HUYGEN'S SECONDARY SOURCE IN 2-D

First we calculate the density of curves as a function of time. The density goes to infinity as the hyperbola separation $\Delta x$ tends to zero in Figure 2. Then the hyperbolas sum to a plane wave which carries the waveform that we want to know.

 

hyplay Figure 2 Many hyperbolas tangent to a line.

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

$$\begin{eqnarray} t_1 &=& v^{-1}\ \sqrt{z^2+x^2} \\ t_2 &=& v^{-1}\ \sqrt{z^2+x^2} +\Delta t\end{eqnarray}$$ (5) (6)

 

hyparea Figure 3 Detailed view of one of the many hyperbolas in figure 2. Two hyperbolas are separated by $\Delta t$. An impulsive signal is defined as one that is zero everywhere but between the two hyperbolas where its amplitude is $1/\Delta t$.

In the limit $\Delta t \rightarrow 0$, 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, $\Delta x$ is

$$\Delta x \quad =\quad \sqrt{ t ^2v^2-z^2} \ -\ \sqrt{(t-\Delta t)^2v^2-z^2}$$ (7)

The area in Figure 3 is a length, integrated over time. Between the two hyperbolas of separation $\Delta t$,we take a constant signal of strength $1/\Delta t$ which adds up to an amplitude at time t of $A_{\rm hyp}(t)$. 

$$A_{\rm hyp}(t) \quad =\quad {\Delta x \over \Delta t} \quad ... ... v^2\over \sqrt{t^2 v^2-z^2} } \quad \quad {\rm for} ~ t\gt z/v$$ (8)

To avoid much clutter that belongs in a mathematics book rather than in a seismology book, we concentrate our attention on the discontinuity itself. Mathematically, this amounts to ignoring all but the highest frequencies. Seismologically, we say that after a signal like equation (8) passes through the filters and gain control processes typical of data collection, the numerator t does not warrant comment, but the power of t at the pole itself is important.

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. $x(t)=\sqrt{t-t_0}$.(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 $A_{\rm par}(t)= 1/\sqrt{t-t_0}$.Notice that the divisor in equation (8) for the hyperbola $\sqrt{t^2 v^2-z^2}=\sqrt{tv-z}\sqrt{tv+z}$has the same half power discontinuity at z=tv as the parabola. Thus the amplitude spectrum at high frequencies for either the parabola or the hyperbola will drop off as the inverse half power of frequency according to equation (2). To get a plane wave with the constant spectrum of an impulse instead of this inverse-square-root spectrum, we must use hyperbolas (or parabolas), not carrying an impulse function, but instead carrying a square-root spectrum (to cancel the inverse-square-root spectrum that arises from superposing the hyperbolas). Equation (4) gives us a causal waveform with the desired spectrum. Thus the decomposition of an impulsive 2-D plane wave into hyperbolas requires the hyperbolas to carry the ``half-order differential'' waveform given in equation (4).