Indicator variograms revisited

Continuous variograms (Part 3) describe how grade smoothly varies across space. INDICATOR variograms describe a different aspect: how do HIGH-VALUE REGIONS connect spatially? Defined via the indicator transform $I(x; c) = \mathbb{1}[z(x) > c]$, the indicator variogram $\gamma_I(h)$ characterises the patch-level structure of values above cutoff $c$. Different cutoffs reveal different scales of spatial connectivity.

The indicator transform

For a continuous random field $z(x)$ and a chosen cutoff $c$:

The transformed field $I(x; c)$ takes only values 0 and 1. It "indicates" whether the original field exceeds the cutoff at each location.

The indicator variogram

The indicator variogram is the variogram of the indicator field:

Since $I$ takes values 0/1, the squared difference is 0 (if both are equal) or 1 (if they differ). So:

$\gamma_I(h; c) = \frac{1}{2} P[I(x + h; c) \ne I(x; c)] = P[I(x + h; c) = 1, I(x; c) = 0] = \text{Cov}(I(x+h), I(x))/p (1-p)$ etc.

Practically: the indicator variogram measures the probability that nearby points differ in their indicator value. As h grows, this probability grows from 0 (perfectly connected at h=0) to a sill (independence).

What indicator variograms reveal

The indicator variogram's range reveals the SCALE of high-value connectivity:

• Short indicator range: high-value patches are SMALL. Frequent transitions between high and low values.
• Long indicator range: high-value zones are LARGE CONNECTED REGIONS. Sustained high-grade areas.

Critical insight: the indicator variogram CAN HAVE A DIFFERENT RANGE than the continuous variogram. A field can have smooth long-range continuous variation but short-range indicator structure (small connected patches separated by gaps). This dual perspective is essential for understanding spatial structure.

Cutoff-dependence

The indicator variogram depends on the chosen cutoff c. At different cutoffs, you see different spatial-connectivity patterns:

• Low cutoff: most data exceed it; indicator is mostly 1s; range close to the continuous variogram range.
• Median cutoff: balanced 0s and 1s; reveals "middle-grade" connectivity.
• High cutoff: rare 1s; indicates concentration of extreme high-grade zones.

Modern practice: compute indicator variograms at multiple cutoffs (e.g., deciles) for full cutoff-dependent characterisation.

Applications

• Indicator kriging (§8.2): kriging on indicators to estimate P(z > c) at unsampled locations.
• Indicator simulation (SISIM, §8.3): generates facies or category realisations.
• Connectivity analysis: identifying connected high-grade zones for development planning.
• Categorical data modelling: facies models for reservoirs.

Try it

• Defaults: range = 2.0, cutoff = 3.0. The continuous field z(x) (blue) has smooth correlation. The indicator I(x; 3.0) is binary; red shaded regions show where z > 3.0.
• Compare the continuous variogram (blue) and indicator variogram (red) at the bottom. They have similar SHAPES but the indicator sill is lower (max ≈ 0.5 for balanced binary).
• Drag cutoff to 4.5 (high). Fewer above-cutoff points; indicator variogram has lower sill (small fraction × large fraction). Indicates rare high-grade events.
• Drop cutoff to 1.5 (low). Most points exceed; indicator is mostly 1s; indicator variogram has very low sill.
• Drag range to 5.0. Continuous field smoother. Indicator variogram range is also longer (connected high-grade zones). The two variograms move together as the underlying spatial structure changes.

A copper-mining deposit has continuous variogram range = 50 m but indicator variogram range at 1.0% Cu cutoff = 20 m. What does this difference mean for the deposit's structure?

What you now know

Indicator variograms describe the spatial-connectivity structure of high-value regions, separate from the continuous variogram's view. Cutoff-dependent: each cutoff reveals different connectivity scales. Foundation for indicator kriging (§8.2), SISIM (§8.3), facies modelling (§8.4). Comprehensive cutoff-by-cutoff characterisation is modern best practice.

Mark section complete →

References

• Journel, A.G. (1983). "Nonparametric estimation of spatial distributions." Mathematical Geology 15(3), 445–468. (Original indicator framework.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford.
• Deutsch, C.V., Journel, A.G. (1998). GSLIB, 2nd ed. Oxford.
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford.
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty, 2nd ed. Wiley.