Kriging

Kriging is a family of geostatistical interpolation techniques named after South African mining engineer Danie Krige, who pioneered the use of statistical methods to estimate ore grades from sparse drill-hole samples. Unlike deterministic interpolation methods, kriging leverages the spatial autocorrelationSpatial AutocorrelationSpatial autocorrelation measures the degree to which values at nearby locations are similar (positive) or dissimilar ... structure of the data, modeled through the variogram, to produce predictions that are statistically optimal in the sense of minimizing prediction variance. This dual output of predicted values and uncertainty estimates distinguishes kriging from all deterministic alternatives.

The foundation of kriging is the variogram (or semivariogramSemivariogramA semivariogram is a graphical and mathematical tool that depicts how spatial dependence changes as a function of dis...), which quantifies how data similarity decreases with increasing distance between sample points. The experimental variogram is calculated from sample data by plotting the semivariance of value differences against the separation distance (lag). A mathematical model, such as spherical, exponential, or Gaussian, is then fitted to the experimental variogram. The fitted model captures three key parameters: the nugget (microscale variation and measurement error), the sill (the semivariance at which spatial autocorrelationSpatial AutocorrelationSpatial autocorrelation measures the degree to which values at nearby locations are similar (positive) or dissimilar ... ceases), and the range (the distance beyond which observations are essentially independent). These parameters directly control the kriging weights and prediction behavior.

Several kriging variants address different data characteristics and analytical needs. Ordinary Kriging is the most widely used form, assuming a constant but unknown mean across the study area. Simple Kriging assumes a known, constant mean. Universal Kriging (Kriging with a Trend) incorporates a deterministic trend component, suitable for data with systematic spatial trends. Indicator Kriging works with binary or categorical data, predicting probabilities of exceeding threshold values. Co-kriging uses correlated secondary variables to improve predictions of the primary variable. Block Kriging predicts average values over defined areas rather than point values.

Kriging is the method of choice for spatial prediction in numerous scientific and engineering fields. Mining and petroleum engineering use kriging to estimate mineral grades and reservoir properties from borehole data, guiding extraction decisions worth millions of dollars. Environmental scientists apply kriging to map soil contamination, creating surfaces that identify areas requiring remediation. Meteorologists use kriging to produce precipitation and temperature maps from station networks. Precision agriculturePrecision AgriculturePrecision Agriculture uses geospatial data, remote sensing, and IoT sensors to optimize farming practices at a sub-fi... employs kriging to map soil nutrient levels, optimizing fertilizer application and reducing environmental impact. Groundwater hydrologists krige water table measurements to model aquifer geometry and flow patterns.

Kriging is the Best Linear Unbiased Predictor (BLUP) under its model assumptions, providing optimal predictions that cannot be improved upon by any other linear method. The kriging variance provides spatially explicit uncertainty estimates, showing exactly where predictions are most and least reliable. The variogram analysis required for kriging also provides valuable insights into the spatial structure of the phenomenon under study. Kriging honors the original sample values exactly, predicting observed values at sample locations.

Kriging requires sufficient sample data to reliably estimate the variogram, and poorly estimated variograms lead to suboptimal predictions. The assumption of stationarity, meaning the spatial structure is consistent across the study area, may not hold for complex phenomena. Kriging is computationally intensive for large datasets due to matrix operations that scale cubically with the number of data points. Variogram modeling involves subjective decisions about model type and parameters that can influence results.

Approximate kriging methods like Fixed Rank Kriging and Nearest Neighbor Gaussian Processes scale to massive datasets with millions of observations. Machine learning hybrid approaches combine kriging's statistical rigor with the flexibility of neural networks. Bayesian kriging provides full posterior distributions rather than point estimates and variances. Space-time kriging extends predictions across both spatial and temporal dimensions simultaneously.