Electromagnetic Inverse Problem Research Articles

New advances in three-dimensional controlled-source electromagnetic inversion

SUMMARY New techniques for improving both the computational and imaging performance of the three-dimensional (3-D) electromagnetic inverse problem are presented. A non-linear conjugate gradient algorithm is the framework of the inversion scheme. Full wave equation modelling for controlled sources is utilized for data simulation along with an efficient gradient computation approach for the model update. Improving the modelling efficiency of the 3-D finite difference (FD) method involves the separation of the potentially large modelling mesh, defining the set of model parameters, from the computational FD meshes used for field simulation. Grid spacings and thus overall grid sizes can be reduced and optimized according to source frequencies and source–receiver offsets of a given input data set. Further computational efficiency is obtained by combining different levels of parallelization. While the parallel scheme allows for an arbitrarily large number of parallel tasks, the relative amount of message passing is kept constant. Image enhancement is achieved by model parameter transformation functions, which enforce bounded conductivity parameters and thus prevent parameter overshoots. Further, a remedy for treating distorted data within the inversion process is presented. Data distortions simulated here include positioning errors and a highly conductive overburden, hiding the desired target signal. The methods are demonstrated using both synthetic and field data.

Evidence of a posterior cingulate involvement (Brodmann area 31) in dyslexia: A study based on source localization algorithm of event-related potentials

The study investigates the differences regarding the position of intracranial generators of P50 component of ERPs in 38 dyslexic children aged 11.47 ± 2.12 years compared with their 19 healthy siblings aged 12.21 ± 2.25. The dipoles were extracted by solving the inverse electromagnetic problem according to the recursively applied and projected multiple signal classification (RAP-MUSIC) algorithm approach. For improved localization of the main dipole the solutions were optimized using genetic algorithms. The statistical analysis revealed differences regarding the position of intracranial generators of low frequency of P50. Particularly, dyslexics showed main activity being located at posterior cingulate cortex (Brodmann's area 31) while controls exhibited main activity being located at retrosplenial cortex (Brodmann's area 30). These results may indicate a role for the posterior cingulate cortex in the pre-attentive processing operation of dyslexia beyond of its traditional function in terms of spatial attention and motor intention.

Progress and challenges in multimodal data fusion

Independently, different brain imaging methods provide compromised spatial and temporal resolutions. Anatomical MRI provides highly accurate information about the individual cortical anatomy. In functional imaging, fMRI is temporally limited by the slow time course of the hemodynamic response, but can provide spatial sampling on a millimeter scale. EEG and MEG in turn provide a temporal resolution of milliseconds, but the localization of sources is more complicated because of the electromagnetic inverse problem. Combining information provided by both anatomical and functional MRI with EEG/MEG data thus facilitates the elucidation of the spatial distribution and temporal orchestration of human brain activity.

Fisher information analysis for two-dimensional microwave tomography

In this paper, a Fisher information analysis is employed to establish some important physical performance bounds in microwave tomography. As a canonical problem, the two-dimensional electromagnetic inverse problem of imaging a cylinder with isotropic dielectric losses is considered. A fixed resolution is analysed by introducing a finite basis, i.e., pixels representing the material properties. The corresponding Cramér–Rao bound for estimating the pixel values is computed based on a calculation of the sensitivity field which is obtained by differentiating the observed field with respect to the estimated parameter. An optimum trade-off between the accuracy and the resolution is defined based on the Cramér–Rao bound, and its application to assess a practical resolution limit in the inverse problem is discussed. Numerical examples are included to illustrate how the Fisher information analysis can be used to investigate the significance of measurement distance, operating frequency and losses in the canonical tomography set-up.

A parallel method for large scale time domain electromagnetic inverse problems

In this work we consider the solution of 3D time domain electromagnetic inverse problems. Solving such problems is an open challenge as they require very high computational resources. We therefore explore a method to parallelize the inverse problem by using time decomposition. We show that our approach can reduce the computational time although it does not scale optimally.

Solving Electromagnetic Inverse Scattering Problems by SVRMs: a Case of Study Towards Georadar Applications

In this paper, an heuristic approach based on Support Vector Regression Machines (SVRMs) is presented in order to solve a simple inverse scattering problem. Interesting results have been obtained, with a remarkable reduction of computational time. Future development of this works will interest the evaluation of the performances of SVRMs for detection of buried objects in stratifled media. This is the starting point to develop models for typical Georadar applications.

Recent studies of Schumann resonance and ELF transients

Schumann resonance (SR) is the global electromagnetic phenomenon occurring in the Earth – ionosphere cavity at frequencies of 8, 14, 20 Hz, etc (the ELF band). Phenomenon was named after the author of the pioneer publication on the planetary resonance. Since natural electromagnetic signal originates from lightning strokes distributed worldwide, parameters of oscillations reflect the average condition of ionospheric plasma, spatial distribution and current intensity of thunderstorms. ‘Regular’ lightning strokes occur at a rate of 50 events per second, and their electromagnetic radiation forms the SR background. An irregular or transient component exists caused by relatively rare powerful strokes (the rate is about one event per minute or even less) that provide pulses ten times higher than an ordinary discharge. Three orthogonal field components (vertical electric and two crossed horizontal magnetic fields) are recorded at a Schumann resonance field site, and the power spectra are obtained in the frequency band up to 40 Hz averaged over a few minutes with the typical frequency resolution of 0.1 Hz. The main objective of report is to review the recent results in the Schumann resonance studies. The report consists of two parts; one of them summarizes the experimental data obtained during last 3 – 5 years, while the second describes model studies of ELF radio propagation. We try to mention the most important results in the Schumann resonance. The report surveys the data obtained by using ‘classical’ power spectra acquired at different observatories. Power spectra give information on the properties of the lower ionosphere boundary and on the dynamics of the global thunderstorms. For instance, the sudden changes can be detected in the space weather such as Solar Proton Events (SPE); the regular seasonal motion of thunderstorms is monitored, which is associated with the ground surface temperature; unusual changes might be found in the SR spectra prior to disastrous earthquakes, etc. The cross-spectra of both magnetic fields and magnetic and electric fields (Poynting vector) might be obtained when three field components are recorded. Cross-spectra enhance the coherent fields and reduce the noise thus presenting a more exact information on both the source location and the SR parameters. Elliptical polarization of horizontal magnetic field is measured via the cross-spectra technique, and recent comparison with the model computations proved the detection of Schumann resonance line splitting. Accumulated experimental data present a wide area for resolving the inverse electromagnetic problems directed to establishing an unknown instant distribution of the global lightning activity. Resonance parameters measured worldwide allowed for constructing the effective globally averaged models of the lower ionosphere and the dispersion relation for ELF radio waves. Recent model developments are connected with applications of finite element methods to the resonance problem. Simultaneously, the ‘classical’ spectral field representation was transformed into the time domain providing an effective computational algorithm. Analytical time domain solution allowed generating the quasi–random pulse successions modeling the ELF radio noise. The model time domain ELF signals might be synthesized with predetermined properties. In the conclusion of review, we note a renewal of interest towards the extra-terrestrial Schumann resonance studies. The Huygens probe mission on Titan stimulated interest to distant detection of feasible lightning strokes at the planets and their global location by using the planetary resonance concurrently with studies of ionospheres.

Electric conductivity measurement of oxides melts

Notwithstanding on variety of existing processes of induction skull melting of oxides, electrical and thermal properties of oxides melts are practically unknown. From the other hand, liquid-phase synthesis of Hi-Tech oxides materials such as new ceramics, monocrystals and glasses, requires knowledge of melts properties up to 3700 K. Inasmuch as the main physical property, which influence to the melt power consumption at induction heating is electrical conductivity then study of electrical conductivity temperature dependence is necessary. The principle and technical realization of the electrical conductivity measurement of melts are presented. The approach is based on induction melting in cold crucible technology and inverse electromagnetic problem solution. Analysis of the method accuracy via input data errors is presented. For increasing of Ill-posed problem solution was used over determination of set of equations. Conductivity of aluminum oxide melts for the temperature up to 3223 K is shown too.

Closed loop GPR data inversion for soil hydraulic and electric property determination

We propose an integrated electromagnetic‐hydrodynamic inverse modeling approach for identifying field‐scale unsaturated soil hydraulic properties and electric profiles from off‐ground time‐lapse ground‐penetrating radar data. Hydrodynamic modeling based on the one‐dimensional Richards' equation with homogeneous soil hydraulic parameters is used to physically constrain the full‐wave radar electromagnetic inverse problem in a closed loop. Significant effects can be observed of water dynamics on the time‐lapse radar data and numerical studies demonstrate the uniqueness of the inverse solution. The results suggest that the method is promising for non‐invasive characterization of the shallow subsurface and monitoring of its dynamics.

Reconstruction Quality and Spectral Content of an Electromagnetic Time-Domain Inversion Algorithm

A tomographic time-domain reconstruction algorithm for solving the inverse electromagnetic problem is described. The application we have in mind is dielectric breast cancer detection but the results are of general interest to the field of microwave tomography. Reconstructions have been made from experimental and numerically simulated data for objects of different sizes in order to investigate the relation between the spectral content of the illuminating pulse and the quality of the reconstructed image. We have found that the spectral content is crucial for a successful reconstruction. The work has further shown that when imaging objects with different scale lengths it is an advantage to use a multiple step procedure. Low frequency content in the pulse is used to image the large structures and the reconstruction process then proceed by using higher frequency data to resolve small scale lengths. Good agreement between the results obtained from experimental data and simulated data has been achieved.

Design methods of the spherical quadrupole magnets and sextupole magnets

In high-energy physics, medical treatments, and space detections, quadrupole field distributions and sextupole field distributions are often required over a large spherical volume. Traditionally, these fields can be generated with cylindrical magnets. In comparison with cylindrical magnets, the spherical magnets have the advantages of smaller values of inductances, lower energy storage, and less magnetic flux leakage. To satisfy these special fields requirements, this paper describes the design methods of spherical magnets. The basic method derives from the target field approach that has been used to solve many of the inverse electromagnetic problems. Applying this approach, this paper has accomplished the design of spherical quadrupole magnets and sextupole magnets. In engineering and manufacture, the above calculation results are hard to utilize directly because of the complex magnet structures and, thus, high fabrication costs. After redesign and optimization on the above calculation results, this paper has also completed the construction of a modified spherical quadrupole magnet that has constant current distributions. This novel type of quadrupole magnet can generate nearly linear gradient fields and is easy to fabricate in engineering. Simulation and test results of this magnet are also presented.

BFGS quasi-Newton method for solving electromagnetic inverse problems

Electromagnetic inverse problems have been found in numerous research areas, such as medical imaging, non-destructive testing, geophysical prospecting and microwave tomography. These inverse problems can be considered as a large-scale nonlinear programming problem. In the paper, a quasi-Newton algorithm based on Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian matrix updating formula is introduced and applied to solve two-dimensional (2-D) electromagnetic inverse problems. Numerical results of image reconstruction from the simulation data for some dielectric cylinders are given, and the efficiency of the proposed method for image reconstruction is evaluated.

Solving inverse electromagnetic problems using FDTD and gradient-based minimization

Numerical simulation is an important tool in understanding the electromagnetic field and how it interacts with the environment. Different topics for time-domain finite-difference (FDTD) and finite-element (FETD) methods for Maxwell's equations are treated in this thesis. Subcell models are of vital importance for the efficient modeling of small objects that are not resolved by the grid. A novel model for thin sheets using shell elements is proposed. This approach has the advantage of taking into account discontinuities in the normal component of the electric field, unlike previous models based on impedance boundary conditions (IBCs). Several results are presented to illustrate the capabilities of the shell element approach. Waveguides are of fundamental importance in many microwave applications, for example in antenna feeds. The key issues of excitation and truncation of waveguides are addressed. A complex frequency shifted form of the uniaxial perfectly matched layer (UPML) absorbing boundary condition (ABC) in FETD is developed. Prism elements are used to promote automatic grid generation and enhance the performance. Results are presented where reflection errors below -70dB are obtained for different types of waveguides, including inhomogeneous cases. Excitation and analysis via the scattering parameters are achieved using waveguide modes computed by a general frequency-domain mode solver for the vector Helmholtz equation. Huygens surfaces are used in both FDTD and FETD for excitation in waveguide ports. Inverse problems have received an increased interest due to the availability of powerful computers. An important application is non-destructive evaluation of material. A time-domain, minimization approach is presented where exact gradients are computed using the adjoint problem. The approach is applied to a general form of Maxwell's equations including dispersive media and UPML. Successful reconstruction examples are presented both using synthetic and experimental measurement data. Parameter reduction of complex geometries using simplified models is an interesting topic that leads to an inverse problem. Gradients for subcell parameters are derived and a successful reconstruction example is presented for a combined dielectric sheet and slot geometry.

Three-Dimensional Electromagnetic Modelling and Inversion from Theory to Application

The whole subject of three-dimensional (3-D) electromagnetic (EM) modelling and inversion has experienced a tremendous progress in the last decade. Accordingly there is an increased need for reviewing the recent, and not so recent, achievements in the field. In the first part of this review paper I consider the finite-difference, finite-element and integral equation approaches that are presently applied for the rigorous numerical solution of fully 3-D EM forward problems. I mention the merits and drawbacks of these approaches, and focus on the most essential aspects of numerical implementations, such as preconditioning and solving the resulting systems of linear equations. I refer to some of the most advanced, state-of-the-art, solvers that are today available for such important geophysical applications as induction logging, airborne and controlled-source EM, magnetotellurics, and global induction studies. Then, in the second part of the paper, I review some of the methods that are commonly used to solve 3-D EM inverse problems and analyse current implementations of the methods available. In particular, I also address the important aspects of nonlinear Newton-type optimisation techniques and computation of gradients and sensitivities associated with these problems.

Sequential quadratic programming method for solution of electromagnetic inverse problems

In this paper, a new algorithm, namely, a reduced Hessian sequential quadratic programming (SQP) method, for solving electromagnetic inverse problems is proposed. The electromagnetic inverse problem is considered to be a constrained nonlinear programming. The reduced Hessian SQP method finds the solution of this constrained nonlinear programming by solving a sequential of quadratic programming subproblems. The reduced Hessian scheme is applied to reduce the requirement of computational memory of the basic SQP method for large inverse problems. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method, and some comparisons show that the proposed method has a better convergence and a faster speed than the previous methods.

Spatiotemporal Bayesian inference dipole analysis for MEG neuroimaging data

Recently, we described a Bayesian inference approach to the MEG/EEG inverse problem that used numerical techniques to estimate the full posterior probability distributions of likely solutions upon which all inferences were based [Schmidt, D.M., George, J.S., Wood, C.C., 1999. Bayesian inference applied to the electromagnetic inverse problem. Human Brain Mapping 7, 195; Schmidt, D.M., George, J.S., Ranken, D.M., Wood, C.C., 2001. Spatial-temporal bayesian inference for MEG/EEG. In: Nenonen, J., Ilmoniemi, R. J., Katila, T. (Eds.), Biomag 2000: 12th International Conference on Biomagnetism. Espoo, Norway, p. 671]. Schmidt et al. (1999) focused on the analysis of data at a single point in time employing an extended region source model. They subsequently extended their work to a spatiotemporal Bayesian inference analysis of the full spatiotemporal MEG/EEG data set. Here, we formulate spatiotemporal Bayesian inference analysis using a multi-dipole model of neural activity. This approach is faster than the extended region model, does not require use of the subject's anatomical information, does not require prior determination of the number of dipoles, and yields quantitative probabilistic inferences. In addition, we have incorporated the ability to handle much more complex and realistic estimates of the background noise, which may be represented as a sum of Kronecker products of temporal and spatial noise covariance components. This reduces the effects of undermodeling noise. In order to reduce the rigidity of the multi-dipole formulation which commonly causes problems due to multiple local minima, we treat the given covariance of the background as uncertain and marginalize over it in the analysis. Markov Chain Monte Carlo (MCMC) was used to sample the many possible likely solutions. The spatiotemporal Bayesian dipole analysis is demonstrated using simulated and empirical whole-head MEG data.

Abstracts of the 48th Irish Geological Research Meeting, Trinity College Dublin, February 2005

S OFTHE48THIRISHGEOLOGICAL RESEARCH MEETING, TRINITY COLLEGE DUBLIN, FEBRUARY 2005 IANSANDERS (ED.) I write this shortpreambleto the abstractswearing two hats: one as editorandthe other as organiserof the meeting With regard to the latter I owe an enormous debt of gratitude to those tireless, enthusiasticandevercheerful research studentsand staff from TCD without whose help the meeting would neverhavehappened As in recent years, the Friday and Saturday eveningswere givenoverto talksby speciallyinvited speakers.The meetingbegan on the Friday evening with a remarkable lecture by Professor John McCloskey from the University of Ulster at Coleraine. His talk, 'The great Sumatra-Andaman earthquake and its implications for possible large near-future earthquakes', includedarigorousanalysis of this recent, catastrophicearthquake and a dire prediction (already,sadly, borne out by events) that further devastationwas to come. On the Saturday evening, ProfessorTim Reston (University of Kiel) spoke clearly and with obvious authonty on 'Continental breakup- lessons from the west of IrelandandtheAlps'. This lecturewas sponsoredby the Irish Geological Association and, as a departure from the practiceof previous meetings, it was held outside the university campus, in a hotel--Chief O'Neill's Hotel-following a special conference dinnerthere Afterwards,participantswere involved in social engagementsinto the early hours. The programmeon both full days was busywith the presentationof research. The large numberof poster contributions justified a dedicated poster session, lasting for two hours on the Saturday afternoon.To allow time for this, a numberof oral presentationstookona new format-a five-minuteslot in the auditoriumfor the speakers to introducethe essential purposeandresultsof their work, followed by anopportunity forthemto discussthedetailsof the work beside a posterdisplay.This approachseemed to work well, andit maybecome a featureof future meetings, where the aim must always be to provide ample time for informal discussion. Another cherished aspect of the meeting is the absence of parallelsessions,therebyenablingeveryonetosample the delights of the complete spectrum of the Earth sciences. In fact, the lecture programme this year deliberatelymixedtogethertalks on disparatetopics withinindividualsessions Altogether 150-200 people attendedthemeeting, andthe standardof presentationwas uniformlyhigh. The balance of topics this year saw a healthy expansionin two fieldsthatlie atoppositeendsof the geological realm-palaeontology andgeophysics,the latterreflecting a hvely and growing researchgroup at the Dublin Institutefor Advanced Studies. The prize for the best student talk went to Maria McNamara from UCD (formerly from UCG), who spokeon the taphonomyof Miocene vertebrates from Spain, and the prize for best student poster was awarded to Brian O'Driscoll from TCD (formerly from UCC), whose work encompassed textural developmentof magmaticolivine. The totalnumberof talksandposterswas 60, and the abstractsof 49 of these are printed below. The othereleven abstractsarenot being publishedon the express wishes of the authors Readers arereminded thatnone of the printedabstractshas been subjected to peerreview.Abstracts areorderedalphabeticallyby first author Irrespective of whether they were presentedas talksoras posters. IrishJournalofEarthSciences23 (2005), 123-145 © RoyalIrishAcademy 125 126 IrishJournalof EarthSciences (2005) Implementation of the quasi-Newton method for ID magnetotelluric inversion AnnaAvdeeva(e-mail:aavdeeva@cp.dias.ie),Dublin Institutefor Advanced Studies, 5 Merrion Square, Dublin2. The quasi-Newton (qN) method became a very popular tool for the numerical solution of electromagnetic (EM) inverse problems. The reasoning behind it is that the method requires calculationof gradientsonly, while at the same time avoiding calculations of second-derivative terms. However,the inherentdisadvantageof this approach is that it may converge slowly. An effective way to accelerateitsconvergenceis to calculatethegradients analytically.As a firststepto solvingthe3D EMcase, I have applied the qN method to a simpler ID magnetotelluric(MT)problem.I consideracouple of synthetic tests. In the first test I applied the qN methodto invertthe MT impedancecalculatedfor a seven-layered earth model at fourteen periods. In particular,the resultsdemonstratethatthe analytical calculationof gradientsacceleratesthe qN inversion by fourto six times. In my second test,I appliedthe qN implementationto synthetic MT data, with and withoutartificialnoise, generatedfor a 106-layered model. Computationaltime using the analytically calculated gradients decreases twenty-fold. In addition, I demonstrate that the Occam-type regularisation stabilises the qN inversion result. Further research will include other methods for optimisationsuchasthenonlinearconjugategradient, Gauss-Newtonandfull Newton methods. 3D non-biomineralised arthropods from a Triassic shore face deposit A.K. Braiden (e-mail: aoife.braiden@ucd.ie),1P.J. Orr,1S.L. Kearns,2S.R. Kearns3 IDepartmentof Geology, UniversityCollege Dublin, Belfield, Dublin4. 2Department of EarthSciences, Universityof Bristol, WillsMemorialBuilding,Queen'sRoad,Bristol BS8 1RJ,England. 317...

Quasi-Newton methods for large-scale electromagnetic inverse problems

We develop quasi-Newton (QN) methods for distributed parameter estimation problems which evolve from electromagnetics, where the forward problem is governed by some form of Maxwell's equations. A Tikhonov-style regularization approach yields an optimization problem with a special structure, where the gradients are calculated using the adjoint method. In many cases, standard QN methods (such as L-BFGS) are not very effective and tend to converge slowly. Taking advantage of the special structure of the problem and the quantities that are calculated in typical gradient descent methods, we develop a class of highly effective methods for the solution of the problem. We demonstrate the merits and effectiveness of our algorithm on two realistic model problems.

Reconstruction of two-dimensional buried objects by a differential evolution method

The application of a global optimization procedure to the detection of buried inhomogeneities is studied in the present paper. The electromagnetic inverse problem is recast as a global optimization problem and iteratively solved by a differential evolution method. The approach is applied to two-dimensional configurations. The paper provides the mathematical formulation of the method together with a discussion on the application of the stochastic optimization procedure. Finally, the results of some numerical simulations are reported.

Solution accelerators for large-scale three-dimensional electromagnetic inverse problems

We provide a framework for preconditioning nonlinear three-dimensional electromagnetic inverse scattering problems using nonlinear conjugate gradient (NLCG) and limited memory (LM) quasi-Newton methods. Key to our approach is the use of an approximate adjoint method that allows for an economical approximation of the Hessian that is updated at each inversion iteration. Using this approximate Hessian as a preconditioner, we show that the preconditioned NLCG iteration converges significantly faster than the non-preconditioned iteration, as well as converging to a data misfit level below that observed for the non-preconditioned method. Similar conclusions are also observed for the LM iteration; preconditioned with the approximate Hessian, the LM iteration converges faster than the non-preconditioned version. At this time, however, we see little difference between the convergence performance of the preconditioned LM scheme and the preconditioned NLCG scheme. A possible reason for this outcome is the behaviour of the line search within the LM iteration. It was anticipated that, near convergence, a step size of one would be approached, but what was observed, instead, were step lengths that were nowhere near one. We provide some insights into the reasons for this behaviour and suggest further research that may improve the performance of the LM methods.