FUZZEKS [to Index]
Theoretical variogram fitting

A first idea in fitting a theoretical variogram is of course to make it similar to the experimental variogram.
But this is not the only goal one has to target in order to get reasonable kriging results.

The second element of importance that can be taken into account is the distribution and the maximal absolute value of the estimated kriging variance (which can be displayed by FUZZEKS). One must be very careful with the estimated kriging variance, because this is a value that states about the quality of the kriged values that are calculated based on the theoretical variogram as well as the estimated kriging variance is also calculated based on the theoretical variogram!
However, there are situations where this is acceptable, if e.g. the absolute values of two kriging variances may be compared in the case of the same theoretical variogram but different data sets.
Example: You have areas with an insufficiant number of data points and therefore a high kriging variance. To improve the situation, fuzzy data points - expert knowledge in the form of fuzzy numbers - are added to the input data set. The same theoretical variogram is used for both old and new data set. Then you can compare the absolute values of the two kriging variances.

A third idea is in fitting a theoretical variogram is to check the actual kriging result: If the isolines for the most possible values are not typical for this sort of parameter (judged by an expert), the theoretical variogram should be changed.

The last method to check the theoretical variogram uses cross-validation. This procedure leaves out sequentially any single point of the input and kriges a value at this location (with the investigated variogram) in order to compare real values with estimated ones. The cross-validation result is typically a number calculated by adding the squares of the differences of each of of the two values (which can be minimized by trying out different variogram types and parameters then). In the current version of FUZZEKS support for cross-validation is not implemented.

It has been found out that a decrease of the estimated kriging variance often results in an increase of the cross-validation and vice versa.
So typically one has to find a theoretical variogram that is a compromise between the different criteria.

Remark: It can be possible that it makes no sense to use kriging (because of the mathematical assumptions). A sign for this can be a large nugget-effect (compared to the sill). A large nugget-effect could mean that the range for the investigated parameter is smaller than the smallest distance between two points and can not be found out by using this data set.
In these cases other interpolation methods (especially the simpler and less statistically based methods) tend to have no advantages either. In these situations, one can only get some interpolation, which does not need to have much in common with the spatial character of this parameter. You can also use kriging (FUZZEKS) then. Typically one would use a linear theoretical variogram, because for the kriging result with this, the range parameter makes no difference (for sill and nugget-effect holds the same). If one thinks of drawing the isolines by hand because no suitable method is found, kriging with linear theoretical variogram can be a pretty good method.

### Possible theoretical variograms The fields above and below the display of the variogram curves show the parameters the user has chosen in order to define the theoretical variogram:
• Range: The smallest distance where two data points do not have anything in common. It is displayed as vertical dashed line.
This parameter has a great influence on the result.
• Sill: The variogram value of these points (out of range). It is displayed as horizontal dashed line.
A change in this parameter does not change the result (but it can change the absolute values - not the distribution - of the estimated kriging variance). Of course, if a change of this parameter is accompanied by a change of the range parameter, which typically will be nessecary, then the result will change!
• Nugget effect: The lowest variogram value, i.e. half of the variance of values for the same point.
For changes of this parameter the same remark holds as for sill.
Remark (relates to range, sill and nugget effect): If the values in these input fields need more space to display than avilable in the input field, you must click with the mouse in the rectangle and go through the whole number (with cursor keys left and right and Home/End key) in order to read the value.
Remark (relates to scroll bars): Remember that you possibly can not read the whole number in the fields. If the actual values are important, always follow the instructions in the remark above!
• Variogram type: The actual variogram type can be composed of at most two raw variogram types.
The variogram type has great influence on the result.
The left box contains a percentage for the type displayed in the right (list-)box (the formulas listed here assume a zero nugget-effect):
• Gaussian: Starts with gradient zero. For large distances the gradient again converges to zero (the value is 95% of the sill when range is reached).
Formula: sill * (1-exp(-distance^2/range^2))
• Exponential: It starts with high gradient which converges to zero as distance goes to infinite (the value is 95% of the sill when range is reached).
Formula: sill * (1-exp(-distance/range))
• Spherical: It starts with high gradient which goes to zero when range is encountered.
Formula:
if dist< range: sill * (3*dist/(2*range) - dist^3/(2*range^3))
if dist>=range: sill
• Linear: Is not typical as variogram function, because it rises endlessly with the distance. If a good choice of theoretical variogram curve is not possible, kriging is possibly (mathematically) not allowed to use (see remark above!).
The linear approximation does not depend on the above parameters, not even on the range (with respect to the kriged values)!
Formula: sill * (distance/range)

The next pages deal with