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1.
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   

2.
An iterative solution to the non-linear 3-D electromagnetic inverse problem is obtained by successive linearized model updates using the method of conjugate gradients. Full wave equation modelling for controlled sources is employed to compute model sensitivities and predicted data in the frequency domain with an efficient 3-D finite-difference algorithm. Necessity dictates that the inverse be underdetermined, since realistic reconstructions require the solution for tens of thousands of parameters. In addition, large-scale 3-D forward modelling is required and this can easily involve the solution of over several million electric field unknowns per solve. A massively parallel computing platform has therefore been utilized to obtain reasonable execution times, and results are given for the 1840-node Intel Paragon. The solution is demonstrated with a synthetic example with added Gaussian noise, where the data were produced from an integral equation forward-modelling code, and is different from the finite difference code embedded in the inversion algorithm  相似文献   

3.
When interpreting electromagnetic fields observed at the Earth's surface in a realistic geophysical environment it is often necessary to pay special attention to the effects caused by inhomogeneities of the subsurface sedimentary and/or water layer and by inhomogeneities of the Earth's crust. The inhomogeneities of the Earth's crust are expected to be especially important when the electromagnetic field is generated by a source located in a magma chamber of a volcano. The simulation of such effects can be carried out using generalized thin-sheet models, which were independently introduced by Dmitriev (1969 ) and Ranganayaki & Madden (1980 ). In the first part of the paper, a system of integral equations is derived for the horizontal current that flows in the subsurface inhomogeneous conductive layer and for the vertical current crossing the inhomogeneous resistive layer representing the Earth's mantle. The terms relating to the finite thickness of the laterally inhomogeneous part of the model are retained in the equations. This only marginally complicates the equations, whilst allowing for a significant expansion of the approximation limits.
  The system of integral equations is solved using the iterative dissipative method developed by the authors in the period from 1978 to 1988. The method can be applied to the simulation of the electromagnetic field in an arbitrary inhomogeneous medium that dissipates the electromagnetic energy. When considered on a finite numerical grid, the integral equations are reduced to a system of linear equations that possess the same contraction properties as the original equations. As a result, the rate at which the iterative-perturbation sequence converges to the solution remains independent of the numerical grid used for the calculations. In contrast to previous publications on the method, aspects of the algorithm implementation that guarantee its effectiveness and robustness are discussed here.  相似文献   

4.
Inversion of time domain three-dimensional electromagnetic data   总被引：7，自引：0，他引：7  
We present a general formulation for inverting time domain electromagnetic data to recover a 3-D distribution of electrical conductivity. The forward problem is solved using finite volume methods in the spatial domain and an implicit method (Backward Euler) in the time domain. A modified Gauss–Newton strategy is employed to solve the inverse problem. The modifications include the use of a quasi-Newton method to generate a pre-conditioner for the perturbed system, and implementing an iterative Tikhonov approach in the solution to the inverse problem. In addition, we show how the size of the inverse problem can be reduced through a corrective source procedure. The same procedure can correct for discretization errors that inevidably arise. We also show how the inverse problem can be efficiently carried out even when the decay time for the conductor is significantly larger than the repetition time of the transmitter wave form. This requires a second processor to carry an additional forward modelling. Our inversion algorithm is general and is applicable for any electromagnetic field  ( E , H , d B / dt )  measured in the air, on the ground, or in boreholes, and from an arbitrary grounded or ungrounded source. Three synthetic examples illustrate the basic functionality of the algorithm, and a result from a field example shows applicability in a larger-scale field example.  相似文献   

5.
The relation between the seafloor electric field and the surface magnetic field is studied. It is assumed that the fields are created by a 2-D ionospheric current distribution resulting in the E-polarization. The layered earth below the sea water is characterized by a surface impedance. The electric field at the seafloor can be expressed either as an inverse Fourier transform integral over the wavenumber or as a spatial convolution integral. In both integrals the surface magnetic field is multiplied by a function that depends on the depth and conductivity of the sea water and on the properties of the basement. The fact that surface magnetic data are usually available on land, not at the sea surface, is also considered. Test computations demonstrate that the numerical inaccuracies involved in the convolution method are negligible. The theoretical equations are applied to calculate the seafloor electric fields due to an ionospheric line current or associated with real magnetic data collected by the IMAGE magnetometer array in northern Europe. Two different sea depths are considered: 100 m (the continental shelf) and 5 km (the deep ocean). It is seen that the dependence of the electric field on the oscillation period is weaker in the 5 km case than for 100 m.  相似文献   

6.
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   

7.
We perform analytical and numerical studies of scaling relations of earthquakes and partition of elastic strain energy between seismic and aseismic components using a thermodynamically based continuum damage model. Brittle instabilities occur in the model at critical damage level associated with loss of convexity of the strain energy function. A new procedure is developed for calculating stress drop and plastic strain in regions sustaining brittle instabilities. The formulation connects the damage rheology parameters with dynamic friction of simpler frameworks, and the plastic strain accumulation is governed by a procedure that is equivalent to Drucker–Prager plasticity. The numerical simulations use variable boundary forces proportional to the slip-deficit between the assumed far field plate motion and displacement of the boundary nodes. These boundary conditions account for the evolution of elastic properties and plastic strain in the model region. 3-D simulations of earthquakes in a model with a large strike-slip fault produce scaling relations between the scalar seismic potency, rupture area, and stress drop values that are in good agreement with observations and other theoretical studies. The area and potency of the simulated earthquakes generally follow a linear log–log relation with a slope of 2/3, and are associated with stress drop values between 1 and 10 MPa. A parameter-space study shows that the area-potency scaling is shifted to higher stress drops in simulations with parameters corresponding to lower dynamic friction, more efficient healing, and higher degree of seismic coupling.  相似文献   

8.
We present a new approach of the Indirect Boundary Element Method (IBEM) for 3-D topographic problems which can be used to deal with an infinitely spread free surface owing to the introduction of a reference solution, that is the analytical solution for the half-space with a flat free surface. This approach is an efficient countermeasure for the non-physical waves owing to the domain truncation which contaminates the computed results in the ordinary approach. Theoretical consideration shows that this newly proposed approach is a higher-grade approximation than some existing ones and achieves a higher efficacy and accuracy than those of existing ones. The discretization of the resulting boundary integral equation for this formulation is carried out with triangular elements. Their contributions to the solution are calculated by Gaussian numerical integration except in the case where the wavefield is evaluated on the source element itself. For this case, we present an analytical formula based on the reasonable assumption that the elements are much smaller than the wavelengths appearing in the calculation. Several numerical examples used for validation show acceptably precise results.  相似文献   

9.
Scattering of surface waves modelled by the integral equation method   总被引：1，自引：0，他引：1  
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.
The perfectly matched layer (PML) absorbing boundary condition is incorporated into an irregular-grid elastic-wave modelling scheme, thus resulting in an irregular-grid PML method. We develop the irregular-grid PML method using the local coordinate system based PML splitting equations and integral formulation of the PML equations. The irregular-grid PML method is implemented under a discretization of triangular grid cells, which has the ability to absorb incident waves in arbitrary directions. This allows the PML absorbing layer to be imposed along arbitrary geometrical boundaries. As a result, the computational domain can be constructed with smaller nodes, for instance, to represent the 2-D half-space by a semi-circle rather than a rectangle. By using a smooth artificial boundary, the irregular-grid PML method can also avoid the special treatments to the corners, which lead to complex computer implementations in the conventional PML method. We implement the irregular-grid PML method in both 2-D elastic isotropic and anisotropic media. The numerical simulations of a VTI lamb's problem, wave propagation in an isotropic elastic medium with curved surface and in a TTI medium demonstrate the good behaviour of the irregular-grid PML method.  相似文献   

11.
Time-domain electromagnetic migration in the solution of inverse problems   总被引：5，自引：0，他引：5  
Time-domain electromagnetic (TDEM) migration is based on downward extrapolation of the observed field in reverse time. In fact, the migrated EM field is the solution of the boundary-value problem for the adjoint Maxwell's equations. The important question is how this imaging technique can be related to the solution of the geoelectrical inverse problem. In this paper we introduce a new formulation of the inverse problem, based on the minimization of the residual-field energy flow through the surface or profile of observations. We demonstrate that TDEM migration can be interpreted as the first step in the solution of this specially formulated TDEM inverse problem. However, in many practical situations this first step produces a very efficient approximation to the geoelectrical model, which makes electromagnetic migration so attractive for practical applications. We demonstrate the effectiveness of this approach in inverting synthetic and practical TDEM data.  相似文献   

12.
The construction of effective methods for electromagnetic modelling   总被引：1，自引：0，他引：1  
Summary. This paper deals with the further development of finite-difference methods for electromagnetic field modelling in two-and three-dimensional cases. The main feature of the approach suggested here is the application of generalized asymptotic boundary conditions valid with the accuracy (1/ρN), where ρ is the distance from the heterogeneities. The finite-difference approximation of problems under solution is made using the balance method, which results in 5-point difference schemes in the 2-D case and 7-point difference schemes in the 3-D case. To solve the linear system of difference equations the successive over-relaxation (SOR) method is used, the relaxation factor being chosen during the iteration procedure. In view of the vectorial character of the problem for the 3-D case, a successive blocked over-relaxation method (SBOR) is applied.
The model's validity is based on the comparison of the fields accounted at the ground surface with those computed by the integral transformation of excessive currents, determined in the heterogeneity region using the finite-difference scheme.  相似文献   

13.
The diffraction of P, S and Rayleigh waves by 3-D topographies in an elastic half-space is studied using a simplified indirect boundary element method (IBEM). This technique is based on the integral representation of the diffracted elastic fields in terms of single-layer boundary sources. It can be seen as a numerical realization of Huygens principle because diffracted waves are constructed at the boundaries from where they are radiated by means of boundary sources. A Fredholm integral equation of the second kind for such sources is obtained from the stress-free boundary conditions. A simplified discretization scheme for the numerical and analytical integration of the exact Green's functions, which employs circles of various sizes to cover most of the boundary surface, is used.
The incidence of elastic waves on 3-D topographical profiles is studied. We analyse the displacement amplitudes in the frequency, space and time domains. The results show that the vertical walls of a cylindrical cavity are strong diffractors producing emission of energy in all directions. In the case of a mountain and incident P, SV and SH waves the results show a great variability of the surface ground motion. These spatial variations are due to the interference between locally generated diffracted waves. A polarization analysis of the surface displacement at different locations shows that the diffracted waves are mostly surface and creeping waves.  相似文献   

14.
We derive a set of non-hypersingular boundary integral equations, both elastodynamic and elastostatic, for the analysis of arbitrarily shaped 2-D anti-plane and in-plane cracks located in an infinite homogeneous isotropic medium, rendered in a unified nomenclature for all cases. The hypersingularities that appear in the usual formulations for the dynamic cases, existent both at the source point and at the wavefront, are removed by way of a regularization technique based on integration by parts. The equations for the in-plane cases are presented in terms of a local Cartesian coordinate system, one of the axes of which is always held locally tangential to the crack trace. The expressions for the elastic field at any point on the model plane are also given.
Our formulations are shown to yield accurate numerical results, as long as appropriate stabilization measures are taken in the numerical scheme. The numerical applicability of our method to non-planar crack problems is illustrated by simulations of dynamic growth of a hackly crack which has small off-plane side-branches. The results imply that the branching of a crack brings about a significant decrease in the crack-tip stress concentration level and consequently may play an essential role in the arrest of earthquake rupturing.  相似文献   

15.
Generalized Born scattering of elastic waves in 3-D media   总被引：1，自引：0，他引：1  
It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.  相似文献   

16.
Many geophysical inverse problems derive from governing partial differential equations with unknown coefficients. Alternatively, inverse problems often arise from integral equations associated with a Green's function solution to a governing differential equation. In their discrete form such equations reduce to systems of polynomial equations, known as algebraic equations. Using techniques from computational algebra one can address questions of the existence of solutions to such equations as well as the uniqueness of the solutions. The techniques are enumerative and exhaustive, requiring a finite number of computer operations. For example, calculating a bound to the total number of solutions reduces to computing the dimension of a linear vector space. The solution set itself may be constructed through the solution of an eigenvalue problem. The techniques are applied to a set of synthetic magnetotelluric values generated by conductivity variations within a layer. We find that the estimation of the conductivity and the electric field in the subsurface, based upon single-frequency magnetotelluric field values, is equivalent to a linear inverse problem. The techniques are also illustrated by an application to a magnetotelluric data set gathered at Battle Mountain, Nevada. Surface observations of the electric ( E y ) and magnetic ( H x ) fields are used to construct a model of subsurface electrical structure. Using techniques for algebraic equations it is shown that solutions exist, and that the set of solutions is finite. The total number of solutions is bounded above at 134 217 728. A numerical solution of the algebraic equations generates a conductivity structure in accordance with the current geological model for the area.  相似文献   

17.
Electromagnetic investigations are usually intended to examine regional structures where induction takes place at a given period range. However, the regional information is often distorted by galvanic effects at local conductivity boundaries. Bahr (1985) and Groom & Bailey (1989) developed a physical distortion model for decomposing the MT impedance tensor, based upon local galvanic distortion of a regional 2-D electromagnetic field. We have extended their method to predict the magnetic variation fields created at an array of sites. The magnetic response functions at periods around 1000 s may be distorted by large-scale inhomogeneities in the upper or middle crust. In this period range, the data measured by a magnetometer array contain common information that can be extracted if the data set is treated as a unit, for example by using hypothetical event analysis. With this technique it is always possible to recover the regional strike direction from distorted data, even if a strong, spatially varying regional vertical field component is present in the data set. The determination of the regional impedance phases, on the other hand, is far more sensitive to deviations from the physical distortion model.
The approach has been used to investigate the Iapetus data set. For the array, which covers an area of 200  km × 300  km in northern England/southern Scotland, the technique revealed a common regional strike azimuth of ca . N125° E in the period range 500–2000  s. This direction differs from the strike indicated by the induction arrows, which seem influenced mainly by local current concentrations along the east–west-striking Northumberland Trough and a NE–SW-striking mid-crustal conductor. Both impedance phases are positive and differ by ca . 10°, which supports the assumptions of distortion fields in the data set and that the regional structure is 2-D.  相似文献   

18.
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19.
The C -response connects the magnetic vertical component and the horizontal gradient of the horizontal components of electromagnetic variations and forms the basis for deriving the conductivitydepth profile of the Earth. Time-series of daily mean values at 42 observatories typically with 50 years of data are used to estimate C -responses for periods between 1 month and 1  yr. The Z : Y method is applied, which means that the vertical component is taken locally whereas the horizontal components are used globally by expansion in a series of spherical harmonics.
In combination with results from previous analyses, the method yields consistent results for European observatories in the entire period range from a few hours to 1  yr, corresponding to penetration depths between 300 and 1800  km.
1-D conductivity models derived from these results show an increase in conductivity with depth z to about 2  S  m-1 at z =800  km, and almost constant conductivity between z =800 and z =2000  km with values of 310  S  m-1, in good agreement with laboratory measurements of mantle material. Below 2000  km the conductivity is poorly resolved. However, the best-fitting models indicate a further increase in conductivity to values between 50 and 150  S  m-1.  相似文献   

20.
We describe finite-difference approximations to the equations of 2-D electromagnetic induction that permit discrete boundaries to have arbitrary geometrical relationships to the nodes. This allows finite-difference modelling with the flexibility normally ascribed to finite-element modelling. Accuracy is demonstrated by comparison with finite-element computations. We also show that related approximations lead to substantially improved accuracy in regions of steep, but not discontinuous, conductivity gradient.  相似文献   

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