Spatial Correlation
SADA characterizes spatial correlation through the use of the semi-variogram model, which provides a measure of variance as a function of distance between data points. This measure is defined as half of the average squared difference between two values separated by vector h: (Deutsch and Journel, GSLIB Geostatistical Software Library and User’s Guide, 1992).
where N(h) is the number of pairs separated by vector h, xi is the starting point (tail) and yi is the ending point (head).
In practice, the number of points separated exactly by vector h will be very small or none at all. Therefore, a distance and direction tolerance is allowed to capture more data points in the calculation of (h). In the figure below, all data points within the blue shaded area will be used.
The distance component of h is referred to as the lag. The tolerance associated with the lag is called the lag tolerance. In practice, for a specified direction, the semivariogram may be calculated for a number of lags. The tolerance associated with a direction is referred to as the angle tolerance. These components together form a cone which can be constrained by the bandwidth factor. The bandwidth controls the maximum width across the cone and allows the cone to focus more on the specified direction.
To calculate semivariogram values, enter the appropriate information on the Cov tab of the Control Panel.
At the top of the tab in the drop down box next to Type, users may select the default ‘OK’ option or create indicator cutoff threshold values. (See Spatial Correlation and Indicator Kriging.)
SADA Version 3 allows the results of two separate cones to be viewed at once to provide visual comparison and check for anisotropic correlation. (It is recommended that the user plot the major and minor axes; however, the cones do not need to be orthogonal to each other for simple viewing. If the model is used in OK or IK, they are considered orthogonal.)
Enter the appropriate parameters in the Variogram table. Each cone is identified by its Name.The Angle parameter refers to the direction of the cone (measured clockwise from the positive y axis), the Tol parameter refers to the angle tolerance, and Band refers to the constraining bandwidth parameter. For the case of three dimensional data, the position of the cone in space is further specified by: Dip - the angle below the plane that is always negative, ZTol - the tolerance angle for Dip, and ZBand - constrains the cone in the vertical direction. (Note: the 3d parameters are disabled for 2d data.)
In the drop down box next to Variogram, select ‘Major’, ‘Minor’, ‘Both’, or ‘Neither’ to define the cones that are included in the graph of semivariogram results.
By pressing the Variography button on the main toolbar, the semivariogram results are calculated and plotted for each cone. The result of calculating the semivariogram for every lag is a series of semivariogram points. The major direction will appear blue while the monor direction is green. These variography results may be modeled separately or combined with a correlation model.
A check for isotropic correlation is possible using the Omnidirectional variogram. The corresponding cone for this variogram allows the data in all directions to be used in calculation. The omni directional variogram is available by setting the Tol (Ztol) equal to 90 degrees. Under the omni directional case, the bandwidth and angle parameters can still have an effect on results. Setting the angle equal to 45 degrees and the bandwidth equal to 100 feet, with an angle tolerance of 90 degrees produces the following cone.
In order to include every point, the bandwidth must be made large enough to encompass all datapoints.
Spatial Correlation and Indicator Kriging