Proportional Effect and Quasi-stationarity

Let ${/boma x}$ and ${/boma x}'$ be two points with coordinates $(x_u, x_v, x_w)$ and $(x_u', x_v', x_w')$. Without any hypothesis of stationarity,

(i)

the expectation of the random function $Z({/boma x})$ depends on the position ${/boma x}$, i.e., on the three coordinates $(x_u, x_v, x_w)$;

(ii)

the semi-variogram $/gamma({/boma x}, {/boma x}')$ or the covariance $C({/boma x}, {/boma x}')$ depends on the two locations ${/boma x}$ and ${/boma x}'$ of the random variables $Z({/boma x})$ and $Z({/boma x}')$, i.e., on the six coordinates $(x_u, x_v, x_w)$ and $(x_u', x_v', x_w')$,

It was indicated that a hypothesis of quasi-stationarity is generally sufficient for geostatistical applications. This hypothesis amounts to assuming that:

(i)

the expectation is quasi-constant over limited neighborhoods and, thus, $m({/boma x}) /cong m({/boma x}') /cong m({/boma x}_0)$ when the two points ${/boma x}$ and ${/boma x}'$ are inside the neighborhood $V({/boma x}_0)$ centered on the point ${/boma x}_0$;

(ii)

inside such a neighborhood $V({/boma x}_0)$, the structural functions $/gamma$ or $C$ depend only on the vector of the separating distance ${/boma h} ={/boma x} -{/boma x}'$ and not on the two locations ${/boma x}$ and ${/boma x}'$; however, this structural function depends on the particular neighborhood $V({/boma x}_0)$, i.e., on the point ${/boma x}_0$,

The construction of a model of quasi-stationarity thus amounts to building a model of the structural function $/gamma({/boma h}, {/boma x}_0)$ which depends on the argument ${/boma x}_0$.