{{notability|date=April 2013}}
In seismology, the '''standard linear solid Q model (SLS Q model)''' for attenuation and dispersion, also known as the '''Zener Q model''', is one of many Q models that gives a definition of how the earth responds to seismic waves. 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.
Transmission losses may 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. 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> compared different Q models including the above model (SLS-model).
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 the SLS model. This model was compared with 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>
==Kolsky's attenuation-dispersion model==
The Kolsky 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)</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 (2)</math>
==SLS model==
The standard linear solid model is developed from the stress-strain relation. Using a linear combination of springs and dashpots to represent elastic and viscous components, Ursin and Toverud used one relaxation time.<ref>Ursin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models. ''Studia Geophysica et Geodaetica'' 46, 293–320.</ref> The model was first developed by Zener.<ref>Zener C. 1948 ''Elasticity and anelasticity of Metals''. University of Chicago Press, Chicago.</ref> The attenuation is given by:
:<math>\alpha=\frac {(w\tau_r)^2}{c_0 Q_c \tau_r[1+(w\tau_r)^2]} \quad (3)</math>
And defines the phase velocity as:
:<math>\frac {1}{c(w)} =\frac {1}{c_0} [1- \frac {(w\tau_r)^2}{Q_c[1+(w\tau_r)^2]}] \quad (4)</math>
==Computations== For each of the Q models, Ursin and Toverud computed the attenuation (1)(3) in the frequency band 0–300 Hz. Figure 1. presents the graph for the Kolsky model (blue) with two datasets (left and right) and same data – attenuation with c<sub>r</sub>=2000 m/s, Q<sub>r</sub>=100 and w<sub>r</sub>=2π100 Hz.
The SLS model (green) has two different datasets,
left c<sub>0</sub>=1990 m/s, Q<sub>c</sub>=100 and τ<sub>r</sub><sup>−1</sup>=2π100
right c<sub>0</sub>=1985 m/s, Q<sub>c</sub>=84.71 and τ<sub>r</sub><sup>−1</sup>=6.75x100
<gallery widths="900px" heights="600px">
File:Zenermodel_attenuation.png|Fig.1.Attenuation – Kolsky model and Zener model (Standard Linear Solid)
</gallery>
== Notes == {{Reflist}}
==References== *{{cite book|last=Wang|first=Yanghua|title=Seismic inverse Q filtering|url=https://books.google.com/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=https://books.google.com/books?id=Jc3z3VHqRe0C|year=1963|publisher=Courier Dover Publications|isbn=9780486495347}}
Category:Seismology measurement Category:Geophysics