This email is a follow up to my two emails to the list
on the subject of trends. I hope that email will answer
any questions which were directed to me by list members.
My original email:
I have recently been looking at the issue of dealing
with a non-stationary mean in kriging. Attendence at
the recent Geostats-UK meeting indicated that there was
some disagreement concerning the use of terminology. In
particular, universal kriging (or kriging with a trend,
KT) is outlined in many major texts as a technique
which may estimate the local trend (or drift) on a
moving window basis and the variogram is not estimated
as part of this process (unlike IRF-k kriging where the
generalised variogram is estimated as a part of the
whole process). I would be interested to know if this
definition of KT is actually disputed between
For KT, many researchers consider it unwise to use the
variogram of residuals from a global polynomial trend.
This is presumably because of (i) bias in the estimated
variogram and (ii) the fact that the modelled trend
used for the variogram estimation is different to that
used for KT. However, I am under the impression that
some researchers use the variogram of residuals from a
polynomial trend in this manner. I would be interested
to know what approaches list members are using to
estimate variograms where there is a marked trend
effect for KT. A more specific issue concerns the
perceived problems with using median polishing (as
outlined in Cressie's 'Statistics for Spatial Data') to
estimate the variogram in the presense of a trend for
use with KT (as opposed to median polish kriging).
This email is really a general attempt at creating a
discussion about what is clearly a major issue in
In reply to this, Andrew Lister commented on the wide
variety of potential approaches that may be used to
deal with a trend. Andrew mentioned the use of cross-
validation and estimation from a sample of a data set
as ways to assess the success of different approaches.
Paulo Ribeiro also commented on the variety of
approaches that may be used to deal with a trend. Paulo
wrote "I advocate a more frequent use of a model-based
approach to geostatistics where tools like likelihood
estimation can be used and mean (constant or not) and
covariance parameters can be jointly estimated... I
believe that such model-based approach together with
the expertise of the researcher (to built sensible
models) and eventually using Bayesian inference tools
can make a better use of the information available
using methods based on theoretically justifiable
My follow-up email:
This email is a follow-up to my email to the list of a
week ago. I would like to thank Andrew Lister and Paulo
Ribeiro who commented on (i) the need to adopt
different approaches on the basis of the data and other
issues and (ii) the potential of model-based
geostatistics in dealing with non-stationarity.
On the same theme, but more specifically, I would be
interested to know what software list members are using
for IRF-k kriging. I am dealing with quite large data
sets (more than 50,000 observations in some cases). I
am aware only of ISATIS, which I am unlikely to be able
to access. If all else fails I will have to write
something myself. Also, I have yet to encounter a
satisfactory approach to estimation of the variogram
for universal kriging (where the trend clearly affects
the variogram in all directions), so any suggestions
would be gratefully received.
As a result of this second email I exchanged several
emails with Edzer Pebesma, primarily concerning
universal kriging (kriging with a trend, KT) and
estimation of the variogram for use with KT.
In relation to my concern with the use of the variogram
of residuals from a polynomial trend Edzer cited:
Kitanidis, P.K., 1993. Generalized Covariance Functions
in Estimation. Math. Geol. 25 (5), pp. 525-540.
Edzer noted that Gstat offers full KT functionality.
Edzer's suggested approach was to use OLS to estimate
the form of the trend as a starting point for GLS.
Following this, given the residual variogram, WLS may
be used as an initial step for fitting the variogram
model followed by REML for estimation of the range. All
of these stages may be implemented in Gstat.
As a result of these emails, one approach I am now
1. Estimate the form of the trend with OLS
2. Modify the coefficients of the trend model with GLS
(= hopefully more robust estimator of the trend)
3. Estimate the variogram from the GLS residuals
4. Fit a model to the variogram with WLS
5. Estimate (using the raw data) with KT
This approach is not universally accepted but it seems
reasonable and is used by many practioners. The
approach is recommended in some text books but it is
clearly rejected in others. This reservation is due in
part to the fact that the trend used to obtain the
residuals for the variogram is different to that used
in KT (the trend being estimated for a moving window).
Thus, many researchers believe IRF-k is a better
alternative to KT...
IRF-k kriging seems to be little used by list members
and I didn't receive any suggestions (in addition to
ISATIS) as to any packages which offer IRK-k kriging
which will work with large data sets. Modification of
existing code or writing new code would seem to be the
only way forward.
I had presumed that there was some dispute as to what
KT actually is, as I have encountered disagreement in
discussions with other researchers (see my original
email above). The definition that seems to be almost
universally accepted (as clearly stated in many text
books) is that KT estimates the trend locally as part
of the kriging process. Some people seem to have used
the term KT to describe the process whereby the form of
the trend is estimated and the residuals used for
estimation of the variogram and for kriging, after
which the trend is added back. It seems there are
grounds for clearer definitions in addition to those in
the conventional literature since disagreement remains,
perhaps the AI-Geostats FAQ would be a good place...
My thanks again to all those who offered comments. If
any members of the list have any queries about any of
the above I will try to answer them.
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