Indicator kriging for facies probabilities

Indicator kriging (IK) replaces the continuous variable $z(x)$ with its INDICATOR transform $I(x;c) = \mathbb{1}[z(x) > c]$ and then kriges that 0/1 field. Because the kriged value of a binary variable is bounded in $[0,1]$ and equals the conditional expectation of the indicator, IK gives the conditional PROBABILITY $P(z(x) > c \mid \text{data})$ directly — without assuming Gaussianity, lognormality, or any specific distributional form. This is its central advantage over (multi)Gaussian methods.

The IK estimator

Given hard data ${z(x_i)}${i=1}^N{z(xi​)}i=1N​, fix a cutoff $c$ and form the indicators $i$\alpha = \mathbb{1}[z(x_\alpha) > c]. With sample fraction $p = \frac{1}{N}\sum_\alpha i_\alpha$ as a prior mean, simple indicator kriging at an unsampled location $x_0$ solves:

where $C_I(\cdot; c)$ is the indicator covariance derived from the indicator variogram $\gamma_I(\cdot; c)$ (§8.1). The estimator is then:

Crucially the RIGHT-HAND SIDE IS A PROBABILITY (in theory bounded in $[0,1]$; in practice clipped). With ordinary IK the kriging system carries the usual unbiasedness constraint $\sum_\alpha w_\alpha = 1$.

From a single cutoff to the full conditional CDF

Run IK at $K$ cutoffs $c_1 < c_2 < \cdots < c_K$. At each unsampled location $x_0$ you then have a discrete CDF:

From this you can extract any conditional quantile, the conditional mean, prediction intervals, exceedance probability for any economic/regulatory cutoff — all WITHOUT a parametric distributional model. This is the original motivation for IK (Journel, 1983).

Order-relation violations and corrections

Raw multi-cutoff IK does NOT enforce monotonicity in $c$. You can find $\hat{F}(c_1 \mid x_0) < \hat{F}(c_2 \mid x_0)$ even when $c_1 < c_2$. This is impossible for a true conditional CDF and arises because IK at each cutoff uses a SEPARATE variogram and separate kriging system — nothing ties the cutoffs together algebraically.

Standard fix: an ORDER-RELATION CORRECTION step. The simplest sweep computes a monotone non-increasing envelope (downward pass then upward pass, average the two). More elaborate corrections solve a constrained least-squares or quadratic-programming problem to project the raw probabilities onto the simplex of valid CDFs. Production codes (GSLIB's ik3d, e.g.) always apply such a step before any downstream use.

IK vs Gaussian methods — when does IK shine?

• Non-Gaussian / strongly skewed distributions where the conditional CDF is not well captured by a multiGaussian model.
• Multi-modal or threshold-driven variables (e.g., facies indicators, ore vs waste, net vs gross).
• When the connectivity structure of high-value zones differs by cutoff — IK lets each cutoff use its OWN variogram (§8.1).
• When PROBABILITIES, not single-point estimates, are the deliverable: exceedance maps, facies probabilities, risk of contamination above MCL.

Categorical / facies IK

For categorical data with $K$ facies, define one indicator per facies: $I_k(x) = \mathbb{1}[\text{facies}(x) = k]$. Krige each indicator field separately, then enforce $\sum_k \hat{P}_k(x_0) = 1$ via a normalisation step (or constrained kriging). The result is a per-location DISCRETE PROBABILITY MASS FUNCTION over facies — the input to sequential indicator simulation (SISIM, §8.3) and to facies-conditioned modelling (§8.4).

Try it

• Defaults: cutoff = 5.0, indicator-variogram range = 5.0. The TOP canvas shows the true continuous field (blue) and the 12 hard-data points (red dots above cutoff, dark below). The MIDDLE canvas shows the IK probability P(z > c | data) as a blue curve vs the true 0/1 indicator (red step). Where the field is far above/below the cutoff, IK approaches 1/0; in transition zones IK gives intermediate probabilities.
• Drag the cutoff slider up to 6.0. Fewer hard data exceed the cutoff; the IK probability flattens toward 0 nearly everywhere with peaks only where the field was strongly above the cutoff. The conditional CDF moves to the right.
• Drag the cutoff down to 4.0. Most hard data are above; IK probability is near 1 everywhere except in the deepest troughs of the field. Note the asymmetry between the cutoff = 4.0 and cutoff = 6.0 regimes — that asymmetry IS the local conditional CDF.
• Drag the indicator-range slider to 1.5 (short). The IK probability becomes very local: it hugs each hard-data indicator value and flattens to the prior p between data. Now drag to 10.0 (long). The IK estimate becomes very smooth and barely varies along x.
• Look at the BOTTOM canvas: P(z > c | data) at five cutoffs c ∈ {4.0, 4.5, 5.0, 5.5, 6.0}. With order-relation correction ON, the curves should DECREASE in c at every x. Toggle the correction OFF — the readout reports the count of violations. Inspect locations where the raw curves cross.
• Click Resample. Different hard-data configurations change the IK estimates substantially; the conditional probability is genuinely data-dependent.

A reservoir engineer asks for the probability that net-to-gross exceeds 0.6 at every unsampled location, given 40 well NTG measurements. Why is IK on the indicator I(x; 0.6) a defensible answer, and what is the single biggest weakness you should disclose in the report?

What you now know

Indicator kriging krieges the binary indicator field $\mathbb{1}[z > c]$ and interprets the kriged value DIRECTLY as the conditional probability $P(z > c \mid \text{data})$. Stacked over multiple cutoffs it estimates the full conditional CDF without any distributional assumption — the price is potential order-relation violations that require a corrective sweep. IK is the natural tool for facies probabilities, exceedance-probability maps, and as the building block of sequential indicator simulation (§8.3) and categorical-facies modelling (§8.4).

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References

• Journel, A.G. (1983). "Nonparametric estimation of spatial distributions." Mathematical Geology 15(3), 445–468. (Original indicator-kriging paper.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford. (Chapter on indicator approaches.)
• Deutsch, C.V., Journel, A.G. (1998). GSLIB, 2nd ed. Oxford. (See ik3d and order-relation correction code.)
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty, 2nd ed. Wiley. (Detailed treatment of disjunctive vs indicator kriging.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford. (Applied IK and SISIM workflows.)