Elastic wavefield directionality vectors |
coil
Figure 1. Sketch of the helix concept - convolution takes place by winding a ``coil'' of filter coefficients over a ``coil'' of data values (Claerbout, 1997). |
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Using spectral factorization, a series of coefficients can be transformed to an alternate set of causal filter coefficients which have a causal inverse. The Wilson-Burg spectral factorization method (Fomel et al., 2003) ensures that the filter is minimum-phase. The autocorrelation of this new set of filter coefficients recreates the original values of the input series. The upshot of this is that application of the original series' coefficients to a dataset is akin to convolving the data with the spectrally factorized filter coefficients in one direction, and then convolving again in the other direction (``coiling'' and then ``uncoiling'' the filter coefficients over the data). This effectively applies the filter and its time reverse (adjoint) to the data, which amounts to multiplying the data by the original input series' coefficients. In the case of finite differencing, the ``input'' series might be the Laplacian, which when made to traverse over the data has the effect of a
derivative approximation.
Application of a derivative finite-difference operator to a dataset is done by:
(19) |
where is the spectrally factorized filter coefficients and is the time-reversed filter. However, since equations 16 - 18 denote division of the displacement fields by a derivative operator, a deconvolution with the filter coefficients is required:
(20) |
The method of performing Zhang and McMechan (2010)'s displacement decomposition in the space domain is to first decide on the order of the derivative finite difference operator, and then use spectral factorization to produce the filter coefficients of this operator. Then, deconvolution with these filter coefficients must be applied to each displacement field, and the derivatives shown in equations 16 - 18 must then be performed on these deconvolved displacement fields:
I use the SEPlib polydiv module to perform the helical deconvolution of the spectrally factorized filter coefficients with the displacement wavefields, and then apply the spatial derivative operators, saving the decomposed results in separate arrays.
Elastic wavefield directionality vectors |