Kriging is an alternative to many other point interpolation techniques. Unlike straightforward methods, such as Nearest Point, Trend Surface, Moving Average or Moving Surface; Kriging is based on a statistical method. Kriging is the only interpolation method available in ILWIS that gives you an an interpolated map and output error map with the standard errors of the estimates. As Kriging is an advanced technique that heavily relies on statistical theory, the novice user of GIS may need to ask advice from an expert.
Before you are going to use the Kriging method you should have thought about things like:
Once you have decided that Kriging is the method you want to use, you should continue with the following steps.
1.1 Visual and statistical data inspection
In the Catalog, select the point map with the sample points. Press the right mouse button and select Properties from the context-sensitive menu. The Point Map - Properties dialog box is opened.
Double-click the point map in the Catalog. The Display Options - Point Map dialog box is presented. Accept the defaults by clicking the OK button. The point map is opened. Check the distribution of the sample points visually. Another way to investigate whether your points are randomly distributed, or appear clustered, regular, or paired, etc., is by doing a Pattern Analysis.
If you decide that there are enough sample points available and that the distribution of points is good enough to do a Kriging interpolation, measure the shortest and longest distance between two sample points in the point map.
You can measure the length of both the shortest and the longest point-pair vector with the Measure Distance button (the pair of compasses) from the toolbar of the map window. You need this distance information later on for defining the number of lags and the lag spacing in the Variogram Surface operation (step 1.2), the Spatial Correlation or the Cross Variogram operation (step 2.1).
If you have sufficient sample points to do a proper Kriging interpolation, split up your data set in two parts. Use one part for interpolation and the other part for verification of the interpolated map.
Next, calculate the variance of the sample data set.
When the variation of the variable under study is not the same in all directions, then anisotropy is present. In case of suspected anisotropy, calculate a Variogram Surface with the Variogram Surface operation.
The output map can best be viewed in a map window using representation Pseudo while a histogram has been calculated. To view the coordinates and the position of the origin in the output raster map, you can add grid lines, where the grid distance equals the specified lag spacing. It is important to recognize the origin of the plot/output map.
You can measure the direction of anisotropy with the Measure Distance button from the toolbar of the map window, e.g. by following a 'line' of blue pixels going through the origin of the plot. You need this angle later on in step 2.1 (Spatial Correlation; bidirectional method) and step 4.3 (Anisotropic Kriging).
One can perform a Kriging operation (i.e. Universal Kriging) while taking into account a local drift or trend that is supposed to exist within the limiting distance defined around each pixel to be interpolated. Very often you know the trend already:
If a global trend is present in the sample set, subtract the trend for the input data with a TabCalc statement. Perform Ordinary Kriging on the de-trended data set and use MapCalc to add both output maps again. This method is an alternative for Universal Kriging. However, a major disadvantage of this alternative is that the error map is incorrect.
1.4 Multiple variables
If you decide that the variable under study is sparsely sampled, find out if there is another variable that is better sampled and has many corresponding sample points (identical XY-coordinates).
Open the attribute table that is linked to the point map. Find out if there are two columns with value domains and corresponding XY- coordinates. If there is a second variable, calculate the variance of both variables individually and the correlation between the two columns.
When the two variables are highly correlated, you can use the better-sampled variable and the relationship between the two variables to help to interpolate the sparsely sampled one with CoKriging.
If the correlation between the two variables is low it is advised to use another interpolation technique or not to interpolate at all.
2.1 Spatial Correlation and Cross Variogram
Kriging assumes a certain degree of spatial correlation between the input point values. To investigate whether your point values are spatially correlated and until which distance from any point this correlation occurs, you can use the Spatial Correlation operation. When you want to use CoKriging, you can investigate spatial correlation with Cross Variogram.
You must use the bidirectional method when anisotropy is present in your data set and you have to do an Anisotropic Kriging interpolation. Use the same anisotropy angle as measured in step 1.2.
Then, display the output table of the Spatial Correlation operation or the output table of the Cross variogram operation in a table window. Inspect in the output table the following columns:
columns Distance, AvgLag, NrPairs;
columns Distance, AvgLag1/AvgLag2, NrPairs1/NrPairs2.
columns: Distance, AvgLag, NrPairsA/NrPairsB/NrPairsAB and verify the Cauchy-Schwarz condition: g2AB £ gA * gB , for all values of the distance h.
2.2 Displaying experimental variograms
The next step is to create point graphs, i.e. experimental semi- and cross-variogram(s), from the columns of the Spatial Correlation or Cross Variogram output table.
When you used the omnidirectional Spatial Correlation method, you can draw one graph:
The discrete experimental semi- or cross-variogram values will by default be displayed as points in a new graph window.
When you used the bidirectional Spatial Correlation method, you can draw two graphs:
When you used the Cross Variogram operation, you can draw three graphs:
To add a second, third etc. graph layer to the graph window, choose Add Graph from Columns from the Edit menu in the graph window, or click the Add Graph button in the toolbar of the graph window.
In the graph window, 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. You may also wish to adapt the Y-axis from 0 to more or less the expected variance of your input sample values (s2, calculated in step 1.1).
3.1 Adding models
Then, you need to model the discrete values of your experimental semi- or cross-variogram by a continuous function, which will give an expected value for any desired distance h.
When you used the bidirectional Spatial Correlation method keep in mind that the model in the specified direction should be the same as the one in the perpendicular direction. ILWIS supports only geometric anisotropy: only the ranges of the models can be different.
3.2 Editing models
You are advised to visually experiment with models and with the values for sill, range, and nugget to find the best line through your discrete experimental semi- or cross variogram values.
If you like, you can also add other layers, i.e. other semi-variaogram models, for instance by choosing Add Graph, Semi-variogram Model from the Edit menu in the graph window, or by clicking the arrow next to the Add Graph button and selecting Semi-variogram Model.
3.3 Checking models
You can use the Column SemiVariogram operation to find which model fits your experimental semi- or cross-variogram values best. This operation calculates semi- or cross-variogram values according to a user-specified model and stores calculated semi- or cross-variogram values in an output column.
are the experimental semi- or cross-variogram values calculated by the Spatial Correlation or by the Cross Variogram operation;
are the semi- or cross-variogram values calculated by the Column SemiVariogram operation;
is the total number of distance classes/intervals.
The numerator of the fraction gives the sum of the squared differences between the experimental semi- or cross-variogram values and the semi- or cross-variogram values calculated by a user-specified semi- or cross-variogram model.
The denominator of the fraction gives the sum of the squared differences between the experimental values and the average experimental semi- or cross-variogram value of all distance classes/intervals.
The Goodness of Fit indicator can be calculated using some TabCalc statements:
g_gam_sqr = SQ(SemiVar - SemiCol)
is the name of the column with the experimental semi- or cross-variogram values
is the name of the column with the semi- or cross-variogram values calculated with the Column SemiVariogram operation
mean_g = AVG(SemiVar)
g_avg_sqr = SQ(SemiVar - mean_g)
R2 = 1 - SUM(g_gam_sqr) / SUM(g_avg_sqr)
The maximum value of R2 is 1, meaning an exact match of semi-or cross-variogram values calculated using a certain semi- or cross-variogram model and parameters, and experimental semi- or cross-variogram values. Once, you have decided which model fits your data best, you can continue with the Kriging operation.
Perform one of the 5 Kriging operations available in ILWIS:
4.1 Simple Kriging
Simple Kriging assumes that the randomized spatial function is stationary and that the mean is constant over the area of interest. This assumption is often unrealistic; the more robust Ordinary Kriging is thus frequently used instead. In Simple Kriging, all input points are used to calculate each output pixel value. For more information, see Kriging : functionality.
4.2 Ordinary Kriging
In Ordinary Kriging the randomized spatial function is non-stationary and the mean varies over the area of interest. Ordinary Kriging amounts to re-estimating the mean at each new location. In Ordinary Kriging, you can influence the number of points that should be taken into account in the calculation of an output pixel value by specifying a limiting distance and a minimum and maximum number of points. Only the points that fall within the limiting distance to an output pixel will be used in the calculation for that output pixel value. For more information, see Kriging : functionality.
4.3 Anisotropic Kriging
Anisotropic Kriging is a variant of Ordinary Kriging. Anisotropic Kriging incorporates the influence of direction dependency , i.e. Anisotropy in the Kriging operation. For more information, see Anisotropic Kriging : functionality.
4.4 Universal Kriging
Universal Kriging is another variant of the Ordinary Kriging operation: Universal Kriging is Kriging with a local trend. The local trend or drift is a continuous and slowly varying trend surface on top of which the variation to be interpolated is superimposed. The assumption of a 'constant' mean in Ordinary Kriging is replaced by a prior trend model that is incorporated in the predictions. For more information, see Universal Kriging : functionality.
CoKriging is a multivariate extension of the Ordinary Kriging method. One parameter, which is too sparsely sampled to carry out an accurate interpolation, is highly correlated with another parameter of which enough samples are available to do Kriging. Based on the well sampled data set Kriging is carried out. For more information, see CoKriging : functionality.
5.1 Confidence interval maps
From the combination of a Kriged output map containing the estimates and its output error map, you can create confidence interval maps by using some MapCalc statements. For more information, see How to calculate confidence interval maps.