Relationship to Pratt's approach to waveform inversion

Equation 27 is a direct statement of the waveform inversion procedure of Pratt and Worthington (1989) and Sirgue and Pratt (2004). However, the use of one-way operators leads to the definition of a explicit scattering operator and a slightly different gradient operator:

$$\begin{eqnarray} g({\bf x}) \approx & - \omega^2 \sum_{{\mathbf s}} \sum_{{\math... ... r}) \Delta \Psi({\mathbf r},{\mathbf s}) \right]. & {\rm (WEMVA)}\end{eqnarray}$$ (27) (28)

Note that the two approaches are similar: the wavefield residuals are back-projected from the source point through the model and correlated with the source Green's function. This approach, though, has the scattering matrix chained between the source and receiver Green's functions. This derives from the application of a differential operator directly on the phase of the extrapolation operator.