The process of determining an appropriate semivariogram stands out as one of the most significant discriminators between EVS and all other analysis and visualization programs. Anyone who has ever struggled with trying to fit an appropriate semivariogram to real world data understands how tedious and difficult it truly is. EVS applies the experience and knowledge of a team of experts to that task in the form of expert system algorithms that assess the data and assure the appropriateness of the results. EVS also offers a technological breakthrough by extending the concept of the semivariogram. A traditional semivariogram plots one-half of the square of the differences between samples versus their distance from one another. EVS extends this concept by also considering the direction (vector distance).
Symmetry
Before beginning the process of best-fitting the semivariogram, the user can set an allowable semivariogram symmetry. If that symmetry is set to one (1.00) the direction or vector component of distance is ignored during semivariogram fitting. If, however, the symmetry is reduced, EVS will analyze trends in the data pairs and determine a "major axis" orientation. Instead of a semivariogram that can be represented by a line in two-space, EVS produces a semivariogram that must be described as a surface in three-space. Major and minor axis semivariogram parameters are determined and these are blended together in the regions between. The resulting multi-dimensional semivariogram is capable of better representing anisotropies or trends in the data such as a ground water plume being driven by a distinct flow direction.
The symmetry is defined as the allowable quotient between the minor axis range and sill and the major axis range and sill.
Viewing Semivariance Cloud Graph
With more complex semivariogram models comes the need to visualize and assess the quality of those models. EVS provides this capability. All semivariograms (whether symmetric or asymmetric) are displayed in three-space as a surface, and representations of the data pairs are displayed as colored lines. The lines are drawn from the pair location (two quadrant vector distance between pairs) to the closest point on the semivariogram surface. Pairs that are fit very well by the surface are represented as short lines (because they are close) and pairs that do not fit as well are longer. Lines above the surface are drawn in yellow and lines below the surface are drawn in magenta. The following examples demonstrate the difference in the semivariogram surface when the allowable symmetry is modified.
The Semivariance cloud is a graph of the semivariances for all pairs of points within the Pair Search Range defined in the Semivariogram Parameters (or calculated by EVS's expert system).The formula to calculate Semivariance for any given pair of points at locations i and j is:
SVar(i,j) = ([zi – zj]^2) / 2
Where:
SVar(i,j) = Semivariance for pair (i,j);zi = measured sample value at point i; and zj = measured sample value at point j.
Any given pair(i,j) are separated by a specific distance; this distance is plotted in the x-y plane of the Semivariance cloud graph.
We will begin by creating a simple network using Krig_2D and the Viewer. However, make no connections yet.
Before we can connect Krig_2D and the Viewer, we need to set some of the Semivariogram Parameters. To do this, open the Kriging Parameters window in Krig_2D, then check Semivariogram Parameters. First we will examine a symmetric semivariogram and then we will allow the symmetry to be reduced. In order to view the semivariogram, you must select the Plot Semivariogram toggle. When you do, the Z_Scale parameter will appear and three additional output ports will be added to the module. Z_Scale is the "z" exaggeration for the semivariogram display and will change the display in "real time". Don't worry about this value now; EVS will adjust it to a reasonable value after running.
The Plot Semivariogram toggle causes a plot to be generated showing the semivariogram that has been selected and how the data pairs fit to that surface. EVS uses a Least Squares best fit routine to select the optimal semivariogram within the criteria discussed above. For this case, what is plotted is one-half of the squares of the differences (semivariance) and vector distance of the pairs. When the semivariance is plotted, the total length of lines above the surface and below should be equal. However in general, the population of points below the surface will be greater (since there will be some large, squared differences balancing).
Now we can connect these modules. Unlike Workbook 1, we will connect the red/white (middle output port) of Krig_2D to the Viewer's red input port. This allows us to visualize the semivariogram.
Once you have constructed the network, click on the Read Data File button. When the browser (shown below) appears, select initial_soil_investigation_subsite.apdv and choose Open.
Run Krig_2D and your EVS Message Console should have the following messages. Note that the semivariogram model has a single set of Range and Sill values since the model is symmetric. Note that the Semivariogram window has been updated and the Z_Scale parameter has been updated.
Your console should show:
Select Az-El and set your view parameters to Scale = 1.10; Elevation = 10; Azimuth = 225 and your display should match.
What is being plotted in this case is the square of the difference between each of the pairs. The total length of yellow lines above the surface and magenta lines below should be equal. As you can see, the total number of points below the surface is greater but the lengths of some of the yellow lines are much longer than the longest magenta lines.
Let's enhance this display by adding axes with the axes module. Instance axes and connect the ports as shown below.
For the picture below, we also changed the Z axes settings to lighten the blue color for the vertical (Z) axis and renamed the axis to "Semivariance". The scale was changed to 0.9. Your viewer should now show something like:
You may want to rotate this model and examine it from all directions. If you set the elevation to zero, it will be more obvious that this surface (although plotted as a rectangle in the x-y projection) is axisymmetric. EVS will tell you how many data pairs there were and how many were plotted. If the number of data pairs exceeds 1,000, EVS will plot a subset of those pairs up to a maximum of 1,000. The subsetting is performed by taking every nth sample after the pairs are sorted. This assures that the plotted pairs will represent the full range of your semivariogram. To keep execution times reasonable EVS uses up to 50,000 pairs to best fit the semivariogram. If the total possible pairs exceed 50,000, EVS will subset the pairs using a deterministic random selection process. This process will select the same pairs for the same data set and settings, assuring repeatability of results.
Asymmetric Semivariogram
Now let's relax the symmetry and see what changes. We will change the symmetry to 0.50 that will allow the minor axis semivariogram to be as small as one-half of the major axis.
Run Krig_2D and your EVS Message Console should have the following messages. Note that the semivariogram model now has two sets of Range and Sill values since the model is asymmetric. EVS also reports the orientation of the major axis (relative to the "x" axis, measured counterclockwise).
The actual symmetry was about 2/3. If it had been close to 0.50, we would have known that our symmetry value of 0.50 had constrained the best-fitting algorithms. The view of the semivariogram cloud will be updated and should now be:
Note that this surface is clearly not axisymmetric. If you study this figure, you may see that it better fits the data.