Geostats - Variogram Model

This page is part of the Geostatistical Mapping of Anomaly Density design.

This page is used to fine-tune the variogram which is used to krig the data. The red dots on the graph are the empirical variogram (also called the experimental variogram) that is computed from the actual data to show how data values are correlated with respect to distance. Each dot is a lag of the experimental variogram. The x-axis represents the distance between pairs of points, and the y-axis represents the calculated value of the variogram, where a greater value indicates less correlation between pairs of points. This particular variogram shows a spatial relationship well suited for geostatistical analysis since pairs of points are more correlated the closer they are together and become less correlated the greater the distance between points.

Because the kriging algorithm that is used to interpolate data values across the site requires a positive definite model of spatial variability, the empirical variogram cannot be used directly. Instead, a model must be fitted to the data to approximately describe the spatial continuity of the data. Certain models (i.e., mathematical functions) that are known to be positive definite are used in the modeling step. The continuous red line on the graph above is a model that has been fitted to the empirical variogram. In most cases, it is more important to fit the shorter distances than the longer distances.

This page contains controls to help you create a model that can be used by the kriging process that closely matches the empirical variogram of your site. These controls are:

Variogram Type	This is the type of variogram used. Certain designs in VSP require a Semivariogram, other designs allow other types such as a Semivariogram of Normal Scores. The discussion of variogram types is beyond the scope of this help document - please consult geostatistical literature for more help choosing a variogram type.
Number of Lags	This is the number of discrete distances used for comparing data. The lags can be thought of as bins which contain the comparisons between points at a specific distance.
Distance between Lags	This number is how far apart the lags are spaced (in map units). A lag distance of 30 feet means that data points that are 30 feet apart will be compared, then data points that are 60 feet apart will be compared, then 90 feet, etc. - up to the number of lags specified above.
Lag Tolerance	This number is tolerance on the distance between lags (in map units). It is usually set to half the lag distance. In this example, it is unlikely that two data points will be spaced exactly 30 feet (or 60 feet, or 90 feet) apart, so this allows points that are spaced within a range to be compared. In this example, with the tolerance set at 15 feet, points that are spaced between 15 and 45 feet apart are compared, then points that are spaced between 45 and 75 feet are compared, etc.
Variogram Coverage	Is a display of the distance covered by the variogram, which is the lag distance multiplied by the number of lags. The site size is also displayed for comparison purposes. It is usually considered a good rule of thumb to try to cover at least half of the site. However, in this example you can see that the variogram flattens out past about 450 feet, so there is no additional useful correlation beyond that distance. The disadvantage of increasing the variogram coverage is that automatic model fitting would try to fit the longer distances when it is more important to fit the shorter distances.
Compute Variogram	Click this button to compute the empirical variogram using the specified variogram type, number of lags, lag distance and lag tolerance. After computing the empirical variogram, VSP will try to automatically fit a model to the variogram. A continuous red line on the graph will indicate success at fitting a model.

Nugget	Is the Y-intercept of the variogram graph. In practical terms, the nugget represents the small-scale variability of the data. A portion of that short range variability can be the result of measurement error.
Fix (Nugget)	Check this box to use the Nugget value entered above rather than trying to automatically fit a Nugget value when the Auto Fit button is clicked.
Model Tabs	Click on the + tab to enter additional models. VSP allows you to fit up to three nested models. When more than one model is used, the kriging process uses them additively.
Model Type	This drop list allows you to choose which mathematical model to fit to your empirical variogram.
Fix (Model)	Check this box to use the Model Type selected above rather than trying a different model type when the Auto Fit button is clicked.
Range	The range is the distance after which the variogram levels off. The physical meaning of the range is that pairs of points that are this distance or greater apart are not spatially correlated. The range can be entered manually or set with the slider control.
Sill	The sill is the variance where the empirical variogram appears to level off. The total sill is the sum of the nugget plus the sills of each nested model. (Note: you can use the variance of data as a reasonable default for the total sill. To see the variance of the data, check the Show Variance box described below.) Variogram points above the sill indicate negative spatial correlation, while points below the sill indicate positive correlation. The variogram may not exhibit a sill if trends are present in the data. In that case, geostatistical analysis should proceed with caution, and at the least, ordinary kriging should be used for mapping (see Kriging Options for help with Ordinary Kriging). The sill can be entered manually or set with the slider control.
Use # of Pairs	Check this box to weight the variogram points with the number of pairs when the Auto Fit button is clicked. This option uses Cressie weights for the model fit.
Auto Fit	Click this button to have VSP try to automatically fit a model to the empirical variogram. The model types that VSP uses are: Spherical, Exponential and Gaussian. VSP will also honor the fixed nugget and fixed model types is specified above. If VSP is unable to fit a model within the given constraints, it will leave the current model unchanged.

Show Variance	Check this box to show the data variance on the graph. It will be plotted as a horizontal blue line. The data variance is a reasonable default value for the sum of the nugget and all the sill values.
Show # of Pairs	Check this box to display the number of data pairs that occur with each lag distance. Lags with fewer than 30 pairs of data might reasonably carry less weight when fitting a model than lags with significantly more pairs.

References:

Cameron, K, and P Hunter. 2002. Using Spatial Models and Kriging Techniques to Optimize Long-Term Ground-Water Monitoring Networks: A Case Study. Environmetrics 13:629-59.

Cressie, N.A.C (1985) Mathematical Geology. 17, 563-586.

Deutsch, C.V. and A.G. Journel. 1998. GSLIB Geostatistical Software Library and User's Guide, 2nd Edition, Applied Geostatistics Series, Oxford University Press, Inc. New York, NY.

Gilbert, RO. 1987. Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York.