Nonlinear inversion and tomography by James G. Berryman

By James G. Berryman

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Then we present some basic theorems about convex programming in this context. 1 Let (s) be any convex functional of s. Then the convex nonlinear pro- gramming problem associated with is to minimize (s) subject to the global feasibility constraints i (s) ti, for i = 1 : : : m. 20) for some positive weights fwi g and some set of ray paths P = fP1 : : : Pmg. Then, the convex linear programming problem associated with P is to minimize P (s) subject to the local feasibility constraints iPi (s) ti , for i = 1 : : : m.

2). 34) CHAPTER 4. ALGORITHMS 47 To nd the needed relation for the nonsymmetric/nonsquare matrix M, again consider the square matrix H. 25). Thus, we have completed one derivation of the pseudoinverse. 34) by writing down the equivalent expansion for Hy. 25) again follows, thus completing another derivation. We observe two special cases in which M is of full rank. If r = n m so the problem is either determined or overdetermined but M is of full rank, then s0 | the vector from the right null space | is necessarily zero.

8) is a concave functional. Proof: Given slowness models s1 and s2 and 2 0 1], let s = s1 + (1 ; )s2. 10) Since minimizes P for any xed model, it must be the case that P (s) (s1) (s1) and similarly for s2. Further, and (1 ; ) are non-negative. 11) CHAPTER 2. 13) Now we show that the concavity of iP and i implies the convexity of F P and F . 1 F P is a convex set. Proof: Suppose s1 s2 2 F P and let s = s1 + (1 ; )s2 where 0 each i, iP is a concave (actually linear) functional, we have P i (s P i (s1) + (1 ; ) ) iP (s2): 1.

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