AMO amplitudes

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 ${\bf k}$ integral in equation amo_freq.eq as in Bleistein and Handelsman (1975),

$$A \approx \frac{2\pi J_1 J_2}{{\left\vert det\left({\bf C} \... ...\raisebox{-.22cm}{$\sim$}\right)\frac{\pi}{4}}. \EQNLABEL{eq56}$$ (81)

Therefore we need to evaluate the determinant and the signature of the curvature matrix ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}$, which is defined as

$${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}=\left\vert \beg... ...hi}{{\partial{k_y}}^2} \end{array} \right\vert. \EQNLABEL{eq60}$$ (82)

Taking the second-order partial derivatives of $\Phi$ with respect to k_x and k_y and using the definitions of $\beta_1$ and $\beta_2$ yields the following expressions for $\frac{\partial^2\Phi}{{\partial{k_x}}^2}$, $\frac{\partial^2\Phi}{{\partial{k_y}}^2}$ and $\frac{\partial^2\Phi}{\partial{k_x}\partial{k_y}}$:

$$\begin{eqnarray} \frac{\partial^2\Phi}{{\partial{k_x}}^2} & = & \frac{{h_{1x}}^2... ...{h_{2x}h_{2y}}{\omega_ot_2}{(1-\nu_{02}^2)}^{3/2}. \EQNLABEL{eq72}\end{eqnarray}$$ (83) (84) (85)

With a little algebra, one may verify that the determinant of the curvature matrix is

$$\begin{eqnarray} det(C) & = & -\frac{\Delta^2}{\left\vert\omega_o\right\vert^2t... ...^2t_0^2}{(1-\nu_{01}^2)}^{2}{(1-\nu_{02}^2)}^{2}. \EQNLABEL{eq76}\end{eqnarray}$$ (86)

Notice that the determinant of ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}$,which is the product of the two eigenvalues of ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}$,is always negative. This means the two eigenvalues have opposite signs and consequently the signature of ${\bf C} \!\!\!\!\! \raisebox{-.22cm}{$\sim$}$,defined as the number of positive eigenvalues minus the number of negative eigenvalues, is always null. Therefore, the second term of the phase shift in equation eq56 vanishes.

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

$$J_1= \frac{(1+\nu_{01}^2)}{\sqrt{1-\nu_{01}^2}}, \;\;\;\;\;\;\; J_2=1. \EQNLABEL{j1j2.eq}$$ (87)

after taking into account the Jacobian of the transformation from t₁ to t₀ ($dt_1 =dt_0 {\sqrt{1-\nu_{01}^2}}$)in the first integral of equation amo_freq_change.eq, we can write the amplitude term for the AMO integral:

$$\begin{eqnarray} A & \approx & \frac{\left\vert\omega_o\right\vert t_0}{2\pi\De... ...1+\nu_{01}^2)}{{(1-\nu_{01}^2)}{(1-\nu_{02}^2)}}. \EQNLABEL{amp2}\end{eqnarray}$$ (88) (89)

The last substitution, $\left\vert\omega_o\right\vert t_0=\left\vert\omega_2\right\vert t_2$,enables us to apply the differentiation operator $\left\vert\omega_2\right\vert$ to the output data; it is correct because t₀ and t₂ are linked by the linear relationship $t_0 =t_2 {\sqrt{1-\nu_{02}^2}}$.

The expression for the amplitudes presented in equation amo_amp.eq of the main text follows by simple substitution of the expressions for $\Delta$, $\nu_{01}$,and $\nu_{02}$, from equations eq40, eq44 and eq48 into equation amp2.