Computational analysis of extended full waveform inversion |

and the extended Green's function satisfies:

where is subsurface offset and denotes a deconvolution operator over subsurface offset. This extended wave equation convolves each time slice by all subsurface offsets of velocity. The cost of extended forward modeling becomes:

where and are number of subsurface offsets along the and axes, respectively. By linearizing equation 7 over the velocity squared, we can compute the adjoint as:

Therefore, the cost of computing the adjoint of EFWI can be written as

and the total cost of one iteration of EFWI becomes

We can see that the computational cost becomes extremely high when we include the subsurface offsets in the velocity model. One way to reduce the cost is presented in Biondi (2012) where the velocity model is extended over time instead of horizontal offset. In that case, the cost becomes a function of one time-lag parameter instead of two horizontal lags in 3D:

where is the number of time lags. The computational disadvantage is that several time slices need to be held in memory for each time instead of the conventional two slices.

Computational analysis of extended full waveform inversion |

2012-10-29