Semivariogram and Kriging

Exploring Spatial Variation

This long and complex section builds on the previous material by first giving a more complete account of the semivariogram. Particularly noteworthy are:

• The discussion beginning on page 198 (section 7.4.4 and its associated subjections) demands careful study. It describes how the (semi-)variogram summarizes three aspects of the spatial structure of the data:
  1. The local variability in observations, in a sense, the uncertainty of measurements, regardless of spatial aspects, is represented by the nugget value;
  2. The characteristic spatial scale of the data is represented by the range. At distances greater than the range, observations are of little use in estimating an unknown value;
  3. The underlying variance in the data is represented by the sill value.

Kriging

Make sure that you read through the discussion of the different forms of kriging. There isn't one form of kriging, but several. Each form has definite assumptions, and those assumptions need to be met for the interpolation to give accurate results. For example, universal kriging is a form of kriging that is used when the data exhibit a strong first order trend. You would be able to see a trend, for example, in a semivariogram as the sill value would not be 'leveling off' but continuously rising. Because of the trend, the further apart are any two observations, the more different their data values will be. We cope with this by modeling the trend using trend surface analysis, subtracting the trend from the base data to get residuals, and then fitting a semivariogram to the residuals. This form of kriging is more complex than ordinary kriging where the local mean of the data are unknown but assumed to be equal. There is co-kriging, simple kriging, block kriging, punctual kriging, and the list continues. 

If you have a strong background in mathematics, you may relish the discussion of kriging, otherwise you will most likely be thinking, "Huh?!" If that's the case, don't panic! It is possible to carry out kriging without fully understanding the mathematical details, as we will see in this week's project. If you are likely to use kriging a lot in your work, I would recommend finding out more from one of the references in the text (Isaaks and Srivastava's Introduction to Geostatistics) is particularly good, and amazingly readable given the complexities involved.