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The Kjartansson constant Q model uses Mathematical Q models to explain how the earth responds to seismic waves and is widely used in seismic geophysical applications. Because these models satisfies the Krämers-Krönig relations they should be preferable to the Kolsky model in seismic inverse Q filtering. Kjartanssons model is a simplification of the first of Azimi Q models (1968).

Kjartansson constant Q model

Kjartanssons model is a simplification of the first of Azimi Q models (1968). Azimi proposed his first model together with Strick (1967) and has the attenuation proportional to |w|^1-γ| and is:

$\alpha (w)=a_1|w{|}^{1-\gamma }(1.1)$

The phase velocity is written:

$\frac{1}{c(w)}=\frac{1}{c_{\mathrm{\infty }}}+a_1|w{|}^{-\gamma }+cot(\frac{\pi \gamma }{2})(1.2)$

If the phase velocity goes to infinity in the first term on the right, we simply has:

$\frac{1}{c(w)}=a_1|w{|}^{-\gamma }+cot(\frac{\pi \gamma }{2})(1.2)$

This is Kjartansson constant Q model.

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):

c_r=2000 m/s, Q_r=100, w_r=2π100

Data for Kjartansson constant Q model (green):

a₁=2.5 x 10 ⁻⁶, γ=0.0031

• 
  Fig.1.Kjartansson constant Q model and the Kolsky model

Notes

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