Tomographic full waveform inversion and linear modeling of multiple scattering

Gradient computation with FWI

The efficient solution of the optimization problem expressed in equation 1 is performed by gradient based methods, and thus requires the evaluation of the linear operator ${\bf L}$ , which is the linearization of ${\cal L}$ with respect to velocity perturbations $\delta{{\bf {v}}}^2$ . This linear operator can be derived by perturbing equation 3 as follows

$$\left[ {\bf D_2}- \left({\bf {v}_o}^2 + \delta{{\bf {v}}}^2\right)\nabla^2 \right] \left({\bf P_o}+{\bf {\delta P_o}}\right) ={\bf f},$$ (4)

where ${\bf P_o}$ and ${\bf {v}_o}$ are the background wavefield and velocity, respectively, and ${\bf {\delta P_o}}$ is the scattered wavefield.

Equation 4 can be rewritten as the following two equations:

$$\left[ {\bf D_2}- {\bf {v}_o}^2\nabla^2 \right] {\bf P_o} = {\bf f},$$ (5)

$$\left[ {\bf D_2}- {\bf {v}_o}^2\nabla^2 \right] {\bf {\delta P_o}} = \delta{{\bf {v}}}^2{\nabla^2\left({\bf P_o}+{\bf {\delta P_o}}\right)},$$ (6)

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:

$$\left[ {\bf D_2}- {\bf {v}_o}^2\nabla^2 \right] {\bf P_o} = {\bf f},$$ (7)

$$\left[ {\bf D_2}- {\bf {v}_o}^2\nabla^2 \right] {\bf {\delta P_o}} = \delta{{\bf {v}}}^2{\nabla^2{\bf P_o}}.$$ (8)

The linear operator ${\bf L}$ used to compute the gradient of the FWI objective function 2 is evaluated by recursively propagating the background wavefield ${\bf P_o}$ and the scattered wavefield ${\bf {\delta P_o}}$ by solving equations 7-8.

The scattered wavefield ${\bf {\delta P_o}}$ is a linear function of the velocity perturbations $\delta{{\bf {v}}}^2$ because equation 8 takes into account only fist order scattering. Notice that the linear operator ${\bf L}\left({\bf {v}_o}^2\right)$ 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