Convex Duality and Separable Inverse Problems in Wave Propagation

A number of inverse problems in wave propagation admit division of the model space into kinematic or propagator variables and dynamic or amplitude (or reeectivity, or source, or...) variables. The dynamic variables have a linear or essentially linear innuence on the simulated data, hence the best t formulations of inverse problems are separable in the sense of nonlinear programming. Mean square objective functions for the analysis of such problems may be studied either in their original, primal form or, via the Fenschel-Rockefellar Theorem of convex analysis, in their dual form. The duality principle displays connections between apparently unrelated approaches: for example the semblance function of Migration Velocity Analysis in reeection seismol-ogy is dual to the Output Least Squares objective, under certain circumstances. An abstract setting allows a uniform approach to the study of a wide variety of objective functions for waveform inversion. For a subset of these objectives (the diierential semblance class) the duality relation permits a simple proof of smoothness of the objective as function of the kinematic variables, and suggests an accurate algorithm for computation of approximate gradients. For a still smaller subset of objective functions and problems, it is possible to demonstrate that only one critical point exists in a large domain of kinematic variables. For such problems and formulations, global solution of the inverse problem is possible using local smooth optimization techniques. Author's Note: These notes form a draft of joint work in progress with Prof. Guy Chavent. I wish to record my indebtedness to Prof. Chavent for provision of the key insight, for his collaboration in working out its implications, and for his hospitality, and that of Universit e de Paris IX and INRIA Roquencourt, where this work was carried out.