Least-squares reverse-time migration with a multiplicative Cauchy constraint

One of reverse-time migration’s main limitations is that an unscaled adjoint operator is prone to produce images with low resolution, inaccurate amplitudes, and even artifacts. Least-squares reverse-time migration (LSRTM) was subsequently introduced to mitigate this inadequacy via the use of an approximation to an inverse operator. LSRTM suffers from its own limitations, most importantly from poor condition, which often manifests itself as image artifacts. One approach to ameliorate this issue is to constrain the optimization problem by introducing a penalty term to the cost function. Penalizing estimated parameters for sparsity is one such constraint that has been shown to be effective. A drawback of this technique is that it introduces a trade-off between data fitting and image sparsity. Furthermore, if using the Cauchy constraint, an additional trade-off is introduced due to the requirement to estimate a hyper-parameter. We introduce an alternative approach that mitigates these trade-offs by combining a multiplicative cost function with an effective means for determining the Cauchy hyper-parameter. We also introduce a new formulation of the multiplicative cost function that avoids over-penalization by the constraint via the introduction of a relaxation term. Finally, we seek to improve the computational efficiency by introducing a new approach for computing the step length. As such, our method introduces three novel aspects to constrained LSRTM: (i) a relaxed multiplicative cost function, (ii) semi-automatic estimation of the Cauchy hyper-parameter, and (iii) efficient computation of the step length. We discuss the theory and implementation, followed by application to three synthetic datasets and a real ultrasonic dataset. Given the presence of large salt bodies, elasticity, and noise, along with the directivity of piezoelectric ultrasonic transducers, these datasets provide a challenging test of the approach outlined. Results demonstrate that our method is robust in handling the challenges imposed by these scenarios.