At the moment I am examining methods for estimating the mean and the

variance of the mean of irregularly shaped blocks, in this case soil

units and agricultural fields, based on an irregular soil sampling

scheme.

The simplest solution is block kriging based on a fine mesh

discretization of the block. However when the range of the variogram is

less than the lengths of the block being estimated this is not

recommended. In this situation conditional simulation is recommended,

e.g the GSLIB manual and some papers by Goovaerts. This involves

simulating values, conditional to the samples, onto a fine grid covering

the block. Over many realizations, the mean and its variance of the

region may be estimated from the simulated values on the fine grid. In

particular, LU Decompsition has been mentioned.

This issue has been raised on the list a few years ago but it was not

clear which way was best.

Other than getting a pdf, some reasons for preferring simulation

include:

(i) numerical instabilities when block kriging with large matrices

(ii) the semivariogram is most accurate at short lags so to using

modelled semivariance values beyond the range is unwise.

I have used both methods (I used LU Decomposition for simualtion) and

get very similar results which is probably as expected. It seems to me

that problems with block kriging the regional mean equally apply to

conditional simulation when using LU Decomposiiton. LU Decomposiiotn

involves a larger matrix than kriging, and like kriging it uses the

estimates of the semivariance at longer lags to fill out the covariance

matrix between the condtioning data and non-conditioning locations.

I could use other simulation algorithms such as Sequential Gaussian

Simulation but it all seems an overkill when I just want an estimate of

the mean and the variance. Especially if I want to extend my work to

cokrige 4 variables simultaneously, each being equally important. I

imagine this would take a long time with Sequential Gaussian Simulation

let alone the time to code this!

Finally, a good estimate of the variance is equally important as the

mean in my work so finding the mean value based on dividing the blocks

into smaller blocks is not appropriate as it does not give an estimate

of the variance

So given that I just want the mean and variance, and not the pdf, why

should I use simulation, especially using LU Decompsition, when the

results are the same?

Would it be wise to state that if you only want the mean and variance =

use block kriging, if you want a pdf = use condtional simulation?

Thanks for your help in advance.

Tom

Thomas Bishop

Biomathematics and Bioinformatics Division

Rothamsted Research

Harpenden

Hertfordshire

AL5 2JQ

United Kingdom

Tel: + 44 (0) 1582 763 133 ext 2574

Fax: + 44 (0) 1582 760 981