Geoscience Reference
In-Depth Information
h e techniques of variogram modeling are still very much under discussion.
Some advocate
objective
variogram modeling by automated curve i tting,
using a weighted least squares, maximum likelihood, or maximum entropy
method. In contrast, it is ot en argued that geological knowledge should be
included in the modeling process and visual i tting is therefore recommended.
In many cases the problem with variogram modeling is much less a question
of whether the appropriate procedure has been used than a question of the
quality of the experimental variogram. If the experimental variogram is
good, both procedures will yield similar results.
Another important question in variogram modeling is the intended use
of the model. In our case the linear model does not at i rst appear to be
appropriate (Fig. 7.21). Following a closer look, however, we can see that the
linear model i ts reasonably well over the i rst three lags. h is can be sui cient
if we use the variogram model only for kriging, because in kriging the
nearby points are the most important points for the estimate (see discussion
of kriging below). Dif erent variogram models with similar i ts close to the
origin will therefore yield similar kriging results if the sampling points are
regularly distributed. If, however, the objective is to describe the spatial
structures then the situation is quite dif erent. It then becomes important to
i nd a model that is suitable over all lags and to accurately determine the sill
and the range. A collection of geological case studies in Rendu and Readdy
(1982) show how process knowledge and variography can be interlinked.
Good guidelines for variogram modeling are provided by Gringarten and
Deutsch (2001) and Webster and Oliver (2001).
We will now briel y discuss a number of other aspects of variography:
•
Sample size
- As in any statistical procedure, as large a sample as possible
is required in order to obtain a reliable estimate. For variography it is
recommended that the number of samples should be in excess of 100 to
150 (Webster and Oliver 2001). For smaller sample numbers a maximum
likelihood variogram should be computed (Pardo-Igúzquiza and Dowd
1997).
•
Sampling design
- In order to obtain a good estimation close to the origin
of the variogram, the sampling design should include observations
over small distances. h is can be achieved by means of a nested design
(Webster and Oliver 2001). Other possible designs have been evaluated by
Olea (1984).
•
Anisotropy
- h us far we have assumed that the structure of spatial
correlation is independent of direction. We have calculated
omnidirectional
