Wave-equation tomography by beam focusing

Theory

In this section I develop the general theory for a transmission tomography problem because it is simpler than reflection tomography. In transmission tomography, the data are recorded after only one propagation path through the medium, as opposed to the downgoing and upgoing paths of a typical reflection tomography problems. Furthermore, in transmission tomography there is no need to image and locate reflectors in depth, which is a major hurdle in reflection tomography. However, the application to reflection tomography of the theory presented in this paper should be fairly straightforward. I propose to solve the transmission tomography problem by maximizing an objective function based on the correlation between recorded data and modeled data. This correlation is analogous to the correlation between source and receiver wavefields required by migration imaging condition.

To further simplify the theoretical development, I define an objective function that rewards consistency of the correlation computed independently for each source location. The objective function measures correlation consistency along the receiver axis. However, I use here the receiver axis as a proxy for the offset axis or the aperture-angle axis in reflection tomography. The application of the concepts developed in this paper to objective functions useful in reflection tomography should be straightforward, although it will require more complex notation and result in expressions for the gradients even more complex than the ones presented here.

I define the recorded data as ${{P}_{D} }\left({t},x_{g},x_{s}\right)$ , and the modeled data as ${\widetilde{{P}} }\left({t},x_{g},x_{s};{s}\right)$ , where ${t}$ is the recording time, $x_{g}$ is the receiver coordinate, $x_{s}$ is the source coordinate, and ${s}\left(z,x\right)$ is the slowness model defined in depth ${z}$ and along the horizontal coordinate ${x}$ .

The cross-correlation $C\left({\tau}\right)$ between the recorded data and the modeled data is defined as a function of the correlation time lag ${\tau}$ as

$$C\left({\tau}\right)\left[ {\widetilde{{P}} }\left({t}\right) , {... ...\sum_{t} {\widetilde{{P}} }\left({t}-{\tau}\right) {{P}_{D} }\left({t}\right) .$$ (A-1)

I introduce an objective function that maximizes the flatness of the correlation function along the receiver axis for all values of the lag ${\tau}$ . In particular, I aim to maximize local correlation flatness after subdividing the receiver array into local subarrays. To extract the correlation for each subarray centered at ${\overline{x_{g}}}$ , I apply a local beam-decomposition operators ${\bf B}_{\overline{x}}$ . Within in each subarray, traces are defined by the local offset $\Delta {x_{g}}$ . The dimensions of each ${\bf B}_{\overline{x}}$ are thus $\left(N_{\Delta {x_{g}}}N_{{\tau}}\times N_{x_{g}}N_{{\tau}}\right)$ .

Given a background slowness ${s}_{0}$ we can compute the correlation in equation 1. In each subarray, the correlation can be flattened by the application of $N_{{\overline{x_{g}}}}$ moveout operators $\mathcal M_{\overline{x}}$ ; that is

$$C\left({\tau}+ {\theta}\left({\boldsymbol \mu}_{\overline{x}}\rig... ...rline{x}}\right) ,{\bf B}_{\overline{x}} C\left({\tau};{s}_{0}\right) \right] ,$$ (A-2)

where ${\boldsymbol \mu}_{\overline{x}}$ are the moveout parameters and ${\theta}$ are the corresponding time shifts. I further define the local stacking operator ${\bf S}_{\overline{x}}$ that sums the correlation traces along the local offset axis $\Delta {x_{g}}$ .

I can now introduce the first, and local, term of the objective function that measures the flatness of the correlation within each subarray as:

$${J_{\rm Local}}\left( {\boldsymbol \mu}_{\overline{x}}\left({s}\r... ...) ,{\bf B}_{\overline{x}} C\left({\tau};{s}_{0}\right) \right] \right\Vert^2_2.$$ (A-3)

This objective function is not a direct function of the slowness ${s}$ , but it depends indirectly from it through the moveout parameters ${\boldsymbol \mu}_{\overline{x}}$ . These parameters are the solutions of $N_{x_{s}}\times N_{{\overline{x_{g}}}}$ independent fitting problems, one for each subarray and source location. These auxiliary objective functions measure the zero lag of the cross-correlation between the correlation computed for a realization of the slowness function ${s}$ and and the moved-out correlation computed with the background slowness ${s}_{0}$ ,

$${J_{\rm FL}}\left({\boldsymbol \mu}_{\overline{x}}\right) = C\left({\tau}=0\right)\left[ \mathcal M_{\overline{x}}\left[ {\th... ...{s}_{0}\right) \right] ,{\bf B}_{\overline{x}} C\left({\tau};{s}\right) \right]$$

$$= \langle\mathcal M_{\overline{x}}\left[ {\theta}\left({\boldsymbol... ...{0}\right) \right], {{\bf B}_{\overline{x}} C\left({\tau};{s}\right) } \rangle,$$ (A-4)

where with the notation $\langle {\bf x}, {\bf y}\rangle$ I indicate the inner product of the vectors ${\bf x}$ and ${\bf y}$ . This inner product spans the time-lag axis ${\tau}$ and the local offset axis $\Delta {x_{g}}$ . The local moveout parameters are the solutions of the following maximization problem:

$$\max_{{\boldsymbol \mu}_{\overline{x}}} {J_{\rm FL}}\left({\boldsymbol \mu}_{\overline{x}}\right) .$$ (A-5)

For velocity estimation, the most effective parametrization of the moveout within each beam is the curvature ${\mu _C}$ , that defines the following moveout equation

$${\theta}\left({\boldsymbol \mu}_{\overline{x}}\right) ={\mu_C}\Delta {x_{g}}^2.$$ (A-6)

Notice that when the slowness is equal to the background slowness ${s}_{0}$ , the corresponding best-fitting moveout parameters ${\bar{{\mu}}}_{\overline{x}}$ are obviously the ones corresponding to no moveout; that is, ${ {\theta}\left(\bar{{\boldsymbol \mu}}_{\overline{x}}\right) =0}$ .

As the numerical examples I show in the next section demonstrate, the beam curvature is effective to capture the long-wavelength perturbations in the velocity model, but is less effective to capture the short-wavelength perturbations. Accordingly, a wave-equation tomography based solely on the objective function 3 may have difficulties to estimate short-wavelength velocity perturbations.

To address this shortcoming I introduce a second, and global, term to the objective function. This term measures flatness across the subarrays, after the local moveouts have been applied, and is defined as,

$${J_{\rm Global}}\left( {\boldsymbol \mu}\left({s}\right) \right) ... ...}_{\overline{x}} C\left({\tau};{s}_{0}\right) \right] \right\} \right\Vert^2_2,$$ (A-7)

where ${\boldsymbol \Sigma}_{\overline{x}}$ assembles all the results of the stacking over the subarrays into a global array, ${\bf S}$ is a global stacking operator, and $\mathcal M$ is a global moveout operator function of the vector of parameter ${\boldsymbol \mu}$ . They both operate on the result of the local stacking of the subarrays. As in the previous case, the moveout parameters are solutions of $N_{x_{s}}$ independent fitting problems, one for each source location. Similarly, these auxiliary objective functions measure the zero lag of the cross-correlation between the local stack of the correlation computed using the current slowness function and the local stack of the moved-out correlation computed using the background slowness; that is,

$${{J_{\rm FG}}\left({\boldsymbol \mu}\right)= }$$

$$\langle { \mathcal M\left\{ {\theta}\left({\boldsymbol \mu}\right... ...{x}}\right) ,{\bf B}_{\overline{x}} C\left({\tau};{s}\right) \right] } \rangle.$$ (A-8)

In this case the inner product spans only the time-lag axis ${\tau}$ .

The global moveout parameters are the solutions of the following $N_{x_{s}}$ maximization problems

$$\max_{{\boldsymbol \mu}} {J_{\rm FG}}\left({\boldsymbol \mu}\right) .$$ (A-9)

I chose to parametrize the global moveout as simple time shifts for each beam center ${\overline {x}}$ that is, the moveout equation is

$${\theta}\left({\boldsymbol \mu}\right) ={\mu_{\theta}}.$$ (A-10)

Notice that with this choice of moveout parameters each maximization problem in 9 is an ensemble of $N_{{\overline{x_{g}}}}$ independent problems. This consideration becomes important when computing the gradient of the objective function.

Combining the objective function in 3 and in 7 we define the maximization problem that we solve to estimate slowness:

$$\max_{{s}} \left[ {J_{\rm Local}}\left( {\boldsymbol \mu}_{\overl... ...silon {J_{\rm Global}}\left( {\boldsymbol \mu}\left({s}\right) \right) \right],$$ (A-11)

where the parameter $\epsilon$ can be tuned to find an optimal relative scaling between the local and global components, although in principle $\epsilon=1$ should be effective.

Wave-equation tomography by beam focusing