|Waves in strata|
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 . Figure 3.8 shows this in an idealized geometry: waves radiated from the horizontal flight of a supersonic airplane.
Figure 8. 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 as it is at depth . This leads (in isotropic media) to Snell's law.
Figure 3.9 illustrates the differential geometry of the wave. Notice that triangles have their hypotenuse on the -axis and the -axis but not along the ray. That's because this figure refers to wave fronts. (If you were thinking the hypotenuse would measure , it could be you were thinking of the tip of a ray and its projection onto the and axes.)
Figure 9. Downgoing fronts and rays in stratified medium . The wavefronts are horizontal translations of one another.
Snell's law relates the angle of a wave in one layer with the angle in another. The constancy of equation (3.8) 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 . A plane wave that happens to enter a medium of depth-variable velocity 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
observable at the surface,
whereas neither nor is directly observable.
is not only observable,
but constant in depth, it is customary to use it
to eliminate from equations (3.8) and (3.9):
|Waves in strata|