🎓 Lesson 2
D2
Variograms, Anisotropy, and Stationarity — What Still Holds?
A variogram shows how much ore grade changes as distance increases, helping us decide how far apart drill holes need to be to reliably estimate the resource.
🎯 Learning Objectives
- ✓ Calculate experimental variogram values from drill hole assay data using lag tolerance and azimuth sectors
- ✓ Analyze directional variograms to identify and quantify geometric anisotropy (e.g., horizontal vs. vertical range ratios)
- ✓ Explain the implications of violating intrinsic stationarity on kriging estimation variance and grade control reliability
- ✓ Design sampling strategies (spacing, orientation, density) based on variogram-derived ranges and anisotropy ratios
- ✓ Apply variogram modeling best practices (e.g., nested structures, permissible models) to ensure numerical stability in AI-driven grade modeling workflows
📖 Why This Matters
In modern AI-powered orebody delineation, the variogram isn’t just legacy theory—it’s the DNA of spatial uncertainty. If your variogram misrepresents continuity (e.g., underestimating vertical range), your AI grade model will over-smooth high-grade shoots or misplace stopes—costing millions in dilution or missed ore. At Rio Tinto’s Koodaideri, a 15% error in vertical range estimation led to 8% grade reconciliation loss in first-year production. Understanding what still holds—and what doesn’t—in variogram assumptions is critical for trustworthy AI integration.
📘 Core Principles
Variograms rest on two key assumptions: intrinsic stationarity (mean constant, variance of differences depends only on separation vector h) and ergodicity (spatial averages ≈ ensemble averages). While classical geostatistics treats these as prerequisites, modern practice acknowledges their frequent violation—especially in structurally complex or hydrothermally altered zones. Anisotropy arises from geological processes (e.g., bedding, faulting, fluid flow) and manifests as differing ranges or sills along azimuths; it must be modeled explicitly (e.g., using geometric anisotropy with stretch factors) rather than ignored. Stationarity is now assessed *locally*: grade control domains are subdivided into geologically homogeneous units where intrinsic stationarity is defensible—validated via residual analysis and cross-validation diagnostics, not assumed globally.
📐 Experimental Variogram Calculation
The experimental (empirical) variogram γ̂(h) estimates spatial variability at lag h by averaging squared differences of sample pairs separated by approximately h. It is computed for multiple lags and directions to detect anisotropy. Proper binning (lag tolerance, azimuth tolerance) ensures robustness against sparse or irregular sampling.
💡 Worked Example
Problem: Given 6 assay pairs at ~20 m east-west separation: (0.8, 1.2), (1.5, 1.1), (2.3, 2.6), (0.9, 1.0), (3.1, 2.9), (1.7, 1.9) g/t Au. Lag tolerance = ±2 m, azimuth tolerance = ±15°.
1.
Step 1: Compute squared differences: (0.4)²=0.16, (0.4)²=0.16, (0.3)²=0.09, (0.1)²=0.01, (0.2)²=0.04, (0.2)²=0.04
2.
Step 2: Sum and divide by 2N(h): sum = 0.50; N(h) = 6 → γ̂(20 m) = 0.50 / (2 × 6) = 0.0417 g²/t²
3.
Step 3: Compare to typical Au variogram sill (0.1–0.8 g²/t²): 0.0417 is plausible for short lag — consistent with strong local continuity.
Answer:
γ̂(20 m) = 0.042 g²/t², indicating low variability at this scale—supporting 20 m E-W drilling spacing for near-term grade control.
🏗️ Real-World Application
At Newmont’s Boddington Mine (WA), structural anisotropy was identified via azimuthal variograms showing 45-m horizontal range (NW-SE) vs. only 12-m vertical range in the oxide cap. Traditional isotropic kriging overestimated continuity vertically, causing 12% overestimation of minable tonnage in pit shells. By applying geometric anisotropy (vertical stretch factor = 3.75) and locally validating stationarity within lithological domains (saprolite vs. claystone), they reduced grade reconciliation error from ±22% to ±9% and increased stope recovery by 7% in the first year of AI-assisted longhole planning.
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