共查询到20条相似文献，搜索用时 31 毫秒
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A conservative staggered-grid finite difference method is presented for computing the electromagnetic induction response of an arbitrary heterogeneous conducting sphere by external current excitation. This method is appropriate as the forward solution for the problem of determining the electrical conductivity of the Earth's deep interior. This solution in spherical geometry is derived from that originally presented by Mackie et al. (1994 ) for Cartesian geometry. The difference equations that we solve are second order in the magnetic field H , and are derived from the integral form of Maxwell's equations on a staggered grid in spherical coordinates. The resulting matrix system of equations is sparse, symmetric, real everywhere except along the diagonal and ill-conditioned. The system is solved using the minimum residual conjugate gradient method with preconditioning by incomplete Cholesky decomposition of the diagonal sub-blocks of the coefficient matrix. In order to ensure there is zero H divergence in the solution, corrections are made to the H field every few iterations. In order to validate the code, we compare our results against an integral equation solution for an azimuthally symmetric, buried thin spherical shell model ( Kuvshinov & Pankratov 1994 ), and against a quasi-analytic solution for an azimuthally asymmetric configuration of eccentrically nested spheres ( Martinec 1998 ). 相似文献
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Abel Amirbekyan Volker Michel Frederik J. Simons † 《Geophysical Journal International》2008,174(2):617-628
We present a mathematical framework and a new methodology for the parametrization of surface wave phase-speed models, based on traveltime data. Our method is neither purely local, like block-based approaches, nor is it purely global, like those based on spherical harmonic basis functions. Rather, it combines the well-known theory and practical utility of the spherical harmonics with the spatial localization properties of spline basis functions. We derive the theoretical foundations for the application of harmonic spherical splines to surface wave tomography and summarize the results of numerous numerical tests illustrating the performance of a practical inversion scheme based upon them. Our presentation is based on the notion of reproducing-kernel Hilbert spaces, which lends itself to the parametrization of fully 3-D tomographic earth models that include body waves as well. 相似文献
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Recovering magnetic susceptibility from electromagnetic data over a one-dimensional earth 总被引：1，自引：0，他引：1
While the inversion of electromagnetic data to recover electrical conductivity has received much attention, the inversion of those data to recover magnetic susceptibility has not been fully studied. In this paper we invert frequency-domain electromagnetic (EM) data from a horizontal coplanar system to recover a 1-D distribution of magnetic susceptibility under the assumption that the electrical conductivity is known. The inversion is carried out by dividing the earth into layers of constant susceptibility and minimizing an objective function of the susceptibility subject to fitting the data. An adjoint Green's function solution is used in the calculation of sensitivities, and it is apparent that the sensitivity problem is driven by three sources. One of the sources is the scaled electric field in the layer of interest, and the other two, related to effective magnetic charges, are located at the upper and lower boundaries of the layer. These charges give rise to a frequency-independent term in the sensitivities. Because different frequencies penetrate to different depths in the earth, the EM data contain inherent information about the depth distribution of susceptibility. This contrasts with static field measurements, which can be reproduced by a surface layer of magnetization. We illustrate the effectiveness of the inversion algorithm on synthetic and field data and show also the importance of knowing the background conductivity. In practical circumstances, where there is no a priori information about conductivity distribution, a simultaneous inversion of EM data to recover both electrical conductivity and susceptibility will be required. 相似文献
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Geophysical observables are generally related to earth structure and source parameters in a complicated non-linear way. Consequently, a large number of forward modelling processes are commonly necessary to obtain a satisfactory estimate of such parameters from observed data. The most time-consuming part of the forward modelling is the computation of the Green's functions of the different earth models to be tested. In this study, we present a fast converging algorithm: the differential transform method for the computation of Green's functions in terms of spherical or cylindrical harmonics. In this method, a deconvolutable high-pass filter is used to enhance the numerical significance of the far-field spectrum of Green's functions. Compared with existing fast converging algorithms such as the Kummer's transformation and the disc factor method, the differential transform method is more efficient except for the extremely near-source region. The new method can be used to suppress numerical phases (non-physical seismic signals) associated with the aliasing effect that may arise in synthetic seismograms when the latter are computed from a windowed wavenumber (or slowness) spectrum. The numerical efficiency of the new method is demonstrated via two representative tests. 相似文献
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Summary. A method for the determination of the electrical conductivity of the Earth is developed when the components of the response function on the basis of spherical harmonics for a fixed frequency are known. By writing the differential equation for the field inside a spherically symmetric conductor as a finite difference equation, it is shown that the formal solution of the latter for the response function has a Thiele representation in the degree of the harmonics. This property enables one to calculate the conductivity at a finite number of points using a continued-fraction expansion of the response function. 相似文献
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Beverley J. Moore 《Geophysical Journal International》1993,115(3):1137-1142
The total Green's function for two-point boundary-value problems can be related to the propagator for initial-value problems. A very simple expression for the Green's function is obtained when the unperturbed medium may be described by material with a constant gradient in quadratic slowness. The derivation requires a correct understanding of assumptions made in the propagator solution. Expressions are also obtained for Green's function in multilayered media. 相似文献
9.
Laiyu Lu Valerie Maupin Rongsheng Zeng Zhifeng Ding 《Geophysical Journal International》2008,174(3):857-872
The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at R = 0 , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources. 相似文献
10.
Summary. The motion of a phase boundary in the Earth caused by temperature and pressure excitations at the Earth's surface is determined under a linear approximation. The solution is found as a sum of convolutions of pressure and temperature Green's functions with the corresponding excitations. The Green's functions are given under the form of Laplace transforms that can be inverted either by numerical evaluation of a branch cut integral or by inversion of a series expansion. This solution is a generalization of a solution previously derived by Gjevik. This latter solution is the first term in the series expansion. The relaxation times associated with the phase boundary motion are of the order of 105 –107 yr for the olivine—spinel phase transition and of 106 –107 yr for the basalt—eclogite transition. The linear approximation remains valid for long times only if the phase boundary moves slowly. 相似文献
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Summary. A new method for solving problems in three-dimensional electromagnetic induction in which the Earth is represented by a uniformly conducting half-space overlain by a surface layer of variable conductance is presented. Unlike previous treatments of this type of problem the method does not require the fields to be separated into their normal and anomalous parts, nor is it necessary to assume that the anomalous region is surrounded by a uniform structure; the model may approach either an E- or a B -polarization configuration at infinity. The solution is expressed as a vector integral equation in the horizontal electric field at the surface. The kernel of the integral is a Green's tensor which is expressed in terms of elementary functions that are independent of the conductance. The method is applied to an illustrative model representing an island near a bent coastline which extends to infinity in perpendicular directions. 相似文献
12.
Summary. The propagation of a pulsed elastic wave in the following geometry is considered. An elastic half-space has a surface layer of a different material and the layer furthermore contains a bounded 3-D inhomogeneity. The exciting source is an explosion, modelled as an isotropic pressure point source with Gaussian behaviour in time.
The time-harmonic problem is solved using the null field approach (the T matrix method), and a frequency integral then gives the time-domain response. The main tools of the null field approach are integral representations containing the free space Green's dyadic, expansions in plane and spherical vector wave functions, and transformations between plane and spherical vector wave functions. It should be noted that the null field approach gives the solution to the full elastodynamic equations with, in principle, an arbitrarily high accuracy. Thus no ray approximations or the like are used. The main numerical limitation is that only low and intermediate frequencies, in the sense that the diameter of the inhomogeneity can only be a few wavelengths, can be considered.
The numerical examples show synthetic seismograms consisting of data from 15 observation points at increasing distances from the source. The normal component of the velocity field is computed and the anomalous field due to the inhomogeneity is sometimes shown separately. The shape of the inhomogeneity, the location and depth of the source, and the material parameters are all varied to illustrate the relative importance of the various parameters. Several specific wave types can be identified in the seismograms: Rayleigh waves, direct and reflected P -waves, and head waves. 相似文献
The time-harmonic problem is solved using the null field approach (the T matrix method), and a frequency integral then gives the time-domain response. The main tools of the null field approach are integral representations containing the free space Green's dyadic, expansions in plane and spherical vector wave functions, and transformations between plane and spherical vector wave functions. It should be noted that the null field approach gives the solution to the full elastodynamic equations with, in principle, an arbitrarily high accuracy. Thus no ray approximations or the like are used. The main numerical limitation is that only low and intermediate frequencies, in the sense that the diameter of the inhomogeneity can only be a few wavelengths, can be considered.
The numerical examples show synthetic seismograms consisting of data from 15 observation points at increasing distances from the source. The normal component of the velocity field is computed and the anomalous field due to the inhomogeneity is sometimes shown separately. The shape of the inhomogeneity, the location and depth of the source, and the material parameters are all varied to illustrate the relative importance of the various parameters. Several specific wave types can be identified in the seismograms: Rayleigh waves, direct and reflected P -waves, and head waves. 相似文献
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C. H. Chapman 《Geophysical Journal International》1981,66(2):445-453
Summary. The Radon transform or slant stack is becoming a widely used technique for analysing high-quality reflection and refraction data. The transform normally used is applicable to data from a line source in a plane model, that is, one Cartesian coordinate. The theoretical basis for the Radon transform pair for one Cartesian coordinate has appeared in the seismological literature. For a point source in plane or spherical geometry, or a line source in cylindrical geometry only the Radon transform for the direct problem (computation of synthetic seismograms) has been published. To analyse data an approximate inverse transform has been used. In this research note, the exact forms of the generalized Radon transform pairs are completed for a point source in plane or spherical geometry, and for a line source in cylindrical geometry. The differences will be important if the waveforms are being interpreted, and are most significant for near-vertical reflections—the type of data most commonly slant stacked. 相似文献
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J. B. Minster 《Geophysical Journal International》1978,52(3):479-501
Summary. The transient and impulse responses (Green's function) for onedimensional wave propagation in a standard linear solid are calculated using a Laplace Transform method. The spectrum of relaxation times is chosen so as to model a constant Q medium within an absorption band covering a broad frequency range which may be chosen so as to include the seismic frequencies. The inverse transform may be evaluated asymptotically in the limit of very long propagation times using the saddle point method. For shorter propagation times the method of steepest descent may be modified so as to yield an accurate first motion approximation. The character of the small amplitude precursor to the large amplitude Visible' signal is investigated analytically. It is shown that the signal velocity is intermediate between the high-frequency ('unrelaxed') and the low-frequency ('relaxed') limits of the phase velocity. 相似文献
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Summary. Analytical results are presented for Love waves generated by sudden changes of the rate of advance of a curved rupture front in an inclined fault plane that is embedded in an elastic half-space. The boundary condition at the surface of the half-space approximates the presence of an overlying layer. The calculation consists of two parts. First, ray theory is used to calculate far-field approximations to the horizontally polarized wavefields which are emitted when the speed of the rupture front suddenly changes. These fields can be expressed as products of emission coefficients (which govern the angular dependence) and propagation terms. Secondly, a representation integral for the Love wave over a surface enclosing the rupture front is constructed, using the emitted signal and an appropriate Green's function. This integral is evaluated asymptotically. The resulting approximate Love-wave spectrum shows an explicit dependence on the nature of the rupture process, on the rupture-front and fault-plane geometry, and on the magnitude of a sudden change in the rate of advance of the rupture front. 相似文献
17.
Z. Martinec 《Geophysical Journal International》1997,130(3):583-594
We present a spectral-finite-element approach to the 2-D forward problem for electromagnetic induction in a spherical earth. It represents an alternative to a variety of numerical methods for 2-D global electromagnetic modelling introduced recently (e.g. the perturbation expansion approach, the finite difference scheme). It may be used to estimate the effect of a possible axisymmetric structure of electrical conductivity of the mantle on surface observations, or it may serve as a tool for testing methods and codes for 3-D global electromagnetic modelling. The ultimate goal of these electromagnetic studies is to learn about the Earth's 3-D electrical structure.
Since the spectral-finite-element approach comes from the variational formulation, we formulate the 2-D electromagnetic induction problem in a variational sense. The boundary data used in this formulation consist of the horizontal components of the total magnetic intensity measured on the Earth's surface. In this the variational approach differs from other methods, which usually use spherical harmonic coefficients of external magnetic sources as input data. We verify the assumptions of the Lax-Milgram theorem and show that the variational solution exists and is unique. The spectral-finite-element approach then means that the problem is parametrized by spherical harmonics in the angular direction, whereas finite elements span the radial direction. The solution is searched for by the Galerkin method, which leads to the solving of a system of linear algebraic equations. The method and code have been tested for Everett & Schultz's (1995) model of two eccentrically nested spheres, and good agreement has been obtained. 相似文献
Since the spectral-finite-element approach comes from the variational formulation, we formulate the 2-D electromagnetic induction problem in a variational sense. The boundary data used in this formulation consist of the horizontal components of the total magnetic intensity measured on the Earth's surface. In this the variational approach differs from other methods, which usually use spherical harmonic coefficients of external magnetic sources as input data. We verify the assumptions of the Lax-Milgram theorem and show that the variational solution exists and is unique. The spectral-finite-element approach then means that the problem is parametrized by spherical harmonics in the angular direction, whereas finite elements span the radial direction. The solution is searched for by the Galerkin method, which leads to the solving of a system of linear algebraic equations. The method and code have been tested for Everett & Schultz's (1995) model of two eccentrically nested spheres, and good agreement has been obtained. 相似文献
18.
3-D linearized scattering of surface waves and a formalism for surface wave holography 总被引：3，自引：1，他引：2
Roel Snieder 《Geophysical Journal International》1986,84(3):581-605
Summary. Scattering of surface waves by lateral heterogeneities is analysed in the Born approximation. It is assumed that the background medium is either laterally homogeneous, or smoothly varying in the horizontal direction. A dyadic representation of the Green's function simplifies the theory tremendously. Several examples of the theory are presented. The scattering and mode conversion coefficients are shown for scattering of surface waves by the root of an Alpine-like crustal structure. Furthermore a 'great circle theorem'in a plane geometry is derived. A new proof of Snell's law is given for surface wave scattering by a quarter-space. It is shown how a stationary phase approximation can be used to simplify the Fourier synthesis of the scattered wave in the time domain. Finally a procedure is suggested to do 'surface wave holography'. 相似文献
19.
Spectral analysis using orthonormal functions with a case study on the sea surface topography 总被引：1，自引：0，他引：1
Cheinway Hwang 《Geophysical Journal International》1993,115(3):1148-1160
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Spherical harmonics are orthonormalized using the Gram-Schmidt process in a function space. The problem of linear dependence of spherical harmonics over the oceans is studied using the Gram matrices and consequently three sets of orthonormal (ON) functions have been constructed. For the process an efficient formula for computing inner products of spherical harmonics has been developed. Important spectral properties of the ON functions are addressed. The ON functions may be used for representing the sea surface topography (SST) in the analysis of satellite altimeter data. The geoid error can be transformed to a representation by the ON functions and hence the comparison of powers of the geoid error and the SST signal only over the oceans is possible, leading to a better way of determining the cut-off frequency of the SST in the simultaneous solution using satellite altimeter data. As a case study, the modified Levitus SST is expanded into the ON functions. The results show that 99.90 per cent of that signal's energy is contained within degree 24 of the orthonormal functions. Such expansions also render better spectral behaviour of oceanic signals as compared to that from spherical harmonic expansions. The study shows that these generalized Fourier functions are suitable for spectral analyses of oceanic signals and they can be applied to future altimetric mission where the geoid and the SST are to be recovered. 相似文献
Spherical harmonics are orthonormalized using the Gram-Schmidt process in a function space. The problem of linear dependence of spherical harmonics over the oceans is studied using the Gram matrices and consequently three sets of orthonormal (ON) functions have been constructed. For the process an efficient formula for computing inner products of spherical harmonics has been developed. Important spectral properties of the ON functions are addressed. The ON functions may be used for representing the sea surface topography (SST) in the analysis of satellite altimeter data. The geoid error can be transformed to a representation by the ON functions and hence the comparison of powers of the geoid error and the SST signal only over the oceans is possible, leading to a better way of determining the cut-off frequency of the SST in the simultaneous solution using satellite altimeter data. As a case study, the modified Levitus SST is expanded into the ON functions. The results show that 99.90 per cent of that signal's energy is contained within degree 24 of the orthonormal functions. Such expansions also render better spectral behaviour of oceanic signals as compared to that from spherical harmonic expansions. The study shows that these generalized Fourier functions are suitable for spectral analyses of oceanic signals and they can be applied to future altimetric mission where the geoid and the SST are to be recovered. 相似文献
20.
J. Arkani-Hamed 《Geophysical Journal International》1979,56(1):63-80
Summary. Equations governing non-linear and finite-amplitude convection in a heterogeneous planetary interior are developed. Using spherical harmonic expressions of variables, together with Green's function of Laplacian operator in a spherical coordinate, the equations are reduced to one-dimensional integro-differential equations and their numerical solutions are obtained by a finite-difference scheme. The theory is then applied to several lunar models and the following conclusions are obtained.
(1) The mean temperatures and velocities of convecting zones of variable viscosity models are higher than those of constant viscosity ones. This is due to the development of lithospheres with 400–500 km thicknesses in the former models, which reduce heat loss considerably.
(2) Molten regions are continuous shells in variable viscosity models whereas they become discontinuous and localized in a constant viscosity model. The continuous molten shells decrease lateral variations of temperature significantly and tend to stabilize convection.
(3) Lateral variations of viscosity have negligible effects on the thermal evolution of the models considered. 相似文献
(1) The mean temperatures and velocities of convecting zones of variable viscosity models are higher than those of constant viscosity ones. This is due to the development of lithospheres with 400–500 km thicknesses in the former models, which reduce heat loss considerably.
(2) Molten regions are continuous shells in variable viscosity models whereas they become discontinuous and localized in a constant viscosity model. The continuous molten shells decrease lateral variations of temperature significantly and tend to stabilize convection.
(3) Lateral variations of viscosity have negligible effects on the thermal evolution of the models considered. 相似文献