Wavefront healing: a banana–doughnut perspective

SUMMARY Wavefront healing is a ubiquitous diffraction phenomenon that affects cross‐correlation traveltime measurements, whenever the scale of the 3‐D variations in wave speed is comparable to the characteristic wavelength of the waves. We conduct a theoretical and numerical analysis of this finite‐frequency phenomenon, using a 3‐D pseudospectral code to compute and measure synthetic pressure‐response waveforms and ‘ground truth’ cross‐correlation traveltimes at various distances behind a smooth, spherical anomaly in an otherwise homogeneous acoustic medium. Wavefront healing is ignored in traveltime tomographic inversions based upon linearized geometrical ray theory, in as much as it is strictly an infinite‐frequency approximation. In contrast, a 3‐D banana–doughnut Fréchet kernel does account for wavefront healing because it is cored by a tubular region of negligible traveltime sensitivity along the source–receiver geometrical ray. The cross‐path width of the 3‐D kernel varies as the square root of the wavelength λ times the source–receiver distance L , so that as a wave propagates, an anomaly at a fixed location finds itself increasingly able to ‘hide’ within the growing doughnut ‘hole’. The results of our numerical investigations indicate that banana–doughnut traveltime predictions are generally in excellent agreement with measured ground truth traveltimes over a wide range of propagation distances and anomaly dimensions and magnitudes. Linearized ray theory is, on the other hand, only valid for large 3‐D anomalies that are smooth on the kernel width scale √(λ L). In detail, there is an asymmetry in the wavefront healing behaviour behind a fast and slow anomaly that cannot be adequately modelled by any theory that posits a linear relationship between the measured traveltime shift and the wave‐speed perturbation.