Born wave-equation MVA

We define the wavefield perturbation $\Delta \mathcal W\left({z+\Delta z} \right)$ as the difference between the wavefield propagated through the medium with correct velocity $\mathcal W\left({z+\Delta z} \right)$ and the wavefield propagated through the background medium $\mathcal W_b \left({z+\Delta z} \right)$. With these definitions, we can write

$$\Delta \mathcal W\left({z+\Delta z} \right)= \mathcal W\left({z+\Delta z} \right)- \mathcal W_b \left({z+\Delta z} \right),$$ (6)

or  

$$\Delta \mathcal W\left({z+\Delta z} \right)= \mathcal W_b \l... ... \k_z} {d s} \right\vert _{s=s_r}\Delta s\Delta z} -1 \right].$$ (7)

Equation (7) represents the foundation of the wave-equation migration velocity analysis method Biondi and Sava (1999). The major problem with Equation (7) is that the wavefield $\Delta \mathcal W$ and slowness perturbations $\Delta s$ are not related through a linear relation, therefore, for inversion purposes, we need to further approximate it by linearizing the equation around the reference slowness (s_r)

Biondi and Sava (1999) choose to linearize Equation (7) using the Born approximation ($e^{i \phi} \approx 1 + i \phi$), from which the WEMVA equation becomes  

$$\Delta \mathcal W\left({z+\Delta z} \right)= \mathcal W_b \l... ...rac{d \k_z} {d s} \right\vert _{s=s_r}\Delta s\Delta z\right].$$ (8)

The problem with the Born linearization, Equation (8), is that it is is based on an assumption of small phase perturbation,

$$1+i \phi \approx \lim_{\phi\to 0} e^{i \phi}$$

which mainly translates into small slowness perturbations. This fact is more apparent if we recall that the linearization $e^{i \phi} \approx 1 + i \phi$ corresponds to an explicit numerical solution of the differential equation (2), a numerical solution which is notoriously unstable unless precautions are taken to consider small propagation steps. The main consequence of the limitations imposed by the Born approximation is that WEMVA can only consider small perturbations in the slowness model, which are likely too small relative to the demands of real problems. Since non-linear inversion is still not feasible for large size problems like the ones typical for seismic imaging, we seek other ways of linearizing Equation (7) which would still enable us to solve our inversion problem within the framework of linear optimization theory.