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![]() | Omnidirectional plane-wave destruction | ![]() |
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opwd
Figure 1. Interpolation in plane-wave construction: line-interpolating PWC interpolates the wavefield at point ![]() ![]() |
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Considering the wavefield
observed in a 2D sampling system
and following Fomel (2002), plane-wave
destruction can be represented in the
-transform domain as
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:
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.
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![]() | Omnidirectional plane-wave destruction | ![]() |
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