Eikonal tomography: surface wave tomography by phase front tracking across a regional broad-band seismic array

We present a new method of surface wave tomography based on applying the eikonal equation to observed phase traveltime surfaces computed from seismic ambient noise. The source-receiver reciprocity in the ambient noise method implies that each station can be considered to be an effective source and the phase traveltime between that source and all other stations is used to track the phase front and construct the phase traveltime surface. Assuming that the amplitude of the waveform varies smoothly, the eikonal equation states that the gradient of the phase traveltime surface can be used to estimate both the local phase speed and the direction of wave propagation. For each location, we statistically summarize the distribution of azimuthally dependent phase speed measurements based on the phase traveltime surfaces centred on different effective source locations to estimate both the isotropic and azimuthally anisotropic phase speeds and their uncertainties. Examples are presented for the 12 and 24 s Rayleigh waves for the EarthScope/USArray Transportable Array stations in the western USA. We show that (1) the major resulting tomographic features are consistent with traditional inversion methods, (2) reliable uncertainties can be estimated for both the isotropic and anisotropic phase speeds, (3) 'resolution' can be approximated by the coherence length of the phase speed measurements and is about equal to the station spacing, (4) no explicit regularization is required in the inversion process and (5) azimuthally dependent phase speed anisotropy can be observed directly without assuming its functional form.