Transformation of midpoint axes

The appropriate midpoint-coordinate transformation to be applied to the AMO impulse response is described by the following chain of transformations  

$$\left[ \begin{array} {c} \xi_1\ \xi_2\ \end{array}... ...{array} {c} \Delta m_x \ \Delta m_y \end{array} \right]\;,$$ (5)

where $\xi_1$, and $\xi_2$ are the transformed midpoint coordinates. Figure  shows a schematic of the relationship between the input and output offset vectors ${\bf h}_{1}$ and ${\bf h}_{2}$, and the transformed midpoint-coordinate unit vectors ${\bf \xi}_1$and ${\bf \xi}_2$. Notice that the $\xi$ axes are dual with respect to ${\bf h}_{1}$ and ${\bf h}_{2}$, but they define a new coordinate system for the midpoint axes of the AMO operator. The right matrix represents a space-invariant rotational squeezing of the coordinate, while the left matrix is a simple rescaling of the axes by a factor dependent on the azimuth rotation $\Delta \theta$, and by the length of the dual offset vectors. When the azimuth rotation is zero, the transformation described in equation (5) becomes singular. In this case the AMO operator degenerates into the 2-D offset continuation operator, as discussed in a previous section. In practice, a simple pragmatic method to avoid the singularity is to set a lower limit for the product $h_{1}h_{2}\sin\left(\Delta \theta\right)$. Because the 3-D AMO operator converges smoothly to the 2-D offset continuation operator Fomel and Biondi (1995b), the error introduced by this approximation is negligible.

 

amonewcoord
Figure 4 The geometric relationship between the unit vectors ${\bf \xi}_1$ and ${\bf \xi}_2$of the transformed midpoint-coordinate axes, and the input offset ${\bf h}_{1}$ and the output offset ${\bf h}_{2}$.

In this new coordinate system, the kinematics of AMO are described by the following simple relationship between the input time t₁ and the output time t₂:  

$$t_2\left(\xi_1, \xi_2\right)=t_1\,\sqrt{\frac{1-{\xi_2}^2}{1-{\xi_1}^2}}\;,$$ (6)

and the amplitudes (based on Zhang-Black amplitudes for DMO) are described by the following equation  

$$A\left(\xi_1, \xi_2\right)=\frac{t_2 \left\vert\omega_2\righ... ...\right)}{{\left(1-{\xi_1}^2\right)\left(1-{\xi_2}^2\right)}}\;.$$ (7)

Notice that this expression for the amplitudes does already take into account the Jacobian of the transformation described in equation (5).