Factorizing plane waves

Let us denote the coordinates of a three-dimensional space by t, x, and y. A theoretical plane wave is described by the equation  

$$P(t,x,y) = f (t - \sigma_x x - \sigma_y y)\;,$$ (116)

where f is an arbitrary function, and $\sigma_x$ and $\sigma_y$ are the plane slopes in the corresponding direction. It is easy to verify that a plane wave of the form () satisfies the following system of partial differential equations:  

$$\left\{\begin{array} {rcl}\displaystyle \left(\frac{\partial... ...\frac{\partial}{\partial t}\right)\,P & = & 0\end{array}\right.$$ (117)

The first equation in () describes plane waves on the $\{t,x\}$ slices and is completely equivalent to equation (). In its discrete form, it is represented as a convolution with the two-dimensional finite-difference filter $\bold{C}_{x}$ from equation (). Similarly, the second equation transforms into a convolution with filter $\bold{C}_y$, which acts on the $\{t,y\}$ slices. The discrete (finite-difference) form of equations () involves a blocked convolution operator:  

$$\left[\begin{array} {c}\displaystyle \bold{C}_x \\ \bold{C}_y\end{array}\right]\,\bold{m} = \bold{0}\;,$$ (118)

where $\bold{m}$ is the model vector corresponding to P(t,x,y).

As follows from the theoretical analysis of the data regularization problem in Chapter , regularization implicitly deals with the spectrum of the regularization filter, which approximates the inverse model covariance. In other words, it involves the square operator  

$$\left[\begin{array} {cc}\displaystyle \bold{C}_x^T & \bold{C... ...\right] = \bold{C}_x^T \bold{C}_x + \bold{C}_y^T \bold{C}_y\;.$$ (119)

If we were able to transform this operator to the form $\bold{C}^T \bold{C}$, where $\bold{C}$ is a three-dimensional minimum-phase convolution, we could use the three-dimensional filter $\bold{C}$ in place of the inconvenient pair $\bold{C}_{x}$ and $\bold{C}_y$.

The problem of finding $\bold{C}$ from its spectrum is the familiar spectral factorization problem. In fact, we already encountered a problem analogous to () in the previous section in the factorization of the discrete two-dimensional Laplacian operator:  

$$\Delta = \nabla^T \nabla = \left[\begin{array} {cc}\display... ...tial_x \\ \partial_y\end{array}\right] = \bold{H}^T \bold{H}\;,$$ (120)

where $\partial_x$ and $\partial_y$ represent the partial derivative operators along the x and y directions, respectively, and the two-dimensional filter $\bold{H}$ is known as helix derivative Claerbout (1999); Zhao (1999).

If we represent the filter $\bold{C}_{x}$ with the help of a simple first-order upwind finite-difference scheme  

$$P(t,x+1) - P(t,x) + \sigma_x \left[P(t+1,x+1) - P(t,x+1)\right] = 0\;,$$ (121)

then, after the helical mapping to 1-D, it becomes a one-dimensional filter with the Z-transform  

$$C_x (Z) = 1 - \sigma_x Z^{N_t + 1} + (\sigma_x - 1) Z^{N_t}\;,$$ (122)

where N_t is the number of samples on the t-axis. Similarly, the filter $\bold{C}_y$ takes the form  

$$C_y (Z) = 1 - \sigma_y Z^{N_t N_x + 1} + (\sigma_y - 1) Z^{N_t N_x}\;.$$ (123)

The problem is reduced to a 1-D spectral factorization of

$$\begin{eqnarray} \nonumber & C_x (1/Z) C_x (Z) + C_y (1/Z) C_y (Z) = - \sigma_... ... + 1} + (\sigma_y - 1) Z^{N_t N_x} - \sigma_y Z^{N_t N_x + 1}\;. &\end{eqnarray}$$ (124)

The spectral factorization of () produces a minimum-phase filter applicable for 3-D forward and inverse convolution. Equation () is shown here just to illustrate the concept. In practice, I use the longer and much more accurate plane-wave filters of equation () in place of the simplified filters () and ().

 

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Figure 25 3-D plane wave construction with the factorized 3-D filter. Left: $\sigma_x=0.75$, $\sigma_y=0.5$. Right: $\sigma_x=-0.75$, $\sigma_y=0.5$.

Figure  shows examples of plane-wave construction. The two plots in the figure are outputs of a spike, divided recursively (on a helix) by $\bold{C}^T \bold{C}$, where $\bold{C}$ is a 3-D minimum-phase filter, obtained by the Wilson-Burg factorization.

Clapp (2000a) has proposed constructing 3-D plane-wave destruction (steering) filters by splitting. In Clapp's method, the two orthogonal 2-D filters $\bold{C}_{x}$ and $\bold{C}_y$ are simply convolved with each other instead of forming the autocorrelation (). While being a much more efficient approach, splitting suffers from induced anisotropy in the inverse impulse response. Figure  illustrates this effect in the 2-D plane by comparing the inverse impulse responses of plane-wave filters obtained by spectral factorization and splitting. The splitting response is evidently much less isotropic.

 

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Figure 26 Two-dimensional inverse impulse responses for filters constructed with spectral factorization (left) and splitting (right). The splitting response is evidently much less isotropic.