Models without a Sill

These models correspond to random functions $Z({/boma x})$ with an unlimited capacity for spatial dispersion; neither their a priori variances nor their covariance can be defined. The random functions $Z({/boma x})$ are only intrinsic.

(a)

Models in $r^/theta$:

the limits 0 and 2 being excluded.

These models have a particular theoretical and pedagogical importance (they show a whole range of behaviors at the origin when the parameter $/theta$ varies and they are easy to integrate).

In practice, only the linear model is currently used:

with $/omega$ the slope at the origin.

Remark 1 For small distances ( $r /rightarrow 0$), the linear model can be fitted to any model that has a linear behavior at the origin (e.g., spherical and exponential models).

Remark 2 As $/theta$ increases, the behavior of $/gamma(r)=r^/theta$ at the origin becomes more regular and corresponds to a random function $Z({/boma x})$ of more and more regular spatial variability.

Experimentally, models in $r^/theta$ for $/theta /in (1,2)$ are often indistinguishable from a parabolic drift effect. The choice of interpretation, as a drift (non-stationarity) or a stationary model in $r^/theta$ with $/theta$ close to 2, depends on whether or not it is desired to make a drift function $m({/boma x})= E/{Z({/boma x})/}$ obvious.

Remark 3 For $/theta /geq 2$, the function $(-r^/theta)$ is no longer a conditional positive definite function and, in particular, the increase of $(r^/theta)$ at infinity is no longer slower than that of $r^2$. This strict limitation $/theta < 2$ prohibits, in particular, any blind fitting of a variogram by least-square polynomials, even though some polynomials of degree higher than two do satisfy the conditional positive definite property, cf. the expression of the spherical model.

(b)

Logarithmic model:

Note that $/log r /rightarrow -/infty$ as $r /rightarrow 0$, so that the logarithmic model cannot be used to describe regionalizations on a strict point support. On the other hand, this model, once regularized on a non-zero support $v$, can be used as the model of a regularized semi-variogram $/gamma_v(h)$. In particular, the regularization of a point logarithmic model is a function which is null at the origin: $/gamma_v(0) = 0$.

Logarithmic model or nested transition models? The logarithmic model has been studied exhaustively by the initial workers in geostatistics (D.G. Krige, 1951; G. Matheron, 1955, 1962; Ph. Formery and G. Matheron, 1963; A. Carlier, 1964). Indeed, up until around 1964-66 this model was practically the only theoretical model used; this is explained by certain analytical properties which make this model a very convenient one, and also by the fact that the first geostatistical applications were carried out on deposits (gold on the Rand, uranium) for which the variograms of the characteristic variables had no sill. Since that time, however, geostatistics has been applied to a great number of other ore bodies for which the variograms show more or less clear sills. J. Serra, using a very detailed study of the different scales of variability of the mineralization of the ferriferous basin of Lorraine (France), showed that a logarithmic model is the limit model of a nested succession of transition models, the ranges of which increase geometrically, cf. J. Serra (1976b, 1968).

As the nested model has more parameters, it is more flexible and lends itself more readily to a geological interpretation. Moreover, the particular analytical properties of the logarithmic model are less attractive now that computers are readily available and it is a relatively simple task to program the evaluation of any integral $/bar/gamma$ of any nested structure $/gamma({/boma h})$.

This explains why the various sill models have increased in use since around 1966 at the expense of the logarithmic model (except perhaps in South Africa). Journel and Huijbregts say, that the use of a logarithmic model or of various nested spherical models is more or less a question of habit and does not alter the result of geostatistical calculations. What is important is that the chosen theoretical model $/gamma({/boma h})$ should be a good fit to the experimental semi-variogram within its limits of reliability ( $/vert{/boma h}/vert / < b$).

In other words, the results of the geostatistical calculations prove to be robust in relation to the choice of the model-provided that the parameters of this model are correctly estimated.