Abstract
SUMMARYThe phase velocity of surface waves can be directly determined from the amplitude and phase of the regional wavefield using the Helmholtz equation. However, the Helmholtz equation involves estimating the Laplacian of the amplitude field, a challenging operation to perform on noisy and sparsely sampled seismic data. For this reason, the amplitude information is often discarded. In that case, phase-velocity maps are reconstructed with the eikonal equation, which relates the local phase slowness to the gradient of the phase. Here, we derive analytical expression of the errors arising from neglecting the amplitude of the wavefield in eikonal tomography. In general, these errors are quite strong but they vary sinusoidally with the wave propagation direction. Consequently, if the azimuthal coverage is good, they will average out, and unbiased phase-velocity maps can be obtained with eikonal tomography. We numerically validate these results with a synthetic tomography experiment.
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