Measurement and implications of frequency dependence of attenuation

Vedran Lekic, Jan Matas, Mark Panning (Princeton University), and Barbara Romanowicz

Introduction

As they propagate through the Earth, seismic waves experience energy loss, which is summarized by $q = - \Delta E / 2\pi E_{max}$, where $\Delta E$ is the internal energy lost by a seismic wave in one cycle. Attenuation is high and nearly constant within a certain frequency band and rapidly falls off with frequency away from this band. The frequency dependence of $q$ can be described using a power law, $q \propto \omega^{\alpha}$, with a model-dependent $\alpha$ that is usually thought to be smaller than 0.5 within the absorption band.

Though seismological efforts at constraining globally-averaged $\alpha$ within the absorption band have benefited from numerous measurements of surface wave or normal mode attenuation (see http://mahi.ucsd.edu/Gabi/rem.html), the determination of $\alpha$ has been confounded by the fact that oscillations at different frequencies can have very different depth sensitivities to elastic and anelastic properties of the earth. As a result of this tradeoff between frequency and depth effects, radial variations of attenuation can obscure the $\alpha$ signal. The only studies attempting to obtain $\alpha$ within the absorption band have found $\alpha$ ranging from 0.1 to 0.3 while emphasizing the lack of resolution on the deduced values. More recent studies have relied upon analysis of body waves to argue for values of $\alpha$ in the 0.1-0.4 range (see Romanowicz and Mitchell, 2007). However, these studies were restricted to frequencies higher than 40mHz and were of regional extent, leaving unanswered the question of the average mantle $\alpha$.

In light of these difficulties, seismic studies routinely assume that, within the seismic band, $\alpha$ cannot be resolved and thus implicitly rely on the frequency-independent attenuation model of Kanamori and Anderson (1977). We re-examine the model's applicability to the mantle using a new method based on the standard analysis of Backus and Gilbert (1970) that allows us to separate the effects of the radial $q$ profile from those due to frequency dependence of $q$ as described by $\alpha$.

Method

We can relate a mode attenuation measurement $q$ to material properties within the Earth via sensitivity (Fréchet) kernels $K_\mu$ and $K_\kappa$:

$$q = \frac{2}{\omega}\int_{0}^R dr \ \kappa_{0} q_{\kappa} K_{\kappa} + \mu_{0} q_{\mu} K_{\mu},$$ (2.11)

where $R$ is the radius of the Earth, $\kappa_0$ and $\mu_0$ are the reference radial profiles of bulk and shear moduli, and $q_\kappa$ and $q_\mu$ are values of radial bulk and shear attenuation.

The sensitivity kernels of fundamental modes with similar frequencies are very similar, implying that the $q$ datasets are highly redundant. We seek to exploit this redundancy and divide modes into a low and high frequency bin, denoted by superscript $l$ and $h$, respectively. Each linear combination of Fréchet kernels of modes in each bin defines a new ``hyperkernel'':

$$\mathbf{H_{\mu,\kappa}^\mathrm{low}} = \sum_{l=1}^{N_l} \gam... ...igh}} = \sum_{h=1}^{N_h} \gamma^h \mathbf{K_{\mu,\kappa}}^h\,,$$ (2.12)

where $N_l$ and $N_h$ are the number of modes in each bin, and the subscripts $\mu,\kappa$ denote that the kernels refer to either shear or bulk attenuation.

Each particular choice of $\gamma^l$ and $\gamma^h$ will yield hyperkernels with different depth sensitivities. Therefore, by requiring that $\gamma^l$ and $\gamma^h$ yield hyperkernels with identical sensitivities to the radial attenuation profile, it is possible to remove the trade-off between depth and frequency dependence of attenuation measurements. Since we focus on the effective $\alpha$ in the mantle, we require the hyperkernels to be zero outside the mantle while providing maximally uniform sensitivity in the mantle. In order to eliminate the contribution from poorly-constrained mantle bulk attenuation, we seek hyperkernels that are insensitive to $q_\kappa$.

To each hyperkernel corresponds a $q$ value, which is a weighted average of the $q$ measurements of its constituent normal modes:

$$q^\mathrm{low} = \sum_{l=1}^{N_l} \gamma^l q^l \ \ \ \mathrm{and} \ \ \ q^\mathrm{high} = \sum_{l=1}^{N_h} \gamma^h q^h.$$ (2.13)

Since the two hyperkernels have identical sensitivity to radial attenuation structure but differing frequency content, differences in $q^\mathrm{low}$ and $q^\mathrm{high}$ can be attributed to frequency dependence of attenuation. These effects of frequency dependence can be accounted for by projecting the individual mode $q$'s to a reference value $q_0$ using:

$$q_{0_i} = q_i \left( \frac{\omega_i}{\omega_0} \right) ^\alpha.$$ (2.14)

In the absence of systematic measurement error, $q^\mathrm{low}$ and $q^\mathrm{high}$ will be reconciled at the reference frequency for the value of $\alpha$ that corresponds to the effective $\alpha$ of the mantle.

Figure 2.46: Preferred model of frequency dependence of attenuation within the absorption band. $\alpha$ is approximately 0.3 at periods shorter than 200 s, decreasing to 0.1 in the period range 300-800s, and becoming negative (-0.4) at periods longer than 1000s.

Results

After validating our method on a synthetic dataset, we apply it to existing attenuation measurements of free oscillations and surface waves spanning the period range 3200s-50s. We observe that effective $\alpha$ is likely to be frequency dependent. Specifically, $\alpha$ is negative at periods longer than 1000s and positive and increasing at shorter periods (see Figure 2.46). This conclusion runs against both the assumption of frequency-independent attenuation often used in seismology, and the constant, positive $\alpha$ model suggested by laboratory studies (Jackson et al., 2005). A frequency-dependent effective $\alpha$ is nevertheless physically plausible. This is because the effective $\alpha$ that one would obtain by analyzing the normal-mode and surface-wave attenuation measurements is the result of an interplay of the actual $\alpha$ associated with the material at a given depth and the position of the absorption band. This interplay can give rise to a negative effective $\alpha$ as long as the actual $\alpha$ is negative at long periods at some depths. Furthermore, our preferred model of frequency dependence of attenuation is consistent with earlier studies that have relied upon body waves and have focused on higher frequencies (see Figure 2.46).

A non-zero value of $\alpha$ carries important implications for the construction of radial profiles of attenuation. Efforts at determining the radial profile of attenuation in the Earth have routinely assumed that attenuation is frequency independent. The resulting models have, therefore, mapped the signal of frequency-dependence of $q$ into its depth profile.

Relating lateral variations of attenuation in terms of temperature requires knowing $\alpha$, since when when $\alpha$ is zero, $\delta q/\delta T$ is a constant, whereas when $\alpha$ is positive, $\delta q/\delta T$ is exponentially dependent on temperature (Minster and Anderson, 1981). Recent studies of lateral attenuation variations rely on data with periods shorter than $\sim$ 300s (e.g. Gung and Romanowicz, 2004), at which periods our preferred model suggests that $\alpha \sim$ 0.3. This value implies an exponential temperature dependence of attenuation, and justifies the interpretation of lateral attenuation variations in terms of temperature variations.

Intrinsic attenuation causes dispersion of seismic velocities, which must be corrected when datasets with different frequency content are used to simultaneously constrain Earth structure. Both the values of $\alpha$ and of q significantly affect the magnitude of the dispersion correction. For an $\alpha$ value of 0.3, the assumption of frequency-independent attenuation will result in 25% error for a frequency ratio of 10 and a 50% error for a frequency ratio of 100.

Finally, the precise knowledge of seismic velocity, its dispersion and associated attenuation is important for meaningful comparisons with other geophysical observables, such as the geoid. Future work should thus be aimed at improving the precision of $q$ measurements and the development of radial $q$ profiles that properly account for the frequency dependence of $q$.

Acknowledgements

This project was supported by NSF grants EAR-0336951 and EAR-0738284, NSF fellowship held by VL, and by the French CNRS-SEDIT program.

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