Common kriging pathologies and how to spot them

Part 5 has built up the kriging machinery from §5.1's simple kriging through §5.7's cokriging. §5.8 closes by cataloging the things that GO WRONG in real workflows. None of these are theoretical defects — they're manifestations of the kriging system colliding with real data. Knowing the failure modes is what separates a textbook kriging operator from a defensible production geostatistician.

Pathology 1: Negative weights from the screen effect

Symptom: some kriging weights are negative. The estimate is a difference of weighted sums rather than purely a weighted average.

Why: the screen effect — when an outer sample is partially "screened" by closer samples between it and the query, the kriging system assigns it a NEGATIVE weight to correct for the over-counting. This is mathematically correct but can produce ESTIMATES OUTSIDE THE DATA RANGE (e.g., negative grade when all samples are positive).

Diagnostic: report (a) total positive weight Σw_i^+, (b) total negative weight Σw_i^-, (c) the "weight extremity" max(|w_i|). Estimates with |w_i^-| > 0.2 or weights beyond ±2 deserve scrutiny.

Fix: (i) increase the nugget effect of the variogram (downweights distant samples); (ii) restrict to non-negative weights via constrained kriging (preview §6); (iii) accept and document — sometimes negative weights are legitimate.

Pathology 2: Smoothing artefacts

Symptom: the kriged map has LESS variability than the data; extreme values are damped; the histogram is narrower than the data's.

Why: kriging is a low-pass filter. It minimises $E[(Z - Z^*)^2]$, which involves both bias (centring) AND variance (spread). The variance-minimising estimator always shrinks toward the mean — the "smoothing" is intrinsic, not a bug.

Diagnostic: compare kriged map histogram vs declustered data histogram (§1.2-§1.5). The kriged distribution should be NARROWER. If it's as wide as the data, something else is going on.

Fix: this is NOT a bug to fix — it's the kriging variance behaving exactly as it should. The fix for "I want realisations with the right histogram" is CONDITIONAL SIMULATION (Part 7), not kriging.

Pathology 3: Singular / near-singular kriging systems

Symptom: the LU decomposition fails or produces wildly extreme weights; condition number of K is enormous (10^12 or more); estimates fluctuate dramatically with tiny data perturbations.

Why: causes include (i) near-duplicate samples (§5.6's clustering pathology), (ii) Gaussian variogram with very short range giving near-singular K (Gaussian is mathematically permissible but numerically poor), (iii) all samples concentrated in a tiny region of search space.

Diagnostic: check K's condition number before solving; check weight magnitudes; check that no two samples are within numerical precision distance.

Fix: (i) add a small nugget to the diagonal (Tikhonov regularisation); (ii) decluster (remove redundant samples); (iii) switch from Gaussian to Spherical or Exponential variogram (numerically tamer near origin).

Pathology 4: Universal kriging extrapolation explosion

Symptom: with universal kriging (§5.3) using polynomial trend, estimates outside the convex hull of data are wildly extreme — sometimes orders of magnitude larger than any sample.

Why: polynomials extrapolate linearly (or worse, quadratically) beyond data range. Add a trend like $\beta_0 + \beta_1 x + \beta_2 x^2$; far from the data, the $\beta_2 x^2$ term dominates. The UK system honours the trend perfectly inside the data but cannot constrain it outside.

Diagnostic: krige at locations far from data and visually inspect the result. Extreme values = extrapolation explosion.

Fix: (i) restrict kriging to the convex hull of data; (ii) use OK or SK instead of UK if a trend is not essential; (iii) use a lower-order trend (linear instead of quadratic) to reduce extrapolation magnitude.

Pathology 5: Anisotropy mismatch

Symptom: kriged map has visible artefacts — geometric bands or streaks oriented differently from the geology.

Why: the variogram's anisotropy direction or ratio is mis-specified (§3.4 / §4.3). Kriging weights match the WRONG correlation structure.

Diagnostic: compute the variogram in 4-6 directions and verify it MATCHES what you fitted. Check the kriged map for visual streak / banding patterns inconsistent with known geology.

Fix: re-fit the anisotropic variogram (§3.4); re-orient the search ellipse to match (§5.6).

Pathology 6: Neighbourhood-induced discontinuities

Symptom: kriged map has discontinuities — abrupt jumps or breaks where the underlying field should be continuous.

Why: with a moving neighbourhood, the sample set changes as the query moves. When a sample drops out of the neighbourhood (because it falls outside the search radius), the kriging system changes structurally and the estimate jumps.

Diagnostic: difference the kriged map with itself shifted by one cell — abrupt large values indicate discontinuities.

Fix: use a LARGER neighbourhood (smoother sample-set transition at the cost of compute); use smoothing of the kriging weights along the moving neighbourhood; or use UNIQUE NEIGHBOURHOOD (all samples) at small datasets where compute is fine.

The defensive QA/QC checklist

Before signing off on a kriging deliverable, verify:

• Variogram model: visually fitted (§4.5), cross-validated (§6), supported by 4-direction analysis (§3.4).
• Neighbourhood: search-ellipse aligned with anisotropy, min/max counts documented, sector rules applied.
• Weights: maximum |w_i| reasonable (< 2 typically); negative weights documented if > 0.2 in absolute sum.
• Kriging variance: cross-validated calibration (mean and SD of standardized residuals near 0 and 1; §5.4).
• Maps: visual inspection at multiple zoom levels; histogram comparison to data; no extrapolation explosions.

Try it

• Reproduce the negative-weights pathology: 4 samples in a line, query just past the most distant. Observe negative weight on the closest sample.
• Add a 6th sample very close to an existing sample. Notice the condition number spike and weight extremity.
• Switch to Gaussian variogram with short range. Watch the kriging system become ill-conditioned.
• Move the query progressively far from the data. With a quadratic UK trend, watch the estimate diverge.
• Walk the QA/QC checklist on the current kriging configuration — is it defensible?

You receive a kriged porosity map from a colleague. What three documentation items would you ask for first before trusting the map?

What you now know

Production kriging fails in characteristic ways — negative weights, smoothing artefacts, singular systems, extrapolation explosions, anisotropy mismatch, neighbourhood discontinuities. Each has a specific diagnostic signature and a known fix. The defensive QA/QC checklist closes Part 5 with a defensible workflow. Part 5 — Kriging — is complete. Part 6 (Cross-validation and QC) systematises the diagnostic step; Part 7 (Sequential Gaussian Simulation) extends from estimation to UNCERTAINTY-quantification via multiple realisations.

Mark section complete →

References

• Deutsch, C.V., Journel, A.G. (1998). GSLIB: Geostatistical Software Library and User's Guide, 2nd ed. Oxford University Press. (Cataloges the GSLIB-era kriging-quality controls and the kt3d diagnostic outputs.)
• Vann, J., Jackson, S., Bertoli, O. (2003). "Quantitative kriging neighbourhood analysis for the mining geologist." AusIMM, 215–223. (THE applied reference on neighbourhood-induced pathologies.)
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford University Press. (Chapter 4 has a comprehensive treatment of kriging failure modes in petroleum applications.)
• Rossi, M.E., Deutsch, C.V. (2014). Mineral Resource Estimation. Springer. (Mining-context failure modes with explicit case studies and regulatory implications.)
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty, 2nd ed. Wiley. (The rigorous treatment of the screen effect and other theoretical foundations of kriging pathologies.)