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

Gradient computation

To compute the gradient of the objective function expressed in equation 5, we need to linearize the extended wave equation 4. Usually equation 3 is linearized over slowness or velocity. However, the extended wave equation 4 includes a deconvolution over the offset axis that is not easy to implement. Hence, we will first rearrange equation 4 to a form that facilitates the computation of the gradient, and that is actually solved numerically in the propagation. This can be achieved by convolving both sides of the equation by the square of velocity then rearranging the terms as follows:

$$\omega^2 G(\mathbf x_s, \mathbf x, \omega, \mathbf v) = v^2(\math... ...s - \mathbf x) - \nabla^2 G(\mathbf x_s, \mathbf x, \omega, \mathbf v) \right).$$

We now can linearize the relationship between the Green's function and the model by perturbing the model around a background value as follows:

$$v^2(\mathbf x, \mathbf h) = v_0^2(\mathbf x, \mathbf h) + \Delta v^2(\mathbf x, \mathbf h),$$

where $v_0(\mathbf x, \mathbf h)$ is the background component and $\Delta v(\mathbf x, \mathbf h)$ is the perturbation component, i.e. the model update. After this separation, the first-order Born approximation can be used to define the gradient as follows:

$$g_d(\mathbf x, \mathbf h) =$$

$$\sum_{\mathbf x_s, \mathbf x_r, \omega} \left[ \nabla^2 f(\mathbf... ...athbf x_s, \mathbf x -\mathbf h, \omega; \mathbf v_0(\mathbf h) \right) \right.$$

$$\left. G \left( \mathbf x_r, \mathbf x +\mathbf h, \omega; \mathb... ... \ \Delta d^*(\mathbf x_s, \mathbf x_r, \omega; \mathbf v_0(\mathbf h))\right],$$

where $\Delta d$ are the data residuals and $^*$ indicates the complex conjugate. Unlike the usual linearization of the wave equation, the scattering term includes a Laplacian operator instead of the second derivative in time. The Laplacian operator does not add to the computational cost since it is already computed in the propagation of the background wavefield.

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