For a random field or Stochastic process ${\displaystyle Z(x)}$ on a domain ${\displaystyle D}$, a covariance function ${\displaystyle C(x,y)}$ gives the covariance of the values of the random field at the two locations ${\displaystyle x}$ and ${\displaystyle y}$:

${\displaystyle C(x,y):=Cov(Z(x),Z(y)).}$

The same ${\displaystyle C(x,y)}$ is called autocovariance in two instances: in time series (to denote exactly the same concept, but where ${\displaystyle x}$ is time), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different locations, ${\displaystyle Cov[Z(x_{1}),Y(x_{2})]}$). ^[1]

For locations ${\displaystyle x_{1},x_{2},\ldots ,x_{N}\in D}$ the variance of every linear combinations

${\displaystyle X=\sum _{i=1}^{N}w_{i}Z(x_{i})}$

can be computed by

${\displaystyle var(X)=\sum _{i=1}^{N}\sum _{i=1}^{N}w_{i}C(x_{i},x_{j})w_{j}}$

A function is a valid covariance function if and only if ^[2] this variance is non-negative for all possible choices of N and weights ${\displaystyle w_{1},\ldots ,w_{N}}$. A function with this property is called positive definite.

In case of a second order stationary random field, where

${\displaystyle C(x_{i},x_{j})=C(x_{i}+h,x_{j}+h)}$

for any lag ${\displaystyle h}$, the covariance function can represented by a one parameter function

${\displaystyle C_{s}(h)=C(0,h)=C(x,x+h)}$

which is called covariogram or also covariance function. Implicitly the ${\displaystyle C(x_{i},x_{j})}$ can be computed from ${\displaystyle C_{s}(h)}$ by:

${\displaystyle C(x,y)=C_{s}(y-x)}$

The positive definitness of the single argument version of the covariance function can be checked by Bochner's theorem. ^[3]

Variogram Random Field Stochastic Process Kriging