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Time-shifting equations

An important task is to predict the wavefield inside the earth given the waveform at the surface. For a downgoing plane wave this can be done by the time-shifting partial differential equation  
{\partial P(t,z) \over \partial z} \quad =\quad-\ {1 \over v }\ {\partial P(t,z) 
\over \partial t}\end{displaymath} (43)
as may be readily verified by substituting either of the trial solutions
P(t,z) &=& f \left( \ t\ -\ {z \over v }\ \right)\ \ \ \ \ \ \ ...
 ...int_0^z\ {dz \over v (z) } \ \right)
 \ \ \ \ \ \ {\rm for} \ v(z)\end{eqnarray} (44)

This also works for nonvertically incident waves with the partial differential equation  
{\partial P(t,x,z) \over \partial z} \quad =\quad
-\ {\partial t_0 \over \partial z} \ {\partial P(t,x,z) \over \partial t}\end{displaymath} (46)
which has the solution  
P(t,x,z) \quad =\quad f ( t\ -\ p\,x\ -\ \int_0^z\ 
{\partial t_0 \over \partial z} \ dz )\end{displaymath} (47)
In interpreting (46) and (47) recall that $1/({\partial t_0}/{\partial z}$ is the apparent velocity in a borehole. The partial derivative of wavefield P(t,x,z) with respect to depth z is taken at constant x, i.e., the wave is extrapolated down the borehole. The idea that downward extrapolation can be achieved by merely time shifting holds only when a single Snell wave is present; that is, the same arbitrary time function must be seen at all locations.

Substitution from (38) and (39) also enables us to rewrite (46) in the various forms
{\partial P(t,x,z) \over \partial z}\ \ \ &=&\ \ \ -\ {\cos\,\t...
 ...partial x} \right)}^2 } \ \ 
 {\partial P(t,x,z) \over \partial t}\end{eqnarray} (48)
Equations (48), (49) and (50) are paraxial wave equations. Since ${\partial t_0}/{\partial x} = p$ can be measured along the surface of the earth, it seems that equation (50), along with an assumed velocity v(z) and some observed data P(t,x,z=0), would enable us to determine ${\partial P}/{\partial z}$, which is the necessary first step of downward continuation. But the presumption was that there was only a single Snell wave and not a superposition of several Snell waves. Superposition of different waveforms on different Snell paths would cause different time functions to be seen at different places. Then a mere time shift would not achieve downward continuation. Luckily, a complicated wavefield that is variable from place to place may be decomposed into many Snell waves, each of which can be downward extrapolated with the differential equations (48), (49) and (50) or their solution (47). One such decomposition technique is Fourier analysis.

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Next: Fourier decomposition Up: THE PARAXIAL WAVE EQUATION Previous: Snell waves
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