample/block geometry; no grade data enters the matrix itself, only the coordinates.

Outputs. Solving the system yields the n kriging weights λ_i (applied to the sample GRADES, not used in the matrix itself, to compute the estimated block grade Z₀=Σλ_iZ_i) and the Lagrange multiplier μ. From these, the kriging (estimation) variance is recovered as σ²_OK=Σλ_iγ_i0+μ – a data-configuration-dependent measure of local estimation confidence (used for reserve-classification cut-offs and to flag blocks needing infill drilling) that, unlike the weights themselves, does NOT depend on the sample grades at all, only on the sample geometry relative to the block.