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1678Re: GEOSTATS: kriging weighted values?

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  • Yetta Jager
    Oct 24, 2000
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      Dear Dr. Sandefur:

      I thought about this problem quite a bit for work with finite populations
      (lake and stream survey data that had a stratified random sampling design). 
      I don't think including sample weights as you have is valid (i.e, it won't
      give the optimal solution).  The point is, you don't know where the other
      members of the population represented by the weight on a particular sample
      are geographically.  We found two solutions.

      In one paper, we cokriged to estimate for a finite population of stream
      nodes from a unequal-probability sample of nodes.  Here we didn't make
      any use of sample weights and compared population estimates obtained from
      a Horowitz-Thompson estimator to those obtained by cokriging, which brought
      in ancillary spatial information (elevation).

      In another paper, we kriged by stratum (see the report on Pattern-plus
      on my webpage).  This means calculating separate variograms, etc.
      I never finished, but I was working on creating a full variance-
      covariance matrix by assuming that the strata had different sills (total
      variances) but the same model form and nugget.  The sample data can
      then be standardized to have the same diagonal (sill) and the VC matrix can
      be filled with entries from the standardized variogram (correlogram).
      Then it should be possible to krig the whole system together and back
      transform to get the final interpolated estimates.  This should work
      if you have samples that belong to sub-populations that differ in their
      means and variances, but not the degree of autocorrelation.  I have a FORTRAN
      program written to do this that I've never gotten around to debugging, if
      anyone wants to take it on :-).

      Good luck, and I'd like to hear about it if you find another solution.

      Yetta

      At 06:27 AM 10/24/00 -0700, you wrote:
      >Hi-
      >
      > Suppose I have some spatial samples with weights (w1 w2 ....) 
      and I want to use these weights in addition to  kriging weights and also suppose (and a big suppose it is) that I have the (a) variogram(gamma).  If I  krige ignoring (w1 w2...) and weight the kriging weights (wk1 wk1 ...) with  (w1 w2) ie
      >
      >  Answer=
      Sum(Value1*w1*wk1+Value2*w1*wk2+...)/Sum(w1*wk1+w1*wk2+...)
      >
      >I get I some cases (zero nugget variogram and some negative weights)
      unreasonable results e.g.
      >
      >V1  w1 wk1
      >30   .1  1.2
      >50   .9  -.2
      > answer=90
      >
      >Unreasonable results are a well known result with negative weights
      and usually indicate a (local) inconsistency between the data and the variogram but I think the problem is exacerbated by not allowing for the weights w1.. in the kriging matrix
      >
      >My guess is that  the sample weights should be  coupled
      WITHIN  the kriging equations something like:
      >  w1w1Gamma11 w1w2Gamma12 ..... .....
      1               wk1              w1Gamma1b
      >  w2w1Gamma21 w2w2Gamma22 ..... ....  
      1               wk2       =   w2Gamma2b
      ......                  ......                  .....  ..... 1                 ..               .....
      >    
      1                        1                     ..... ......0                 mu                      1
      >
      >Before I work thru the math for my guess and code a solution  I
      would like input on
      >   1) Has this problem been solved before?
      >   2) Is a C or Fortran or public .exe version of  the
      solution available (3d preferred)
      >
      >thanx
      >
      >bob sandefur
      >
      >Principal Geostatistician
      >Pincock Allen & Holt
      >International Minerals Consultants
      >274 Union Suite 200
      >Lakewood CO 80228
      >USA
      >303 914-4467  v
      >303 987-8907 f
      >
      >
      >
      >
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      ------------------------------------------------------
      Yetta Jager                                            
      Environmental Sciences Division  
      Oak Ridge National Laboratory        
      P.O. Box 2008, MS 6036            
      Oak Ridge, TN 37831-6036
      U.S.A.

      OFFICE: 865/574-8143 
      FAX:    865/576-8543
      Work email: jagerhi@...
      Home email: hjager@...
      WEBpage: http://www.esd.ornl.gov/~zij/
      -----------------------------------------------------

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