You are perfectly right that, IF the sample histogram is normal,

the ensemble of all (conditional) realisations at any unsampled

grid node should have a normal distribution with for mean the simple

kriging estimate and with variance the simple kriging variance.

In many situations, however, the sample histogram is non-normal

and we have to transform the data first.

As I mentioned in my original e-mail, you could use a lognormal

transform, but then there is the problem of backtransforming

the kriging estimate. More importantly, what do you do with

the lognormal kriging variance??

Stochastic simulation is more appropriate than kriging whenever

the sample histogram is non-normal. You proceed as follows:

1. A normal score transform is first used to normalize the

histogram, that is each data is replaced by the corresponding

quantile of the standard normal distribution. Unlike the lognormal

transform, that kind of graphical transform ensures that the

resulting histogram is normal, regardless the shape of the

original histogram.

2. You perform the simulation (sequential or others) within the

multiGaussian framework.

3. You back-transform your simulated normal scores using the

inverse of the normal score transform.

As a result, although you use the multiGaussian formalism, the

distribution of simulated values at each grid node is not normal!

I agree with you that the incorporation of uncertainty into

decision-making requires not only a measure of the spread of

the distribution of possible values but also the values

themselves. In other words, it might not be worth sampling

a location with a large uncertainty if it appears that the

probability of exceeding a regulatory threshold is negligeable

at that location.

Pierre

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

________ ________

| \ / | Pierre Goovaerts

|_ \ / _| Assistant professor

__|________\/________|__ Dept of Civil & Environmental Engineering

| | The University of Michigan

| M I C H I G A N | EWRE Building, Room 117

|________________________| Ann Arbor, Michigan, 48109-2125, U.S.A

_| |_\ /_| |_

| |\ /| | E-mail: goovaert@...

|________| \/ |________| Phone: (734) 936-0141

Fax: (734) 763-2275

http://www-personal.engin.umich.edu/~goovaert/

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

On Mon, 7 Dec 1998, Edzer J. Pebesma wrote:

> Pierre Goovaerts wrote:

> > Cross validation has also its limitations. For example, it can not

> > be used to derive the location of additional monitoring sites.

> > For that type of applications, I would use a simulation approach

> > whereby a set of realizations, which reproduce the distribution

> > and pattern of variability of the data, is generated conditionally

> > to the measurements. For example, the generation of 100 realizations

> > would provide, at each grid node, a distribution of 100 simulated values,

> > the spread of which could be used as a measure of uncertainty.

> > In other words, locate your station where the uncertainty is the

> > largest (widest distribution of simulated values).

> >

> When you're using Gaussian simulation here, you're back at the kriging

> variance: the ensemble of all (conditional) realisations should have a

> normal distribution with mean the kriging estimate and with variance the

> kriging variance.

>

> Personally, I often find it hard to believe that kriging variance only can

> provide a sensible measure for deciding where to place additional sampling.

> Most often, one will be interested in high values, in low values, or in the

> decision whether interpolated values are below/above a critical threshold.

> Then, both kriging estimate and kriging variance should be taken into account,

> even when you're kriging indicators.

> --

> Edzer

> --

> *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!