Variogram Models and Their Fitting

The four main operations of linear geostatistics (variances of estimation and dispersion, regularization and kriging) involve only the structural function of the random function (covariance or variogram). Thus, every geostatistical study begins with the construction of a model designed to characterize the spatial structure of the regionalized variable studied.

Let $Z({/boma x})$ be an intrinsic random function with semi-variogram $/gamma({/boma h})$. The two main characteristics of a stationary variogram are:

(i)

its behavior at the origin, the three types of which are shown in Figure 4.3 (parabolic, linear and nugget effect);

(ii)

the presence or absence of a sill in the increase of $/gamma({/boma h})$, i.e., $/gamma({/boma h}) =$ constant when $/vert{/boma h}/vert > a$.

Thus, the currently used theoretical models can be classified as:

Models with a sill (or transition models) and a linear behavior at the origin

(a)

spherical model

(b)

exponential model
and a parabolic behavior at the origin

(c)

Gaussian model

Models without a sill (the corresponding random function is then only intrinsic and has neither covariance nor finite a priori variance)

(a)

models in $/vert{/boma h}/vert^/theta$, with $/theta /in (0,2)$

(b)

logarithmic model

Nugget effect: It has been seen that an apparent discontinuity at the origin of the semi-variogram, i.e., a nugget constant $C_o$, can be interpreted as a transition structure reaching its sill value $C_o$ at a very small range compared with the available distances of observation.

Remark: For the moment, only isotropic models will be considered, i.e., random functions which have the same spatial variability in all directions of space. Thus, in the three-dimensional space, the isotropic notation $/gamma({/boma h})=/gamma(r)$ means

where $(h_u,h_v,h_w)$ are the three coordinates of the vector ${/boma h}$.