## 1678Re: GEOSTATS: kriging weighted values?

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• Oct 24, 2000
Dear Dr. Sandefur:

(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
>
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
>
>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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