Tomographic full waveform inversion and linear modeling of multiple scattering

Conventional Full Waveform Inversion (FWI)

Conventional full waveform inversion is performed by solving the following optimization problem

$$\min_{{\bf {v}}^2} {J_{\rm FWI}}\left({\bf {v}}^2\right)$$ (1)

where:

$${J_{\rm FWI}}\left({\bf {v}}^2\right) = \frac{1}{2} \left\Vert {\cal L}\left({\bf {v}}^2\right) - {\bf d} \right\Vert^2_2,$$ (2)

${\bf {v}}= {v}\left(\vec x\right)$ is the velocity vector, ${\cal L}$ is a wave-equation operator non linear with respect to velocity perturbations and the data vector ${\bf d}$ is the pressure field ${\bf P}= {P}\left({t},\vec x\right)$ measured at the surface.

The wave-equation operator is evaluated by recursively solving the following finite difference equation

$$\left[ {\bf D_2}- {\bf {v}}^2\nabla^2 \right] {\bf P} ={\bf f},$$ (3)

where ${\bf D_2}$ is a finite-difference representation of the second derivative in time, $\nabla^2$ is a finite-difference representation of the Laplacian, and ${\bf f}$ is the source function.

Tomographic full waveform inversion and linear modeling of multiple scattering