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),
(81) |
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With a little algebra, one may verify that the determinant of the curvature matrix is
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To obtain expressions for the AMO amplitude, we need to substitute equation eq76 in equation eq56, together with the corresponding expressions for J_{1} and J_{2}. Given a forward DMO with a Jacobian term J_{1}, 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
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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.