Adaptive Wasserstein Distance-driven Least-squares Reverse Time Migration with Mini-batch Strategy

Least-squares reverse time migration (LSRTM) involves three essential components: objective function, gradient computation, and update strategy. Typically, the L2-norm misfit is used with gradients computed via adjoint-state method. However, the L2-norm misfit only emphasizes pointwise amplitude differences, ignoring structural information of seismic signals and lacking the ability to capture global waveform similarities. Moreover, it also tends to overemphasize data points with large residuals, which inadvertently diminishes the influence of small misfit data during inversion, thereby potentially degrading the overall imaging quality. To alleviate these issues, based on optimal transport (OT) theory, we propose novel adaptive Wasserstein distance-driven LSRTM inversion methods with Mini-batch strategy. Specifically, we integrate the Wasserstein1 distance and its coupled form into the objective function. Then the Wasserstein gradient flow is computed by novel automatic differentiation (AD) framework instead of adjoint-state method, thus avoiding heuristic approximations. For gradient updates, we perform batch sampling to original shots, forming small batches to enable Mini-batch optimization. This strategy reduces memory consumption and enhances the exploration capability for imaging in localized regions. In optimization process, we replace the computationally expensive line-search iteration method with an adaptive moment estimation (Adam) algorithm, which approximates adaptive step correction through momentum and bias correction, thereby ultimately establishing adaptive LSRTM inversion framework. Numerical experiments demonstrate that Wasserstein distance-driven LSRTM methods, including Wasserstein1-LSRTM and Coupled-LSRTM, significantly outperform L2-LSRTM in recovering deep structures. Overall, it shows notable advantages in edge preservation, convergence speed, and improved imaging quality, offering a promising alternative for geophysical imaging.