Tomographic full waveform inversion: Practical and computationally feasible approach

Scale Mixing

A direct use of the gradients is to update their corresponding models directly. However, this would hinders the simultaneous inversion of different wavelengths of the model. Hence, we first mix the two gradients and then separate them in Fourier domain to get the update of each model as follows:

$$s_{\mathbf b}(\mathbf x) = \mathbf C_{\mathbf b}(g_{\mathbf b}(\mathbf x)+g_{\mathbf r}(\mathbf x, \mathbf h=0)),$$ (16)

where $s_{\mathbf b}(\mathbf x)$ is the search direction of the background model and $\mathbf C_{\mathbf b}$ is a low-pass filter. Similarly, we can compute the update of the perturbation model as:

$$s_{\mathbf r}(\mathbf x, \mathbf h) = \mathbf C_{\mathbf r}(g_{\mathbf b}(\mathbf x)+g_{\mathbf r}(\mathbf x, \mathbf h)),$$ (17)

where $s_{\mathbf r}(\mathbf x, \mathbf h)$ is the search direction of the perturbation model and $\mathbf C_{\mathbf r}$ is a high-pass filter. In order to sum the two gradients properly, both of them need to have the same units as well as the same scale. This requires careful implementation of each operator at each linearization.

For the examples shown in this paper, we used a radial cut-off in the Fourier domain with a cosine squared taper. The wavelength cut-off is based on the dominant frequency in the data as well as the average velocity of the initial model. Also, the two filters were designed such that they always sum to one at all wavelength to maintain the energy of the gradients.

Tomographic full waveform inversion: Practical and computationally feasible approach