Consensus optimization of total variation–based reverse time migration

Abstract

A common method of imaging with seismic data is reverse time migration (RTM) in which recorded data are back-propagated to form an image of the subsurface. Least-squares RTM (LSRTM) extends this method to an iterative method that minimizes a data misfit term. An appropriate regularization term such as a sparsity-promoting functional (e.g., total variation (TV)) is required to stabilize the LSRTM solution. In this paper, in order to efficiently solve such regularized LSRTM via distributed optimization algorithms, we first reformulate the problem into a consensus form. The alternating direction method of multipliers (ADMM) allows us to split the problem into separate subproblems, resulting to a two-step iteration which efficiently solves the original problem. The most time-consuming step is due to LSRTM of the data associated with each source, separately, which is carried out in parallel via a set of workers. The resulting local images are sent to a master that is responsible for generating (via a proximal mapping) a unique global image that is close to the mean of local images while minimizing the regularizing function. This loop gives a general framework for the parallel computation of regularized LSRTM and allows different regularizing functions to be employed by using appropriate proximal operators. We demonstrate the performance of the proposed algorithm with a set of numerical examples. The results show that TV methods can generate more accurate images compared with the l1-norm sparse LSRTM, l2-norm LSRTM, and traditional RTM. Furthermore, the new algorithm shows improvements over the traditional formulations due to the consensus parallelization.