Stationarity, ergodicity, and the practical compromise

§0.1 introduced spatial data and the regionalised-variable framework $Z(\mathbf{s})$. Now we tackle the foundational assumption that makes geostatistics computable: STATIONARITY. Without it, we couldn't pool variogram estimates across the field, kriging weights would have no defensible interpretation, and a single field couldn't inform us about its own population. With it, geostatistics works.

Strict (strong) stationarity

A random field ${Z(\mathbf{s}) : \mathbf{s} \in D}$ is STRICTLY STATIONARY if the joint distribution of any finite set $(Z(\mathbf{s}_1), \ldots, Z(\mathbf{s}_k))$ is INVARIANT under translation:

This is a strong condition — the entire joint distribution structure is invariant in space. Used in theoretical results.

Weak (second-order) stationarity

Most of geostatistics relies on a WEAKER version:

• $E[Z(\mathbf{s})] = \mu$ (constant mean across space).
• $\mathrm{Cov}(Z(\mathbf{s}), Z(\mathbf{s} + \mathbf{h})) = C(\mathbf{h})$ (covariance depends only on the lag, not on absolute location).

This is what kriging and variogram-modeling actually need. The covariance function C(h) is the foundation; the variogram γ(h) = C(0) - C(h) is its complement.

The INTRINSIC HYPOTHESIS — when even weak stationarity fails

For some processes, mean E[Z] is constant but variance Var[Z] is infinite (e.g., Brownian motion). The intrinsic hypothesis relaxes weak stationarity:

• $E[Z(\mathbf{s}) - Z(\mathbf{s} + \mathbf{h})] = 0$.
• $\mathrm{Var}[Z(\mathbf{s}) - Z(\mathbf{s} + \mathbf{h})] = 2\gamma(\mathbf{h})$ depends only on h.

Weakly stationary ⇒ intrinsic; intrinsic ⇏ weakly stationary. Intrinsic is enough for variogram-based estimation; it's the workable default in mining and reservoir geostatistics.

Ergodicity: one realisation is enough

ERGODICITY says: spatial averages over a single (large) realisation converge to ensemble averages over many realisations.

This matters because in practice we have ONE earth — one field — and we'd like its variogram estimate to inform us about the population, not just this realisation. Ergodicity is what makes this leap valid. Practical test: if your field is "large enough" relative to the correlation range, spatial sampling is informative about the ensemble.

What goes wrong without stationarity

• Trend: mean varies with location (e.g., porosity increases northward). Variogram estimate is contaminated by trend; kriging weights are biased. Fix: detrend first, krige residuals (universal kriging or KED — §5.3).
• Regime change: different statistics in different zones (e.g., different facies). Pooling them gives a chimaera variogram fitting no zone. Fix: zone-by-zone modeling.
• Local heteroscedasticity: variance changes spatially. Variogram's sill is misleading; kriging variance miscalibrated.

Try it

• Strictly Stationary scenario: heat-map looks uniformly textured. Local-window mean profile is FLAT (within sampling noise). Left-half mean ≈ right-half mean. Verdict: PLAUSIBLY stationary.
• Trend scenario: heat-map shows visible left-to-right gradient (cool → warm). Local-window mean profile slopes UP. Left-half mean ≈ 3, right-half ≈ 7. NON-STATIONARY. → universal kriging needed.
• Regime-change scenario: heat-map shows two zones with different colour palettes. Local-window mean has a jump. Variances may differ too. → zone-by-zone modeling needed.
• Run the same scenario at multiple seeds. For Stationary, the qualitative diagnosis stays the same. For Trend / Regime, the structure stays the same even though specific values shift.
• Look at the heat-map alone (without the profile) for the regime-change case. Sometimes the eye doesn't catch the regime change in a noisy realisation; the profile makes it unmistakable. Always check stationarity QUANTITATIVELY, not just visually.

A team has well data with porosity values 0.1-0.3 in the northern half and 0.05-0.15 in the southern half. They fit ONE variogram to the whole field. What are the two consequences of this stationarity violation, and what should they do instead?

What you now know

Stationarity assumptions: strictly stationary (full distributional invariance) → weakly stationary (constant mean + lag-only covariance) → intrinsic (constant mean of differences + lag-only semivariance). Geostatistics works under the WEAKEST you can defend. Ergodicity lets one realisation stand in for many. Trend and regime change are the common stationarity failures; universal kriging and zone-by-zone modeling are the standard fixes. §0.3 turns to a related issue — the volume problem (support and scale).

Mark section complete →

References

• Matheron, G. (1971). The Theory of Regionalized Variables and Its Applications. Cahiers du CMM, Fascicule 5, Paris. (Stationarity definitions in their original form.)
• Cressie, N. (1993). Statistics for Spatial Data. Wiley. (Chapter 2 — stationarity formalism.)
• Chilès, J.-P., Delfiner, P. (2012). Geostatistics: Modeling Spatial Uncertainty, 2nd ed. Wiley.
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press. (Chapter 4 — stationarity + intrinsic hypothesis in mining context.)
• Wackernagel, H. (2003). Multivariate Geostatistics, 3rd ed. Springer. (Section 2.3 — practical stationarity diagnostics.)