Strategies in Adjoint Tomography

We investigate issues of convergence, resolution, and nonlinearity related to the feasibility of adjoint tomography in regional and global tomography and exploration geophysics. Most current methods of adjoint tomography, whether based on adjoint methods or other formulations, suffer from slow convergence in that only the gradient (not the Hessian) is readily available for computing model updates. As an alternative to working with the unpreconditioned gradients, we examine the speed-up offered by various preconditioners that can be computed in the framework of adjoint methods. We show that each preconditioner bears some similarity to the Hessian, thus motivating and justifying its use for accelerating convergence. Next, we examine the role of the Hessian in resolution analysis. Recalling that the action of the Hessian on an arbitrary model perturbation relates to the classical point spread function concept, we introduce a scalar quantity termed the average eigenvalue that provides a good overall representation of resolution. Whereas a point-spread function reveals the orientation of misfit contours, the average eigenvalue describes the sharpness of the misfit function along the direction of the chosen model perturbation. Finally, we provide an example in which we directly compare the results of travel time and waveform tomography, illustrating the resolution limits of the former and the nonlinearity pitfalls of the latter.