Browsing by Author "Liu, Qing Huo"

In this thesis, a 3D microwave imaging method is developed for a microwave imaging system with an arbitrary background medium. In the previous study on the breast cancer detection of our research group, a full wave inverse method, the Diagonal Tensor approximation combined with Born Iterative Method (DTA-BIM), was proposed to reconstruct the electrical profile of the inversion domain in a homogenous background medium and a layered background medium. In order to evaluate the performance of the DTA-BIM method in a realistic microwave imaging system, an experimental prototype of an active 3D microwave imaging system with movable antennas is constructed. For the objects immersed in a homogenous background medium or a layered background medium, the inversion results based on the experimental data show that the resolution of the DTA-BIM method can reach finely to a quarter of wavelength of the background medium, and the system's signal-noise-ratio (SNR) requirement is 10 dB. Moreover, the defects of this system make it difficult to be implemented in a realistic application. Thus, another active 3D microwave imaging system is proposed to overcome the problems in the previous system. The new system employs a fix patch antenna array with electric switch to record the data. However, the antenna array makes the inversion system become a non-canonical inhomogeneous background. The analytical Greens' functions used in the original DTA-BIM method become unavailable. Thus, a modified DTA-BIM method, which use the numerical Green's functions combined with measured voltage, is proposed. This modified DTA-BIM method can be used to the inversion in a non-canonical inhomogeneous background with the measured voltages (or $S_{21}$ parameters). In order to verify the performance of this proposed inversion method, we investigate a prototype 3D microwave imaging system with a fix antenna array. The inversion results from the synthetic data show that this method works well with a fix antenna array, and the resolution of reconstructed images can reach to a quarter wavelength even in the presence of a strongly inhomogeneous background medium and antenna couplings. A time-reversal method is introduced as a pre-processing step to reduce the region of interest (ROI) in our inversion. In addition, a Multi-Domain DTA-BIM method is proposed to fit the discontinue inversion regions. With these improvements, the size of the inversion domain and the computational cost can be significantly reduced, and make the DTA-BIM method more feasible for rapid response applications.

This dissertation focuses on a FFT based forward EM solver and its application in inverse problems. The main contributions of this work are two folded. On the one hand, it presents the first scaled lab experiment system in the oil and gas industry for through casing hydraulic fracture evaluation. This system is established to validate the feasibility of contrasts enhanced fractures evaluation. On the other hand, this work proposes a FFT based VIE solver for hydraulic fracture evaluation. This efficient solver is needed for numerical analysis of such problem. The solver is then generalized to accommodate scattering simulations for anisotropic inhomogeneous magnetodielectric objects. The inverse problem on anisotropic objects are also studied.

Before going into details of specific applications, some background knowledge is presented. This dissertation starts with an introduction to inverse problems. Then algorithms for forward and inverse problems are discussed. The discussion on forward problem focuses on the VIE formulation and a frequency domain solver. Discussion on inverse problems focuses on iterative methods.

The rest of the dissertation is organized by the two categories of inverse problems, namely the inverse source problem and the inverse scattering problem.

The inverse source problem is studied via an application in microelectronics. In this application, a FFT based inverse source solver is applied to process near field data obtained by near field scanners. Examples show that, with the help of this inverse source solver, the resolution of unknown current source images on a device under test is greatly improved. Due to the improvement in resolution, more flexibility is given to the near field scan system.

Both the forward and inverse solver for inverse scattering problems are studied in detail. As a forward solver for inverse scattering problems, a fast FFT based method for solving VIE of magnetodielectric objects with large electromagnetic contrasts are presented due to the increasing interest in contrasts enhanced full wave EM imaging. This newly developed VIE solver assigns different basis functions of different orders to expand flux densities and vector potentials. Thus, it is called the mixed ordered BCGS-FFT method. The mixed order BCGS-FFT method maintains benefits of high order basis functions for VIE while keeping correct boundary conditions for flux densities and vector potentials. Examples show that this method has an excellent performance on both isotropic and anisotropic objects with high contrasts. Examples also verify that this method is valid in both high and low frequencies. Based on the mixed order BCGS-FFT method, an inverse scattering solver for anisotropic objects is studied. The inverse solver is formulated and solved by the variational born iterative method. An example given in this section shows a successful inversion on an anisotropic magnetodielectric object.

Finally, a lab scale hydraulic fractures evaluation system for oil/gas reservoir based on previous discussed inverse solver is presented. This system has been setup to verify the numerical results obtained from previously described inverse solvers. These scaled experiments verify the accuracy of the forward solver as well as the performance of the inverse solver. Examples show that the inverse scattering model is able to evaluate contrasts enhanced hydraulic fractures in a shale formation. Furthermore, this system, for the first time in the oil and gas industry, verifies that hydraulic fractures can be imaged through a metallic casing.

In this study we propose a fast hybrid spectral-element time-domain (SETD) / finite-element time-domain (FETD) method for transient analysis of multiscale electromagnetic problems, where electrically fine structures with details much smaller than a typical wavelength and electrically coarse structures comparable to or larger than a typical wavelength coexist.

Simulations of multiscale electromagnetic problems, such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and electronic packaging, can be very challenging for conventional numerical methods. In terms of spatial discretization, conventional methods use a single mesh for the whole structure, thus a high discretization density required to capture the geometric characteristics of electrically fine structures will inevitably lead to a large number of wasted unknowns in the electrically coarse parts. This issue will become especially severe for orthogonal grids used by the popular finite-difference time-domain (FDTD) method. In terms of temporal integration, dense meshes in electrically fine domains will make the time step size extremely small for numerical methods with explicit time-stepping schemes. Implicit schemes can surpass stability criterion limited by the Courant-Friedrichs-Levy (CFL) condition. However, due to the large system matrices generated by conventional methods, it is almost impossible to employ implicit schemes to the whole structure for time-stepping.

To address these challenges, we propose an efficient hybrid SETD/FETD method for transient electromagnetic simulations by taking advantages of the strengths of these two methods while avoiding their weaknesses in multiscale problems. More specifically, a multiscale structure is divided into several subdomains based on the electrical size of each part, and a hybrid spectral-element / finite-element scheme is proposed for spatial discretization. The hexahedron-based spectral elements with higher interpolation degrees are efficient in modeling electrically coarse structures, and the tetrahedron-based finite elements with lower interpolation degrees are flexible in discretizing electrically fine structures with complex shapes. A non-spurious finite element method (FEM) as well as a non-spurious spectral element method (SEM) is proposed to make the hybrid SEM/FEM discretization work. For time integration we employ hybrid implicit / explicit (IMEX) time-stepping schemes, where explicit schemes are used for electrically coarse subdomains discretized by coarse spectral element meshes, and implicit schemes are used to overcome the CFL limit for electrically fine subdomains discretized by dense finite element meshes. Numerical examples show that the proposed hybrid SETD/FETD method is free of spurious modes, is flexible in discretizing sophisticated structure, and is more efficient than conventional methods for multiscale electromagnetic simulations.

Electromagnetic scattering in a layered medium (LM) is important for many engineering applications, including the hydrocarbon exploration. Various computational methods for tackling well logging simulations are summarized. Given their advantages and limitations, main attention is devoted to the surface integral equation (SIE) and its hybridization with the finite element method (FEM).

The thin dielectric sheet (TDS) based SIE, i.e., TDS-SIE, is introduced to the simulation of fractures. Its accuracy and efficiency are extensively demonstrated by simulating both conductive and resistive fractures. Fractures of variant apertures, conductivities, dipping angles, and extensions are also simulated and analyzed. With the aid of layered medium Green's functions (LMGFs), TDS-SIE is extended into the LM, which results in the solver entitled LM-TDS-SIE.

In order to consider the borehole effect, the well-known loop and tree basis functions are utilized to overcome low-frequency breakdown of the Poggio, Miller, Chang, Harrington, Wu, and Tsai (PMCHWT) formulation. This leads to the loop-tree (LT) enhanced PMCHWT, which can be hybridized with TDS-SIE to simulate borehole and fracture together. The resultant solver referred to as LT-TDS is further extended into the LM, which leads to the solver entitled LM-LT-TDS.

For inhomogeneous or complex structures, SIE is not suitable for their scattering simulations. It becomes advantageous to hybridize FEM with SIE in the framework of domain decomposition method (DDM), which allows independent treatment of each subdomain and nonconformal meshes between them. This hybridization can be substantially enhanced by the adoption of LMGFs and loop-tree bases, leading to the solver entitled LM-LT-DDM. In comparison with LM-LT-TDS, this solver is more powerful and able to handle more general low-frequency scattering problems in layered media.

Advanced numerical algorithms should be amenable to the scalability in the increasingly powerful supercomputer architectures, the adaptivity in the intricately multi-scale engineering problems, the efficiency in the extremely large-scale wave simulations, and the stability in the dynamically multi-phase coupling interfaces.

In this study, I will present a multi-scale \& multi-physics 3D wave propagation simulator to tackle these grand scientific challenges. This simulator is based on a unified high-order discontinuous Galerkin (DG) method, with adaptive nonconformal meshes, for efficient wave propagation modeling. This algorithm is compatible with a diverse portfolio of real-world geophysical/biomedical applications, ranging from longstanding tough problems: such as arbitrary anisotropic elastic/electromagnetic materials, viscoelastic materials, poroelastic materials, piezoelectric materials, and fluid-solid coupling, to recent challenging topics: such as fracture-wave interactions.

Meanwhile, I will also present some important theoretical improvements. Especially, I will show innovative Riemann solvers, inspired by physical meanings, in a unified mathematical framework, which are the key to guaranteeing the stability and accuracy of the DG methods and domain decomposition methods.

Domain decomposition method provides a solution for the very large electromagnetic

system which are impossible for single domain methods. Discontinuous Galerkin

(DG) method can be viewed as an extreme version of the domain decomposition,

i.e., each element is regarded as one subdomain. The whole system is solved element

by element, thus the inversion of the large global system matrix is no longer necessary,

and much larger system can be solved with the DG method compared to the

continuous Galerkin (CG) method.

In this work, the DG method is implemented on a subdomain level, that is, each subdomain contains multiple elements. The numerical flux only applies on the

interfaces between adjacent subdomains. The subodmain level DG method divides

the original large global system into a few smaller ones, which are easier to solve,

and it also provides the possibility of parallelization. Compared to the conventional

element level DG method, the subdomain level DG has the advantage of less total

DoFs and fexibility in interface choice. In addition, the implicit time stepping is

relatively much easier for the subdomain level DG, and the total CPU time can be

much less for the electrically small or multiscale problems.

The hybrid of elements are employed to reduce the total DoF of the system.

Low-order tetrahedrons are used to catch the geometry ne parts and high-order

hexahedrons are used to discretize the homogeneous and/or geometry coarse parts.

In addition, the non-conformal mesh not only allow dierent kinds of elements but

also sharp change of the element size, therefore the DoF can be further decreased.

The DGTD method in this research is based on the EB scheme to replace the

previous EH scheme. Dierent from the requirement of mixed order basis functions

for the led variables E and H in the EH scheme, the EB scheme can suppress the

spurious modes with same order of basis functions for E and B. One order lower in

the basis functions in B brings great benets because the DoFs can be signicantly

reduced, especially for the tetrahedrons parts.

With the basis functions for both E and B, the EB scheme upwind

ux and

EB scheme Maxwellian PML, the eigen-analysis and numerical results shows the

eectiveness of the proposed DGTD method, and multiscale problems are solved

eciently combined with the implicit-explicit hybrid time stepping scheme and multiple

kinds of elements.

The EB scheme DGTD method is further developed to allow arbitrary anisotropic

media via new anisotropic EB scheme upwind

ux and anisotropic EB scheme

Maxwellian PML. The anisotropic M-PML is long time stable and absorb the outgoing

wave eectively. A new TF/SF boundary condition is brought forward to

simulate the half space case. The negative refraction in YVO4 bicrystal is simulated

with the anisotropic DGTD and half space TF/SF condition for the rst time with

numerical methods.

Based on the compressed sensing theorem, we present the integrated software and hardware platform for developing a total-variation based image restoration algorithm by applying prior image information and free-form deformation fields for image guided therapy. The core algorithm we developed solves the image restoration problem for handling missing structures in one image set with prior information, and it enhances the quality of the image and the anatomical information of the volume of the on-board computed tomographic (CT) with limited-angle projections. Through the use of the algorithm, prior anatomical CT scans were used to provide additional information to help reduce radiation doses associated with the improved quality of the image volume produced by on-board Cone-Beam CT, thus reducing the total radiation doses that patients receive and removing distortion artifacts in 3D Digital Tomosynthesis (DTS) and 4D-DTS. The proposed restoration algorithm enables the enhanced resolution of temporal image and provides more anatomical information than conventional reconstructed images.

The performance of the algorithm was determined and evaluated by two built-in parameters in the algorithm, i.e., B-spline resolution and the regularization factor. These parameters can be adjusted to meet different requirements in different imaging applications. Adjustments also can determine the flexibility and accuracy during the restoration of images. Preliminary results have been generated to evaluate the image similarity and deformation effect for phantoms and real patient's case using shifting deformation window. We incorporated a graphics processing unit (GPU) and visualization interface into the calculation platform, as the acceleration tools for medical image processing and analysis. By combining the imaging algorithm with a GPU implementation, we can make the restoration calculation within a reasonable time to enable real-time on-board visualization, and the platform potentially can be applied to solve complicated, clinical-imaging algorithms.

Discontinuous Galerkin (DG) methods provide an efficient option for modeling multiscale problems. With the help of the Riemann solver (upwind flux), a discontinuous Galerkin based multi-domain multi-solver technique is introduced in this work for multiscale electromagnetic modeling. Specifically, the proposed technique allows multiple subdomains and multiple solvers. Through multiple subdomains, the original large linear system is reduced into multiple subsystems to solve, thus reducing computational complexity. Different subdomains can be non-conformal with each other. Different element types (tetrahedron, hexahedron, Yee's cell) and element sizes (h-refinement) can be used with different orders of basis functions (p-refinement). For different solvers, the finite element method, spectral element method and finite difference method are incorporated into the proposed technique. As is well known, the finite element method features great mesh flexibility for arbitrarily shaped objects, the spectral element method shows spectral accuracy with high order basis functions, and the finite difference method has great computational efficiency for time domain modeling. With multiple solvers the proposed technique can provide efficient solution for multiscale problems based on different element types. Considering the model geometry, for irregular and complicated structures the finite element method is used with small tetrahedron elements for local refinement, for simple structures the spectral method is used with high order basis functions based on hexahedron elements to exploit its spectral accuracy, and for perfectly matched layer (PML) and layered structures the finite different method is used to improve computational efficiency.

For time domain modeling, firstly hybrid spectral element-finite element method in time domain (hybrid SETD-FETD) is implemented based on the first-order Maxwell's curl equations. To facilitate modeling of electrically small problems, an efficient implicit non-iterative time integration method is proposed based the EB scheme for sequentially ordered systems. Compared with the previous Block-Thomas algorithm, the proposed block Lower-Diagonal-Upper (LDU) decomposition algorithm shows better performance in terms of CPU time and memory, due to the separation of surface unknowns from the volume unknowns.

Then a second-order wave equation based discontinuous Galerkin time domain (DGTD) framework is proposed with a modified Riemann solver to evaluate the flux. Compared with the first-order Maxwell's curl equations based DGTD methods, the new DGTD framework reduces the degrees of freedom for each subdomain by solving the E unknowns plus only surface H unknowns. By contrast, the first-order Maxwell's curl equations based DGTD methods require to solve all the E and B unknowns in each subdomain. To model open problems, a novel coupling method is proposed to incorporate PML into the wave equation based DGTD framework. The PML region is based on the first-order Maxwell's curl equations based DGTD framework with implicit Crank-Nicholson time integration while the physical region follows the wave equation based DGTD framework with implicit Newmark-beta time integration.

To further extend the existing hybrid methods, the hybrid FDTD-SETD-FETD method is proposed by incorporating the finite different time domain method (FDTD). In this work, completely non-conformal mesh is implemented for the first time to hybridize FDTD, SETD and FETD for 3D modeling. Based on the DGTD framework, a buffer zone is introduced between FDTD and SETD/FETD to facilitate the coupling procedure. A global leapfrog time integration is implemented to validate the proposed hybrid FDTD-SETD-FETD method, and the implicit-explicit time integration is proposed to improve its performance for practical applications. To remove the buffer zone, a more advanced hybridization technique is introduced, which shows better performance in terms of CPU time and memory. The corresponding explicit leapfrog and implicit-explicit time integration scheme are also given for the new hybrid FDTD-SETD-FETD method without buffer.

For frequency domain modeling, based on the second-order wave equation and time harmonic assumption, a discontinuous Galerkin frequency domain (DGFD) method is introduced with the Riemann solver for anisotropic media. To improve the accuracy and efficiency, a mixed total field/scattered field DGFD (TF/SF DGFD) formulation is given. For subdomains with sources and receivers, the scattered field based DGFD is used to improve accuracy while for the remaining subdomains the total field DGFD is used to improve efficiency. With TF/SF DGFD, the scattered field at the receivers can be directly obtained. In addition, some useful boundary conditions, including scattering boundary condition (SCBC), surface impedance boundary condition (SIBC) and impedance transition boundary condition (ITBC), are incorporated into the proposed DGFD framework to further improve its performance for geophysical exploration problems. SCBC is used to truncate the physical region, SIBC is for approximating the effect of thick imperfect conductor, and ITBC is for approximating thin imperfect conductor as a surface. ITBC is used for the first time in this work for fracture modeling, and shows good agreement with the reference.

Finally, a domain decomposition based inversion method is proposed based on the DGFD forward solver for inverse scattering problems. The inversion algorithm is given based on the Gauss-Newton method, and more importantly, the formulation related to domain decomposition is derived. With the advantage of domain decomposition, an adaptive inversion procedure is introduced to gradually improve the inversion resolution and accuracy.

In this study, we developed and improved the numerical mode matching (NMM) method which has previously been shown to be a fast and robust semi-analytical solver to investigate the propagation of electromagnetic (EM) waves in an isotropic layered medium. The applicable models, such as cylindrical waveguide, optical fiber, and borehole with earth geological formation, are generally modeled as an axisymmetric structure which is an orthogonal-plano-cylindrically layered (OPCL) medium consisting of materials stratified planarly and layered concentrically in the orthogonal directions.

In this report, several important improvements have been made to extend applications of this efficient solver to the anisotropic OCPL medium. The formulas for anisotropic media with three different diagonal elements in the cylindrical coordinate system are deduced to expand its application to more general materials. The perfectly matched layer (PML) is incorporated along the radial direction as an absorbing boundary condition (ABC) to make the NMM method more accurate and efficient for wave diffusion problems in unbounded media and applicable to scattering problems with lossless media. We manipulate the weak form of Maxwell's equations and impose the correct boundary conditions at the cylindrical axis to solve the singularity problem which is ignored by all previous researchers. The spectral element method (SEM) is introduced to more efficiently compute the eigenmodes of higher accuracy with less unknowns, achieving a faster mode matching procedure between different horizontal layers. We also prove the relationship of the field between opposite mode indices for different types of excitations, which can reduce the computational time by half. The formulas for computing EM fields excited by an electric or magnetic dipole located at any position with an arbitrary orientation are deduced. And the excitation are generalized to line and surface current sources which can extend the application of NMM to the simulations of controlled source electromagnetic techniques. Numerical simulations have demonstrated the efficiency and accuracy of this method.

Finally, the improved numerical mode matching (NMM) method is introduced to efficiently compute the electromagnetic response of the induction tool from orthogonal transverse hydraulic fractures in open or cased boreholes in hydrocarbon exploration. The hydraulic fracture is modeled as a slim circular disk which is symmetric with respect to the borehole axis and filled with electrically conductive or magnetic proppant. The NMM solver is first validated by comparing the normalized secondary field with experimental measurements and a commercial software. Then we analyze quantitatively the induction response sensitivity of the fracture with different parameters, such as length, conductivity and permeability of the filled proppant, to evaluate the effectiveness of the induction logging tool for fracture detection and mapping. Casings with different thicknesses, conductivities and permeabilities are modeled together with the fractures in boreholes to investigate their effects for fracture detection. It reveals that the normalized secondary field will not be weakened at low frequencies, ensuring the induction tool is still applicable for fracture detection, though the attenuation of electromagnetic field through the casing is significant. A hybrid approach combining the NMM method and BCGS-FFT solver based integral equation has been proposed to efficiently simulate the open or cased borehole with tilted fractures which is a non-axisymmetric model.

Hydraulic fracturing is an essential way to improve the production of unconventional oil and gas, especially shale gas. Therefore, it is important to characterize the produced fractures using either acoustic or electromagnetic (EM) methods, and evaluation of hydraulic fractures has been under intensive study since last decade. Electromagnetic techniques, including induction logging and galvanic techniques, have the advantages of nondestructive measurements and high sensitivity to the formation and fracture resistivity. They are widely used for produced fracture evaluation.

However, conventional forward and inverse methods in low frequency range face significant challenges by such multiscale problems where the fracture width (<1cm) is orders of magnitude smaller than its diameters (>100 m). The problem becomes much more complicated when the effects of borehole, casing, and planar stratified medium need to be considered for realistic oil field application.

This dissertation focuses on three aspects. First of all, the application of newly developed efficient forward electromagnetic solvers, hybrid distorted Born approximation and mixed ordered stabilized bi-conjugate gradient FFT (DBA-BCGS-FFT) method, and hybrid numerical mode matching with the stabilized bi-conjugate gradient FFT (NMM-BCGS-FFT) method, are illustrated. For the DBA-BCGS-FFT method, the two components of the solver, distorted Born approximation (DBA) and mixed ordered stabilized bi-conjugate gradient FFT (BCGS-FFT), are separately discussed with their advantages and disadvantages. Then the hybrid DBA-BCGS-FFT will be introduced and explained, including how the combination of the advantages of the two solvers and overcome their disadvantages. For the second forward method, the numerical mode matching (NMM) method is introduced with the procedures of the NMM-BCGS-FFT method for analyzing the effects of the complex cased borehole and planar stratified medium.

Second, the inverse solver, variational Born iterative method (VBIM), is introduced for hydraulic fracture reconstruction. The box constraints in the inversion process is introduced to enhance the fracture reconstruction resolution and avoid unrealistic parameter in the inversion. In the procedures of the inverse solver, the forward solvers are applied to construct the system matrix. In this application, the inverse solvers are applied to process the secondary field data obtained by field scanners and laboratory detectors.

The results will be separate into three sections. First, the validation of the forward and inverse solvers is demonstrated. The commercial software, COMSOL, is used for the validation. Then, induction logging detection and galvanic detection model results show the capability of the forward and inverse solvers. Last, two established experiment systems will be described with details. The laboratory scaled experimental system is established for feasibility study of the electromagnetic induction detection, and the field test control source electromagnetic system is designed and built for hydraulic fracture evaluation. In induction logging detection model, experimental results show that the inverse scattering algorithm is effective for electromagnetic contrast enhanced through-casing hydraulic fracture evaluation. In galvanic detection model, the impact of different hydraulic fracturing material and choices of transmitter/receiver locations on signal response will be discussed to show the application of the forward solvers in field configuration design. Both fracture reconstruction results by the inverse solvers with the experimental data will be discussed in these two chapters.

For the petroleum industry, layered medium subsurface detection plays an important role in discovering reservoirs and drilling wells. In geophysics, resistivity is an essential property for distinguishing formation layers or even small fractures. Well logging with electromagnetic induction tools can measure the subsurface resistivity. This measurement includes two steps: 1) directly measure the low-frequency response signals using the tool and 2) determine the subsurface geometric model and resistivity. The problem is that no formula can directly calculate the resistivity from the measured tool responses. A systematic solution is to combine forward electromagnetic simulations and inversion of the subsurface model. In this dissertation, two categories of inversion are investigated: Determine the proper subsurface model by 1) optimizing the objective function, such as data misfit, and 2) training a surrogate model for the inverse mapping. Many forward simulations are demanded for either estimating the data misfit of new candidate models or collecting data for training. Therefore, efficient electromagnetic simulation is critical for resistivity logging. From complex to simple, three types of simulation are discussed: 1) borehole simulation with real tool configuration, 2) borehole simulation with point sources as the virtual tool, and 3) simplified layered medium simulation with virtual tool. Three optimal methods are implemented, respectively: the domain decomposition method, the finite element boundary integral method, and the analytical method. The tool calibration and the borehole effects are studied in the comparison of these simulations. Ideally, the simplest forward simulation should be used in the inversion, and the additional effects can be extracted as correction terms. The optimization-based inversion of the formation model uses simulations of a virtual tool in the layered medium. The Occam inversion or Monte Carlo Markov chain can minimize the data misfit. Another special simulation for small fractures using the thin dielectric sheet approximate method collects the dataset of fracture models. Fracture parameters such as resistivity, extension, and tilt angle are accurately determined by machine learning methods. The surrogate model also tends to predict fracture properties correctly, even for the complete simulation result.

Electromagnetic forward modeling and inversion methods have extensive applications in geophysical exploration, and large-scale controlled-source electromagnetic method has recently drawed lots of attention. However, to obtain a rigorous and efficient forward solver for this large-scale three-dimensional problem is difficult, since it usually requires to solve for a large number of unknowns from a system of equations describing the complicate scattering behavior of electromagnetic waves that happened within inhomogeneous media. As for the development of an efficient inversion solver, because of the nonlinear, non-unique and ill-posed properties of the problem, it is also a very challenging task.

In the first part of this dissertation, a fast three-dimensional nonlinear reconstruction method is proposed for controlled-source electromagnetic method. The borehole-to-surface and airborne electromagnetic survey methods are investigated using synthetic data. In this work, it is assumed that there is only electric contrast between the inhomogeneous object and the layered background medium, for the reason that the electric contrast is much more dominant than magnetic contrast in most cases of the earth formation. Therefore, only the EFIE is needed to solve. While the forward scattering problem is solved by the stabilized bi-conjugate gradient FFT (BCGS-FFT) method to give a rigorous and efficient modeling, the Bore iterative method along with the multiplicative regularization technique is used in the inversion through frequency hopping. In the inversion, to speed up the expensive computation of the sensitivity matrix relating every receiver station to every unknown element, a fast field evaluation (FFE) technique is proposed using the symmetry property of the layered medium Green's function combined with a database strategy. The conjugate-gradient method is then applied to minimize the cost function after each iteration. Due to the benefits of using 3D FFT acceleration, the proposed FFE technique as well as the recursive matrix method combined with an interpolation technique to evaluate the LMGF, the developed inversion solver is highly efficient, and requires very low computation time and memory. Numerical experiments for both 3D forward modeling and conductivity inversion are presented to show the accuracy and efficiency of the method.

Some recent research on artificial nanoparticles have demonstrated the improved performance in geophysical imaging using magnetodielectric materials with enhanced electric and magnetic contrasts. This gives a promising perspective to the future geophysical exploration by infusing well-designed artificial magnetodielectric materials into the subsurface objects, so that a significantly improved imaging can be achieved. As a preparation for this promising application, the second part of the dissertation presents an efficient method to solve the scattering problem of magnetodielectric materials with general anisotropy that are embedded in layered media. In this work, the volume integral equation is chosen as the target equation to solve, since it solves for fields in inhomogeneous media with less number of unknowns than the finite element method. However, for complicated materials as magnetodielectric materials with general anisotropy, it is a very challenging task, because it requires to simultaneously solve the electric field integral equation (EFIE) and magnetic field integral equation (MFIE). Besides that, the numerous evaluation of the layered medium Green's function (LMGF) for the stratified background formation adds on the difficulty and complexity of the problem. To my knowledge, there is no existing fast solver for the similar problem. In this dissertation, using the mixed order stabilized biconjugate-gradient fast Fourier transform (mixed-order BCGS-FFT) method, a fast forward modeling method is developed to solve this challenging problem. Several numerical examples are performed to validate the accuracy and efficiency of the proposed method.

Besides the above mentioned two topics, one-dimensional inversion method is presented in the third part to determine the tilted triaxial conductivity tensor in a dipping layered formation using triaxial induction measurements. The tilted triaxial conductivity tensor is described by three conductivity components and three Euler angles. Based on my knowledge, due to the highly nonlinear and ill-posed nature of the inverse problem, this study serves as the first work that investigates on the subject. There are six principal coordinate systems that can give the same conductivity tensor. Permutation is performed to eliminate the ambiguity of inversion results caused by the ambiguity of the principal coordinate system. Three new Euler angles after permutation for each layer can be found by solving a nonlinear equation. Numerical experiments are conducted on synthetic models to study the feasibility of determining triaxially anisotropic conductivity tensor from triaxial induction data. This project is accomplished during my internship in the Houston Formation Evaluation Integration Center (HFE) at Schlumberger, a world-leading oilfield service company.

This work focuses on the electromagnetic simulation of doubly periodic structures embedded in layered media, which can be commonly found in extreme ultraviolet (EUV) lithography, metasurfaces, and frequency selective surfaces. Such problems can be solved by rigorous numerical methods like finite-difference time-domain (FDTD) method and finite element method (FEM). However, FDTD and FEM are universal methods far from achieving the best efficiency for the target problem. To exploit the problem property and facilitate the problem solving of large size in low complexity, two approaches are proposed.

The first approach, Calder\'{o}n preconditioned spectral-element spectral-integral (CP-SESI) method, is an improvement over the existing finite-element boundary-integral method. By introducing the Calder\'{o}n preconditioner, domain decomposition and the fast Fourier transform technique, the time and memory complexity of CP-SESI is reduced to O(N\textsuperscript{1.30}) and O(N\textsuperscript{1.07}), respectively.

The second approach, based on modified U-Net, introduces two stages of problem solving: the training stage and the inference stage. In the training stage, accurate data generated by the rigorous CP-SESI solver is fed to the U-Net. In the inference stage, the U-Net can be applied to solve unseen problems in real time. Particularly, the EUV problem with mask size of 4 um by 4 um can be solved on a personal desktop within 5 min on CPU or 30 s on GPU.

Besides, two types of equivalent boundary conditions to replace thin structures are developed and incorporated into the framework of CP-SESI. The first one, surface current boundary condition, has better accuracy for resistive materials. The second one, impedance transition boundary condition, is more accurate for conductive materials. The accuracy comparison between the above two boundary conditions are compared.

Crosswell electromagnetic (EM) fields measurement has been widely applied in oil industry which has a reservoir scale detection range. One of the limitation of this technique is that the EM singals are usually too weak to be detected. In order to overcome the disadvantage, nanoparticles (NP) designed with high contrast EM properties (conductivity, permittivity and magnetic permeability) are introduced to enhance the signals. They are injected into the formation and moving with the fluids. The movement of NP with the flow in a porous medium is modeled by solving the flow transport equations. The 3-D spectral-element time domain method (SETD) based on Gauss-Lobatto-Legendre (GLL) polynomials is employed to solve the flow equation to obtain the NP concentration distribution as a function of time. Since the method shows a spectral accuracy, i.e., the error decreases exponentially with the order of basis functions, less unknown is needed to achieve a given accuracy with a high order basis function.

The injected fluids with high contrast NP increase the electric conductivity and magnetic permeability in the flooded zone. The effective EM properties of the mixed fluids are calculated by the mixing theory, e.g., Bruggeman mixing rule. The increased EM property values produce higher EM signals in the receivers. The EM fields are then modeled by the volume integral equations (VIE), thus realizing the coupling of fluid flow and EM measurements. Based on the coupling, the detection range of the high contrast NP can be analyzed. Different types of NP are investigated under both electric and magnetic dipole sources.

The magnetic contrast NP excited by a magnetic dipole source can generate a detectable signal, while the electric contrast NP is more sensitive to an electric dipole. Using a magnetic dipole source, it is hard to generate detectable signals at receivers with high dielectric particles, however, increasing the frequency will improve the signals.

The coupling technique can also be used to evaluate the heterogeneity of the formation. When the high contrast agents are injected into a heterogeneous medium, e.g., with a low permeable region. The EM responses at the four producers are different. The signals at the producer near the barrier are lower than the other producers, since the fluids containing the high contrast NP is blocked. The proposed multiphysics coupling technique of fluid flow and EM measurements provides guidance for NP field application and help monitor the flow movement in reservoirs.

One of the applications of the high contrast agents is used for hydraulic fractures detection. Hydraulic fracturing is a technique to crack rocks by pumping high pressure fluid into a segment of a well. The created fractures serve as a pathway to release the hydrocarbon resource such as oil or natural gas from the rock. It is an efficient technology to increase the oil/gas production in tight formations. Successful fracture imaging is important to evaluate the created fractures. This is a part of a large project of the Advanced Energy Consortium (AEC) to image large-scale hydraulic fractures in deep underground with high contrast proppants injection. A group of small-scaled fracturing field tests are performed by AEC to investigate the feasibility of injecting high contrast proppants to detect fractures. The injected proppants are designed with high conductivity and permittivity to generate detectable signals at electrode-type sensors. To map the created fractures, an efficient 3-D EM inversion method with physical constraint on the inverted unknowns is developed to simultaneously reconstruct conductivity and permittivity profiles.

The inversion solver is firstly applied to a theoretical model with the noise-polluted synthetic data to reconstruct the fracture, and then applied to two hydraulic fracturing field tests with injected high conductive proppants, Loresco Coke Breeze and steel shot. The fracture conductivity and permittivity are reconstructed based on the scattered voltage signals which are the difference between the post-fracturing and pre-fracturing data. The post-fracturing data are the signals measured after the fracturing and the pre-fracturing data are measured before the fracturing. The difference signals are regarded as from the created fracture. The reconstructed fracture profiles are compared with the coring data to show the reliability of the inversion results. Their good agreement demonstrates the effectiveness of the inverse solver to estimate the fracture size and location.

In recent several decades, EM fields from layered media have attracted considerable attention concerning various applications including geophysical exploration, microwave remote sensing, wave propagation, microstrip circuits, antennas, etc. Especially, the EM waves in anisotropic laminates are of much concern. For geophysical problems, anisotropy happens commonly in many formations, e.g., shale formation. To accurately evaluate the anisotropic medium, a forward solver capable of handling arbitrary anisotropy is needed.

In this work, the formulations for EM fields in multilayered general anisotropic media are derived. Maxwell's equations in the spectral domain are written into a first-order differential (in $z$) equation concerning the transverse electric and magnetic field components in the spectral domain. The equation can be solved to obtain the EM fields in a homogeneous anisotropic medium. For fields in layered anisotropic media, the local transmission and reflection matrices, the global reflection matrices and the recursion relations of the wave amplitudes at interfaces are derived and used to express the EM fields in arbitrary layers. The electric and magnetic dipole sources can locate in arbitrary layers and the medium can possess an arbitrary anisotropy.

To transform the spectral domain solution into the spatial domain involves the inverse Fourier transform which needs integration from $-\infty$ to $+\infty$. The speed of integration calculation depends on the decaying of the integrands. The singular behavior of the fields in the close vicinity of the dipole source needs to be considered since the integrand usually converges very slowly. In this work, it is handled by subtracting the direct fields in the spectral domain, since the direct fields contribute most of the singular problem. The contributions of the subtracted part in the spatial domain are calculated and added afterwards. An example is modeled to show the convergence of the integrands with / without the singularity subtraction. The subtraction makes the integrands decaying rapidly as functions of $k_x$ and $k_y$.

To validate the algorithm, a multilayer full anisotropic medium is modeled and compared with the finite element method (FEM) results. It is also applied to the geophysical EM well logging by modeling the triaxial induction logging tool. The responses in vertical and deviated wells are computed and compared with FEM results. The good agreement between the two results further validates the algorithm and shows the capability of modeling induction logging tools in multilayered general anisotropic media.

The scattering of EM fields from anisotropic objects has been studied intensively in recent years. Most of the work studies the inhomogeneities in homogeneous isotropic background media and a few work has been done on uniaxial anisotropic media. This work extends to inhomogeneities embedded in layered general anisotropic media. The volume integral equation based on the electric dyadic Green's function is derived and solved efficiently with the fast Fourier transform (FFT) based BCGS method. The FFT technique is employed to calculate the convolution and correlation efficiently involved in the integral equation which reduces the computation cost from $O(N^2)$ to $O (N log N)$. A series of numerical examples are modeled and compared with FEM results to validate the algorithm.

In this dissertation, four topics will be presented: (1) The volume integral eqaution method with the domain decomposition method (VIE-DDM) and the inversion using VIE-DDM as forward solver; (2)The numerical mode matching method with surface current boundary condition (NMM-SCBC); (3) NMM-VIE-DDM.VIE-DDM and the inversion: In many applications, electromagnetic scattering from inhomogeneous objects embedded in multiple layers needs to be simulated numerically. The straightforward solution by the method of moments (MoM) for the volume integral equation method is computationally expensive. Due to the shift-invariance and correlation properties of the layered-medium Green's functions, the Stabilized Bi-Conjugate Gradient Fast Fourier Transform (BCGS-FFT) has been developed to greatly reduce the computational complexity of the MoM, but so far this method is limited to objects located in a homogeneous background or in the same layer of a layered medium background. For those problems with objects located in different layers, FFT cannot be applied directly in the direction normal to the layer interfaces, thus the MoM solution requires huge computer memory and CPU time. To overcome these difficulties, the BCGS-FFT method combined with the domain decomposition method (DDM) is proposed in this work. With the BCGS-FFT-DDM, the objects or different parts of an object are first treated separately in several subdomains, each of which satisfies the 3D shift-invariance and correlation properties; the couplings among the different objects/parts are then taken into account, where the coupling matrices can be built to satisfy the 2D shift-invariance property if the objects/subdomains have the same mesh size on the xy-plane. Hence, 3D FFT and 2D FFT can respectively be applied to accelerate the self- and mutual-coupling matrix-vector multiplications. By doing so, the impedance matrix is explicitly formed as one including both the self- and mutual-coupling parts, and the solver converges well for problems with considerable conductivity contrasts. The computational complexity in memory and CPU time can be significantly reduced. Using the BCGS-FFT-DDM as the forward solver, the inversion algorithm based on the Born approximation method and Born iterative method are developed to reconstructed the size, location and properties of the targets buried underground.

NMM-SCBC: The NMM method is widely employed in well-logging problems, because it can transform the original 2.5D problem to a 1D eigenvalue problem at the radial direction, which is usually treated with finite element method (FEM), and a semi-analytical problem at the z direction, which can be easily dealt with the mode matching strategy, and therefore the computational load is significantly reduced. However, more and more well-logging problems are equipped with carbon steel casing, which is extremely thin but with extremely high conductivity. With the conventional NMM method, the extremely thin casing will make the mesh for the FEM tremendously dense and the extremely high conductivity will make the matrices for the eigenvalue problem near ill-posed, both of which will make the solution of the eigenvalue problem inefficient and inaccurate. To overcome this problem, we proposed to apply the SCBC to substitute the extremely thin and highly conductive casing. To employ the NMM-SCBC, the mixed-order FEM isdeveloped to treat the 1D eigenvalue problem, in which the SCBC is deliberately applied. After the eigenvalues and the eigenvectors are solved for each horizontal layer, the mode matching strategy will be applied across the horizontal layers as the conventional NMM method.

MM-VIE-DDM: In many applications, electromagnetic scattering from inhomogeneous objects embedded in multiple layers with cylindrical geometry needs to be simulated numerically. The Numerical Mode Matching (NMM) method has long been The numerical mode matching (NMM) method has long been demonstrated to be the most efficient algorithm for the application under the axial symmetric background, compared with the full 2D methods (such as finite element, finite difference, integral equation method etc.) However, it only works well for the problems with axial symmetric background. For the problem with both cylindrical geometry and 3D objects (such as borehole with reservoirs, fractures), full 3D solvers (such as BCGS-FFT-DDM) can be applied, but will require tremendously large memory and CPU time. The combination of the NMM method and the BCGS-FFT-DDM, or NMM-BCGS-FFT-DDM, is proposed in this paper to deal with the limitations of each method alone. Following the general work flow of the conventional BCGS-FFT-DDM, we substitute the incident field and the dyadic Green's function from the objects to the receivers in the layer media with those considering the cylindrical structures, which are obtained from the NMM method, and neglect the impacts from the cylindrical structures when calculating the total fields inside the objects. Reciprocity and interpolation will be utilized to speed up the calculation when obtaining the Green's function from the objects to the receivers. With the proposed method, the problems with objects in layer media with cylindrical structure can be treated efficiently and accurately. Some numerical results are presented to show the capability of this method.

This thesis introduces two related lines of study on classification of hyperspectral images with nonlinear methods. First, it describes a quantitative and systematic evaluation, by the author, of each major component in a pipeline for classifying hyperspectral images (HSI) developed earlier in a joint collaboration [23]. The pipeline, with novel use of nonlinear classification methods, has reached beyond the state of the art in classification accuracy on commonly used benchmarking HSI data [6], [13]. More importantly, it provides a clutter map, with respect to a predetermined set of classes, toward the real application situations where the image pixels not necessarily fall into a predetermined set of classes to be identified, detected or classified with.

The particular components evaluated are a) band selection with band-wise entropy spread, b) feature transformation with spatial filters and spectral expansion with derivatives c) graph spectral transformation via locally linear embedding for dimension reduction, and d) statistical ensemble for clutter detection. The quantitative evaluation of the pipeline verifies that these components are indispensable to high-accuracy classification.

Secondly, the work extends the HSI classification pipeline with a single HSI data cube to multiple HSI data cubes. Each cube, with feature variation, is to be classified of multiple classes. The main challenge is deriving the cube-wise classification from pixel-wise classification. The thesis presents the initial attempt to circumvent it, and discuss the potential for further improvement.

Complex nonlinear dynamical systems have been appeared in many fields of science and engineering. We are curious about two specific instances of those systems. Those two instances connect memristors and Josephson junctions to the electromagnetic fields. The first instance investigated microstrip patch antenna embedding dual memristors. This hybrid system produces broadband radiation in a narrow band radiation structure. The second one studies the novel ultra-sensitive magnetic field receiver implemented by superconducting quantum interference filters (SQIFs).

For the first instance, we notice that memristor has been proposed as the fourth passive element. We start with investigating the circuit model of this memristive element. Then, we embedded this circuit model into an EM radiation structure. We first report an efficient broadband electromagnetic radiation from a narrowband microstrip patch antenna. The directly modulated microstrip patch antenna system with dual memristors is calculated by using an integrated full-wave finite-difference time-domain solver and an embedded SPICE3 solver. Nonlinear transient electromagnetic responses are analyzed. The radiation frequency spectrum demonstrates the broadband radiation performance from the narrowband antenna system. We predict that the conceptual challenge of high frequency memristors will stimulate pioneering work in the fields of microwave and memristors.

For the second one, we predict that superconducting quantum interference filters (SQIFs) might play a key role in future quantum wireless communication systems. We analyze the dynamic behavior of this large-scale 2D DC SQIF (two-dimensional superconducting direct current quantum interference filter) array in a dynamic electromagnetic environment. The investigation under this framework starts from the SPICE circuit description of a RCSJ (resistively and capacitively shunted junction) model of a Josephson junction and then extends to the 2D SQIF with few device parameters. We separate the interface and the implementation of 2D DC SQIF. This approach can significantly improve circuit-level design efficiency of 2D SQIF array and ultimately allows us to accelerate the hybrid design with an electromagnetic radiation structure. Our findings on the average voltage response of this device offer compelling evidence that the bias static magnetic field plays a key role in designing an effective far-field magnetic field sensor. Since this device can function as both a robust and sensitive low noise pre-amplifier as well as a receiving antenna which only senses the magnetic field component of far-field electromagnetic wave signals, we call it magnetic-antenna or B-antenna. We believe that our research not only directly benefits the sensor design for Information Operations/Signals Intelligence (IO/SIGINT) applications in Very High Frequency/Ultra High Frequency (VHF/UHF) bands, but also opens new dimension of novel ultra-sensitive receiving antenna technology.

A fast electromagnetic (EM) forward solver has been developed for the subsurface detection, with application includes producing synthetic logging data and instructing large-scale field test and inversion. A deep learning based full wave inversion method has also been developed to reconstruct the underground anomaly.

Since the gas and oil industry has very high demands for the forward modeling speed when doing inversion, the inversion model is usually simplified to a 1D or 2D problem by supposing the geometry of object invariant in two or one direction. The full 3D inversion is still a hot topic for research, which requires both fast 3D forward solver and efficient inversion method. The bottleneck for the forward solver is how to solve the large-scale linear system efficiently; the bottleneck for the inversion is how to pick the global minimum from lots of local minimums efficiently for the inverse problem.

For the forward part, the domain decomposition method (DDM) inspired discontinuous Galerkin frequency domain (DGFD) method has been extended to model the vertical open borehole resistivity measurement with structured gradient meshes; besides, the DGFD method has been extended to model the logging-while-drilling (LWD) resistivity measurement in high-angle and horizontal (HA/HZ) well and curved layers with a flipped total field/scattered field (TF/SF) mixed solver. An approximated casing model has also been proposed to accelerate the large-scale curved casing modeling with borehole-to-surface measurements.

For the inversion part, a convolutional neural network based inversion has been developed to reconstruct the lateral extent and direction of the hydraulic fracture through scattered electromagnetic field data under borehole-to-surface measurements; further, the deep transfer learning is applied in the same scenario to improve the performance of the inversion. Additionally, a fully connected neural network has been developed for the Devine field data and successfully reconstruct the shape of the hydraulic fracture with good agreement to the conventional inversion.

In this work, a hybrid mixed order numerical framework is proposed for multiscale linear/ nonlinear nano optical computation. Starting from the principle of the spectral element boundary integral (SEBI) method, the mixed-order SEBI solver with homogeneous Green's function is first developed for the nano-scale linear and nonlinear electromagnetic scattering analyses. The SEBI realizes the exact radiation boundary condition with a set of surface integral equations (SIE's), and discretize the whole computation domain with the fast convergent Gauss-Lobatto-Legendre (GLL) basis function. The Bloch periodic boundary condition is applied for efficient simulation of structures with periodicities in one or two directions.

For nonlinear optical simulation, full-wave solver is developed self-consistently by iteratively solving the vector Helmholtz equations at each harmonic frequency. To further address the multiscale scattering analysis in nano optics, a hybrid framework is developed by combing the SEBI solver with the dyadic periodic layered medium Green's function (PLMGF) and the domain decomposition method (DDM). Formulating the SIE's with the dyadic PLMGF, all unknowns in the background layered medium are pushed to the radiation boundaries. Thus, the whole planar layered background can be truncated from the computation domain. Considering its highly singular analytical properties, the PLMGF is carefully and systematically formulated under matrix representation. A feasible and effective technique is proposed for the on-interface PLMGF singularity extractions. By extracting the primary and secondary terms' singularities separately, all PLMGF-related SIE components can be efficiently evaluated. The DDM further reduces the memory cost for electrically large problems and enhances the framework's flexibility. Finally, a scattered field perfectly matched layer - surface integral equation (PML-SIE) radiation boundary condition is proposed to enable the non-periodic modeling. With the hybrid radiation boundary condition, the periodic and non-periodic solvers are maximumly integrated together with the minimum maintenance cost.

Benefiting from the exponential convergence and flexibility of the SEBI, computationally challenging problem can be solved with considerably reduced number of samplings. As a typical application, the multilayer defects analysis in extreme ultraviolet (EUV) lithography is studied for both 2-D and 3-D models. The light absorption engineering of graphene is also investigated around the visible spectra. Benefiting from the accuracy of the full-wave nonlinear solver, couplings between the fundamental frequency (FF) field and the higher harmonic (HH) field ignored my most previous studies can also be self-consistently analyzed in nonlinear optical simulation. With this tool, the investigation is extended to the engineering of graphene's visible spectra absorption tuning and third harmonic generation (THG) enhancement. Graphene's Kerr effects are also studied under strong surface plasmonic resonance. The hybrid higher order method's efficiency and accuracy are further validated through various multiscale nano-optical cases.