Question 12 of 23: Nested Spherical Variogram – Gamma Values at Four Lags

Question 3.2: Nested Spherical Variogram – Gamma Values at Four Lags (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.

Given.

Find. γ(h) at h = 0, 50, 250 and 1000 m for the nested model $$\gamma(h)=C_0+C_1\cdot Sph(h/a_1)+C_2\cdot Sph(h/a_2)$$

Approach. Evaluate each structure's own spherical function Sph(x) = 1.5x−0.5x³ for x≤1, else 1, at x = h/a₁ and h/a₂, scale by that structure's sill contribution, and sum with the nugget (added only for h>0).

1. h = 0 m. By definition γ(0)=0 (the nugget is the DISCONTINUITY as h→0⁺, not the value exactly at h=0): $$\gamma(0)=\boxed{0}$$
2. h = 50 m. Structure 1: x₁=50/100=0.5≤1, Sph(0.5)=1.5(0.5)−0.5(0.5)³=0.75−0.0625=0.6875, contributing 0.5×0.6875=0.34375. Structure 2: x₂=50/500=0.1≤1, Sph(0.1)=1.5(0.1)−0.5(0.1)³=0.15−0.0005=0.1495, contributing 0.4×0.1495=0.0598. $$\gamma(50)=0.1+0.34375+0.0598=\boxed{0.504}$$
3. h = 250 m. Structure 1: h>a₁(100 m) ⇒ full sill contribution 0.5. Structure 2: x₂=250/500=0.5≤1, Sph(0.5)=0.6875, contributing 0.4×0.6875=0.275. $$\gamma(250)=0.1+0.5+0.275=\boxed{0.875}$$
4. h = 1000 m. h exceeds BOTH ranges (100 m and 500 m), so both structures contribute their full sill: $$\gamma(1000)=C_0+C_1+C_2=0.1+0.5+0.4=\boxed{1.000}$$ — the total sill of the nested model, and the value γ(h) approaches (from 500 m onward) for any larger lag.