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The Azimi Q models used Mathematical Q models to explain how the earth responds to seismic waves. Because these models satisfies the Krämers-Krönig relations they should be preferable to the Kolsky model in seismic inverse Q filtering.
Azimi's first model
editAzimi's first model [1] (1968), which he proposed together with [2] Strick (1967) has the attenuation proportional to |w|1−γ and is:
The phase velocity is written:
Azimi's second model
editAzimi's second model is defined by:
where a2 and a3 are constants. Now we can use the Krämers-Krönig dispersion relation and get a phase velocity:
Computations
editStudying 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.[3]
Data for the Kolsky model (blue):
upper: cr=2000 m/s, Qr=100, wr=2π100
lower: cr=2000 m/s, Qr=100, wr=2π100
Data for Azimis first model (green):
upper: c∞=2000 m/s, a=2.5 x 10 −6, β=0.155
lower: c∞=2065 m/s, a=4.76 x 10 −6, β=0.1
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Azimis 1 model - the power law
Data for Azimis second model (green):
upper: c∞=2000 m/s, a=2.5 x 10 −6, a2=1.6 x 10 −3
lower: c∞=2018 m/s, a=2.86 x 10 −6, a2=1.51 x 10 −4
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Fig.1.Attenuation - phase velocity Azimi's second and Kolsky model
Notes
edit- ^ 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
- ^ Strick: 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
- ^ Ursin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models. Studia Geophysica et Geodaetica 46, 293-320.
References
edit- Wang, Yanghua (2008). Seismic inverse Q filtering. Blackwell Pub. ISBN 978-1-4051-8540-0.