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Is the following a straight-forward extension?
Underlying Gaussian process Y(s);
Observations z(s_i) being - in the simplest case - an average of Y(s) in a region centred on s_i; that is the 'support' of the observations is a region.
More generally the observations might be 'kernel-smoothed' versions of Y(s)
It is related to equation (37) on
http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9868.2011.00777.x/full