    Omnidirectional plane-wave destruction  Next: Anti-aliasing ability Up: Chen, Fomel & Lu: Previous: Introduction

# From line interpolation to circle interpolation opwd
Figure 1.
Interpolation in plane-wave construction: line-interpolating PWC interpolates the wavefield at point , while circle-interpolating PWC interpolates at point .   Considering the wavefield observed in a 2D sampling system and following Fomel (2002), plane-wave destruction can be represented in the -transform domain as (1)

where are the unit shift operators in the first and second dimensions. (or for convenience) denotes the transform of . is the local slope. We call and plane-wave constructor and destructor, respectively. The slope has the following relationship with the dip angle : .

Applying at one point, for example, point in Figure 1, PWC obtains the wavefield at the point with a unit shift in the second dimension and unit shifts in the first dimension, denoted by . As , can be any value from to . That is to say, the forward plane-wave constructor interpolates the wavefield along the vertical line at . Similarly, the backward PWC interpolates the wavefield along the vertical line at .

In order to handle both vertical and horizontal structures, we propose to modify the plane-wave destruction in equation 1 into the following form: (2)

where are parameters related to the trial dip angle, as follows: , .

In other words, we consider a circle in polar coordinates, parameterized by the radius and the dip angle . Applying the new PWC at point , it obtains the wavefield at the point with unit shifts in the first dimension and unit shifts in the second dimension. That is point . As changes, the new PWC interpolates the wavefield along a circle with radius . We draw the interpolating circle with in Figure 1. The circle-interpolating PWC corresponds to a 2D interpolation. Equation 1 can also be seen as a special case of equation 2 when . Compared with the 1D line-interpolating method, the main benefit of circle interpolation is its antialiasing ability.

Subsections    Omnidirectional plane-wave destruction  Next: Anti-aliasing ability Up: Chen, Fomel & Lu: Previous: Introduction

2013-08-09