Misplaced Pages

Azimi Q models

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (July 2013) (Learn how and when to remove this message)
This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (July 2013) (Learn how and when to remove this message)
(Learn how and when to remove this message)

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

Azimi's first model (1968), which he proposed together with Strick (1967) has the attenuation proportional to |w| and is:

α ( w ) = a 1 | w | 1 − γ ( 1.1 ) {\displaystyle \alpha (w)=a_{1}|w|^{1-\gamma }\quad (1.1)}

The phase velocity is written:

1 c ( w ) = 1 c ∞ + a 1 | w | − γ + c o t ( π γ 2 ) ( 1.2 ) {\displaystyle {\frac {1}{c(w)}}={\frac {1}{c_{\infty }}}+a_{1}|w|^{-\gamma }+cot({\frac {\pi \gamma }{2}})\quad (1.2)}

Azimi's second model

Azimi's second model is defined by:

α ( w ) = a 2 | w | 1 + a 3 | w | ( 2.1 ) {\displaystyle \alpha (w)={\frac {a_{2}|w|}{1+a_{3}|w|}}\quad (2.1)}

where a2 and a3 are constants. Now we can use the Krämers-Krönig dispersion relation and get a phase velocity:

1 c ( w ) = 1 c ∞ − 2 a 2 l n ( a 3 w ) π ( 1 − a 3 2 w 2 ) ( 1.2 ) {\displaystyle {\frac {1}{c(w)}}={\frac {1}{c_{\infty }}}-{\frac {2a_{2}ln(a_{3}w)}{\pi (1-a_{3}^{2}w^{2})}}\quad (1.2)}

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.

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 , β=0.155

lower: c∞=2065 m/s, a=4.76 x 10 , β=0.1

  • Azimis 1 model - the power law Azimis 1 model - the power law

Data for Azimis second model (green):

upper: c∞=2000 m/s, a=2.5 x 10 , a2=1.6 x 10

lower: c∞=2018 m/s, a=2.86 x 10 , a2=1.51 x 10

  • Fig.1.Attenuation - phase velocity Azimi's second and Kolsky model Fig.1.Attenuation - phase velocity Azimi's second and Kolsky model

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. 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
  3. Ursin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models. Studia Geophysica et Geodaetica 46, 293-320.

References

Categories: