Elastic wavefield directionality vectors

Discussion and Conclusion

I have verified that the displacement decomposition operator (Zhang and McMechan, 2010) works for an isotropic elastic wavefield in a homogeneous medium. Currently, I have implemented it in the wavenumber domain, as stated in equations 12 and 14, and in the space domain, as stated in equation 24. Since the wave propagation itself is in the time-space domain, executing transforms to wavenumber domain at each time step may be more costly in the eventual 3D implementation than using the space domain decomposition by filter deconvolution. For the moment, it appears that both methods produce similar results as far as understanding of the polarization direction is concerned. This is despite the fact that the error introduced by the wavenumber domain displacement decomposition is six orders of magnitude smaller than the one introduced by space domain displacement decomposition. It appears that the space domain error is large but smooth, while the wavenumber domain error is small, but highly variable.

The size of the sliding time window within which the polarization coefficient is calculated is a critical decision to the derivation of polarization direction, and therefore to the calculation of the wave propagation direction. The ability to calculate the propagation direction during propagation is the cornerstone of the method I wish to eventually implement for constructing angle gathers. Therefore, a robust formulation of the time window size as a function of the wavefield parameters is required.

The cut-off value for the correlation between the displacements on the two axes, which I use as a determination of the level of local polarization, is another critical factor. Obviously, where the displacement values are very small or where they are not correlated there is no point in trying to estimate polarization. At those locations in the wavefield, either there is no significant energy anyway, or there is more than one wave at the same time. It should be noted that this same limitation exists for current angle gather construction methodology, but it is not addressed until the result is viewed. The use of statistical correlation (equation 25) does not seem to be the most robust way of measuring the linearity of polarization in the propagating wavefield. There are other methods, such as executing SVD on a matrix consisting of the displacement components of an image point over within a time window (Meersman et al., 2006), which may provide a more stable polarization estimation.

I calculate the slope of the line that best fits, in the least-square sense, the crossplotted displacements $U_x$ and $U_z$ . This slope is then used to determine the angle between the polarization direction and the vertical. According to the results (Figures 19 - 25), this method is imperfect. I would expect that along the line radiating from the source in Figure 19, the polarization angle would be constant. I have a nagging suspicion that the cause of this is the staggered grid methodology, which in effect means that the $U_x$ and $U_z$ displacements are not calculated at the same model location. I will need to find a way to test this possibility. If this is a critical issue, some correction to the displacements will be necessary so they'll be calculated at the same grid points. Zhang and McMechan (2010) mention that they interpolate the values from one staggered grid to another, to ensure that displacement information for different axes is located at the same grid points.

As for the use of this directionality information - I believe knowing the wavefield's direction at each time during propagation incorporates the advantages of wavefield propagation methods and ray-based methods. It may thus enable a method of angle gather creation in the style shown in (Koren et al., 2008), but without the restrictions imposed by ray theory. Furthermore, knowing the wavefield's direction will make it unnecessary to ``shoot'' rays in many directions from various CDP's in the model space, and then create the angle gather using only the data from those rays that reached the surface near shot-receiver points, where data exists. This can result in a significant saving of computational and I/O resources.

Using the displacement information inherent in elastic wavefield propagation can also make redundant the angle gather creation method by spatial and temporal lag cross-correlation (Sava and Fomel, 2003). The extended imaging condition there requires cross-correlation at various spatial lags between the receiver and source wavefields, but the reason for doing this is exactly because there is no information as to the wavefield's propagation direction. Therefore, a ``scan'' of a CIG's local neighbourhood is executed, to cover all directions where the source and receiver wave energy could have arrived from. Just as for the ray-based method, a possible reduction in computational resources can be achieved by using the pre-existing wavefield directionality information in the displacement fields.

Another use for wavefield directionality information is for the cross-correlation imaging condition itself. If we have the source and receiver propagation direction, we can ensure that we cross-correlate only those events traveling in opposite directions, such as the upgoing receiver wavefield with the downgoing source wavefield. By doing that, we remain within the assumptions of the cross-correlation imaging condition, and the resulting image should contain fewer artifacts. Many authors acknowledge the need for this separation, and useful methods exist to carry it out. What I am suggesting is that the same directionality information acquired by displacement decomposition, which I wish to use for angle gather construction, can also be used in conjunction with the imaging condition, in order to image the events which the imaging condition was designed for.

Elastic wavefield directionality vectors