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1.
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   

2.
Summary. Approximate methods of solution for induction in arbitrarily shaped oceans, derived earlier for 'oceans of infinite conductivity, are extended to cover cases where the oceans are of finite conductivity. The method enables the magnetic potential to be evaluated without undue effort. It is further shown how these ideas link up with integro-differential formulations involving either the oceanic electric current density or its associated current function. The problem of uniqueness of solution when there are two or more land masses is finally considered.  相似文献   

3.
Summary. The problem of determination of the electrical conductivity of the Earth from the geomagnetic induction data is formulated as that of finding the coefficients of the continued fraction expansion of a certain rational fraction representation of the total response (or impedance) of the medium at the surface when this quantity is given for N frequencies. The coefficients of expansion are related to the conductivities of N layers of constant attenuation, where, within each layer the conductivity is assumed to be constant. Thus in this approach the conductivity profile resulting from the inversion of the response function is given as a series of step functions.  相似文献   

4.
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   

5.
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   

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.
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.  相似文献   

8.
Summary. This paper proves that, subject to certain assumptions, the one-dimensional magnetotelluric response function and the associated electric fields are Fréchet differentiable with respect to the conductivity function. The flaw in the proof offered by an earlier author is explained.  相似文献   

9.
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   

10.
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.  相似文献   

11.
We introduce two sets of fully normalized harmonics for the spectral analysis of functions defined on a spherical cap. The harmonics are the products of Fourier functions and the fully normalized associated Legendre functions of non-integer degree. Using Sturm-Liouville theory for boundary-value problems, we present two convenient and stable formulae for computing the zeros of the associated Legendre functions that form two sets of orthogonal functions. Formulae for the stable numerical evaluation of the fully normalized associated Legendre functions of non-integer degree that avoid the gamma function are also derived. The result from the expansions of sea-level anomaly from altimetry into Set 2 fully normalized cap harmonics shows fast convergence of the series, and the degree variances decay rapidly without aliasing effects. The zero-degree coefficients (Set 2) of sea-level anomaly from TOPEX/ POSEIDON (T/P) and ERS-1 indicate an El Niño event during 1993 January-1993 July, and a La Niño event during 1993 November-1994 July, although the ERS-1 result is less obvious. Ocean circulations over the South China Sea and the Kuroshio area are clearly identified with the low-degree expansions of sea-surface topography (SST) from T/P and ERS-1. A cold-core eddy of 4° in diameter centred at 17.5°N, 118 E was detected with the expansion of SST from T/P cycle 47, and a property of the cap harmonics is used to compute this eddy's kinetic energy. The kinetic energy is at a low in winter and high in summer, and its variation seems to be periodic with an amplitude of 0.4 m2 S-2.  相似文献   

12.
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13.
塔克拉玛干沙漠地下水矿化度与电导率关系的研究   总被引：5，自引：0，他引：5  
吴诗怡 《中国沙漠》1996,16(4):374-378
用数理统计的方法推导水质矿化度与电导率的回归方程, 并绘制其关系曲线。经测定, 新疆塔克拉玛干沙漠84°E沿线水化学类型主要为Cl-·SO42--Na+, 通过推导得出, 25℃情况下, 当电导率为4.5~12ms/cm, 矿化度依电导率的回归方程为: y=-0.3016+0.6936x; 当电导率<5.4ms/cm, y=-0.171+0.6833x, 建立矿化度与电导率的关系后, 测定地下水的电导率, 即可获得待测样的矿化度, 为野外考察或定点观测提供获得地下水矿化度的简便、快速而准确的方法。  相似文献   

14.
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.  相似文献   

15.
b
Spherical harmonics are orthonormalized using the Gram-Schmidt process in a function space. The problem of linear dependence of spherical harmonics over the oceans is studied using the Gram matrices and consequently three sets of orthonormal (ON) functions have been constructed. For the process an efficient formula for computing inner products of spherical harmonics has been developed. Important spectral properties of the ON functions are addressed. The ON functions may be used for representing the sea surface topography (SST) in the analysis of satellite altimeter data. The geoid error can be transformed to a representation by the ON functions and hence the comparison of powers of the geoid error and the SST signal only over the oceans is possible, leading to a better way of determining the cut-off frequency of the SST in the simultaneous solution using satellite altimeter data. As a case study, the modified Levitus SST is expanded into the ON functions. The results show that 99.90 per cent of that signal's energy is contained within degree 24 of the orthonormal functions. Such expansions also render better spectral behaviour of oceanic signals as compared to that from spherical harmonic expansions. The study shows that these generalized Fourier functions are suitable for spectral analyses of oceanic signals and they can be applied to future altimetric mission where the geoid and the SST are to be recovered.  相似文献   

16.
Prediction on the coupled thermal-hydraulic fields of embankment and cutting slopes is essential to the assessment on evolution of melting zone and natural permafrost table, which is usually a key factor for permafrost embankment designin frozen ground regions. The prediction may be further complicated due to the inherent uncertainties of materialproperties. Hence, stochastic analyses should be conducted. Firstly, Karhunen-Loeve expansion is applied to attain the random fields for hydraulic and thermal conductions. Next, the mixed-form modified Richards equation for mass transfer (i.e., mass equation) and the heat transport equation for heat transient flow in a variably saturated frozen soil are combined into one equation with temperature unknown. Furthermore, the finite element formulation for the coupled thermal-hydraulic fields is derived. Based on the random fields, the stochastic finite element analyses on stability of embankment are carried out. Numerical results show that stochastic analyses of embankment stability may provide a more rational picture for the distribution of factors of safety (FOS), which is definitely useful forembankment design in frozen ground regions.  相似文献   

17.
Summary. A striking feature of the day-side response of the Moon to periodic fluctuations in the solar wind is the rapid rise, and subsequent fall, in the amplitude of the transfer function as the inducing field frequency increases. This behaviour can be characterized by the amplitude values at the two frequencies 24 and 40 mHz. Before the response of a conductivity model representing the Moon can be calculated at a given frequency, the parameters (ν, θ) (where ν is the solar wind speed and θ is the angle between the solar wind velocity and the magnetic field propagation direction) have to be specified. By applying some results due to Parker (1972) to the above two data points, we have determined constraints on the parameter space (ν, θ). In particular, we determine the region of the (ν, θ) space in which conductivity models may be found that satisfy our data pair. Outside this region, there are no conductivity models satisfying the data pair, and hence many (ν, θ) values are inconsistent with the original data and the model assumptions.  相似文献   

18.
Classical treatment of mass conservation of solids in growing sediments states an advection-diffusion equation for the bulk sediment density. Thus, the diffusion coefficient has to account for elemental processes of exchange of solid particles by pore water or reciprocally. Nevertheless, in a gravity field, these exchanges are forced and cannot be treated as diffusion but as mass flow. A compaction potential energy is defined so that its spatial gradients force a mass flow involving a conductivity function. This leads to a more consistent definition of mass sedimentation rates and to a writing of the continuity equation for density involving only an advection term. Typical bulk density profiles show an asymptotic increase with depth. With the present formulation, this can be obtained as a steady-state solution under constant sedimentation rate and constant conductivity, while the classical formulation fails to do it. Alternatively, it can be found that under these conditions, the compaction potential is a linear function of the bulk density. The mass flow due to compaction and the compaction potential are found for several sediment cores from literature data. From this basis, the advection-diffusion equation for a particle-associated tracer is rewritten. In particular, when the mass flow term due to compaction is considered, the resulting sub-grid scale processes lead to a different formulation of the diffusive fluxes: they are proportional to the gradients of specific concentration in solids instead of the concentration per unit bulk volume. This new formulation is most suitable to find out analytical solutions for radiometric dating models involving mixing and compaction. Numerical solutions are found for the new and the classical treatment for some particular cases to illustrate differences.  相似文献   

19.
Long-period geomagnetic data can resolve large-scale 3-D mantle electrical conductivity heterogeneities which are indicators of physiochemical variations found in the Earth's dynamic mantle. A prerequisite for mapping such heterogeneity is the ability to model accurately electromagnetic induction in a heterogeneous sphere. A previously developed finite element method solution to the geomagnetic induction problem is validated against an analytic solution for a fully 3-D geometry: an off-axis spherical inclusion embedded in a uniform sphere. Geomagnetic induction is then modelled in a uniform spherical mantle overlain by a realistic distribution of oceanic and continental conductances. Our results indicate that the contrast in electrical conductivity between oceans and continents is not primarily responsible for the observed geographic variability of long-period geomagnetic data. In the absence of persistent high-wavenumber magnetospheric disturbances, this argues strongly for the existence of large-scale, high-contrast electrical conductivity heterogeneities in the mid-mantle. Lastly, for several periods the geomagnetic anomaly associated with a mid-mantle spherical inclusion is calculated. A high-contrast inclusion can be readily detected beneath the outer shell of oceans and continents. A comparison between observed and computed c responses suggests that the mid-mantle contains more than one order of magnitude of lateral variability in electrical conductivity, while the upper mantle contains at least two orders of magnitude of lateral variability in electrical conductivity.  相似文献   

20.
Summary. Compressional and shear wave velocities and electrical conductivity have been measured simultaneously on three saturated samples of granite up to 0.4 GPa effective pressure and one of chalk up to 0.04 GPa effective pressure. The velocity–pressure data have been inverted using the non-interactive theory of Kuster & Toksöz to produce pore aspect ratio spectra. Archie's Law, and a model devised by Hoening have been used to calculate a theoretical electrical conductivity from the derived pore spectra. In general there is good agreement between the observed conductivity and the conductivity calculated using Hoening's equation applied to 'cigar-shaped' cracks. Predicted conductivities obtained via Archie's Law are in approximate agreement using a power of two and are much less than the observed values using a power of one. The results suggest that simultaneous electrical conductivity measurements might provide an important constraint in interpreting velocity—pressure data in terms of pore spectra.  相似文献   

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