Neighbourhood selection and search ellipsoids

The §§5.1–5.5 kriging derivations assumed all N samples enter the system. In real datasets N is large enough (thousands of drill holes; millions of seismic traces) that solving the full kriging system at every grid cell is computationally impossible AND numerically unstable. The KRIGING NEIGHBOURHOOD is the subset of samples actually used for each estimate. Choosing it is a practical art on top of the algebraic theory.

Why a moving neighbourhood matters

Three reasons to restrict the sample set:

• Computational: solving a 5000×5000 kriging system at every grid cell is hours per realisation. Restricting to the nearest 20-50 samples reduces this to milliseconds.
• Numerical stability: large covariance matrices with samples far beyond the variogram range have rows / columns of nearly zero off-diagonals — near-singular systems with large condition numbers.
• Local stationarity: real geology violates global stationarity. A moving neighbourhood implicitly assumes only LOCAL stationarity — the mean and variogram are constant within the neighbourhood, not globally. This is much more defensible.

The search ellipse / ellipsoid

The neighbourhood is defined by a search ellipse (2D) or ellipsoid (3D), typically aligned with the variogram anisotropy from §3.4 / §4.3:

• Major axis = variogram's major direction.
• Major search radius ≈ 1.5 to 3 times the variogram's major range.
• Minor axes scaled by the anisotropy ratio.

The "1.5–3× range" rule: samples beyond 1.5× range contribute essentially zero kriging weight under most variogram models (Spherical reaches sill at the range itself; Exponential at 3×range). Including them adds rows of mostly-zero covariance to the system — computationally wasteful and numerically destabilising.

Min and max sample counts

Within the search ellipse, an explicit cap on the number of samples is standard. Typical defaults (GSLIB kt3d):

• Min = 4–8 samples. Below this, the kriging system is too poorly conditioned to trust; skip the location (output "missing") or krige with the global mean as fallback.
• Max = 20–40 samples. Above this, the system becomes large enough to slow down computation while adding very little new information (the screen effect makes distant samples redundant once close ones are included).

Quadrant and octant rules

Even within a properly-sized search ellipse, samples can be ALL ON ONE SIDE of the query point. The kriging system then has poor angular coverage; the estimate is one-sided. Sector rules force spatial spread:

• Quadrant rule (2D): divide the ellipse into 4 quadrants around the query; max N/4 samples per quadrant. Ensures angular spread.
• Octant rule (3D): 8 octants; max N/8 per octant.

Without sector rules, dense sampling on one side of the query (typical near edges of a domain) creates "lopsided" estimates where the kriged map has visible directional bias toward the dense region.

The clustering pathology inside the neighbourhood

If multiple samples are very close together inside the neighbourhood (e.g., twinned drillholes, repeated measurements), the K matrix has rows / columns that are nearly identical — near-rank-deficient. The kriging weights can become extreme: large positive on one of the cluster, large negative on the other, fighting each other. This is the "kriging fights itself" pathology.

Two responses:

• Add a small numerical "nugget" to the diagonal of K to regularise (computational fix, not physical).
• Decluster before kriging — remove redundant samples or downweight them.

Documenting the neighbourhood

The kriging output depends materially on the neighbourhood choice. Defensible reporting always includes:

• Search ellipse: major/medium/minor radii, orientation (azimuth, dip, plunge).
• Min and max sample counts.
• Sector rule (if any) and per-sector max.
• How "missing" cells (insufficient samples) are handled.

Without this, the kriging is not reproducible — the same dataset can produce different maps with different neighbourhood settings.

Try it

• In the widget, drag the query point. Watch the search ellipse follow it and the highlighted neighbourhood samples change. Observe the kriging weights rebalance.
• Shrink the search radius to 0.5× the variogram range. Notice the kriging variance grows substantially — you're ignoring samples that DO carry information.
• Grow the search radius to 5× the range. Notice the kriging weights on the most distant samples are essentially zero; you're paying compute cost for no benefit.
• Move the query toward one edge of the domain where samples are dense on one side only. Watch the directional bias in the estimate.
• Activate the quadrant rule. Observe how the selected sample set becomes more angularly balanced and the directional bias decreases.

If the variogram has a major range of 200 m and a minor range of 50 m (anisotropy ratio 4:1), what search-ellipse dimensions would you propose for a kriging study with target neighbourhood size 25 samples? Why?

What you now know

Kriging neighbourhoods are the practical bridge between the textbook kriging system and production geostatistics. The search ellipse aligns with variogram anisotropy; min/max sample counts balance numerical stability against compute time; sector rules force angular spread. Documenting these choices is essential for reproducibility. §5.7 introduces cokriging, where the search ellipse must accommodate samples of BOTH the primary and the secondary variable. §5.8 catalogues common kriging pathologies, several of which trace back to neighbourhood mis-specification.

Mark section complete →

References

• Deutsch, C.V., Journel, A.G. (1998). GSLIB: Geostatistical Software Library and User's Guide, 2nd ed. New York: Oxford University Press. (The kt3d program documentation specifies the canonical search-ellipse + min/max/sector control conventions used industry-wide.)
• Vann, J., Jackson, S., Bertoli, O. (2003). "Quantitative kriging neighbourhood analysis for the mining geologist." Proc. 5th Int. Mining Geology Conf., AusIMM, 215–223. (THE practical reference on neighbourhood tuning for resource estimation.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. New York: Oxford University Press. (§5.4 covers neighbourhood selection in the context of moving-window kriging.)
• Isaaks, E.H., Srivastava, R.M. (1989). An Introduction to Applied Geostatistics. Oxford University Press. (Practical neighbourhood selection chapter with worked-through examples on mining datasets.)
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty, 2nd ed. Wiley. (§3.4.3 develops the screen effect rigorously, justifying why distant samples can be safely excluded.)