In this chapter, I present my development of a ``true-amplitude'' function for AMO so that amplitude variations as a function of offset and azimuth are not distorted by the transformation. Beyond the derivation of AMO amplitudes, the Chapter serves as a good example for using asymptotic theory to derive ``true amplitude'' functions for partial or full prestack imaging operators. The discussions focus on the DMO and AMO operators, however, the same concepts have been used in literature to derive amplitude weights for migration and demigration.
Given that AMO is derived by chaining DMO and inverse DMO, I compare the amplitude behavior of kinematically equivalent DMO operators with various amplitude functions adopted through different definition of ``true-amplitude''. I also compare the asymptotic inverse to the adjoint, and derive an amplitude preserving inverse for Zhang's 1988 DMO and Bleistein's 1990 Born DMO. Finally, I present a true-amplitude function for AMO and demonstrate the preservation of the reflectivity by the transformation through synthetic examples.
Since the term ``true-amplitude'' for wave-equation operators is an ill-defined concept, I restrict its definition to be consistent with most interpreters for preserving the peak amplitude of reflection events. The derivations of AMO amplitudes assume continuous wavefields and the synthetic experiments are conducted on regularly sampled input data.