OBTAINING SMOOTH SOLUTIONS TO LARGE, LINEAR, INVERSE PROBLEMS

It is not uncommon now for geophysical inverse problems to be paramete rized by 10(4) to 10(5) unknowns associated with upwards of 10(6) to 1 0(7) data constraints. The matrix problem defining the linearization o f such a system (e.g., Am = b) is usually solved with a least-squares criterion (m = (A(t)A) - 1 A(t)b). The size of the matrix, however, di scourages the direct solution of the system and researchers often turn to iterative techniques such as the method of conjugate gradients to obtain an estimate of the least-squares solution. These iterative meth ods take advantage of the sparseness of A, which often has as few as 2 -3 percent of its elements nonzero, and do not require the calculation (or storage) of the matrix A(t)A. Although there are usually many mor e data constraints than unknowns, these problems are, in general, unde rdetermined and therefore require some sort of regularization to obtai n a solution. When the regularization is simple damping, the conjugate gradients method tends to converge in relatively few iterations. Howe ver, when derivative-type regularization is applied (first derivative constraints to obtain the flattest model that fits the data; second de rivative to obtain the smoothest), the convergence of parts of the sol ution may be drastically inhibited. In a series of 1-D examples and a synthetic 2-D crosshole tomography example, we demonstrate this proble m and also suggest a method of accelerating the convergence through th e preconditioning of the conjugate gradient search directions. We deri ve a 1-D preconditioning operator for the case of first derivative reg ularization using a WKBJ approximation. We have found that preconditio ning can reduce the number of iterations necessary to obtain satisfact ory convergence by up to an order of magnitude. The conclusions we pre sent are also relevant to Bayesian inversion, where a smoothness const raint is imposed through an a priori covariance of the model.