Direct and inverse problems in electromagnetic sounding of three-dimensional heterogeneous medium

The methods for solving three-dimensional (3D) direct and inverse problems of electromagnetic sounding are considered. It is shown that the method of integral equations is an efficient instrument for mathematical modeling of electromagnetic fields in 3D heterogeneous media. Adaptation of the integral equation technique to the solution of inverse problems is demonstrated.     

Modern trends in the solution of forward and inverse 3D electromagnetic induction problems

The main results obtained during the last 5–8 yr in the solution of forward and inverse problems of 3D induction studies are summarized. The up-to-date status of 3D modelling is presented and prospective improvements in the formulation and numerical solution of forward problems are discussed. Approximate techniques and practical aspects of 3D modelling are specially considered. The general scheme of 3D interpretation of electromagnetic geophysical observations is outlined and realistic formalized approaches to solving 3D inverse problems, namely direct inversion and formalized model fitting, are studied.     

Analytical solutions to global and local problems of electromagnetic induction in the earth

This review of analytic solutions is divided into two parts. The first part reviews electromagnetic induction in radially symmetric distributions of conductivity, and is appropriate to the study of global problems. In the second part, local problems of a specific nature are considered, the model being a half-space conductor with at least one lateral discontinuity separating regions of different uniform conductivities. In some problems, an approximate surface-boundary condition is used, and it is shown that the accuracy of the solutions has yet to be determined satisfactorily.     

Global electromagnetic induction

Methods of analysis of long period geomagnetic variations (periods over a few hours), the available electromagnetic response function estimates, and the effect of lateral inhomogeneity within the Earth are reviewed. Recent advances in the inversion of response function data to produce conductivitydepth profiles are mentioned, and aspects of the inverse problem specific to global (spherical Earth) induction are discussed.There is a rapid rise in electrical conductivity between about 400 km and 800 km, but whether this is a gradual change or consists of one or several steps is not resolvable with the presently available data and naive inversion algorithm used here. At the greatest depths to which presently analysed variations penetrate (1000–1500 km), available data have some difficulty in resolving finer structure, but there are indications that the electrical structure of the continents becomes more laterally homogeneous as depth increases. Previously published inversions see lateral variations in electrical structure down to at least 500 km, and at shallower depths these variations are unambiguously resolved.     

Marine electromagnetic induction studies

In reviewing seafloor induction studies conducted over the last seven years, we observe a decline in single-station magnetotelluric (MT) experiments in favour of large, multinational, array experiments with a strong oceanographic component. However, better instrumentation, processing techniques and interpretational tools are improving the quality of MT experiments in spite of the physical limitations of the band limited seafloor environment, and oceanographic array deployments are allowing geomagnetic depth sounding studies to be conducted. Oceanographic objectives are met by the sensitivity of the horizontal electric field to vertically averaged motional currents, providing the same information, at much greater reliability and much lower cost, as an array of continuously operating current meter moorings.The seafloor controlled source method has now become, if not routine, at least viable. Prior to 1982, only one seafloor controlled source experiment has been conducted; now at least three groups are involved in the experimental aspects of this field. The horizontal dipole-dipole configuration is favoured, although a variant of the magnetometric resistivity method utilising a vertical electric transmitter has been developed and deployed. By exploiting the characteristics of the seafloor environment, source receiver spacings unimaginable on land can be achieved; on a recent deployment dipole spacings of 90 km were used with a clear 24 Hz signal transmitted through the seafloor. This, and prior experiments, show that the oceanic upper mantle is characteristically very resistive, 10⁵ m at least. This resistive zone is becoming apparent from other experiments as well, such as studies of the MT response in coastal areas on land.Mid-ocean ridge environments are likely to be the target of many future electromagnetic studies. By taking available laboratory data on mineral, melt and water conductivity we predict to first order the kinds of structures the EM method will help us explore.     

Global electromagnetic induction in the Moon and planets

A summary of experiments and analyses concerning electromagnetic induction in the Moon and other extraterrestrial bodies is presented. Magnetic step-transient measurements made on the lunar dark side show the eddy current response to be the dominant induction mode of the Moon. Analysis of the poloidal field decay of the eddy currents has yielded a range of monotonic conductivity profiles for the lunar interior: the conductivity rises from 3·10^?4 mho/m at a depth of 170 km to 10^?2 mho/m at 1000 km depth. The static magnetization field induction has been measured and the whole-Moon relative magnetic permeability has been calculated to be $\text{μ}\text{μ}{}_0\text{= 1.01 ± 0.06}$. The remanent magnetic fields, measured at Apollo landing sites, range from 3 to 327 γ. Simultaneous magnetometer and solar wind spectrometer measurements show that the 38-γ remanent field at the Apollo 12 site is compressed to 54 γ by a solar wind pressure increase of 7·10^?8 dyn/cm². The solar wind confines the induced lunar poloidal field; the field is compressed to the surface on the lunar subsolar side and extends out into a cylindrical cavity on the lunar antisolar side. This solar wind confinement is modeled in the laboratory by a magnetic dipole enclosed in a superconducting lead cylinder; results show that the induced poloidal field geometry is modified in a manner similar to that measured on the Moon. Induction concepts developed for the Moon are extended to estimate the electromagnetic response of other bodies in the solar system.     

Spectral techniques in inverse stokes and Overdetermined problems

This paper deals with spectral techniques applied to geodetic problems. The solutions of the Inverse Stokes problem and of the Overdetermined Boundary Value Problem have been obtained applying The Wiener principle directly in the spectral domain. The resulting estimator for the Inverse Stokes problem is a low pass filter which is tuned by the covariance structure of the data while the one solving the Overdetermined Boundary Value Problem is nearly a weighted mean of the two spectra of the boundary data. Numerical examples on simulated data have been carried out to test the derived estimators.     

Inversion problem in regional electromagnetic induction studies

For completeness of the theory proposed by Oni (1972), the inversion problem is examined. It is shown that parameters which constrain distinct solutions of the inverse problem can be determined in the application of the theory. The fundamental basis of the inversion and the strategy to be adopted are discussed.     

An introduction to electromagnetic induction in the ocean

The governing equations for the induction of electromagnetic fields in the ocean by ionospheric and oceanic sources are presented. A uniformly conducting layered model and a nonuniformly conducting thin-sheet model are discussed with reference to the interpretation of fields observed in the ocean. A procedure for the separation of the electric field continuum into parts of ionospheric and oceanic origin is presented.     

Theoretical models for electromagnetic induction in the oceans

The different methods and techniques employed in the theory of electromagnetic induction in thin sheets are reviewed and the methods for approximation to the solution are indicated. These depend on whether the sheet is closed or finite and on whether the integrated conductivity and/or the frequency of variations is high or low.Results for induction in finite sheets which are suitable for ocean modelling are given. These include sheets of perfect conductivity and sheets of finite conductivity which is either discontinuous or continuous at the boundary. The dependence of the “coastline effect” for a global ocean on the location of the edge of the continental shelf, the period of variation of the external field and the conductivity of the underlying earth is explained.     

Minimax approach to inverse problems of geophysics

A new approach is suggested for solving the inverse problems that arise in the different fields of applied geophysics (gravity, magnetic, and electrical prospecting, geothermy) and require assessing the spatial region occupied by the anomaly-generating masses in the presence of different types of a priori information. The interpretation which provides the maximum guaranteed proximity of the model field sources to the real perturbing object is treated as the best interpretation. In some fields of science (game theory, economics, operations research), the decision-making principle that lies in minimizing the probable losses which cannot be prevented if the situation develops by the worst-case scenario is referred to as minimax. The minimax criterion of choice is interesting as, instead of being confined to the indirect (and sometimes doubtful) signs of the “optimal” solution, it relies on the actual properties of the information in the results of a particular interpretation. In the hierarchy of the approaches to the solution of the inverse problems of geophysics ordered by the volume and quality of the retrieved information about the sources of the field, the minimax approach should take special place.     

航空瞬变电磁法一维正反演研究

优化了航空瞬变电磁法一维正演算法,采用新的汉克尔变换系数,理论上提高了正演的精度.提出了航空瞬变电磁法一维反演算法——模型交替调整反演算法,阐述了该算法的原理和计算方法,编写出反演程序,以已知模型正演响应作为实测数据,对若干典型模型进行了反演计算,取得了较理想的反演效果,与Zohdy法相比,该方法有更高的精度.     

Energy Group optimization for forward and inverse problems in nuclear engineering: application to downwell-logging problems

Simulating radiation transport of neutral particles (neutrons and γ‐ray photons) within subsurface formations has been an area of research in the nuclear well‐logging community since the 1960s, with many researchers exploiting existing computational tools already available within the nuclear reactor community. Deterministic codes became a popular tool, with the radiation transport equation being solved using a discretization of phase‐space of the problem (energy, angle, space and time). The energy discretization in such codes is based on the multigroup approximation, or equivalently the discrete finite‐difference energy approximation. One of the uncertainties, therefore, of simulating radiation transport problems, has become the multigroup energy structure. The nuclear reactor community has tackled the problem by optimizing existing nuclear cross‐sectional libraries using a variety of group‐collapsing codes, whilst the nuclear well‐logging community has relied, until now, on libraries used in the nuclear reactor community. However, although the utilization of such libraries has been extremely useful in the past, it has also become clear that a larger number of energy groups were available than was necessary for the well‐logging problems. It was obvious, therefore, that a multigroup energy structure specific to the needs of the nuclear well‐logging community needed to be established. This would have the benefit of reducing computational time (the ultimate aim of this work) for both the stochastic and deterministic calculations since computational time increases with the number of energy groups. We, therefore, present in this study two methodologies that enable the optimization of any multigroup neutron–γ energy structure. Although we test our theoretical approaches on nuclear well‐logging synthetic data, the methodologies can be applied to other radiation transport problems that use the multigroup energy approximation. The first approach considers the effect of collapsing the neutron groups by solving the forward transport problem directly using the deterministic code EVENT, and obtaining neutron and γ‐ray fluxes deterministically for the different group‐collapsing options. The best collapsing option is chosen as the one which minimizes the effect on the γ‐ray spectrum. During this methodology, parallel processing is implemented to reduce computational times. The second approach uses the uncollapsed output from neural network simulations in order to estimate the new, collapsed fluxes for the different collapsing cases. Subsequently, an inversion technique is used which calculates the properties of the subsurface, based on the collapsed fluxes. The best collapsing option is chosen as the one that predicts the subsurface properties with a minimal error. The fundamental difference between the two methodologies relates to their effect on the generated γ‐rays. The first methodology takes the generation of γ‐rays fully into account by solving the transport equation directly. The second methodology assumes that the reduction of the neutron groups has no effect on the γ‐ray fluxes. It does, however, utilize an inversion scheme to predict the subsurface properties reliably, and it looks at the effect of collapsing the neutron groups on these predictions. Although the second procedure is favoured because of (a) the speed with which a solution can be obtained and (b) the application of an inversion scheme, its results need to be validated against a physically more stringent methodology. A comparison of the two methodologies is therefore given.     

On the inversion of global electromagnetic induction data

The initial phase of any inversion of geophysical data must examine the question of the existence of globally distinct solutions. Previous inversion st point of view. A basic inversion strategy for geophysical data is considered. It is concluded that future progress depends on the use of synthetic data to resolve questions about the potential constraining power of GEMI data.     

Numerical solution of the two-dimensional inverse geomagnetic induction problem

Summary An effective numerical approach to the solution of the two-dimensional inverse geomagnetic induction problem using the linearization method is presented. The numerical realization of the inversion is based on Marquardt's algorithm, for which the solution of the direct problem and the partial derivatives of this solution with respect to the electrical parameters of the medium are computed by the finite difference method. Theoretical models are studied and numerical results are presented.     

Unexploded ordnance discrimination using time-domain electromagnetic induction and self-organizing maps

Self-organizing maps (SOM) are implemented for discrimination of geologic noise, buried metal objects and unexploded ordnance using the geophysical method of time-domain electromagnetic induction. The learning and misfit measures are based on a Euclidean metric. The U*-matrix method is shown to be a reliable tool for determining data clusters and cluster boundaries. The performance of SOM for data-type discrimination was tested using three synthetic, idealized geophysical datasets consisting of exponential, multi-exponential and stretched-exponential decaying transients. In addition, experimental data were acquired using a modified Geonics EM63 instrument. Results from the synthetic examples show that SOM clusters the data based on their functional origin, when represented using U*-matrices. The percentage of correct classification is 100%. Unsupervised learning using the field dataset obtained with the Geonics EM63 succeeded in producing a multi-clustered map in which the background transients cluster themselves and are separated from clusters associated with metal clutter objects and UXO. Even though in some cases the SOM did not produce a single cluster for each type of causative body, it was able to separate clutter data from target data by producing several small clusters. The results are encouraging in view of the heterogeneity and sparsity of the training dataset.