Amplitude preserving AMO from true amplitude DMO and inverse DMO
Starting from the definition of Azimuth Moveout ( AM O) as the cascade of D M O and inverse D M O at different offsets and azimuths, we derive an amplitude-preserving function for the AM O operator. This amplitude function is based on the FK definition of D M O and the definition of its true inverse. Similar to Liner’s formalism of a true inverse for Hale’s D M O, we derive an asymptotically true inverse for Black/Zhang’s D M O and Bleinstein’s Born D M O. A numerical test is given that compares amplitude preservation using kinematically equivalent D M O operators cascaded with their true inverses. We define amplitude-preserved processing as the preservation of the offset-dependent reflectivity after AM O transformation, where the reflectivity is considered to be proportional to the peak amplitude of each event. We found that an AM O operator defined using Zhang’s D M O cascaded with its true inverse best reconstructs data amplitudes after transformation to a new offset and azimuth. The new amplitude function represents an amplitudepreserving azimuth moveout.
The Time And Space Formulation of Azimuth Moveout
Azimuth moveout (AMO) transforms 3-D prestack seismic data from one common azimuth and offset to different azimuths and offsets. AMO in the time-space domain is an integral operator defined in the time and midpoint coordinates. The operator components are the summation path, the weighting function, and the aperture (range of integration). The summation path and the weighting function can be determined by cascading dip moveout (DMO) and inverse DMO for different azimuths the time-space domain. To evaluate the aperture, we apply a geometric approach, defining AMO as the result of cascading prestack migration (inversion) and modeling in three dimensions. The AMO spatial extent found by the proposed procedure depends on the reflection wave velocity and decreases substantially with a decrease of the azimuth rotation. When the azimuth rotation angle approaches zero, AMO converges to the twodimensional offset continuation operator. Thus the aperture limitations provide a consistent description of AMO for small azimuth rotations and justify the economic efficiency of the method in practical applications, such as cable feathering corrections in marine seismics.
Time and Heritage
As a historian studying what can be considered a form of intellectual history, I have slowly come to adopt Michel de Certeau’s observation made at the end of the 1980s that ‘objectifying the past, for the last three centuries, has undoubtedly left unconsidered time within a discipline that has continued to use it as a taxonomic instrument’. To a certain degree, time has become commonplace for the historian who has preserved or instrumentalized it. It is not considered because it is inconceivable, but because we do not think of it or, more simply, we do not think about it. As a historian attempting to pay attention to the time I’m living in, I have thus, like many others, noticed the swift development of the category of the present until it has become obvious that the present is omnipresent. This is what I refer to here as ‘presentism’.