Tomographic full waveform inversion and linear modeling of multiple scattering

Tomographic Full Waveform Inversion (TFWI)

We can rewrite equations 5-6 by performing the following substitution

$${\bf {\delta P_o}}= {\bf P_o}\stackrel{{t'}}{\ast} \left({\bf T}-{\bf I}\right),$$ (12)

and consequently

$${\bf P_o}+{\bf {\delta P_o}}= {\bf P_o}\stackrel{{t'}}{\ast}{\bf T},$$ (13)

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

$$\left[ {\bf D_2}- {\bf {v}_o}^2\nabla^2 \right] {\bf {\delta P_o}... ... {v}}}^2\left(\widetilde{\bf T}\stackrel{{t'}}{\ast}{\nabla^2{\bf P_o}}\right),$$ (14)

where the substitution of ${\bf T}$ with $\widetilde{\bf T}$ takes into account of the Laplacian.

If we define an velocity model extended in time $\delta{{\bf {\tilde{{v}}}}}^2\left({t},{t'}\right) = \delta{{\bf {v}}}^2\stackrel{{t'}}{\ast}\widetilde{\bf T}$ , we can rewrite equation 14 as

$$\left[ {\bf D_2}- {\bf {v}_o}^2\nabla^2 \right] {\bf {\delta P_o}... ...\tilde{{v}}}}}^2\left({t},{t'}\right) \stackrel{{t'}}{\ast}{\nabla^2{\bf P_o}}.$$ (15)

The estimation of an extended velocity as a function of both ${t}$ and ${t'}$ , 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, ${\tau}={t}-{t'}$ and the same for each seismic experiment. The approximation of equation 15 can be written as

$$\left[ {\bf D_2}- {\bf {\tilde{{v}}}}^2\left({\tau}=0\right) \nab... ...\tilde{{v}}}}}^2\left({\tau}\right) \stackrel{{\tau}}{\ast}{\nabla^2{\bf P_o}},$$ (16)

where the change of notation from ${\bf {\delta P_o}}$ to ${\bf {\Delta P}}$ indicates that the scattered wavefield ${\bf {\Delta P}}$ is now an approximation of the true multiple-scattered wavefield ${\bf {\delta P_o}}$ .

Formally solving equation 16 we obtain

$${\bf {\Delta P}} = \left[ {\bf D_2}- {\bf {\tilde{{v}}}}^2\left({... ...{v}}}}}^2\left({\tau}\right) \stackrel{{\tau}}{\ast}{\nabla^2{\bf P_o}} \right]$$ (17)

that is a linear relationship between $\delta{{\bf {\tilde{{v}}}}}^2$ and ${\bf {\Delta P}}$ defined by the linear operator $\widehat{\bf L}$ such as ${\bf {\Delta P}}=\widehat{\bf L}\delta{{\bf {\tilde{{v}}}}}^2$ .

If we define the total wavefield to be

$${\bf P}={\bf P_o}+{\bf {\Delta P}},$$ (18)

and the extended non-linear modeling operator as

$$\tilde{\cal L}\left({\bf {\tilde{{v}}}}\right) = {\cal L}\left({ ... ... \widehat{\bf L}\left({\bf {\tilde{{v}}}}\right) \delta{{\bf {\tilde{{v}}}}}^2,$$ (19)

the objective function

$${J_{\rm EFWI}}\left({\bf {\tilde{{v}}}}\right) = \frac{1}{2} \left\Vert \tilde{\cal L}\left({\bf {\tilde{{v}}}}\right) - {\bf d} \right\Vert^2_2$$ (20)

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,

$$\min_{{\bf {\tilde{{v}}}}} {J_{\rm TFWI}}\left({\bf {\tilde{{v}}}}\right) ,$$ (21)

with

$${J_{\rm TFWI}}\left({\bf {\tilde{{v}}}}\right) = \frac{1}{2} \lef... ...- {\bf d} \right\Vert^2_2 - \epsilon {\cal F}\left({\bf {\tilde{{v}}}}\right) ,$$ (22)

where ${\cal F}$ is an operator that measure the focusing of the model at zero time lag. A straightforward example of such operator is

$${\cal F}\left({\bf {\tilde{{v}}}}\right) = -\left\Vert \left\vert{\tau}\right\vert {\bf {\tilde{{v}}}} \right\Vert^2_2.$$ (23)

Tomographic full waveform inversion and linear modeling of multiple scattering