Iterative Helmholtz Surface Wave Tomography Using Generalized Wave Equation Smoothing Splines—Application to Western Europe

Abstract Over the last decade, the expansion and densification of broadband seismic networks has led to an increased interest in tomographic approaches based on gradiometry, which relies on the spatial derivatives of the phase and amplitude of surface wavefronts. Nevertheless, the use of amplitude information remains challenging. In this study, we present a novel approach to Helmholtz tomography based on the use of generalized wave equation smoothing splines. The application of these generalized smoothing splines results in the generation of smooth amplitude and phase fields that satisfy the Helmholtz and transport equations, and hence the wave equation. This enables the direct derivation of phase velocity maps using the Helmholtz equation. We apply this new Helmholtz tomography approach to Rayleigh waves recorded by permanent and temporary networks in Western Europe from 2008 to 2022 to obtain phase velocity maps for periods between 25 and 120 s. These maps reveal the detailed structure of the crust and upper mantle in this region. Compared to eikonal tomography, the phase velocity anomalies obtained by Helmholtz tomography are stronger in amplitude and more sharply defined.