# Bulletin of the Seismological Society of America

- Copyright © 1995, by the Seismological Society of America

## 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 τ̅

_{m}_{1}, τ̅

_{2}, Q̅

_{m}, and

*M*in the creep-function approach (s̅

_{R}_{1}= 1/τ̅

_{1}and s̅

_{2}= 1/τ̅

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

_{m}is approximately the midband quality factor when Q̅

_{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}; s̅

_{1}< ω

_{0}< s̅

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

_{1}and τ

_{2}(τ

_{2}< τ

_{1}), or to τ̅

_{1}and τ̅

_{2}(τ̅

_{2}< τ̅

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

*Q*and

_{m}*M*, or Q̅

_{R}_{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 (τ̅

_{1}, τ̅

_{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 (τ̅

_{1}/τ̅

_{2}) − ln [(1 + ω

^{2}

_{0}τ̅

^{2}

_{1})/(1 + ω

^{2}

_{0}τ̅

^{2}

_{2})]}/{2 arctan [ω

_{0}(τ̅

_{1}− τ

_{2})/(1 + ω

^{2}

_{0}τ̅

_{1}τ̅

_{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.