Geometrical interpretation of PS-AMO

A trace with input offset vector 79#79 and midpoint position 73#73 is first transformed to its corresponding CCP position and zero offset. By defining the new offset and azimuth position and by applying inverse PS-DMO, we transform the data to a new CCP position and its corresponding CMP position.

Here, we follow the same procedure as (, ) for the derivation of the PS-AMO operator.

First, we refer to equations () and () in order to understand the relationship between CMP and CCP for the 3D case. We rewrite equation () as

80#80

where 81#81 is the angle between the midpoint vector (73#73) and the transformation vector (82#82).

We can then rewrite equation () as  

 83#83 (35)

82#82 is an extension of 74#74 and lies in the CCP space. Figure  shows both 74#74 and 82#82 in the same plane. Since the vectors are parallel, the angle between 73#73 and 82#82 is the same as the angle between 73#73 and 74#74.If the coordinate system is aligned with the midpoint coordinates, then the angle 81#81 is the same as the azimuth (84#84). 81#81 changes after and before PS-AMO. This variation is responsible for the event movement along the common conversion point.

 

rot Figure 1 Definition of offset vector 74#74 and transformation vector 82#82, before and after PS-AMO

Figure  shows how event movement along CCP changes with depth. This is due to the dependence of 82#82 with respect to v_p, 66#66 and t_n. This variance with depth will persist even in a constant velocity media. Figure  also illustrates that the time after PS-AMO (t₂) has a new 74#74 and 82#82, therefore, a new CCP position.

 

plane2 Figure 2 Comparison between the CMP and CCP position in the PS-AMO operator

Continuing with the procedure presented by () to obtain the PS-AMO operator, we cascade PS-DMO [equation ()] with its inverse. Figure  shows a scheme of the PS-AMO transformation. A trace with input offset vector 79#79 and midpoint at the origin is transformed into equivalent data with output offset vector 85#85 and midpoint position 73#73. The data is first transformed to its corresponding CCP position and 77#77. Subsequently, the inverse PS-DMO repositions the data to a new midpoint position 73#73 with a new offset vector 85#85.

 

plane Figure 3 CMP-CCP plane, PS-AMO geometrical interpretation.

The new trace position is defined by

86#86 (36)

Both 87#87 and 88#88 can be expressed as terms of the final midpoint position 73#73 by using the rule of sines in the triangle (73#73,87#87,88#88)in Figure  as The final expression takes the form of

75#75 (37)

where This expression represents the azimuth rotation in both the CCP domain and the CMP domain.