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weighting CA neighbors by distance


I am doing some variations of cellular automata.
Using only the actual value of the neighbors in determining the value of
the current cell, I can use:

Apply[Plus, Map[RotateRight[mat1, ##] &, nbrs]];

where mat1 is the matrix of values for whatever the CA is modelling.
and nbrs is a list of pairs (eg.
{{-1,-1},{-1,0},{-1,1},{0,-1},{0,1},{1,-1},{1,0},
{1,1}} for the 8 cell moore neighborhood.)  defining the neighborhood I
am interested in.

The line above sums all neighbors as they are but, I wish to weight the
neighbors
according to their distance (eg. {-1,-1} is farther away than {-1,0}
when using the
RotateRight function.
I think I may be able to do this with ListConvolve, or Inner but I can't
figure it out.
Anyone have any ideas?



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