The time and space formulation of azimuth moveout |

Let's consider the general symmetric ellipsoid equation

where is the wave velocity. The center of the ellipsoid is at the initial midpoint .

This section addresses the kinematic problem of reflection
from the ellipsoid defined by (15). In particular, we are looking for
the answer to the following question: *For a given elliptic
reflector defined by the input midpoint, offset, and time coordinates,
what points on the surface can form a source-receiver pair valid for a
reflection?* If a point in the output midpoint-offset space
cannot be related to a reflection pattern, we should exclude it from the AMO
impulse response defined in (1).

Fermat's principle provides a general method of solving the kinematic reflection problems. Consider a formal expression for the two-point
reflection traveltime

Solving the system of equations (18) for and allows us to find the reflection ray path for a given source-receiver pair on the surface. The solution is derived in Appendix B to be

where has the same meaning as in the preceding section and is defined by (7).

Since the reflection point is contained inside the ellipsoid,
its projection obeys the evident inequality

amoapp
The AMO impulse
response traveltime. Parameters:
m,
m, sec. The top plots illustrate
the case of an unrealistically low velocity ( m/s); on the bottom,
m/s. On the left side the azimuth rotation
; on the right,
Figure 2. |
---|

The AMO operator's contours for different azimuth rotation angles are shown in Figure 2. Comparing the results for the case of an unrealistically low velocity (the top two plots in Figure 2) and the case of a realistic velocity (the bottom two plots) clearly demonstrates the gain in the reduction of the aperture size achieved by the aperture limitation. The gain is especially spectacular for small azimuths. When the azimuth rotation approaches zero, the area of the 3-D aperture monotonously shrinks to a line, and the limit of the traveltime of the AMO impulse response (the inverse of (9)) approaches the offset continuation operator (14) (Figures 3). This means that taking into account the aperture limitations of AMO provides a consistent description valid for small azimuth rotations including zero (the offset continuation case). Obviously, the cost of an integral operator is proportional to its size. The size of the offset continuation operator cannot extend the difference between the offsets . If we applied DMO and inverse DMO explicitly, the total size of the two operators would be about , which is substantially greater. This fact proves that in the case of small azimuth rotations the AMO price is less than those of not only 3-D prestack migration, but also 3-D DMO and inverse DMO combined (Canning and Gardner, 1992). Figure 4 shows the saddle shape of the AMO operator impulse response in a 3-D AVS display.

amocom
Traveltime curves
of the impulse responses. The dashed lines indicate the AMO impulse
response with an azimuth rotation of 3 degrees (projection on the
plane); the solid lines, the 2-D offset continuation impulse response.
Figure 3. | |
---|---|

amoavs
AMO impulse
response traveltime in three dimensions (the AVS display). Parameters:
m,
m, sec, m/s,
.
Figure 4. |
---|

The time and space formulation of azimuth moveout |

2014-12-03