variogram

The variogram is a function used in geostatistics to describe the spatial dependence of a random field. For a second-order stationary process Z(x), the semivariogram γ(h) is defined as one half the expected squared difference between values separated by a lag vector h: γ(h) = 1/2 E[(Z(x) - Z(x + h))^2]. If the process is homogeneous and isotropic, γ depends only on the distance h = ||h||, not the direction.

An empirical or experimental variogram estimates γ(h) from observed data by averaging the squared differences of pairs of observations separated by lag h: γ̂(h) = (1/2N(h)) Σ [Z(x_i) - Z(x_i + h)]^2. Variograms can be computed in a directional form to assess anisotropy. In typical spatial fields, γ(h) increases with distance and approaches a sill, the plateau value. A nugget effect, an abrupt jump near zero, reflects measurement error or micro-scale variability.

Modeling and use: The variogram is often represented by a theoretical model (such as spherical, exponential, Gaussian, or Matérn) fitted to the empirical variogram. Key parameters are the nugget, sill, and range (distance where the variogram reaches the sill). The variogram is related to the covariogram by γ(h) = C(0) - C(h) under second-order stationarity.

Applications: Variograms underpin kriging and spatial simulation, providing a quantitative summary of spatial structure. They are widely used in geology, mining, hydrology, environmental science, and meteorology, with attention to anisotropy, non-stationarity, and measurement error.