Irrespective of which interpolation method you use remember that there are
two parts; one is the interpolation method and two is the data. Ideally one
would like to tailor the method to the data, that is one of the potential
strengths of geostatistics, i.e., kriging. The data can be used to model
the variogram but the user still must make some choices and one must be
aware that in some cases the interpolation results are affected more by the
choices than by the data.
To contrast kriging with thin plate splines for example, the thin plate
spline is in fact a form of kriging with a particular choice of a variogram
(the choice being made without any reference to the data but rather with
respect to the characteristics of the interpolating function, the thin
plate spline forces the interpolating function to be smooth, i.e., have a
continuous second derivative). It is possible to adjust Inverse Distance
Weighting to the data in the sense of changing the exponent but this
requires splitting the data into a test data set and a control data set (
which obviously does not work for small data sets). See Kane et al
(Computers and Geosciences, 1982 ) for an example of this and also for
examples of how sensitive or insensitive the interpolation fitting is to
the choice of the exponent.
Finally note that the geometry/pattern of the data locations and the
pattern of interpolation locations can override nearly everything else.
Suppose the data were on a regular grid and the objective is to interpolate
at the mid point of each cell, then essentially any (isotropic)
interpolation method will give the same results if only the nearest four
data locations are used, namely the arithmetic average of the four. In the
case of kriging, it doesn't matter which variogram model is used.
Another point, if a valid variogram model is used (no matter how it is
obtained, whether by fitting to the data or simply inspiration) kriging
will produce the best linear interpolation, note
that "best" is in the context of the particular variogram or covariance
that is used. Similarly IWD will produce results.
A previous comment about smoothing is pertinent but again essentially any
interpolation method will smooth the data. Using the indicator transform is
one way to reduce this effect.
Donald E. Myers
Department of Mathematics
University of Arizona
Tucson, AZ 85721
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