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.     

The traveltime perturbations for seismic body waves in factorized anisotropic inhomogeneous media

The traveltime perturbation equations for the quasi-compressional and the two quasi-shear waves propagating in a factorized anisotropic inhomogeneous (FAI) media are derived. The concept of FAI media simplifies considerably these equations. In the FAI medium, the density normalized elastic parameters a _ijkl ( X _i ) can be described by the relation a _ijkl ( X _i) = f ²( x _i ) A _ijkl, where A _ijkl are constants, independent of coordinates x _i and f ²( x _i) is a continuous smooth function of x _i . The types of anisotropy ( A _ijkl ) and inhomogeneity [ f ( x _i)] are not restricted. The traveltime perturbations of individual seismic body waves ( q ^P , qS 1 and qS 2) propagating in the FAI medium depend, of course, both on the structural pertubations [δ f ²( x _i)] and on the anisotropy perturbations (δ A _ijkl ), but both these effects are fully separated. The perturbation equations for the time delay between the two qS -waves propagating in the FAI medium are simplified even more. If the unperturbed (background) medium is isotropic, the perturbation of the time delay does not depend on the structural perturbations (δ f ²( x _i) at all. This striking result, valid of course only in the framework of first-order perturbation theory, will simplify considerably the interpretation of the time delay between the two split qS -waves in inhomogeneous anisotropic media. Numerical examples are presented.     

Three-dimensional VP and VP /VS models of the upper crust in the Friuli area (northeastern Italy)

3-D images of P velocity and P - to S -velocity ratio have been produced for the upper crust of the Friuli area (northeastern Italy) using local earthquake tomography. The data consist of 2565 P and 930 S arrival times of high quality. The best-fitting V _P and V _P / V _S 1-D models were computed before the 3-D inversion. V _P was measured on two rock samples representative of the investigated upper layers of the Friuli crust. The tomographic V _P model was used for modelling the gravity anomalies, by converting the velocity values into densities along three vertical cross-sections. The computed gravity anomalies were optimized with respect to the observed gravity anomalies. The crust investigated is characterized by sharp lateral and deep V _P and V _P / V _S anomalies that are associated with the complex geological structure. High V _P / V _S values are associated with highly fractured zones related to the main faulting pattern. The relocated seismicity is generally associated with sharp variations in the V _P / V _S anomalies. The V _P images show a high-velocity body below 6 km depth in the central part of the Friuli area, marked also by strong V _P / V _S heterogeneities, and this is interpreted as a tectonic wedge. Comparison with the distribution of earthquakes supports the hypothesis that the tectonic wedge controls most of the seismicity and can be considered to be the main seismogenic zone in the Friuli area.     

A uniformly valid asymptotic representation of normal mode multiplet spectra on a laterally heterogeneous Earth

Summary. A new asymptotic formula is obtained for the spectrum of an isolated normal mode multiplet _nS_l or _nT_l , with n ≪ l , on a laterally heterogeneous Earth. The principal feature of this formula is that it is uniformly valid on the Earth's surface, including near the epicentre and its antipode. The formal conditions for its validity are that | δm / m ₀|≪ 1 and s _max≪ l ≪ s _min| δm / m ₀|^–1, where | δm / m ₀| is the relative magnitude of the lateral heterogeneity, and s _min and s _max are the minimum and maximum significant degrees in its spherical harmonic expansion. As well as providing a basis for the geographical interpretation of near-epicentral or near-antipodal long-period recordings, the new formula also unifies the asymptotic theory and adds insight into the phenomena which govern the details of multiplet spectra in general.     

Observation of Coriolis coupled modes below 1 mHz

We present observations of spectral energy at toroidal mode frequencies in vertical seismic recordings of the 1998 Balleny Islands earthquake. Since toroidal modes on a spherically symmetric, nonrotating Earth have horizontally polarized particle motion these observations call for an explanation. We first rule out local and instrumental effects as being responsible for the verticalcomponent signal of the toroidal modes ₀ T ₃ (0.59 mHz) and ₀ T ₄ (0.77 mHz). The global effects that we consider are general heterogeneous mantle structure, ellipticity of figure and rotation. We find that rotation through Coriolis coupling of loworder spheroidal and toroidal oscillations is the dominant mechanism.     

Surface motion of a fluid planet induced by impacts

In order to approximate the free-surface motion of an Earth-sized planet subjected to a giant impact, we have described the excitation of body and surface waves in a spherical compressible fluid planet without gravity or intrinsic material attenuation for a buried explosion source. Using the mode summation method, we obtained an analytical solution for the surface motion of the fluid planet in terms of an infinite series involving the products of spherical Bessel functions and Legendre polynomials. We established a closed form expression for the mode summation excitation coefficient for a spherical buried explosion source, and then calculated the surface motion for different spherical explosion source radii a (for cases of   a / R = 0.001  to 0.035, R is the radius of the Earth) We also studied the effect of placing the explosion source at different radii r ₀ (for cases of   r ₀/ R = 0.90  to 0.96) from the centre of the planet. The amplitude of the quasi-surface waves depends substantially on a / R , and slightly on   r ₀/ R   . For example, in our base-line case,   a / R = 0.03, r ₀/ R = 0.96  , the free-surface velocity above the source is 0.26 c , whereas antipodal to the source, the peak free surface velocity is 0.19 c . Here c is the acoustic velocity of the fluid planet. These results can then be applied to studies of atmosphere erosion via blow-off caused by asteroid impacts.     

Absolute S-velocity estimation from receiver functions

We present a novel method to recover absolute S velocities from receiver functions.
For a homogeneous half-space the S velocity can be calculated from the horizontal slowness and the angle of surface particle motion for an incident P wave. Generally, the calculated S velocity is an apparent half-space value which depends on model inhomogeneity and P -waveform. We therefore, suggest to calculate such apparent half-space S velocities from low-pass filtered (smoothed) receiver functions using a suite of filter-parameters, T . The use of receiver functions neutralize the influence of the P -waveform, and the successive low-pass filterings emphasize the variation of S velocity with depth.
We apply this   V _{S ,app.}( T )  technique to teleseismic data from three stations: FUR, BFO and SUM, situated on thick sediments, bedrock and the Greenland ice cap, respectively. The observed   V _{S ,app.}( T )  curves indicate the absolute S velocities from the near surface to the uppermost mantle beneath each station, clearly revealing the different geological environments. Application of linearized, iterative inversion quantify these observations into   V _S ( z )  models, practically independent of the S -velocity starting model. The obtained models show high consistency with independent geoscientific results. These cases provide also a general validation of the   V _{S ,app.}( T )  method.
We propose the computation of   V _{S ,app.}( T )  curves for individual three-component broad-band stations, both for direct indication of the S velocities and for inverse modelling.     

On high-frequency spheroidal modes and the structure of the upper mantle

Summary. The asymptotic properties of spheroidal mode dispersion at high frequency for fixed phase velocity are related to the intercept times τ_β( p ) for P and S waves. If the mode eigenfrequency and the ratio of horizontal to vertical displacement at the surface for the mode are known τ_α( p ) and τ_β( p ) may be separately estimated. If discontinuities exist in the velocity model then 'solotone' effects occur, in frequency at fixed slowness, and in τ_α( p ), τ_β( p ) estimated from the mode dispersion as a function of slowness. The coupling of P and S waves in the spheroidal modes means that the interaction of P waves with upper-mantle discontinuities affects also the estimates of the S wave τ_β( p ) values for which the corresponding turning points lie in the lower mantle. The asymptotic formalism also shows that sharp pulses formed by superposition of spheroidal modes correspond to multiple PS reflections.
A study of τ_α( p ), τ_β( p ) estimates derived from spheroidal modes with periods from 45–50s, calculated for model 1066B, shows that even in the presence of strong upper-mantle discontinuities the errors in intercept time are only about one-tenth of a period. The asymptotic properties may there-for provide a useful means of estimating intercept times from modes with a few seconds period as a supplement to travel-time methods.     

Some comments on the descriptions of the polarization states of waves

Summary. Procedures are developed for specifying the polarization characteristics of n -dimensional waves, and in particular three-dimensional waves of geophysical interest. We show that when a wave is in a pure state or is totally polarized, all the polarization information can be represented by a single vector u in an n -dimensional unitary space. Simple measures of the degree of polarization of the wave are constructed from the characteristic equation of the spectral matrix S . These measures are functions only of the scalar invariants of S and consequently S need not be diagonalized. If S represents a purely polarized wave, the unitary vector which contains the polarization information about the wave can be obtained directly from S using any 2 n – 1 equations of n ² possible equations. By multiplying by a phase-factor this unitary vector can be written in the form u = r ₁+ i r ₂ where r ₁ and r ₂ are orthogonal vectors in a real space. For an elliptically polarized wave, r ₁ and r ₂ locate the major and minor axes of the ellipse, and the ellipticity is given by the ratio of their magnitudes. The polarization parameters of ULF magnetic waves at the Earth's surface are computed from one set of five equations ( n = 3) and compared with parameters calculated using established techniques.     

The tidal dynamics of the Irish and Celtic Seas

Summary. Current meter data collected over periods of more than 14 day from the Irish and Celtic Seas are harmonically analysed and presented in maps of tidal stream information. Making use of the analysed current data, and by constructing time series of frictional and inertial stresses which are also harmonically analysed, harmonic constituents of the surface tidal slopes at current meter stations are obtained. Using these with data collected from offshore tide gauges, and in conjunction with coastal tide data, cotidal maps are drawn with some confidence for M ₂, S ₂, O ₁ and K ₁, the M ₂ chart resolving the discrepancy which exists between the different charts of the Celtic Sea already produced. Cotidal maps for M ₃ and M ₄ are also presented.
The mean over a tidal cycle of the energy flux for M ₂, S ₂ and O ₁ is also presented in the form of the total energy flux in these constituents which crosses different sectional lines. A flux of 44 × 10⁶ kW is observed to enter the Celtic Sea from the Atlantic and this is compared with previous estimates. An energy budget is also performed for M ₂, including all the effects of astronomical forcing and Earth tides to enable comparison to be made between the true energy inflow and the estimated frictional dissipation. Finally, comparison is made between the mean of the instantaneous energy flux and the sum of the energy fluxes associated with the major harmonics.     

On the nature of regional seismic phases–II. On the influence of structural barriers

The blockage of the L _g wave by crustal barriers such as continental margins and graben structures has long been recognized as providing a very useful tool for mapping large-scale lateral crustal variations along the propagation path. Numerical simulation of L _g -wave propagation in complex anelastic media using the pseudospectral method provides insight into the nature of the propagation process using both snapshots of the wavefield and synthetic seismograms. A variety of 2-D structures have been investigated, including the influence of sediments, crustal thickness and attenuation.
Thick sedimentary basins covering a graben structure can have a major influence, since they remove L _g energy by generating P conversion and scattering–the principal mechanisms for strong L _g attenuation across a graben. The reduction of the L _g energy is reinforced by anelastic attenuation in the sediments as well as the influence of the gradually thinning crustal waveguide associated with an elevated Moho.
The extinction of L _g in a sequence of explosions fired across the central graben of the North Sea can be simulated by numerical calculations for the structure derived from refraction experiments.     

Focal mechanisms of small earthquakes in the southeastern Brazilian shield: a test of stress models of the South American plate

Focal mechanisms of small earthquakes with magnitudes of about 3 in the SE Brazilian shield are calculated using S / P amplitude ratios. Low attenuation ( Q _p from 400 to 800) in the shield upper-crustal layers allowed sharp S arrivals to be recorded up to distances of 100 km. Besides P -wave polarities, SH -wave first motions were also used to constrain the nodal-plane orientations. Normal and reverse faulting mechanisms with strike-slip components were found. The inversion of four mechanisms to estimate the stress tensor indicated a strike-slip stress regime with roughly E–W-orientated σ ₁ and N–S σ ₃. Both the orientations and the shape factor ( φ =0.7) of the inverted stress are in excellent agreement with theoretical predictions for that part of Brazil from the driving-force model of Coblentz & Richardson (1996) . Good agreement with the nature of the stress, as well as its orientation, was also found for the model of Meijer (1995) . Both of these theoretical models include spreading stresses along the continent/ocean lithospheric transition. Because the earthquakes are more than 300 km from the continental shelf they should not be affected by the local flexural forces caused by sediment load in the marginal basins. The agreement between observed and theoretical stresses then confirms the importance of continental spreading forces in modelling intraplate stresses.     

Approaches to Modelling the Surface Albedo of a High Arctic Glacier

Broadband surface albedo measurements, made during the summer melt season at three weather stations on John Evans Glacier (79°40 ' N, 74°00 ' W), varied strongly with the solar zenith angle ( θ _z ). Tests were carried out to assess the impact of diurnal variations in surface albedo on seasonal net shortwave radiation ( K ^* ) totals. Removing the diurnal signal from albedo measurements by daily averaging of hourly measurements, or by applying midday measurements to all hours of the day, changed K ^* by up to 16%. Ignoring measurements made at θ _z & 75°, to account for measurement (cosine) error at high θ _z , decreased K ^* by between 5 and 18%. Given the sensitivity of K ^* to diurnal patterns in surface albedo, experiments were carried out with two albedo models. One model accounted for albedo variations with θ _z and one did not. The model driven by θ _z , when implemented within a surface energy balance model for John Evans Glacier, produced better melt estimates. This suggests that diurnal variations in surface albedo should be accounted for in energy balance models of glacier melt.     

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.     

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.     

Linearity spectrum analysis of multi-component magnetizations and its application to some igneous rocks from south-eastern Australia

Summary. A method that enables the objective resolution of almost parallel multi-component magnetizations is described and demonstrated. A feature distinguishing this method from others is its simultaneous analysis of demagnetization data from a group of specimens, rather than the analysis of data from one specimen at a time. The only prerequisite is that the specimens are derived from a homogeneous source. Thus for a formation carrying a simple single component magnetization, all specimens from the formation may be simultaneously reduced. For a more complicated two component magnetization it is shown that only specimens from a particular site can be considered homogeneous, and for a complex three component system each sample often requires undivided attention. Thus the workload is proportionally increased to achieve analyses of comparable reliability from data of variable quality.
New pole positions from Mesozoic intrusions of the Sydney Basin, NSW are: from the Marsden Park Breccia pipe 48°S, 127°E ( A ₉₅= 6°); the St Marys Breccia pipe 46°S, 150°E ( A ₉₅= 8°); the Prospect Dolerite 60°S, 142°E ( A ₉₅= 13°) and 53°S, 180°E ( A ₉₅= 6°); and from the Dundas Breccia pipe 58°S, 162°E ( A ₉₅= 36°) and 31°S, 195°E ( A ₉₅= 16°). The last two formations possess multi-component magnetizations. These pole positions are consistent with previous results from south-eastern Australia.     

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.     

Rapid computation of magnetic anomalies with demagnetization included, for arbitrarily shaped magnetic bodies

Summary. The potential function ø for a magnetic body of susceptibility μ in a medium of susceptibility μ* satisfies the integral equation
Here Φ* is the potential function for the region without the heterogeneity and R is the distance from the point of observation to the point on the surface, s , of the body. δΦ /δn is the normal derivative, in the direction of the outward normal. The equation allows for the effects of demagnetization. For numerical purposes the surfaces can be divided into N facets over which δΦ/δ n is a constant. The unknown quantities δΦ/δn_j can be found from the system of equations defined by:
The prime on the summation sign denotes that the summation does not include the i th element. The magnetic field in the direction of the unit vector P( P ₁, P ₂, P₃ ) is given by:      

Estimation of the Bingham distribution function on nearly two-dimensional data sets

Summary. In cases where directional data, such as palaeomagnetic directions, lie nearly along a great circle, a good approximation to the maximum likelihood estimate of the intermediate concentration parameter k ₂ in the Bingham probability distribution is given by: 2( t ₂/ N ) – 1 = I ₁(1/2 k ₂)/ I ₀(1/2 k ₂), where t ₂ is the intermediate eigenvalue, N is the number of samples, and the I_i are the appropriate modified Bessel functions of the first kind. This estimate, the asymptotic limit as the smallest eigenvalue t ₁→ 0, corresponds to restricting all points to lie on a great circle. The limit is also useful as an endpoint for interpolation, especially since numerical calculation in this region is difficult.