AMO APERTURE: CASCADING MIGRATION AND MODELING

Cascading DMO and inverse DMO allowed us to evaluate the AMO operator's summation path and the corresponding weighting function. However, this procedure is not sufficient for evaluating the third major component of the integral operator, that is, its aperture (range of integration). To solve this problem, we apply an alternative approach, defining AMO as the cascade of the 3-D common-offset common-azimuth migration and the 3-D modeling for a different azimuth and offset. The impulse response of the common-offset common-azimuth migration is a symmetric ellipsoid with the center in the input midpoint and axis of symmetry along the input-offset direction. Such an ellipsoid is described by the general formula  

$$z({\bf m})=\sqrt{R^2- \Delta m^2 + \gamma\, \frac{\left({\bf \Delta m}\cdot{\bf h}_{1}\right)^2}{h_{1}^2}}\;,$$ (51)

where z stands for the depth coordinate, ${\bf m}$ is the surface coordinate, ${\bf \Delta m}= {\bf m}- {\bf m}_1$, R is the small semi-axis of the ellipsoid, and $\gamma$ is a nondimensional parameter describing the stretching of the ellipse $(\gamma < 1)$. Deregowski and Rocca 1981 derived the following connections between the geometric properties of the reflector and the coordinates of the corresponding impulse in the data:  

$$R={{v\,t_1}\over 2}\;;\; \gamma={{{4\,{h_{1}}^2}\over v^2} \over t_1^2+ {{4\,{h_{1}}^2}\over v^2}}\;,$$ (52)

where v is the propagation velocity. The impulse response of the AMO operator corresponds kinematically to the reflections from the ellipsoid defined by equation (52) to a different azimuth and different offset. To constrain the AMO aperture, we should look for the answer to the following question: For a given elliptic reflector defined by the input midpoint, offset, and time coordinates, what points on the surface can form a source-receiver pair valid for a reflection? If a point in the output midpoint-offset space cannot be related to a reflection pattern, it should be excluded from the AMO aperture.

Fermat's principle provides a general method of solving the kinematic reflection problem Goldin (1986). The formal expression for the two-point reflection traveltime is given by  

$$t_2={\sqrt{({\bf s}_2- {\bf m})^2+z^2({\bf m})} \over v}+ {\sqrt{({\bf r}_2- {\bf m})^2+z^2({\bf m})} \over v}\;,$$ (53)

where ${\bf m}$ is the vertical projection of the reflection point to the surface, ${\bf s}_2$ is the source location, and ${\bf r}_2$ is the receiver location for the output trace. According to Fermat's principle, the reflection raypath between two fixed points must correspond to the extremum value of the traveltime. Hence, in the vicinity of a reflected ray,  

$${\partial t_2 \over \partial {\bf m}}=0\;.$$ (54)

Solving equation (55) for ${\bf m}$ allows us to find the reflection raypath for a given source-receiver pair on the surface. To find the solution of (55), it is convenient to decompose the reflection-point projection ${\bf m}$ into three components: ${\bf m}= {\bf m}_1+ {\bf m}_{\parallel}+ {\bf m}_{\perp}$, where ${\bf m}_{\parallel}$ is parallel to the input offset vector ${\bf h}_{1}$, and ${\bf m}_{\perp}$ is perpendicular to ${\bf h}_{1}$. The plane, drawn through the reflection point and the central line of ellipsoid (52), must contain the zero-offset (normally reflected) ray because of the cylindrical symmetry of the reflector. The fact that the zero-offset ray is normal to the reflector gives us the following connection between the zero-offset midpoint ${\bf m}_0$ and the ${\bf m}_{\parallel}$ component of the reflection point ${\bf m}$: 

$${\bf m}_0= ({\bf m}_1+ {\bf m}_{\parallel}) + z \left({\bf ... ...\parallel}\right)} = {\bf m}_1+ \gamma\,{\bf m}_{\parallel}\,.$$ (55)

Equation (56) evaluates ${\bf m}_{\parallel}$ in terms of ${\bf m}_0$, as follows:  

$${\bf m}_{\parallel}={{{\bf \Delta m}_{10}}\over{\gamma}}\;.$$ (56)

where the length of the vector ${\bf \Delta m}_{10} = {\bf m}_0- {\bf m}_1$ can be determined from equation (44) for any given input and output midpoints ${\bf m}_1$ and ${\bf m}_2$ and azimuths $\theta_{1}$ and $\theta_{2}$.To find the third component of the reflection point projection ${\bf m}_{\perp}$, we substitute expression (57) into (54). Choosing a convenient parameterization ${\bf s}_2= {\bf m}_0+ {\bf h}_{2}^s$, ${\bf r}_2= {\bf m}_0+ {\bf h}_{2}^r$, where ${\bf h}_{2}^r-{\bf h}_{2}^s=2\,{\bf h}_{2}$, and ${\bf h}_{2}^r+{\bf h}_{2}^s = 2\,{\bf \Delta m}_{02} = 2\,\left({\bf m}_2-{\bf m}_0\right)$, we can rewrite the two-point traveltime function from (54) in the form

$$t_2 = {\sqrt{R^2 - \gamma\,(1- \gamma) m_{\parallel}^2 + (h_... ..._{\perp}+ (1 - \gamma)\,{\bf m}_{\parallel}\right)} \over v}\;+$$

$${\sqrt{R^2 - \gamma\,(1- \gamma) m_{\parallel}^2 + (h_{2}^r)... ... m}_{\perp}+ (1 - \gamma)\,{\bf m}_{\parallel}\right)} \over v}$$ (57)

Fermat's principle (55) leads to a simple linear equation for the length of ${\bf m}_{\perp}$, which has the explicit solution  

$$m_{\perp}=(\gamma -1)\,m_{\parallel}\,\cot{\left(\theta_2 - ... ...}\right)^2\right)\, \sin{\left(\theta_2 - \theta_1\right)}}}\;,$$ (58)

where $m_{\parallel}$ is defined by (57), and $\Delta m_{02}$satisfies relationship (44). Because the reflection point is contained inside the ellipsoid, its projection obeys the evident inequality  

$$z^2 ({\bf m}) = R^2 - m_{\perp}^2 - (1 - \gamma)\,m_{\parallel}^2 \geq 0\;.$$ (59)

It is inequality (60) that defines the aperture of the AMO operator. After transformation (5) and algebraic simplifications, it takes the form of inequality (12), which is convenient for an efficient implementation of AMO.