Seismic motion in urban sites consisting of blocks in welded contact with a soft layer overlying a hard half-space

We address the problem of the response to a seismic wave of an urban site consisting of   N _b   blocks overlying a soft layer underlain by a hard substratum. The results of a theoretical analysis, appealing to a space–frequency mode-matching (MM) technique, are compared to those obtained by a space–time finite-element (FE) technique. The two methods are shown to give rise to the same prediction of the seismic response for   N _b = 1  , 2 and 40 blocks. The mechanism of the interaction between blocks and the ground, as well as that of the mutual interaction between blocks, are studied. It is shown, in the first part of this paper, that the presence of a small number of blocks modifies the seismic disturbance in a manner which evokes qualitatively, but not quantitatively, what was observed during the 1985 Michoacan earthquake in Mexico City. Anomalous earthquake response at a much greater level, in terms of duration, peak and cumulative amplitude of motion, is shown, by a theoretical and numerical analysis in the second part of this paper, to be induced by the presence of a large (≥10) number of identical equi-spaced blocks that are present in certain districts of many cities.     

Geoelectromagnetic induction in a heterogeneous sphere:a new three-dimensional forward solver using a conservative staggered-grid finite difference method

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 ).     

E-polarization induction in two thin half-sheets

Summary. An analytical solution is obtained for the E-polarization problem of electromagnetic induction in two adjacent half-sheets underlain by a uniform conducting half-space. In this mode the inducing magnetic field is assumed horizontal, uniform and perpendicular to the discontinuity. The same model was previously solved under B-polarization by Dawson & Weaver. The present solution then completes the study of two-dimensional induction in the described model. Further, it extends both the analytic E-polarization solution of Weidelt by the inclusion of an underlying conductor and that of Raval, Weaver & Dawson by the inclusion of arbitrary conductance values for the two surface sheets. The solution may be used as an idealized model of the coast effect and allows detailed study of the field behaviour near the discontinuity. The horizontal magnetic field on each side of the surface layer has a finite jump discontinuity at the interface and the vertical magnetic field exhibits a logarithmic singularity there. If the right-hand conductance (say) becomes infinite, the horizontal magnetic field exhibits an algebraic singularity as the coastline is approached from the right, while the vertical magnetic field does likewise from the left. Calculations are presented for the same two models as discussed in B-polarization by Dawson & Weaver and the results are compared to values obtained from a more general numerical scheme. The electric current distribution inside the conducting half-space is depicted for the second model.     

Recovering magnetic susceptibility from electromagnetic data over a one-dimensional earth

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.     

Effect of the presence of a conducting channel between India and Sri Lanka Island on the features of the equatorial electrojet

Summary. Transient geomagnetic variations like SSCs, Bays, Sq and storm-time variations show anomalously large Z amplitudes at the three permanent magnetic observatories in India under the equatorial electrojet. Our earlier studies have shown that these anomalies cannot be explained in terms of the usual coastal effect. Another unique feature of this area is the small equatorial enhancement of all magnetic fluctuations, which still remains to be completely understood. It is also noticed that Z/H ratio at Annamalainagar is very large for night-time variations but becomes insignificant for day-time fluctuations, whereas no such large difference is seen at Trivandrum, the other coastal station. All the above features have been explained here by introducing the presence of a conducting channel in the lower crust or upper mantle between India and Sri Lanka Island. The anomalies of the equatorial electrojet in the Indian region thus do not necessitate the conducting surface of the mantle to be deeper in this area as has been suggested by earlier workers. A less conducting mantle in the Indian region otherwise could be difficult to incorporate in the present theories of mantle convection when this area is known to be tectonically active. The presence of the conducting channel is further confirmed by analogue model experiments of Papamastorakis. We further observe that the equatorial electrojet does have an associated internal part in the Indian region.     

三西地区可持续发展水平综合定量评价研究

以三西地区为背景，针对区域可持续发展水平综合评价问题，建立多层次、多要素评价指标体系，以及多级模糊综合评价的定量化计算模型，借助于该模型，运用有关实地调查资料和统计数据，对三西地区各县（市、区）可持续发展水平进行综合排序计算，有关结果为三西地区可持续发展决策提供了依据。     

Three-dimensional induction in a non-uniform thin sheet at the surface of a uniformly conducting earth

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.     

Modelling of electromagnetic fields in thin heterogeneous layers with application to field generation by volcanoes—theory and example

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.     

A comparison of Sq analyses with model calculations

Summary. Three analyses of Sq are examined in detail. The analyses separate Sq into external and internal parts. The external part is used as an inducing field in a model earth consisting of a perfectly conducting inner sphere (con-ductosphere) surrounded by a thin non-uniformly conducting shell representing in some detail the distribution of oceans and continents. The calculated induced part is compared to the internal part deduced in the analysis. The comparison is poor and the reason is shown to be that the magnetic observatories are not located in regions influenced by the main induced oceanic currents. It is further shown that some Sq analyses are unacceptable on physical grounds.     

Analytical approach for the toroidal relaxation of viscoelastic earth

This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core–mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes—unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core–mantle boundary and Earth's surface—however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works.     

Three-dimensional massively parallel electromagnetic inversion—I. Theory

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     

Interpretation of long-offset transient electromagnetic data from the Odenwald area, Germany, using two-dimensional modelling

The standard 1-D inversion approach for the interpretation of transient electromagnetic (TEM) data usually fails in the presence of near-surface conductivity anomalies. Since multidimensional inversion codes are not routinely available, the only alternative to discarding the data may be trial-and-error forward modelling. We interpret data from a long-offset transient electromagnetic (LOTEM) survey which was carried out in 1995 in the Odenwald area, using 2-D finite-difference modelling. We focus on a subsegment of the LOTEM profile, which was shot with two different electric dipole transmitters. A model is found which consistently explains the electric and magnetic field data at eight locations for both transmitters. First, we introduce a conductive dyke under the receiver spread to explain sign reversals in the magnetic field transients. A conductive slab under one of the transmitters is required to obtain a reasonable quantitative fit for that transmitter. Consideration of the electric field data then requires a modification of the layered earth background. Finally, we study the response of a crustal conductor, which was the original target of the survey. The data are sensitive to the conductor, and for the investigated subset of the data the fits are slightly better without the conductive layer.     

Numerical experiments on the onset of convective instability in the Earth's mantle

Summary. A simplified model of convection in the mantle is used to investigate the transient effect of cooling a fluid layer from above, The model, representing the mantle overlain by the lithosphere, consists of a two-dimensional fluid layer overlain by a solid conducting lid. The initial temperature of both layers is the same, with the top surface of the lid kept at 0°C throughout. We observe the onset of small-scale flow in the model. In the absence of internal heating the behaviour of the system is controlled by the Rayleigh number, R , and the ratio of the thicknesses of the two layers, a . The onset time of convection as defined by reference to conduction temperature profiles is related simply to a boundary layer critical Rayleigh number. The mean temperature profiles for the convection model are also compared with the observed depth—age relation for oceanic lithosphere and the results are used to estimate the viscosity of the mantle.     

Time-domain electromagnetic migration in the solution of inverse problems

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.     

H-polarization induction over an ocean edge coupled to the mantle by a conducting crust

Summary. The Wiener—Hopf technique is used to obtain an exact analytical solution for the problem of H -polarization induction over the edge of a perfectly conducting thin sheet, representing an ocean, electrically connected to a perfectly conducting mantle through a slab of finite conductivity and thickness, which represents the Earths crust. It is shown that the induced currents resulting from this type of induction process are drawn up into the sea from the cust and mantle with the greatest concentration of current near the ocean edge. The surface impedance over the land surface, is calculated for various mantle depths and is shown to increase sharply as the coastline is approached. The magnetic field along the ocean floor is also plotted as a function of distance from the coastline, and the results are found to agree very well with those calculated previously by approximate and numerical methods.     

On the non-linear evolution of sand dunes

Summary. Adopting a multiple scale method, the non-linear evolution of fully developed dunes of non-cohesive sand is studied using Kennedy's model. It is shown that both the two- and three-dimensional problems are, in general, governed by three coupled second-order partial differential equations. However, for boundary conditions appropriate to localized solutions these equations reduce to one for the two-dimensional problem and to two for the three-dimensional problem. It is also found that the uniform Stokes 'wave' is a solution of the general problem, and the stability of this solution (on the long time-scale of the non-linear theory) is analysed for various values of the parameters of the problem. The significance of the soliton solution of the single equation for the two-dimensional problem is discussed and the relevance of the localized solutions of the two coupled equations to the formation and interaction of fully developed dunes is indicated. The results are in broad agreement with the cumulative effects of dune movement over long periods.     

B-polarization induction in two generalized thin sheets at the surface of a conducting half-space

Summary. A new closed-form solution is obtained analytically for a B- polarization induction problem of geophysical interest, in which a local region of the Earth is represented by a generalized thin sheet at the surface of and in electrical contact with a uniformly conducting half-space. The generalized sheet, first introduced by Ranganayaki & Madden, is a mathematical idealization of a double layer which consists, in this problem, of two adjacent half-planes with distinct conductances representing a surface conductivity discontinuity such as an ocean—coast boundary, underlain by a uniform sheet of finite integrated resistivity representing the lower crust. The resistive sheet exerts a considerable mathematical influence on the solution causing, under certain conditions, an additional pole to appear in one of the forms of contour integral by which the solution can be expressed; it also weakens or eliminates field singularities that would otherwise occur at the conductance discontinuity. A numerical calculation is made for model parameters typifying an ocean—coast boundary underlain by a highly resistive crust. It is found that the residue of the pole associated with the resistive sheet dominates the solution for this example, the main consequence of which is a huge increase in the horizontal range over which the induced currents adjust themselves between the different 'skin-effect' distributions at infinity on either side of the model. Moreover the solution shows that this 'adjustment distance' has a more complicated dependence on the conductance and integrated resistivity of the sheet than that given simply by the square root of their product which was the length parameter proposed by Ranganayaki & Madden.     

Model of the geoelectrical structure of the mid- and lower mantle in the Europe–Asia region

The conductivity structure of the Earth's mantle was estimated using the induction method down to 2100  km depth for the Europe–Asia region. For this purpose, the responses obtained at seven geomagnetic observatories (IRT, KIV, MOS, NVS, HLP, WIT and NGK) were analysed, together with reliable published results for 11  yr variations. 1-D spherical modelling has shown that, beneath the mid-mantle conductive layer (600–800  km), the conductivity increases slowly from about 1  S  m⁻¹ at 1000  km depth to 10  S  m⁻¹ at 1900  km, while further down (1900–2100  km) this increase is faster. Published models of the lower mantle conductivity obtained using the secular, 30–60  yr variations were also considered, in order to estimate the conductivity at depths down to the core. The new regional model of the lower mantle conductivity does not contradict most modern geoelectrical sounding results. This model supports the idea that the mantle base, situated below 2100  km depth, has a very high conductivity.     

Transient electromagnetic responses in seafloor with triaxial anisotropy

Electrical anisotropy of young oceanic crust at mid-ocean ridges is detectable by observation of the rate and geometry of the diffusion of electromagnetic fields. The anisotropy in electrical properties arises from the presence of conductive seawater in an interconnected network of mostly ridge-parallel cracks. In this paper, we first justify the choice of a triaxial model to represent young oceanic crust, with three distinct electrical conductivities in the vertical, strike and spreading directions. We then present an algorithm to calculate the transient electromagnetic responses generated by an electric dipole source over such a triaxially anisotropic seafloor. We show that if the transient passages are measured with three distinct electric dipole-dipole configurations, it is possible to discern all three unknown conductivities independently of each other.     

ON TAKING MEANS OF DENSITY DISTRIBUTIONS IN THE EARTH

Summary. The error is investigated when the mean of two solutions of a non-linear second-order differential equation which gives the variation of density with depth in certain layers of the Earth is used as a solution.
A solution (Model M) of the equation for layer E is given, which has a density of 9.385 g/m³ at the top of layer E.
Identifying, at the top of layer E, the mean of two numerically estimated solutions with a similarly estimated solution, it is shown that the error is a maximum at the bottom of layer E, where it is of the order of 0.04 per cent.