(1) |

The splitting approximation of the 3-D dispersion relation Brown (1983) is given by

(2) |

The 45-degree approximation of the 3-D dispersion relation with splitting is

(3) |

In Figure 1, the solid line shows the contours of for the exact dispersion relation of equation (1); the dashed line, the splitting approximation for the 3-D dispersion relation of equation (2); and the dotted line, the 45-degree splitting approximation for the 3-D dispersion relation of equation (3). The splitting approximations show the familiar anisotropy in relation to the azimuth. The greatest error appears at 45 degree azimuth.

exac-split-45
Contours of constant amplitude and phase for the exact 3-D
dispersion relation described in equation (1) (solid line),
the splitting approximation in equation (2) (dashed line),
and the 45-degree approximation to splitting
in equation (3) (dotted line).
Figure 1 |

After an azimuth rotation of 45 degrees, as shown in Figure , the 45-degree approximation for the split 3-D dispersion relation can be obtained as follows:

Figure 2

(4) |

(5) |

where *k*_{x}' and *k*_{y}' are the new axes after
a rotation by angle . For degrees, substituting the
following relations

(6) |

(7) |

into Equation (3) gives

(8) |

Figure 3 shows the contours of for the
45-degree approximation, with an azimuth rotation of 45 degrees (the
dashed line)
and without (the solid line).
The curves of the two approximations have
their greatest errors at different
places. For the one without rotation,
the worst error is at the 45 degree azimuth;
for the one with 45 degree rotation,
the worst error is along *x* and *y* axes.
Therefore, the average of the two should
give us a more isotropic operator.

rot-45
Contours of constant amplitude and phase for the
45-degree splitting approximation described in equation (3)
(solid line) and the rotated
45-degree splitting approximation in equation
(8)(dashed line).
Figure 3 |

The operator obtained by averaging equation (3) and (8) gives

(9) |

In Figure , the solid line indicates the contour for the exact 3-D dispersion relation equation (1); the dot-dashed line, the four-pass operator of equation (9). The dispersion relation for the four-pass method looks isotropic but has an error that increases with dip, which is caused by the 45-degree approximation.

exac-rotavg
Contours of constant amplitude and phase for the exact 3-D
dispersion relation described in equation (1) (solid line)
and the averaged 45-degree
splitting approximation obtained from the four-pass operator,
as in equation (9)(dotted line).
Figure 4 |

11/16/1997