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1.
Summary. We reduce the problem of constructing a smooth, 1-D, monotoni-cally increasing velocity profile consistent with discrete, inexact τ ( p ) and X( p ) data to a quadratic programming problem with linear inequality constraints. For a finite-dimensional realization of the problem it is possible to find a smooth velocity profile consistent with the data whenever such a profile exists. We introduce an unusual functional measure of roughness equivalent to the second central moment or 'Variance' of the derivative of depth with respect to velocity for smooth profiles, and we prove that its minimal value is unique. In our experience, solutions minimizing this functional are very smooth in the sense of the two-norm of the second derivative and can be constructed inexpensively by solving one quadratic programming problem. Still smoother models (in more traditional measures) may be generated iteratively with additional quadratic programs. All the resulting models satisfy the τ ( p ) and X( p ) data and reproduce travel-time data remarkably well, although sometimes τ ( p ) data alone are insufficient to ensure arrivals at large X; then an X( p ) datum must be included.  相似文献   

2.
Summary. The asymptotic properties of spheroidal mode dispersion at high frequency for fixed phase velocity are related to the intercept times τβ( 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 τα( p ) and τβ( p ) may be separately estimated. If discontinuities exist in the velocity model then 'solotone' effects occur, in frequency at fixed slowness, and in τα( p ), τβ( 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 τβ( 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 τα( p ), τβ( 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.  相似文献   

3.
Summary An extension of the Love-Larmor theory to a low-loss unelastic earth model, leads to the surprisingly simple approximation
   
where τs= 447.4 sidereal day is the static wobble period, τR= 306 sidereal day is the rigid-earth wobble period and τw= 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 τW. The known numerical factor F is only slightly dependent on the Earth structure.  相似文献   

4.
Summary. The slowness-distance ( p, x ) plane is an alternative to the slowness-time intercept ( p, τ ) plane as the intermediate image space in inversion of seismic refraction data. The production of a ( p, x ) image from travel time-distance ( T, x ) data has been presented elsewhere so emphasis here is on ( p, x ) to velocity-depth ( v, z ) transformation. Iterative downward continuation of a ( p, x ) image converges to the correct ( v, z ) image in a manner similar to that in the widely used ( p, τ ) to ( v, z ) process. Application to a real refraction data set from the Imperial Valley of southern California gives a similar ( v, z ) solution via both ( p, x ) and ( p, τ ) images.  相似文献   

5.
Summary. The transformation of a set of seismograms to the delay time-slowness, τ—p, domain is presented as a sequence of Fourier and Bessel transforms, For a horizontally layered medium, this sequence gives an exact cylindrical wave decomposition of the response to a point source; correctly compensating for the phase shifting and geometrical spreading associated with transmission through the Earth. The resultant τ—p map or 'slant stack' contains true amplitude and phase information. The spatial aliasing properties of the transformation, when applied to a dataset, are greatly improved by the use of only outgoing waves in the Bessel transform. This is equivalent to using Hankel functions rather than Bessel functions, and is justified by the absence of incoming waves from most datasets. The WKBJ approximation to the medium response enables predictions to be made about the shape and amplitude variation with slowness of truncation effects. Theoretically the τ—p transformation is reversible, thus the τ—p domain is a suitable one in which to perform filtering operations before seismogram reconstruction.  相似文献   

6.
This is the second paper of a series of two concerning strong ground motion in SW Iberia due to earthquakes originating from the adjacent Atlantic area. The aim of this paper is to use the velocity model that was proposed and validated in the companion paper for seismic intensity modelling of the 1969 ( M s= 8.0) and 1755 ( M = 8.5–8.7) earthquakes.
First, we propose a regression to convert simulated values of Peak Ground Velocity (PGV) into Modified Mercalli Intensity (MMI) in SW Iberia, and using this regression, we build synthetic isoseismal maps for a large ( M s= 8.0) earthquake that occurred in 1969. Based on information on the seismic source provided by various authors, we show that the velocity model effectively reproduces macroseismic observations in the whole region. We also confirm that seismic intensity distribution is very sensitive to a small number of source parameters: rupture directivity, fault strike and fault dimensions. Then, we extrapolate the method to the case of the great ( M = 8.5–8.7) 1755 earthquake, for a series of hypotheses recently proposed by three authors about the location of the epicentral region. The model involving a subduction-related rupture in the Gulf of Cádiz results in excessive ground motion in northern Morocco, suggesting that the source of the 1755 earthquake should be located further west. A rupture along the western coast of Portugal, compatible with an activation of the passive western Iberian margin, would imply a relatively low average slip, which, alone, would could not account for the large tsunami observed in the whole northern Atlantic ocean. A seismic source located below the Gorringe Bank seems the most likely since it is more efficient in reproducing the distribution of high intensities in SW Iberia due to the 1755 earthquake.  相似文献   

7.
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 2Lτ≤ 1 (where τ is the relaxation time associated with the one-dimensional phase boundary motion and κ is the thermal diffusivity). If κ k 2Lτ > 1, then the relaxation time for the phase boundary motion in two dimensions is of the order of κ−1 k −2L.
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.  相似文献   

8.
We describe a method of expressing azimuthally anisotropic surface wave velocities on the Earth using a local and smooth spherical-spline parametrization. Anisotropy in the Earth leads to azimuthally varying Love and Rayleigh wave velocities that can be expressed as (cos 2ζ, sin 2ζ) and (cos 4ζ, sin 4ζ) perturbations to the isotropic velocities, where ζ is the direction of surface-wave propagation. The strength of the perturbations varies laterally, and a current goal of seismic tomography is the detailed global mapping of these variations. Several parametrizations have previously been used to describe azimuthally varying velocities. The representation proposed here uses spherical splines and is designed to describe smooth variations in both the strength and geometry of azimuthal anisotropy. The method builds on a simple geometrical approximation for the local azimuth of propagation expressed at the defining spline knot points. It avoids the singularities at the poles that result when azimuthal variations are parametrized using traditional scalar spherical harmonics. Compared with a generalized spherical-harmonic expansion of the tensor fields that represent 2ζ and 4ζ azimuthal variations smoothly on a sphere, the new method offers the advantages of local geographical support and simplicity of implementation.  相似文献   

9.
Summary. This note reports on the remarkable focusing of seismic body waves at or near the antipode (Δ= 180°) of an earthquake's epicentre. The particular seismic velocity structure and sphericity of the Earth cause body-wave phases such as P (diff), PKP, PP, PPP, PcPPKP, SKSSKS, SS , etc. to converge individually at antipodal distances after being diffracted, reflected or refracted at discontinuities. This focusing strongly amplifies each signal up to almost one order of magnitude with respect to the normal phase recorded two or more degrees away. Since the signal/noise ratio is enhanced in the same proportion, seismograms at antipodal distances provide clear and strong arrivals of otherwise weak phases. Antipodal monitoring of seismic waves is suggested as a powerful means of exploring the Earth's interior. The study of these 'seismic images' generated at focal points of seismic rays will yield information on the departures from lateral homogeneity and sphericity of the core, as well as stronger constraints on earth models. To interpret the observations correctly, the data must be compared with theoretically generated seismograms. Since the appropriate ray theory equations (see, e.g. Scholte; Gilbert & Helmberger; Richards) are singular at Δ=180°, a corrective measure is taken which provides a formal expression for the wave amplitude that remains finite at the antipode, and reproduces the usual expressions at other distances.  相似文献   

10.
A general tomographic technique is designed in order (i) to operate in anisotropic media; (ii) to account for the uneven seismic sampling and (iii) to handle massive data sets in a reasonable computing time. One modus operandi to compute a 3-D body wave velocity model relies on surface wave phase velocity measurements. An intermediate step, shared by other approaches, consists in translating, for each period of a given mode branch, the phase velocities integrated along ray paths into local velocity perturbations. To this end, we develop a method, which accounts for the azimuthal anisotropy in its comprehensive form. The weakly non-linear forward problem allows to use a conjugate gradient optimization. The Earth's surface is regularly discretized and the partial derivatives are assigned to the individual grid points. Possible lack of lateral resolution, due to the inescapable uneven ray path coverage, is taken into account through the a priori covariances on parameters with laterally variable correlation lengths. This method allows to efficiently separate the 2ψ and the 4ψ anisotropic effects from the isotropic perturbations. Fundamental mode and overtone phase velocity maps, derived with real Rayleigh wave data sets, are presented and compared with previous maps. The isotropic models concur well with the results of Trampert & Woodhouse. Large 4ψ heterogeneities are located in the tectonically active regions and over the continental lithospheres such as North America, Antarctica or Australia. At various periods, a significant 4ψ signature is correlated with the Hawaii hotspot track. Finally, concurring with the conclusions of Trampert & Woodhouse, our phase velocity maps show that Rayleigh wave data sets do need both 2ψ and 4ψ anisotropic terms.  相似文献   

11.
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.  相似文献   

12.
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*= tu/Qm where t u is the travel time and Qm 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.  相似文献   

13.
Properties of iron at the Earth's core conditions   总被引：2，自引：0，他引：2  
Summary. The phase diagram of iron up to 330 GPa is solved using the experimental data of static high pressure (up to 11 GPa) and the experimental data of shock wave data (up to 250 GPa). A solution for the highest triple point is found ( P = 280 GPa and T = 5760 K) by imposing the thermodynamic constraints of triple points. This pressure of the triple point is less than the pressure of the inner core–outer core boundary of the Earth. These results indicate that the density of iron at the inner core–outer core boundary pressure is close to 13 g cm−3, which lies close to the seismic solutions of the Earth at that pressure. It is thus concluded that the Earth's inner core is very likely to be virtually pure iron in its hexagonal close packed (hcp) phase.
It is shown that four properties of the Earth's inner core determined from seismology are close in value to the corresponding properties of hcp iron at inner core conditions: density, bulk modulus, longitudinal velocity, and Poisson's ratio. The density–pressure profile of hcp iron at inner core conditions matches the density–pressure profile of the inner core as determined by seismic methods, within the spread of values given by recent seismic models.
This indicates that the Earth is slowly cooling, the Earth's inner core is growing by crystallization, and the impurities of the core are concentrated in the outer core. The calculated temperature at the Earth's centre is 6450 K.  相似文献   

14.
We have been developing an accurate and efficient numerical scheme, which uses the finite-difference method (FDM) in spherical coordinates, for the computation of global seismic wave propagation through laterally heterogeneous realistic Earth models. In the field of global seismology, traditional axisymmetric modeling has been used widely as an efficient approach since it can solve the 3-D elastodynamic equation in spherical coordinates on a 2-D cross-section of the Earth, assuming structures to be invariant with respect to the axis through the seismic source. However, it has the severe disadvantages that asymmetric structures about the axis cannot be incorporated and the source mechanisms with arbitrary shear dislocation have not been attempted for a long time. Our scheme is based on the framework of axisymmetric modeling but has been extended to treat asymmetric structures, arbitrary moment-tensor point sources, anelastic attenuation, and the Earth center which is a singularity of wave equations in spherical coordinates. All these types of schemes which solve 3-D wavefields on a 2-D model cross-section are classified as 2.5-D modeling, so we have named our scheme the spherical 2.5-D FDM. In this study, we compare synthetic seismograms calculated using our FDM scheme with three-component observed long-period seismograms including data from stations newly installed in Antarctica in conjunction with the International Polar Year (IPY) 2007–2008. Seismic data from inland Antarctica are expected to reveal images of the Earth's deep interior with enhanced resolution because of the high signal-to-noise ratio and wide extent of this region, in addition to the rarity of sampling paths along the rotation axis of the Earth. We calculate synthetic seismograms through the preliminary reference earth model (PREM) including attenuation using a moment-tensor point source for the November 9, 2009 Fiji earthquake. Our results show quite good agreement between synthetic and observed seismograms, which indicates the accuracy of observations in the Antarctica, as well as the feasibility of the spherical 2.5-D modeling scheme.  相似文献   

15.
We describe a waveform modelling technique and demonstrate its application to determine the crust- and upper-mantle velocity structure beneath Africa. Our technique uses a parallelized reflectivity method to compute synthetic seismograms and fits the observed waveforms by a global optimization technique based on a Very Fast Simulated Annealing (VFSA). We match the S , Sp, SsPmP and shear-coupled PL phases in seismograms of deep (200–800 km), moderate-to-large magnitude (5.5–7.0) earthquakes recorded teleseismically at permanent broad-band seismic stations in Africa. Using our technique we produce P - and S -wave velocity models of crust and upper mantle beneath Africa. Additionally, our use of the shear-coupled PL phase, wherever observed, improves the constraints for lower crust- and upper-mantle velocity structure beneath the corresponding seismic stations. Our technique retains the advantages of receiver function methods, uses a different part of the seismogram, is sensitive to both P - and S -wave velocities directly, and obtains helpful constraints in model parameters in the vicinity of the Moho. The resulting range of crustal thicknesses beneath Africa (21–46 km) indicates that the crust is thicker in south Africa, thinner in east Africa and intermediate in north and west Africa. Crustal P - (4.7–8 km s−1) and S -wave velocities (2.5–4.7  km s−1) obtained in this study show that in some parts of the models, these are slower in east Africa and faster in north, west and south Africa. Anomalous crustal low-velocity zones are also observed in the models for seismic stations in the cratonic regions of north, west and south Africa. Overall, the results of our study are consistent with earlier models and regional tectonics of Africa.  相似文献   

16.
Summary. Three Fast Fourier Transform numerical methods for computing the Hilbert transform have been evaluated for their accuracy by numerical examples. All three methods employ the property that the Hdbert transform is a convolution. The first method uses the result that the Fourier transform of 1/π x is — isgn(ω). The second method is based on a discrete Hilbert transform introduced by Saito. The third method, introduced in this research note, uses linear interpolation to transform the Hilbert transform integral into a discrete convolution. The last method is shown by numerical examples from fault dislocation models to be more accurate than the other two methods when the Hilbert transform integral has high-frequency components.  相似文献   

17.
Summary. Within the framework of Newtonian kinematics VLBI observations are analysed with respect to estimability of geodetic and astrometric quantities. An Earth model of either rigid or deformable type is designed; instrumental clock offsets and clock drifts are included. Observational patterns are studied in all detail reviewed in seven tables. Appendix A is an introduction to the set-up of the observational model for a deformable Earth both in terms of coordinate-free and coordinate-related geometry. Appendix B is a study of the invariance characteristics of VLBI observations. Interrelations of three fundamental quantities, length unit, time unit and velocity of light are discussed. An overall result of an Earth model of deformable type is the need of simultaneous observations to more than one source; VLBI time delay observations cannot distinguish between secular changes of network size (expansion or shrinking) and a common secular drift (deceleration or acceleration) of clocks used.  相似文献   

18.
Summary. The three-dimensional (3-D) shear wave structure of the mantle, down to the depth of about 900 km, is obtained by inverting waveforms of radial component seismograms. Radial component seismograms contain large amplitude overtone signals which circle the Earth as wave packets and are sometimes called X1, X2, X3, … We use data which contain R1, X1 and X2 and filtered between 2 and 10mHz. It is shown that, unless each seismogram is weighted, all seismograms are not fitted uniformly. Only data from large earthquakes are fitted and the final velocity anomalies are biased by the small number of large earthquake data. Resolution is good at shallow depths, becomes worse in the intermediate depth range between about 400 and 500 km and then becomes better at greater depth ranges (600–900km). Even though we use only spheroidal mode data, velocity anomalies in the shallow structure show excellent correlation with the age of the surface rocks of the Earth. In the deeper regions, between about 600 and 900km, South America shows a fast velocity anomaly which may indicate the slab penetration beyond 700 km there. Another region which shows a fast velocity anomaly is the Mariana trench, but other subduction regions do not show such features.  相似文献   

19.
The fractal dimension (D) was estimated for nine tidal channels depicted in thematic mapper (TM) Landsat-5 imagery to derive information about the degree of geomorphological control on a tidal channel network characteristic of the Bahía Blanca Estuary (Argentina). Two methods, box counting and contiguity, were used to estimate fractal dimensions for each tidal channel. All channels produced D values close to 1, meaning that they are self-affine fractal features. However, these fractal dimensions do not represent the meandering pattern complexity characteristic of the tidal channels analysed. Although both methods allowed for estimation of D, the contiguity method showed that three of the channels actually are not fractal but have sinusoidal characteristics, a condition that was not detected by the former method.  相似文献   

20.
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