Angles of departure and arrival

The previous sections presented a method to compute the traveltime from a given source point to a given receiver point. Using the image point reflection method has eschewed the need for traditional ray tracing. However, in order to estimate the effect of the acquisition arrays on the amplitudes, or to graphically display the raypaths, we need to compute the angles of departure of the rays from the source and of arrival to the receiver.

Shah (1973) shows that if we denote with $\alpha_s$ the smallest angle between the raypath departing from the source and the vertical, with $\alpha_g$ the similarly defined arrival angle, with s the coordinate of the source and with g the coordinate of the receiver, the two angles can be found from the relations:  

$$\frac{\sin \alpha_s}{v}=\frac{\partial t}{\partial s},$$ (52)

 

$$\frac{\sin \alpha_g}{v}=\frac{\partial t}{\partial g}.$$ (53)

Writing (32) as  

$$t^2= {\bf u}_1 \cdot {\bf u}_1 + s^2 {\bf u}_2 \cdot {\bf u}... ...{\bf u}_2+2g{\bf u}_1\cdot{\bf u}_3+2sg{\bf u}_2\cdot{\bf u}_3,$$ (54)

we obtain  

$$\frac{\partial t}{\partial s}=\frac{1}{t}{\bf u}_2 \cdot \left({\bf u}_1+{\bf u}_2+g{\bf u}_3\right),$$ (55)

 

$$\frac{\partial t}{\partial g}=\frac{1}{t}{\bf u}_3 \cdot \left({\bf u}_1+s{\bf u}_2+{\bf u}_3\right).$$ (56)

Replacing now s with -h and g with h, the angles are given by:  

$$\sin \alpha_s = \frac{v}{t}{\bf u}_2 \cdot \left({\bf u}_1+{\bf u}_2+h{\bf u}_3\right),$$ (57)

 

$$\sin \alpha_g = \frac{v}{t}{\bf u}_3 \cdot \left({\bf u}_1-h{\bf u}_2+{\bf u}_3\right),$$ (58)

where t is computed as a function of h as given by (35).