- 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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DO NOT SEND Subscribe/Unsubscribe requests to the list! - 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

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

>

>

>

>

>

>--

>*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!

>

*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!