To derive an expression for the amplitudes of the AMO impulse response, we start from the general expression for the stationary-phase approximation of the integral in equation amo_freq.eq as in Bleistein and Handelsman (1975),
With a little algebra, one may verify that the determinant of the curvature matrix is
To obtain expressions for the AMO amplitude, we need to substitute equation eq76 in equation eq56, together with the corresponding expressions for J1 and J2. Given a forward DMO with a Jacobian term J1, I showed in Chapter 3 of the main text that an asymptotic inverse provides a better representation for the inverse DMO operator than the approximate adjoint. Also, by restricting the definition of ``true-amplitude'' to preserving the peak amplitude of reflection events, I also showed that an amplitude-preserving function for AMO can be defined by cascading Zhang and Black 1988 DMO with its asymptotic ``true-inverse''; therefore given
The expression for the amplitudes presented in equation amo_amp.eq of the main text follows by simple substitution of the expressions for , ,and , from equations eq40, eq44 and eq48 into equation amp2.