It Is Only a Banana - Traveltime Sensitivity Kernels Using the Unwrapped Phase

Traveltime sensitivity kernels for finite-frequency traveltimes computed using the Born or Rytov approximations admits hallow banana shaped responses in the plane of propagation and a circular doughnut shaped responses in the cross section. This suggests that finite-frequency traveltimes are insensitive to velocity information along the infinite-frequency ray path, which is obviously inaccurate and creates a disconnect in the traveltime dependency on frequency. Using the instantaneous traveltime of the wavefield, which is capable of unwrapping the phase function, we obtain traveltime sensitivity kernels that have plain banana shape responses, with the thickness of the banana governed by the investigated frequency. This result confirms that the hallow banana shape is simply a result of the wrapping of the phase of the wavefield, in which Born nor Rytov approximations can properly deal with. The instantaneous traveltime can, thus, mitigate the nonlinearity problem encountered in finite-frequency traveltime inversions that may arise from these hallow banana sensitivity kernels.