Computational analysis of extended full waveform inversion |

where is the background component and is the perturbation component. The linearized FWI forward operator can be written as follows:

As shown in equation 15, the cost of the linearized forward operator is equal to the adjoint cost of conventional FWI. Also, the adjoint of the linearized operator with respect to the perturbation is the same as the conventional FWI adjoint and has the same cost as well. The adjoint with respect to the background is:

Although equation 16 has six Green's functions, only four propagations are required since each background wavefield is the same for two Green's functions. In addition, these background wavefields are the same for the adjoint of perturbation. Hence, the total cost of the linearized FWI per iteration, assuming complete reuse of background wavefields, is

This shows that scale separation by itself increases the cost of the original nonlinear operator since it adds several propagations, imaging and scattering steps. However, a significant cost cutting for extended inversions is possible by extending only the perturbation component without extending background component. By following the same derivation for linearized FWI, I find the cost of linearized extended FWI in subsurface offset when I extend perturbation only to be

By extending only the perturbation, we remove the subsurface offset multiplication factor . This results in a large reduction in cost because the number of propagation time steps is much larger than the number of imaging time steps . Similarly, the cost of linearized extended FWI in time can be written as

For the extended FWI in time, the cost reduction by linearization is less dramatic than the extended FWI in subsurface offset since there is only one additional convolution axis.

Computational analysis of extended full waveform inversion |

2012-10-29