In general, the synthetic data for one frequency, a shot positioned at and a receiver positioned at can be given by a linear operator acting on the full model space with ( in 2D) as
In equation (6), we use two important properties Ehinger et al. (1996): first, the Green functions are computed by means of the one-way wave equation, and second, the extrapolation is performed by using the adequate paraxial wave equations (flux conservation) Bamberger et al. (1988).
The quadratic cost function is
Notice that computing in equation (8) needs only the precomputed Green functions at model points and . Thus, the size of the problem can be considerably reduced by computing the Green functions only at the target location , reducing equation (8) to