Moment-area theorem

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:

$\alpha (w)=a_{1}|w|^{1-\gamma }\quad (1.1)$

The phase velocity is written:

${\frac {1}{c(w)}}={\frac {1}{c_{\infty }}}+a_{1}|w|^{-\gamma }+cot({\frac {\pi \gamma }{2}})\quad (1.2)$

Azimi's second model

Azimi's second model is defined by:

$\alpha (w)={\frac {a_{2}|w|}{1+a_{3}|w|}}\quad (2.1)$

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

${\frac {1}{c(w)}}={\frac {1}{c_{\infty }}}-{\frac {2a_{2}ln(a_{3}w)}{\pi (1-a_{3}^{2}w^{2})}}\quad (1.2)$

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

• 

  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

• 

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

Notes

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References

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