variogrammodel

In geostatistics, a variogram model (variogrammodel) is a parametric function that describes how the spatial dependence of a regionalized variable changes with separation distance. It provides the theoretical variogram γ(h) used in kriging and related methods. The empirical variogram is estimated from sample data and then fitted with a chosen model to obtain a smooth, usable representation of spatial structure.

Common models include spherical, exponential, Gaussian, and Matérn. All share parameters such as the nugget, sill, and range. The nugget represents short-scale variation or measurement error, the sill is the variance explained by spatial structure, and the range is the lag distance at which the variogram levels off. Anisotropy can be represented by direction-dependent ranges or by rotating and scaling coordinates to reflect directional continuity.

Model fitting is typically done by adjusting parameters to minimize the difference between the empirical and theoretical variograms, using methods such as least squares or maximum likelihood. Practitioners may combine models (nested or blended) to capture different spatial scales, and may constrain the range to avoid overfitting. The choice of model influences kriging weights and prediction uncertainty.

Applications include ordinary, simple, and universal kriging, as well as cokriging and stochastic simulation. Variogram models are implemented in geostatistical software and libraries such as gstat, geoR, and SGeMS. The model is extrapolated beyond the observed range with caution, and its validity is often assessed by cross-validation.

Limitations include assumptions of second-order stationarity and isotropy (unless explicitly modeled). Poorly estimated models can bias predictions, and non-stationary fields may require detrending or more complex non-stationary approaches.