Kriging VariantsEdit
Kriging variants are a family of geostatistical tools that extend the basic idea of kriging to handle a wider range of real-world data patterns. At their core, these methods estimate values at unsampled locations by leveraging the spatial correlation structure of observed measurements. The approach blends theory and practical intuition: measurements nearby in space tend to be more alike than those farther apart, and this local similarity can be codified into a set of weights that yield best linear unbiased predictions along with uncertainty estimates. Over decades, practitioners in mining, environmental management, hydrology, agriculture, and infrastructure planning have adapted kriging to accommodate trends, auxiliary information, multiple variables, and computational realities.
Kriging variants are distinguished by how they treat the mean function, incorporate external information, handle non-stationarity, and manage multiple variables. While all variants share the same probabilistic backbone, the choice among them depends on data characteristics, the decision context, and the tolerance for bias or variance in the predictions. Proponents emphasize that carefully chosen kriging variants deliver more accurate estimates and honest uncertainty quantification, especially when data are scarce or when decisions hinge on spatial risk. Critics tend to stress the importance of model simplicity, transparent assumptions, and rigorous validation to avoid overfitting or misinterpretation. In practice, the best approach is often a transparent blend of methods tailored to the problem at hand, validated against held-out data and aligned with the decision-maker’s objectives.
Kriging Variants
Simple Kriging
Simple Kriging assumes that the local mean is known and constant across the domain. In practice, this variant is most appropriate when there is reliable prior information about the overall level of the variable of interest and the data are moderately dense. The method yields predictions that minimize squared error given the known mean and the spatial covariance structure. It is straightforward to implement, but the requirement of a known mean limits its applicability in many field situations.
Ordinary Kriging
Ordinary Kriging is the workhorse of geostatistics. It assumes the local mean is unknown but constant within the neighborhood used for prediction. This makes the method widely applicable because it requires less prior information about the field. The resulting predictor is linear, unbiased, and, under second-order stationarity, optimal in the sense of minimum variance among all unbiased linear estimators. Ordinary Kriging balances practicality with statistical rigor and remains a default choice when there is no strong trend or covariate information to drive the mean.
Universal Kriging (Kriging with Trend)
Universal Kriging extends the basic framework to allow deterministic trends in the data. By modeling the mean as a known or estimable function of location (for example, a linear or polynomial drift), it accommodates non-stationarity in the first moment. This is useful in fields where systematic gradients exist, such as long pipelines, road networks, or ore bodies with changing geometry. The approach can be more data-hungry than ordinary kriging, but it reduces bias when a trend is present.
Regression Kriging
Regression Kriging combines a regression on auxiliary variables with a kriging of the residuals. The idea is to capture large-scale patterns through a deterministic model (the regression part) and to recover local structure through spatial interpolation of the residuals. This approach is particularly advantageous when strong, interpretable covariates explain substantial variation, such as soil properties influenced by known land-use factors or environmental covariates.
Kriging with External Drift (KED)
Kriging with External Drift generalizes Regression Kriging by allowing the drift to be any function of covariates or spatial coordinates, including complex, non-linear relationships. It blends a regression-like component with kriging of the residuals, where the drift itself is informed by external information. KED is valuable when auxiliary data (e.g., topography, climatic variables, industrial measurements) are informative about the target field.
Co-kriging
Co-kriging uses multiple related variables that are spatially correlated to improve predictions of a primary variable. By exploiting cross-covariances between variables (for example, ore grade and mineral density, or soil contaminant and soil moisture), co-kriging can achieve smaller prediction errors than treating each variable independently. This method is common in mining, environmental monitoring, and groundwater studies where related measurements inform each other.
Indicator Kriging
Indicator Kriging is designed for categorical or ordinal data, or for estimating probability maps of categories (e.g., presence/absence, high/medium/low classes). The data are converted to binary indicators, and kriging is applied to each indicator set to obtain category probabilities. This variant is useful when decisions hinge on thresholds or risk classifications rather than on exact numeric values.
Block Kriging
Block Kriging estimates spatial averages over a block or polygon rather than a point value. This is especially relevant in resource planning, land-use assessment, or environmental impact studies where decisions are made on areas rather than pinpoint locations. It smooths out micro-scale variability and aligns predictions with the scale of the decision problem.
Local vs Global Kriging (Non-stationary Kriging)
Non-stationary kriging acknowledges that spatial relationships can change over space. Local kriging builds the prediction model in neighborhood windows, allowing covariance structures to vary with location. This approach improves accuracy in heterogeneous landscapes where a single global variogram would misrepresent local patterns.
Kriging with Transforms
Transformations, such as logarithmic or Box-Cox transforms, are applied to the data before kriging to stabilize variance or to handle skewness. Predictions are made in the transformed space and then back-transformed, with appropriate bias correction. This variant helps when the original data violate normality assumptions underlying many kriging models.
Separable and Large-Scale Kriging Approaches
For large datasets, exact kriging can be computationally prohibitive. Separable models, reduced-rank approximations, and partitioned or local schemes (sometimes called fast kriging) aim to retain accuracy while improving scalability. These approaches are critical in modern environmental monitoring, mineral exploration, and infrastructure planning where data volumes are substantial.
Bayesian Kriging
Bayesian Kriging treats the kriging problem within a probabilistic framework that explicitly encodes uncertainty about model parameters, such as variograms and mean structures. The outcome is a full posterior predictive distribution at each location, which supports risk-aware decision-making. Variants include Gaussian-process kriging and hierarchical models that blend prior information with observed data.
Applications and Debates
Kriging variants find use in a broad spectrum of domains. In mining and mineral economics, they underpin resource estimation and reserve classification, guiding investment and development decisions. In hydrology and environmental science, they aid in mapping groundwater recharge, contaminant plumes, and soil properties. In agriculture and land-use planning, kriging informs precision management by providing spatially explicit risk and yield estimates.
A central debate centers on the balance between model complexity and practical decision-making. Proponents of advanced variants argue that incorporating trends, covariates, or multiple variables yields more accurate and credible predictions, especially when data are sparse or when decision thresholds depend on local conditions. Critics push for simplicity, emphasizing the importance of transparent assumptions, robust cross-validation, and the danger of overfitting or misinterpreting uncertainty. The consensus in many professional settings is that model selection should be driven by predictive performance, validated against independent data, and aligned with the economic and regulatory context of the problem.
Another area of discussion concerns non-stationarity and how best to handle it. Real-world fields often exhibit changing mean structures or shifting covariance patterns. Local kriging and regression-based drift models address these realities, but they demand careful design choices about neighborhoods, covariate selection, and computational resources. In applications where covariates carry physical meaning (e.g., topography for soil properties, climate variables for hydrology), external drift and co-kriging are powerful, provided that covariate measurements are reliable and avoid introducing spurious correlations.
Uncertainty quantification remains central. The probabilistic outputs from Bayesian kriging and related approaches are appealing for risk assessment and decision support, but they require careful specification of priors and model hierarchies. Standard cross-validation and out-of-sample testing help ensure that the chosen variant generalizes well to unseen locations.
In practice, many practitioners favor a pragmatic workflow: start with a robust, interpretable baseline like ordinary kriging, assess potential improvements from incorporating drift or covariates, test multiple variants on held-out data, and select the approach that offers the best predictive performance with transparent uncertainty estimates. This mindset emphasizes accountability, reproducibility, and clear communication of what the models do and do not imply for decision-makers.