Physical Sensitivity Kernels Can Emerge in Data-Driven Forward Models: Evidence From Surface-Wave Dispersion

Abstract

Data-driven neural networks are increasingly used as surrogate forward models in geophysics, but it remains unclear whether they recover only the data mapping or also the underlying physical sensitivity structure. Here we test this question using surface-wave dispersion. By comparing automatically differentiated gradients from a neural-network surrogate with theoretical sensitivity kernels, we show that the learned gradients can recover the main depth-dependent structure of physical kernels across a broad range of periods. This indicates that neural surrogate models can learn physically meaningful differential information, rather than acting as purely black-box predictors. At the same time, strong structural priors in the training distribution can introduce systematic artifacts into the inferred sensitivities. Our results show that neural forward surrogates can recover useful physical information for inversion and uncertainty analysis, while clarifying the conditions under which this differential structure remains physically consistent.

Figures (2)

• Figure 1: Influence of training priors on surrogate-derived sensitivity kernels for a representative velocity model. (a) Weak-prior ensemble, in which velocity models are broadly distributed without imposing specific structural features. The representative model used for all kernel calculations is shown by the black curve; shading indicates the P10–P90 and P25–P75 ranges. (b)–(c) Rayleigh- and Love-wave sensitivity kernels computed from the surrogate trained on the weak-prior ensemble, compared with theoretical kernels. The surrogate gradients closely follow the theoretical kernels, with peak sensitivity shifting systematically with period, consistent with physical expectations. (d) Strong-LVZ-prior ensemble, in which a low-velocity zone (LVZ) is imposed in the upper mantle. Horizontal bands and lines indicate the mean and ±1$\sigma$ ranges of the Moho depth and LVZ center inferred from the prior. (e)–(f) Rayleigh- and Love-wave sensitivity kernels from the surrogate trained on the strong-prior ensemble, using the same representative model as in (a). In contrast to (b)–(c), the surrogate gradients exhibit persistent depth-localized anomalies near the imposed LVZ depth that remain approximately fixed across periods, deviating from theoretical kernels whose sensitivity shifts with period. This comparison demonstrates that the physical consistency of surrogate-derived gradients depends on the training prior, with strong structural priors introducing non-physical features in the inferred sensitivity kernels.
• Figure 2: Surrogate-based gradient inversion and local posterior uncertainty estimated from the Fisher information matrix. Left panels show recovered shear-wave velocity models for four representative samples, including the true model (black), initial model (gray dashed), and inverted model (red). Shaded regions denote $\pm 2\sigma$ uncertainty derived from the inverse Fisher matrix. Right panels show corresponding dispersion fits for Rayleigh and Love waves before and after inversion. The inversion successfully recovers the main structural features of the true model and achieves close agreement with observed dispersion curves. The posterior uncertainty increases with depth, reflecting reduced sensitivity and increased non-uniqueness in deeper regions. These results demonstrate that gradients derived from the neural surrogate provide a physically meaningful approximation of the forward operator Jacobian and can be used for both inversion and uncertainty quantification.