Variograms

A variogram is a function that quantifies spatial dependence of a random field Z across locations. The semivariogram gamma(h) is defined as gamma(h) = 1/2 E[(Z(x+h) - Z(x))^2], where h is a spatial lag. It describes how dissimilarity between observations increases with distance.

In practice, the empirical variogram is estimated from data by averaging squared differences of pairs separated by lag h: gamma_hat(h) = (1/2N(h)) sum [Z(x_i) - Z(x_i + h)]^2 over all pairs with separation in a bin around h. Key features are the sill (the plateau value), the range (the distance where the variogram reaches the sill), and the nugget (the value at h approaching zero, often indicating measurement error or microscale variation).

The variogram is tied to second-order stationarity; if the underlying field is second-order stationary or intrinsic, the variogram exists and describes the dependence structure. Theoretical models often fitted to empirical variograms include spherical, exponential, Gaussian, and Matérn forms, characterized by parameters such as nugget, sill, and range. Anisotropy can be captured by directional variograms, which are computed along different directions to reveal varying spatial dependence.

Use in kriging: the fitted variogram (or its corresponding covariance model) is used to construct kriging weights, enabling optimal spatial interpolation and uncertainty quantification. The variogram thus links observed data to predictions at unsampled locations.

Practical considerations include removing deterministic trends, ensuring adequate sampling geometry, handling outliers, and validating the model via cross-validation. Software packages commonly implement variogram estimation, model fitting, and variogram-based kriging.