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Review of PWD

The local plane wave can be represented by the following differential equation (Claerbout, 1992):
\begin{displaymath}
\frac{\partial u}{\partial x}+\sigma\frac{\partial u}{\partial t} =0\;,
\end{displaymath} (1)

where $\sigma$ is the local slope in continuous space, with dimension time/length. The wavefields observed at the two positions $x_1,x_2$ have a time delay which is proportional to their distance, $\sigma\vert x_1-x_2\vert$. In the sampled system with space and time intervals $\Delta x$ and $\Delta t$, we define the discrete space slope in the unit of $\Delta t/\Delta x$, as $p=\sigma\Delta x/\Delta t$. As $p$ is independent of the sampling interval, it can be directly used in irregular dataset (in this case, the unit of the slopes becomes space variant). The time delay between two adjacent positions is then the slope $p\Delta t$:
\begin{displaymath}
u(x,t)=u(x+\Delta x, t+p\Delta t)\;.
\end{displaymath} (2)

With the $Z$ transform applied along both time and space directions, the above equation becomes

\begin{displaymath}
(1-Z_xZ_t^p)U(Z_x,Z_t)=0\;,
\end{displaymath} (3)

where $Z_t$ is the unit time-shift operator, $Z_x$ denotes the unit space-shift operator and $U(Z_x,Z_t)$ is the $Z$ transform of $u(x,t)$. The operator $1-Z_xZ_t^p$ is the plane-wave destructor. Using Thiran's fractional delay filter $H(Z_t)=\displaystyle{\frac{B(1/Z_t)}{B(Z_t)}}$ (Thiran, 1971) to approximate the time-shift operator $Z_t^p=e^{j\omega p}$, where $\omega$ is the circular frequency, the plane-wave destructor can be expressed as (Fomel, 2002),
\begin{displaymath}
C(p)=B(Z_t)-Z_xB(\frac{1}{Z_t}),
\end{displaymath} (4)

where
\begin{displaymath}
B(Z_t)=\sum_{k=-N}^N b_k(p) Z_t^{-k},
\end{displaymath} (5)

$N$ is the order of the noncausal temporal filter and $b_k(p)$ are functions of the local slope $p$.

Equation 4 is a 2D filter. Applying the filter at an arbitrary point in the wavefield, the plane-wave destruction equation 3 becomes a nonlinear equation for the local slope $p$:

\begin{displaymath}
C(p, Z_x, Z_t)U(Z_x,Z_t) \approx 0\;.
\end{displaymath} (6)

An iterative method, such as Newton's method, can be applied to find the slope. In practice, wavefields are polluted by noise and the plane wave assumption may not hold true where faults and conflicting boundaries exist. To obtain a stable slope estimation, an additional smoothing regularization process (Fomel, 2007a) is needed at each step. The total computational cost of slope estimation by plane-wave destruction becomes $O(N_dN_fN_lN_n)$, where $N_d$ is the size of the data, $N_f=2N+1$ is the size of the filter, $N_l$ is the number of linear iterations for regularization, and $N_n$ is the number of nonlinear iterations for solving equation 6. Typical values are $N_f=3,5$, $N_l=10$-50, and $N_n=5$-10.


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Next: Accelerated PWD Up: Theory Previous: Theory

2013-03-02