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.
Azimuth moveout: the operator parameterization and antialiasing
A practical implementation of azimuth moveout (AMO) must be both computationally efficient and accurate. We achieve computational efficiency by parameterizing the AMO operator with the help of a transformed midpoint coordinate system. To achieve accuracy, the AMO operator needs to be carefully designed for antialiasing. We propose a modified version of Hale’s antialiasing algorithm, which switches between interpolation in time and interpolation in space depending on the operator dips. The method is applicable to a vide variety of integral operators and compares favorably with the triangle filter technique. A simple synthetic example tests the applicability of the method to the AMO case.
3-D Migration to Zero Offset For a Constant Velocity Gradient
DMO processes are designed to move energy to where it would have been recorded by a zero-offset experiment conducted on the datum. For a constant velocity model, the spatial correction is always on a line between the source and the receiver to a position updip from the trace midpoint, and the temporal correction is a shift towards zero time additional to NMO, there being no DMO effects for pure strike shooting. In the case of a non-constant velocity field refraction effects complicate the situation, and the appropriate operator for a given reflection time becomes an area1 surface whose characteristics may be counter-intuitive; for example, there can be marked “DMO” effects even for strike shooting, with spatial and temporal corrections occurring in a direction opposite to the corrections for dip shooting. In this paper, we derive the MZO (Migration to Zero Offset) operator for a velocity which increases linearly with depth, and we exhibit the surprising and important effects of this operator when applied to data for which the source-receiver orientation is strike to the reflectors.