Moments, Variograms

A random function $Z$ is described by the distribution of the random variable $Z=Z({/boma x})$ at each place ${/boma x}$ in a certain domain, together with all dependencies. The so defined probability distribution is also called spatial distribution law. In environmental and earth sciences we never need the entire law, it usually suffice to know the first two moments for finding acceptable, approximate solutions. The description of the law therefore mostly is done by the first two moments, such that there is no distinction between $Z({/boma x}_1)$ and $Z({/boma x}_2)$ if these moments are the same.

If the distribution function of $Z=Z({/boma x})$ has an expectation (and we shall suppose that it has), then this expectation is generally a function of ${/boma x}$, and is written as

It is a weighted mean of all possible realizations at ${/boma x}$.

The variance, or more precisely the ``a priori'' variance of $Z({/boma x})$: If this variance exists, it is defined as the second-order moment about the expectation $m({/boma x})$ of the random variable $Z({/boma x})$, i.e.,

As with the expectation $m({/boma x})$, the variance is generally a function of x.

The covariance: It can be shown that if the two random variables $Z({/boma x}_1)$ and $Z({/boma x}_2)$ have variances at the points ${/boma x}$ and ${/boma x}_{2} = {/boma x}+{/boma h}$ (with distance ${/boma h}$), then they also have a covariance which is a function of the two locations ${/boma x}$ and ${/boma x+/boma h}$, and is written as

The variogram: The variogram function is defined as the variance of the increment $[Z({/boma x}) - Z({/boma x}+{/boma h})]$, and is written as

The function $/gamma({/boma x},{/boma x}+{/boma h})$ is then called semi-variogram.