Transient and impulse responses of a one-dimensional linearly attenuating medium—II. A parametric study

Summary. We investigate one-dimensional waves in a standard linear solid for geophysically relevant ranges of the parameters. The critical parameters are shown to be T*= t_u/Q_m where t _u is the travel time and Q_m the quality factor in the absorption band, and τ⁻¹ _m , the high-frequency cut-off of the relaxation spectrum. The visual onset time, rise time, peak time, and peak amplitude are studied as functions of T* and τ _m. For very small τ _m , this model is shown to be very similar to previously proposed attenuation models. As τ _m grows past a critical value which depends on T* , the character of the attenuated pulse changes. Seismological implications of this model may be inferred by comparing body wave travel times with a'one second'earth model derived from long-period observations and corrected for attenuation effects assuming a frequency independent Q over the seismic band. From such a comparison we speculate that there may be a gap in the relaxation spectrum of the Earth's mantle for relaxation times shorter than about one second. However, observational constraints from the attenuation of body waves suggest that such a gap might in fact occur at higher frequencies. Such a hypothesis would imply a frequency dependence of Q in the Earth's mantle for short-period body waves.     

t*— an unsuitable parameter to characterize anelastic attenuation in the Eastern Carpathians

Investigation of teleseismic P -wave recordings at a temporary network in the Eastern Carpathians, equipped with predominantly short-period sensors, is compared with synthetic modelling of anelastic attenuation of teleseismic waves in the upper mantle. Using the t * approach, we examine variations of amplitude decrease over frequency for teleseismic recordings in the frequency band 0.5–1.5 Hz. The results reveal a consistent pattern of increased t * values in the centre of the network, in the Vrancea region at the bend of the Carpathian Arc, although the magnitude of the observed variation in t * is much higher than expected. Synthetic t * parameter computations for the same event-receiver configurations reproduce the observed pattern in terms of relative variations. However, the amplitude of the synthetic t * values explains only 10–20 per cent of the observed variation in t *. t * is not a direct measure for anelastic attenuation but rather for a combination of anelastic and other attenuating effects such as scattering and amplitude fluctuation related to velocity inhomogeneities. If regional amplitude variations are solely attributed to anelastic attenuation, all other effects are mapped into Q . We discuss the role of anelastic attenuation and other effects in the case of the Eastern Carpathians and conclude that t * is an unsuitable parameter to characterize anelastic attenuation in the Eastern Carpathians.     

Methodologies for estimating t*(f) from short-period body waves and regional variations of t*(f) in the United States

Summary. In this paper we discuss some aspects of estimating t * from short-period body waves and present some limits on t* (f) models for the central and south-western United States (CUS and SWUS). We find that for short-period data, with frequencies above 1 or 2 Hz, while the average spectral shape is stable, the smaller details of the spectra are not; thus, only an average t *, and not a frequency-dependent t *, can be derived from such information. Also, amplitudes are extremely variable for short-period data, and thus a great deal of data from many stations and azimuths must be used when amplitudes are included in attenuation studies.
The predictions of three pairs of models for t* (f) in the central and south-western United States are compared with time domain observations of amplitudes and waveforms and frequency domain observations of spectral slopes to put bounds on the attenuation under the different parts of the country. A model with the t * values of the CUS and SWUS converging at low frequencies and differing slightly at high frequencies matches the spectral domain characteristics, but not the time domain amplitudes and waveforms of short-period body waves. A model with t * curves converging at low frequencies, but diverging strongly at high frequencies, matches the time domain observations, but not the spectral shapes. A model with nearly-parallel t* (f) curves for the central and south-western United States satisfies both the time and frequency domain observations.
We conclude that use of both time and frequency domain information is essential in determining t* (f) models. For the central and south-western United States, a model with nearly-parallel t* (f) curves, where Δ t *∼ 0.2 s, satisfies both kinds of data in the 0.3–2 Hz frequency range.     

Lateral variation and frequency dependence of coda-Q in the southern part of Iberia

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A method based on the coda attenuation law: Q = Q ₀( f/f ₀)^v leads to the determination of the lateral variation of coda- Q in the southern part of the Iberian Peninsula using seismograms belonging to the seismological network of the Cartuja Observatory, located in Granada. The lateral variation of Q ₀ ( Q value corresponding to a reference frequency f ₀ of 1 Hz) and its frequency dependence for the 1 to 5 Hz frequency range are, in general, in agreement with coda- Q values for frequencies less than about 1 Hz, previously determined in the region under study.
To determine the coda- Q values analytical functions have been used to fit the magnification curves of the vertical component short-period seismographs belonging to the Cartuja network. The problem is solved by using least-squares techniques and non-linear inversion. The determined coda- Q ₀ values and its frequency dependence correlate well with several known geophysical parameters in the southern part of the Iberian Peninsula.     

Anelasticity of the Crust and Upper Mantle of South America From the Inversion of Observed Surface Wave Attenuation

Fundamental-mode Rayleigh and Love waves generated by several earthquakes situated along great-circle paths between pairs of seismograph stations have been analysed to obtain coefficients of attenuation, group velocities, phase velocities, and specific quality factors in the period range 18–80s in two regions of the South American continent. One set of paths crosses the shield region which lies on the eastern coast and another set traverses the mountainous region inland. the average attenuation coefficient values are clearly higher in the tectonically active western region throughout the entire period range than in the eastern or shield region.
Inversion of the attenuation data yielded shear wave internal friction ( Q ^-1_β) models as a function of depth in the crust and upper mantle in both regions. A low- Q zone below the lithosphere is prominent in both regions. the results show that substantial variations of Q _β occur in the two regions of South America. the Q_β values were found to be inversely related to the heat flow values or to the temperature.     

Anisotropy of heterogeneity scale lengths in the lower mantle from PKIKP precursors

The anisotropy of heterogeneity scale lengths in the lower mantle is investigated by modelling its effect on the high-frequency precursors of PKIKP scattered by the heterogeneities. Although models having either an isotropic or an anisotropic distribution of scale lengths can fit the observed coda shapes of short-period precursors, the frequency content of broad-band PKIKP precursors favours a dominantly isotropic distribution of scale lengths. Precursor coda shapes are consistent with 1 per cent fluctuations in P velocity in the wavenumber band 0.05–0.5  km⁻¹ extending to 1000  km above the core–mantle boundary, and with a D" region open to circulation throughout the lower mantle. The level of excitation of PKIKP precursors observed in the frequency band 0.02–2  Hz requires a power spectrum of heterogeneity that is nearly white or slowly increasing with wavenumber. Anisotropy of scale lengths may exist in a D" layer having larger horizontal than vertical scale lengths and produce little or no detectable effects on PKIKP precursors for P -velocity perturbations as high as 3 per cent when averaged over a vertical scale of several kilometres, and much higher when averaged over scales of hundreds of metres or less.     

Evidence of spatial variations of attenuation in the western Pyrenean range

Summary. Spectral attenuation of coda waves has been studied in the range 2–40 Hz from local events recorded in the western Pyrenean range from 1980 to 1982. Q _c was obtained using a single scattering model of S -waves for different segments of the coda. An increase of Q _c with lapse time was found and attributed to a rapid increase of Q _β with depth.
Three groups of events were selected from distinct focal areas. Two data sets are mainly composed of aftershocks of moderate earthquakes of magnitude 5.1 and 4.8, respectively. No moderate earthquake occurred in the third area in the few years preceding or following the selected events. Use of stations close to epicentres allowed sampling of the coda at very short lapse times and then study of small, distinct scattering volumes. Noticeable differences were found between the three studied areas and attributed to spatial rather than temporal variations.
The Q _c frequency dependence was studied according to Q _c= qf ^α. α is found to range from 0.7 to 1.1 and q from 30 to 140. These values are in agreement with those found in other tectonic areas. It is shown that scattering is the dominant attenuation process below 10Hz.     

Rayleigh-wave amplitudes from earthquakes in the range 0–150°

Summary . Vertical component Rayleigh-wave amplitudes from 1461 shallow earthquakes recorded in the distance range 0–150° are analysed to separate the effects of earthquake size, epicentral distance (Δ) and recording station.
The estimated decay of amplitude with distance has the form of a theoretical curve for the decay of Rayleigh waves with distance if the assumption is made that the decay due to dispersion for the data analysed is that of an Airy phase. Writing the decay due to anelastic attenuation as exp (- k Δ), k is estimated to be 0.676/rad over the whole range of distance. If the distance effects are represented by a straight line of the form h log Δ+ constant, h is estimated to be 1.15. The calibration function for computing M _s derived from the estimated distance effects is very similar to that of Marshall & Basham.
Station effects on Rayleigh-wave amplitudes though statistically significant are small, and can probably be ignored in the computation of M _s.
Comparing the estimated surface-wave magnitudes (earthquake size) obtained in this study with the long and short period body-wave magnitudes ( m ^LP_b and m ^SP_b respectively) obtained by Booth, Marshall & Young for the same earthquake shows that m ^LP_b is about equal to M _s over the magnitude range of interest (˜4.0–7.0). The m ^LP_b and M_s relationship shows that the greater the long-period energy radiated by an earthquake the smaller proportionately is the short-period energy.     

Frequency dependence of the quality factor in the upper crust: a deep seismic sounding approach

Summary. Ray amplitudes are computed in a one-dimensional velocity structure where the quality factor Q varies continuously with depth. An iterative process is then proposed to derive the quality factor distribution in the upper crust from deep sounding data. Results for compressional waves in the French Massif Central and for a signal frequency close to 20 Hz show that Q increases in a rather linear way from about 40 in surface up to 600 at 7 km depth. Q seems to be higher in the Central Alps: 180 in surface, 1600 at 5 km depth (at 20 Hz). Using proper signal processing, the frequency dependence of Q is finally investigated in the frequency range 10–25 Hz. The results indicate a dependence of the form Q = Q ₀ f ^α, where α= 0.25 ± 0.1.     

Magnitude corrections for attenuation in the upper mantle

Summary. The m _b: M _s relation for explosions at the Nevada Test Site (NTS) differs from those for explosions in other parts of the world. There is considerable evidence that this results mostly from high body-wave attenuation in the upper mantle beneath the western US. The authors have developed an empirical magnitude correction for body-wave attenuation and applied it to both source and receiver ends of the teleseismic body-wave path. The results imply that m _b values are lower for NTS explosions than for Soviet explosions of comparable yield and seismic coupling. The authors have also developed and applied a source-depth correction to account for _pP-P interference in the P -wave arrival.
The body-wave magnitude resulting from these corrections is designated m_o to distinguish it from other definitions of m _b. Values of m_Q determined for a world-wide set of large explosions show that a single m_Q : yield relation is a fair fit to the data for the explosions with high seismic coupling. However, grouping the explosions under two m_Q :yield relations gives a better fit to the data.
All the studied explosions in salt or granite or below the water table fit a common M _s:yield relation. Explosions from North America, Eurasia and Africa have a common m_Q : M _s relation.     

Comments on the Chandler wobble Q

Summary. The Chandler wobble Q _w, as obtained from the astronomical data cannot be equated with the Q _m of the source of damping, as an examination of Chandler wobble energetics reveals. We find that if dissipation occurs in the mantle then Q _w≃ 9 Q _m, implying that either the mantle Q is frequency dependent or the wobble Q is much larger than 100. If the dissipation is in the oceans then Q _w≃ 20 Q _o, and the pole tide must be far from equilibrium.     

Average attenuation of 0.7–5.0 Hz Lg waves and magnitude scale determination for the region bounding the western branch of the East African Rift

The investigation of L g attenuation characteristics in the region bounding the western branch of the East African rift system using digital recordings from a seismic network located along the rift between Lake Rukwa and Lake Malawi is reported. A set of 24 recordings of L g waves from 12 regional earthquakes has been used for the determination of anelastic attenuation, Q _Lg , and regional body-wave magnitude, m _{b Lg} , scale. The events used have body-wave magnitudes, m _b , between 4.6 and 5.5, which have been determined teleseismically and listed in ISC bulletins. The data were time-domain displacement amplitudes measured at 10 different frequencies (0.7–5.0  Hz). Q _Lg and its frequency dependence, η , in the region can be represented in the form Q _Lg = (186.2 ± 6.5)  f  ^(0.78±0.05). This model is in agreement with models established in other active tectonic regions. The L g -wave-based magnitude formula for the region is given by m _{b Lg} = log   A + (3.76 ± 0.38)  log   D − (5.72 ± 1.06), where A is a half-peak-to-peak maximum amplitude of the 1  s L g wave amplitude in microns and D is the epicentral distance in kilometres. Magnitude results for the 12 regional earthquakes tested are in good agreement with the ISC body-wave magnitude scale.     

Stacking for the frequencies and Qs of 0S0 and 1S0

Summary. Using nine IDA records for the Indonesian earthquake of 1977 August 19, we have formed an optimal linear combination of the records and have measured the frequency and Q of ₀ S ₀ and ₁ S ₀. The frequency was measured using the moment ratio method. The attenuation was measured by the minimum width method and by the time-lapse method. The frequency and attenuation were measured simultaneously by varying them to obtain a best fit to the data. A 2000-hr stack, the sum of nine individual records, for ₀ S ₀ gave a frequency of 0.814664 mHz±4 ppm. The values for the Q of ₀ S ₀ for the three different methods of measurement were 5600,5833 and 5700, respectively. The error in the estimates of Q ^-1 is about 5 per cent for the minimum power method. For ₁ S ₀ a 300-hr stack yielded a frequency of 1.63151 mHz±30 ppm. The values of Q for this mode were 1960, 1800 and 1850, respectively, with an error in Q ^-1 of about 12 per cent for the minimum power method.     

Earthquake intensity attenuation pattern in the United States

Summary. Attenuation of earthquake intensities with epicentral distance was studied by analysing the intensity data for 39 earthquakes in the United States. Attenuation of MM intensity ( I ) with distance (Δ km) follows a simple relation of the type log I = log I ₀ - m Δ, where I ₀ is the intensity at the epicentre and m is a constant. Slope m is found to be inversely proportional to the square of the focal depth. Intensity attenuation pattern in the United States in general can be represented by a unified relation I/I ₀= exp [-(0.8999/ h ²+ 0.0014)Δ] where 16km ≤ h ≤ 60km. Intensities were calculated with the help of this equation and a good agreement with the observed intensities were found. A comparative study has also been made between the attenuation relations applicable to India and the United States.     

Pn anisotropy studies in northern Britain and the eastern and western United States

Summary. Available seismic refraction data from three different continental areas, northern Britain and the eastern and western United States, has been studied for possible P_n , velocity anisotropy using the methods described by Bamford. There are various deficiencies in the time—distance data used in each case but, while the uppermost mantle beneath northern Britain and the eastern United States seems to be isotropic within the limits of measurement error, there is a small but significant anisotropy beneath the western United States.
Both the amount (up to 3 per cent) and the direction (70–80° east of north) of this anisotropy are very similar to the results obtained in the Pacific Ocean off California. We tentatively conclude that this anisotropy is present as a consequence of the subduction of oceanic lithosphere beneath the western United States.     

Crust and upper mantle structure of the central Iberian Meseta (Spain)

Summary. Quarry blasts recorded along three lines on the central Iberian Meseta are used in an attempt to interpret the crustal structure. The results of the interpretation of the data, together with published surface wave and earthquake data, suggest a layered structure of the crust having the following features: the basement, in some areas covered by up to 4 km of sediments, has a P -velocity of 6.1 km s⁻¹; a low-velocity layer, between 7 and 11 km depth, seems to exist on the basis of both P and S interpretation of seismic data; a thick middle crust of 12 km has a P -velocity of 6.4 km s⁻¹ and overlies a lower crust with a mean P -velocity of 6.9 km s⁻¹ and a possible slight negative gradient; the mean v _p/ v _s ratio for the crust is about 1.75; the Moho is reached at about 31 km depth and consists of a transition zone at least 1.5 km thick. The P -velocity of the upper mantle is close to 8.1 km s⁻¹ and the S -velocity about 4.5 km s⁻¹, which gives a v _p /v _s ratio of 1.8 for the uppermost mantle. A tentative petrological interpretation of the velocities and composition of the layers is given.     

Seismic amplitude tomography for crustal attenuation beneath China

Amplitude tomography reconstructs seismic attenuation directly from recorded wave amplitudes. We have applied the tomography to amplitude data reported in the 'Annual Bulletin of Chinese Earthquakes' and interpreted the regionally varying crustal attenuation in terms of tectonics. The seismic amplitudes were originally recorded for determining the M _L and M _S magnitudes. They generally correspond to the maximum amplitudes of the horizontal components of the short-period S waves and intermediate-period Rayleigh waves. Both sets of measurements are sensitive to crustal structure. The peak amplitudes from M _L amplitudes spread spherically with significant dispersion and scattering. M _S amplitudes show cylindrical spreading with little dispersion. Average crustal Q values for attenuation at 1 Hz are 737 and 505 for M _L and M _S, respectively, with substantial regional variations. Frequency dependence in the attenuation is also indicated. Regions with the lowest attenuation (high Q values) are beneath the south China Block, Sichuan Basin, Ordos Platform, the Daxinganling and the Korea Craton. These tend to be tectonically inactive regions, which are generally dominated by intrusive and cratonic rocks in the upper crust. Regions with the highest attenuation (low Q values) are beneath Bohai Basin, Yunnan, eastern Songpan-Ganzi Terrain, margins of the Ordos platform and the Qilian Shan. These are predominantly active basins, grabens and fold belts. The continental margin also highly attenuates both S and surface waves.     

A three-dimensional radially anisotropic model of shear velocity in the whole mantle

We present a 3-D radially anisotropic S velocity model of the whole mantle (SAW642AN), obtained using a large three component surface and body waveform data set and an iterative inversion for structure and source parameters based on Non-linear Asymptotic Coupling Theory (NACT). The model is parametrized in level 4 spherical splines, which have a spacing of ∼ 8°. The model shows a link between mantle flow and anisotropy in a variety of depth ranges. In the uppermost mantle, we confirm observations of regions with   V_SH > V_SV   starting at ∼80 km under oceanic regions and ∼200 km under stable continental lithosphere, suggesting horizontal flow beneath the lithosphere. We also observe a   V_SV > V_SH   signature at ∼150–300 km depth beneath major ridge systems with amplitude correlated with spreading rate for fast-spreading segments. In the transition zone (400–700 km depth), regions of subducted slab material are associated with   V_SV > V_SH   , while the ridge signal decreases. While the mid-mantle has lower amplitude anisotropy (<1 per cent), we also confirm the observation of radially symmetric   V_SH > V_SV   in the lowermost 300 km, which appears to be a robust conclusion, despite an error in our previous paper which has been corrected here. The 3-D deviations from this signature are associated with the large-scale low-velocity superplumes under the central Pacific and Africa, suggesting that   V_SH > V_SV   is generated in the predominant horizontal flow of a mechanical boundary layer, with a change in signature related to transition to upwelling at the superplumes.     

The attenuation mechanism of seismic waves in northwestern Himalayas

We analysed local earthquake waveforms recorded on a broad-band seismic network in northwestern Himalayas to compute the intrinsic and scattered attenuation parameters from coda waves. Similar to other tectonically active and heterogeneous regions, attenuation-frequency relation for western Himalaya is   Q ⁻¹ _c = (113 ± 7)  f ^(1.01±0.05)  where   Q_c   is the coda Q parameter. Intrinsic  ( Q ⁻¹ _i )  and scattering  ( Q ⁻¹ _s )  attenuations was separated using   Q_c   and direct S -wave Q data  ( Q_d )  . It is observed that estimated   Q ⁻¹ _c   is close to   Q ⁻¹ _i   and both of them are much larger than   Q ⁻¹ _s   suggesting that coda decay is predominantly caused by intrinsic attenuation. At higher frequencies, both the attenuation parameters   Q_c   and,   Q_d   are similar indicating that coda is predominantly composed of back-scattered S waves at these frequencies.     

A new derivation of the ratio Q(wobble)/Qμ (mantle)

Summary An extension of the Love-Larmor theory to a low-loss unelastic earth model, leads to the surprisingly simple approximation
   
where τ_s= 447.4 sidereal day is the static wobble period, τ_R= 306 sidereal day is the rigid-earth wobble period and τ_w= 433 sidereal day is the observed Chandler period. Q _W, Q _μ are the respective average Q values of the wobble and the Earth's mantle at τ_W. The known numerical factor F is only slightly dependent on the Earth structure.