Question 11 of 23: Variogram Model Types, Their Basis, and Terminology

Question 3.1: Variogram Model Types, Their Basis, and Terminology (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.

Model types. The spherical model rises with an S-shaped curve, hits its sill exactly at a finite range a (a genuine cut-off distance beyond which samples are uncorrelated), and is the most common choice for mineral deposits because most grade distributions do have a definite zone of influence. The exponential model approaches its sill asymptotically, never quite reaching it (a “practical range” is defined at ~95% of sill, conventionally 3× the model's distance parameter), suited to deposits with a less sharply defined zone of influence. The Gaussian model is parabolic near the origin (very smooth, highly continuous close in) before flattening toward its sill, appropriate only for genuinely smoothly-varying phenomena (e.g. some geophysical or environmental fields) and used cautiously in ore grade work because its over-smooth origin behaviour can produce unrealistic (artifactually smooth) kriged estimates.

Why a fitted mathematical model is used, rather than the raw experimental points. The raw experimental variogram is noisy (each lag is averaged from a finite, unequal number of sample pairs) and is only known at the discrete lags actually sampled; a kriging system needs γ(h) evaluated at EVERY distance and direction between samples and estimation points, including distances shorter than the smallest experimental lag. A smooth, mathematically well-behaved (positive-definite) model fitted through the experimental points supplies a continuous, always-solvable covariance function for the kriging equations, which the raw scattered points cannot.

Data and calculation. A semi-variogram is built from all available paired samples (drillhole or channel assays) that share the same relative separation h in a chosen direction/tolerance: $$\gamma(h) = \frac{1}{2N(h)}\sum_{i=1}^{N(h)} \left[z(x_i) - z(x_i+h)\right]^2$$ averaged over all N(h) pairs at that lag, then plotted against h to give the experimental variogram, which the model curve is fitted to. The experimental curve reveals how quickly grade similarity decays with distance (short vs. long-range continuity), whether continuity differs by direction (anisotropy), and whether there is unexplained short-scale variability (nugget).

Units of γ. γ(h) has units of the SQUARED grade variable (e.g. %², (g/t)², ppm²) — it is a variance, not a grade — because it is built from squared differences of the grade variable.

3.1.1 Spherical Model. A specific mathematical form, $$\gamma(h)=C_0+C\left[1.5\tfrac{h}{a}-0.5\left(\tfrac{h}{a}\right)^3\right]$$ for h ≤ a, sill C₀+C for h > a — the standard model for a deposit with a genuine, finite zone of spatial influence.

3.1.2 Nugget. The apparent discontinuity/jump in γ(h) as h→0⁺ (a variogram is defined to be exactly 0 at h=0 by construction, but the fitted model's intercept as h→0⁺ is C₀>0); it represents unresolved short-scale variability — sampling/assay error plus true grade variation at a scale finer than the sample spacing — and directly limits how well any interpolator, however good, can predict an unsampled point.

3.1.3 Sill. The plateau value C₀+C that γ(h) reaches (or approaches) at large h, equal to the overall variance of the data when the field is stationary; it is the variance a kriging estimate approaches once samples are too far away to be informative.

3.1.4 Range. The lag distance a at which γ(h) first reaches the sill (spherical) or its practical equivalent (exponential/Gaussian) — beyond the range, samples are effectively uncorrelated and contribute negligible information (and, in kriging, near-zero weight) to an estimate.

3.1.5 Step or Lag. The fixed increment of separation distance h used to bin sample pairs when computing the experimental variogram; lag size and tolerance control the trade-off between resolution (small lag, many bins, but each based on fewer, noisier pairs) and stability (larger lag, smoother curve, coarser resolution).