## Abstract

Because of its simple form, a bandlimited, four-parameter anelastic model that yields nearly constant midband *Q* for low-loss materials is often used for calculating synthetic seismograms. The four parameters used in the literature to characterize anelastic behavior are τ_{1}, τ_{2}, *Q _{m}*, and

*M*in the relaxation-function approach (

_{R}*s*

_{1}= 1/τ

_{1}and

*s*

_{2}= 1/τ

_{2}are angular frequencies defining the bandwidth,

*M*is the relaxed modulus, and

_{R}*Q*is approximately the midband quality factor when

_{m}*Q*≫ 1); or

_{m}τ

_{1},

τ

_{2},

*Q*

_{m}, and

*M*in the creep-function approach (

_{R}*s*

_{1}= 1/

τ

_{1}and

*s*

_{2}= 1/

τ

_{2}are angular frequencies defining the bandwidth, and

*Q*

_{m}is approximately the midband quality factor when

*Q*

_{m}≫ 1). In practice, it is often the case that, for a particular medium, the quality factor

*Q*(ω

_{0}) and phase velocity

*c*(ω

_{0}) at an angular frequency ω

_{0}(

*s*

_{1}< ω

_{0}<

*s*

_{2};

*s*

_{1}< ω

_{0}<

*s*

_{2}) are known from field measurements. If values are assigned to τ

_{1}and τ

_{2}(τ

_{2}< τ

_{1}), or to

τ

_{1}and

τ

_{2}(

τ

_{2}<

τ

_{1}), then the two remaining parameters,

*Q*and

_{m}*M*, or

_{R}*Q*

_{m}and

*M*, can be obtained from

_{R}*Q*(ω

_{0}). However, for highly attenuative media, e.g.,

*Q*(ω

_{0}) ≦ 5,

*Q*(ω) can become highly skewed and negative at low frequencies (for the relaxation-function approach) or at high frequencies (for the creep-function approach) if this procedure is followed. A negative

*Q*(ω) is unacceptable because it implies an increase in energy for waves propagating in a homogeneous and attenuative medium. This article shows that given (τ

_{1}, τ

_{2}, ω

_{0}) or (

τ

_{1},

τ

_{2}, ω

_{0}), a lower limit of

*Q*(ω

_{0}) exists for a bandlimited, four-parameter anelastic model. In the relaxation-function approach, the minimum permissible

*Q*(ω

_{0}) is given by ln [(1 + ω

^{2}

_{0}τ

^{2}

_{1})/(1 + ω

^{2}

_{0}τ

^{2}

_{2})]/{2 arctan [ω

_{0}(τ

_{1}− τ

_{2})/(1 + ω

^{2}

_{0}τ

_{1}τ

_{2})]}. In the creep-function approach, the minimum permissible

*Q*(ω

_{0}) is given by {2 ln (

τ

_{1}/

τ

_{2}) − ln [(1 + ω

^{2}

_{0}

τ

^{2}

_{1})/(1 + ω

^{2}

_{0}

τ

^{2}

_{2})]}/{2 arctan [ω

_{0}(

τ

_{1}− τ

_{2})/(1 + ω

^{2}

_{0}

τ

_{1}

τ

_{2})]}. The more general statement that, for a given set of relaxation mechanisms, a lower limit exists for

*Q*(ω

_{0}) is also shown to hold. Because a nearly constant midband

*Q*cannot be achieved for highly attenuative media using a four-parameter anelastic model, a bandlimited, six-parameter anelastic model that yields a nearly constant midband

*Q*for such media is devised; an expression for the minimum permissible

*Q*(ω

_{0}) is given. Six-parameter anelastic models with quality factors

*Q*∼ 5 and

*Q*∼ 16, constant to 6% over the frequency range 0.5 to 200 Hz, illustrate this result. In conformity with field observations that

*Q*(ω) for near-surface earth materials is approximately constant over a wide frequency range, the bandlimited, six-parameter anelastic models are suitable for modeling wave propagation in highly attenuative media for bandlimited time functions in engineering and exploration seismology.

You do not currently have access to this article.