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

instead of the standard deviation.

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.

Main references about BME are:

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

weights under the uncertainty admitted for your data

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.

For more information about this method, check the books by G. Christakos,

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