Question 9 of 13: Spherical Variogram — Model Terms, Point Calculation and Anisotropy

Question 3: Spherical Variogram — Model Terms, Point Calculation and Anisotropy (15 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 Definitions. The spherical model is the most common permissible variogram model in mining geostatistics: it rises from the nugget in a cubic curve and reaches the sill exactly at the range, after which it stays flat – $$\gamma(h) = C_0 + C\left[1.5\frac{h}{a} - 0.5\left(\frac{h}{a}\right)^3\right],\ h\le a; \qquad \gamma(h)=C_0+C,\ h>a$$ The nugget, C₀, is the discontinuity at the origin – the semivariance that any two arbitrarily close samples would still show, arising from micro-scale grade variability plus sampling/analytical error; a large nugget signals noisy, poorly-continuous data. The sill, C₀+C, is the plateau semivariance the model reaches once samples are far enough apart to be spatially uncorrelated, and equals the a priori (population) variance of the grade. The range, a, is the lag distance at which the sill is reached – beyond it, samples carry no useful information about each other, so it directly sets the kriging search-neighbourhood radius. The step or lag, h, is simply the separation distance (and direction) between a pair of samples used to compute one point of the experimental variogram. Gamma, γ(h), is the semivariance itself, expressed in units of (grade)² – e.g. (%)² or (g/t)² – since it is defined from the squared difference of two grade values.

Given.

Find. γ(h) at h = 0.0, 100 and 300 m (Fig. 3).

Approach. The structure's own sill contribution is C = sill − nugget = 0.8; evaluate the cubic spherical form for h ≤ a and clamp at the full sill once h exceeds the range.

1. γ(0.0 m). By definition γ(0)=0 exactly – zero separation means zero variance; the nugget is the value the model jumps to for any h however small, not a value AT h=0. $$\boxed{\gamma(0)=0.000}$$
2. γ(100 m). h=100 m lies inside the range (100<200), so x=h/a=0.500 and Sph(x)=1.5(0.500)−0.5(0.500)³=0.750−0.0625=0.6875. Substituting: $$\gamma(100)=0.2+0.8(0.6875)=0.2+0.55=\boxed{0.750}$$
3. γ(300 m). h=300 m exceeds the range (300>200), so the structure has fully saturated and the variogram has reached the total sill: $$\gamma(300)=C_0+C=\boxed{1.000=\text{Sill}}$$

3.3 Further terms. A nested spherical model sums two or more spherical structures with different ranges (and possibly different sills) into a single variogram model – each structure is itself a valid model, and the sum of valid models is always itself valid, so nesting adds fitting flexibility (capturing both a short-range and a longer-range component of continuity) without breaking the kriging system's solvability. Anisotropy means the variogram's range (geometric anisotropy) or sill (zonal anisotropy) changes with direction, reflecting that grade continuity in a real deposit is rarely equal in all directions (e.g. much longer range along a mineralized trend than across it). Tolerance is the allowed angular deviation (azimuth and dip tolerance) and lag-distance tolerance within which a sample pair is still counted as belonging to a given direction/lag bin when computing the experimental variogram, since real drilling almost never lines up on an exact direction or exact spacing. Band width is a perpendicular-distance limit applied alongside the angular tolerance, capping how far off the nominal search line a pair may fall even if it satisfies the angular tolerance – without it, the angular tolerance cone would admit distant, off-axis pairs that are not really representative of the intended direction.

3.4 Why azimuth 90°/dip 45° need not equal azimuth 270°/dip 45°. It is tempting to assume that reversing the azimuth by 180° simply reverses the direction vector, and that γ(h) in the reverse of a direction always equals γ(h) in the forward direction (which IS true – γ is symmetric under a true vector reversal). But azimuth 270° with the SAME downward dip of 45° is not the true reverse of azimuth 90°/dip 45° down: the true reverse of "down-and-to-the-East at 45°" is "up-and-to-the-West at 45°" – i.e. azimuth 270° with dip 45° up, not down. Azimuth 270°/dip 45° down is a genuinely different, non-collinear direction (Fig. 4), forming a "V" with the 90°/45°-down direction rather than a straight line through the origin.

Since these are physically different directions through the rock mass, there is no requirement that they sample the same geological fabric: if the deposit has geometric or zonal anisotropy – for example a plunging ore shoot, bedding-controlled mineralization, or a structural fabric that is not symmetric about the horizontal – the range and/or sill measured along each direction can genuinely differ. The variogram is only guaranteed to be symmetric for a direction and its own true 180°-reversal (dip sign flipped along with azimuth), not for any two directions that merely share the same dip magnitude.