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1.
Inversion of time domain three-dimensional electromagnetic data   总被引：7，自引：0，他引：7  
We present a general formulation for inverting time domain electromagnetic data to recover a 3-D distribution of electrical conductivity. The forward problem is solved using finite volume methods in the spatial domain and an implicit method (Backward Euler) in the time domain. A modified Gauss–Newton strategy is employed to solve the inverse problem. The modifications include the use of a quasi-Newton method to generate a pre-conditioner for the perturbed system, and implementing an iterative Tikhonov approach in the solution to the inverse problem. In addition, we show how the size of the inverse problem can be reduced through a corrective source procedure. The same procedure can correct for discretization errors that inevidably arise. We also show how the inverse problem can be efficiently carried out even when the decay time for the conductor is significantly larger than the repetition time of the transmitter wave form. This requires a second processor to carry an additional forward modelling. Our inversion algorithm is general and is applicable for any electromagnetic field  ( E , H , d B / dt )  measured in the air, on the ground, or in boreholes, and from an arbitrary grounded or ungrounded source. Three synthetic examples illustrate the basic functionality of the algorithm, and a result from a field example shows applicability in a larger-scale field example.  相似文献   

2.
We investigate the use of general, non- l 2 measures of data misfit and model structure in the solution of the non-linear inverse problem. Of particular interest are robust measures of data misfit, and measures of model structure which enable piecewise-constant models to be constructed. General measures can be incorporated into traditional linearized, iterative solutions to the non-linear problem through the use of an iteratively reweighted least-squares (IRLS) algorithm. We show how such an algorithm can be used to solve the linear inverse problem when general measures of misfit and structure are considered. The magnetic stripe example of Parker (1994 ) is used as an illustration. This example also emphasizes the benefits of using a robust measure of misfit when outliers are present in the data. We then show how the IRLS algorithm can be used within a linearized, iterative solution to the non-linear problem. The relevant procedure contains two iterative loops which can be combined in a number of ways. We present two possibilities. The first involves a line search to determine the most appropriate value of the trade-off parameter and the complete solution, via the IRLS algorithm, of the linearized inverse problem for each value of the trade-off parameter. In the second approach, a schedule of prescribed values for the trade-off parameter is used and the iterations required by the IRLS algorithm are combined with those for the linearized, iterative inversion procedure. These two variations are then applied to the 1-D inversion of both synthetic and field time-domain electromagnetic data.  相似文献   

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

4.
Summary . Born inverse methods give accurate and stable results when the source wavelet is impulsive. However, in many practical applications (reflection seismology) an impulsive source cannot be realized and the inversion needs to be generalized to include an arbitrary source function. In this paper, we present a Born solution to the seismic inverse problem which can accommodate an arbitrary source function and give accurate and stable results. It is shown that the form of the generalized inversion algorithm reduces to a Wiener shaping ***filter, which is solved efficiently using a Levinson recursion algorithm. Numerical examples of synthetic and real field data illustrate the validity of our method.  相似文献   

5.

The use of spontaneous potential (SP) anomalies is well known in the geophysical literatures because of its effectiveness and significance in solving many complex problems in mineral exploration. The inverse problem of self-potential data interpretation is generally ill-posed and nonlinear. Methods based on derivative analysis usually fail to reach the optimal solution (global minimum) and trapped in a local minimum. A new simple heuristic solution to SP anomalies due to 2D inclined sheet of infinite horizontal length is investigated in this study to solve these problems. This method is based on utilizing whale optimization algorithm (WOA) as an effective heuristic solution to the inverse problem of self-potential field due to a 2D inclined sheet. In this context, the WOA was applied first to synthetic example, where the effect of the random noise was examined and the method revealed good results using proper MATLAB code. The technique was then applied on several real field profiles from different localities aiming to determine the parameters of mineralized zones or the associated shear zones. The inversion parameters revealed that WOA detected accurately the unknown parameters and showed a good validation when compared with the published inversion methods.

  相似文献   

6.
We explore the inner dynamics of daily geoelectrical time series measured in a seismic area of the southern Apennine chain (southern Italy). Autoregressive models and the Higuchi fractal method are applied to extract maximum quantitative information about the time dynamics from these geoelectrical signals. First, the predictability of the geoelectrical measurements is investigated using autoregressive models. The procedure is based on two forecasting approaches: the global and the local autoregressive approximations. The first views the data as a realization of a linear stochastic process, whereas the second considers the data points as a realization of a deterministic process, which may be non-linear. Comparison of the predictive skills of the two techniques allows discrimination between low-dimensional chaos and stochastic dynamics. Our findings suggest that the physical systems governing electrical phenomena are characterized by a very large number of degrees of freedom and can be described only with statistical laws. Second, we investigate the stochastic properties of the same geoelectrical signals, searching for scaling laws in the power spectrum. The spectrum fits a power law P (  f )∝  f  −α , with the scaling exponent α a typical fingerprint of fractional Brownian processes. In this analysis we apply the Higuchi method, which gives a linear relationship between the fractal dimension D Σ and the spectral power law scaling index α : D Σ=(3− α )/2. This analysis highlights the stochastic nature of geoelectrical signals recorded in this seismic area of southern Italy.  相似文献   

7.
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   

8.
Summary. In the present paper, we give an approximate analytic solution of the Dirichlet and Neumann problems in the two-dimensional case for a contour of any shape. The formalism is based on a representation of potential functions as a sum of elementary interpolating functions and uses the theory of generalized inverse matrices. Formulae are given in the cases of Cartesian, polar and elliptic co-ordinates. The fact that an analytic expression for continuation is obtained enables one to compute and draw equipotential lines as well as field lines; besides, any further computations that one might want to perform on either the field or the potential can be handled in an analytic way. The formalism can be extended from the Laplace to the Helmholtz equation.
We give examples in the case of magnetostatics, treating in more detail the problem of the distortion of magnetic field lines by an inclusion. We also show how the method allows the computation of conformal mappings in otherwise intricate situations.  相似文献   

9.
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   

10.
Spherical Slepian functions and the polar gap in geodesy   总被引：4，自引：0，他引：4  
The estimation of potential fields such as the gravitational or magnetic potential at the surface of a spherical planet from noisy observations taken at an altitude over an incomplete portion of the globe is a classic example of an ill-posed inverse problem. We show that this potential-field estimation problem has deep-seated connections to Slepian's spatiospectral localization problem which seeks bandlimited spherical functions whose energy is optimally concentrated in some closed portion of the unit sphere. This allows us to formulate an alternative solution to the traditional damped least-squares spherical harmonic approach in geodesy, whereby the source field is now expanded in a truncated Slepian function basis set. We discuss the relative performance of both methods with regard to standard statistical measures such as bias, variance and mean squared error, and pay special attention to the algorithmic efficiency of computing the Slepian functions on the region complementary to the axisymmetric polar gap characteristic of satellite surveys. The ease, speed, and accuracy of our method make the use of spherical Slepian functions in earth and planetary geodesy practical.  相似文献   

11.
Summary. We investigate the issues of stability and conditioning for the one-dimensional seismic inverse problem. We show that these issues are distinct; i.e. that numerically stable implementations of solutions to the inverse problem will not give accurate results if the problem is ill-conditioned. In addition, we identify the factors which determine the condition of the inverse problem. These are the total variation of the acoustic impedance profile being sought and the accuracy of the low-frequency content of the reflection data. We illustrate these results on implementations of two numerically stable algorithms for the inverse problem, one of which has a reputation for being unstable. A comparison shows nearly identical results for the two methods on noise-contaminated and frequency band limited reflection data. In fact, we conjecture that all of the well-known 'layer-stripping' inverse scattering methods share the same mathematical stability characteristics. On the other hand, we also show that ill-conditioning can lead to failure of such algorithms, through amplification of error due either to inaccurate data or to discretization or roundoff. Finally, we observe that appropriate smoothing of the seismic data for an ill-conditioned inverse problem (high-variation impedance profile) can cause the problem to become well-conditioned (lower-variation profile). As is typical with regularizations, the price paid for the newly-acquired ability to solve the problem is a loss of accuracy in the solution.  相似文献   

12.
A new method for computing synthetic seismograms   总被引：10，自引：0，他引：10  
Summary. The computation of theoretical seismograms for models in which the elastic parameters and density vary only with depth (in a plane, cylindrical or spherical geometry) reduces to the solution of an ordinary differential equation plus the evaluation of inverse transformations. In principle, the problem is straightforward. In practice, many techniques and approximations can be used at each stage and many combinations and variants are possible. In this paper, we discuss a new method of evaluating the inverse transforms. Any method can be used to solve the differential equation and we only discuss a few analytic approximations to illustrate the new method. The inverse transformations are a frequency and wavenumber integral. Essentially four techniques can be used to evaluate these depending on the order of integration and whether the wavenumber integral is distorted from the real axis. Three of these have been widely used, but the technique of evaluating the frequency integral first and keeping the wavenumber real is new. In this paper, we discuss some of the advantages of this combination.  相似文献   

13.
Summary. A problem in modelling electromagnetic fields used in exploration geophysics is treated mathematically. Analytical expressions are obtained for the electric field due to a harmonic current in a horizontal loop on or above a conducting ground in which is buried a conductive and permeable sphere (ore body). The loop is coaxial with the sphere. For a general time-varying current in the loop, the analysis is carried to the stage where a Fourier inversion can be used to obtain readily the electric field in the time-domain. A new relationship between spherical and cylindrical wave functions is obtained as a transformation of local elements.
Solution of this problem has not been presented before in this form. Lee's solution of 1975 which uses an integral-equation formulation treats a similar problem without taking account of differences in magnetic permeability. The effects of magnetic permeability may have important and useful implications for geophysical explorations.  相似文献   

14.
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   

15.
Existing algorithms of geomorphometry can be applied to digital elevation models (DEMs) given with plane square grids or spheroidal equal angular grids on the surface of an ellipsoid of revolution or a sphere. Computations on spheroidal equal angular grids are trivial for modelling of the Earth, Mars, the Moon, Venus, and Mercury. This is because: (a) forms of these celestial bodies can be described by an ellipsoid of revolution or a sphere and (b) for these surfaces, there are well-developed theory and algorithms to solve the inverse geodetic problem as well as to determine spheroidal trapezoidal areas. It is advisable to apply a triaxial ellipsoid for describing the forms of small moons and asteroids. However, there are no geomorphometric algorithms intended for such a surface. In this article, first, we formulate the problem of geomorphometric modelling on a triaxial ellipsoid surface. Then, we recall definitions and formulae for coordinate systems of a triaxial ellipsoid and their transformation. Next, we present analytical and computational solutions, which provide the basis for geomorphometric modelling on the surface of a triaxial ellipsoid. The Jacobi solution for the inverse geodetic problem has a fundamental mathematical character. The Bespalov solutions for determination of the length of meridian/parallel arcs and the spheroidal trapezoidal areas are computationally efficient. Finally, we describe easy-to-code algorithms for derivation of local and non-local morphometric variables from DEMs based on a spheroidal equal angular grid of a triaxial ellipsoid.  相似文献   

16.
We investigate the reconstruction of a conductive target using crosswell time-domain electromagnetic tomography in the diffusive limit. The work is a natural extension of our ongoing research in the modification of time-domain methods for the rugged marine mid-ocean-ridge environment, an environment characterized by extreme topography and pronounced variations in crustal conductivity on all scales. We have proved both in theory and in practice that 'traveltime', the time taken for an electromagnetic signal to be identified at a receiver following a change of current in the transmitter, is an excellent, robust estimator of average conductivity on a path between transmitter and receiver. A simple estimate of the traveltime for a parallel electric dipole-dipole system is the time at which the derivative of the electric field with respect to logarithmic time at the receiver reaches its maximum. We have derived the fundamental relationship between the traveltime and the conductivity of the medium for a uniform whole-space. We have applied the concept of the traveltime inversion to the related crosswell problem and demonstrated reconstructions of finite targets based on tomographic analyses. Results show that the crosswell time-domain electromagnetic tomography can supply useful information, such as the location and shape of a conductive target.  相似文献   

17.
Summary. An iterative algorithm is presented to be used in the search for the shape of a 2-D local deep geoelectric inhomogeneity lying in a layered medium; an anomaly having been identified in the usual way by observing an alternating time-harmonic electromagnetic field along the surface of the Earth. The normal section parameters (conductivities and thicknesses) and the excess electrical conductivity (inside inhomogeneity) are assumed to be known. The shape of the inhomogeneity is determined by means of a misfit functional minimization technique. A gradient minimization algorithm is constructed and Tikhonov's regularization scheme is applied to achieve stability of the solution. The effectiveness of such an approach is demonstrated by model calculations and by the interpretation of the Carpathian geomagnetic anomaly. Finally, a brief discussion of the problems of the practical application of this formalized trial procedure is presented. Because of the lack of reliable estimates of the excess conductivity, it is proposed to consider a family of models selected for the set of probable values of model parameters. This family can be treated as a generalized solution of the interpretation problem.  相似文献   

18.
Measured changes in the Earth's length of day on a decadal timescale are usually attributed to the exchange of angular momentum between the solid mantle and fluid core. One of several possible mechanisms for this exchange is electromagnetic coupling between the core and a weakly conducting mantle. This mechanism is included in recent numerical models of the geodynamo. The 'advective torque', associated with the mantle toroidal field produced by flux rearrangement at the core–mantle boundary (CMB), is likely to be an important part of the torque for matching variations in length of day. This can be calculated from a model of the fluid flow at the top of the outer core; however, results have generally shown little correspondence between the observed and calculated torques. There is a formal non-uniqueness in the determination of the flow from measurements of magnetic secular variation, and unfortunately the part of the flow contributing to the torque is precisely that which is not constrained by the data. Thus, the forward modelling approach is unlikely to be useful. Instead, we solve an inverse problem: assuming that mantle conductivity is concentrated in a thin layer at the CMB (perhaps D"), we seek flows that both explain the observed secular variation and generate the observed changes in length of day. We obtain flows that satisfy both constraints and are also almost steady and almost geostrophic, and therefore assert that electromagnetic coupling is capable of explaining the observed changes in length of day.  相似文献   

19.
We use Monte Carlo Markov chains to solve the Bayesian MT inverse problem in layered situations. The domain under study is divided into homogeneous layers, and the model parameters are the conductivity of each layer. We use an a priori distribution of the parameters which favours smooth models. For each layer, the a priori and a posteriori distributions are digitized over a limited set of conductivity values.
  The Markov chain relies on updating the model parameters during successive scanning of the domain under study. For each step of the scanning, the conductivity is updated in one layer given the actual value of the conductivity in the other layers. Thus we designed an ergodic Markov chain, the invariant distribution of which is the a posteriori distribution of the parameters, provided the forward problem is completely solved at each step.
  We have estimated the a posteriori marginal probability distributions from the simulated successive values of the Markov chain. In addition, we give examples of complex magnetotelluric impedance inversion in tabular situations, for both synthetic models and field situations, and discuss the influence of the smoothing parameter.  相似文献   

20.
The continuation inverse problem revisited   总被引：1，自引：0，他引：1  
The non-uniqueness of the continuation of a finite collection of harmonic potential field data to a level surface in the source-free region forces its treatment as an inverse problem. A formalism is proposed for the construction of continuation functions which are extremal by various measures. The problem is cast in such a form that the inverse problem solution is the potential function on the lowest horizontal surface above all sources, serving as the boundary function for the Dirichlet problem in the upper half-plane. The desired continuation, at the higher level of interest, must then be in the range of the upward continuation operator acting on this boundary function, rather than being allowed the full freedom of itself being part of a Dirichlet problem boundary function. Extremal solutions minimize non-linear functionals of the continuation function, which are re-expressed as different functionals of the boundary function. A crux of the method is that there is no essential distinction between the upward and downward continuation inverse problems to levels above or below data locations. Casting the optimization as a Lagrange multiplier problem leads to an integral equation for the boundary function, which is readily solved in the Fourier domain for a certain class of functionals. The desired extremal continuation is then given by upward continuation. It is found that for some functionals, application of the Lagrange multiplier theorem requires a further restriction on the set of allowable boundary functions: bandlimitedness is a natural choice for the continuation problem. With this imposition, the theory is developed in detail for semi-norm functionals penalizing departure from a constant potential, in the 2-norm and Sobelev norm senses, and illustrated by application for a small synthetic Deep Tow magnetic field data set.  相似文献   

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