Loading ...
Sorry, an error occurred while loading the content.

GEOSTATS: Trends summary

Expand Messages
  • c.lloyd@Queens-Belfast.AC.UK
    Hello, 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
    Message 1 of 1 , Dec 15, 1999
    • 0 Attachment
      Hello,

      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
      researchers.

      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
      geostatistics.

      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
      inferential principles".

      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
      using is:

      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.

      Chris Lloyd
      --
      *To post a message to the list, send it to ai-geostats@....
      *As a general service to list users, please remember to post a summary
      of any useful responses to your questions.
      *To unsubscribe, send email to majordomo@... with no subject and
      "unsubscribe ai-geostats" in the message body.
      DO NOT SEND Subscribe/Unsubscribe requests to the list!
    Your message has been successfully submitted and would be delivered to recipients shortly.