Hazardous cross terms

Suppose that we had not gotten rid of the layered media terms. The observations would be u+w and we would be autocorrelating that. Both u and w have essentially the same autocorrelation, but their cross-term is dangerous. It has a different form (as you can verify). This form multiplies the expectation of $\xi(g)$which might be theoretically zero but in practice might often not be zero. In principle we eliminate the cross term by eliminating u from the observation u+w, but in practice this difficulty remains to haunt us wherever we find the earth is not sufficiently horizontally layered.