Tomographic full waveform inversion (TFWI) by combining full waveform inversion with wave-equation migration velocity analysis

Tomographic full waveform inversion (TFWI)

The conventional FWI objective function $J_{\rm FWI}$ can be written as:

$$J_{\rm FWI}(\mathbf v) = \lVert \mathbf d (\mathbf v) - \mathbf d_{\rm obs} \rVert^2_2,$$ (1)

where $\mathbf v$ is the velocity model, $\mathbf d (\mathbf v)$ is the modeled data, and $\mathbf d_{\rm obs}$ is the observed data.

The modeled data is computed as:

$$d(\mathbf x_s, \mathbf x_r, \omega; \mathbf v) = f(\mathbf x_s, \... ...) G(\mathbf x_s, \mathbf x, \omega; \mathbf v) \delta(\mathbf x_r - \mathbf x),$$ (2)

where $f(\mathbf x_s, \omega)$ is the source function, $\omega$ is frequency, $\mathbf x_s$ and $\mathbf x_r$ are the source and receiver coordinates, and $\mathbf x$ is the model coordinate. In the acoustic, constant-density case the Green's function $G(\mathbf x_s, \mathbf x, \omega; \mathbf v)$ satisfies:

$$\left( \frac{\omega^2}{v^2(\mathbf x)} + \nabla^2 \right) G(\mathbf x_s, \mathbf x, \omega; \mathbf v) = \delta(\mathbf x_s - \mathbf x).$$ (3)

For the sake of compact notation, in the rest of the paper we present the expressions for computing the data and the gradient of objective functions in the frequency domain. However, we perform the computation in the time domain.

We can extend the velocity in the subsurface-offset dimension $\mathbf h$ which changes the wave equation into the following form

$$\left( v^2(\mathbf x, \mathbf h) *^{-1} \omega^2 + \nabla^2 \right) G(\mathbf x_s, \mathbf x, \omega; \mathbf v) = \delta(\mathbf x_s - \mathbf x),$$ (4)

where with $*^{-1}$ we indicate deconvolution. Notice that the division by velocity in equation 4 becomes a deconvolution over the offset axis. Once we define the Green's function, the data could be computed similarly to equation 2. We now write the new objective function as follows:

$$J_{\rm EFWI}\left(\mathbf v(\mathbf h)\right) = \left\lVert \mathbf d \left( \mathbf v(\mathbf h) \right) - \mathbf d_{\rm obs} \right\rVert^2_2.$$ (5)

The long-wavelength components of the solution of the optimization problem defined by equation 5 are not likely to be substantially different from the long-wavelength components of the initial model. The extension of the model, and in particular of its reflectivity component, to non-zero subsurface offset causes the kinematics of the modeled data to match the kinematics of the recorded data independently from the accuracy of the long-wavelength components.

Another term must be added to the objective function to drive the solution towards a model that focuses the image. Symes (2008) suggests the addition of a differential semblance penalty function (DSO); that is,

$$J_{\rm DSO}(\mathbf v(\mathbf h)) = \lVert \vert \mathbf h \vert \mathbf v(\mathbf h) \rVert^2_2.$$ (6)

We use this focusing term in the numerical experiments described in this paper. Another valid choice would be the maximization of the normalized power of the stack over reflection angles as a function of a residual moveout parameter $\rho$ , as suggested by Zhang and Biondi (2012).

The important characteristic of the second term is that its gradient ``imposes'' on the current model only a phase shift, and not a bulk vertical shift. This assures that the corresponding perturbations on the modeled data are mere phase shifts, and not bulk time shifts. The absence of bulk time shifts in the modeled data avoids large discrepancies between the kinematics of modeled data and recorded data. These large discrepancies are at the root of the convergence problems in conventional FWI.

Another practically important consideration is that in the proposed formulation the computation of the gradient of a term like the one presented in Zhang and Biondi (2012) is straightforward because it does not require back-projection of image perturbations. This is in contrast with WEMVA-like methods, where the computation of the gradient must take into account the constraint that the image is the result of migrating the recorded data. Therefore, at least in principle, it would be equally easy to add to the objective function other terms that reward focusing of the model along the midpoint spatial axis, in addition to the subsurface offset or reflection angle (Biondi, 2010).

Tomographic full waveform inversion (TFWI) by combining full waveform inversion with wave-equation migration velocity analysis