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24-MMP-A4 Mine Valuation and Mineral Resource Estimation · May 2016

Question 6 of 29: The Ordinary Kriging Matrix

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

EGBC National Exam — Mining and Mineral Processing Engineering, 09-Mmp-A4 Mine Valuation and Mineral Resource Estimation, 2016-May. 3 hours duration; one handwritten 8.5×11 in reference sheet permitted (not an open-book exam); only approved Sharp or Casio calculators allowed. Question 1 is compulsory (40 marks, parts 1.1–1.6); candidates then select THREE of the six optional Questions 2–7 (20 marks each) to complete the paper.

Reference texts: Isaaks & Srivastava, An Introduction to Applied Geostatistics (variogram modelling, kriging estimators, volume–variance relations); Hustrulid, Kuchta & Martin, Open Pit Mine Planning and Design (mine valuation, NPV and cut-off grade methodology, mineable reserves); Gentry & O'Neil, Mine Investment Analysis (Canadian mining taxation, inflation and financing effects on DCF yield, smelter/refining contract terms, net smelter return); SME Mining Engineering Handbook, 3rd ed. (mineral exploration/evaluation stages, ore reserve classification, ore deposit models); CIM Best Practice Guidelines and NI 43-101 (Canadian Securities Administrators).

Question 1.6: The Ordinary Kriging Matrix (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.

Ordinary kriging (OK) estimates an unknown block/point grade as a weighted linear combination of n nearby samples, Z*OK = ΣλiZ(xi), where the weights λi are chosen to be both unbiased (E[Z*−Z]=0, enforced by the constraint Σλi=1, which frees OK from needing to know the population mean) and minimum-variance (minimizing the estimation variance Var[Z*−Z] subject to that constraint).

Minimizing the estimation variance subject to the unbiasedness constraint, via a Lagrange multiplier μ, produces the (n+1)×(n+1) "ordinary kriging matrix" system:

$$\begin{bmatrix} \gamma(x_1,x_1) & \cdots & \gamma(x_1,x_n) & 1 \\ \vdots & \ddots & \vdots & \vdots \\ \gamma(x_n,x_1) & \cdots & \gamma(x_n,x_n) & 1 \\ 1 & \cdots & 1 & 0 \end{bmatrix} \begin{bmatrix}\lambda_1\\ \vdots\\ \lambda_n\\ \mu\end{bmatrix} = \begin{bmatrix}\gamma(x_1,x_0)\\ \vdots\\ \gamma(x_n,x_0)\\ 1\end{bmatrix}$$

Determining the inputs. The left-hand matrix entries γ(xi,xj) are the modelled (fitted spherical/exponential/etc.) semi-variogram value evaluated at the separation distance between every pair of the n sample points – read directly off the variogram model fitted in advance from the sample data (Question 1.3/3.2). The right-hand side γ(xi,x0) is the same variogram model evaluated between each sample and the point/block being estimated (x0); for a BLOCK estimate this is replaced by the average variogram value between sample i and every point discretising the block (γ-bar), not a single point-to-point value. The bottom row/right-most column of 1's and the trailing 0 encode the Σλi=1 unbiasedness constraint and its Lagrange multiplier.

Solving the system. The matrix is inverted (or solved by Gaussian elimination/LU decomposition) once per unique sample configuration – in practice the search neighbourhood and hence the sample set changes block to block, so the system is re-solved for every block being estimated.

Outputs. Two quantities come out of every solve: (1) the n kriging weights λi, combined with the sample grades to give the estimated block/point grade Z*OK = ΣλiZ(xi); and (2) the ordinary kriging variance, σ²OK = Σλiγ(xi,x0) + μ, a data-configuration-dependent measure of estimation uncertainty (independent of the actual sample grades) used for resource classification, confidence intervals, and conditional-simulation calibration.