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A 3D plane-wave source is specified by two ray-parameters, *p*_{x} and *p*_{y}. Given the velocity at the surface *v*_{z0}, we
can calculate the propagation direction of the plane-wave source at the surface from the two ray-parameters.
For each plane-wave source, we rotate the Cartesian coordinates, so that the extrapolation direction
of the new coordinates is close to the propagation direction of the plane-wave.
In 3D, the propagation direction of a plane-wave source is defined by two angles: the azimuth angle and the take-off angle .
Given the velocity at the surface *v*_{z0} and a plane-wave source with a ray parameter pair (*p*_{x} *p*_{y}), its propagation direction at the surface is defined by the vector (*q*_{x},*q*_{y},*q*_{z}), where
| |
(13) |

| (14) |

| (15) |

The azimuth angle and take-off angle of the plane-wave source are calculated
from the vector (*q*_{x},*q*_{y},*q*_{z}) as follows:
| |
(16) |

| (17) |

Rotations in 3D are specified by the axis of rotation and rotation angle. They can be described by a rotation matrix.
For example, a rotation about the *z*-axis by an angle (Figure ) is

| |
(18) |

To design a coordinate system with an extrapolation direction parallel
the propagation direction of the plane-wave source (*q*_{x},*q*_{y},*q*_{z}),
we rotate the coordinates in two steps.
First we rotate about the *z*-axis by an angle
(equation 18) shown in Figure .
Second we rotate about the
*y*_{1}-axis by an angle (Figure ) as follow:

**rottilt
**
Figure 2 Rotation about the axis *y*_{1} by |
| |

| |
(19) |

Combining the two rotations, we have the rotation from original coordinates
to the new tilted coordinates as follows,
| |
(20) |

It is easy to verify that the depth axis of the new coordinates parallels the propagation
direction (*q*_{x},*q*_{y},*q*_{z}).
In practice, we do not use the direction exactly paralleling the propagation of the plane-wave source at the surface.
Considering that the velocity usually increases with the depth,
the propagation direction of a plane-wave source becomes
increasingly horizontal. So we usually choose a tilting angle that is a little bigger than .
**coordinates
**

Figure 3 A plane-wave source and its coordinates to image the salt body.(*x*,*y*,*z*)
is the original Cartesian coordinates. (*x*_{2},*y*_{2},*z*_{2}) is the new tilted coordinates. The dashed line with
arrows are the source and receiver rays of the plane-wave source. The extrapolation direction of the new
coordinates *z*_{2} axis is closer to the propagation direction of the plane-wave source.

Figure shows
a typical coordinate system used for a plane-wave source to image a salt dome.
The dashed lines with arrows show the propagation direction of the source and receiver waves for the plane-wave source. (*x*,*y*,*z*) is the
Cartesian coordinate system, and (*x*_{2},*y*_{2},*z*_{2}) is the tilted coordinates for the plane-wave source. The extrapolation direction of
the new coordinates, which parallels *z*_{2}-axis, is much closer to the propagation direction of the plane-wave source
than the conventional vertical extrapolation direction.

For conical-wave source migration, we can similarly design tilted coordinates for
each conical source. In contrast to the 3D plane-wave source migration,
we apply the second rotation directly without rotating the azimuth
of the data. Given the conical-wave source with a ray parameter *p*_{x} and the surface
velocity *v*_{z0}, we rotate the velocity and the surface data along *y*-axis by a angle of
and migrate the data in the new coordinates.
This usually works when the in-line direction is the predominant
dip direction in the subsurface. When there are steep dips in the cross-line direction,
it is still difficult to image these dips with conical-wave migration in tilted coordinates.
In contrast, 3D plane-wave migration in tilted coordinates rotate the data and model into a general
direction dependent on the propagation direction and it can image these cross-line direction dips.

**ppanel
**

Figure 4 The dots represent all possible plane-wave sources. The circles are the contour of their take-off angles and the smallest circle represents plane-wave sources with a take-off angle of . The radial lines show the contour of azimuth angles. All possible plane-wave sources are divided into small cells. The plane-wave sources in a cell have a similar take-off and azimuth angle and share a coordinate system.

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Stanford Exploration Project

5/6/2007