Multi-frequency autocorrelation mapping for shear wave elastography in reverberant wavefields

The estimation of shear-wave speed (SWS) is essential in elastography, where wave dynamics act as a direct indicator of the mechanical stiffness and viscoelastic profile of the medium. While traditional estimation frameworks generally rely on simple plane-wave assumptions, the interplay of boundary reflections, structural heterogeneities, and mode conversions induces a spatiotemporally intricate wavefield. In reverberant fields, phase singularities inherently associated with interference nodes invalidate the assumption of wavefield differentiability, rendering conventional phase-gradient mappings fundamentally ill-posed and susceptible to unbounded errors. Thus, we explore the mathematics of multi-component shear-wave fields to derive the fundamental properties required for an efficient estimation strategy. We introduce a stable autocorrelation mapping approach based on a phase-accumulation operator, Ψ, constructed from finite-lag spatial statistics. This non-differential, non-iterative inversion exploits the curvature of the normalized spatial autocorrelation to extract SWS with high statistical stability. Numerical experiments illustrate the characteristics of reverberant fields and confirm the framework, enabling the accurate estimation of Kelvin-Voigt fractional-derivative (KVFD) parameters (E₀,α,η). Ultimately, these findings suggest that the Ψ-operator provides a promising and theoretically grounded formulation for viscoelastic mapping, which appears well-behaved in the investigated reverberant scenarios, with experimental validation remaining an important next step.