Geostatistics

Core Idea

Geostatistics provides a framework for analyzing and predicting spatially distributed phenomena based on the principle that nearby measurements are more similar than distant ones. The variogram (or semivariogram) quantifies this spatial dependence by measuring how the variance between paired observations increases with separation distance. Kriging uses the variogram model to produce optimal, unbiased spatial predictions at unsampled locations, along with prediction uncertainty estimates. Unlike simple interpolation methods (inverse distance weighting, splines), kriging is grounded in the theory of regionalized variables and produces not just predicted values but confidence intervals -- telling you both what the estimate is and how reliable it is.

Explainer

Most environmental, geological, and resource variables are not randomly distributed -- they exhibit spatial structure. Soil properties, ore grades, groundwater levels, and pollutant concentrations all show patterns where nearby locations tend to have similar values. Geostatistics provides the mathematical framework to describe, model, and exploit this spatial structure for prediction.

The variogram is the central diagnostic tool. It plots the average squared difference between paired observations against their separation distance. A typical variogram rises from a nugget (near-zero distance variance) to a sill (maximum variance) over a characteristic range (the distance at which spatial correlation disappears). The shape of this rise (linear, exponential, spherical, Gaussian) describes how quickly spatial similarity decays with distance -- steep rises indicate rapid decorrelation (patchy phenomena), while gradual rises indicate broad spatial continuity (smooth phenomena).

Kriging is the prediction engine. Given a variogram model and a set of sample data, ordinary kriging computes the optimal linear prediction at any unsampled location by weighting nearby samples according to their spatial configuration and the variogram. Samples closer to the prediction point and in less-redundant configurations receive higher weights. The result is the Best Linear Unbiased Predictor (BLUP) -- it minimizes prediction variance while remaining unbiased. Crucially, kriging also produces a prediction variance at each location, enabling probabilistic statements ("there is a 95% probability that contamination exceeds the threshold here").

Variants include simple kriging (known mean), universal kriging (models spatial trends), indicator kriging (predicts probabilities of exceeding thresholds), and co-kriging (uses correlated secondary variables). Geostatistics underpins mineral resource estimation, environmental site assessment, precision agriculture, and any application where spatial prediction with quantified uncertainty is needed.