Fractional integration and constant Q

By equation (61) and theorem 6, fractional powers of integration and differentiation are also impedance functions. Kjartansson [1979] has advocated the fractional power as a stress-strain law for rocks. See also Madden [1976]. Classical studies in rock mechanics begin with a stress-strain law such as

$$\rm{stress} \ \ = \ \ \rm{stiffness} \ \ \times \ \ \rm{strain} \ \ + \ \ \rm{viscosity} \ \times \ \ \hbox{strain-rate}$$

which in the transform domain is  

$$\rm{stress} \ \ =\ \ [ (- \, i \omega )^0 \ \ \times \ \ \rm... ...mega )^1 \ \ \times \ \ \rm{viscosity} ] \ \times\ \rm{strain}$$ (72)

Experimentally, the viscoelastic law (72) does a poor job of describing real rocks. Let us try another mathematical form that is like (72) in its limiting behavior at high and low viscosity:

$$\begin{eqnarray} \ \ \ \ \ \ \ \ \ \ \rm{stress}\ \ \ &=&\ \ \ \rm{const} \ \ ... ... ( - \, i \omega )^{ \epsilon -1 } \ \times \ \ \hbox{strain-rate}\end{eqnarray}$$ (73) (74)

Here $\epsilon$ close to zero gives elastic behavior and $\epsilon$ close to one gives viscous behavior. The fact that $( - \, i \omega )^{ \epsilon -1 }$ is an impedance function meshes nicely with the concepts that (17) stress may be determined from strain history and strain may be determined from stress history, and (18) stress times strain-rate is dissipated power. Kjartansson [1979] points out that $( - \, i \omega )^{\gamma}$ exhibits the mathematical property called constant Q, so that as a stress/strain law for fitting experimental data on rocks, it is far superior to (72). To see the constant Q property more clearly, express $( - \, i \omega )^{\gamma}$ in real and imaginary parts:

$$\begin{eqnarray} ( - \,i \omega )^{\gamma} \ \ \ &=&\ \ \ \vert \omega \vert^{\... ...(\omega )\ \sin \left( {\pi \, \gamma \over 2 }\ \right) \ \right]\end{eqnarray}$$ (75) (76) (77) (78)

The constant Q property follows from the constant ratio between the real and imaginary parts of this function. Q itself is defined by  

$${1 \over Q } \eq \tan \, \pi \epsilon \ \ \ \approx \ \ \ \pi \epsilon$$ (79)

A pulse with a Q of about 10 is shown in Figure 8.

 

Qq
Figure 8 The constant Q pulse given by $e^{{-\,(-\,} i \omega )^{.97} t_0 }$. The frequency axis is represented by a discrete Fourier transform over 256 points. Zero time and zero frequency are on the left end of their respective axes.

EXERCISES:

1. Take $\epsilon < 0$ and expand the integration operator for negative powers of Z. Explain the sign difference.
2. Let $\alpha \gt 0$ be a real, positive scaling constant, and let C be a reflectance function. Without using Muir's rules, prove that C ' is a reflectance, where

   $${1\ -\ C ' \over 1\ +\ C ' } \eq \alpha \ {1\ -\ C \over 1\ +\ C }$$

   Note that you have proven Muir's first rule. Muir's third rule can also be proven in an analogous way, but with much more algebraic detail.
3. The word isomorphism means not only that any impedance R₁ , R₂ , R ' can be mapped into a reflectance C₁ , C₂ , C ' , but also that Muir's three rules will be mapped into three rules for combining reflectances.

   a.

   What are these three rules?

   b.

   Although C ' = C₁ C₂ does not turn out to be one of the three rules, it is obviously true. Either show that it is a consequence of the three rules or conclude that it is an independent rule that can be mapped back into the domain of the impedances to constitute a fourth rule.

4. Show that the log of the discrete causal integration operator, $\ln [(1+Z)/(1-Z)]$, is one side of the discrete Hilbert transform. Show that the reflected pulse from a boundary between two media with the same velocity but slightly different Q is one side of the Hilbert transform.
5. Consider the fourth-order Taylor expansion for square root in an extrapolation equation

   $${dP \over dz } \eq i \omega \left[ 1 \ -\ {1 \over 2 }\ \lef... ... {1 \over 8 }\ \left( {vk \over \omega }\ \right)^4 \right] \ P$$

   a.

   Will this equation be stable for the complex frequency $- i \omega = - i \omega_0 + \epsilon$? Why?

   b.

   Consider causal and anticausal time-domain calculations with the equation. Which, if any, is stable?

6. Consider a material velocity that may depend on the frequency $\omega$ and on the horizontal x-coordinate as well. Suppose that, luckily, the velocity can be expressed in the factored form $v ( x , \omega ) = v_1 (x) \ v_2 ( \omega )$.Obtain a stable 45$^\circ$ wave-extrapolation equation. Hints: try

   $$s \ \ \ =\ \ \ -\ { i \omega \over v_2 }$$

   $$X^2 \ \ \ =\ \ \ \ \hbox{positive eigenvalue of} \ \ (v_1 \partial_x )( v_1 \partial_x )^T$$

7. Is the Levinson Recursion described in FGDP related to the rules in this section? If so, how? Hint: see Jones and Thron [1980].
8. Show the converse to theorem 4, namely, that if the phase curve of a causal function does not enclose the origin, then the inverse is causal.