Tomographic full waveform inversion and linear modeling of multiple scattering |

(12) |

and consequently

(13) |

where is a convolutional operator in time that may vary both in space and time; is the identity operator. For example, when the perturbed wavefield is a time-shifted version of the background wavefield, the operator is a shifted delta function. With this substitution equation 6 can be rewritten as

where the substitution of with takes into account of the Laplacian.

If we define an velocity model *extended* in time
,
we can rewrite equation 14
as

The estimation of an extended velocity as a function of both and , and for each seismic experiment (e.g. shot), can be unpractical. We can approximate equation 15 by making the velocity dependent only from the convolutional time lag; that is, and the same for each seismic experiment. The approximation of equation 15 can be written as

where the change of notation from to indicates that the scattered wavefield is now an approximation of the true multiple-scattered wavefield .

Formally solving equation 16 we obtain

that is a linear relationship between and defined by the linear operator such as .

If we define the total wavefield to be

(18) |

and the extended non-linear modeling operator as

(19) |

the objective function

has the same local minima of the original FWI objective function, but it also provides smooth descending paths to the global minimum in the additional dimensions. The problem is now under constrained because many solutions fit the data equally well. Among all these possible solutions we are interested in the solutions for which the extended velocity model is as focused as possible around the zero time lag of the model.

To converge towards a desirable solution we can add an additional term to the objective function that penalizes extended velocity model with significant energy at non-zero time lag; that is,

(21) |

with

where is an operator that measure the focusing of the model at zero time lag. A straightforward example of such operator is

Tomographic full waveform inversion and linear modeling of multiple scattering |

2012-10-29