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Snell waves

It is natural to begin studies of waves with equations that describe plane waves in a medium of constant velocity. However, in reflection seismic surveys the velocity contrast between shallowest and deepest reflectors ordinarily exceeds a factor of two. Thus depth variation of velocity is almost always included in the analysis of field data. Seismological theory needs to consider waves that are just like plane waves except that they bend to accommodate the velocity stratification v(z). Figure 17 shows this in an idealized geometry: waves radiated from the horizontal flight of a supersonic airplane.

Figure 17
Fast airplane radiating a sound wave into the earth. From the figure you can deduce that the horizontal speed of the wavefront is the same at depth z1 as it is at depth z2. This leads (in isotropic media) to Snell's law.


The airplane passes location x at time t0(x) flying horizontally at a constant speed. Imagine an earth of horizontal plane layers. In this model there is nothing to distinguish any point on the x-axis from any other point on the x-axis. But the seismic velocity varies from layer to layer. There may be reflections, head waves, shear waves, and multiple reflections. Whatever the picture is, it moves along with the airplane. A picture of the wavefronts near the airplane moves along with the airplane. The top of the picture and the bottom of the picture both move laterally at the same speed even if the earth velocity increases with depth. If the top and bottom didn't go at the same speed, the picture would become distorted, contradicting the presumed symmetry of translation. This horizontal speed, or rather its inverse ${\partial t_0}/{\partial x}$,has several names. In practical work it is called the stepout. In theoretical work it is called the ray parameter. It is very important to note that ${\partial t_0}/{\partial x}$does not change with depth, even though the seismic velocity does change with depth. In a constant-velocity medium, the angle of a wave does not change with depth. In a stratified medium, ${\partial t_0}/{\partial x}$ does not change with depth.

Figure 18 illustrates the differential geometry of the wave.

Figure 18
Downgoing fronts and rays in stratified medium v(z). The wavefronts are horizontal translations of one another.


The diagram shows that
{\partial t_0 \over \partial x} \ \ \ &=&\ \ \ { \sin \, \theta... t_0 \over \partial z} \ \ \ &=&\ \ \ { \cos \, \theta \over v }\end{eqnarray} (36)
These two equations define two (inverse) speeds. The first is a horizontal speed, measured along the earth's surface, called the horizontal phase velocity. The second is a vertical speed, measurable in a borehole, called the vertical phase velocity. Notice that both these speeds exceed the velocity v of wave propagation in the medium. Projection of wave fronts onto coordinate axes gives speeds larger than v, whereas projection of rays onto coordinate axes gives speeds smaller than v. The inverse of the phase velocities is called the stepout or the slowness.

Snell's law relates the angle of a wave in one layer with the angle in another. The constancy of equation (36) in depth is really just the statement of Snell's law. Indeed, we have just derived Snell's law. All waves in seismology propagate in a velocity-stratified medium. So they cannot be called plane waves. But we need a name for waves that are near to plane waves. A Snell wave will be defined to be the generalization of a plane wave to a stratified medium v(z). A plane wave that happens to enter a medium of depth-variable velocity v(z) gets changed into a Snell wave. While a plane wave has an angle of propagation, a Snell wave has instead a Snell parameter $p\,=\,{\partial t_0}/{\partial x}$.

It is noteworthy that Snell's parameter $p\,=\,{\partial t_0}/{\partial x}$ is directly observable at the surface, whereas neither v nor $ \theta $ is directly observable. Since $p\ =\ {\partial t_0}/{\partial x}$ is not only observable, but constant in depth, it is customary to use it to eliminate $ \theta $ from equations (36) and (37):
{\partial t_0 \over \partial x} \ \ \ &=&\ \ \ {\sin\,\theta \o...
 ...s\,\theta \over v }\quad =\quad\sqrt{
 {1 \over v (z)^2}\ -\ p^2 }\end{eqnarray} (38)

Taking the Snell wave to go through the origin at time zero, an expression for the arrival time of the Snell wave at any other location is given by
t(x,z) \ \ \ &=&\ \ \ {\sin\,\theta \over v }\ x\ +\ \int_0^z\ ...
 ...\,x\ +\ \int_0^z\ 
\sqrt{ {1 \over v ( z ) ^2 }\ -\ 
p^2 } \ \ d z\end{eqnarray} (40)
The validity of (41) is readily checked by computing ${\partial t_0}/{\partial x}$ and $\partial t_0 / \partial z $,then comparing with (38) and (39).

An arbitrary waveform f(t) may be carried by the Snell wave. Use (40) and (41) to define the time t0 for a delayed wave f[t-t0 (x,z)] at the location (x,z).  
\hbox{SnellWave}(t,x,z)\quad =\quad f \, \left( \ t\ -\ 
\sqrt{ {1 \over v ( z )^2}\ -\ p^2 } \ \ dz \ \right)\end{displaymath} (42)

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