🎓 Lesson 3
D2
Calculating Experimental Variogram Parameters
An experimental variogram shows how ore grade changes with distance between drill holes, helping us understand how predictable the grade is across the deposit.
🎯 Learning Objectives
- ✓ Calculate the experimental variogram γ(h) for a set of drill hole assays at specified lag distances
- ✓ Interpret variogram parameters (nugget, sill, range) to assess data quality and spatial continuity
- ✓ Diagnose common variogram artifacts (e.g., poor azimuthal sampling, compositing bias) from plot morphology
- ✓ Apply directional variogram analysis to identify geological anisotropy in ore distribution
📖 Why This Matters
In grade control and orebody modeling, assuming grades are randomly distributed leads to over- or under-estimation of resource confidence—and costly misallocation of mill feed. The experimental variogram is your first diagnostic tool: it tells you *how far* grade remains predictable, *how much* uncertainty is inherent at zero distance (nugget), and *whether* the deposit behaves isotropically. Without it, AI-powered delineation models train on flawed spatial assumptions—turning sophisticated algorithms into expensive curve-fitting exercises.
📘 Core Principles
Spatial variability is not random—it reflects geologic processes (e.g., hydrothermal flow, sedimentary layering). The experimental variogram formalizes this by measuring dissimilarity as a function of separation vector h = |x_i − x_j|. At h = 0, γ(0) estimates the combined effect of measurement error and micro-scale heterogeneity (nugget). As h increases, γ(h) rises until it levels off at the sill—the total variance of the attribute. The lag distance where γ(h) reaches ~95% of the sill is the range: beyond this, samples are effectively uncorrelated. Directional variograms reveal anisotropy when ranges differ significantly across azimuths (e.g., longer range along bedding than across it), directly informing drill spacing and block model support.
📐 Key Calculation
The experimental variogram is computed pairwise for all sample pairs within a lag tolerance. It requires careful binning (lag tolerance ±δh) and directional constraints to avoid smearing. The estimator is unbiased only if stationarity holds—but robust interpretation focuses on relative shape, not absolute fit.
💡 Worked Example
Problem: Given 6 assay pairs at lag h = 10 m: (0.8, 1.2), (1.5, 1.3), (2.1, 2.4), (0.9, 1.1), (1.7, 1.9), (3.0, 2.8) — all in g/t Au. Lag tolerance δh = ±1 m. Calculate γ(10).
1.
Step 1: Compute squared differences: (0.4)²=0.16, (0.2)²=0.04, (0.3)²=0.09, (0.2)²=0.04, (0.2)²=0.04, (0.2)²=0.04
2.
Step 2: Sum squared differences = 0.16 + 0.04 + 0.09 + 0.04 + 0.04 + 0.04 = 0.41
3.
Step 3: Apply formula γ(h) = 1/(2N) × Σ(z(x_i) − z(x_j))² = 1/(2×6) × 0.41 = 0.41 / 12 = 0.0342 g²/t²
Answer:
γ(10) = 0.034 g²/t². This falls within typical nugget-influenced range for gold in shear-hosted deposits (0.02–0.05 g²/t²), suggesting moderate micro-variability and acceptable assay precision.
🏗️ Real-World Application
At the Cadia East porphyry copper-gold deposit (NSW, Australia), geologists computed east-west vs. north-south experimental variograms from 20,000 m of RC drilling. The east-west variogram showed a range of 85 m for Cu%, while north-south range was only 42 m—confirming structural control along NW-trending faults. This anisotropy directly informed the 40 m × 80 m drill grid orientation for infill drilling and constrained the search ellipsoid in subsequent ordinary kriging, reducing grade estimation variance by 22% compared to isotropic modeling (Newcrest Mining, 2021 Technical Report).
✏️ Student Exercise
Using the provided dataset of 12 copper assay pairs (in % Cu) at lag h = 25 m: [(0.42, 0.39), (0.51, 0.53), (0.67, 0.62), (0.38, 0.41), (0.74, 0.76), (0.49, 0.47), (0.55, 0.58), (0.63, 0.60), (0.44, 0.46), (0.71, 0.69), (0.52, 0.54), (0.68, 0.65)], calculate γ(25). Then interpret: (a) Is the nugget effect dominant? (b) What does the magnitude suggest about sampling representativeness?
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