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Then, show the following: 1. Cnn ' Ctn 2. Cnn = Ctt ; MCst ; Cts MT + MCssMT 3. Cts ' MCss 4. Ctt = MCst + Cnt. 31) ;1 : Contrast this result with the damped least-squares method assuming that the correlation matrix Cnn is diagonal. 31) is known as the \stochastic inverse" Franklin, 1970 Jordan and Franklin, 1971]. 5 Physical Basis for Weighted Least-Squares So far we have treated the weights wi as if they are arbitrary positive constants. But are they arbitrary? If they are not arbitrary, then what physical or mathematical feature of the inversion problem determines the weights?

3, we show that an argument based on stability and regularization leads to the same choice of weight matrices. , j tij for some small threshold , or ray path i corresponds to a feasible ray path with ti 0). Then, we may want to make corrections using only the ray paths that are CHAPTER 3. LEAST SQUARES 39 considered unsatisfactory. We renumber the ray paths so the unsatisfactory ones are the rst m0 of the m total paths and suppose m0 < n. 53) n matrix and t0 is the corresponding m0 -vector of unsatisfactory where M0 is an m0 traveltime errors.

53) n matrix and t0 is the corresponding m0 -vector of unsatisfactory where M0 is an m0 traveltime errors. The problem of solving for the sj s is underdetermined as stated. 53) using a type of backprojection. We argue that the correction vector component sj should be a sum whose terms are proportional to lij (so that rays not passing through cell j make no contribution) and it should be a linear combination of the traveltime errors ti. 50). 50). 53). 53). 58) in general requires that the inverse of the m0 m0 matrix M0C;1(M0 )T must always exist.