Here are a few things to be aware of when looking for automatic semi

variogram modeling software.

1. Generally, one is faced with the problem of modeling several directional

sample variograms calculated from a particular data set.

2. However, modeling the directional sample variograms from a two

dimensional data set is relatively simple. If a 2D anisotropy is suspected,

one should calculate multiple directional sample variogram on azimuths of at

least 30 degree intervals. Four directional sample variograms at 90 degree

intervals are generally insufficient for accurately modeling an anisotropy.

The task at hand is to "fit" a model to all of the directional sample

variograms. Thus, any automatic fitting software that only models one

directional sample variogram at a time is likely not very useful.

3. The problem is considerably more difficult when working with 3

dimensional data. For example, experience suggests that in general, one

cannot obtain a reasonable sample of the inherent 3 dimensional anisotropic

spatial continuity with less than 19 directional sample variograms, e.g. 30

degree increments on the azimuth combined with 30 degree increments on dip.

(Recall, that the turning bands algorithm is based on the "icosahedron

approximation" which requires a minimum of 15 lines to simulate an isotropic

covariance in 3D space). The task at hand is to "fit" a model to all 19

directional sample variograms. To appreciate the difficulty, consider

fitting a simple variogram model with two structures to the 19 directional

sample variograms. One must determine values for 15 parameters, e.g.,

1. The nugget.

2. The coefficient for the first structure.

3. The coefficient for the second structure.

4. The range along the major axis of the anisotropy model associated with

the first structure.

5. The range along the semi-major axis of the anisotropy model associated

with the first structure.

6. The range along the minor axis of the anisotropy model associated with

first structure.

7. The range along the major axis of the anisotropy model associated with

the second structure.

8. The range along the semi-major axis of the anisotropy model associated

with the second structure.

9. The range along the minor axis of the anisotropy model associated with

second structure.

10. The rotation angle around the Z axis of the anisotropy model for the

first structure.

11. The rotation angle around the rotated X axis of the anisotropy model for

the first structure.

12. The rotation angle around the rotated Y axis of the anisotropy model for

the first structure.

13. The rotation angle around the Z axis of the anisotropy model for the

second structure.

14. The rotation angle around the rotated X axis of the anisotropy model for

the second structure.

15. The rotation angle around the rotated Y axis of the anisotropy model for

the second structure.

Thus, although the problem can be stated quite simply in terms of least

squares, e.g.;

Find the set of 15 parameters that;

Minimize [ Sum_i w_i * (VariogramModel(15 parameters, lag_i) -

sample_variogram_point_i)^2 ] where "Sum_i" is the sum over all sample

variogram points from all 19 directional sample variograms. Note that w_i

may be a set of weights such as "number of pairs" etc.

The solution is not straightforward. However, the software product

"SAGE2001" solves the problem above quite quickly by determining the optimum

set of 15 parameters. The result is a weighted least squares fit to all

directional sample variogram points simultaneously.

SAGE2001 is used throughout the mining industry by many companies including

Newmont, Barrick, Placer, AMEC, and so on.

Unfortunately, SAGE2001 development costs were steep. Thus, it is not

freeware or shareware.

But, SAGE2001 can be downloaded at www.isaaks.com for free and is fully

functional with no restrictions for 20 days. So if you have a difficult

modeling problem - feel free to use it.

Edward Isaaks