Derivation of energy migration equation

Energy migration in (x,t)-space is analyzed in a fashion similar to the way the group velocity was derived. Take depth to be large in the integral  

$$\int \int \ e^{ i\,z\, [ k_z ( \omega , k_x ) \ -\ \omega \ t / z \ +\ k_x \ x / z ] } \ d \omega \ dk_x$$ (28)

The result is that the energy goes to  

$$(x,t) \ \eq \ z \ \left( \, { - \ { \partial k_z \over \part... ...x } \ ,\ { \partial k_z \over \partial \omega \ } } \, \right)$$ (29)

This justifies our previous assertion that (20) and (21) can be used to analyze energy propagation errors. Equation (29) was also used to calculate the curve in Figure 9. The validity of the stationary phase concept is confirmed by Figure 16, which was produced using inverse Fourier transformation.

 

aniso45 Figure 16 Impulse response of the 45$^\circ$ wave-extrapolation equation. The arrival before t0 is a wraparound.