Linearization

The simplest linearization of equation (7) is done by the Born approximation, which involves an approximation of the exponential function by a linear function $e^{i\phi} = 1+i\phi$. With this approximation, we obtain  

$$\Delta \mathcal U\approx \mathcal U_o\;i \Delta \Phi\;.$$ (9)

We can, therefore, compute a linear wavefield perturbation $\Delta \mathcal U\left(\omega, z \right)$ using a Born WEMVA operator:  

$$\Delta \mathcal U= {\bf B}\left(\mathcal U_o\right)\left[\Delta s\right]\;,$$ (10)

from which we can compute an image perturbation by summation over frequency:

$$\Delta \mathcal R= \sum_\omega\Delta \mathcal U\;.$$ (11)

For wave-equation MVA, we are interested in applying an inverse WEMVA operator to a given image perturbation. Therefore, the main challenge of the linearized WEMVA is to estimate correctly $\Delta \mathcal R$, i.e. an image perturbation corresponding to the accumulated phase differences given by all slowness anomalies above each image point.

Given an image perturbation $\Delta \mathcal R$,we can compute a wavefield perturbation $\Delta \mathcal U$ by the adjoint of the imaging operator, from which we can compute a slowness perturbation based on the background wavefield $\mathcal U_o$: 

$$\Delta s= {\bf B}^* \left(\mathcal U_o\right)\left[\Delta \mathcal U\right]\;.$$ (12)