Standard linear solid Q model

A standard linear solid Q model for attenuation and dispersion is one of many mathematical 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. Bjørn Ursin and Tommy Toverud [1] published an article where they compared different Q models of which the above model (SLS-model) was one of them.

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 the SLS model. This model was 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).

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:

Kolsky's attenuation-dispersion model

The Kolsky model assumes the attenuation α(w) to be strictly linear with frequency over the range of measurement:

$\alpha ={\frac {|w|}{(2c_{r}Q_{r})}}\quad (1)$

And defines the phase velocity as:

${\frac {1}{c(w)}}={\frac {1}{c_{r}}}(1-{\frac {1}{\pi Q_{r}}}ln|{\frac {w}{w_{r}}}|)\quad (2)$

The SLS-model

The Standard Linear Solid model is developed from the stress-strain relation standard linear solid model.Using a linear combination of springs and dashpots to represent elastic and viscous components Ursin and Toverud used one relaxation time . The model was first developed by Zener. The attenuation is given by:

$\alpha ={\frac {|w\tau _{r}|^{2}}{c_{0}Q_{c}\tau _{r}[1+(w\tau _{r})^{2}]}}\quad (3)$

And defines the phase velocity as:

${\frac {1}{c(w)}}={\frac {1}{c_{0}}}[1-{\frac {|w\tau _{r}|^{2}}{c_{0}Q_{c}\tau _{r}[1+(w\tau _{r})^{2}]}}\quad (4)$

Computations

For each of the Q models Ursin B. and Toverud T. presented in their article they computed the attenuation (1)(3) in the frequency band 0-300 Hz. Fig.1. presents the graph for the Kolsky model (blue) with two datasets (left and right)and same data - attenuation with c_r=2000m/s, Q_r=100 and w_r=2π100 Hz. SLS model (green) has two different datasets,

left c₀=1990 m/s, Q_c=100 and τ_r^-1=2π100

right c₀=1985 m/s, Q_c=84.71 and τ_r^-1=6.75x100

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  Fig.1.Attenuation - Kolsky model and Zener model

References

• Wang, Yanghua (2008). Seismic inverse Q filtering. Blackwell Pub. ISBN 978-1-4051-8540-0.
• Kolsky, Herbert (1963). Stress Waves in Solids. Courier Dover Publications.

External links

• example.com