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Mathematical Q models provide a model of the earth's response to seismic waves. In reflection seismology, the anelastic attenuation factor, often expressed as seismic quality factor or Q, which is inversely proportional to attenuation factor, quantifies the effects of anelastic attenuation on the seismic wavelet caused by fluid movement and grain boundary friction. When a plane wave propagates through a homogeneous viscoelastic medium, the effects of amplitude attenuation and velocity dispersion may be combined conveniently into the single dimensionless parameter, Q. As a seismic wave propagates through a medium, the elastic energy associated with the wave is gradually absorbed by the medium, eventually ending up as heat energy. This is known as absorption (or anelastic attenuation) and will eventually cause the total disappearance of the seismic wave.〔Toksoz, W.M., & Johnston, D.H. 1981. Seismic Wave Attenuation. SEG.〕 The frequency-dependent attenuation of seismic waves leads to decreased resolution of seismic images with depth. Transmission losses may also occur due to friction or fluid movement, and for a given 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 〔Ursin B. and Toverud T. 2002 Comparison of seismic dispersion and attenuation models. Studia Geophysica et Geodaetica 46, 293-320.〕 published an article where they compared different Q models. ==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 several other models. These other models were compared with the behavior of the Kolsky-Futterman model. The Kolsky-Futterman model was first described in the article ‘Dispersive body waves’ by Futterman (1962).〔Futterman (1962) ‘Dispersive body waves’. Journal of Geophysical Research 67. p.5279-91〕 'Seismic inverse Q-filtering' by Yanghua Wang (2008) contains an outline discussing the theory of Futterman, beginning with the wave equation:〔Wang 2008, p. 60〕 : where U(r,w) is the plane wave of radial frequency w at travel distance r, k is the wavenumber and i is the imaginary unit. Reflection seismograms record the reflection wave along the propagation path r from the source to reflector and back to the surface. Equation (1.1) has an analytical solution given by: : where k is the wave number. When the wave propagates in inhomogeneous seismic media the propagation constant k must be a complex value that includes not only an imaginary part, the frequency-dependent attenuation coefficient, but also a real part, the dispersive wave number. We can call this K(w) a propagation constant in line with Futterman.〔Futterman (1962) p.5280〕 : k(w) can be linked to the phase velocity of the wave with the formula: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mathematical Q models」の詳細全文を読む スポンサード リンク
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