Improved resolution of complex eigenfrequencies in analytically continued seismic spectra

Summary. The response of the Earth to an earthquake is a transient that is effectively zero several days after the event. A recording of the event, of finite duration in time, has a Fourier spectrum that is an entire, or integral, analytic function of frequency. We present a very simple procedure for computing the Fourier spectrum as a function of complex frequency; the analytically continued spectrum. By investigating the properties of the analytically continued spectrum we show how to extract high- Q modes, how to estimate Q either from the amplitude or from the width of a resonance function, and how to improve the resolution of splitting to the theoretical maximum. Examples of these procedures, using observed data, are presented.     

On the density distribution within the Earth

The distribution of density as a function of position within the Earth is much less well constrained than the seismic velocities. The primary information comes from the mass and moment of inertia of the Earth and this information alone requires that there be a concentration of mass towards the centre of the globe. Additional information is to be found in the frequencies of the graver normal modes of the Earth which are sensitive to density through self-gravitation effects induced in deformation.
  The present generation of density models has been constructed using linearized inversion techniques from earlier models, which ultimately relate back to models developed by Bullen and based in large part on physical arguments. A number of experiments in non-linear inversion have been conducted using the PREM reference model, with fixed velocity and attenuation, but with the density model constrained to lie within fixed bounds on both density and density gradient. A set of models is constructed from a uniform probability density within the bound and slope constraints. Each of the resultant density models is tested against the mass and moment of inertia of the Earth, and for successful models a comparison is made with observed normal mode frequencies. From the misfit properties of the ensemble of models the robustness of the density profile in different portions of the Earth can be assessed, which can help with the design of parametrization for future reference models. In both the lower mantle and the outer core it would be desirable to allow a more flexible representation than the single cubic polynomial employed in PREM.     

Rigorous velocity bounds from soft τ (p) and X(p) data

Summary. The convergence of two methods of inferring bounds on seismic velocity in the Earth from finite sets of inexact observations of τ ( p ) and X( p ) are examined: the linear programming (LP) method of Garmany, Orcutt & Parker and the quadratic programming (QP) method of Stark & Parker. The LP method uses strict limits on the observations of τ and X as its data, while QP uses estimated means and variances of τ and X. The approaches are quite similar and involve only one inherent approximation: they use a finite-dimensional representation of seismic velocity within the Earth. Clearly, not every Earth model can be written this way. It is proved that this does not hinder the methods - they may be made as accurate as desired by increasing the number of dimensions in a specified way. It is shown how to get the highest accuracy with a given number of dimensions.     

Four barriers to the development of effective exploratory visualisation tools for the geosciences

This paper outlines four specific problems that appear to represent considerable obstacles to the development of visualisation strategies for use within the domain of geography and the Earth sciences. These are: (1) the speed of graphical rendering, (2) the management of perceptual anomalies due to visual combination effects, (3) the vast range of potential approaches and mappings (the complexity of the visual assignment process), and (4) the orientation of the user into an artificial or virtual reality. Each problem is discussed in terms of the visualisation of geographical data for the purpose of exploratory visual analysis. The specific underlying research issues and questions are described, with particular emphasis to how these relate to the geographical domain. Where possible, some potential solutions are suggested. Specific examples of geographical data visualisation are given to substantiate the arguments presented. The discussion highlights the need for further research in a number of key areas, and stresses the weaknesses of current visualisation theory and technology when applied to non-trivial geographical datasets.     

Is the seismic moment–frequency relation universal?

We evaluate the seismic moment–frequency relation for the Harvard catalogue in the period 1977–1994. This catalogue is composed of about 12 000 earthquakes. After selection of events in terms of depth and energy, we retain about 8000 data points. We estimate two parameters of the seismic moment distribution: the power exponent β and the cut-off value M _m . The method used is a least-squares linear fit on a log–log scale performed over a range selected on the basis of the standard deviation from the histogram. The analysis is carried out for different subdivisions of the Earth in square grids of different sizes. Neither parameter exhibits a dependence on cell size, suggesting the universality of their values and the interpretation of the existence of a cut-off as a finite size effect linked to a finite catalogue length. The variations of the parameters are investigated as a function of time (duration of the catalogue) and versus the number of events used for building up the distribution. Again, β and M _m do not depend on time, but M _m depends on the number of events, reaching a stable value for N ≈ 1000. The only significant change in the parameters is observed for different values of M _0upper in the catalogue, revealing the existence of universality classes.     

The free oscillations of an anelastic aspherical earth

Summary. Some century-old results, due to Rayleigh and Routh, have been adapted to investigate the normal mode eigenfrequencies and eigenfunctions of an earth with laterally variable anelasticity and to determine the transient response of such an earth to earthquakes. Using degenerate perturbation theory, the eigenfrequencies are found to first order and the associated eigenfunctions to zeroth order in the small deviations of the Earth away from a spherical perfectly elastic reference earth model. Both the eigenfrequencies and the eigenfunctions are complex and, in addition, the latter are not mutually orthogonal, reflecting the non-Hermitian character of the normal mode eigenvalue problem. The effect of laterally heterogeneous attenuation on the shape of an unresolved multiplet spectrum has been investigated in the surface-wave geometrical-optics limit. Singlet cancellation leads in that limit to the appearance of a single resonance peak whose decay rate or apparent Q ⁻¹ depends only on the average attenuation structure underlying the source—receiver great-circle path.     

分形理论在地理信息科学研究中的应用

该文简介分形理论 ,并讨论分形理论在地理信息科学研究中的价值和地位。针对地理信息科学发展的趋势 ,结合相应的实例 ,提出分形理论在地理信息科学中的地理数据挖掘和空间决策支持、空间数据压缩和传递、制图概括、虚拟现实等方面目前的主要研究进展和方向。     

Stability of Planetary Interiors

Summary Many accept thermal convection within the mantle of the Earth as the driving mechanism for continental drift. It is also of considerable interest to determine whether thermal convection is occurring within Venus, Mars, and the Moon. In this paper a systematic treatment of the stability of planetary interiors is given. The thermal stability problem for a layer of fluid heated from below is solved when the viscosity of the fluid increases exponentially with depth. For a semi-infinite fluid with exponentially increasing viscosity the critical Rayleigh number based on the surface viscosity and the scale length of the viscosity increase is found to be 30 for a fixed surface boundary condition and 23 for a free surface boundary condition. This stability analysis is also extended to include volume heat release. The thermal stability of the interiors of the Earth, Venus, Mars and the Moon is examined. Using temperature-depth profiles in the literature and a theoretical expression for the viscosity of a crystalline solid based on diffusion creep, viscosity depth profiles for planetary interiors are obtained. Because of the strong pressure effect the viscosity within the Earth and Venus increases greatly from a near surface minimum. For Mars the increase is less pronounced and within the interior of the Moon the viscosity is nearly constant. For all the cases considered the planetary interiors are found to be thermally unstable. Because of the dependence of viscosity on depth the interiors of the Moon and Mars are found to be considerably less stable than Venus and the Earth. It is concluded that thermal convection is occurring within the planetary bodies considered.     

The effect of underground structure on seismic motions of the ground surface

Summary. Two-dimensional finite element modelling of underground structural anomalies at shallow depths has been done to obtain the response at the ground surface to damped, vertically incident, SH -waves. Power spectral ratios are examined to determine what effects the position, shape, depth and size of the anomaly have on the surface seismograms. Based on the results gathered from a number of models, inferences are made with respect to the inverse problem: given the seismic motion of the ground surface, what can be said about the underground structure?     

The Backus–Gilbert theory for piecewise continuous structures with variable discontinuity levels and its application to the magnetotelluric inverse problem

Summary. The Backus–Gilbert theory is extended to the case when the models are piecewise continuous vector functions of depth with variable discontinuity locations. In addition, some of the layers may be represented by linear combinations of known functions. For such layers only a finite number of discrete parameters is to be determined. The iterative process for obtaining a model satisfying the data is convergent, the numerical procedure by which the iterations are performed being equivalent to the method of spectral decomposition for continuous structures. The method of obtaining the Fréchet kernels by using the first perturbation of the differential system satisfied by the corresponding functionals is shown to be valid.
The theory is applied to the magnetotelluric problem, Fréchet kernels being calculated for isotropic and non-isotropic structures.
A few numerical examples are described.     

Geocentre motion from the DORIS space system and laser data to the Lageos satellites: comparison with surface loading data

Surface mass redistribution within the Earth system, especially in the atmosphere, oceans, continents and ice sheets, causes the position of the centre of mass to vary in a reference frame attached to the solid Earth. Space techniques are now precise enough to measure the centre of mass motion. Here we present a determination of the centre of mass coordinates at regular monthly intervals using DORIS data on SPOT‐2, SPOT‐3 and Topex–Poseidon (1993–1997) and laser data on Lageos‐1 and Lageos‐2 (1993–1996). The amplitude and phase of the space‐geodesy‐derived annual cycle for each coordinate are further compared to estimates based on surface mass redistribution at the Earth surface derived from various climatic data sources: surface pressure, soil moisture, snow depth and ocean mass variations.     

Discrete visibility problems and graph algorithms

Many problems of practical interest involve line-of-sight on a topographic surface. Some such problems can be successfully studied on the basis of the mutual visibility among a finite number of representative points. Such visibility problems can be formalized and resolved as graph problems. In this paper, we show that graph algorithms can be useful to find efficient solutions for discrete visibility problems in several cases. On the basis of results from the theory of complexity, we give some practical rules to apply such an approach. We further investigate the solution of some relevant visibility problems under this perspective.     

Normal modes of the viscoelastic earth

Summary A uniformly valid linear viscoelastic rheology is described which takes the form of a 'generalized' Burgers' body and which appears capable of reconciling the behaviour of the Earth's mantle across the complete spectrum of geodynamic time-scales. This spectrum is bracketed by the short time-scales of body wave and free oscillation seismology on which anelastic effects are dominant, and the long time-scale of mantle convection on which the Earth behaves viscously. The parameters of the model which control the viscous response are fixed by post-glacial rebound data whereas those which govern the anelasticity are to be determined by fitting the model to observations of seismic Q. The paper is concerned primarily with a discussion of the normal mode spectrum of the Earth as a generalized Burgers' body. Focusing upon the homogeneous model, it includes an initial analysis of the accuracy of first-order perturbation theory as a method of calculating the respective Q s of the elastic gravitational free oscillations. Also considered are the quasi-static modes of relaxation which only exact eigenanalysis can reveal. The importance of these modes is assessed within the context of a discussion of the effect of viscoelasticity upon the efficiency of Chandler wobble excitation.     

Inversion of the travel-time curve of a reflected wave

Summary. A general theory of inversion of the reflected travel-time curve is developed. The properties of the travel-time curve and its analytical continuation are examined. The inversion formulae are presented and the structure of the set of velocity—depth functions is studied in detail. The inverse problem or a finite number of reflectors and the problem of separation of the reflected and multiple travel-time curves are both considered.     

Determination of Lg-wave attenuation using single-station seismograms: a case study for Zimbabwe

In this paper we show how the quality factor Q may be calculated using a single seismograph station and a number of events recorded on analogue seismograms. We followed Nuttli's (1973) method and extended it to one seismograph station. Using the single station Bulawayo (BUL), we determined a mean Q value of 650 for Zimbabwe. Furthermore, we considered different propagation paths over Zimbabwe as the seismic waves travelled to BUL and found a low Q value of 350 for the Deka fault zone. the Q value of 650 obtained in this study agrees well within error with that of 603 reported by Chow et al. (1980) using the multistation, multi-event method, and lies within the range of Q values (454–759) reported by Xie & Mitchell (1990) using a 'back-project'method to image large-scale lateral variations of Lg coda Q . the results obtained are important as the Q value constitutes part of the input data in seismic hazard calculations. the method may be used to determine Q in regions where there may be only one station with reliable analogue seismogram data.     

A continued fraction approach to the inverse problem of electrical conductivity

Summary. The problem of determination of the electrical conductivity of the Earth from the geomagnetic induction data is formulated as that of finding the coefficients of the continued fraction expansion of a certain rational fraction representation of the total response (or impedance) of the medium at the surface when this quantity is given for N frequencies. The coefficients of expansion are related to the conductivities of N layers of constant attenuation, where, within each layer the conductivity is assumed to be constant. Thus in this approach the conductivity profile resulting from the inversion of the response function is given as a series of step functions.     

Three-dimensional inversion without blocks

Summary. We propose a method for solving non-linear inverse problems in the case where the unknown is a function of the spatial coordinates and the data set is discrete and finite. The method is based on a generalized leastsquares criterion, it is defined directly for non-linear problems (without previous linearization of the forward problem), and in the particular linear case it gives the same results (although slightly more general) than the Backus & Gilbert approach. As an example, we apply the method to the three-dimensional seismic velocity inverse problem, using as data the arrival times of seismic waves. The following paper (Nercessian et al .) shows some esults obtained using the present method.     

Isoparametric and isozenithal-type representations of the actual gravity field of the Earth

Summary. The first-order theory of three-dimensional representations of the actual gravity field of the Earth on the normal field given by Bocchio has been applied to the class of isoparametric-and of isozenithal-type mappings. The resulting linear and angular deformations together with the expressions for the gravity anomalies and disturbances have been compared and discussed and some invariant relations have been put into evidence.     

The classical analysis of the response of a long baseline radio interferometer

summary . A general derivation of the classical response of a long baseline interferometer is presented including the effects of finite bandwidth of the receiver systems, instrumental delays and phase shifts, instabilities in the separate local oscillators, as well as the effects of the retarded baseline. The general theory is then applied to a pair of antennae attached to the Earth and discussed in the light of a number of geophysical phenomena including the effects of precession, nutation, rotation, polar motion, annual aberration and Earth tides.     

A direct analytic method of calculating the quadrupole parameters of a planetary magnetic field

Summary. Attention is drawn to a direct analytic method of calculating the quadrupole parameters of a planetary main magnetic field. Following a brief survey of the general theory of magnetic multipoles, an explicit algorithm is derived for calculating the quadrupole moment and the directions of the two quadrupole axes, given the five spherical harmonic coefficients of the second degree. It is shown that the direct analytic method of calculating the quadrupole parameters yield results for the geomagnetic quadrupole that are in exact agreement with those obtained by the more usual iterative procedure. Moreover, the direct analytic method has the distinct advantage that it does not require approximate values of the quadrupole parameters to be specified in advance. The direct analytic method of solution is used to calculate the quadrupole parameters for three different models of the main field of the Earth at epoch 1975; these computations provide some indication of the current uncertainties in the geomagnetic quadrupole parameters. It is also pointed out that a pseudo-quadrupole moment, which has been used to compare the quadrupole strengths of different planetary magnetic fields, is not strictly consistent with Maxwell's classical definition of a quadrupole moment. A precise physical definition of this pseudo-quadrupole moment is propounded.