Electromagnetic wave scattering by discrete random media with remote sensing applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2001.Includes bibliographical references (p. 171-182).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.The scattering of electromagnetic waves in medium with randomly distributed discrete scatterers is studied. Analytical and numerical solutions to several problems with implications for the active and passive remote sensing of the Earth environment are obtained. The quasi-magnetostatic (QMS) solution for a conducting and permeable spheroid under arbitrary excitation is presented. The spheroid is surrounded by a weakly conducting background medium. The magnetic field inside the spheroid satisfies the vector wave equation, while the magnetic field outside can be expressed as the gradient of the Laplace solution. We solve this problem exactly using the separation of variables method in spheroidal coordinates by expanding the internal field in terms of vector spheroidal wavefunctions. The exact formulation works well for low to moderate frequencies; however, the solution breaks down at high frequency due to numerical difficulty in computing the spheroidal wavefunctions. To circumvent this difficulty, an approximate theory known as the small penetration-depth approximation (SPA) is developed. The SPA relates the internal field in terms of the external field by making use of the fact that at high frequency, the external field can only penetrate slightly into a thin skin layer below the surface of the spheroid. For spheroids with general permeability, the SPA works well at high frequency and complements the exact formulation. However, for high permeability, the SPA is found to give accurate broadband results. By neglecting mutual interactions, the QMS frequency response from a collection of conducting and permeable spheroids is also studied.(cont.) In a dense medium, the failure to properly take into account of multiple scattering effects could lead to significant errors. This has been demonstrated in the past from extensive theoretical, numerical, and experimental studies of electromagnetic wave scattering by densely packed dielectric spheres. Here, electromagnetic wave scattering by dense packed dielectric spheroids is studied both numerically through Monte Carlo simulations and analytically through the quasi-crystalline approximation (QCA) and QCA with coherent potential (QCA-CP). We assume that the spheroids are electrically small so that single-particle scattering is simple. In the numerical simulations, the Metropolis shuffling method is used to generate realizations of configurations for non-interpenetrable spheroids. The multiple scattering problem is formulated with the volume integral equation and solved using the method of moments with electrostatic basis functions. General expressions for the self-interaction elements are obtained using the low-frequency expansion of the dyadic Green's function, and radiative correction terms are included. Results of scattering coefficient, absorption coefficient, and scattering matrix for spheroids in random and aligned orientation configurations are presented. It is shown that independent scattering approximation can give grossly incorrect results when the fractional volume of the spheroids is appreciable.(cont.) In the analytical approach, only spheroids in the aligned configuration are solved. Low-frequency QCA and QCA-CP solutions are obtained for the average Green's function and the effective permittivity tensor. For QCA-CP, the low-frequency expansion of the uniaxial dyadic Green's function is required. The real parts of the effective permittivities from QCA and QCA-CP are compared with the Maxwell-Garnett mixing formula. ...by Chi On Ao.Ph.D

Understanding the magnetic polarizability tensor

The aim of this paper is to provide new insights into the properties of the rank 2 polarizability tensor M̆ proposed by Ledger and Lionheart for describing the perturbation in the magnetic field caused by the presence of a conducting object in the eddy-current regime. In particular, we explore its connection with the magnetic polarizability tensor and the Pólya-Szegö tensor and how, by introducing new splittings of M̆, they form a family of rank 2 tensors for describing the response from different categories of conducting (permeable) objects. We include new bounds on the invariants of the Pólya-Szegö tensor and expressions for the low-frequency and high-conductivity limiting coefficients of M̆. We show, for the high-conductivity case (and for frequencies at the limit of the quasi-static approximation), that it is important to consider whether the object is simply or multiply connected but, for the low-frequency case, the coefficients are independent of the connectedness of the object. Furthermore, we explore the frequency response of the coefficients of M̆ for a range of simply and multiply connected objects

An Explicit Formula for the Magnetic Polarizability Tensor for Object Characterization

The magnetic polarizability tensor (MPT) has attracted considerable interest due to the possibility it offers for characterizing conducting objects and assisting with the identification and location of hidden targets in metal detection. An explicit formula for its calculation for arbitrary-shaped objects is missing in the electrical engineering literature. Furthermore, the circumstances for the validity of the magnetic dipole approximation of the perturbed field, induced by the presence of the object, are not fully understood. On the other hand, in the applied mathematics community, an asymptotic expansion of the perturbed magnetic field has been derived for small objects and a rigorous formula for the calculation of the MPT has been obtained. The purpose of this paper is to relate the results of the two communities, to provide a rigorous justification for the MPT, and to explain the situations in which the approximation is valid

Methods in wave propagation and scattering

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Vita.Includes bibliographical references (p. 195-213).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Aspects of wave propagation and scattering with an emphasis on specific applications in engineering and physics are examined. Frequency-domain methods prevail. Both forward and inverse problems are considered. Typical applications of the method of moments to rough surface three-dimensional (3-D) electromagnetic scattering require a truncation of the surface considered and call for a tapered incident wave. It is shown how such wave can be constructed as a superposition of plane waves, avoiding problems near both normal and grazing incidence and providing clean footprints and clear polarization at all angles of incidence. The proposed special choice of polarization vectors removes an irregularity at the origin of the wavenumber space and leads to a wave that is optimal in a least squared error sense. Issues in the application to 3-D scattering from an object over a rough surface are discussed. Approximate 3-D scalar and vector tapered waves are derived which can be evaluated without resorting to any numerical integrations. Important limitations to the accuracy and applicability of these approximations are pointed out. An analytical solution is presented for the electromagnetic induction problem of magnetic diffusion into and scattering from a permeable, highly but not perfectly conducting prolate spheroid under axial excitation, expressed in terms of an infinite matrix equation. The spheroid is assumed to be embedded in a homogeneous non-conducting medium as appropriate for low-frequency, high-contrast scattering governed by magnetoquasistatics. The solution is based on separation of variables and matching boundary conditions where the prolate spheroidal wavefunctions with complex wavenumber parameter are expanded in terms of spherical harmonics. For small skin depths, an approximate solution is constructed, which avoids any reference to the spheroidal wavefunctions. The problem of long spheroids and long circular cylinders is solved by using an infinite cylinder approximation. In some cases, our ability to evaluate the spheroidal wavefunctions breaks down at intermediate frequencies. To deal with this, a general broadband rational function approximation technique is developed and demonstrated. We treat special cases and provide numerical reference data for the induced magnetic dipole moment or, equivalently, the magnetic polarizability factor. The magnetoquasistatic response of a distribution of an arbitrary number of interacting small conducting and permeable objects is also investigated. Useful formulations are provided for expressing the magnetic dipole moment of conducting and permeable objects of general shape. An alternative to Tikhonov regularization for deblurring and inverse diffraction, based on a local extrapolation scheme, is described, analyzed, and illustrated numerically for the cases of continuation of fields obeying Laplace and Helmholtz equations. At the outset of the development, a special deconvolution problem, where a parameter describes the degree of additive blurring, is considered. No a priori knowledge on the unblurred data is assumed. A standard solution based on an output least squares formulation includes a regularization parameter into a linear, shift-invariant filter. The proposed alternative approach takes advantage of the analyticity of the smoothing process with respect to the blurring parameter. Here a simple local extrapolation scheme is employed. The problem is encountered in applications involving potential theories dealing with magnetostatics, electrostatics, and gravity data. As a generalization to the dynamic case, inverse diffraction of scalar waves is considered. Examples are presented and the two methods compared numerically. The problem of inferring unknown geometry and material parameters of a waveguide model from noisy samples of the associated modal dispersion curves is considered. In a significant reduction of the complexity of a common inversion methodology, the inner of two nested iterations is eliminated: The approach described does not employ explicit fitting of the data to computed dispersion curves. Instead, the unknown parameters are adjusted to minimize a cost function derived directly from the determinant of the boundary condition system matrix. This results in a very efficient inversion scheme that, in the case of noise-free data, yields exact results. Multi-mode data can be simultaneously processed without extra complications. Furthermore, the inversion scheme can accommodate an arbitrary number of unknown parameters, provided that the data have sufficient sensitivity to these parameters. As an important application, the sonic guidance condition for a fluid-filled borehole in an elastic, homogeneous, and isotropic rock formation is considered for numerical forward and inverse dispersion analysis. The parametric inversion with uncertain model parameters and the influence of bandwidth and noise are investigated numerically. The cases of multi-frequency and multi-mode data are examined. Finally, the borehole leaky-wave modes are classified according to the location of the roots of the characteristic equation on a multi-sheeted Riemann surface. A comprehensive set of dipole leaky-wave modal dispersions is computed. In an independent numerical experiment the excitation of some of these modes is demonstrated. The utilization of leaky-wave dispersion data for inversion is discussed.by Henning Braunisch.Ph.D

Methods in wave propagation and scattering

Supervised by Jin A. Kong.Also issued as Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 195-213).by Henning Braunisch

Low-frequency dipolar excitation of a perfect ellipsoidal conductor

International audienceThis paper deals with the scattering by a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency. The source and the ellipsoid are embedded in an infinite homogeneous conducting ground. The main idea is to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion. Maxwell equations and boundary conditions, describing the problem, are firstly expanded using low-frequency expansion of the fields up to order three. It will be shown that fields have to be found incrementally. The static one (term of order zero) satisfies the Laplace equation. The next non-zero term (term of order two) is more complicated and satisfies the Poisson equation. The order-three term is independent of the previous ones and is described by the Laplace equation. They constitute three different scattering problems that are solved using the separated variables method in the ellipsoidal coordinate system. Solutions are written as expansions on the few analytically known scalar ellipsoidal harmonics. Details are given to explain how those solutions are achieved with an example of numerical results

Numerical methods for computing the modal decomposition of the magnetic polarizability of conducting objects

Numerical methods are presented for characterizing the wideband responses of conducting objects to excitation by electromagnetic induction sensors. Electromagnetic induction sensors can be used to measure the magnetic polarizability tensor of conducting targets, a tensor that encapsulates the entire scattering interaction between target and sensor. Wideband characterization of the magnetic polarizability tensor can be achieved by expanding the frequency response in pole-expansion form. The pole-expansion coefficients may then be used as a signature, which can be used for subsurface target detection. To derive the coefficients numerically, the interaction between a target of interest and the sensor is modeled as a linear system, which can then be set up as generalized eigenvalue problem. The eigenvalues of the system correspond to the pole locations of the pole expansion. The remaining coefficients can be derived from the eigenvectors of the system, which correspond to the that natural modes that are excited by the sensor. Integral methods are presented for numerically computing the pole-expansion coefficients of thin conducting sheets and shells, conducting volumes, and bodies of revolution. A differential method is also presented for conducting volumes. Computational results are compared to known analytical solutions when possible as well as to experimental measurements.Ph.D

Classification, identification, and modeling of unexploded ordnance in realistic environments

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 205-218).Recovery of buried unexploded ordnance (UXO) is very slow and expensive due to the high false alarm rate created by clutter. Electromagnetic induction (EMI) has been shown to be a promising technique for UXO detection and discrimination. This thesis uses the EMI response of buried targets to identify or classify them. To perform such discrimination, accurate forward models of buried UXO are needed. This thesis provides a survey of existing target models: the dipole model, the spheroid model, and the fundamental mode model. Then the implementation of a new model, the spheroidal mode model, is described and validated against measurements of a UXO. Furthermore, an in-depth study of the effects of permeable soil, modeled as a permeable half space, is presented. This study concludes that the discontinuity created by the air to permeable soil interface produces minimal effect in the response of a buried object. The change is limited to a magnitude shift of the real portion of the EMI response and can be reproduced by superposition of a permeable half space response on the response of the same object in frees pace. Accurate soil modeling also allows one to invert for soil permeability values from measured data if such data are in known units. However, the EMI sensor used in this study provides measurements in consistent but unknown units. Furthermore, the instrument is from a third party and is proprietary. Therefore, this thesis describes the development of a non-invasive method to model and calibrate non-adaptive instruments so that all measurements can be converted into units consistent with modeled data. This conversion factor is shown to be a constant value across various conditions, thus demonstrating its validity.(cont.) Given that now a more complete model of the measurable response of a buried UXO is implemented, this study proceeds to demonstrate that EMI responses from UXO and clutter objects can be used to identify the objects through the application of Differential Evolution (DE), a type of Genetic Algorithm. DE is used to optimize the parameters of the UXO fundamental mode model to produce a match between the modeled response and the measured response of an unknown object. When this optimization procedure is applied across a library of models for possible UXO, the correct identity of the unknown object can be ascertained because the corresponding library member will produce the closest match. Furthermore, responses from clutter objects are shown to produce very poor matches to library objects, thus providing a method to discriminate UXO from clutter. These optimization experiments are conducted on measurements of UXO in air, UXO in air but obscured by clutter fragments, buried UXO, and buried UXO obscured by clutter fragments. It is shown that the optimization procedure is successful for shallow buried objects obscured by light clutter contributing to roughly 20 dB SNR, but is limited in applicability towards very deeply buried UXO or those in dense clutter environments. The DE algorithm implemented in this study is parallelized and the optimization results are computed with a multi-processor supercomputer. Thus, the computational requirement of DE is a considerable drawback, and the method cannot be used for real time, on-site inversion of measured UXO data. To address this concern, a different approach to inversion is also implemented in this study. Rather than identifying particular UXO, one may do a discrimination between general UXO and general clutter items. Previous work has shown that the expansion coefficients of EMI responses in the spheroidal coordinate system can uniquely characterize the corresponding targets.(cont.) Therefore, these coefficients readily lend themselves for use as features by which objects can be classified as likely to be UXO or unlikely to be UXO. To do such classification, the relationship between these coefficients and the physical properties of UXO and clutter, such as differences in size or body-of-revolution properties or material heterogeneity properties, must be found. This thesis shows that such relationships are complex and require the use of the automated pattern recognition capability of machine learning. Two machine learning algorithms, Support Vector Machines and Neural Networks, are used to identify whether objects are likely to be UXO. Furthermore, the effects of small diffuse clutter fragments and uncertainty about the target position are investigated. This discrimination procedure is applied on both synthetic data from models and measurements of UXO and clutter. It is found that good discrimination is possible for up to 20 dB SNR. But the discrimination is sensitive to inaccurate estimations of a target's depth. It is found that the accuracy must be within a 10 cm deviation of an object's true depth. The general conclusion forwarded by this work is that while increasingly accurate discrimination capabilities can be produced through more detailed forward modeling and application of robust optimization and learning algorithms, the presence of noise and clutter is still of great concern. Minimization or filtering of such noise is necessary before field deployable discrimination techniques can be realized.by Beijia Zhang.Ph.D

Characterizing the shape and material properties of hidden targets from magnetic induction data

The aim of this paper is to show that, for the eddy current model, the leading order term for the perturbation in the magnetic field, due to the presence of a small conducting magnetic inclusion, can be expressed in terms of a symmetric rank 2 polarization tensor. This tensor contains information about the shape and material properties of the object and is independent of position. We apply a recently derived asymptotic formula for the perturbed magnetic field, due to the presence of a conducting inclusion, which is expressed in terms of a new class of rank 4 polarization tensors (Ammari, H., Chen, J., Chen, Z., Garnier, J. & Volkov, D. (2014) Target detection and characterization from electromagnetic induction data. J. Math. Pures Appl., 101, 54–75.) and show that their result can be written in an alternative form involving a symmetric rank 2 tensor involving 6 instead of 81 complex components in an orthonormal coordinate frame. For objects with rotational and mirror symmetries we show that the number of coefficients is still smaller. We include numerical examples to demonstrate that the new polarization tensors can be accurately computed by solving a vector-valued transmission problem by hp-finite elements and include examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical predictions