Ordinary KrigingEdit
Ordinary kriging is a foundational method in geostatistics for estimating unknown values at locations where data have not been measured directly. It rests on the idea that spatially close observations tend to be more alike than distant ones, and it uses a modeled sense of spatial dependence to weight nearby measurements in making predictions. The method is local in nature: for each target location, it focuses on a neighborhood of measured values and finds the combination of those values that yields the best linear predictor under an unbiasedness constraint. This makes ordinary kriging both intuitive and mathematically tractable, and it has become a standard tool in fields ranging from mining and groundwater management to environmental monitoring and agricultural planning. It is closely associated with the broader framework of geostatistics and sits within the family of techniques that includes kriging as a general concept.
From a practical viewpoint, ordinary kriging is valued for its explicit handling of uncertainty. The predicted value is accompanied by an estimation variance that reflects the amount of information available in the neighborhood and the strength of spatial correlation. This allows decision-makers to weigh predicted values against the costs and risks of acting on them. In commercial settings, such as mineral resource estimation and site remediation planning, kriging-based estimates can translate into more efficient exploration, better risk assessment, and clearer ownership of property and development rights. In public-sector contexts, the method can support planning and environmental monitoring when applied with transparent documentation of the assumptions and data used. The technique relies on several key ingredients: a sample of observations point data with measured values, a model of how those values correlate with distance (the variogram), and a neighborhood search that defines which nearby observations influence the estimate for a given location. For more on the underlying concepts, see semivariogram and spatial interpolation.
Theory and method
Basic idea and assumptions
Ordinary kriging estimates a value at an unmeasured location by forming a weighted sum of nearby measured values. The weights are chosen to minimize the mean squared prediction error while ensuring the predictor is unbiased (its expected value equals the true value). Two core assumptions underlie the standard formulation: second-order stationarity (the statistical properties are constant within the neighborhood considered) and isotropy (spatial dependence depends only on the separation distance, not on direction). When these conditions are not met, practitioners may adapt the approach into variants such as universal kriging (which allows a deterministic trend to be modeled) or other non-stationary schemes. The chosen assumptions influence both the estimator and its reported uncertainty, so they are matters of engineering judgment as well as statistics. See stationarity and isotropy for deeper discussions, and variogram for how spatial dependence is quantified.
Variogram modeling
A central object in ordinary kriging is the variogram (or semivariogram), which characterizes how related values are as a function of spatial separation. Empirically, the semivariogram is estimated from data by computing half the average squared difference between pairs of observations at given lag distances. A theoretical variogram model is then fitted to these empirical points; common models include spherical, exponential, and Gaussian forms. The fitted variogram guides the kriging weights and the estimation variance. For detailed treatment, see variogram and semivariogram.
The kriging system and the estimator
For a target location, the ordinary kriging weights are obtained by solving a linear system that arises from minimizing the prediction variance subject to the unbiasedness constraint. The system pairs the neighboring data values with a set of Lagrange multipliers that enforce the constraint, producing weights that sum to one and reflect both the spatial arrangement of measurements and the chosen variogram model. The resulting predictor is a linear combination of the observed values, and the associated kriging variance provides a measure of uncertainty in the estimate. Practical implementation often involves choosing a search neighborhood, handling data gaps, and performing cross-validation to assess predictive performance. See kriging for a broader discussion of these ideas and cross-validation for methods used to evaluate predictive accuracy.
Practical implementation and considerations
In applied work, several practical choices affect performance: - Neighborhood definition and search strategy: how many nearby points to include, and whether to allow anisotropy (direction-dependent correlation). - Trend and drift handling: if a clear regional trend exists, practitioners may remove it before kriging (as in universal kriging) or incorporate it within the model. - Data pre-processing: normalization, outlier treatment, and measurement error considerations. - Software and tooling: common implementations appear in packages like gstat (an R-based framework), PyKrige (Python), and commercial platforms such as ArcGIS and QGIS plug-ins. - Validation: cross-validation and backcasting help assess sensitivity to variogram choice and neighborhood size. See spatial statistics and resource estimation for broader methodological context.
Variants and related methods
Ordinary kriging is one member of the broader family of kriging techniques. Other variants address non-stationarity, multiple variables, or different data types: - Universal kriging allows a deterministic trend to be modeled as part of the prediction, which can be important in landscapes with systematic variation. - Kriging with external drift/co-kriging uses auxiliary variables that are correlated with the target variable to inform the prediction. - Indicator kriging is used for categorical or thresholded variables, such as presence/absence. - Co-kriging extends the framework to multiple correlated variables. - Non-stationary or locally adaptive approaches seek to relax stationarity assumptions by allowing model parameters to vary over space. See non-stationarity for a broader discussion.
In practice, the choice among these methods depends on the data-generating process, the intended use of predictions, and the acceptable balance between model complexity and interpretability. The core idea remains the same: leverage spatial correlation to interpolate values while providing an explicit expression of uncertainty.
Applications and impact
Ordinary kriging finds use in a wide range of professional domains: - Resource estimation in the minerals sector, where owners and investors rely on estimates of ore grade and tonnage to guide development and financing decisions. See mineral exploration and resource estimation. - Groundwater and environmental management, where kriging helps map contaminant concentrations, salinity, or moisture content across unsampled areas. See hydrogeology and environmental monitoring. - Agriculture and soil science, where spatially informed interpolation supports precision farming, irrigation planning, and yield estimation. See soil science and precision agriculture. - Public health and epidemiology in some contexts, where spatial interpolation is used for smoothing surveillance data, though care is needed to avoid masking local outbreaks or biases. See epidemiology and spatial analysis.
Critically, ordinary kriging provides a transparent, model-based approach to estimation with an accompanying quantified uncertainty. Proponents emphasize that, when used with careful variogram modeling, neighborhood selection, and validation, the method supports efficient decision-making under uncertainty. Critics point out that the method relies on assumptions that may not hold in all landscapes or data-generating processes, and that mis-specification of the variogram or overfitting to noisy data can lead to optimistic uncertainty estimates or biased predictions. In debates about data collection, model transparency, and regulatory overhead, the balance tends to favor practical, market-oriented use that emphasizes clear documentation, reproducibility, and accountability for predictions and their consequences.
From this vantage, ordinary kriging is often evaluated not only by predictive accuracy but also by how well its results align with cost-benefit considerations, property rights, and the efficiency of resource allocation. The method’s appeal lies in its explicit handling of uncertainty and its compatibility with decision frameworks that reward clarity, traceability, and scalability.