**3. Zener or standard linear
solid model**

* *

*The figure shows the standard
model used in Sørsdal (2008) taken from Horton. In this outline we will see
that what we call model 4 in Hortons figure can be simplified to one rel. time* *τ _{c}*

* *

We will discuss the linear solid viscoelastic model and follow the lines of Wang (2008):

.

The Zener (1948) or standard linear solid model is the most general linear equation that links the stress and strain as we have seen in Sørsdal (2008). It defines the attenuation coefficient as:

_{} (3.3)

and the phase velocity as

_{} (3.4)

where c_{0} is the
phase velocity c(w) for w* *-> 0, and τ_{c} and *Q _{c}
*are two constant parameters describing the attenuation property in the
standard linear solid model: the attenuation at the peak of the attenuation
function with respect to the frequency is Q

* *

*Mathematical outline is as following:*

* *

The Zener (1948) or standard
linear solid model is the most general linear equation in stress *a, *strain
*e *and their first time derivatives *a *and *s . *In the
Fourier transform domain (equation 3.4), the stress ∑*(w) *and the
strain *E(w) *are linked by the complex modulus defined as (Ben-Menahem
and Singh, 1981):

_{} (3.5)

where _{}* *is the strain relaxation time, _{}* *is the stress relaxation time and *M _{R}
*is a deformation modulus with sub-index

The mechanical 'relaxation' means that the strain produced by the sudden application of a fixed stress to a material increases asymptotically with time. Similarly, the stress produced when the material is suddenly strained relaxes asymptotically (Kolsky, 1953). It is found that stress waves whose periods are close to the 'relaxation times' of such a medium are severely attenuated when passing through it.

Physically, the 'relaxed'
elastic modulus *M _{R} *is the final value of the ratio of stress
to strain after relaxation has taken place, whereas a so-called 'unrelaxed'
elastic modulus,

The unrelaxed modulus can be given by

* *

*M _{U} = *lim

Thus, *M _{R} *and

A special case of the standard
linear solid model is the Kelvin-Voigt viscoelastic model obtained when _{}* = *0, so that

*M(w) = M _{R} *(1 + iw
τ

In the Kelvin-Voigt
viscoelastic model, the stress *a *in the medium for a given strain *s *and
strain rate *s *is the superposition of linear elastic and linear viscous
stresses

The real part of the complex
modulus *M(w) *in equation (3.5) may be written as

_{} (3.8)

and the attenuation *Q ^{-1}
*is given by

_{} (3.9)

which measures the lag of the
strain behind the stress. We assume here τ_{3 }> τ_{4}.
Substituting them into the equations for attenuation and dispersion given earlier,
we may obtain the attenuation

_{} (3.10)

and the phase velocity

_{} (3.11)

From (3.10) and (3.11) we can
draw a very important conclusion about this model. Since τ_{3 }> τ_{4
}the paranthesis on right side of expression
(3.11) will always be less than one. That means that the wave number k always
will be less than one. Because of that we will never get a causal solution of
the wave equation using this model.

We proceed with Wang introducing
c_{0} as the phase velocity for w* *-> 0, and we have

*c _{0}= *

Attempting to use a single relaxation time, we may define two parameters as follows:

_{} and _{} (3.13)

For the standard linear solid
model, considering equation (3.6), this can be written as *Q _{c}*

_{} (3.14)

that is, the normalized difference between the unrelaxed and relaxed moduli. For the standard linear solid
model, the attenuation at the peak of the attenuation function with respect to
frequency is 1/2∆M and the (angular) frequency at this location is
l/τ_{c}.

The parameters τ_{3}* *and τ_{4}* *in
(3.10) and (3.11) may be expressed as (Casula and Carcione, 1992):

and

_{} (3.15)

And

_{} (3.16)

obtained from the *Q _{c}
*and τ

_{} (3.17)

** **

To compare this standard linear solid model with the Kolsky model, may first set

_{} (3.18)

since the highest frequency of the seismic band has the strongest attenuation.

We then use the following two approximations:

_{}

Right side of equation 3.6 can be written:

_{} (3.19)

and

_{} (3.20)