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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$ .
[pdf] [png] [gnuplot]

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

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

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



Subsections
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Next: Anti-aliasing ability Up: Chen, Fomel & Lu: Previous: Introduction

2013-08-09