Azimi Q models

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:


 * $$ \frac {1}{c(w)} = \frac {1}{c_\infty} -\frac{2a_2ln(a_3w)}{\pi (1-a_3^2w^2)} \quad (1.2)$$

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: cr=2000 m/s, Qr=100, wr=2π100

lower: cr=2000 m/s, Qr=100, wr=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