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24-MMP-A4 Mine Valuation and Mineral Resource Estimation · Undated paper

Question 10 of 19: The Volume–Variance Relationship

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Notes on this paper

EGBC National Exam — Mining and Mineral Processing Engineering, 09-MMP-A4 Mine Valuation and Mineral Resource Estimation, undated sitting. 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 (parts 1.1–1.5); candidates then select THREE of the five optional Questions 2–6 (20 marks each) to complete the paper.

Reference texts: Isaaks & Srivastava, An Introduction to Applied Geostatistics (variogram modelling, anisotropy, volume–variance relations); Hustrulid, Kuchta & Martin, Open Pit Mine Planning and Design (mine scheduling, NPV/valuation methods, stripping-ratio economics); Gentry & O'Neil, Mine Investment Analysis (Canadian mining taxation, CCA classes, smelter/refining contract terms, net smelter return); SME Mining Engineering Handbook, 3rd ed. (mineral exploration/evaluation stages, ore reserve classification); Guilbert & Park, The Geology of Ore Deposits (volcanogenic massive sulphide genesis); CIM Best Practice Guidelines and NI 43-101 (Canadian Securities Administrators).

Some question wording is assumed where the paper is unclear. Several tables in the paper do not reconcile arithmetically (the Q1.4.3 reserve table, the Q4 ore/waste schedule totals, the Q5.5 earnings-split percentages), and some sub-part mark values do not add to the question totals. This solution answers the conceptual and methodological content in full and works the self-consistent numeric sub-parts (NPV in 1.3, the nested variogram in 3.2, the depreciation schedule in 5.1, the NSV/NSR chain in 6.3–6.5), flagging every place an inconsistency is carried forward.

Question 3.3: The Volume–Variance Relationship (5 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.

3.3.1 — The volume–variance relationship. As the support (physical volume) over which a grade value is defined increases, the variance of that grade decreases — a single point/core sample has the highest possible variance, while the average grade of a large block, integrating many internal point-scale fluctuations, is inherently smoother and less variable, provided the block dimension stays inside the range of spatial correlation (beyond the range, averaging no longer smooths anything further because the internal points are already independent). Given two blocks of different sizes, both with dimensions less than the range, the larger block has the lower internal grade variance — more internal averaging occurs within it — and this relationship is quantified by Krige's relationship / the dispersion variance formula, which decomposes point variance into a sum of block-to-block and within-block variance components as support size increases.

3.3.2 — Pseudo samples. A pseudo sample is not a real physical sample but a synthetic, regularly-spaced point (or small sub-volume) placed on a grid within a block purely as a computational device, used to numerically estimate the average variogram value between "the block" and a point (or between two blocks) by discretizing the continuous block-averaging integral into a finite, tractable sum. Their layout is typically a regular internal grid (e.g. a $4\times4$ or $5\times5\times5$ array of points spanning the block's full extent) dense enough that the discretized average closely approximates the true continuous block integral, but not so dense that the calculation becomes computationally unwieldy.

3.3.3 — Information needed for pseudo-sample variance. The essential information is the fitted variogram model itself (nugget, sill, range and structure type) — since the "distance" between pseudo samples is known purely from their geometric layout, evaluating the variance/covariance between any pair of pseudo samples (or between a pseudo sample and the block) requires reading $\gamma(h)$ off the modelled variogram at each pairwise separation distance, then averaging all pairwise values to obtain the discretized block variance.