Re: AI-GEOSTATS: (co)kriging predictions in the linear model of (co)regionalization
- Hi Bernard,
Here is a stab at an answer to your questions. I think that my reasoning is more or less correct - I hope it's not too obscure!
1) There is no reason for the estimates Y1* and Y2* to be uncorrelated. In fact they would only be under unusual circumstances. The easiest way to see this is to think of kriging as a projection (this is kind of hard to draw on this sort of screen - but see, for example, Journel + Huijbrets (ch 8).
Simple Kriging of Z can be interpreted as the projection of the unknown Z unto the linear space, S, generated by the data Zi (+ the constant vector 1) using the covariance as a metric. The characteristic of the kriged result is that the error Z-Z* is orthogonal to each vector in the space S.
Now you have Z = Y1 + Y2 + Y3. likewise the kriging of each component Yi can be done by projecting onto the same linear space and is characterised by the fact that the error Yi - Yi* is orthogonal to each vector in S. Therefore the error vector is 'perpendicular' to the S space.
As you noted, the overall space of random variables can be decomposed into 3 orthogonal subspaces (the Y1 Y2 and Y3 are orthogonal by virtue of the fact that they 'live' in these orthogonal subspaces). For the projections Yi*, which live in the space S to remain orthogonal, the projections onto S would have to take place parallel to these orthogonal subspaces. However, we have already noted that the error is 'perpendicular' to the S space. Thus, the only way this can happen is that S is congruent to the decomposition of the space into the 3 components. However this is most unlikely - it would mean that all of the observed data was missing some of the Yi components - in general the S space will be 'at an angle' to all three subspaces and the Yi* will not be orthogonal.
2) If I understand correctly here you wish to estimate Z1 from just Z2 and Z3. There is nothing impossible about doing this. The most important thing to note is that there is no way just using information about Z2 and Z3 to estimate the mean of Z1 - so all you can hope to estimate is the residual Z1 - m1. This will mean that the sum of weights must be zero.
You can krige by z1* = sum_i ( lambda_i * z2(x_i)) + sum_i ( mu_i * z3(x_i))
(the kriging equations can be found in the usual way) and your weights must satisy
sum_i lambda_i = 0
sum_i mu_i = 0
----- Original Message -----
From: Bernard Pelletier
Sent: Monday, July 16, 2001 11:15 PM
Subject: AI-GEOSTATS: (co)kriging predictions in the linear model of (co)regionalization
I am presently using geostatistical tools to analyze my data and would like to clarify a few things about the interpretation of (co)kriging estimates or predictions in the linear model of (co)regionalization. I would greatly appreciate any feedback, comment or suggestion on the following issues. I am planning to send to the LIST a summary of the answers received.
Thank you in advance for your help
Natural Resource Sciences
Macdonald Campus, McGill University
Using a linear model of regionalization, consider a random function Z that is a linear combination of three independent random functions (Y1, Y2, Y3) corresponding to three semivariogram models: a nugget (Y1)and two spherical models (Y2,Y3). We can then compute, by kriging, the estimates Y1*, Y2*, Y3* at each point on the original sampling grid. In theory, the random functions or spatial components Y1, Y2, and Y3 are uncorrelated. In practice, however, I get significant correlations (sometimes as high as 0.4) among Y1*, Y2*, and Y3*.
Are Y1*, Y2*, and Y3* supposed to be orthogonal (by some mathematical construction) or is it acceptable to observe a certain degree of correlation between them? I assume that this is due to some bias in the correlation estimation as a consequence of the presence of autocorrelation in Y2* and Y3*. What would be the consequences of an orthogonalization of the estimates after kriging? Note that I use all the original sampling points in the kriging equations and the sum of the three estimates does give me the initial Z back (exact interpolation). Since I have modified a MATLAB kriging module to calculate these estimates, I want to verify whether the presence of correlation between them is due to a mistake in the modified module.
Consider three regionalized variables: Z1, Z2 and Z3. Using a linear model of coregionalization and ordinary co-kriging, the predictions of Z1at unsampled locations (Z1*) are based on the information contained in both the fitted auto-variogram model for Z1 and the two fitted cross-variograms models (Z2-Z1 and Z3-Z1). Therefore, a part of Z1* is predicted from the autocorrelation structure of Z1, while another part (not exclusive) is predicted from the spatial dependency of Z1 on Z2 and Z3.
Is there any way to calculate the component of Z1* that is specifically related to the spatial dependency of Z1 on Z2 and Z3 ? Would it be possible to modify the system of co-kriging equations in order to include only the information contained in the fitted cross-variograms? What constraints should be put on the co-kriging weights? I am not sure about the theoretical ramifications of trying to calculate a new Z1* based only on cross-variograms. I am aware, however, that this could not really be called "kriging" since we would not find Z1 back when interpolating at sampling grid points.
Thank you again...
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