The solution of the inverse gravity potential problem for cylindrical bodies

Summary. The inverse gravity potential problem consists in the determination of the form and the density of the body by its exterior gravity potential. We describe two similar classes of bodies for which this problem has a unique constructive solution.
(1) The first class contains the cylindrical bodies with finite length, arbitrary form of section and ρ( R , ø, z) =ρ₁( z )ρ₂( R , ø) density distribution, where z is the cylindrical coordinate; R , ø are the polar coordinates in a section plane. This class is important for prospecting geophysics in that it allows us to determine in a unique and constructive way, the function ρ₁( R , ø), the length, form and orientation of the cylinder if we know the function ρ₁( z ) and the exterior potential. The classical moment problem of functions is the basis for the solution of this problem.
(2) The analogous problem for the class of the spherical cylinders, or bodies bounded by arbitrary similar sections of two different concentric spheres and the radial lateral surface, appears when bodies of planetary size are studied. (An example of these bodies would be the Moon mascons.) The density distribution of these cylinders is ρ(τ, θ, ø) =ρ₁(τ)ρ₂(θ, ø) where τ, θ, ø are the spherical coordinates. The function ρ₁(θ, ø), length and form of spherical sections can be uniquely determined by exterior potential if we know the function ρ₁(τ). We propose a new constructive method for harmonic continuation of the gravity potential into the region containing the perturbing masses for the solution of the problem.     

Determination of earthquake early warning parameters, τc and Pd, for southern California

We explore a practical approach to earthquake early warning in southern California by determining a ground-motion period parameter  τ _c   and a high-pass filtered displacement amplitude parameter Pd from the initial 3 s of the P waveforms recorded at the Southern California Seismic Network stations for earthquakes with M > 4.0. At a given site, we estimate the magnitude of an event from  τ _c   and the peak ground-motion velocity ( PGV ) from Pd . The incoming three-component signals are recursively converted to ground acceleration, velocity and displacement. The displacements are recursively filtered with a one-way Butterworth high-pass filter with a cut-off frequency of 0.075 Hz, and a P -wave trigger is constantly monitored. When a trigger occurs,  τ _c   and Pd are computed. We found the relationship between  τ _c   and magnitude ( M ) for southern California, and between Pd and PGV for both southern California and Taiwan. These two relationships can be used to detect the occurrence of a major earthquake and provide onsite warning in the area around the station where onset of strong ground motion is expected within seconds after the arrival of the P wave. When the station density is high, the methods can be applied to multistation data to increase the robustness of onsite early warning and to add the regional warning approach. In an ideal situation, such warnings would be available within 10 s of the origin time of a large earthquake whose subsequent ground motion may last for tens of seconds.     

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.     

Coupled normal-mode sensitivity to inner-core shear velocity and attenuation

We have studied the response of normal modes to perturbations in inner-core shear velocity and attenuation, using fully coupled mode synthetics. Our results indicate that (i) mode pairs   _n S _l – _{n ±1} S _l   are strongly coupled by anelasticity, (ii) this coupling causes shear velocity perturbations to strongly affect the Q values of modes through exchange of inner-core characteristics, (iii) there is no evidence for a weakly attenuating inner core in shear, and (iv) the discrepancy between attenuation models returned from normal modes and body waves is small. These results suggest that inversions for inner-core attenuation and shear velocity should be performed simultaneously and should take account of the strong cross-coupling due to attenuation.     

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.     

Regional variations of heat flow differences with depth in Alberta, Canada

Summary. Differences between estimated average heat flow values for the Mesozoic and Cenozoic formations ( Q ₁) and estimated average heat flow values for the Palaeozoic formations below the erosional unconformity ( Q ₂) are calculated for the Alberta part of the western Canadian sedimentary basin. Significant heat flow differences exist for these two intervals and the map of Δ Q = Q ₁– Q ₂ shows that Q ₂ is generally greater than Q ₁ in the western and south-western part of Alberta, while in the northern part of the province Q ₂ is generally less than Q ₁. The regional variations of Δ Q are large, with standard deviation of 26 mW m⁻² and average value –13.5 mW m⁻². A regional trend of Δ Q correlates with topographic relief and the hydraulic head variations in the basin. It is shown that there is a heat flow increase with depth in water recharge areas and a decrease in heat flow with depth in the low topographic elevation water discharge areas when comparing the average heat flow in Mesozoic + Cenozoic and Palaeozoic formations.     

A note on the surface volume change of shallow earthquakes

Summary. This note presents an exact analytical formula for determining the magnitude of coseismic surface volume change (δ V ) of earthquake faults in a half-space. For a Poisson solid, the formula is remarkably simple; δ V = M _zz |8μ, where M _zz is one of the moment tensor elements of the source. Maximum δ V values derive from dip slip on faults plunging 45°. For these events, surface volume changes of 0.0001 and 4.3 km³ can be expected for magnitude 5 and 8 earthquakes respectively. All of the coseismic surface volume change is recovered in the interseismic period through relaxation of the Earth and rebound of the surface. A useful rule of thumb for estimating the magnitude of vertical rebound in 45° dip slip events is δ h _p=Δ s /24, where Δ s is the coseismic slip on the fault.     

Focal mechanisms of recent earthquakes in the Southern Korean Peninsula

We evaluate the stress field in and around the southern Korean Peninsula with focal mechanism solutions, using the data collected from 71 earthquakes ( M_L = 1.9–5.2) between 1999 and 2004. For this, the hypocentres were relocated and well-constrained fault plane solutions were obtained from the data set of 1270 clear P -wave polarities and 46 SH / P amplitude ratios. The focal mechanism solutions indicate that the prevailing faulting types in South Korea are strike-slip-dominant-oblique-slip faultings with minor reverse-slip component. The maximum principal stresses (σ₁) estimated from fault-slip inversion analysis of the focal mechanism solutions show a similar orientation with E–W trend (269°–275°) and low-angle plunge (10°–25°) for all tectonic provinces in South Korea, consistent with the E–W trending maximum horizontal stress (σ_Hmax) of the Amurian microplate reported from in situ stress measurements and earthquake focal mechanisms. The directions of the intermediate (σ₂) and minimum (σ₃) principal stresses of the Gyeongsang Basin are, however, about 90 deg off from those of the other tectonic provinces on a common σ₂–σ₃ plane, suggesting a permutation of σ₂ and σ₃. Our results incorporated with those from the kinematic studies of the Quaternary faults imply that NNW- to NE-striking faults (dextral strike-slip or oblique-slip with a reverse-slip component) are highly likely to generate earthquakes in South Korea.     

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.     

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.     

Dynamic behaviour of a phase boundary under non-uniform surface loads

Summary. A method is outlined to determine the dynamic behaviour of a phase boundary in the Earth when non-uniform time-varying pressure and temperature conditions are assumed at the Earth's surface. An integral equation describing the phase boundary motion is derived and it is solved under a linearizing assumption. The solution is obtained in the form of a double integral transform. Short and long time-expansions of the solution can be obtained from series expansion and integration of the Laplace transform along a branch cut. The method is illustrated by considering a stepwise change in surface pressure conditions.
For short times, the solution exhibits the same type of time dependence (i.e. the first-order term is in t ^1/2) as the one obtained in the one-dimensional case (i.e. uniform pressure perturbation at the Earth's surface).
For long times, it is shown that the time dependence of the phase boundary motion is almost identical to the one derived for the one- dimensional case if the wavenumber k _L of the surface excitation is such that κ k ²_Lτ≤ 1 (where τ is the relaxation time associated with the one-dimensional phase boundary motion and κ is the thermal diffusivity). If κ k ²_Lτ > 1, then the relaxation time for the phase boundary motion in two dimensions is of the order of κ⁻¹ k ⁻²_L.
When considering parameters that would be appropriate for a basalt to eclogite phase transition at Moho depth, the latter situation is met only when the load wavelength is smaller than 35 km.     

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.     

Further evidence for oceanic excitation of polar motion

While the role of the atmosphere in driving variations in polar motion is well established, the importance of the oceans has been recognized only recently. Further evidence for the role of the oceans in the excitation of polar motion is presented. To estimate the equatorial excitation functions, χ ₁ and χ ₂ , for the ocean, we use velocity and mass fields from a constant-density ocean model, driven by observed surface wind stresses and atmospheric pressure, for the period 1993–1995; comparison with similar functions derived from a more complex density-stratified ocean model indicates the effectiveness of the simple constant-density modelling approach. Corresponding atmospheric excitation functions are computed from NCEP/NCAR re-analyses. Results indicate significant improvements in the agreement with the observed polar motion excitation when the simulated oceanic effects are added to atmospheric excitation. Correlations between the polar motion and the geophysical signals at periods of 15–150 days increase from 0.53 to 0.80 and from 0.75 to 0.88 for χ ₁ and χ ₂ , respectively. The oceanic signals are particularly important for seasonal variations in χ ₁ (correlation increases from 0.28 to 0.85 when oceanic excitation is included). A positive impact of the oceans on more rapid polar motion is also observed, up to periods as short as 5 days. The sensitivity of the results to different forcing fields and different amounts of friction in the oceans is also discussed.     

The influence of water on the stress supported by experimentally faulted Westerly granite

Summary. Fault zones in wet Westerly granite deformed at temperatures of 300° and 400°C require markedly lower shear stresses for sliding than when dry, even when the effective confining pressure is held constant between the wet and dry tests, provided that the strain rate is lower than 10⁻⁷s⁻¹. The rate of strength reduction is enhanced by increasing the pore water pressure. The deformation rate is a power function of the applied stress where the stress exponent is approximately 7 for pore water pressure of 100 MPa and 21 for pore water pressure of 20 MPa.
The experimental results are extrapolated to conditions believed to occur at depths of 10 km along the San Andreas Fault Zone. It is suggested that for slow tectonic deformation at strain rates of 10⁻¹¹ and 10⁻¹⁴s⁻¹ the shear stress for sliding on faults in granite would be approximately 60 and 20 MPa, respectively, at pore water pressures equal to the hydrostatic head. Fluid overpressures of c. 0.8 lithostatic pressure are required to lower the shear stress for sliding into the 10 MPa range at the slower strain rate.     

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.     

The Influence of Surface Inhomogeneities On Deep Electromagnetic Soundings of the Earth

Summary. Asymptotic expressions for components of the electromagnetic field of a grounded electric dipole are considered for the model consisting of a thin surface-layer overlapping a stratified medium with a highly resistive screen on the roof. It is shown that the method of spatial derivatives makes it possible to obtain proper estimates of the impedance at distances of r ≥|λ₀| from the nearest edge of the surface anomaly (|λ₀| being the effective depth of the field penetration in the underlying section). the magnetotelluric methods allow one to obtain the true values of impedance, provided r ≥ max {|λ₀|, |/( S ⁻¹+ Z ⁻₀|^1/2} where S is the integrated conductivity of the surface layer, is the transverse resistance of the screen, and Z ⁻₀ is the Tikhonov—Cagniard impedance for the medium underlying the surface layer.     

Core–mantle boundary deformations and J2 variations resulting from the 2004 Sumatra earthquake

The deformation at the core–mantle boundary produced by the 2004 Sumatra earthquake is investigated by means of a semi-analytic theoretical model of global coseismic and postseismic deformation, predicting a millimetric coseismic perturbation over a large portion of the core–mantle boundary. Spectral features of such deformations are analysed and discussed. The time-dependent postseismic evolution of the elliptical part of the gravity field ( J ₂) is also computed for different asthenosphere viscosity models. Our results show that, for asthenospheric viscosities smaller than 10¹⁸ Pa s, the postseismic J ₂ variation in the next years is expected to leave a detectable signal in geodetic observations.     

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.     

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.     

The influence of barometric-pressure variations on gravity

Summary. The superconducting gravimeter has been used to measure the influence of barometric pressure on gravity in the frequency range 0.1–10 cycles/day. These measurements show that the incoherent barometric fluctuations are the major cause of random fluctuations in local gravity and account for much of the 'noise' on our gravimeter records. A simple model has been constructed which adequately explains the response of gravity to the local pressure fluctuations. These measurements also show a response to the global atmospheric tides at S ₁ and S ₂ which is much larger than the response to local fluctuations. Although this behaviour is most likely due to the response of the world-wide oceans to the atmospheric tides, no theoretical model has yet been constructed.