AI-GEOSTATS: (co)kriging predictions in the linear model of (co)regionalization

[Bernard Pelletier]

Hello, 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 Yours truly, Bernard Pelletier Natural Resource Sciences Macdonald Campus, McGill University Question 1 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. Question 2 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... Bernard Pelletier