# The Kjartansson constant Q model

The Kjartansson constant Q model uses Mathematical Q models to explain how the earth responds to seismic waves and is widely used in seismic geophysical applications. Because these models satisfies the Krämers-Krönig relations they should be preferable to the Kolsky model in seismic inverse Q filtering. Kjartanssons model is a simplification of the first of Azimi Q models [1] (1968).

## Kjartansson constant Q model

Kjartanssons model is a simplification of the first of Azimi Q models [2] (1968). Azimi proposed his first model together with [3] Strick (1967) and has the attenuation proportional to |w|1-γ| and is:

$\alpha (w)=a_1|w|^{1-\gamma} \quad (1.1)$

The phase velocity is written:

$\frac {1}{c(w)} = \frac {1}{c_\infty} +a_1|w|^{-\gamma} +cot(\frac{\pi \gamma}{2}) \quad (1.2)$

If the phase velocity goes to infinity in the first term on the right, we simply has:

$\frac {1}{c(w)} = a_1|w|^{-\gamma} +cot(\frac{\pi \gamma}{2}) \quad (1.2)$

This is Kjartansson constant Q model.

## Computations

Studying the attenuation coefficient and phase velocity, and compare them with Kolskys Q model we have plotted the result on fig.1. The data for the models are taken from Ursin and Toverud.[4]

Data for the Kolsky model (blue):

cr=2000 m/s, Qr=100, wr=2π100

Data for Kjartansson constant Q model (green):

a1=2.5 x 10 −6, γ=0.0031

## Notes

1. ^ Azimi S.A.Kalinin A.V. Kalinin V.V and Pivovarov B.L.1968. Impulse and transient characteristics of media with linear and quadratic absorption laws. Izvestiya - Physics of the Solid Earth 2. p.88-93
2. ^ Azimi S.A.Kalinin A.V. Kalinin V.V and Pivovarov B.L.1968. Impulse and transient characteristics of media with linear and quadratic absorption laws. Izvestiya - Physics of the Solid Earth 2. p.88-93
3. ^ Strick E. The determination of Q, dynamic viscosity and transient creep curves from wave propagation measurements. Geophysical Journal of the Royal Astronomical Society 13, p.197-218
4. ^ Ursin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models. Studia Geophysica et Geodaetica 46, 293-320.