Inexact Scaled-Sobolev Gradient Projection Least-Squares Reverse Time Migration

Wave equation-based inversion, such as Least-Squares Reverse Time Migration (LSRTM), is a powerful seismic imaging technique. However, the performance of LSRTM in constructing high-resolution subsurface models is closely tied to the design of the gradient. The commonly used <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gradient, derived within the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> space framework, often leads to insufficient illumination and slow convergence in inversion. To confront these issues, we introduce a novel Sobolev gradient based on Sobolev space and propose a novel inexact scaled-Sobolev gradient projection LSRTM (ISGP-LSRTM), which can correct the conventional <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gradient, accelerates convergence and mitigates illumination deficiencies. Specifically, we construct a scaled-Sobolev gradient operator to optimize the direction of the conventional adjoint gradient, addressing the low-resolution caused by insufficient gradient information during iterations. Futhermore, by incorporating an inexact backtracking strategy, a proximal backtracking projection operator based on scaled- Sobolev gradient is constructed to refine the traditional update process in LSRTM. Numerical tests on multiple models, including the wide-angle Valley model, the truncated Marmousi model, and the fault-rich truncated Sigsbee2B model, demonstrate the superior performance of the ISGP-LSRTM inversion. The proposed approach significantly improves imaging illumination and event continuity, achieves faster convergence in iterative curves, and effectively enhances migration imaging quality.