A preconditioning scheme for full waveform inversion

Examples

We illustrate the geologically-constrained waveform inversion method on a synthetic dataset. We modified the Marmousi 2-D velocity model by adding a 250 m. thick water layer (Figure 2a). We created 184 shots 50 m. apart with a fixed receiver array (369 in total) at the surface using the scalar wave equation (no density). The source is a Ricker-2 wavelet with a fundamental frequency of 8Hz. To illustrate that our inversion works (without preconditioning), we show in Figure 2c the result of waveform inversion with four frequency scales (0-4Hz, 0-8Hz, 0-12Hz, and 0-16Hz) using the starting guess in Figure 2b (obtained by smoothing the true model in Figure 2a) and using all the shots. There is a very good match between the inverted and true model. Because all the data was used, little would be gained by using preconditioning.

To make a compelling case, we kept only four shots, 2.5 Km. apart. First, we show in Figure 3 a comparison between the gradient without preconditioning $\nabla f(\textbf {m}_n)$ and the gradient with preconditioning back in the velocity space $\mathbf{S}\nabla \tilde{f}(\textbf{p}_n)$ . Because only 4 shots are present, the unpreconditioned gradient looks noisy and resemble geology in very few locations only. Some authors suggest attenuating the high wavenumbers in the gradient by smoothing it after each iteration (Ravaut et al., 2004), where the size of the smoothing operator in the horizontal and vertical directions is a function of an average wavelength at a given frequency. This bears a resemblance with our proposed scheme but doesn't allow for directional smoothing. On the contrary, thanks to the preconditioning with directional Laplacian filters, the reprojected gradient in Figure 3b shows the geology captured in the dip field of Figure 1b very well.

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Figure 3. (a) Gradient $\nabla f(\textbf {m}_n)$ of the unpreconditioned inversion after 4 iterations of the 0-8Hz scale (4 shots, 2.5 Km. apart). (b) Reprojected gradient $\mathbf{S}\nabla \tilde{f}(\textbf{p}_n)$ of the preconditioned inversion after 4 iterations of the 0-8Hz scale (4 shots, 2.5 Km. apart). With preconditioning, the gradient follows the information captured in the dip field and looks more geologically appealing than in (a).

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Figure 4. (a) Inversion result for the unpreconditioned inversion. (b) Inversion result for the preconditioned inversion with directional Laplacian filters. The geology at the reservoir level is recovered very well in (b).

Now, we show in Figure 4 the inversion results for the unpreconditioned (Figure 4a) and preconditioned inversion (Figure 4b). Although quite noisy, the unpreconditioned result shows the geology very well: the presence of low frequencies in the data, along with the multi-scale approach, act as regularization operators. This effect will be less pronounced with real data where low frequencies are often missing. The preconditioned inversion result in Figure 4b is much cleaner: the geology at the reservoir level is recovered very well.

A preconditioning scheme for full waveform inversion