Moment-area theorem

2013-11-14T07:02:12Z

115.111.221.150:

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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 [[Kramers-Kronig relations|Krämers-Krönig relations]] they should be preferable to the [[The Kolsky basic model and modified model for attenuation and dispersion|Kolsky model]] in [[seismic inverse Q filtering]].

==Azimi's first model==

Azimi's first model <ref>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</ref> (1968), which he proposed together with <ref>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</ref> Strick (1967) has the attenuation proportional to |w|1-γ| and is:

:<math> \alpha (w)=a_1|w|^{1-\gamma} \quad (1.1)</math>

The phase velocity is written:

:<math> \frac {1}{c(w)} = \frac {1}{c_\infty} +a_1|w|^{-\gamma} +cot(\frac{\pi \gamma}{2}) \quad (1.2)</math>

==Azimi's second model==

Azimi's second model is defined by:

:<math> \alpha (w)= \frac {a_2|w|}{1+a_3 |w|} \quad (2.1)</math>

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

:<math> \frac {1}{c(w)} = \frac {1}{c_\infty} -\frac{2a_2ln(a_3w)}{\pi (1-a_3^2w^2)} \quad (1.2)</math>

==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.<ref>Ursin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models. Studia Geophysica et Geodaetica 46, 293-320.</ref>

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

<gallery widths="900px" heights="600px">

File:Powerlawb.png|Azimis 1 model - the power law

</gallery>

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

<gallery widths="900px" heights="600px">

File:Azimi1 second.png|Fig.1.Attenuation - phase velocity Azimi's second and Kolsky model

</gallery>

== Notes ==
{{Reflist}}

== References ==
*{{cite book|last=Wang|first=Yanghua|title=Seismic inverse Q filtering|url=http://books.google.no/books?id=IpwAjT-F_TgC|year=2008|publisher=Blackwell Pub.|isbn=978-1-4051-8540-0}}


[[Category:Seismology measurement]]
[[Category:Geophysics]]

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Moment-area theorem