en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-21T00:51:33Z

Fix CS1 deprecated date parameter errors

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'''Mathematical Q models''' gives a definition of how the earth responds to [[seismic wave]]s. When a plane wave propagates through a homogeneous viscoelastic medium, the effects of amplitude attenuation and velocity dispersion may be combined conveniently into a single dimensionless parameter, Q, the medium-quality factor.

The frequency-dependent attenuation of seismic waves causes decreased resolution of seismic images with depth. Transmission losses may also occur due to friction or fluid movement, and whatever the physical mechanism, they can be conveniently described with an empirical formulation where elastic moduli and propagation velocity are complex functions of frequency. Bjørn Ursin and Tommy Toverud <ref>Ursin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models. Studia Geophysica et Geodaetica 46, 293-320.</ref> published an article where they compared different Q models.

==Basics==
In order to compare the different models they considered plane-wave propagation in a homogeneous viscoelastic medium. They used the Kolsky-Futterman model as a reference and studied several other models. These other models were compared with the behaviour of the Kolsky-Futterman model.

The Kolsky-Futterman model was first described in the article ‘Dispersive body waves’ by Futterman (1962).<ref>Futterman (1962) ‘Dispersive body waves’. Journal of Geophysical Research 67. p.5279-91</ref>

However, I would recommend the outline in the book 'Seismic inverse Q-filtering' by Yanghua Wang (2008). He discuss the theory of Futterman and starts with the wave equation:<ref>Wang 2008, p. 60</ref>

:<math>\frac { dU(r,w)}{ dr} - ikU(r,w)=0 \quad (1.1)</math>

where U(r,w) is the plane wave of radial frequency w at travel distance r, k is the wavenumber and i is the imaginary unit. Reflection seismograms record the reflection wave along the propagation path r from the source to reflector and back to the surface.

Equation (1.1) has an analytical solution given by

:<math>U(r+\bigtriangleup r,w) =U(r,w)\exp (ik\bigtriangleup r) \quad (1.2)</math>

Where k is the wave number. When the wave propagates in inhomogeneous seismic media the propagation constant k must be a complex value that includes not only an imaginary part, the frequency-dependent attenuation coefficient, but also a real part, the dispersive wavenumber. We can call this K(w) a propagation constant in line with Futterman.<ref name="Futterman">Futterman (1962) p.5280</ref>

:<math>K(iw) =k(w)+ i a(w) \quad (1.3)</math>

k(w) can be linked to the phase velocity of the wave with the formula:

:<math> c(w)=\frac {w}{k(w)} \quad (1.4)</math>

==Kolsky's attenuation-dispersion model==

To obtain a solution that can be applied to seismic k(w) must be connected to a function that represent the way U(r,w) propagates in the seismic media. This functions can be regarded as a Q-model.

In his outline Wang calls the Kolsky-Futterman model the Kolsky model. The model assumes the attenuation α(w) to be strictly linear with frequency over the range of measurement:<ref>Wang 2008, p. 18, sec. 2.1: Kolsky's attenuation-dispersion model</ref>

:<math>\alpha=\frac {|w|}{(2 c_r Q_r)} \quad (1.5)</math>

And defines the phase velocity as:

:<math>\frac {1}{c(w)} =\frac {1}{c_r} (1-\frac {1}{\pi Q_r} ln |\frac{w}{w_r}|) \quad (1.6)</math>

Where cr and Qr are the phase velocity and the Q value at a reference frequency wr.

For a large value of Qr >>1 the solution (1.6) can be approximated to

:<math>\frac {1}{c(w)} =\frac {1}{c_r} |\frac{w}{w_r}|^{-\gamma} \quad (1.7)</math>

where

:<math> \gamma =(\pi Q_r)^{-1} </math>

Kolsky’s model was derived from and fitted well with experimental observations. A requirement in the theory for materials satisfying the linear attenuation assumption is that the reference frequency wr is a finite (arbitrarily small but nonzero) cut-off on the absorption. According to Kolsky, we are free to choose wr following the phenomenological criterion that it be small compared with the lowest measured frequency w in the frequency band.<ref>Wang 2008, p.19</ref> Those who want a deeper insight into this concept can go to Futterman (1962)<ref>Futterman W.I. 1962. Dispersive body waves. Journal of Geophysical Research 67. p.5279-91</ref>

==Computations==
For each of the Q models Ursin B. and Toverud T. presented in their article they computed the attenuation (1.5) and phase velocity (1.6) in the frequency band 0–300 Hz. Fig.1. presents the graph for the Kolsky model - attenuation (left) and phase velocity (right) with cr=2000 m/s, Qr=100 and wr=2π100 Hz.

<gallery widths="900px" heights="300px">

File:kolsky1.png|Fig.1.Attenuation - dispersion Kolsky model

</gallery>

==Q models==

Wang listed the different Q models that Ursin B. and Toverud T. applied in their study. The list of Wang was classified into two groups. The first group consists of model 1-5 below, and the other group includes model 6-8. The main difference between these two groups is the behaviour of the phase velocity when the frequency approaches zero, where the first group has a zero-valued phase velocity, and the second group has a finite, nonzero phase velocity.

1) the Kolsky model (linear attenuation)

2) [[Azimi Q models|the Strick-Azimi model]] (power-law attenuation)

3) [[The Kjartansson constant Q model|the Kjartansson model (constant Q)]]

4) [[Azimi Q models|Azimi's second and third models]] (non-linear attenuation)

5) Müller's model (power-law Q)

6) [[Standard linear solid Q model for attenuation and dispersion]] the Zener model (the standard linear solid)

7) the Cole-Cole model (a general linear-solid)

8) a new general linear model

==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}}
*{{cite book|last=Kolsky|first=Herbert|title=Stress Waves in Solids|url=http://books.google.no/books/about/Stress_Waves_in_Solids.html?id=Jc3z3VHqRe0C&redir_esc=y|year=1963|publisher=Courier Dover Publications}}

==External links==
*[http://bki.net/ricc/inverseQfilter Some aspects of seismic inverse Q-filtering theory] by Knut Sørsdal



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

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