A preconditioning scheme for full waveform inversion

Defining the preconditioning operator S

Preconditioning amounts to a change of the gradient direction. For waveform inversion, a gradient that embeds some geological information could result in more meaningful velocity models. To this end, we follow the approach of Hale (2007) for the construction of the operator $\mathbf{S}$ . Doing so, this operator becomes a non-stationary deconvolution with directional Laplacian filters.

Directional Laplacian filters are built from small wavekill filters $\mathbf{A}$ , similar to those of Claerbout (1995). Wavekill filters have the ability to anihilate local planar-events with a given dip. From these filters, new operators $\mathbf{A'A}$ are formed by autocorrelation. These new operators are then factorized into minimum-phase filters $\mathbf{\tilde{A}}$ such that $\mathbf{\tilde{A}'\tilde{A}} \approx \mathbf{A'A}$ . Having minimum-phase filters, we can build a stable non-stationary deconvolution operator $\mathbf{S}= \mathbf{\tilde{A}^{-1}\tilde{A}'^{-1}}$ and its inverse $\mathbf{S}^{-1}=\mathbf{\tilde{A}'\tilde{A}}$ . Because the wavekill filter $\mathbf{A}$ is dip dependent, the operator $\mathbf{S}$ has the ability to smooth along a given direction as well. Therefore, if we can estimate a dip field that contains some desired geological features, the directional Laplacian filters can make the solution of an inverse problem resemble such features.

In practice, we estimate and use the directional Laplacian filters as follows: first, we estimate a dip field following the approach of Fomel (2002); then we estimate a bank of directional Laplacian filters for a range of angles; finally we apply the appropriate inverse Laplacian filter on each sample according to the local dip. One added feature of our preconditioning scheme is that the strength of the smoothing can be controlled spatially: different areas with similar dips can be smoothed across different distances. These areas are identified by a weighting function which varies from high values (i.e., little smoothing) to low values (i.e., strong smoothing).

To illustrate the preconditioning operator $\mathbf{S}$ , we show in Figure 1a the migration result of a synthetic dataset based on the 2-D Marmousi model. This result is obtained with Reverse Time Migration (RTM). In real data cases, the dip field could be re-estimated iteratively from a migrated image estimated with the updated velocity field, adding a third outside loop to our waveform inversion algorithm (one for frequency band and one for the muting/masking operator). This possibility is not investigated in this paper.

From the RTM image, we can estimate the local dip field (Figure 1b). This dip field is obtained iteratively with some smoothing using the method of Fomel (2002). We also picked by hand the location of three faults. From these picks, we estimated the dip on the fault and smoothed the local dip horizontally. These three faults were picked to get sharper velocity contrasts. Now, we apply the operator $\mathbf{S}$ to a random field the size of the migrated image in Figure 1a to obtain Figure 1c. We notice that the "texture" of the original migrated image is recovered and that no smoothing is applied in the water layer: for this example, our weigthing function had only two values separated by the water bottom. Finally, we can clearly identify the fault locations. In the next section, we demonstrate that this operator can be used to constrain the solution of waveform inversion.

A preconditioning scheme for full waveform inversion