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Kriging VarianceEdit

Kriging variance is the quantitative expression of uncertainty that arises when predicting the value of a spatial process at unsampled locations using kriging. It sits alongside the kriging predictor as a companion measure: the predictor gives the best linear unbiased estimate under a specified spatial model, while the variance conveys how much doubt should be attached to that estimate given the same model and sampling design. In practice, the kriging variance is a deterministic quantity computed from the chosen variogram (or covariance function) and the locations of observed samples, and it tends to shrink as data become denser or the spatial structure is captured more accurately. However, it is not a blanket approval for perfect foresight; misspecification of the model or unaccounted nonstationarity can distort what the variance truly implies about future error.

For many applied fields, the kriging variance is a core tool in risk assessment and decision making. Resource estimation in mining Kriging projects, environmental monitoring geostatistics, and land-use planning often rely on a clear, quantitative sense of how uncertain the predicted values are at locations where no data exist. The variance complements the actual prediction by providing a disciplined way to construct confidence intervals and to compare alternative sampling schemes or models. The elegance of the method lies in its explicit accounting for spatial correlation: the same data that drive the estimate also inform the projected reliability of that estimate through the kriging variance.

Fundamentals

What kriging tries to solve

Kriging seeks to predict Z(s0), the value of a spatial process at a location s0, by forming a weighted sum of observed values Z(si) at nearby sample sites. The weights are chosen to minimize the mean squared prediction error under an assumed spatial model, while often enforcing an unbiasedness constraint. The result is a predictor Z*(s0) = ∑i λi Z(si) with a set of weights λi that depend on the spacing and the assumed spatial dependence structure. The same structure that determines the weights also yields a quantity called the kriging variance, denoted σ_k^2, which quantifies the expected squared difference between Z(s0) and Z*(s0) given the model and the observed data.

Core ideas and components

Variogram and covariance: The spatial dependence is summarized by a variogram γ(h) or a covariance function C(h). These functions describe how similarity between observations decays with distance and direction. See variogram and covariance function for details.
Stationarity and drift: Standard kriging relies on some form of stationarity (often second-order stationarity) so that spatial structure is stable across the region of interest. When the mean varies systematically, methods like universal kriging incorporate a drift term.
Types of kriging: Different formulations handle drift, trends, and multiple variables in varied ways. Common variants include ordinary kriging, universal kriging, and cokriging; there are also block-based and local approaches such as block kriging for resolution-aligned predictions.
The kriging system and weights: The weights λi solve a linear system derived from the variogram/covariance model, sometimes with Lagrange multipliers to enforce unbiasedness. The solutions yield both Z*(s0) and σ_k^2.

What the kriging variance represents

Conditional uncertainty given the model: σ_k^2 is the predicted mean squared error of Z(s0) conditional on the observed data and the assumed spatial model. It can be used to form prediction intervals around Z*(s0).
Dependence on sampling design: The configuration of sample sites (density, proximity, and anisotropy) directly affects σ_k^2. Densely sampling or aligning sites with strong directional structure typically reduces the variance.
Dependence on the structure model: The chosen variogram or covariance function governs how much the nearby observations “pull” the prediction at s0, and thereby determines σ_k^2. Misspecification can lead to under- or overestimation of uncertainty.

Practical interpretation and limitations

Not a universal error bound: σ_k^2 assumes the model is correct. If the field shows nonstationarity, outliers, or non-Gaussian behavior, the variance may misrepresent real predictive risk.
Distinction from empirical error: σ_k^2 reflects conditional uncertainty under the model, not the actual error observed in past predictions unless the model is well-calibrated. Cross-validation and diagnostic checks are important complements.
Role in decisions: In resource management or environmental policy, σ_k^2 feeds into risk-informed decisions, sensitivity analyses, and the design of future sampling campaigns. It is most informative when paired with transparent model assumptions and a clear accounting of potential model error.

Variants and extensions

Local kriging and neighborhood schemes: For large datasets, restricting calculations to a neighborhood of s0 or using approximate methods can control computational cost while preserving useful uncertainty estimates.
Block kriging: When the quantity of interest is defined over a region (a block) rather than a point, variance is propagated to the block solution, which is crucial for applications in mining and environmental assessment.
Cokriging and multi-variable cases: When multiple related spatial fields are observed, cokriging can improve predictions and their associated variances by leveraging cross-covariances. See cokriging and Bayesian kriging for related ideas.
Bayesian approaches: Some practitioners incorporate parameter uncertainty directly via Bayesian kriging or other probabilistic frameworks, treating the variogram or covariance parameters as random and propagating that uncertainty into the predictive distribution.

Controversies and debates

Model risk versus data-driven claims: A central debate concerns how much one should trust the kriging variance when the underlying variogram is inferred from data with limited support. Critics warn that overconfident variance estimates can mislead decisions if the spatial model is misspecified. Proponents emphasize that, when properly validated, the kriging variance remains a principled, model-based quantification of uncertainty.
Accounting for uncertainty in the model itself: Some argue that the standard kriging variance omits uncertainty about the model parameters. In response, Bayesian and ensemble approaches seek to incorporate variogram-parameter uncertainty and to present predictive distributions that reflect both sampling variability and model uncertainty.
Nonstationarity and drift: In fields where spatial dependence changes across the region, the assumption of stationarity becomes questionable. Advocates for nonstationary approaches contend that kriging variance under a rigid stationary model can be misleading, while supporters of simpler, stationary models value transparency and interpretability and argue that drift terms (as in universal kriging) offer a pragmatic middle ground.
Computational and design trade-offs: As data volumes grow, some critique the balance between computational burden and the quality of uncertainty quantification. Methods that approximate the full kriging system can save time but may also distort variance estimates if not carefully calibrated.
Practical versus theoretical purity: A conservative stance often emphasizes the clarity and traceability of variance estimates—explicit assumptions, straightforward validation, and clear communication to decision-makers—over highly flexible but opaque modeling frameworks. In many applications, such an approach is valued for its defensibility and reproducibility.