Full Waveform Inversion with a Pseudotime Approach

Formulating full waveform inversion with earth models parameterized in vertical time can reduce, without removing it, the non-linearity of the inverse problem when hard contrasts are present in the initial models. The change of coordinates between the depth and pseudotime coordinates depends on the vertical velocity because the vertical time is equal to the integral over depth of the inverse of the vertical velocity. This transformation requires no assumption about the lateral variations of the earth model. The pseudotime full waveform inversion implementation is achieved through change rules before and after solving the wave equations. This allows the use of standard frequency-domain solvers or time marching schemes. The dependency of the change of coordinates on the earth parameters we invert for needs to be accounted for in the change rules. We illustrate the relevance of this formulation with a synthetic and a real example when hard velocity contrasts (e.g., due to salt) are included in the initial earth model.