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P-wave and S-wave decomposition in the space domain

The numerators in equations 16 - 18 can be calculated in the space domain by a finite-difference approximation to the $ 2^{nd}$ spatial derivative operator $ \frac{\partial^2}{\partial x^2_i}$ . This finite-difference operator can be convolved over the displacement field $ \bold U$ in the space domain. However, the denominator in equations 16 - 18, a division in the wavenumber domain, can only be approximated in the space domain by a deconvolution. I chose to perform this deconvolution using spectral factorization and helical deconvolution, following the implementation in Claerbout (1997). This method has the advantage of treating multidimensional problems as one dimensional problems. Specifically, it enables execution of multidimensional deconvolutions as 1-D deconvolutions.

coil
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).
[pdf] [png]

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 $ 2^{nd}$ derivative approximation.

Application of a $ 2^{nd}$ derivative finite-difference operator to a dataset $ \bold U$ is done by:

$\displaystyle \frac{\partial^2}{\partial x^2_{i}} U_i = h^{'} * h * \bold U,$ (19)

where $ h$ is the spectrally factorized filter coefficients and $ h^{'}$ is the time-reversed filter. However, since equations 16 - 18 denote division of the displacement fields by a $ 2^{nd}$ derivative operator, a deconvolution with the filter coefficients is required:

$\displaystyle \frac{1}{ \frac{\partial^2}{\partial x^2_{i}} } U_i = \left( h^{'} * h \right )^{-1} \bold U = h^{-1} * (h^{'})^{-1} * \bold U.$ (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 $ 2^{nd}$ derivative finite difference operator, and then use spectral factorization to produce the filter coefficients $ h$ 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:


$\displaystyle U^P_x$ $\displaystyle =$ $\displaystyle \frac{\partial^2}{\partial x^2} \left( h^{-1} * (h^{'})^{-1} * U_...
...{\partial^2}{\partial x \partial z} \left( h^{-1} * (h^{'})^{-1} * U_z \right),$ (21)
$\displaystyle U^P_z$ $\displaystyle =$ $\displaystyle \frac{\partial^2}{\partial z^2} \left( h^{-1} * (h^{'})^{-1} * U_...
...{\partial^2}{\partial z \partial x} \left( h^{-1} * (h^{'})^{-1} * U_x \right),$ (22)
$\displaystyle U^S_x$ $\displaystyle =$ $\displaystyle \frac{\partial^2}{\partial z^2} \left( h^{-1} * (h^{'})^{-1} * U_...
...{\partial^2}{\partial x \partial z} \left( h^{-1} * (h^{'})^{-1} * U_z \right),$ (23)
$\displaystyle U^S_z$ $\displaystyle =$ $\displaystyle \frac{\partial^2}{\partial x^2} \left( h^{-1} * (h^{'})^{-1} * U_...
...{\partial^2}{\partial z \partial x} \left( h^{-1} * (h^{'})^{-1} * U_x \right).$ (24)

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 $ \left( U^P_x, U^P_z, U^S_x, U^S_z \right)$ in separate arrays.


next up previous [pdf]

Next: Determination of polarity and Up: Theoretical background Previous: P-wave and S-wave amplitude

2011-05-24