Question 2 of 19: Variogram Fundamentals and Anisotropy

Question 1.2: Variogram Fundamentals and Anisotropy (13 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.

1.2.1 — What a variogram is and how it is used. A (semi-)variogram quantifies spatial continuity by measuring how dissimilar pairs of sample grades become as the separation distance ("lag," $h$) between them increases: $\gamma(h)=\dfrac{1}{2N(h)}\displaystyle\sum_{i=1}^{N(h)}\left[z(x_i)-z(x_i+h)\right]^2$, averaged over all sample pairs a distance $h$ apart. Close-together samples are geologically similar (low $\gamma$), and $\gamma$ rises with $h$ until samples become statistically independent, at which point $\gamma$ flattens at the sill — the separation at which this happens is the range $a$, and any y-intercept above zero as $h\to0$ is the nugget effect $C_0$ (short-scale variability/sampling error). Fitting a model (spherical, exponential, Gaussian) to the experimental points gives the kriging weights used to interpolate grade at unsampled points, and directly quantifies the deposit's structure: the range tells the estimator how far a sample's influence extends, and the nugget-to-sill ratio tells the geologist how "noisy" vs. "structured" the mineralization is.

1.2.2 — Directional equivalence. A variogram computed at azimuth 035°/dip 0° is the same variogram as azimuth 215°/dip 0°, because 215° = 035° + 180° — these are simply the two opposite look-directions along one horizontal line, and a variogram (like distance itself) is symmetric in $h$ and $-h$; every pair counted "outward" at 035° is the identical pair counted "inward" at 215°. The second case is not the same: azimuth 060°/dip −55° and azimuth 240°/dip −35° are not $180^\circ$ opposed in true 3-D orientation once dip is included — reversing azimuth by 180° (060→240) must be paired with reversing the sign of dip (−55→+55) to describe the same line; pairing it instead with a different dip magnitude (−35°) defines a genuinely different spatial direction, so these two variograms are independent and, in an anisotropic deposit, would be expected to differ.

1.2.3 — Included angle and band width. When searching for pairs to bin into a directional variogram, no real dataset has points falling on an exact bearing, so a tolerance cone is used: the included angle (angular tolerance) is the half-angle of that cone around the target azimuth/dip within which a pair is still accepted (e.g. ±22.5° either side of 035°). The band width is a linear (not angular) limit placed on how far off the centre-line a pair may fall once the cone would otherwise widen indefinitely with distance — it caps the perpendicular offset so that at large lag distances the search does not balloon into an effectively omni-directional bin.

1.2.3–1.2.5 (continuation) — Raw omni-directional variograms and drill-core treatment. Using assay results from raw, variable-length drill-core runs directly in an omni-directional variogram is expedient (fast, no compositing decisions needed) but statistically weak: because sample support (the physical length each assay represents) varies run-to-run, the resulting variance mixes true geological variability with pure support-size variability, inflating the nugget and blurring the true range — omni-directional pooling then further hides any real anisotropy the deposit has. Correct treatment separates the drill core first by geological domain (economic mineralization, sub-economic, barren waste) so unlike populations are never averaged together, and then composites each economic interval to a constant support length (a length-weighted average grade over fixed intervals, e.g. 2 m or bench-height composites) before any variogram is computed, removing the support-size artifact. A raw, uncomposited omni-directional variogram will therefore show a noisier, higher-nugget, shorter-apparent-range structure than the same data properly composited and domained. This distinction matters differently for the two deposit styles in the question: bulk, disseminated deposits (porphyry, epithermal), typically mined by open pit or block cave on a regular blast-hole/production grid, tolerate omni-directional or broad-sector variograms reasonably well because their grade fields are diffuse and only mildly anisotropic, so a coarser compositing/domaining pass is often adequate; rich, compact deposits (narrow vein/lens, VMS), mined selectively underground, have strongly anisotropic, geometry-controlled grade continuity (along strike and dip of the vein, sharply truncated across it), so raw or omni-directional variograms are essentially useless — directional variograms computed strictly within the domained vein envelope, with tight included angle and band-width limits, are required or the true along-structure continuity is masked by mixing vein and wall-rock samples.