In general, c(ω), M(ω) and Q(ω) are all functions of frequency. All the elastic moduli are complex and each wave type has its own Qand velocity.
2.2
Frequency dependence of attenuation
In a perfectly elastic homogeneous body, the elastic wave velocities are independent of frequency. In an anelastic body, the velocities are dispersive, they depend on frequency. A number of processes contribute to seismic wave attenuation in a crys- talline material. These processes all involve a high-frequency, or unrelaxed, modulus and a low-frequency, or relaxed modulus. Relaxed modulus is a low value of Young’s modulus measured in the slow-strain-rate regime. In this regime, the load or stress is applied slowly over time to the material, and total strain is a sum of the elastic strain and anelastic strain. Elastic strains are an instantaneous response to the ap- plied high stress, while anelastic strains are additional strains in the material caused by the time-dependent migration of atom lattices to favoured sites because of the prolonged load. Unrelaxed modulus, on the other hand, is a high value of Young’s modulus measured in the high-strain-rate regime. In this regime, the high stress is applied rapidly, over short amount of time, and the total strain is exactly equal to the elastic strain only. Anelastic strain in this case is equal to zero due to insufficient time for the rise of anelastic effects.
At high frequencies, the defects in the material, which are characterised by a time constant, do not have time to contribute and the body behaves as a perfectly elastic body. Attenuation is low and Q is high in the high-frequency limit. At very low frequencies the defects have enough time to respond to the applied stress and they contribute an additional strain. Because the stress cycle time is long compared to the defects response time, the stress and strain are in phase and attenuation is again low (Q is high). Because of the additional relaxed strain, however, the modulus is low and the relaxed velocity is low. When the frequency of the oscillatory stress is comparable to the characteristic time of the defects, attenuation reaches its maxi- mum and velocity changes rapidly with frequency.
These are the characteristics of the standard linear solid, which is analogous to a spring and a dashpot arranged in a parallel circuit, which is then attached to another spring. In high frequencies, only the second, or series, spring responds to the applied force and it is this spring’s constant that is the effective modulus and controls the total extension. At low frequencies the spring and dashpot in parallel both extend with a time constant τ characteristic of the dashpot, the total extension is larger
and the effective modulus is lower. This is also described as a viscoelastic solid. The
Q−1 of such a system is given by
Q−1(ω) = k2
k1 ωτ
1 + (ωτ)2, (2.2.1)
where k2 and k1 are spring constants of the series and parallel spring, respectively,
and τ is the relaxation time, τ = η/k2, where η is viscosity. Attenuation will be
maximum when ωτ = 1,
Q−1(ω) = 2Qmin−1 ωτ
1 + (ωτ)2, (2.2.2)
and Q−1(ω) → 0 as ωτ when ω → 0, and as (ωτ)−1 when ω → ∞. The resulting
absorption peak is a characteristic bell-shaped curve centered at ω = (τ)−1, and
this is common for a medium with a single characteristic frequency. However, solids in general are not characterised by a single relaxation time and a single absorption peak. A distribution of relaxation times broadens the peak and gives rise to an ab- sorption band. Qis only weakly dependent on frequency in such a band. SeismicQ
values are nearly constant with frequency over much of the seismic band. A possible explanation for a relatively constant Q can be that seismic waves of different fre- quencies traveling through Earth’s interior feel the net effect of the absorption bands with different relaxation times thus producing a flat absorption spectrum (Figures 2.2.1 2.2.2). The absorption spectrum (or absorption band) width is governed by the intrinsic attenuation specified by the relaxation time. In case of scattering, the absorption spectrum has a peak at a frequency of wavelength comparable to the size of scatterers. The presence of a flat absorption spectrum is a sign that the distribution of scatterers is homogeneous over a particular scale range. Far from the corners of the absorption band Q=Qmin and the attenuation is constant and equal to Qmin−1. Relaxation mechanisms hence lead to an internal peak of the following
form: Q−1(ω) = ∆ Z ∞ −∞ D(τ) ωτ 1 +ω2τ2 dτ, (2.2.3)
where ω is the applied frequency, τ is a characteristic time,D(τ) is the retardation spectrum and ∆ is the modulus defect, the relative difference between the high- frequency, unrelaxed shear modulus, and the low-frequency, relaxed modulus. The modulus defect is also a measure of total reduction in modulus obtained in going from low temperature to high temperature.
§2.2 Frequency dependence of attenuation 9
body waves to study the inner core. Their dominant frequency is between 0.5 and 2.0 Hz, which is well within the frequency band of nearly constantQ(Figure 2.2.2), and so we will assume a frequency-independent Q model in our inversions.
Figure 2.2.1: Relaxation spectrum for a polycrystalline material showing attenuation peaks at different frequencies due to different microscopic mechanisms (Stein and Wyses- sion, 2002)
Figure 2.2.2: Schematic model explaining the observation that Q is roughly constant
over a wide range of frequencies. The superposition of absorption peaks for different compositions at different temperatures and pressures yields a flat absorption band (Stein and Wysession, 2002)