Question 8 of 19: Variogram Construction – Fundamentals

Question 3.1: Variogram Construction – Fundamentals (10 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.

3.1.1 — Calculating an individual variance value. For every sample pair $(x_i,\,x_i+h)$ separated by (approximately) lag distance $h$ in a given direction, the squared difference of their grades, $\left[z(x_i)-z(x_i+h)\right]^2$, is calculated — this single squared difference is the "individual variance value" contributed by that one pair.

3.1.2 — Combining values into one variogram point. All pairs whose separation falls within one lag-distance bin (and, for a directional variogram, within the azimuth/dip tolerance and band width) are averaged: $\gamma(h)=\dfrac{1}{2N(h)}\sum\left[z(x_i)-z(x_i+h)\right]^2$, giving one plotted point at that lag. Yes, the pair count $N(h)$ behind each point should always be stored and reported (often directly on the plot) — a point built from only a handful of pairs is statistically unreliable and can be a misleading outlier on the fitted model, while a point built from hundreds of pairs is robust; without $N(h)$ recorded, a modeller cannot tell a genuine structural feature from noise in a sparsely-populated lag bin.

3.1.3 — Variogram axes and units. The horizontal axis is lag distance $h$ (the separation between sample pairs), in the same linear distance units as the sample coordinates (e.g. metres or feet). The vertical axis is semi-variance $\gamma(h)$, in the squared units of the grade variable being modelled (e.g. (g/t)² for gold grade, or %² for a base-metal percentage grade) — a point of frequent confusion, since $\gamma$ has different physical units than the grade itself.

3.1.4 — The three main variogram parameters. Nugget effect $C_0$ — the apparent $y$-intercept of $\gamma(h)$ as $h\to0$, representing variability at scales smaller than the sample spacing plus pure sampling/analytical error; it never truly reaches zero because no two samples occupy the exact same point. Range $a$ — the lag distance at which $\gamma(h)$ reaches the sill and samples become spatially independent (uncorrelated); it defines the maximum useful search radius for interpolation. Sill $C_0+C$ — the plateau value of $\gamma(h)$ beyond the range, which for a stationary variable equals the total (a priori) variance of the dataset; the sill minus the nugget ($C$, the "partial sill" or "structured" variance) is the portion of total variability that is spatially structured (predictable) rather than random.

3.1.5 — Smoothing a sparse variogram. With too few sample pairs per lag, the raw experimental variogram is a noisy, erratic scatterplot rather than a clean curve. It is "smoothed" by widening the lag-bin tolerance (accepting a broader range of separation distances into each plotted point), increasing the angular/band-width tolerance to pool more directional pairs together, or fitting a smooth theoretical model (spherical, exponential, Gaussian) through the noisy points by least-squares/visual fit rather than connecting the raw points directly. What is lost in doing so is directional resolution and short-scale detail: widening tolerances pools together pairs that may sample genuinely different (anisotropic) directions or different short-range structural behaviour, so a heavily smoothed variogram can mask real anisotropy or a subtle nested short-range structure that a denser, more tightly-tolerant dataset would have revealed.