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

http://www.arizona.edu/~donaldm

At 03:56 PM 7/1/98 -0500, you wrote:>Greetings,

--

>

>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

>any suggestion.

>

>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

>Rice University

>Statistics Department, MS138

>6100 Main Street

>Houston, TX 77005-1892

>

>

>

>

>

>--

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