Earth:Azimi Q models

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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 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|^1−γ and is:

[math]\displaystyle{ \alpha (w)=a_1|w|^{1-\gamma} \quad (1.1) }[/math]

The phase velocity is written:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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.

Data for the Kolsky model (blue):

upper: c_r=2000 m/s, Q_r=100, w_r=2π100

lower: c_r=2000 m/s, Q_r=100, w_r=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

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  Azimis 1 model - the power law

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

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

References

• Wang, Yanghua (2008). Seismic inverse Q filtering. Blackwell Pub.. ISBN 978-1-4051-8540-0. https://books.google.com/books?id=IpwAjT-F_TgC.