Linearization

Minimizing the objective function J in Equation (19) involves solving a non-linear least-squares problem using the gradient given by Equation (25).

Alternatively, we can linearize the wavefield $\u$ with respect to a reference wavefield $\v$

$$\u = \v + \delta \u= \v + {\bf G}\delta s$$ (30)

and optimize  

$$J\left(s\right) = \frac{1}{2} \left\vert{\bf I}\left(\AA \v - {\bf B}\mathcal T+ \AA {\bf G}\delta s\right)\right\vert^2,$$ (31)

which is a linear least-squares problem.

Equation (31) can also be represented by fitting goals using the usual SEP terminology as:

$$- {\bf I}\left(\AA \v - {\bf B}\mathcal T\right)\approx {\bf I}\AA {\bf G}\delta s,$$ (32)

which takes special forms depending on our choice of the operators $\AA$ and ${\bf B}$:

WEMVA by TIF WEMVA by DSO

$$- {\bf I}\left(\v - \mathcal T\right)\approx {\bf I}{\bf G}\delta s - {\bf I}\left({\bf D}\v \right)\approx {\bf I}{\bf D}{\bf G}\delta s$$