In reflection seismology, Q models are mathematical models used to study how the Earth affects seismic waves by measuring energy loss and speed changes as the waves travel through materials like rock. These models focus on the Q factor (seismic quality factor, where higher Q means less energy loss) to capture anelastic attenuation (or absorption)—the gradual loss of wave energy into heat due to fluid movement and friction in the subsurface, eventually causing the wave to disappear completely. Introduced as a single parameter to combine amplitude weakening and velocity dispersion, Q helps explain why deeper seismic images lose clarity as wave effects worsen deeper down. Researchers like Bjørn Ursin and Tommy Toverud have compared different Q models to better understand these transmission losses, using equations that adapt to the medium’s changing properties.
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). 'Seismic inverse Q-filtering' by Yanghua Wang (2008) contains an outline discussing the theory of Futterman, beginning with the wave equation:
d U ( r , w ) d r − i k U ( r , w ) = 0 ( 1.1 ) {\displaystyle {\frac {dU(r,w)}{dr}}-ikU(r,w)=0\quad (1.1)}
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:
U ( r + △ r , w ) = U ( r , w ) exp ( i k △ r ) ( 1.2 ) {\displaystyle U(r+\bigtriangleup r,w)=U(r,w)\exp(ik\bigtriangleup r)\quad (1.2)}
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.
K ( i w ) = k ( w ) + i a ( w ) ( 1.3 ) {\displaystyle K(iw)=k(w)+ia(w)\quad (1.3)}
k(w) can be linked to the phase velocity of the wave with the formula:
c ( w ) = w k ( w ) ( 1.4 ) {\displaystyle c(w)={\frac {w}{k(w)}}\quad (1.4)}
Kolsky's attenuation-dispersion model
To obtain a solution that can be applied to seismic k(w) must be connected to a function that represents the way in which U(r,w) propagates in the seismic media. This function can be regarded as a Q-model. In his outline Wang calls the Kolsky–Futterman model the Kolsky model. The model assumes the attenuation α(w) to be strictly linear with frequency over the range of measurement:
α = | w | ( 2 c r Q r ) ( 1.5 ) {\displaystyle \alpha ={\frac {|w|}{(2c_{r}Q_{r})}}\quad (1.5)}
And defines the phase velocity as:
1 c ( w ) = 1 c r ( 1 − 1 π Q r ln | w w r | ) ( 1.6 ) {\displaystyle {\frac {1}{c(w)}}={\frac {1}{c_{r}}}(1-{\frac {1}{\pi Q_{r}}}\ln |{\frac {w}{w_{r}}}|)\quad (1.6)}
where cr and Qr are the phase velocity and the Q value at a reference frequency wr. For a large value of Qr >> 1 the solution (1.6) can be approximated to
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