Hessian-based elastic reflection waveform inversion with scale fusion strategy

Full waveform inversion (FWI) theoretically has the potential to recover high-resolution and accurate velocity models. However, FWI primarily relies on refractions and turning waves, and therefore often fails to update the deep background velocity under limited-offset acquisition geometries. Reflection waveform inversion (RWI) can recover the low-to intermediate-wavenumber components of the velocity model from deeply penetrating reflected waves, thereby improving the background velocity model. Elastic reflection waveform inversion (ERWI) exploits multicomponent seismic data to simultaneously invert for P-wave and S-wave velocities, providing enriched information about subsurface structures. The construction of perturbation models plays a critical role in ERWI, as these models not only influence the background model updating process but also constitute an integral part of the final inversion results. However, conventional ERWI methods rely on gradient-based optimization and neglect the contribution of the Hessian matrix, leading to lower accuracy of the perturbation models. The Hessian matrix accounts for the effects of amplitude imbalance, uneven illumination, band-limited wavelets, and multi-parameter trade-offs. To overcome these limitations, we propose a truncated Gauss-Newton ERWI method in which the perturbation models are updated using efficient Hessian-vector products and a matrix-free conjugate gradient algorithm. Furthermore, based on the analysis of the Fréchet derivatives and gradient expressions, we introduce a scale-fusion strategy for ERWI. This strategy enables effective utilization of the perturbation models while progressively improving their accuracy throughout the iterative inversion process, thereby establishing a mutually reinforcing feedback mechanism between the perturbation and background models. Numerical experiments demonstrate that the proposed method achieves faster convergence, lower data misfit and model errors, and significantly improved inversion quality, while exhibiting strong robustness to noise.