PARAMETERIZATION

Azimuth moveout in the time-and-space domain is a three-dimensional integral operator Fomel and Biondi (1995a). Change (substitution) of the integrable variables as a method of integral simplification is well known in classic calculus. In the case of AMO, a convenient choice of the parameters of integration is of particular value because of the complicated shape of the operator aperture.

 

amox12
Figure 1 Schematic geometry of AMO and the transformed coordinate system.

In order to simplify the form of the AMO operator, thus reducing the cost of its computation, we propose the following substitution of variables. Let $\alpha_1$ be the input offset azimuth, and $\alpha_2$be the output offset azimuth with respect to the midpoint coordinate system. Draw one axis (y₁) perpendicular to the direction of $\alpha_1$, and the other axis (y₂) perpendicular to $\alpha_2$. This defines a non-orthogonal coordinate system on the midpoint plane (Figure 1). The transformation of variables, written in the matrix form, is

 

$$\left[ \begin{array} {c} y_1 \\ y_2 \\ \end{array} \... ... \; \left[ \begin{array} {c} x \\ y \end{array} \right]\;,$$ (1)

where x and y are the Cartesian coordinates of a midpoint in the original coordinate system. The Jacobian of transformation (1) is simply $\vert\sin\alpha_1\,\cos\alpha_2-\sin\alpha_2\,\cos\alpha_1\vert = \vert\sin\alpha\vert$, where $\alpha=\alpha_2-\alpha_1$ is the angle of azimuth rotation. Assuming that $\alpha$ is greater than zero, transformation (1) defines a spatially invariant rotational squeezing of the midpoint space. The special case of $\alpha$ equal to zero corresponds to the two-dimensional version of AMO, known as offset continuation Bolondi et al. (1982); Chemingui and Biondi (1994); Fomel (1995), which can be handled separately.

 

amotta
Figure 2 Traveltime and amplitude of the AMO impulse response in the transformed coordinate system.

The expression for the traveltime of the AMO impulse response (formula (4) in Biondi and Chemingui (1994a)) transforms to

 

$$t_2=t_1\,{\left\vert{\bf h_2 \over h_1}\right\vert}\, \sqrt{... ...\alpha}- y_2^2} \over {{\bf h_2^2}\,\sin^2{\alpha}- y_1^2}}\;,$$ (2)

where $\{y_1, y_2\}$ is the midpoint separation in the transformed coordinate system. In the notation of Biondi and Chemingui, y₁ corresponds to $X \sin(\varphi-\theta_1)$, and y₂ corresponds to $X \sin(\varphi-\theta_2)$. One can see that the axes of the transformed coordinate system are now aligned along the axes of the AMO "saddle". The amplitude equations Chemingui and Biondi (1995); Fomel and Biondi (1995a) are also simplified (Figure 2). What is more important, the transformation (1) affects the shape of the AMO aperture. The aperture limitation (21) from Fomel and Biondi (1995a) transforms after some heavy algebra to

 

$$\left({2 \over {v\,t_1}}\right)^2 \leq {{{\bf h_1^2}\,\sin^2... ...amma_1^2+\gamma_2^2-2\,\gamma_1\,\gamma_2\,\cos\alpha\right)\;,$$ (3)

where  

$$\gamma_1={y_1 \over {{\bf h_2^2}\,\sin^2{\alpha}- y_1^2}}={1 \over t_2}\,{\partial t_2 \over \partial y_1}\;,$$ (4)

and  

$$\gamma_2={y_2 \over {{\bf h_1^2}\,\sin^2{\alpha}- y_2^2}}= -{1 \over t_2}\,{\partial t_2 \over \partial y_2}.$$ (5)

The largest possible aperture corresponds to the zero input time (or zero velocity) and coincides with the interior of a rectangle centered at $\{\Delta y_1,\Delta y_2\}=\{0,0\}$ with the sides of the rectangle equal to $2\,{\bf \vert h_2\vert}\,\sin{\alpha}$ and $2\,{\bf \vert h_1\vert}\,\sin{\alpha}$. With the time increase, the aperture gradually decreases in size, and its shape approaches a quasi-elliptical form (Figure 3).

 

amoapp
Figure 3 AMO aperture in the transformed coordinate system as a function of the input time. The different plots correspond to different geometries of AMO. From top to bottom: the angle of azimuth rotation $\alpha$ changes from 10 degrees (top) to 30 degrees (middle) and 60 degrees (bottom). From left to right: the ratio of offsets $\left\vert{\bf h_2/ h_1}\right\vert$ changes from 1/2 (left) to 1 (middle) and 2 (right).

From the computational point of view, it is convenient to evaluate the right-hand side of inequality (3) outside of the input time loop and use this inequality to limit the range of times for each point of the operator.