Helmholtz surface wave tomography for isotropic and azimuthally anisotropic structure

The growth of the Earthscope/USArray Transportable Array (TA) has prompted the development of new methods in surface wave tomography that track phase fronts across the array and map the traveltime field for each earthquake or for each station from ambient noise. Directionally dependent phase velocities are determined locally by measuring the gradient of the observed traveltime field without the performance of a formal inversion. This method is based on the eikonal equation and is, therefore, referred to as 'eikonal tomography'. Eikonal tomography is a bent-ray theoretic method, but does not account for finite frequency effects such as wave interference, wave front healing, or backward scattering. This shortcoming potentially may lead to both systematic bias and random error in the phase velocity measurements, which would be particularly important at the longer periods studied with earthquakes. It is shown here that eikonal tomography can be improved by using amplitude measurements to construct a geographically localized correction via the Helmholtz equation. This procedure should be thought of as a finite-frequency correction that does not require the construction of finite-frequency kernels and is referred to as 'Helmholtz tomography'. We demonstrate the method with Rayleigh wave measurements following earthquakes between periods of 30 and 100 s in the western US using data from the TA. With Helmholtz tomography at long periods (>50 s): (1) resolution of small-scale isotropic structures, which correspond to known geological features, is improved, (2) uncertainties in the isotropic phase velocity maps are reduced, (3) the directionally dependent phase velocity measurements are less scattered, (4) spurious 1-psi azimuthal anisotropy near significant isotropic structural contrasts is reduced, and (5) estimates of 2-psi anisotropy are better correlated across periods.