Re: GEOSTATS: Kriging Instability
- I am wondering about your modeling of the spatial temporal variograms, this
may be the problem.
a. Did you attempt to use a metric in space time?
b. In general using a zonal anisotropy will not work, i.e., writing the
space-time variogram as the sum of a spatial variogram and a temporal
variogram. See Myers and Journel, Math Geology circa 1990.
c. One could use the product of two covariances and then convert to a
variogram, see De Cesare, Myers and Posa in the proceedings of the 1996
Geostatistics meeting in Wollongong.
d. A better choice however is not only the product but also a sum of
covariances but one has to be careful about the coefficients to ensure the
conditional negative definiteness of the resulting variogram. De Cesare,
Posa and I have a paper we will be presenting at the conference in Valencia
in November that uses this model.
e. Posa and Journel have a paper in Math Geology on ill-conditioning, i.e.,
the conditioning number of the coefficient matrix. Also see some work of
Narcowich and Ward (Texas A & M) on radial basis functions. For the
connection see a couple of papers of mine.
Donald E. Myers
At 03:56 PM 7/1/98 -0500, you wrote:
>I'm working with a dataset of irregularly sampled sea-surface temperatures
>following seasonal and gross E-W and N-S trend removal. The estimated
>variogram is a linear combination of admissible isotropic variograms as
>functions of lag space and time. The kriging step involves estimation of
>about 250 gridded points through almost 50 years.
>I've found that theoretical efforts to define a kriging neighborhood are
>overshadowed by numerical instabilities in the solution of the (ordinary)
>linear kriging equation G*b=g, where G contains the variogram estimates of
>the observed differences, g is a vector containing variogram estimates of
>the observed - predicted location, and b is the vector of kriging weights
>(and of course single LaGrange multiplier).
>Even with iterative refinement, I've found that kriging neighborhoods of
>about 10 observations are the the largest I can use before G becomes
>ill-conditioned. My only measure of reliability is the condition number
>obtained by taking the ratio of the largest to smallest singular values in
>an SVD. My intuition is that this is a more common problem than has been
>addressed in the literature. I have not yet received a copy of McCarn and
>Carr (1992) and am in hopes this helps. In the meantime, my questions to
>the group are as follows;
>1. Is there a way to estimate numerical precision of the kriging weights
>using the condition number, or something else for that matter? I've seen
>this done using condition numbers calculated from norms of the inverses
>but I question that approach since the inverse is inaccurate in
>ill-conditioned cases. My ultimate goal is to identify and eliminate
>imprecise kriging weights.
>2. Is there a more optimal technique than gaussian elimination with
>partial pivoting combined with iterative improvement? Can someone
>recommend a package or subroutine? I typically cannot use a "canned"
>package because of the spatiotemporal nature of the problem but am open to
>3. How have others dealt with this problem? I would be most interested in
>hearing of other experiences with kriging instability. Hopefully, there
>are ways I have not thought of in getting around this.
>Thank you for your comments. I will post the responses.
>L. Scott Baggett
>Statistics Department, MS138
>6100 Main Street
>Houston, TX 77005-1892
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