Upcoming waves

All the above equations are for downgoing waves. To get equations for upcoming waves you need only change the signs of z and $\partial / \partial z$.Letting D denote a downgoing wavefield and U an upcoming wavefield, equation (21), for example, is found in Table .6.

 

$${\strut\partial^2 \over \partial z \partial t} \ D \eq +\ \disp... ...tyle {\strut v\over 2} \displaystyle {\strut\partial^2 \over\partial x^2} \ D$$

$${\strut\partial^2 \over \partial z \partial t} \ U \eq -\ \disp... ...tyle {\strut v\over 2} \displaystyle {\strut\partial^2 \over\partial x^2} \ U$$

Using the exploding-reflector concept, it is the upcoming wave equation that is found in both migration and diffraction programs. The downgoing wave equation is useful for modeling and migration procedures that are more elaborate than those based on the exploding-reflector concept (chapter ).

EXERCISES:

1. Consider a tilted straight line tangent to a circle, as shown in figure 2 . Use this line to initialize the Muir square-root expansion. State equations and plot them $(-2 \ \le \ X \ \le \ + 2)$ for the next two Muir semicircle approximations.
    

   tanexer Figure 2