Chaining DMO and inverse DMO

The derivation of AMO as the cascade of DMO and ${\rm DMO^{-1}}$ results into the following expression for the amplitude term,

$$A({\bf x},{\bf h}_1,{\bf h}_2,t_1) \approx \frac{2\pi {J_1} {J_2}}{\sqrt{\Delta}}, \EQNLABEL{amo.amp}$$ (13)

where $\bf x$ is the output location vector in midpoint coordinates, J₁ and J₂ are the Jacobians in the FK definition of DMO and ${\rm DMO^{-1}}$ and $\Delta$ is the determinant of a Hessian matrix that is independent of the particular DMO of choice. For given input half-offset and time (${\bf h}_{1}$, t₁) and output half-offset $({\bf h}_{2})$,equation amo.amp describes the weights along the impulse response of AMO in the time-space domain. The Jacobians J₁ and J₂ and the determinant $\Delta$ in equation amo.amp are evaluated at the stationary point of the phase function in the integral kernel of AMO.

The amplitude behavior of AMO is thoroughly controlled by the Jacobian terms of DMO and its inverse. Liner and Cohen 1988 argued that the adjoint is a poor representation for an inverse DMO. They showed that the application of Hale's DMO followed by its adjoint inverse results in serious amplitude degradation and therefore they proposed an asymptotic ``true-inverse'' for Hale's DMO. Similar to their formulation, I derive asymptotic inverses for Zhang's and Bleistein's DMO operators which, as demonstrated later in the text, do preserve the amplitudes better.