hyplay
Many hyperbolas tangent to a line.
Figure 2 |

When a hyperbolic event carries a wavelet,
the wavelet covers an area in the (*t*,*x*)-plane.
Define this area as that between the two hyperbolas
shown in Figure 3.
These two curves are

(5) | ||

(6) |

hyparea
Detailed view of one of the many hyperbolas
in figure 2.
Two hyperbolas are separated by . An impulsive signal is defined as one that
is zero everywhere but between the two hyperbolas
where its amplitude is .Figure 3 |

In the limit , the waveform is an impulse function whose temporal spectrum is constant. In that limit, the many waveforms in Figure 2 superpose giving a great concentration at the apex. Since the many hyperbolas are identical with one another, the time dependence of their sum is (within a scale factor) the same time dependence of the shaded area of the single hyperbola in Figure 3.

The separation between the two curves, is

(7) |

(8) |

By making this high frequency approximation,
we are setting aside some calculations
that we should be able to do for a constant velocity medium
but which would not apply to a medium with velocity as a function of depth.
Divergence considerations, for example,
suggest that the amplitude of each hyperbola should not be
a constant, but should drop off proportional to *t ^{-1/2}*.
Another example is that of parabolas instead
of hyperbolas, i.e. .(When velocity changes with depth, we no longer have hyperbolas,
but the tops of the traveltime curve could be approximated by a parabola.)
Then the amplitude is .Notice that the divisor in equation (8) for the hyperbola
has the same half power discontinuity at

11/17/1997