Omnidirectional plane-wave destruction

From line interpolation to circle interpolation

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Figure 1. Interpolation in plane-wave construction: line-interpolating PWC interpolates the wavefield at point $A$ , while circle-interpolating PWC interpolates at point $B$ .

Considering the wavefield $u(x_1,x_2)$ observed in a 2D sampling system and following Fomel (2002), plane-wave destruction can be represented in the $Z$ -transform domain as

$$(1-Z_2Z_1^p)U(Z_1,Z_2)=0\;,$$ (1)

where $Z_1, Z_2$ are the unit shift operators in the first and second dimensions. $U(Z_1,Z_2)$ (or $U$ for convenience) denotes the $Z$ transform of $u(x_1,x_2)$ . $p$ is the local slope. We call $Z_2Z_1^p$ and $1-Z_2Z_1^p$ plane-wave constructor and destructor, respectively. The slope $p$ has the following relationship with the dip angle $\theta$ : $p=\tan \theta$ .

Applying $Z_2Z_1^p$ at one point, for example, point $O$ in Figure 1, PWC obtains the wavefield at the point with a unit shift in the second dimension and $p$ unit shifts in the first dimension, denoted by $A(x_1+p\Delta x_1,x_2+\Delta x_2)$ . As $-\pi/2 \leq \theta \leq \pi/2$ , $p$ can be any value from $-\infty$ to $+\infty$ . That is to say, the forward plane-wave constructor $Z_2Z_1^p$ interpolates the wavefield along the vertical line at $x_2+\Delta x_2$ . Similarly, the backward PWC interpolates the wavefield along the vertical line at $x_2-\Delta x_2$ .

In order to handle both vertical and horizontal structures, we propose to modify the plane-wave destruction in equation 1 into the following form:

$$(1-Z_2^{p_2}Z_1^{p_1})U=0,$$ (2)

where $p_1,p_2$ are parameters related to the trial dip angle, as follows: $p_1=r\sin \theta$ , $p_2=r\cos \theta$ .

In other words, we consider a circle in polar coordinates, parameterized by the radius $r$ and the dip angle $\theta$ . Applying the new PWC $Z_2^{p_2}Z_1^{p_1}$ at point $O$ , it obtains the wavefield at the point with $p_1$ unit shifts in the first dimension and $p_2$ unit shifts in the second dimension. That is point $B(x_1+p_1\Delta x_1,x_2+p_2\Delta x_2)$ . As $\theta$ changes, the new PWC $Z_1^{p_1}Z_2^{p_2}$ interpolates the wavefield along a circle with radius $r$ . We draw the interpolating circle with $r=1$ in Figure 1. The circle-interpolating PWC $Z_1^{p_1}Z_2^{p_2}$ corresponds to a 2D interpolation. Equation 1 can also be seen as a special case of equation 2 when $p_2=1$ . Compared with the 1D line-interpolating method, the main benefit of circle interpolation is its antialiasing ability.

Omnidirectional plane-wave destruction