Bob Harrington wrote:
M.J. Abedini wrote:
Another issue regarding detrending concerns with variogram modeling
when trend coefficients are not given (i.e., unknown). I found
literature on this issue very disparate. Any comments regarding
unified approach on variogram modeling will be greatly appreciated.
This is not a unified approach, but considering Sebastiano Trevisani's
suggestion of using topography as an external drift variable for
kriging hydraulic head, the residual is simply depth to water from the
land surface. Depth to water is typically available (or easily
derivable from the DEM and head data) and the residual variogram could
be modeled directly from depth to water.
These are slightly different approaches:
Abedini discusses the use of topography T(s) as a linear component in
the trend, as in Z(s) = a + b * T(s) + e(s), Z(s) being groundwater
head, a and be unknown regression coefficients and e(s) a second order
stationary residual. Bob suggests to set, a priori, a to zero and b to
one, and do ordinary kriging basically on e(s)=Z(s)-T(s). Indeed, this
removes the estimation problem of a and b, but a value of b between 0
and 1 may yield better results; groundwater table behaves in general
much smoother than topography, for one thing. (Coefficient a is
basically present both in external drift/universal kriging and in
ordinary kriging).
The problem of having to estimate a and b before being able to access
residuals, needed for the residual variogram is a well known chicken and
egg problem: you need the residual variogram for "optimal" estimation of
a and b. My feeling is that geostatisticians were much more worrried
about this 10-20 years ago than they are now. A couple of papers that
suggested that the problem is not as large as we think it might be were
written by Peter Kitanitis, try for example
@article{kitanidis93,
author = {P. Kitanidis},
title = {Generalized Covariance Functions in Estimation},
journal = {Mathematical Geology},
volume = {25},
number = {5},
pages = {525--540},
year = {1993}
}
I have never run into problems with using residuals from ordinary least
squares regression for variogram estimation. I did try iterative
approaches with iteratively updated generalized least squares residuals
from time to time and found differences too small to pay attention to,
given the usually "rough" step of fitting some parametric variogram
model. If differences are considerable, you may have to rethink the
regression model; there might be multiple collinearity or
heteroscedasticity.
Best regards,
--
Edzer
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