The helical coordinate

Constant Q medium

From the absorption law of a material, spectral factorization yields its impulse response. The most basic absorption law is the constant Q model. According to it, for a downgoing wave, the absorption is proportional to the frequency $\omega$ , proportional to time in the medium $z/v$ , and inversely proportional to the ``quality'' $Q$ of the medium. Altogether, the spectrum of a wave passing through a thickness $z$ is changed by the factor $e^{-\vert\omega\vert\tau} = e^{-\vert\omega\vert(z/v)/Q}$ . This frequency function is plotted in the top line of Figure 7.

futterman
Figure 7. Autocorrelate the bottom signal to get the middle, then Fourier transform it to get the top. Spectral factorization works the other way, from top to bottom.

The middle function in Figure 7 is the autocorrelation giving on top the spectrum $e^{-\vert\omega\vert\tau}$ . The third function is the factorization. An impulse entering the medium comes out with this shape. There is no physics in this analysis, only mathematics that assumes the broadened pulse is causal with an abrupt arrival. The short wavelengths are concentrated near the sharp corner, while the long wavelengths are spread throughout. A physical system could cause the pulse to spread further (effectively by an additional all-pass filter), but physics cannot make it more compact.

All distances from the source see the same shape, but stretched in proportion to distance. The apparent $Q$ is the traveltime to the source divided by the width of the pulse.

The helical coordinate