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In a random field $$Z(x)$$, a covariance function $$C(x_1,x_2)$$ gives the covariance of the random field between two different locations $$C(x_1,x_2)=Cov[Z(x_1),Z(x_2)].$$

To be a valid covariance function, $$C(x_1,x_2)$$ must deliver a positive variance for any linear combination of $$Z(x_1), Z(x_2), \ldots, Z(x_N)$$, the random field at any finite set of $$N$$ locations, $$x_1, x_2, \ldots, x_N$$. A function must only be a positive definite function to fulfill this criterion.

The same concept is called autocovariance in two instances: in time series (to denote exactly the same concept, but where $$x$$ 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, $$Cov[Z(x_1),Y(x_2)]$$).

Stationarity assumptions
Three common types of stationarity may be taken in order to work with covariance functions from a practical point of view. All of them involve a displacement $$h$$ of the space:


 * strict stationarity, when the joint probability distribution of the random field at a finite set of locations does not change after the same translation is applied to the whole set, $$F(Z(x_1+h), Z(x_2+h), \ldots, Z(x_N+h))=F(Z(x_1), Z(x_2), \ldots, Z(x_N))$$
 * second-order stationarity, when mean and variance (the second-order moments) of the random field at two locations do not change after translation, $$E[Z(x+h)]=E[Z(x)]$$ and $$Cov[Z(x_1+h),Z(x_2+h)]=Cov[Z(x_1),Z(x_2)]$$
 * intrinsic stationarity, when the increment of the random field between two locations has mean and variance which do not change after translation, $$E[Z(x_1+h)-Z(x_2+h)]=E[Z(x_1)-Z(x_2)]$$ and $$Var[Z(x_1+h)-Z(x_2+h)]=Var[Z(x_1)-Z(x_2)]$$. Note that necessarily, $$E[Z(x_1)-Z(x_2)]=0$$.

Under any of the first two assumptions, the covariance function becomes a function of just the difference vector $$h=x_1-x_2$$, thus it can be expressed with $$Cov[Z(x),Z(x+h)]=C(h).$$ Under the last assumption, the covariance function does not necessarily exist, and one has to work with the semivariogram.