Locally varying anisotropy (LVA): letting the direction of continuity change through the deposit

Why one ellipsoid is a compromise

Standard practice gives each domain one anisotropy: one rotation and one set of ranges. This assumes the deposit has the same continuity everywhere in the domain, which geostatisticians call stationarity. In reality continuity swings around a fold hinge, tightens in a shear and opens out in a limb. A single ellipsoid, rotated to a compromise angle, is wrong almost everywhere and least wrong on average.

Geologists often cope by cutting the domain into many small sub-domains, each with its own ellipsoid. That works, but it multiplies the number of variograms to fit, and it creates artificial boundaries in the estimate where the sub-domains meet.

Estimating the local anisotropy

At a chosen location, GEOREKA finds the nearest K samples and builds a moment-of-inertia tensor from them, weighted by grade so that the neighbour cloud is stretched along the directions in which grade persists. The eigenvectors of that tensor give the local directions of continuity, and the spread along each axis gives the ratios between the long, intermediate and short axes. Anisotropy can also be derived from dip and azimuth measurements in a structural data set, when the drilling is too thin to carry the orientation.

Two design choices keep this to a single dial. First, the neighbourhood is defined by a count, not a radius, so it shrinks where drilling is dense and grows where it is sparse. Second, the field is evaluated at a percentage of the sample points and carried to the rest, which both saves time and damps noise from local sampling accidents.

Making the field trustworthy

A raw anisotropy estimate inherits the biases of the drilling. GEOREKA includes three controls for this:

• Grade cut-off. Samples below the cut-off are removed before locations are chosen and neighbours are searched, so that waste or background cannot host anisotropy discs and bias the neighbourhood.
• Correction for sampling geometry. Clustered or patterned drilling stretches the tensor along the direction of the holes. The correction divides out the pattern of all samples, so the field reflects grade continuity and not where the holes were drilled.
• Scale selection. The neighbour count sets the scale of the estimate. Too few neighbours give a noisy field and too many average across different structure. GEOREKA looks at how stable the field is as the scale changes and picks the knee of that curve.

From the field to an estimate

A field of ellipsoids is only geometry until it enters the covariance, the function that decides how strongly two samples inform each other (see the covariance function). Letting the search ellipsoid change from point to point and plugging the distances into an ordinary covariance function does not give a valid model, because the resulting covariance is in general not positive-definite and the system may have no solution.

GEOREKA uses a construction by Paciorek and Schervish (2006) that measures the distance between two samples in the average of their two local ellipsoids and adds a normalisation term. It stays positive-definite for any admissible anisotropy field, and it reduces to ordinary anisotropic kriging when the field is constant. The estimate is made with support vector regression on this covariance, so the intuition from kriging carries over.

The result is an implicit function. There is no grid chosen in advance: the function can be evaluated afterwards at any resolution and on any support, as many times as you like. In our tests, about 180,000 points were evaluated in around 30 seconds on a consumer GPU (RTX 5070 Ti) and in around 45 seconds on a 24-core CPU. Timings depend on your machine and data.

Where it strains

• Faults. The method assumes continuity is locally smooth. Across a fault that offsets grade, that assumption is wrong, and the geology must be domained as usual.
• Sparse data. A fixed neighbour count in sparse ground means distant samples, possibly from different structural domains. The field looks confident but is a long-range average. Where drilling is thin, anchor the orientation to structural measurements or to the implicit model.
• High anisotropy ratios. Very elongated ellipsoids are numerically fragile, so the ratio is capped.
• Cost. Local tensors cost more per point than one global ellipsoid. Parallel and GPU evaluation keep that practical.

Check that it helps

An LVA field that looks more geological is not enough. Validate the field against mapped or logged structure, and cross-validate on held-out samples to confirm that LVA measurably improves prediction compared with a single ellipsoid. See validating a resource estimate.

Keep reading

LVA also feeds classification: distances to drilling can be measured under the local anisotropy, see resource classification methods.