Expand Messages
• Dear list, Some time ago I asked the following question: I am currently working with spatial interpolation of geophysical data. Each observation is associated
Message 1 of 1 , Feb 14, 2003
• 0 Attachment
Dear list,

Some time ago I asked the following question:

I am currently working with spatial interpolation of geophysical
data. Each observation is associated with an individual and known
standard deviation. How should this infomation be incorporated if I
want to use ordinary kriging for interpolation ?? My idea was the
following:

When finding the vector of weights (w) by solving the system of
linear equations A*w=b, I would exchange the zeros in the diagonal
of the A-matrix with the individual observation variances. Does this
sound reasonable ??

and got the following four very useful answers. Thank you very much !!

***********************************************************************
Colin Daly wrote:

That works if your matrix is made up of covariance terms rather than
variogram terms. However you should use the variance of the error term

So in your notation A_ij = C_ij = D - Gamma_ij

where Gamma_ij are the variogram values, D is the sill of the variogram
(or larger than the largest variogram value if your variogram does not have
a sill) and at the diagonal you use C_ii+K_ii where K_ii are the variance
error terms.

This will not interpolate your data! It will filter the noise terms (which
you say that you know the variance of at each point)

***********************************************************************
Rubens Caldeira Monteiro wrote:

I suggest you to incorporate those information as soft data,
i.e., a data that you have some uncertainties related to them. A good
approach to do it is using Bayesian Maximum Entropy. This spatiotemporal
geostatistical method allows you to incorporate hard and soft data and
even deterministic (e.g., physical equations) and stochastic (e.g.,
variograms) general knowledge.

Christakos, G., P. Bogaert, and M.L. Serre, 2002, Temporal GIS,
Springer-Verlag, New York, N.Y., 220 p., CD Rom included.

Christakos, G., 2000, Modern Spatiotemporal Geostatistics, Oxford
University Press, New York, NY, 304 p. (3rd Reprint, 2001).

***********************************************************************
Isobel Clark wrote:

I presume what you have is a sort of 'analytical
error' for each sample? That is, the standard
deviation for two samples at the same location around
the 'true value' at the same location?

In this case, you can put the variance down the
diagonal of your kriging system to obtain optimal
values.

You would need to be careful that the 'analytical
variance' was not greater than the nugget effect of
the semi-variogram model.

The kriging system would be similar to that obtained
when the sample is not treated as a 'point', but
rather as a volume. This results in a lower kriging
variance than using zero on the diagonal, so to
compensate you should probably add the complete
'analytical variance' back on to get realistic
estimation variances.

There seems to be a lot of confusion in the books (and
software) about what happens if you have a significant
replication variance.

***********************************************************************
Dimitri D'OR wrote:

You surely find a proper way of solving your problem in using the
Bayesian Maximum Entropy (BME) approach. This method is especially
dedicated to the incorporation of various types of soft (imprecise)
information as intervals of values, pdf's, etc.

In your case, if you consider a Gaussian distributed error, you may
consider your data as soft data of the pdf-type. The best predictor will
thus be nonlinear, which is not possible if you stay within the class of
kriging predictors. Moreover, BME does not require hard (accurate) data.
It is able to make good use of data sets made of only soft information.

and some references on my web page (you will find the address in the
"people" section of the AI-Geostat Web site). Have also a look at the
BMElib software proposed in the software section (It's a freeware running
with Matlab).

Best regards / Venlig hilsen

SÃ¸ren Lophaven
******************************************************************************
Master of Science in Engineering | Ph.D. student
Informatics and Mathematical Modelling | Building 321, Room 011
Technical University of Denmark | 2800 kgs. Lyngby, Denmark
E-mail: snl@... | http://www.imm.dtu.dk/~snl
Telephone: +45 45253419 |
******************************************************************************

--
* To post a message to the list, send it to ai-geostats@...
* As a general service to the users, please remember to post a summary of any useful responses to your questions.
* To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
* Support to the list is provided at http://www.ai-geostats.org
Your message has been successfully submitted and would be delivered to recipients shortly.