Two-dimensional induction in a thin sheet of variable integrated conductivity at the surface of a uniformly conducting earth

Summary. The forward solution of the general two-dimensional problem of induction in a model earth comprising a uniformly conducting half-space covered by a thin sheet of variable integrated conductivity is obtained. Unlike some previous treatments of similar problems, the method presented here does not require the field to be separated into its normal and anomalous parts. Both the E - and B -polarization modes of induction are considered and in each case the solution is expressed in terms of the horizontal component of the electric field satisfying, on the surface of the conductor, a singular integral equation whose kernel is a well-known analytic function. A recently published solution of the coast effect is included as a special case. The numerical procedure for solving the integral equations is described and some illustrative calculations are presented.     

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 ).     

An algebraic formulation of geophysical inverse problems

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.     

The growing spherical seismic source

Summary. A solution is found for the seismic radiation from an arbitrarily growing spherical source in an inhomogeneously prestressed elastic medium. The general problem of the growing seismic source in a prestressed medium is formulated as a boundary value problem. For the special case of the growing spherical source, an expansion in vector spherical harmonics reduces the problem to a set of one-dimensional Volterra integral equations. The equations can be easily formed through the use of Bessel function recursion relations. The integral equations for a growing spherical cavity are solved numerically. Waveforms are then computed for homogeneous and inhomogeneous stress fields for several growth histories. The resulting waveforms are similar to the waveforms of the corresponding instantaneous problem, but stretched out in time and reduced in amplitude. The effects of diffraction and the overshoot of equilibrium are reduced with a reduction in growth rate. The effects caused by inhomogeneity of the stress field are quite strong for the growing as well as for the instantaneous seismic source.     

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.     

An explicit method for solving the geomagnetic induction problem

Summary. The problem of inferring the Earth conductivity as a function of depth from surface measurements of the magnetic field is considered. Two versions of a method analogous to that of Bailey are presented. The main feature of the method consists in the fact that the conductivity is given explicitly in terms to two auxiliary sequences of functions which are found by integrating a set of first-order non-linear differential equations.     

Spectral-finite-element approach to two-dimensional electromagnetic induction in a spherical earth

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.     

An Integral Equation and its Solution for some Two-and Three-Dimensional Problems in Resistivity and Induced Polarization

Summary. The potential, U , about a point electrode, at the surface of a layered ground in which there is an heterogeneity embedded, satisfies the integral equation:
Here, U * and σ* are the corresponding quantities for the potential and conductivity without the heterogeneity. The integral is taken over the surface of the heterogeneity, ∂ U /∂ n is the normal derivative (in the direction of the outward normal) of U , and G is a Green's function.
Solutions to this equation can readily be found by using the Galerkin method of solving integral equations. The solutions of this equation when the heterogeneity is a sphere or a cylinder in a uniform ground or beneath a conductive overburden are the most readily found.
When the solution of the integral has been found for the potential it is a simple matter to calculate the apparent resistivity or chargeability for any electrode configuration.     

Electromagnetic induction in the Earth in electrical contact with the oceans

Summary. Integral equations are formulated to solve the problem of electromagnetic induction in the Earth in the presence of thin finitely conducting oceans. Unlike earlier models, the oceans are assumed to be in electrical contact with the mantle and leakage of oceanic electric current is therefore possible. A powerful iterative method is developed to solve the very large system of equations and is applied successfully to a model problem. A feature of the integral equations is that they lead to a good first approximation. When fed into the iterative scheme this enhances the rate of convergence. Variable mesh size also appears possible.     

Climatic physical snowpack properties for large-scale modeling examined by observations and a physical model

Here we have conducted an integral study using site observations and a model with detailed snow dynamics, to examine the capability of the model for deriving a simple relationship between the density and thermal conductivity of the snowpack within different climatic zones used in large-scale climate modeling. Snow and meteorological observations were conducted at multiple sites in different climatic regions (two in Interior Alaska, two in Japan). A series of thermal conductivity measurements in snow pit observations done in Alaska provided useful information for constructing the relationship. The one-dimensional snow dynamics model, SNOWPACK, simulated the evolution of the snowpack and compared observations between all sites. Overall, model simulations tended to underestimate the density and overestimate the thermal conductivity, and failed to foster the relationship evident in the observations from the current and previous research. The causes for the deficiency were analyzed and discussed, regarding a low density of the new snow layer and a slow compaction rate. Our working relationships were compared to the equations derived by previous investigators. Discrepancy from the regression for the melting season observations in Alaska was found in common.     

Efficient global matrix approach to the computation of synthetic seismograms

Summary. A numerically efficient global matrix approach to the solution of the wave equation in horizontally stratified environments is presented. The field in each layer is expressed as a superposition of the field produced by the sources within the layer and an unknown field satisfying the homogeneous wave equations, both expressed as integral representations in the horizontal wavenumber. The boundary conditions to be satisfied at each interface then yield a linear system of equations in the unknown wavefield amplitudes, to be satisfied at each horizontal wavenumber. As an alternative to the traditional propagator matrix approaches, the solution technique presented here yields both improved efficiency and versatility. Its global nature makes it well suited to problems involving many receivers in range as well as depth and to calculations of both stresses and particle velocities. The global solution technique is developed in close analogy to the finite element method, thereby reducing the number of arithmetic operations to a minimum and making the resulting computer code very efficient in terms of computation time. These features are illustrated by a number of numerical examples from both crustal and exploration seismology.     

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.     

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.     

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.     

Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modelling

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.     

A model of time-periodic mantle flow

Summary. The instability of a layer consisting of a lighter viscous fluid on top of a heavier less viscous fluid is considered in the case when the heavy fluid is adiabatically stratified and the light fluid contains heat sources and possesses a lower heat conductivity. A perturbation in the thickness of the upper fluid layer causes horizontal temperature variations in the lower fluid. The motions induced by thermal buoyancy can interact with the distortion of the interface in such a way that the initial perturbation is reinforced in the form of an overstable oscillation. It is proposed that this mechanism is relevant to the problem of time-dependent flow in the Earth's mantle.     

The oceanological application of electromagnetic fields induced by synoptic currents

Summary. The determination of induced magnetic fields of synoptic ocean movements is considered in the case of horizontal uniformity of the bottom conductivity. The problem of calculation of hydrodynamic parameters is solved by means of induced fields. These appear to depend closely on the appearance of barotropic or baroclinic parts in the synoptic ocean movement. The different properties of fields from parts of the above permit us to separate hydrodynamic parameters of complex ocean movements into barotropic and baroclinic parts by means of magnetic measurements.     

Electric field at the seafloor due to a two-dimensional ionospheric current

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.     

Bimodal electromagnetic induction in non-uniform thin sheets with an application to the northern Pyrenean induction anomaly

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.     

Electromagnetic 2.5-dimensional forward modelling with a boundary integral formulation

An effective and accurate technique for the numerical solution of 2-D electromagnetic scattering problems with 3-D sources is presented. This solution introduces a set of the usual boundary integral equations and uses a scalar Green's function. In this scalar version, the unknowns of the problem are the boundary values of the longitudinal fields and their normal derivatives in the Fourier domain. A generalization of the usual boundary integral formulation enables us to handle a large class of models composed of piecewise homogeneous domains, including contiguous domains, multiply-connected domains and unbounded domains. This formulation involves the solution of a system of linear equations, and results in a significant saving in computation time in comparison with other rigorous methods.
  The requirements for the numerical implementation of this solution are described in detail. Numerical tests were carried out using the important example of electromagnetic tomography. The specific symmetry properties of the response function in this case are illustrated. Numerical accuracy is verified over a large frequency range, up to 1  MHz.