Tomographic full waveform inversion and linear modeling of multiple scattering |

where and are the background wavefield and velocity, respectively, and is the scattered wavefield.

Equation 4 can be rewritten as the following
two equations:

which represents a nonlinear relationship between velocity perturbations and scattered wavefield. To linearize this relationship we drop the term multiplying the perturbations with each other; that is, we drop the scattered wavefield from the right hand side of equation 6 and obtain the following coupled equations:

The linear operator used to compute the gradient of the FWI objective function 2 is evaluated by recursively propagating the background wavefield and the scattered wavefield by solving equations 7-8.

The scattered wavefield is a linear function of the velocity perturbations because equation 8 takes into account only fist order scattering. Notice that the linear operator is itself a non linear function of the background velocity, both directly by determining the propagation speed of the scattered wavefield (left hand side in equation 8), and indirectly through the background wavefield (right hand side in equation 8).

Tomographic full waveform inversion and linear modeling of multiple scattering |

2012-10-29