1: Two-Structure Nested Spherical Variogram Model: 24-MMP-A4 Mine Valuation and Mineral Resource Estimation, May 2016, Q12

Question 3.2.1: Two-Structure Nested Spherical Variogram Model (8 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.

Given.

Parameter	Value
Nugget, C0	0.1
Structure 1: sill contribution C1, range a1	0.5, 100 m
Structure 2: sill contribution C2, range a2	0.4, 500 m

Find. γ(h) at h = 0, 50, 250 and 1000 m, and the sketch of the nested model and its total sill.

Fig. 3.2.1 – The nested two-structure spherical model, with the four requested evaluation points marked: (0, 0), (50, 0.504), (250, 0.875), (1000, 1.000 = sill).

Approach. Evaluate each structure's spherical function Sph(h/a_i)=1.5(h/a_i)−0.5(h/a_i)³ while h≤a_i, clamped at 1 once h exceeds that structure's own range, then sum γ(h)=C0+C1·Sph(h/a1)+C2·Sph(h/a2).

Total sill. $$\text{Sill} = C_0+C_1+C_2 = 0.1+0.5+0.4 = \boxed{1.00}$$
γ(0). By definition γ(0)=0 exactly (no separation → no variance); C0 is the discontinuous jump the model makes as h→0+, not a value AT h=0. $$\boxed{\gamma(0)=0}$$
γ(50). h=50 lies inside both ranges. Structure 1: x1=50/100=0.500, Sph(x1)=1.5(0.5)−0.5(0.5)³=0.750−0.0625=0.6875. Structure 2: x2=50/500=0.100, Sph(x2)=1.5(0.1)−0.5(0.1)³=0.150−0.0005=0.1495. $$\gamma(50)=0.1+0.5(0.6875)+0.4(0.1495)=0.1+0.34375+0.0598=\boxed{0.504}$$
γ(250). h=250 has passed structure 1's range (250>100, Sph1=1, fully saturated) but is inside structure 2's range (250<500). Structure 2: x2=250/500=0.500, Sph(x2)=1.5(0.5)−0.5(0.5)³=0.6875. $$\gamma(250)=0.1+0.5(1)+0.4(0.6875)=0.1+0.5+0.275=\boxed{0.875}$$
γ(1000). h=1000 exceeds both ranges (1000>100 and 1000>500), so both structures are fully saturated and the variogram has reached the total sill. $$\gamma(1000)=0.1+0.5(1)+0.4(1)=\boxed{1.00 = \text{Sill}}$$

Quantity	Value
Sill (C0+C1+C2)	1.00
γ(0)	0.000
γ(50)	0.504
γ(250)	0.875
γ(1000)	1.000 (sill reached)

Concept & Theory

Topic: Nested spherical variogram models.

Key relations: Sph(x)=1.5x−0.5x³ for x≤1, else 1; total sill = ΣC_i (i=0..n).

Why this works: Each spherical structure is itself a valid, positive-definite covariance model, and a sum of valid variogram models is always itself a valid variogram model, so nesting structures at different ranges is a mathematically safe way to add fitting flexibility – here representing both a short-range (100 m, grain/vein-scale) and a longer-range (500 m, zone-scale) continuity structure – without breaking the kriging system's solvability.

Common pitfall: Forgetting that once h exceeds a structure's OWN range, that structure's Sph term is clamped at 1 rather than continuing to apply the cubic formula (which would overshoot past the sill).

Source: Isaaks & Srivastava, Ch. 7–8 (variogram models, nested structures).

Question 12 of 29: 1: Two-Structure Nested Spherical Variogram Model

Question 3.2.1: Two-Structure Nested Spherical Variogram Model (8 marks)