Cross Variogram

Functionality

The Cross Variogram calculates experimental semi-variogram values for the two input variables and cross-variogram values for the combination of both variables. As two variables are handled simultaneously, the Cross Variogram operation can be seen as the multivariate form of the Spatial correlation operation. As input for the Cross Variogram operation, you can use a point map with a linked attribute table containing at least two value attribute columns.

Often data may be available for more than one attribute per sampled location. One set of samples may be expensive or difficult to measure and is therefore sampled infrequently while another variable may be cheap or easy to measure and has more observations or more accurate ones. If the correlation between the two variables is high (positive or negative), then it may be possible to use the information about the spatial variation of a well-sampled variable (the covariable) to help to interpolate a sparsely sampled variable (the predictand). This estimation method is known as CoKriging.

From the output table of the Cross Variogram operation, you can create semi-variogram models for both variables, and a cross-variogram model for the combination of the variables (see Additional information below or Graph window : Add semi-variogram model. The models do not need to be identical: for the experimental semi-variogram values of variable A, you may use for instance a spherical model; for the experimental semi-variogram values of variable B, an exponential model; and for the cross-variogram values of variables A and B, yet another model may be used (see Figure 1). The models should obey the Cauchy-Schwarz inequality. For more information, see Cross Variogram : algorithm.

All three models are required as input for the CoKriging operation. CoKriging calculates estimates or predictions for a poorly sampled variable (the predictand) with help of a well-sampled variable (the covariable). The variables should be highly correlated (positive or negative).

Spherical distance:

Optionally, you can choose to calculate with spherical distances, i.e. distances calculated over the sphere using the projection that is specified in the coordinate system used by the input point map. It is advised to use this spherical distance option for maps that comprise large areas (countries or regions) and for maps that use LatLon coordinates. In more general terms, spherical distance should be used when there are 'large' scale differences within a map as a consequence of projecting the globe-shaped earth surface onto a plane.

When the spherical distance option is not used, distances will be calculated in a plane as Euclidean distances.

Tip:

When you used the spherical distance option in the Cross Variogram operation, you should also use the spherical distance option in a subsequent CoKriging operation.

Input map requirements:

The input point map should be a point map with an ID domain. The point map should be linked to an attribute table that contains at least 2 columns with a value domain (column A and column B).

Cross-variogram values will only be calculated for point pairs that have values in both Columns A and B. This implies that for some locations, you need to have measured values for both variables.

Output table:

An output table with domain None is created.

The output table will contain 8 columns:

Column Distance lists the middle values of the distance intervals;
Column AvgLag lists for each distance interval, the average distance between points of point pairs in this distance interval;
Column NrPairsA lists for each distance interval, the number of point pairs of variable A found at these distances towards each other;
Column NrPairsB lists for each distance interval, the number of point pairs of variable B found at these distances towards each other;
Column NrPairsAB lists for each distance interval, the number of point pairs with valid A and B value respectively found at these distances towards each other;
Column SemivarA lists for each distance interval, the experimental semi-variogram value of the point pairs in this distance interval for variable A;
Column SemivarB lists for each distance interval, the experimental semi-variogram value of the point pairs in this distance interval for variable B;
Column CrossVarAB lists for each distance interval, the experimental cross-variogram value of the point pairs in this distance interval for the combined variables A and B.

Mind:

when in an distance interval no point pairs are found, then the values in columns AvgLag and SemiVarA, SemivarB and CrossVarAB will be undefined for these distance intervals.

Additional information

Semi-variogram(s) and Cross-variogram(s):

From the results of the Cross Variogram operation, you can make two semi-variograms and one cross-variogram. In the semi-variograms and the cross-variogram, the discrete experimental semi-variogram values and the cross-variogram values, that are the outcome of the Cross Variogram operation, can be modeled by a continuous function so that a semi-variogram value or a cross-variogram value will be available for any desired distance h for the CoKriging operation later on.

How to display Semi-variogram(s) and Cross-variogram(s):

Display the input table of the Cross Variogram operation in a table window. You first have to determine the variance (s²) of your input variables.

From the Columns menu in the table window, choose the Statistics command.

In the Column Statistics dialog box:

select the Variance function;
select the input variable for which you want to calculate the variance.

Display the output table of the Cross Variogram operation in a table window.

Inspect the following columns in the output table:

columns Distance and AvgLag: usually not more than half of the total sampled distance should be taken into account; the larger the distance between the point pairs, the less point pairs, and the less reliable the outcome;
columns NrPairsA, NrPairsB, NrPairsAB: for reliable semi-variogram and cross-variogram values, distance classes should at least contain 30 point pairs. When you find many point pairs per distance class, you may consider to decrease the lag spacing.

In the output table of Cross Variogram operation, create point graphs, i.e. experimental semi-variograms and an experimental cross-variogram, from the Distance, SemiVar and CrossVar columns:

From the File menu in the table window, choose the Create Graph command.
In the Create Graph dialog box:

choose for the X-axis, the Distance column or the AvgLag column;
choose for the Y-axis, a SemiVar or the CrossVar column.

The experimental semi-variogram or cross-variogram values will be automatically displayed as a point graph.
You may wish to adapt the boundaries of the X-axis from 0 to more or less half of the total distance between the samples, and the Y-axis from 0 to more or less the expected variance (s²) of your input sample values.

In this way, you can create three point graphs (e.g. in three graph windows):

Distance or AvgLag against SemiVarA (semi-variogram values of the first variable),
Distance or AvgLag against SemiVarB (semi-variogram values of the second variable),
Distance or AvgLag against CrossVarAB (cross-variogram values).

For more information on semi-variograms, see Spatial correlation : functionality, section Additional information on Semi-variograms, or Graph window : Add Semi-variogram Model, section Additional information.

The next step, before CoKriging, is to model the discrete values of your experimental semi-variograms and your cross-variogram by a continuous function, which will give an expected value for any desired distance.

From the Edit menu in the graph window, choose the Add Graph Semi-variogram Model command.
In the Add Graph Semi-variogram Model dialog box, you can choose the type of semi-variogram model or cross-variogram model (spherical, exponential, etc.), and you can fill out values for the sill, range and nugget. A (continuous) line will then be drawn according to the model you selected and the values you selected for sill, range and nugget.

You are advised to visually experiment a little with models and sill, range, and nugget values to find the best line through your experimental semi-variogram and cross-variogram values. For more information on creating semi-variogram models, refer to the Graph window : Add Semi-variogram Model dialog box.

To save your graphs, you can choose the Save or the Save As commands from the File menu in a graph window.

To find which semi-variogram or cross-variogram models fit your experimental semi-variogram or cross-variogram values best, you can also use the Column Semi-variogram operation. This operation calculates semi-variogram or cross-variogram values according to a user-specified semi-variogram model or cross-variogram model, and it stores calculated semi-variogram values or cross-variogram values in an output column. By calculating the Goodness of Fit (R²), you can test which semi-variogram model or cross-variogram model fits best. For more information, see Column Semi-variogram operation, section Calculating R².

Once you have decided, which semi-variogram or cross-variogram models, and which values for sill, range and nugget fit your data best, you can continue with the CoKriging operation.