The propagation operator correction

Things are different, however, for the amplitude correction which is applied during the propagation. Equation 28 of Vlad et al. (2003) states that the correction to be applied at each downward continuation step is

$$U'_z = U_z e^{-\frac{v_z\Delta z }{2v }} \left[1-\frac{v_z\... ...\Delta z}{4}-v\right)\left(\frac{k_x}{\omega}\right)^4 \right].$$ (5)

Counting a possible linearization of the exponential in front and the presence of v_z, this is an expression of degree 5 in velocity. Moreover, at each depth step this has to be multiplied with the first-order-in-slowness propagation step in Equation 2 . The result would be a expression of at least degree 6. Discarding terms of order higher than 1 would result in losing an important amount accuracy in the propagation step itself, since a part of it will be multiplied with higher-order amplitude correction terms. We conclude that the propagation operator amplitude correction is not applicable to linearized downward continuation because of compounding linearization errors that will affect both the kinematics and the amplitudes.