Automatic Differentiation-Based Borehole Full Waveform Inversion

Accurate characterization of elastic properties near boreholes is critical for reservoir evaluation, well integrity assessment, and high-resolution formation imaging. Traditional borehole inversion methods often suffer from limited resolution and complex implementation, particularly in scenarios involving radial heterogeneity and thin layers. In this study, we present a novel borehole full waveform inversion framework based on automatic differentiation and recurrent neural networks. By reformulating the elastic wave equation in cylindrical coordinates as a recurrent structure, forward modeling becomes naturally embedded in a deep learning framework. Gradient computation is performed using automatic differentiation, which eliminates the need for manually derived adjoint equations and enables flexible implementation of multi-parameter inversion. We validate the proposed method through numerical experiments on a series of synthetic models with increasing geological complexity, including radially layered formations, curved interfaces, and thin low-velocity layers. We further investigate the effects of different acquisition configurations, such as source-receiver offsets, sparse sampling, and unidirectional observation setups. Results show that the proposed method can recover both P-wave and S-wave velocities with high accuracy, even resolving sub-wavelength thin beds beyond the capability of conventional velocity estimation method using logging data. The approach also demonstrates robustness under reduced acquisition coverage and practical logging conditions. Our findings suggest that the proposed framework offers a powerful and flexible solution for high-fidelity borehole imaging, with broad applicability in geophysical exploration and logging interpretation.