Born approximation

Equation (1) exhibits a non-linear relationship between the laterally variable slowness and the propagated wavefield. For the remaining of this paper, I will conventionally refer to the methods in this class as non-linear methods. A second class of methods are found using the Born approximation for the wavefield perturbations. In physical terms, this approximation is only valid for media characterized by weak scattering, that is small velocity variation. Mathematically, the Born approximation is equivalent to a linearization of the exponential $e^x \approx 1+x$. With this new approximation, the expression for the downward-continued wavefield becomes:  

$$U_{z+\Delta z}\approx e^{i {k_z}_o\Delta z} U_z \left\{ 1+ ... ...} \S^{2n} \delta_n \right)\right]\left(s - s_o\right) \right\}.$$ (6)

Next two sections describe the various mixed-domain methods belonging to the two aforementioned classes, linear and non-linear, in relation to the general formula given by Equation (2).