Before tackling this tutorial, you will need to download and install a dataset following these instructions:
Create a folder called kriging somewhere under your personal directory (e.g. C:\Users\jdoe\Documents\Tutorials\kriging\).
Download the data for this exercise then extract the contents of kriging.zip into your newly created kriging folder.
From the kriging/ project folder, open the kriging.aprx map document.
The map consists of two layers: a precipitation point layer, Precipitation.shp (in mm) and the outline of the 48 states, States.shp.
Click on the Analysis tab to bring up its ribbon.
Near the right-hand side of the ribbon, click on the Geostatistical Wizard tool.
If the tool is grayed out, you might not be properly licensed to use the tool.
The ordinary Kriging interpolation leverages spatial autocorrelation in the data when determining the weights of all sampled locations. As such, it assumes that the mean precipitation value is constant across the study extent. We therefore need to check for any overall trend in precipitation values across the 48 states before attempting an ordinary kriging interpolation.
In the geostatistical wizard window, select Global Polynomial Interpolation under the Deterministic methods option.
Set the Precipitation layer as the input dataset and Precip as the data field.
Click Next.
In the next window, you will see a preview of the trend surface using a first order polynomial.
A first order polynomial fits a flat plane to the data (think of a tilted sheet of paper). In this example, the plane is tilted along the southeast-northwest axis.
The complexity of the trend model can be increased by increasing the polynomial order. For example, increase the polynomial order to 2; this adds curvature to the trend as shown in the adjoining figure.
You could increase the polynomial order to capture more of the variability in the data. However, we must remember the purpose of this exercise: That is to identify overall trends in the precipitation values and not to capture localized variability which will be addressed with the kriging operation.
The geostatistical wizard offers a leave-one-out cross validation tool that you can use to help guide you in your interpolation parameter settings. If you completed the IDW interpolation tutorial you already know how to fine-tune the interpolation parameters using this tool. Here, we will only focus on the results of the cross-validation since the goal here is not to necessarily reduce the predictive errors with the trend model.
In the Geostatistical wizard window, click on Next.
This window plots the predicted values at each sampled location against the expected values. The blue line is a slope that best fits the observed point pattern. The root mean square error (RMSE) statistic offers an overall estimated error (in the same units as the input attribute value). In this example, the RMSE is 294.7 mm.
Now, the next logical step would be to generate a raster layer from the trend model, then subtract the estimated precipitation values from the original point precipitation dataset. In other words, we would need to generate a separate point layer with de-trended precipitation values before proceeding with the kriging interpolation.
Fortunately, creating a separate "de-trended" point layer is not needed. The built-in kriging tool provides an option to de-trend the data before generating the variogram model. We will explore this tool next.
In the Geostatistical Wizard window, click on the Back button several times until you return to the first window.
Select the Kriging/CoKriging method and set Precipitation and Precip as the source dataset and data field, respectively.
Click Next.
Select Ordinary Kriging >> Prediction for output type.
Set the trend removal order to Second. This is the trend model we decided to go with in the earlier step. You'll note that this pull-down menu limits you to a third order polynomial for the reasons explained in the previous section.
Click Next.
This next window offers a preview of the trend model. You should not have to change the parameters in this window. But make sure that the exploratory trend surface analysis value is set to 0 (this ensures that we are generating a global model and not a local one).
Click Next.
This next window shows a semi-variogram plot of the de-trended precipitation values and provides you with a plethora of parameters to modify. The plot also shows the binned values and resulting fitted variogram model. For more information on the making of a variogram model, refer to the lecture notes.
Change the model to Spherical (another popular model is Circular).
Here, we'll take advantage of the optimizer option to come up with a decent set of model parameters for the model. Click on the optimizer button. (Note: Be sure to select the spherical model before optimizing).
Click Next.
There are many more parameters that can be tweaked. While kriging interpolation is not for the faint of heart, the few suggested adjustments can go a long ways in improving the interpolation results.
In this next window, you are shown a preview of the interpolated surface. You are also shown the weights assigned to the sampled points surrounding an unsampled location.
This window also offers additional interpolation parameters that pertain to the search window such as the ellipse shape and ellipse sectors from which to draw sampled points. We'll keep the default settings.
Click Next.
This last window provides us with a cross-validation plot and the RMSE value.
A thorough workflow would entail toggling back and forth between this window and tweaking of model parameters in the previous windows. Note that this window provides additional diagnostic plots such as a normal quantile plot (this compares the residuals distribution to a Normal distribution) and a standardized error plot. As noted earlier, kriging interpolation can be quite involved thus requiring these additional diagnostic plots to help fine tune the model.
Click Finish to generate the final interpolation layer.
A window might pop up confirming your choice of parameters. Click OK to proceed.
At this point, the interpolated surface resides only in memory. To make this a permanent raster, you must export it.
In the Contents pane, right-click on the Kriging layer and select Export layer >> to Rasters.
This will bring up the GA Layer to Rasters geoprocessing pane.
Save the output raster to your project folder and name it kriging.tif.
Set the output pixel size to 20000 m.
Before clicking Run, we will need to set the output extent and mask.
Click on the Environments tab.
In the Environments window, set the Extent and the Mask to that of the States layer.
Click Run to generate the final raster.
Feel free to change the symbology of the newly created raster. Preferably one that adopts a stretch scheme. Save and close your project.
This completes the tutorial.