Next: AMO amplitudes Up: Derivation of integral AMO Previous: Derivation of integral AMO

# AMO impulse response

To derive a time-space representation of the AMO impulse response from its frequency-wavenumber representation, we evaluate the stationary-phase approximation of the inverse Fourier transform along the midpoint coordinates.

The operator and its inverse, , can be defined in the zero-offset frequency and the midpoint wavenumber as
 (63)
 (64)
The operator is given by the cascades of and and its impulse response can be written as,
 (65)
The derivation of the stationary-phase approximation of the integral in is similar to the one presented in Black et al. (1993b) for deriving a time-space formulation for the conventional DMO impulse response. We begin by changing the order of the integrals and rewriting amo_freq.eq as
 (66)
The phase of this integral is,
 (67)
where,
 (68)
Next we make the following change of variables and let
 (69)
Therefore, and become
 (70)
The derivatives of and with respect to the in-line component of the wavenumber kx and the cross-line component ky can be written as
 (71)
Making one more change of variables, we let
 (72)
Setting the derivative of the phase to zero yields the system of equations:
 (73)
which we solve for and (i.e., and ) at the stationary path . The determinant of the system is given by
 (74)
and the solutions for and are
 (75)
and
 (76)
Now we need to evaluate the phase function along the stationary path .By respectively multiplying the equations in eq36 by k0x and k0y and summing them together we obtain,
 (77)
Substituting this relationship into the expression of the phase function [equation phase_app.eq] we obtain
 (78)
The phase function along the stationary path is thus peaked for
 (79)
Substituting equations eq44 and eq48 into ratio.eq we obtain amo_surf.eq of the main text:
 (80)

Next: AMO amplitudes Up: Derivation of integral AMO Previous: Derivation of integral AMO
Stanford Exploration Project
1/18/2001