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 τ1, τ2, Qm, and MR in the relaxation-function approach (s1 = 1/τ1 and s2 = 1/τ2 are angular frequencies defining the bandwidth, MR is the relaxed modulus, and Qm is approximately the midband quality factor when Qm ≫ 1); or τ1, τ2, Qm, and MR in the creep-function approach (s1 = 1/τ1 and s2 = 1/τ2 are angular frequencies defining the bandwidth, and Qm is approximately the midband quality factor when Qm ≫ 1). In practice, it is often the case that, for a particular medium, the quality factor Q(ω0) and phase velocity c(ω0) at an angular frequency ω0 (s1 < ω0 < s2; s1 < ω0 < s2) are known from field measurements. If values are assigned to τ1 and τ2 (τ2 < τ1), or to τ1 and τ2 (τ2 < τ1), then the two remaining parameters, Qm and MR, or Qm and MR, can be obtained from Q(ω0). However, for highly attenuative media, e.g., Q(ω0) ≦ 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 (τ1, τ2, ω0) or (τ1, τ2, ω0), a lower limit of Q(ω0) exists for a bandlimited, four-parameter anelastic model. In the relaxation-function approach, the minimum permissible Q(ω0) is given by ln [(1 + ω20τ21)/(1 + ω20τ22)]/{2 arctan [ω0(τ1 − τ2)/(1 + ω20τ1τ2)]}. In the creep-function approach, the minimum permissible Q(ω0) is given by {2 ln (τ1/τ2) − ln [(1 + ω20τ21)/(1 + ω20τ22)]}/{2 arctan [ω0(τ1 − τ2)/(1 + ω20τ1τ2)]}. The more general statement that, for a given set of relaxation mechanisms, a lower limit exists for Q(ω0) 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(ω0) 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.
New general decay result for a system of two singular nonlocal viscoelastic equations with general source terms and a wide class of relaxation functions
