Non-linear migration methods

The non-linear class of methods preserve the downward continuation operator, given by Equation (1), in its exponential form. Starting from Equation (2), we can derive the equations that describe various approximate migration methods. Here is a summary of methods, going from complex to simple:

1.

We can simplify the FFD migration equation by ignoring 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 equation Biondi (1999):

$$k_z\approx {k_z}_o+ \omega\left[1- \left(\sum\limits_{n=1}^... ...ht]^{2n} \left(2n-1\right) \right)\right]\left(s - s_o\right).$$ (7)

2.

In the next simplification, we consider, in addition to the earlier approximations, that the ratio $\frac{s}{s_o}=0$, which leads to the split-step Fourier method a.k.a. phase-screen method Stoffa et al. (1990):

$$k_z\approx {k_z}_o+ \omega\left(s - s_o\right)$$ (8)

3.

Finally, the simplest method of the family is phase-shift Gazdag and Sguazzero (1984); Gazdag (1978), for which we further assume that s-s_o=0, therefore

$$k_z\approx {k_z}_o.$$ (9)

For most of these methods, we can separate approximations of various orders , depending on the number of terms in the sum (n). We can also have versions that use several values of the reference slowness (s_o), followed by interpolation of the continued wavefield.

The bottom line is that all these simplified methods are just particular cases of the Fourier finite-difference method mathematically described by Equation (2). Similarly, we can derive all these simplified methods from the generalized screen method, mathematically described by Equation (5).