Hi Mark,
it seems you have already realized the paradox of sampling in
geostatistics: the more you know about the variable in question, the better
you can optimize a sampling scheme for it. It isn't easy to break out of
this paradox, and that's probably the reason that sampling has received
relatively little attention in the geostatistical literature. You will not
find much about it in most textbooks.
You have probably already found a number of papers by Webster and McBratney
from the beginning of the 80's (mainly in the Journal of Soil Science, I
think). They described an algorithm for calculating the optimum grid
spacing for a sampling scheme, given the maximum allowed kriging variance
and a variogram. These papers, although relatively old, are still often
quoted. Another paper from those days dealing with the optimal type of grid
is Yfantis, E.A., Flatman, G.T. and Behar, J.V., 1987. Efficiency of
kriging estimation for square, triangular and hexagonal grids. Mathematical
Geology, 19(3): 183-205.
I normally don't like to advertize my own work this much, but hey.... this
was my Ph.D. thesis. I developed a simulated annealing - based algorithm
that (among other things) optimizes for the same criterion as the
Webster/McBratney papers, but that optimizes the optimal location of
individual points, rather than optimal grid spacing. Although this might
not be very useful in large, contiguous sampling areas, it considerably
improves your sampling efficiency when you already have preliminary samples
and/or many sampling constraints. Again, you need (to assume) a variogram.
A couple of references to my work:
-Van Groenigen, J.W. and Stein, A., 1998. Constrained optimization of
spatial sampling using continuous simulated annealing. Journal of
Environmental Quality, 27(5): 1078-1086.
-Van Groenigen, J.W., Siderius, W. and Stein, A., 1999. Constrained
optimisation of soil sampling for minimisation of the kriging variance.
Geoderma, 87: 239-259.
-Van Groenigen, J.W., Pieters, G. and Stein, A., 2000. Optimizing spatial
sampling for multivariate contamination in urban areas. Environmetrics, 11:
227-244.
Also, you can download a preliminary software implementation of this
algorithm from my website (see below).
Of course, there is a lot a controversy in the geostatistical community
about the use of kriging variance as a measure for interpolation error,
since it does not take into account the actual values of the measured
variable, which can give you problems when the intrinsic hypothesis doesn't
hold (and it often doesn't). Although this has some truth to it, my
philosophy is that that is exactly what makes it interesting for sampling
optimization, since you don't have those values before sampling anyway....
However, the last of my references used an optimization criterion that
doesn't involve kriging variance.
Cheers,
Jan Willem.
******************************************
Jan Willem van Groenigen
University of California - Davis
Dept. of Agronomy and Range Science
1 Shield Avenue
Davis, CA 95616 - 8515, U.S.A.
------------------------------
e-mail: [EMAIL PROTECTED]
http://agronomy.ucdavis.edu/groenigen
tel. (530) 752-3457
fax. (530) 752-4361
*****************************************
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