Linear migration methods

Another class of screen methods are derived using the Born approximation applied to the scattered wavefield. Mathematically, this is described by a linearization of the downward continuation operator. After linearization, the wavefield at depth $z+\Delta z$ is related to the wavefield at the previous depth level z by Equation (6). As for the non-linear methods, we can simplify this equation, and obtain different methods commonly encountered in the literature:

1.

We can, again, ignore the spatial variability of the slowness function for the terms of the summation, $\frac{s}{s_o}=1$ and $\delta_n = \sum\limits_{l=0 \choose n\ge 1}^{2n-2} 1$,which gives the following linear mixed-domain migration equation, known in the literature as the extended local Born-Fourier method Huang et al. (1999):  

$$U_{z+\Delta z}\approx e^{i {k_z}_o\Delta z} U_z \left\{ 1+... ...\left(2n-1\right) \right)\right]\left(s - s_o\right) \right\}.$$ (10)

2.

We can consider Equation (10) as a Taylor series expansion, therefore we can write it in a more compact, but equivalent, form as  

$$U_{z+\Delta z}\approx e^{i {k_z}_o\Delta z} U_z\left[1+ i\De... ...\left\vert \bf k_m\right\vert^2} } \left(s - s_o\right)\right].$$ (11)

This equation describes the local Born-Fourier a.k.a. pseudo-screen method Huang and Wu (1996).

The extended local Born-Fourier method, Equation (10), is preferable in practice, since Equation (11) can lead to instability when the denominator vanishes. Another way of avoiding the instability is to add a small complex quantity, $i\eta\left\vert \bf k_m\right\vert$, to the denominator, method that is known as the complexified local Born-Fourier or complexified pseudo-screen method de Hoop and Wu (1996):

$$U_{z+\Delta z}\approx e^{i {k_z}_o\Delta z} U_z\left[1+ i\De... ...^2\left\vert \bf k_m\right\vert^2}} \left(s - s_o\right)\right]$$ (12)