  Least-squares reverse time migration is well-known for its capability to
generate artifact-free true-amplitude subsurface images through fitting
observed data in the least-squares sense. However, when applied to realistic
imaging problems, this approach is faced with issues related to overfitting and
excessive computational costs induced by many wave-equation solves. The fact
that the source function is unknown complicates this situation even further.
Motivated by recent results in stochastic optimization and transform-domain
sparsity-promotion, we demonstrate that the computational costs of inversion
can be reduced significantly while avoiding imaging artifacts and restoring
amplitudes. While powerful, these new approaches do require accurate
information on the source-time function, which is often lacking. Without this
information, the imaging quality deteriorates rapidly. We address this issue by
presenting an approach where the source-time function is estimated on the fly
through a technique known as variable projection. Aside from introducing
negligible computational overhead, the proposed method is shown to perform well
on imaging problems with noisy data and problems that involve complex settings
such as salt. In either case, the presented method produces high resolution
high-amplitude fidelity images including an estimates for the source-time
function. In addition, due to its use of stochastic optimization, we arrive at
these images at roughly one to two times the cost of conventional reverse time
migration involving all data.
