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Fourier decomposition

Fourier analyzing the function f(x,t,z=0), seen on the earth's surface, requires the Fourier kernel $\exp ( -\, i \omega t\,+\,i\,k_x x )$.Moving on the earth's surface at an inverse speed of $ {\partial t_0}/{\partial x} \,=\, k_x / \omega $,the phase of the Fourier kernel, hence the kernel itself, remains constant. Only those sinusoidal components that move at the same speed as the Snell wave can have a nonzero correlation with it. So if the disturbance is a single Snell wave, then all Fourier components vanish except for those that satisfy $p\,=\,k_x / \omega$.You should memorize these basic relations:

 
 \begin{displaymath}

\fbox {\rule[-.4cm]{0cm}{1cm}$\displaystyle
{\strut \partia...
 ...{.2in}=\hspace{.2in}
\displaystyle {\strut k_x \over \omega}$}
\end{displaymath} (51)

In theoretical seismology a square-root function often appears as a result of using (51) to make a cosine.

Utilization of this Fourier domain interpretation of Snell's parameter p enables us to write the square-root equations (48), (49) and (50) in an even more useful form. But first the square-root equation must be reexpressed in the Fourier domain. This is done by replacing the $\partial / \partial t$ operator in (48), (49) and (50) by $- i \omega$.The result is  
 \begin{displaymath}
{\partial P( \omega , k_x ,z) \over \partial z} \quad =\quad...
 ...z)^2} \ \ -\ { k_x^2 \over \omega^2}
 } \ \ P( \omega , k_x ,z)\end{displaymath} (52)
At present it is equivalent to specify either the differential equation (52) or its solution (47) with f as the complex exponential:  
 \begin{displaymath}
P( \omega , k_x ,z)
 \quad =\quad\exp \left( \, i \omega
 \i...
 ...1 \over v(z)^2}\ -\ {k_x^2 \over \omega^2 }\ } \ \ dz \ \right)\end{displaymath} (53)
Later, when we consider lateral velocity variation v(x), the solution (53) becomes wrong, whereas the differential equation (50) is a valid description of any local plane-wave behavior. But before going to lateral velocity gradients we should look more carefully at vertical velocity gradients.


previous up next print clean
Next: Velocity gradients Up: THE PARAXIAL WAVE EQUATION Previous: Time-shifting equations
Stanford Exploration Project
10/31/1997