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1.
Summary. In an earlier work, mathematical formulation on computing the electromagnetic response of an arbitrarily shaped three-dimensional inhomogeneity in a layered earth had been worked out using an integral equation technique. The method has been used to show its efficacy by computing numerical results. Introducing suitable changes of variables the secondary contributions to Green's dyadic are put in the form of convolution integrals and are computed using a digital linear filtering scheme. The matrix equation is solved for the unknown electric fields in the inhomogeneity. The scattered fields are then calculated at the surface of the Earth using the appropriate Green's dyadic. The performance of the computations has been shown by comparing the numerical results with those obtained by analogue modelling as well as by other numerical schemes. The use of digital linear filtering saves an enormous amount of computer time.
The effects of varying excitation-frequency, conductivity of the host medium and that of the overburden have been studied in detail for a horizontal loop system traversing over a two-layered earth with a prismatic inhomogeneity situated in the lower conducting half space.  相似文献   

2.
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   

3.
Rays propagating through strongly laterally varying media exhibit chaotic behaviour. This means that initially close rays diverge exponentially, rather than according to a power law. This chaotic behaviour is especially pronounced if the medium contains laterally varying interfaces. By studying simple 2-D and 3-D versions of models with laterally varying interfaces, the importance of chaotic ray behaviour is determined. A model of the Moho below Germany produces sharp variations with epicentral distance of the number of arrivals. In addition, the number of caustics grows dramatically: up to 1200 caustics are present between a distance of 0 and 800 km. Using the theory of Hamiltonian systems, a more in-depth study of the chaotic character of the ray equations is obtained. It is found that for realistic heterogeneous models most of the relevant rays will exhibit chaotic behaviour. The degree of chaos is quantified in terms of predictability horizons. Beyond the predictability horizons ray tracing cannot be carried out accurately. For the models under consideration, the length from the source to the predictability horizon has an order of magnitude of 1000 km. The chaotic behaviour of the rays makes it necessary to use extensions of asymptotic ray theory, such as Maslov theory, to compute seismic waveforms. It is shown that pseudo-caustics, an important obstacle in computing Maslov synthetics, are a generic feature of the 2-D laterally varying models that are studied. Eventually, the use of asymptotic methods is restricted because of the inaccuracy in the computation of the ray paths.  相似文献   

4.
Summary Isotropic earth models are unable to provide uniform fits to the gross Earth normal mode data set or, in many cases, to regional Love-and Rayleigh-wave data. Anisotropic inversion provides a good fit to the data and indicates that the upper 200km of the mantle is anisotropic. The nature and magnitude of the required anisotropy, moreover, is similar to that found in body wave studies and in studies of ultramafic samples from the upper mantle. Pronounced upper mantle low-velocity zones are characteristic of models resulting from isotropic inversion of global or regional data sets. Anisotropic models have more nearly constant velocities in the upper mantle.
Normal mode partial (Frediét) derivatives are calculated for a transversely isotropic earth model with a radial axis of symmetry. For this type of anisotropy there are five elastic constant. The two shear-type moduli can be determined from the toroidal modes. Spheroidal and Rayleigh modes are sensitive to all five elastic constants but are mainly controlled by the two compressional-type moduli, one of the shear-type moduli and the remaining, mixed-mode, modulus. The lack of sensitivity of Rayleigh waves to compressional wave velocities is a characteristic only of the isotropic case. The partial derivatives of the horizontal and vertical components of the compressional velocity are nearly equal and opposite in the region of the mantle where the shear velocity sensitivity is the greatest. The net compressional wave partial derivative, at depth, is therefore very small for isotropic perturbations. Compressional wave anisotropy, however, has a significant effect on Rayleigh-wave dispersion. Once it has been established that transverse anisotropy is important it is necessary to invert for all five elastic constants. If the azimuthal effect has not been averaged out a more general anisotropy may have to be allowed for.  相似文献   

5.
Magnetotelluric data observed in frequency and time domains are expressed as apparent resistivity. the apparent resistivity is a weighted spatial average of the subsurface resistivity distribution. In this paper, we develop analytical expressions to compute the apparent resistivity in the time domain for various three-layer earth models. the present approach to computing the magnetotelluric response in the time domain is found to overcome the problems encountered by the method of images. the magnetotelluric response in the time domain for various three-level models have been computed and shown graphically. the time-domain responses show a characteristic behaviour with a small change in layered parameters (resistivity and thickness of the layers), whereas frequency-domain responses do not show such behaviour. This characteristic behaviour of time-domain magnetotelluric sounding curves will be useful in the qualitative interpretation of field data.  相似文献   

6.
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   

7.
Summary. An array of 26 magnetometers deployed in a tectonically active area of Quebec has mapped a boundary in terrestrial electrical conductivity for 200 km along strike. The contrast in conductivity across the boundary, from previous magnetotelluric soundings, is about one order of magnitude. Anomalous variation fields associated with electric currents flowing along the boundary are readily detected at pulsation periods only when the horizontal field is polarized transverse to the structure (the E -polarization case). The anomaly is hardly visible in transfer functions from substorms, for a number of reasons: a predominant H -polarization orientation of the substorm fields, the single order-of-magnitude contrast in conductivity, and the probably small depth extent of the structure. Attempts were made to estimate the response of a one-dimensional earth via the inductive scale length with gradients evaluated from polynomial surfaces fitted to the smoothly varying substorm fields. The results were inconsistent, owing to vertical fields with strong external components and to horizontal fields with scale lengths too small relative to their penetration distances.  相似文献   

8.
b
The effect of anisotropy on the distribution of Earth's conductivity is evaluated by calculating the electromagnetic response of multilayered 2-D structures. The electric and magnetic fields are expanded in terms of Fourier series, the coefficients being obtained by applying the corresponding boundary conditions on each interface, given by arbitrary analytical functions. Then the results are used to analyse some particular structures.  相似文献   

9.
Controlled-source electromagnetic (CSEM) surveys have the ability to provide tomo-graphic images of electrical conductivity within the Earth. the interpretation of such data sets has long been hampered by inadequate modelling and inversion techniques. In this paper, a subspace inversion technique is described that allows electric dipole-dipole data to be inverted for a 2-D electrical conductivity model more efficiently than with existing techniques. the subspace technique is validated by comparison with conventional inversion methods and by inverting data collected over the East Pacific Rise in 1989. A model study indicates that, with adequate data, a variety of possible mid-ocean-ridge conductivity models could be distinguished on the basis of a CSEM survey.  相似文献   

10.
Summary. Geomagnetic time-variations observed at several sites on the island of Hawaii are analysed for the effects of island bathymetry as well as for the inductive response of the deeper mantle. The data are generally consistent with the deep conductivity profile derived using lower frequency, electromagnetic data from the Island of Oahu. Hawaii data fit better if that model is modified to give the upper 200 km of the mantle a lower conductivity of 0.02 S/m compared to 0.1 S/m for Oahu. The data are represented by a complex, frequency-dependent function of location, T u, relating the vertical variation Z to a component U of the horizontal variation ( Tu = Z/U ). The direction of U is nearly frequency independent at each site but is different for each site. Below a frequency of about 30 cycles per day, the functions, T u, at any two sites are found to be related by a real constant. This suggests that the deeper conductivity structure is the same beneath each site. This result is consistent with quasi-static induction in a non-uniformly conducting thin sheet above a stratified conductivity structure. The response of such a model can be written as T u= Aq , where q is a quasi-uniform, complex, frequency-response function characterizing the effect of the deep conductivity and A is a spatially dependent parameter parameterizing the effect of variable conductivity in the thin sheet. The parameter A may be estimated by fitting observational estimates of T u to models of deep conductivity structure.  相似文献   

11.
Summary . Using a variational formulation for the response function V ( r ), commonly used in the inversion of electromagnetic induction data for a spherically symmetric earth, a number of independent expressions for the total variation of this response function with respect to perturbations in the (electrical) conductivity S o have been derived. These results have been used to indicate:
(1) How the boundary constraints contained in the expressions for the total variation of V ( r ) affect any computational implementation.
(2) How refinement modelling for the inversion of electromagnetic induction data can be implemented iteratively without the use of linearization.
In addition, these results have been used to examine the validity of Parker's linearization proposal by showing that his results depend heavily on the exclusion of certain boundary constraints, and the choice of the L 2 norm as the norm to use.  相似文献   

12.
基于表观电导率与实测光谱的干旱区湿地土壤盐分监测   总被引：2，自引：0，他引：2  
以新疆艾比湖滨盐渍化土壤为对象,利用磁感应电导仪和光谱仪测得的盐渍土表观电导率和可见光/近红外光谱数据,选取与EM38解译的土壤盐分相关性最好的光谱变换形式和特征波长,分别建立多元逐步回归、偏最小二乘回归和支持向量回归的土壤盐分监测模型。结果表明：（1）表观电导率两种模式相结合建立的盐分含量解译模型的拟合优度达到0.91,即在该区域内电磁感应技术可用于土壤盐分含量的间接监测。（2）一阶微分处理优于二阶微分,经一阶微分变换后的光谱可以较好地预测土壤盐分含量。（3）3种建模方法中,支持向量回归的建模精度最高,偏最小二乘回归和多元逐步回归次之。干旱区湖滨湿地土壤盐分含量的估测模型宜选取基于平滑后的原始一阶微分光谱数据建立的支持向量回归模型。  相似文献   

13.
Summary. In examining the effect of discontinuities in the Earth's interior on free oscillations, McNabb, Anderssen & Lapwood derived an equation for the asymptotic behaviour of torsional overtone eigenfrequencies of a discontinuous earth model, the constants in their equation being explicitly determined only for the case of one internal discontinuity. Since Brune's phase correlation method for the evaluation of eigenfrequencies from body-wave data implies a ray-mode duality only for continuous earth models, it is desirable to justify the McNabb et al. formulation from the point of view of ray theory.
By a novel method of ray analysis, Wang, Cleary & Anderssen showed that, for earth models with a single discontinuity between the Earth's surface and the core—mantle boundary, the McNabb et al. formulation can be derived from an adaptation of Brune's method to multiply reflected SH body waves recorded at small epicentral distances. In this paper, the technique of Wang et al. is extended to derive the McNabb et al. formulation (with constants explicitly determined) for the general case of earth models with N discontinuities. This establishes a basis for a ray-mode duality for discontinuous earth models.  相似文献   

14.
Summary. Using a single scattering approximation, we derive equations for the scattering attenuation coefficients of P- and S -body waves. We discuss our results in the light of some recent energy renormalization approaches to seismic wave scattering. Practical methods for calculating the scattering attenuation coefficients for various earth models are emphasized. The conversions of P - to S -waves and S- to P -waves are included in the theory. The earth models are assumed to be randomly inhomogeneous, with their properties known only through their average wavenumber power spectra. We approximate the power spectra with piecewise constant functions, each segment of which contributes to the net, frequency-dependent, scattering attenuation coefficient. The smallest and largest wavenumbers of a segment can be plotted along with the wavevectors of the incident and scattered waves on a wavenumber diagram. This diagram gives a geometric interpretation for the frequency behaviour associated with each spectral segment, including a 'transition' peak that is due entirely to the wavenumber limits of the segment. For regions of the earth where the inhomogeneity spectra are concentrated in a band of wavenumbers, it should be possible to observed such a peak in the apparent attenuation of seismic waves. We give both the frequency and distance limits on the accuracy of the theoretical results.  相似文献   

15.
We investigate the effect of laterally varying earth structure on centroid moment tensor inversions using fundamental mode mantle waves. Theoretical seismograms are calculated using a full formulation of surface wave ray theory. Calculations are made using a variety of global tomographic earth models. Results are compared with those obtained using the so-called great-circle approximation, which assumes that phase corrections are given in terms of mean phase slowness along the great circle, and which neglects amplitude effects of heterogeneity. Synthetic tests suggest that even source parameters which fit the data very well may have large errors due to incomplete knowledge of lateral heterogeneity. The method is applied to 31 shallow, large earthquakes. For a given earthquake, the focal mechanisms calculated using different earth models and different forward modelling techniques can significantly vary. We provide a range of selected solutions based on the fit to the data, rather than one single solution. Difficulties in constraining the dip-slip components of the seismic moment tensor often produce overestimates of seismic moment, leading to near vertical dip-slip mechanisms. This happens more commonly for earth models not fitting the data well, confirming that more accurate modelling of lateral heterogeneity can help to constrain the dip-slip components of the seismic moment tensor.  相似文献   

16.
We portray a dedicated spectral-element method to solve the elastodynamic wave equation upon spherically symmetric earth models at the expense of a 2-D domain. Using this method, 3-D wavefields of arbitrary resolution may be computed to obtain Fréchet sensitivity kernels, especially for diffracted arrivals. The meshing process is presented for varying frequencies in terms of its efficiency as measured by the total number of elements, their spacing variations and stability criteria. We assess the mesh quantitatively by defining these numerical parameters in a general non-dimensionalized form such that comparisons to other grid-based methods are straightforward. Efficient-mesh generation for the PREM example and a minimum-messaging domain decomposition and parallelization strategy lay foundations for waveforms up to frequencies of 1 Hz on moderate PC clusters. The discretization of fluid, solid and respective boundary regions is similar to previous spectral-element implementations, save for a fluid potential formulation that incorporates the density, thereby yielding identical boundary terms on fluid and solid sides. We compare the second-order Newmark time extrapolation scheme with a newly implemented fourth-order symplectic scheme and argue in favour of the latter in cases of propagation over many wavelengths due to drastic accuracy improvements. Various validation examples such as full moment-tensor seismograms, wavefield snapshots, and energy conservation illustrate the favourable behaviour and potential of the method.  相似文献   

17.
Summary. The study of the asymptotic behaviour of eigenfrequencies of torsional modes of the Earth is of interest in the problem of the existence of discontinuities in the Earth's interior and the determination of their depth and scale, since the solotone effect, which is a persistent oscillatory cohponent in the asymptotic overtone structure, is very sensitive to the presence of discontinuities. The asymptotic behaviour of torsional eigenfrequencies of the second order differential equation for the Earth's free oscillations can be compared with that obtained from eigenfrequencies evaluated from synthetic SH-wave seismograms by Brune's phase correlation method, using various earth models. The solotone effect that appears in the former for discontinuous models can be explained in terms of multiple reflections fram the discontinuities, and can be reconstructed from synthetic SH -wave pulses arising from these reflections. Its properties vary systematically with the depth and the scale of discontinuities and can be superposed for several discontinuities.  相似文献   

18.
In order to handle the distortion of large-scale induced electric currents by local conductivity anomalies, the problem of electromagnetic induction in non-uniform thin sheets has been reformulated in terms of an integral equation over the anomalous domain. This formulation considers in the layered substratum in addition to toroidal currents also the poloidal current mode (vertical current loops), at the expense that two scalar functions have to be determined. Simple formulas for the required kernels are derived. The algorithm is applied to model the gross features of the northern Pyrenean induction anomaly. It is suggested that this pronounced anomaly results from a conductive channel between the Atlantic Ocean and the Mediterranean Sea.  相似文献   

19.
Summary. The algorithm of Dawson & Weaver for modelling electromagnetic induction effects in a thin sheet at the surface of a uniform earth is modified to permit the use of a layered earth model. The theory is developed in Fourier space in terms of the toroidal and poloidal transfer functions instead of with the Green's function approach which was used by Dawson & Weaver. The integral equation for the surface electric field and most of the integral formulae for the derived field components are the same as before, except for the inclusion of additional integral the kernel of which has to be calculated numerically with the aid of fast Hankel transforms. The accuracy of the results is tested by comparing solutions with those obtained from a related 2-D algorithm and finally an example of 3-D modelling is presented.  相似文献   

20.
Summary. Amplitude spectra of Rayleigh and Love waves in a layered non-gravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson—Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical- and flat-earth models have been investigated in the range 20–300–s period and expressed in polynomials in the period.  相似文献   

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