Two-way vs. one-way modeling

For efficiency reasons the Green's functions in equations 4, 5, and 6 are computed by means of a one-way wave-equation extrapolator. No upgoing energy can be modeled by following this approach, since the evanescent energy is usually damped Claerbout (1985). This makes the one-way propagator act as a dip filter depending on the velocity model (from the dispersion relation, $\frac{\omega^2}{v^2} < {\bf k}^2$). Also, the conventional one-way wave-equation does not model accurately the the amplitude behavior with the angle of propagation Zhang et al. (2005). Another problem arises when the velocity varies laterally, then getting energy to accurately propagate close to $90^\circ$ is a big challenge.

The previous limitations of the one-way modeling can be mitigated by getting sophisticated when implementing the extrapolator. The dip filter effect should be reduced by including the Jacobian of the change of variable from $\omega$ to k_z Sava and Biondi (2001), thus making ${\bf L}$ closer to be unitary. To properly model the amplitude behavior with the angle of propagation, Zhang et al. (2005) proposed using a modified one-way wave-equation who's solution match the Kirchhoff inversion solution. The effect caused by the lateral variation of the velocity can also be mitigated by using better approximations of the square root operator.

For the example presented in this paper I used the conventional one-way wave equation Claerbout (1985). I did not include the Jacobian, and approximated the square root operator with Split Step Fourier plus interpolation (PSPI).