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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.

**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 *z*_{1} as it is at depth *z*_{2}.
This leads (in isotropic media) to Snell's law.

The airplane passes location *x* at time *t*_{0}(*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 ,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 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,
does not change with depth.

Figure 18 illustrates the differential geometry of the wave.

**frontz
**

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

The diagram shows that

| |
(36) |

| (37) |

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 * .

It is noteworthy that
Snell's parameter is directly
observable at the surface,
whereas neither *v* nor is directly observable.
Since is not only observable,
but constant in depth, it is customary to use it
to eliminate from equations (36) and (37):

| |
(38) |

| (39) |

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

| |
(40) |

| (41) |

The validity of (41) is readily checked by
computing and ,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 *t*_{0} for
a delayed wave *f*[*t*-*t*_{0} (*x*,*z*)] at the location (*x*,*z*).

| |
(42) |

** Next:** Time-shifting equations
** Up:** THE PARAXIAL WAVE EQUATION
** Previous:** Fourier derivation of the
Stanford Exploration Project

10/31/1997