Chiplet employs advanced packaging techniques to heterogeneously integrate chips with different processes and functions, featuring multidie integration and high density. However, the increased power density and thermal coupling within these systems introduce significant electrothermal issues, leading to heat accumulation and power management complexities. Meanwhile, due to the complex multilayer heterogeneous integration and multiscale geometric features of chiplet systems, the number of degrees of freedom increases sharply, resulting in significant computational challenges and high demands for computational resources. Traditional numerical methods, such as finite element analysis (FEA), although capable of providing high-accuracy results, are computationally expensive. This work proposes a parametric dynamic mode decomposition (DMD)-based reduced-order modeling method for the electrothermal analysis, extending DMD to handle variations in parameters such as convection coefficients and thermal conductivity, thereby enabling robust analysis for diverse design requirements. As a matrix decomposition technique based on singular value decomposition (SVD), DMD extracts low-rank structures that capture both temporal dynamics and spatial correlations. The numerical results show that the proposed method provides an efficient and reliable solution for electrothermal analysis of chiplets, achieving a 4–9 times speedup over commercial software like COMSOL Multiphysics, making it an alternative for optimization design and reliability analysis of advanced systems.

A double-layer neural network combining a fully convolutional network (FCN) and U-Net is introduced to improve the accuracy of 2.5-D magnetotelluric (MT) inversion. The initial model obtained through the Bostick inversion method is randomly transformed to generate the dataset for the training of the convolutional neural network (CNN). The training input consists of the apparent resistivity obtained through the forward modeling of transformed models, while the output represents the resistivity of those same models. The proposed method has the local optimization of the neural network inversion method and narrows the range of network optimization by employing an initial solution obtained from the Bostick inversion method. The results of the inversion experiments, including an actual measurements, demonstrate a significant enhancement in the accuracy of the inversion results when employing the neural network method. This demonstrates the efficiency of neural networks in solving 2.5-D magnetotelluric (MT) inversion problems.

Elastic wave propagation in heterogeneous, vertically layered elastic media plays a critical role in fields such as geophysics and acoustic (including sonic and ultrasound) oil well logging. Accurately modeling these interactions is challenging, particularly in complex geometries like perforations in cased-hole acoustic well-logging models. This study extends the Numerical Mode Matching (NMM) method to the field of elastic waves, reducing the original heterogeneous, vertically layered elastic wave problem into a series of two-dimensional (2D) waveguide eigenvalue problems coupled with a 1D layered medium problem. Additionally, it introduces a hybrid numerical method that combines NMM with the 3D Spectral Element Method (SEM) to address perforations in well-logging models. The hybrid NMM-SEM exploits NMM’s strengths for regular layered structures and SEM’s flexibility for irregular geometries. Numerical examples demonstrate significant improvements in both accuracy and efficiency compared to traditional numerical methods. This method offers an effective solution for acoustic well-logging applications and the potential for broader applications in geophysical modeling.

Electromagnetic (EM) scattering of complex 3-D distributions in a layered-medium background is crucial in many remote sensing and geophysical applications, including ground-penetrating radar, microwave tomography, and atmospheric EM analysis. The Volume Integral Equation (VIE) method offers distinct advantages in simulating such problems. In particular, the recently proposed VIE combined with the Domain Decomposition Method (DDM) and accelerated by the Fast Fourier Transform (VIE-DDM-FFT) has shown promise. However, limitations in handling dense meshes along the z-axis led to the development of a low-rank approximation (LRA) algorithm, specifically the Adaptive Cross Approximation (ACA), for mutual matrix-vector multiplications (VIE-DDM-ACA). Despite reducing computational complexity, ACA suffers from premature, unstable, and unpredictable convergence, particularly when dealing with closely spaced objects, resulting in inaccuracies and inefficiencies. In this paper, we propose utilizing the Randomized Matrix Approximation (RMA) method for LRA. RMA provides superior low-rank factorization by requiring fewer ranks and exhibiting more stable and predictable convergence, making it particularly suitable for reconstructing mutual matrices in remote sensing scenarios involving closely spaced scatterers, such as subsurface imaging and layered-medium inversion problems. While RMA involves multiple matrix-vector multiplications (MVMs) during the preparation stage, these MVMs are independent and can be parallelized, mitigating computational costs. Additionally, the iterative MVMs in the equation-solving procedure retain linear complexity with a smaller coefficient compared to ACA due to the reduced rank requirements. Our results demonstrate that the RMA method outperforms ACA in terms of low-rank factorization and accuracy, making VIE-DDM-RMA more efficient than VIE-DDM-FFT and more accurate than VIE-DDM-ACA for EM scattering in remote sensing and geoscience applications involving close objects and complex scattering environments in layered media.

A fast and accurate frequency sweep method combining the mixed finite element method (MFEM) with the second-order Arnoldi algorithm is proposed to characterize the EM material properties for high-loss geoelectromagnetic problems. The MFEM exploits the tree–cotree splitting technique for spatial discretization to eliminate the spurious modes arising from the null space of the curl operator. The modal order reduction method based on the second-order Arnoldi algorithm is employed to generate a set of orthogonal basis of the projection subspace for the quadratic eigenvalue problem (QEP) with both the conduction current and displacement current considered. The efficiency and accuracy of the method are verified by comparing with commercial software HFSS for lossy dielectric characterization in subsurface.

Determination of petrophysical parameters is regarded as a critical task for the exploration and production of oil and gas reservoirs. Traditionally, the joint inversion of electromagnetic (EM) and seismic data under petrophysics constraints can be exploited to reconstruct the distribution of reservoir petrophysical parameters. However, methods involved with such schemes face challenges when solving high-contrast nonlinear inverse scattering problems due to the complexity of the relationship between resistivity/velocity and petrophysical properties and the nonuniqueness of these inverse problems. Here, to resolve these challenges, we have developed a field data transformation-joint inversion method (FDT-JIS) to directly reconstruct the distribution of porosity and water saturation. Specifically, the chain rule to transform geophysical parameters into petrophysical parameters is leveraged, and the field data transformation module is subsequently utilized to transform the scattered field data generated under the test configuration into those under the training configuration. Finally, a joint network is adopted to establish the mapping relationship between EM and acoustics data and petrophysical parameters to attain the inversion of petrophysical parameters. With numerical examples, we demonstrate that FDT-JIS not only allows different transceiver configurations to be used in training and testing, but also accurately reconstructs petrophysical parameters of complex models with high contrast in noisy environments.

The numerical mode matching (NMM) method is a widely used semianalytical approach for modeling the propagation of electromagnetic (EM) waves in orthogonal-plano-cylindrically layered (OPCL) media. In this article, we present four significant enhancements aimed at augmenting the efficiency, accuracy, and applicability of the NMM method. First, we introduce the spectral element method (SEM) to efficiently calculate eigenmodes within each horizontally stratified layer with radial variations. The SEM employs Gauss–Lobatto–Legendre (GLL) polynomials as basis functions to expand the EM field and leverages efficient numerical quadrature for integration. Our study demonstrates that this SEM-based NMM (SNMM) approach offers superior efficiency and accuracy compared with the finite-element method (FEM)-based NMM method, while also requiring lower memory resources. Second, apart from the dipole sources, we expand the capabilities of the SNMM method to simulate EM fields generated by current line sources, broadening its applicability in geophysical exploration. Third, by exploiting field symmetry between two harmonics with opposite mode indices for various excitation scenarios, we achieve a significant reduction in computational time, effectively halving the simulation effort. Fourth, to enhance the condition of the system matrix and improve result accuracy, we introduce a scale factor that reduces the contrast of elements in the mass matrix. This adjustment ensures better numerical stability and fidelity in our simulations. The numerical examples demonstrate the accuracy and efficiency of the proposed SNMM method.

In this work, a hybrid forward-inverse neural network (HFINN) with a transceiver transformation module (TTM) is proposed to increase the generalizability of machine learning-based electromagnetic (EM) inversion methods. The HFINN consists of two parts: a forward module and an inverse module. The inverse module combines the Res-Net with a fully convolutional network (FCN), which is called “Res-FCN,” to show the performance in wideband inversion; however, the data misfit from Res-FCN often remains high because it only minimizes the model misfit; thus, a forward module based on a classical neural network, U-Net, is trained first to alleviate the problem of large data misfit. To train HFINN better, a new loss function is proposed so that the frequency information is used as a prior physical constraint to optimize HFINN. Meanwhile, to further improve the generalizability of HFINN, TTM is incorporated into the HFINN as physical assistance so that it does not need to be retrained for different transceiver configurations. A total of 4000 random test samples are employed to verify the performance of the proposed HFINN, and the average model misfit is 27.15%. Six numerical examples are also provided to verify the inversion performance of HFINN over the whole frequency band. After adding the forward module, the average data misfit of the result is reduced by 7.5%. The numerical results show that HFINN performs well across the whole frequency band, even when the testing transceiver configuration is different from the training transceiver configuration.

In this study, an efficient deep-learning-driven sparse-target imaging (DLSTI) method was developed for array borehole radar to improve the accuracy of target localization in a subsurface nonuniform media. First, by making use of the linear superposition and separable characteristics of the target and background echo, the background echo was generated with an electromagnetic (EM) simulation using prior medium information. The background echo was then removed from the radar receiver echo using a convex-optimization-based front-end target echo extractor (TEE) to obtain the raw sparse target echo. Subsequently, the raw target echoes and true target locations in the simulation dataset were utilized for the training of a back-end stacked autoencoder (SAE) in a data-driven manner, which is capable of illustrating accurate target locations in field tests in nonuniform environments after training. The comparison results in multiple simulations and field scenes show that the proposed DLSTI outperforms other effective imaging methods in terms of target localization accuracy and image sidelobes (SLs), including reverse time migration and back-projection (BP), whose localization error was improved to 0.02 m and the image SL was reduced by 16.09 dB.

In elastic wave inverse scattering problems, the material-property reconstruction, e.g. distributions of mass density, compressional wave speed, and shear wave speed from a limited set of measurement data, has attracted considerable research interest. However, simultaneous inversion of multiple parameters endures high computation costs, and reconstructed compressional and shear speeds may become unrealistic if no physical constraint is imposed. Meanwhile, because large objects with high contrasts over the background medium tend to induce high nonlinearity during the inversion process, it is difficult to obtain high-quality high-contrast material properties. To overcome such difficulties, we have developed a multi-module deep learning scheme with physical constraint for multi-parameter elastic wave inversion of high-contrast objects in inhomogeneous media. This scheme consists of (1) a preliminary imaging module (PIM), in which a deep residual network (ResNet) is employed to convert the scattered field data into the preliminary inversion images, (2) an image-enhancement module (IEM), in which a U-Net is employed to further enhance the image quality, and (3) a convolutional neural network (CNN) that is employed as the physical constraint module (PCM) to allow elastic wave parameters to satisfy actual physical constraint. Numerical examples have demonstrated that the proposed scheme not only accurately achieves multi-parameter elastic wave inversion, but also has good generalizability. Finally, the scheme can be applied to complex objects with high contrasts in both noise-free and noisy environments.

Simulations of 3D electromagnetic (EM) scattering from complicated anomalies in a layered medium are important in geophysical applications. The volume integral equation (VIE) method, specifically its fast solvers such as FFT-based methods, has been widely applied to tackle these problems. Recently, we proposed to combine the domain decomposition method (DDM) with VIE to solve the problems with multiple objects in a layered medium (VIE-DDM). This method leveraged 3D and 2D FFTs to accelerate the self- and mutual-coupling matrices-vector multiplications (MVMs) in the stabilized-biconjugate gradient (BCGS) method when solving the linear system derived from the VIE-DDM. Though significantly extending the applicability and improving the efficiency of VIE in some scenarios, the BCGS-FFT-DDM (or VIE-DDM-FFT) has the limitations of requiring conformal and uniform meshes in thexy-plane and not allowing dense meshes in thezdirection. It turned out both limitations come from the mutual-coupling matrices. In this paper we propose to exploit the rank-deficient property of the mutual-coupling matrices relating well-separated objects, apply the adaptive cross approximation (ACA) method to find the low-rank factors of these matrices, and then achieve the MVMs using the low-rank factors. The ACA is purely algebraic and only requires a few rows and columns from the original matrix, which can be easily obtained by computing the Green’s functions, to find the low-rank factor matrices. With this new method, both the storage and computation complexities become linear. These linear complexities successfully avoid the bottleneck of densezdirection meshes in the BCGS-FFT-DDM. Furthermore, the new method does not strictly require conformal and uniform meshes in any direction for different objects, making it more flexible in treating multiscale problems. We tested the performance of this method in two typical scenarios and compared the results with the BCGS-FFT-DDM and commercial platforms. The results show that the new method achieves significant improvement in memory and CPU time cost and has more flexibility in treating multiscale problems.

For designing or calibrating a triaxial induction well-logging tool, efficient simulations of moving the tool in an arbitrary borehole in an invaded layered medium are essential. The computational time of conventional methods is directly proportional to the number of tool positions. In this work, a domain decomposition method (DDM) that utilizes the layered medium Green’s functions (LMGFs) and loop-tree (LT) basis functions is proposed; it separates stationary subdomains from the variable subdomain where the tool moves. Efforts are saved in calculating the matrix related to the stationary subdomains. A special stationary subdomain is the boundary surface of the finite-element volume. The combination of the finite-element method (FEM) and the layered medium surface integral equation (SIE) method minimizes the number of unknowns. LT basis functions are used to discretize the equivalent current densities on the boundary to improve numerical accuracy at low frequencies. The accuracy and efficiency of this new method are validated in well-logging simulations with a prototype induction tool including the solenoids, mandrels, and sleeves in a deviated borehole and layered media.

This article first presents machine-learning (ML) inversion of resistivity profiles from multifrequency electromagnetic measurements on undulating terrain surfaces based on synthetic data training by the mixed spectral element method (MSEM). The inversion method combines several advanced technologies with various merits. A semiregular mesh generation method is designed and developed for adaption to complex undulating terrain and multifrequency measured data, and the proposed meshing technology is also suitable for modeling different training models under the same undulating terrain. By simulating the application scenarios of measurements, the apparent resistivity data at eight frequencies from 1 to 2048 Hz are simulated with the 2.5-D MSEM to ensure the accuracy and efficiency of the simulation of undulating terrains. Fast simulation of stochastic models for training datasets is achieved by twisting and extruding the initial model obtained by Bostick inversion. Since the unknown weight matrices are solved only once in the training process, the extreme learning machine (ELM) is used for ML inversion to reduce the training cost and obtain high-precision inversion results. Then it is applied to reconstruct a metallogenic model to verify the method’s validity and accuracy and to reconstruct the resistivity profile of underground ore bodies with actual measurements. The results show that the proposed method can be effectively used to detect metal ores at a depth of less than 3000 m underground.

In this letter, a new full-wave inversion (FWI) scheme is proposed to reconstruct multiple dielectric parameters of 3-D arbitrary anisotropic objects buried in layered media. Three inverse solvers, including the isotropic one, biaxial anisotropic one, and the arbitrary anisotropic one, are cascaded sequentially. The dielectric parameters obtained by the first solver are used as the initial values of the next solver. Meanwhile, the inversion domain is synchronously downsized on the basis of discrepancies between the inverted dielectric parameters and the background ones. Numerical simulations show that, compared with the direct arbitrary anisotropic inverse solver, the cascading inversion scheme not only can produce more reliable reconstructed profiles but also significantly lowers the computational cost. In addition, the antinoise ability of the cascaded solvers is also tested.

In this letter, the application of the supervised descent method (SDM) for solving controlled-source electromagnetic inversion is studied. The descent direction in each iteration step of the 1-D full-wave inversion (FWI) is learned from the training data set with certain prior information in the off-line training and then saved. In the online prediction, it is directly combined with the measured data and the forward model to implement the FWI. Compared with the traditional iterative method, the efficiency is significantly enhanced since the computation of the Jacobian matrix is circumvented. Both the synthesized and field-measured grounded electrical-source airborne transient electromagnetic (GREATEM) data are used to verify the feasibility and efficiency of SDM. In addition, the learning ability of the SDM is also studied.

An efficient two-dimensional (2-D) electromagnetic (EM) full-wave inversion (FWI) method based on the hybrid spectral-element spectral-integral (SESI) forward solver is proposed. In the forward scattering computation, the scalar Helmholtz equation is discretized and solved by the spectral element method (SEM). Its computational domain is truncated by a circle on which a 2-D surface integral equation is performed and the spectral-integral method (SIM) is used to accelerate its solution. Transmitters and receivers are placed outside the circular boundary and they are connected to the equivalent current on the circle by Green’s functions. In each iteration of FWI, the sensitivity matrix is updated by the electric field values solved by SESI only in the internal nodes inside the computation domain. Then the conjugate gradient (CG) method is used to solve for the relative permittivity and conductivity values in the corresponding interior elements. Meanwhile, the structural consistency constraint (SCC) algorithm is adopted to gradually compress the inversion domain and guarantee the inversion accuracy. Several numerical experiments are carried out not only to show the computation efficiency of the SESI forward solver but also to verify the correctness of the iterative inversion procedure as well as the effectiveness of SCC.

In electromagnetic (EM) well logging for petroleum exploration, off-centered tools within a noncircular borehole and invasion zones in multilayered media represent a challenging 3-D problem for traditional numerical methods. The 3-D finite-element numerical mode-matching (FNMM) method and the spectral numerical mode-matching (SNMM) method are developed to address this problem in this work. The numerical mode-matching (NMM) methods reduce the original 3-D well logging problem into a series of 2-D open waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically solved by a recursion procedure. These waveguide eigenvalue problems with anisotropic inhomogeneous media in the horizontal directions are solved numerically by the mixed finite-element method (MFEM) with the flexibility for the complex geometry and the mixed spectral element method (MSEM) with the exponential convergence. A general operator form of Maxwell’s equations is also derived to obtain the source excitation vector independent of the$z$-axis. Therefore, the NMM solvers can effectively and accurately simulate the response of off-centered EM well logging tools in an irregular borehole in complex formations. The NMM methods are applied to simulate three formations including a five-layer isotropic homogeneous formation, an off-centered logging tool in a noncircular borehole with anisotropic invasion zones in a five-layer anisotropic formation, and the Oklahoma formation. Numerical experiments indicate that the NMM methods are highly efficient and accurate for the EM well logging models.

The numerical mode matching (NMM) method is widely employed in the simulation of electromagnetic (EM) problems with cylindrical geometries. Its significant reduction of the computational cost is achieved by transforming the original 2.5-D problem to a 1-D eigenvalue problem in the radial direction, which is usually treated with the finite element method (FEM), and a semianalytical problem in the$z$-direction, which can be easily tackled with the mode-matching strategy. However, when an extremely thin vertical layer is encountered, such as the steel casing in well logging, the conventional NMM method will be challenged. The extremely thin layer will cause the mesh for the FEM to be tremendously dense, thus greatly lowering its efficiency. Moreover, if the thin layer is extremely conductive, the matrices for the eigenvalue problem are highly ill-conditioned, making the solution of the eigenvalue/eigenvector inaccurate. To overcome these problems, we propose to apply the surface current boundary condition (SCBC) to substitute the extremely thin vertical layer in the NMM method. To employ the NMM-SCBC method, the mixed-order FEM is first implemented to treat the 1-D eigenvalue problem, and the capability of the corresponding mixed-order NMM method is shown in this article. Furthermore, the expression of the SCBC is derived and applied to the mixed-order NMM method. Several numerical examples show the accuracy and efficiency of the mixed-order NMM-SCBC method.

In this article, an efficient radar fusion-before-imaging (RFBI) method for heterogeneous borehole radar systems is proposed. Different from conventional fusion-after-imaging processes, the sample sets collected from heterogeneous borehole radar systems (monostatic, bistatic, or multiple-input multiple-output) are first merged into one data set before the imaging process in RFBI, and a single imaging operation is demanded to obtain the target space image with high precision. Specifically, the diversity in heterogeneous borehole radar sample sets is taken into consideration, and the radar sample sets are inserted and fused into a high-dimensional sample set before imaging. The target space spectrum is generated according to the echo space–frequency constraint relationship, and the target space is extracted from one imaging process. The influence of clutters in RFBI results is reduced, and the imaging accuracy is satisfactory. Meanwhile, due to the fusion process ahead, the computational time of RFBI hardly increases with the number of radar sample sets. The synthetic and field experiment results show that RFBI demonstrates comparable accuracy as Kirchhoff migration and higher efficiency in processing large amounts of data sets at the cost of large memory requirement, which is suitable for the joint imaging of heterogeneous radar systems.

Electromagnetic simulation plays an essential role in formation conductivity determination by electromagnetic well-logging tools, especially when the nonvertical borehole and invasion zones are present in a layered earth. A hybrid finite-element boundary integral (FEBI) method is suitable for such well-logging simulations. The usage of layered medium Green’s functions allows the simulation background to be a planar stratified medium to characterize the earth formation. However, conventional FEBI using the Rao–Wilton–Glisson (RWG) basis function produces inaccurate results due to the low-frequency breakdown of the boundary integral equation solution. The new contribution of this work is to apply the loop-tree (LT) basis functions in the FEBI method to reduce the numerical error at low frequencies. Such an LT-FEBI method can handle all specific requirements of well-logging simulations at low frequencies. A direct solver is applied to make it more efficient to obtain logging curves with multiple source locations. The LT-FEBI simulation results are validated in several numerical examples, including a challenging well-logging model with a deviated borehole and invasion zones in the Oklahoma formation having 28 layers.

A hybrid method is proposed to reconstruct the subsurface 3-D objects with electromagnetic fields. The frequency-domain reverse time migration (FRTM) is first used to determine the approximate locations and sizes of the objects. Then, based on these results, the full-wave inversion, the variational Born iteration method (VBIM) is used to reconstruct both the shapes and dielectric parameters of the objects. The Monte Carlo method (MCM) is adopted to further refine the reconstructed shapes. Numerical simulations show that the proposed hybrid method can be effectively used for the subsurface imaging and detection.

This article presents an efficient analytic algorithm to calculate the electromagnetic (EM) fields radiated from orthogonal magnetic and electric dipoles in the fully anisotropic planar-stratified media. The EM fields are transformed into the frequency–wavenumber domain by the double Fourier transform, which yields a set of ordinary differential equations, consisting of a system matrix, a field vector, and a source vector. The eigenvalues of the system matrix are uniquely determined, but the corresponding eigenvectors can be multiplied by an arbitrary constant or be normalized by energy. With these eigenvalues and eigenvectors, the field vectors are expressed as mode fields consisting of the up-going and down-going fields. The propagation of the mode fields in each layer is described by the generalized reflection/transmission method. To accelerate the evaluation of the inverse Fourier transform of EM field from spectral domain to spatial domain, an approximated primary field was subtracted from total field. Numerical results from a finite element method (FEM) validate the proposed method. Then, we apply this new algorithm to a variety of real applications in geophysical prospecting.

We propose a hybrid method based on the finite-difference time-domain (FDTD) algorithm and analytical methods to solve the elastic scattering of 3-D objects outside a borehole. In the proposed method, the FDTD algorithm containing the incident waves generated by a borehole acoustic source is first used to calculate the wave fields inside a rectangular volume enclosing the object. Then, the reciprocity theorem for the wave fields in the fluid/solid configurations is employed to transform the near fields around the object into the far fields inside the borehole. This method uses the regional domain which tightly encloses the object, thus avoiding the simulation for the whole domain and leading to high efficiency for the large-scale and multiscale problems containing both the borehole and far-field objects. The validations with both analytical solutions and independent numerical methods demonstrate the accuracy and efficiency of the hybrid method.

We present a 2.5-D finite difference (FD) algorithm to simulate the logging-while-drilling (LWD) electromagnetic (EM) measurements with arbitrarily oriented magnetic dipole sources in 2-D anisotropic media. The real 3-D modeling in the spatial domain is converted to a series of independent 2-D equations in the spectral domain by implementing the Fourier transform (FT). By replacing the dipoles with incident fields, the scattered fields are solved directly to avoid the point source singularity as well as improve the convergence of inverse FT (IFT). The efficiency of the IFT is further enhanced with a modified Gauss-Hermite (GHM) quadrature-based numerical integration scheme. Numerical tests demonstrate that the proposed 2.5-D algorithm is fast and accurate in modeling the LWD EM measurements in complex scenarios. Discussions are also made to compare the performance of the proposed methods with 1-D and 3-D algorithms.

This letter presents a 3-D electromagnetic inversion method based on the Born approximation (BA) and a convolutional neural network (CNN), the 3-D U-Net. In the training stage, the BA is first used to obtain the preliminary 3-D images of a series of homogeneous scatterers with regular shapes that are further improved by the Monte Carlo method. Then, these images are used to train the 3-D U-Net. In the testing stage, inhomogeneous scatterers with complex shapes are reconstructed by both the trained 3-D U-Net and the traditional iterative method, variational Born iteration method (VBIM). Their performance is evaluated and compared.

This letter presents the joint reconstruction of unknown subsurface structures by the full-wave inversion (FWI) and reverse time migration (RTM) imaging. In the FWI, the 1-D distorted Born iteration method (DBIM) is employed to retrieve the dielectric parameters of the layered subsurface medium by minimizing the difference between measured fields and calculated fields via Fréchet derivatives. Based on the inversion results, the RTM is directly performed in the frequency domain to image the buried objects using the ground-penetrating radar (GPR) data. Numerical and laboratory experiments show that the proposed joint method can be used to reconstruct the subsurface structures reliably and efficiently.

This letter presents a fast and memory-efficient 3-D electromagnetic solver for logging-while-drilling (LWD) tools based on the flipped total field/scattered field-based discontinuous Galerkin frequency-domain (TF/SF DGFD) method. The new method inherits the Riemann transmission condition (RTC) with surface current sources from the mixed TF/SF DGFD method to couple nonconformal meshes as well as TF/SF solvers; it then extends to the moving-tool scenarios for the LWD application with a solver flipping technology and delivers a single global matrix of dramatically reduced dimension for all source positions, which can be solved directly on a normal-memory computer. By incorporating the tetrahedral/hexahedral mixed mesh and the curved domain decomposition method, the new solver is applied to two LWD cases and about 70% of unknowns can be saved for the flipped TF/SF DGFD remeshed regions.

An analytic algorithm is developed to solve the electromagnetic (EM) field excited by orthogonal magnetic dipoles in the multilayered biaxial anisotropic medium. It allows forward modeling and inversion of EM well logging in the planar-stratified formation without borehole and invasion zones. First, the transverse components of the EM fields are divided into up/down-going waves in the frequency wavenumber domain. A novel concept of reflection/transmission is introduced to describe the propagation of the EM waves through the boundaries. Different from previous work, those reflection/transmission coefficients are concerning the propagation of up/down-going waves directly rather than the amplitude of the total EM field in each layer, which makes the derivation process clearer and easier to understand. Then, we adopt the cubic spline interpolation to calculate the 2-D Fourier integral that transforms the field from the spectral domain to the spatial domain, and an efficient sampling rule is also provided to achieve high accuracy and stability in numerical calculation. For avoiding the singularity when the field locations are close to the source along the vertical direction, the primary field is subtracted in the spectral domain to make the integrand decay fast. Finally, we validate our new formulas compared with another numerical method and analyze the responses of the triaxial induction tool in biaxial anisotropic formations.

Helicopter transient airborne electromagnetics (HTEM) has become a useful tool in mineral explorations and geological or environmental detection in recent decades. This article presents the frequency-domain 3-D full-wave inversion of the electromagnetic data recorded by a newly built HTEM system. In the forward process, the secondary magnetic field is calculated through the volume electric-field integral equation (EFIE). In the inversion process, the secondary field is first extracted from the total field measured by the HTEM system. Then the Born iterative method (BIM) is adopted to solve the nonlinear inverse scattering problem for the 3-D reconstruction of conductivity. It is first applied to the synthetic models to verify their effectiveness and accuracy. The effects of adjacent underground anomalies on the 3-D inversion performed in a local region within the long flight lines are studied and discussed. Then the BIM solver is used to invert for the underground anomalies using the field measured data recorded by the newly built HTEM system. The 2-D slices from the reconstruction are compared with the nomogram. It is found that the locations of the high-conductivity regions in the BIM results are consistent with locations of the peaks in the nomogram. The reconstructed profiles are also compared with the drilling data obtained near one flight line. The good agreement shows that the 3-D BIM inversion algorithm can be used to reconstruct the underground ore in HTEM surveys.

An hp- and memory-adaptive discontinuous Galerkin time-domain algorithm is presented to efficiently model wave propagation in poroelastic media, with the incorporation of 3-D fully anisotropic intrinsic attenuation. From the perspective of physics, the attenuation from the triclinic rock frame, the loss in the pore fluid, and the friction for their interaction are all considered. From the perspective of implementation, a new frequency-domain constitutive equation is introduced, involving complex-valued poroelasticity matrix. A new Q value is defined as the ratio between its real and imaginary parts for every entry. Then, a generalized Maxwell body is adopted to approximate this frequency-dependent Q in the time domain. Mathematically speaking, the hyperbolicity is preserved for the new viscous poroelastic system. Our results corroborate that the intrinsic attenuation anisotropy makes tangible effects in fluid-saturated porous formations.

Imaging hydraulic fractures is of paramount importance to subsurface resource extraction, geologic storage, and hazardous waste disposal. The use of electrically conductive proppants and current energized steel casing provides a promising approach to monitor the distribution of fractures. In this article, a borehole-to-surface system is employed to energize the steel casing and measure electric and magnetic fields on the ground. A convolutional neural network (CNN) is then trained to learn the relationship between the measured field pattern and the parameterized fracture, namely, the lateral extent and direction. To accelerate the generation of training data with limited accuracy loss, an approximate hollow casing is modeled by the impedance transition boundary condition with a tenfold magnified radius and reduced conductivity. Two training strategies are then presented with a grid search of the network's hyperparameters. The well-trained CNN shows good generalization to unseen fracture conductivity, the true casing model, as well as white Gaussian noise. Finally, we apply the CNN to image irregular fractures and obtain reliable results even under strong noise, indicating a promising imaging technique for more complicated fractures.

Because of strong nonlinearity, it is always a challenge to acquire fine resolution reconstruction results for electric strong scatterers with high contrasts using electromagnetic (EM) data. Many electric strong scatterers have weak nonlinearity in acoustic inversion. To make full use of the advantages of acoustic inversion, two joint inversion methods based on the structural constraints to reconstruct electric strong scatterers using EM and acoustic data are proposed in this article. These two methods utilize the framework of the subspace-based optimization method (SOM). In the inversion process, one of the methods utilizes a cross-gradient function to link the EM and the acoustic inversions, enforcing the structural similarity between the permittivity and the sound velocity. The second method is to utilize the reconstruction result of acoustic data as initial structural information for the EM inversion process. Compared with conventional separate SOM, these two methods achieve finer EM reconstruction for electric strong scatterers. Both structure and contrast values are well reconstructed. The efficiency and the robustness of the methods are validated through numerical experiments.

This letter extends the discontinuous Galerkin frequency domain (DGFD)-based domain decomposition method (DDM) to model the open borehole environment accurately with a newly developed hierarchical sudoku mesh. Both the borehole and mud invasion effects can be modeled by this framework efficiently. Furthermore, deep reading measurements can be modeled conveniently by inserting arbitrarily shaped objects into the hierarchical sudoku mesh; the capability to distinguish a cavity saturated with either oil or water with different borehole-object distances are then studied for a deep reading tool in an open borehole carbonate environment. The DGFD with the hierarchical sudoku mesh shows six times faster speed than the traditional finite-element method (FEM) for the deep reading case.

As one of the most important nondestructive characterization techniques, electromagnetic (EM) methods can be used in the subsurface fracture detection, especially for hydraulic fracture evaluation in unconventional petroleum exploration and development. The multiscale nature of long but extremely thin 3-D fractures is difficult for conventional EM modeling methods such as the finite element method (FEM) in numerical simulation. The problem becomes even more challenging when the effects of tilted borehole, casing, and planar stratified media need to be considered. So far, modeling a tilted borehole in layered media is still a major challenge for conventional methods. In this paper, we present the hybrid numerical mode-matching method with the stabilized biconjugate gradient fast Fourier transform method as the forward modeling algorithm that can efficiently model 3-D fractures in planar stratified media with a cased borehole environment. Numerical results validate the accuracy of the hybrid forward method and show orders of magnitude higher efficiency of this forward solver than the FEM.

Electromagnetic simulation of fractures has a growing importance in geophysical exploration, where the formation is usually inhomogeneous and modeled as a layered medium (LM). In this paper, we have developed the first integral-equation based solver to simulate the scattering of fractures straddling an LM. We refer to this solver as LM-thin dielectric sheet (TDS)-surface integral equation (SIE), since it is built on the TDS-based SIE and the LM Green's function (LMGF). Compared with the traditional finite element method (FEM) and volume integral equation (VIE), LM-TDS-SIE achieves excellent efficiency by taking advantage of the multiscale feature of fracture, rather than being restricted by its volumetric mesh. Good accuracy but the lower computational cost of LM-TDS-SIE is well demonstrated by investigating several typical examples, where reference results are obtained by the commercial software COMSOL. In addition to fractures, LM-TDS-SIE can also be used to simulate the scattering from other types of TDS.

A joint petrophysical and structural inversion method for electromagnetic (EM) and seismic data based on the volume integral equation (VIE) is proposed in this paper. In the forward EM problem, only the contrast of conductivity is solved by the electric field integral equation method. However, in the forward seismic problem, both the contrasts of velocity and mass density are solved by the combined field VIE method. Both forward solvers are accelerated by the fast Fourier transform. In the inversion problem, by using the petrophysical equations about the porosity and saturation and applying the chain rule, we fuse the EM and seismic data and construct the joint petrophysical inversion equations, which can be solved by the variational Born iteration method. Then, in order to further enhance the reconstructed results of the joint petrophysical inversion, we enforce the structural similarity constraint between porosity and water saturation and add the cross-gradient function to the joint petrophysical inversion cost function. Two typical geophysical models based on the remote sensing measurement are used to validate the proposed methods. One is the cross-well model, and the other is the marine surface exploration model. The advantage of the joint inversion compared with the separate inversion is evaluated based on the resolution and the data misfits of the reconstructed profiles as well as the antinoise ability.

This paper presents an efficient multiple-input multiple-output (MIMO) borehole radar imaging method based on a high degree of freedom for subsurface sensing. The variable separations between the different transmitter and receiver pairs in MIMO borehole radar are considered and introduced as imaging coefficients into the expanded unified MIMO sample set, which gives the MIMO sample set desired characteristics of a high degree of freedom. By exploiting the high degree of freedom in the unified MIMO expanded sample set, the sample interpolation and energy migration of target reflections can be done in once efficient imaging processing for all transmitters of the MIMO radar system, and finally, accurate target image with low sidelobe level (SL) can be delivered. The computational cost of the proposed method will barely increase with the number of transmitters in the MIMO radar survey, which enables the proposed method to handle MIMO borehole radar imaging in an accurate and efficient manner. The imaging properties of the proposed method are proven to be superior to the conventional imaging methods in SL and computational cost, which is suitable for large MIMO borehole imaging surveys in subsurface sensing applications.

A newly designed grounded electrical-source airborne transient electromagnetics (GREATEM) system was introduced recently. Detailed data preprocessing techniques to acquire the high-precision measured magnetic field are discussed here. Different from the previous work in which the reconstruction of the underground structure is performed in 1-D, we interpret the GREATEM data in 3-D by the volume integral equation (VIE) method in the frequency domain. Therefore, the VIE in the forward electromagnetic scattering model is formulated in the low-frequency regime. It is solved by using the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method. In the nonlinear inversion, the Born iterative method (BIM) and the conjugate gradient method are adopted to minimize the cost function. A synthetic model of GREATEM survey is used to validate the proposed 3-D forward and inversion algorithms. Then, the field data from two GREATEM surveys are used to test the effectiveness and accuracy of the proposed inversion algorithm. The reconstructed conductivity structures are consistent with geological drilling results, confirming the potential of our method for solving the 3-D GREATEM inversion problems in geophysical engineering applications. This paper represents the first application of the BCGS-FFT and BIM algorithms to a GREATEM system.

To model the responses of electromagnetic surveys for geophysical subsurface sensing, a mixed total field/scattered field-based discontinuous Galerkin frequency-domain (TF/SF DGFD) method is proposed in this paper. The proposed TF/SF DGFD method is implemented at a subdomain level based on the domain decomposition technique. Different subdomains can employ either the TF DGFD framework or the SF DGFD framework, which are then coupled through the Riemann transmission condition. To balance the computation efficiency and accuracy for practical applications, the proposed method prefers to using the SF DGFD framework for subdomains with sources and using the TF DGFD framework for the remaining subdomains. At the interfaces between total field and scattered field subdomains, the Riemann transmission condition is slightly modified by incorporating the background fields due to the physically imposed sources in the background media. In this way, the proposed method only requires surface integrals of the background fields as extra overhead instead of elementwise integration of the scattering objects for the purely scattered field-based method, which can improve the computational efficiency. Also, it is more accurate than the purely TF DGFD method given the same mesh. Numerical examples are studied to examine the performance of the proposed method, which is proven to have better accuracy than the TF DGFD method. The TF/SF DGFD method will facilitate modeling of electromagnetic surveys under complicated geophysical environments for subsurface sensing.

In this paper, full mechanisms of dissipation and dispersion in poro-viscoelastic media are accurately simulated in time domain. Specifically, four Q values are first proposed to depict a poro-viscoelastic medium: two for the attenuation of the bulk and shear moduli in the solid skeleton, one for the bulk modulus in the pore fluid, and the other one for the solid-fluid coupling. By introducing several sets of auxiliary ordinary differential equations, the Q factors are efficiently incorporated in a high-order discontinuous Galerkin algorithm. Consequently, in the mathematical sense, the Riemann problem is exactly solved, with the same form as the inviscid poroelastic material counterpart; in the practical sense, our algorithm requires nearly negligible extra time cost, while keeping the governing equations almost unchanged. Parenthetically, an arbitrarily nonconformal-mesh technique, in terms of both h- and p-adaptivity, is implemented to realize the domain decomposition for a flexible algorithm. Furthermore, our algorithm is verified with an analytical solution for the half-space modeling. A validation with an independent numerical solver, and an application to a large-scale realistic complex topography modeling demonstrate the accuracy, efficiency, flexibility, and capability in realistic subsurface sensing.

Porous materials are ubiquitous in the subsurface formations of the earth where acoustic and seismic waves are used for remote sensing. However, it is not well understood how the dissipation and the dispersion of poroelastic waves are caused by the viscoelastic and viscous properties of the constituents such as solid grains and pore fluid and by the viscoelastic dissipation of the solid frame, as well as the viscodynamic coupling of the pore fluid to the solid frame due to its global and local flows relative to the solid grains. Such attenuation mechanisms have seldom been incorporated in subsurface sensing simulations, although they can be very important to applications. In this paper, we propose a complete attenuation model, including both full stiffness and viscodynamic dissipation, for poroelastic media in seismic wave simulations. Completely based on a generalized Zener model, the effects associated with physical dissipation and frequency-dependent dispersion are accurately simulated by a finite-difference time-domain algorithm. Verifications with analytical solutions show the accuracy, efficiency, and flexibility of our method. Numerical results demonstrate that the attenuation of Biot's model in the sediment of the seafloor has significant effects on acoustic wave scattering from complex geologic structures.

Ground penetrating radar (GPR) has been widely applied to detection of subsurface linear targets, such as underground pipes and reinforced rebars in concrete structures. The azimuth direction of a subsurface linear target can be hardly delineated by a commercial single-polarization GPR system. In this paper, a hybrid dual-polarimetric GPR system is employed to detect buried linear objects. A full-polarimetric scattering matrix is extracted from the double-channel GPR reflection signal. A rotation transformation is applied to the scattering matrix to estimate the target azimuth angle. A laboratory experiment was conducted to detect four metal rebars buried in dry sand at different azimuth angle relative to the GPR scan direction. The maximum error of the estimated azimuth direction is less than 15%. It is concluded that radar polarimetry can provide richer information than single-polarization GPR.

This letter extends the axisymmetric version of the efficient forward and inverse algorithms to characterize and reconstruct arbitrary 3-D fractures in a cased borehole environment. We improve our previous hybrid distorted Born approximation and stabilized biconjugate gradient fast Fourier transform method as the 3-D forward modeling algorithm that can significantly reduce the computational cost in forward modeling. The bounding constraints and model parameter transformation functions are introduced to our previous axisymmetric variational Born iterative inverse method to effectively reconstruct 3-D fractures. Numerical results show orders of magnitude higher efficiency of the forward algorithm than the finite-element method, and the effectiveness of the inverse algorithm for through-casing 3-D fracture reconstruction.

In this letter, we present a set of compact and no-overflow formulations to calculate the electromagnetic (EM) fields from coaxial coil antennas in a concentric cylindrically multilayered medium. It can be applied to fast forward computation and the inverse problem for EM well logging. The derivation is performed by using the tangential component of electric field. In contrast with previous formulations, the adopted novel reflection and transmission coefficients are scalars rather than matrices, which make it easy to get an accurate and efficient Jacobian matrix for inversion problem. The basic unit of the formulations is the ratio of the cylindrical functions so that the notorious overflow problem for numerical computation can be obviated. Numerical results in comparison with those from other approaches have demonstrated the validity and stability of our new formulations for forward modeling. To show the potential of the proposed formulations, an inverse result from a simple formation model is also presented.

The hardware instability of a ground penetrating radar (GPR) system has a severe impact on the quantitative analysis of GPR data, which is aimed for material characterization and subsurface monitoring. In this letter, an instability index is proposed to quantify the stability performance of a GPR system and the influences of the GPR system type, warm-up time, environmental noise, and the antenna vibration on it are evaluated through a series of laboratory experiments on a sandbox model. It is found that the GPR signal recorded by a stepped-frequency GPR system based on a vector network analyzer is much more stable than that by a commercial impulse GPR system at a cost of more sweep time. A warm-up time of several minutes is enough for an impulse GPR system. Environmental noise has a negligible influence on the stability performance of a GPR system. Mechanical vibrations of GPR antennas have a severe impact on the stability performance of the GPR system, and the instability index and timing jitter can be increased by more than one order of magnitude in a vibrating condition over those in a static condition. The instability index of the direct signal has a negligible difference with that of the reflection signal from a metal plate; thus, a simple measurement of direct signal on the ground surface is suggested for the evaluation of the instability of a GPR system in field in the future.

This letter investigates the inversion of rough surface parameters (the root mean square height and the correlation length) from microwave images by using deep convolutional neural networks (CNNs). Training data for the deep CNN are simulated numerically using computational electromagnetic method. As CNN is powerful in extracting image features, scattering field from rough surfaces is first converted to microwave images via interpolated fast Fourier transform and then fed into the CNN. In order to reduce overfitting, the regularization technique and dropout layer are used. The proposed CNN consists of five pairs of convolutional and maxpooling layers and two additional convolution layers for feature extraction and two fully connected layers for parameter regression. The experimental results demonstrated the feasibility using deep neural networks for the parameter inversion of rough surface from electromagnetic scattering fields. It suggests potential application of CNN for rough surface parameter inversion from microwave sensing data.

In this paper, the sensitivity and detectability of an electromagnetic induction tool for hydraulic fracture detection are analyzed by calculating the electromagnetic response from an orthogonal transverse fracture (OTF) in an open or cased borehole using the improved numerical mode matching method. The OTF is modeled as a slim circular disk, which is axially symmetric with respect to the borehole axis and filled with a conductive or magnetic proppant. The feasibility of the induction tool for fracture diagnostics was validated via scaled-down experimental measurements. We then quantitatively analyze the induction response sensitivity to different proppant parameters, such as conductivity and permeability, to evaluate the effectiveness of the induction logging tool for fracture detection. The analysis reveals that the short-spaced receiver can accurately locate the fracture position and distinguish small-sized fractures, while the long-spaced receiver can differentiate fractures with large dimension. Casings with different thicknesses, conductivities, and permeabilities are modeled together with the fracture to investigate their effects in fracture detection. Finally, the application of the induction tool to a fracture network is evaluated.

In this paper, we present a set of compact formulations to model the response of the electromagnetic well-logging tools in eccentric multilayered medium. Different from previous models, the coil antennas are imbedded in an insulating protection layer (IPL) rather than expose to the borehole directly, which agrees much better with some real tools, e.g., the array induction logging tool. The effects of both metal mandrel and an IPL are considered in eccentric scenarios. Our formulations are derived on the basis of the ratios of cylindrical functions, which obviate the notorious overflow issues during numerical computation. Numerical results show that the effect of IPL gradually appears with the increase of the eccentric distance.

Hydraulic fracturing is a technique to fracture rocks by pumping high-pressure fluid into a segment of a well. The created fractures help to release a hydrocarbon resource such as oil or natural gas from the rock. A group of small-scaled fracturing field tests are performed by the Advanced Energy Consortium to investigate the feasibility of using the galvanic electromagnetic (EM) method to map fractures. The injected proppants are designed with high EM contrasts (e.g., conductivity and permittivity) to generate detectable signals at electrode-type sensors. To map the created fractures, an efficient 3-D EM inversion method is introduced to simultaneously reconstruct conductivity and permittivity profiles in fractures. First, to test the capability of the inversion solver and the designed experimental setting for successful fracture mapping, the noise-polluted synthetic data are used to reconstruct the fracture on a theoretical model. It shows that the designed experimental setting can be used to map the fracture and the inversion solver is able to reconstruct the fracture in both conductivity and permittivity. The inversion method is then applied to two hydraulic fracturing field tests with injected high-contrast proppants, Loresco coke breeze and steel shot. The fracture conductivity and permittivity are reconstructed based on the voltage signals difference between the postfracturing and prefracturing data. The reconstructed fracture profiles are compared with the coring samplings to show the reliability of the inversion results. Their good agreement demonstrates that the experimental setting and the galvanic inverse solver are able to estimate the fracture size and location reliably.

This paper presents the formulations for calculating the electromagnetic (EM) fields in multilayered generalized anisotropic media. Maxwell’s equations 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 have both full-tensor magnetic and dielectric anisotropy. The singular behavior of the solution in the close vicinity of the dipole source is subtracted to make the integrands decay rapidly as functions of$k_{x}$and$k_{y}$. The contributions of the subtracted part are calculated analytically. A three-layer anisotropic medium is modeled to show the convergence of the integrals with the singularity subtraction. To validate the algorithm for multilayered generalized anisotropic media, a five-layer medium is modeled and compared with finite element method results. The algorithm is also applied in geophysical EM well logging by modeling the triaxial induction logging tool. The responses in vertical and deviated wells are computed and compared with finite element results. The good agreement between the two results further validates the algorithm and demonstrates its capability to model induction logging tools in multilayered generalized anisotropic media.

An efficient 2-D joint full-waveform inversion method for electromagnetic and seismic data in a layered medium background is developed. The joint inversion method based on the integral equation (IE) method is first proposed in this paper. In forward computation, the IE method is employed, which usually has smaller discretized computation domain and less cumulative error compared with the finite-difference method. In addition, fast Fourier transform is used to accelerate the convolution between Green's functions and induced sources due to the shift invariance property of the layered Green's functions in the horizontal direction. In the inversion model, the cross-gradient function is incorporated into the cost function of the separate inversion to enforce the structure similarity between electric conductivity and seismic-wave velocity. We use the improved variational Born iteration method and two different iteration strategies to minimize the cost function and reconstruct the contrasts. Several typical models in geophysical applications are used to validate our joint inversion method, and the numerical simulation results show that joint inversion can improve the inversion results when compared with those from the separate inversion.

A new iterative inversion algorithm is proposed to reconstruct the electrical conductivity profile in a stratified underground medium for the grounded electrical source airborne transient electromagnetic (GREATEM) system. In forward modeling, we simplify the mathematical expressions of the magnetic fields generated by a finite line source in the layered ground to semianalytical forms in order to save the computation time. The Fréchet derivative is derived for the electromagnetic response at the receivers due to a small perturbation of the conductivity in a certain layer underground. The initial expression of the Fréchet derivative has an expensive triple integral and contains the Bessel function in the integrand. It is simplified by partially eliminating the integration along the source line and deriving the analytical expression for the integration in the vertical direction inside the perturbed layer. In the inverse solution, we use the distorted Born iterative method (DBIM). This is the first time that the DBIM is applied to data measured by the GREATEM system. Besides, the forward and inverse procedures are carried out in the frequency domain and based on the Fréchet derivative of a line source. We demonstrate the validity of our forward model, Fréchet derivative, inverse model, and the precision as well as robustness of the inversion algorithm through numerical computation and comparisons. Finally, we apply the inversion algorithm to the measured data and compare the retrieved conductivity to the actual drilling data.

This paper presents a novel accurate and stable current and charge integral equation (CCIE) solver for the low-frequency scattering of penetrable objects in layered medium (LM). To the best of our knowledge, this is the first time to extend CCIE into LM simulations at low frequency. In order to integrate the matrix-friendly LM Green’s functions (LMGFs) into CCIE, we have rederived them to define new quasi-vector/scalar potentials, which are able to annihilate the frequency singularity in the original LMGFs. Moreover, an effective preconditioner is adopted to improve the conditioning of impedance matrices. In comparison with other perconditioners, this method performs much better. The excellent performance of this new CCIE solver is then demonstrated by numerical experiments.

A high-order discontinuous Galerkin (DG) method with nonconformal meshes is developed to accurately simulate large-scale poroelastic wave propagation in 3-D isotropic media. An exact upwind flux is succinctly derived to serve as an accurate coupling solver for the DG algorithm. Specifically, the eigenvalue problem in the Riemann solution is effectively reduced from the rank of 13 to 4. Furthermore, this new numerical flux gives more explicit physical insight, which indicates three-type waves in poroelastic media: two P waves and one S wave. Validations and verifications with analytical/semianalytical numerical solutions demonstrate the accuracy, robustness, and flexibility of the proposed solver.

The variational Born iterative method (VBIM) is employed here to reconstruct 3-D objects with permittivity contrast buried in spherically multilayered media. The nonlinear inverse problem is solved iteratively via the conjugate-gradient method, and in each iteration, the scattered field is linearized by using the Born approaximation. The forward solver is provided by the method of moments combined with a Krylov subspace method. The dyadic Green's function for spherically layered media is constructed in terms of the spherical vector wave functions by using the scattering superposition in the spherical coordinate system and then transformed into the Cartesian coordinate system. Thus, the inversion region is discretized into N uniform cubic cells and the reconstructed result can be obtained in the Cartesian coordinate system by employing VBIM. Numerical results with high resolution are presented to validate the capability of our method in reconstructing 3-D multiple objects of arbitrary shapes buried in spherically multilayered media.

In this letter, a diagonal approximation has been introduced in the framework of subspace-based optimization method (SOM), for reducing computational complexity. Due to this approximation, the operator which relates the electric field and equivalent current becomes a diagonal one, instead of the nonlinear one in full-wave inversion. Consequently, the proposed method is named as diagonal SOM (DSOM). Compared with the original SOM, DSOM has a more simplified objective function with much less computational cost. DSOM can be applied for solving inverse scattering problems involving not only isotropic objects, but also uniaxial anisotropic objects, which is demonstrated by numerical examples. Furthermore, DSOM provides reconstruction results that are comparable in quality to the ones obtained using SOM, but with much less computation load.

We apply the forward scattering and inverse scattering algorithms to a cold unmagnetized plasma region within a multilayered background medium. Each layer has a different plasma frequency. The disturbed region in the plasma has an arbitrary shape, so it is an electromagnetic wave scatterer and can exist in any layer. The stabilized biconjugate-gradient fast Fourier transform (BCGS-FFT) algorithm is used to compute the scattered field. The scattered fields calculated by the BCGS-FFT yield excellent agreement with simulated results from the commercial software. In the inverse scattering process, the variational Born iterative method is used to reconstruct the relative permittivity, and thus the plasma frequency of the disturbed region. Multiple frequencies are adopted to determine the dispersive property of the plasma medium.

A set of compact and stable formulations is presented to calculate the response of logging tools employing tilted-coil antennas in anisotropic concentrically cylindrical multilayered formation. The presented formulations can be used to investigate the effect of mandrel, borehole, and invasion. Our formulations are different from previous ones in three aspects. First, the tool mandrel can be cylindrically multilayered, and electrical parameters of each layer are arbitrary. That is to say, the mandrel is not limited to being perfectly metallic. Second, the formation can be either anisotropic or isotropic. Finally, the propagation of electromagnetic wave in an inhomogeneous cylindrical medium is described by the generalized reflection and transmission coefficients without the numerical overflow issue. Numerical results in comparison with those from other approaches have demonstrated the validity and stability of our new formulations.

Rapid simulations of large-scale low-frequency subsurface electromagnetic measurements are still a challenge because of the low-frequency breakdown phenomenon that makes the system matrix extremely poor-conditioned. Hence, significant attention has been paid to accelerate the numerical algorithms for Maxwell's equations in both integral and partial differential forms. In this letter, we develop a novel 2.5-D method to overcome the low-frequency breakdown problem by using the mixed spectral element method with the divergence-free constraint and apply it to solve the marine-controlled-source electromagnetic systems. By imposing the divergence-free constraint, the proposed method considers the law of conservation of charges, unlike the conventional governing equation for these problems. Therefore, at low frequencies, the Gauss law guarantees the stability of the solution, and we can obtain a well-conditioned system matrix even as the frequency approaches zero. Several numerical experiments show that the proposed method is well suited for solving low-frequency electromagnetic problems.

Hydraulic fracturing is an essential way to improve the production of unconventional shale oil and gas. It is important to characterize the produced fractures using either acoustic or electromagnetic (EM) methods. Conventional EM solvers in the low-frequency range face significant challenges by such multiscale problems where the fracture width is orders of magnitude smaller than its diameters. Furthermore, the cased borehole environment is extremely difficult to simulate with conventional EM solvers due to meshing difficulties and the multiscale nature of the problem. In this paper, we develop a hybrid distorted Born approximation and 3-D mixed ordered stabilized biconjugate gradient fast Fourier transform (DBA-BCGS-FFT) method to simulate the very challenging 2-D, 2-D-axisymmetric, and 3-D hydraulic fracture models under both open and cased borehole environments. Numerical examples show that this method has orders of magnitude higher efficiency than the finite element method. The capabilities of the DBA-BCGS-FFT method for the induction tool fracture mapping are demonstrated by comparing with laboratory experimental results and other reference results.

For circular trace scanning synthetic aperture radar (CTSSAR) with a circular track, the conventional hyperbolic equation becomes inadequate to express the range history of a point target accurately, and when it comes to the wide swath observation and imaging, the range variance makes it even harder to focus the target on the edge of the scene. Thus, an expression with high-order terms is needed to approximate the range history and the range variance should also be considered in the imaging algorithm. In this paper, based on the method of series reversion, a fourth-order approximated range model is established for the CTSSAR processing and the 2-D spectrum is derived for the echo signal in CTSSAR with circular trajectory. At the same time, in order to deal with the range-variant range cell migration problem in large-area CTSSAR imaging, a modified chirp scaling algorithm is proposed to realize precise wide swath CTSSAR focusing. Experiments and analyses are performed to validate the effectiveness of the proposed algorithm.

A novel hybrid method is developed for the flexible and accurate electromagnetic simulation of penetrable objects in a layered medium (LM). In this method, the original complex simulation domain is first divided into several subdomains, following the spirit of divide-and-conquer. Each subdomain is then meshed and solved independently, where nonconformal mesh is inevitable. The Riemann type transmission condition is utilized at the interfaces of each subdomain to correctly exchange information so that the solutions of all subdomains converge rapidly to the real solution of the original problem. More specifically, in our method, the surface integral equation (SIE) combined with the LM Green’s functions (LMGFs) is adopted for the boundary subdomain, while the finite-element method (FEM) and the spectral element method (SEM) are employed for all the other interior dielectric subdomains. The SIE with LMGFs truncates the simulation domain tightly within the object itself, which drastically decreases the number of unknowns. The interior subdomains are modeled by either FEM or SEM, depending on the geometry and material property of each subdomain. To further enhance the simulation capability, higher order approaches are adopted for all the subdomain solvers in this hybrid method. Several numerical examples are demonstrated, where a high convergence and accuracy of this method is observed. This paper will serve as an efficient and flexible simulation tool for the applications of geophysical exploration.

To facilitate the detection of hydraulic fractures by electromagnetic survey, a discontinuous Galerkin frequency-domain (DGFD) method is introduced in this paper to efficiently model the fracture responses under complicated geophysical environments. In the proposed DGFD method, the computational domain can be split into multiple subdomains with nonconformal meshes. The Riemann solver (upwind flux) is introduced to evaluate the numerical flux. The impedance transition boundary condition (ITBC) is employed to facilitate fracture modeling by approximating fractures as surfaces. Numerical results show that the ITBC works well for different fracture conductivities, dipping angles, operation frequencies, as well as different sources. For both small- and large-scale fractures, it also shows good agreement with the references. The responses of fractures increase as their conductivities become larger. Large dipping angles can cause spikes on the responses in a borehole. For a magnetic source, higher operation frequencies can enhance the signal level, while for an electric source, the sensitivity to frequency is small. When no borehole is considered, the responses due to an electric source are in general larger than those due to a magnetic one. However, when a borehole with conductive mud is included, the responses can be reversed for the electric and magnetic sources. For multiple fractures outside a cased borehole, the signal level of an electric source is significantly reduced, while that of a magnetic source remains at a similar level compared with the scenario without a casing. With the proposed technique, multiscale modeling of hydraulic fractures in complicated geophysical environments becomes possible.

Evaluation of hydraulic fractures has been under intensive study since the last decade. Among published works, only a few have included casing effects. This paper focuses on the through-casing electromagnetic (EM) induction imaging method that evaluates hydraulic fractures with enhanced contrasts. An experimental system and an inverse scattering algorithm are presented. The laboratory scaled experimental system is built for the feasibility study of the EM induction imaging method for the through-casing hydraulic fracture evaluation. To develop the inverse scattering algorithm, fractures outside boreholes with metallic casing are modeled by a novel hybrid approximation method. This method combines the distorted Born approximation and the mixed order stabilized bi-conjugate gradient fast Fourier transform method to solve the forward scattering problem. The variational Born iterative method is applied to solve the nonlinear inverse problem iteratively. Experimental results show that the inverse scattering algorithm is effective for EM contrast enhanced through-casing hydraulic fracture evaluation.

We present a discontinuous Galerkin pseudospectral time-domain (DG-PSTD) algorithm to solve elastic-/acoustic-wave propagation problems. The developed DG-PSTD algorithm combines the merits of flexibility from a finite-element method and spectral accuracy and efficiency from a high-order pseudospectral method, while having a flavor closer to a finite-volume method. This numerical approach not only uses structured/unstructured conformal meshes but also handles nonconformal meshes (h-adaptivity) with nonuniform approximation orders (p-adaptivity) in different regions, thus leading to high flexibility and efficiency for heterogeneous multiscale problems. To implement the discontinuous Galerkin algorithm, a concise but more general heterogeneous Riemann solver is provided to effectively and accurately resolve the coupling of multiple subdomains for both elastic-elastic/fluid-fluid and fluid-solid coupling. Finally, numerical results demonstrate the flexibility, high accuracy, and efficiency of our method for elastic-/acoustic-wave simulation.

We present an efficient nonconformal-mesh discontinuous Galerkin (DG) method for elastic wave propagation in viscous media. To include the attenuation and dispersion due to the quality factor in time domain, several sets of auxiliary ordinary differential equations (AODEs) are added. Unlike the conventional auxiliary partial differential equation-based algorithm, this new method is highly parallel with its lossless counterpart, thus requiring much less time and storage consumption. Another superior property of the AODE-based DG method is that a novel exact Riemann solver can be derived, which allows heterogeneous viscoelastic coupling, in addition to accurate coupling with purely elastic media and fluid. Furthermore, thanks to the nonconformal-mesh technique, adaptive hp-refinement and flexible memory allocation for the auxiliary variables are achieved. Numerical results demonstrate the efficiency and accuracy of our method.

One fundamental difficulty in low-frequency subsurface electromagnetic exploration is the low-frequency breakdown phenomenon in numerical computation. It makes the discretized linear system very poorly conditioned and thus difficult to solve. This issue is present in both integral equation and partial differential equation solution methods, and thus has attracted many researchers who have proposed various methods to overcome this difficulty. In this paper, we propose a new mixed spectral element method (mixed SEM) to eliminate this low-frequency breakdown problem and apply this method to solve the subsurface electromagnetic exploration problem. Since Gauss' law is now explicitly enforced in the mixed SEM to make the system matrix well-conditioned even at extremely low frequency, we can solve the linear system from dc to high frequencies. With the proposed method, we study the surface-to-borehole electromagnetic system for hydrocarbon exploration. Numerical examples show that the mixed SEM is accurate and efficient, and has significant advantages over conventional methods.

This paper presents a multi-input and multi-output (MIMO) ground penetrating radar (GPR) system, which is going to be launched to the moon for imaging shallow regolith structures and estimating the dielectric properties. This system, as an important part of China' Chang-E 5 lunar exploration mission, employs twelve off-ground Vivaldi antennas as transmitters/receivers, and works in a stationary mode. A reverse-time migration algorithm is developed to process the MIMO GPR dataset for obtaining a high-resolution image of the subsurface objects. The results of a laboratory experiment on a volcanic ash pit demonstrate that the upper and lower interfaces of a marble slab of 3 cm thickness buried at a depth up to 2 m can be clearly imaged. A full waveform inversion algorithm based on Born iterative method is applied to invert the dielectric properties of the subsurface objects. The preliminary results of a numerical experiment demonstrate that the dielectric permittivity of a subsurface cubic object can be accurately obtained using the MIMO GPR dataset at only six discrete frequencies.

For circular trace scanning synthetic aperture radar (CTSSAR), the conventional approximated quadratic range equation assumption does not hold due to the impact of the curved flight path. Thus, existing SAR methods cannot focus the target accurately and efficiently in CTSSAR. To address this problem, the stationary phase can be regarded as an implicit function of the Doppler frequency. Therefore, the 2-D spectrum can be consequently achieved by employing the principle of stationary phase. In this letter, an imaging algorithm based on the derived spectrum is developed for CTSSAR. Promising results from simulation demonstrate the validity and effectiveness of the approach.

A sparse and uniform multiple-input-multiple-output array is generally utilized in through-wall radar to implement real-time imaging of moving targets. However, the array sparsity with the interelement spacing much bigger than half a wavelength gives rise to grating lobe interference smearing the images. In order to enhance the signal-to-interference ratio, this letter introduces the sign coherence factor (SCF) to weigh through-wall images to suppress the grating lobes. The SCF is first proposed in medical ultrasound imaging and directly reflects the coherence of sign bits of all the transmit-receive channels in each pixel. Since the SCF calculated only by the sign bits, it has the smallest amount of computations compared with two other weighing factors, namely, the coherence factor (CF) and the phase coherence factor (PCF), which have been applied to through-wall imaging already. Moreover, the SCF has comparable performance in suppressing grating lobes with the PCF, much better than the CF. These two advantages make the SCF the most suitable for real-time imaging of moving targets. The experimental results with a two-transmitting eight-receiving stepped-frequency continuous-wave through-wall radar verifies the excellent performance of the SCF.

Wall penetration of the electromagnetic wave causes target image defocused and displaced from its true position in through-wall radar imaging. To solve this problem, this letter proposes an approximate wall compensation algorithm, named the refraction angle approximation algorithm, which assumes that the actual refraction angle is approximately equal to the one whose incidence angle is the azimuth angle of the target. The assumption is reasonable since the refraction angle is less sensitive to the incidence angle and limited in a small range based on the Snell's law, and the cosine function has a small derivative at a small angle. Theoretical derivation indicates that the time delay estimation error can be ignored if the target is not so close to the wall surface around the radar. Numerical simulation verifies the efficacy of the algorithm.

In this letter, we derive the distribution characteristics of first-order multipath ghosts in a nested multiple-input-multiple-output (MIMO) through-wall radar and evaluate the efficacy of the phase coherence factor (PCF) in ghost suppression. Different from a synthetic aperture radar, the first-order multipath echoes of a nested MIMO through-wall radar generate several ghosts. For example, for a nested MIMO array composed of a compact receiving subarray and M spatially dispersed transmitters, there are M ghosts at the same side of the wall as the array. The mth ghost is supposed to occur near the intersection of the line, connecting the target and the center of the receiving subarray, and the ellipse whose foci are the positions of the target and the mth transmitter. Under the assumption of phase uniform distribution clutter, the PCF can suppress the ghosts up to -20 lg(1 - √(M2- 1)/M2) dB, which is about 17.46 dB when M = 2.

The thin dielectric sheet-based surface integral equation (SIE) method is introduced to the accurate simulation of fracture scattering, which is an important and challenging topic in hydrocarbon exploration. In the proposed method, the equivalent volume current is decomposed into the equivalent surface current and the constant normal current. With this decomposition, the D-field volume integral equation degenerates into the SIE, which makes our proposed method stable, accurate, and efficient at low frequency. Numerical results have demonstrated its excellent performance.

Low-frequency subsurface electromagnetic measurements are important tools for characterizing natural resources and environmental wastes. Rapid simulations of low-frequency subsurface electromagnetic measurements are still a challenge because of the large computational domain and low-frequency breakdown phenomenon. We develop an effective method to simulate these low-frequency subsurface electromagnetic measurements by using the spectral element method together with a domain decomposition method (DDM). A specific mesh has been designed based on the traveling wave nature in the air and the diffusion field nature in the underground space to greatly reduce the number of unknowns. The frequency-domain version of the Riemann solver (upwind flux) is used as an effective transmission condition to simulate the interactions between neighboring subdomains in DDM. Several numerical examples demonstrate the efficiency of the proposed approach in low-frequency subsurface electromagnetics simulations.

We present a set of stable and efficient formulas to compute the electromagnetic radiation of arbitrary magnetic dipoles in a cylindrically layered anisotropic medium. This set of formulas is derived on the basis of the generalized reflection and transmission coefficients. By expressing our new formulas by the ratios of Bessel or Hankel functions, rather than directly by these special functions, the notorious overflow problem for numerical computation is successfully obviated. Our formulas can be easily implemented for any number of layers, as well as for arbitrary locations and orientations of the transmitter and the receiver. Numerical experiments have demonstrated their correctness and stability. They are then utilized to investigate the influence of the eccentricity distance, the eccentricity angle, and the formation anisotropy on the response of a multicomponent induction logging tool.

Due to the nonlinear property of large uniaxial anisotropic scatterers, many iterative optimization methods have a high risk of being trapped in local minima. In this paper, a frequency-hopping subspace-based optimization method (SOM) is proposed to reconstruct the relative permittivity distribution of 2-D large uniaxial anisotropic scatterers with transverse electrical (TE) illumination. This hybrid method utilizes the results obtained at lower frequency to provide good initial guesses for higher frequency reconstruction, which reduces the occurrence of local minima for the inversion at the higher frequency. For lower frequency, it can only obtain coarse resolution image although it is unable to show the details of the scatterers. However, this coarse image provides a priori information for the reconstruction at higher frequencies to get finer resolution. Numerical examples demonstrate that the proposed hybrid method can effectively rebuild large uniaxial anisotropic scatterers (six wavelengths) with higher stability compared with conventional SOM that uses only single-frequency data.

In this paper, the formulation of reverse time migration (RTM) is improved for impulse borehole radar imaging in the subsurface scenarios with layered media. By fully adopting the prior information of surrounding media, the time gating function is designed and applied to the incident wave field and scattering wave field for each imaging point, which strengthens the correlation between the wave fields in the time domain. The clutters partly caused by the multiple reflections between different media layers are suppressed due to the gating function. A normalized zero-offset cross correlation with gated samples is conducted and used to weight the result of RTM. The improved approach is compared with the conventional RTM, the back-projection method, and the Stolt migration algorithm with synthetic data and is then validated by a single-borehole radar experiment in a layered media scenario. The results demonstrate that the developed approach is superior to the conventional methods in locating targets and robust in complex subsurface environments.

In this paper, the formulation of Stolt migration is modified for impulse borehole radar imaging in large nonuniform subsurface scenarios. By applying the nonuniform fast Fourier transform (FFT) to the acquisition of the frequency-wavenumber spectrum (FWS), the efficiency of Stolt migration for nonuniform surveys is improved. First, each nonuniform exponent basis in Fourier transform is approximated with a weighted summation of several uniform exponent bases. Then, the nonuniform samples with the same uniform exponent basis are accumulated to generate a larger virtual uniform sample set. The FWS of nonuniform samples is approximated with the virtual sample set by FFT. Finally, angular frequency interpolation and inverse FFT are performed over a sample FWS to reconstruct the reflectivity map of the imaging area. The selection of approximation parameters is discussed to make a tradeoff between approximation error and computational cost. The improved Stolt migration technique is compared with the conventional backprojection method and the Kirchhoff migration method on synthetic data and validated by a single borehole radar experiment in a subsurface scenario. The results show that the developed Stolt migration is superior to the conventional methods in terms of computational cost, cross-range resolution, and the ability to reconstruct the targets.

A new frequency-domain backprojection method for processing synthetic aperture radar (SAR) data is presented. It forms SAR images by backprojecting the spectrum of the radar echo into the image wavenumber domain and is capable of fully correcting wavefront curvature and handling a general aperture geometry. From each reconstructed spectral pixel, information about the entire scene can be obtained, allowing the formation of interim lower resolution images during the processing. In addition, the method allows for an arbitrary sampling of the image spectrum. It forms SAR images by backprojecting the spectrum of the radar echo into the image wavenumber domain. Processing of simulated and experimental ultrawideband/widebeam SAR raw data demonstrates the efficacy of this new method.

Positioning 3-D targets buried in layered medium with electromagnetic waves has widespread applications, such as the detection of land mines. Most of the current electromagnetic inverse methods need to know the preknowledge of the dielectric properties for layered background medium, in order to accurately reconstruct concealed targets. However, this condition is hardly satisfied in real problems. In this letter, an electromagnetic inverse scattering series method (EISSM) is developed and derived for positioning 3-D targets buried in two-layered medium. The prominent advantage of the proposed method is that it does not need any prior information about the dielectric properties of the layered medium. The position error predicted by the EISSM is analyzed and discussed. Compared with the commonly used time-reversal-mirror technique, numerical simulations show that the EISSM is capable of positioning the target buried in two-layer medium with less error.

A novel data acquisition and imaging method based on compressive sensing is utilized for impulse borehole radar (IBR). With the sparse transform that we present for IBR systems, only 50% or even less samples are needed to be collected and transmitted to reconstruct the target space, which reduces the sampling rate and data transmission rate of IBR systems. The simulation and experiment results show that the proposed method is more robust in noise environment and the reconstructed target spaces have less artifacts compared with the solutions of the traditional Stolt migration method.

This paper presents a near-field 3-D synthetic aperture radar (SAR) imaging system for which the 2-D aperture is radially scanned. Compared to the current SAR imaging systems, the proposed system has several advantages such as quick data collection, full 360° inspection of target, and simple image formation processing. However, in radial scan, the samples do not fall on a Cartesian grid, which prevents us from using the fast Fourier transform (FFT) to form SAR image without calling for interpolation. In this paper, the 2-D nonuniform FFT (NUFFT) is used for dealing with the problem. After 2-D NUFFT of the radial sampled data, the 3-D reflectivity image can be efficiently reconstructed by using the 3-D version of the range migration algorithm. The Stolt mapping has been implemented implicitly by another 1-D NUFFT to reduce the artifacts caused by the conventional interpolation processing. In addition, to alleviate the data sampling burden, a compressed 2-D slow-time sampling strategy is also discussed. Finally, the proposed Rad-SAR system and the imaging method are demonstrated using near-field wideband simulation data.

In this paper, the formulation of Stolt migration is modified for impulse borehole radar near-field imaging in the subsurface scenarios where the transceiver is widely separated with respect to the detection range. The proposed approach consists of the following aspects. First, the locations of the transmitter and receiver in the survey are regarded as independent sample dimensions, and the original sample set is converted to an enlarged virtual sample set. The frequency-wavenumber spectrum (FWS) of the virtual sample set is available via multidimensional fast Fourier transform (FFT). Then, the relation between the angular frequency and wavenumbers of the transmitter and receiver is derived in the frame of the virtual sample set, which provides the basis for the interpolation in angular frequency and weighting process of FWS. By applying multidimensional inverse FFT (IFFT) to the interpolated and weighted FWS of the virtual sample set, the energy of target responses will focus in some profile of the IFFT result, the position of which is related with the separation between the transmitter and receiver. Finally, the desired target space can be extracted from the IFFT result. The improved Stolt migration technique is compared with the conventional Stolt migration algorithm, back-projection method, and Kirchhoff migration algorithm on synthetic data and validated by single-borehole radar experiment in the subsurface scenario. The results show that the developed Stolt migration is superior to the conventional methods in cross-range resolution, computational cost, and the ability to reconstruct locations and shapes of targets.

There has been increasing efforts in enhanced biomedical and geophysical imaging in the recent years exploring the magnetic contrast agent. The handling of both dielectric and magnetic contrasts adds on difficulties to the forward and inverse scattering problems. In this paper, the 3-D scattering and the inverse scattering from objects having simultaneous electric and magnetic contrasts are presented, where the permittivity, conductivity, and permeability of the objects can all be different from the background. To cope with the challenging high computation cost in the 3-D scattering problem, we formulate the combined field volume integral equations and extend the stabilized biconjugate gradient method and fast Fourier transform (BCGS-FFT) method to compute the electromagnetic field incorporating both the electric and magnetic contrasts. The variational Born iterative method for electrical contrast inversion in axisymmetric media is generalized to 3-D and to the simultaneous reconstruction of objects with electric and magnetic contrasts. The BCGS-FFT method provides the predicted scattered field from the 3-D heterogeneous objects and the Fréchet derivatives in the inverse scattering problem. The efficient forward solver also dramatically reduces the computation time of the inverse problem. Numerical results are presented to validate the forward solver and to demonstrate the effectiveness of the inverse scattering method.

In inverse synthetic aperture radar (ISAR) imaging of nonuniformly rotating targets, such as highly maneuvering airplanes and ships fluctuating with oceanic waves, azimuth echoes have to be modeled as cubic phase signals (CPSs) after the range migration compensation and the translational-induced phase error correction. For the CPS model, the chirp rate and the quadratic chirp rate, which deteriorate the azimuth focusing quality due to the Doppler frequency shift, need to be estimated with a parameter estimation algorithm. In this paper, by employing the proposed generalized scaled Fourier transform (GSCFT) and the nonuniform fast Fourier transform (NUFFT), a fast parameter estimation algorithm is presented and utilized in the ISAR imaging of the nonuniformly rotating target. Compared to the scaled Fourier transform-based algorithm, advantages of the fast parameter estimation algorithm include the following: 1) the computational cost is lower due to the utilization of the NUFFT, and 2) the GSCFT has a wider applicability in ISAR imaging applications. The CPS model and the algorithm implementation are verified with the real radar data of a ship target. In addition, the cross-term, which plays an important role in correlation algorithms, is analyzed for the fast parameter estimation algorithm. Through simulations of the synthetic data and the real radar data, we verify the effectiveness of the fast parameter estimation algorithm and the corresponding ISAR imaging algorithm.

Interpolation-free Stolt mapping, based on Fourier transform and phase multiplications, is proposed. Benefiting from this method, an efficient wavenumber domain algorithm (E-ω-k) can be achieved for generic synthetic aperture radar imaging. The method is finally demonstrated using simulated data.

With appropriate geometry configurations, bistatic synthetic aperture radar (SAR) can break through the limitations of monostatic SAR on forward-looking imaging. Thanks to such a capability, bistatic forward-looking SAR (BFSAR) has extensive potential applications, such as self-navigation and self-landing. In the mode of BFSAR with a stationary transmitter (ST-BFSAR), the two-dimensional spatial variation makes it difficult to use traditional data focusing algorithms. In this letter, an imaging algorithm based on keystone transform and nonlinear chirp scaling (NLCS) is proposed to deal with this problem. Keystone transform is used to remove the spatial variation of range cell migration. NLCS can eliminate the variation of azimuth reference function. Numerical simulations show that by combining first-order keystone transform and azimuth NLCS operation, the raw data of ST-BFSAR can be well imaged.

In inverse synthetic aperture radar (ISAR) imaging of targets with complex motions such as fluctuating ships with oceanic waves and high maneuvering airplanes, the azimuth echo signals can be modeled as cubic phase signals (CPSs). In this letter, a new ISAR imaging algorithm based on the keystone time-chirp rate distribution (KTCRD) is proposed for the targets with complex motions. Compared with the recently published algorithms for the CPSs, the KTCRD can estimate the parameters of multicomponent CPSs without searching procedures and can acquire high antinoise performance with a relatively low computational load. With the estimated motion parameters, high-quality ISAR images can be obtained. Several simulation examples on the synthetic model are shown to validate the effectiveness of the new algorithm presented in this letter.

In translationally variant (TV) bistatic synthetic aperture radar (BSAR), 2-D spatial variation is a major problem to be tackled. In this paper, a generalized Omega-K imaging algorithm to deal with this problem is proposed. The method utilizes a point target reference spectrum of the generalized Loffeld's bistatic formula (LBF) (GLBF). Without the bistatic-deformation term, GLBF is the latest development of LBF. Similar to the monostatic case, it has a much simpler form than other point target reference spectra. Based on the spatial linearization of GLBF, the Stolt mapping relationship is derived. Different from the traditional Omega-K algorithms for monostatic SAR and translationally invariant BSAR, this approach uses a 2-D Stolt frequency transformation. Through this transformation, the method can deal with the 2-D spatial variation. It can also consider the linear spatial variation of Doppler parameters, which is usually not considered in the previous publications on bistatic Omega-K algorithms. This method can handle the cases of TV-BSAR with different trajectories, different velocities, high squint angles, and large bistatic angles. In addition, a compensation method for the phase error caused by the linearization is discussed. Numerical simulations and experimental data processing verify the effectiveness of the proposed method.

In this paper, an omega-K imaging algorithm to focus the raw data of translational invariant (TI) bistatic synthetic aperture radar (BSAR) is proposed. The method utilizes a point target reference spectrum of generalized Loffeld's bistatic formula (GLBF). Without the bistatic deformation term, GLBF is the latest development of Loffeld's bistatic formula. It is comparable in precision with the method of series reversion (MSR), but it has a much simpler form than MSR and a similar form to a monostatic case. Based on the spatial linearization of GLBF, the Stolt transformation relationship is derived. The method can consider the linear spatial variation of Doppler parameters, which is always ignored in previous publications about bistatic omega-K algorithms. This method can handle the cases of TI BSAR with high squint angles and large bistatic degrees. In addition, a compensation method for the phase error caused by the linearization is discussed. Numerical simulations and experimental data processing verify the effectiveness of the proposed method.

In inverse synthetic aperture radar (ISAR) imaging of targets with complex motion such as fluctuating ships with oceanic waves and high maneuvering airplanes, the azimuth echo signals can be modeled as cubic phase signals (CPSs) after the migration compensation. The chirp rate (CR) and the quadratic chirp rate (QCR) are two important physical quantities of the CPS, which deteriorate the azimuth focusing quality due to the Doppler frequency shift. With these two quantities, other parameters can be estimated by using the fast Fourier transform (FFT). Therefore, the CPS can be uniquely determined by both CR and QCR. In this paper, based on the proposed generalized keystone transform and the parametric instantaneous autocorrelation function, a novel distribution of the CPS, known as the CR-QCR distribution (CRQCRD), is presented and applied in a newly proposed ISAR imaging algorithm for targets with complex motion. The CRQCRD is simple and only requires the FFT and the nonuniform FFT (NUFFT). Owing to the application of the NUFFT, the computational cost is saved, and the searching procedure is unnecessary for the nonuniformly spaced signal. Compared to other four representative methods for CPSs, the CRQCRD, which can acquire higher antinoise performance and no error propagation, is searching-free and more suitable for the situation of multitargets. Several simulation examples, analyses of the antinoise performance, and ISAR images validate the effectiveness of the CRQCRD and the corresponding ISAR imaging algorithm.

Efficient multiscale electromagnetic simulations require several major challenges that need to be addressed, such as flexible and robust geometric modeling schemes, efficient and stable time-stepping methods, etc. Due to the versatile choices of spatial discretization and temporal integration, discontinuous Galerkin time-domain (DGTD) methods can be very promising in simulating transient multiscale problems. This paper provides a comprehensive review of different DGTD schemes, highlighting the fundamental issues arising in each step of constructing a DGTD system. The issues discussed include the selection of governing equations for transient electromagnetic analysis, different basis functions for spatial discretization, as well as the implementation of different time-stepping schemes. Numerical examples demonstrate the advantages of DGTD for multiscale electromagnetic simulations.

The papers in this special issue are devoted to the topic of large-scale electromagnetic computation methods for modeling and applicatoins.

Nanophotonic applications often involve large-scale problems with excessive demand on computational resources. We develop a domain decomposition method (DDM) to reduce computer memory and central processing unit (CPU) time requirements by combining the spectral element method (SEM) and the spectral integral method (SIM) for large-scale finite periodic structures. The interior scattering subdomains within each period are modeled by the SEM while the exterior scattering problem is modeled by the SIM. The interactions between neighboring subdomains are modeled by the frequency-domain version of the Riemann solver. Numerical convergence of the Riemann solver is fast and weakly dependent on the size of the system. Two sets of examples demonstrate the typical nanophotonic applications: The first periodic system is a vertical coupling waveguide based on a photonic crystal slab which opens a way to construct and simulate optical circuits. The second periodic system is a finite-sized metamaterial with an effective negative refractive index, whose edge effects are visualized and analyzed.

A new ray-tracing method in 3-D heterogeneous isotropic media is proposed based on the bilinear travel-time interpolation and wavefront group marching method (GMM). In this method, an irregular cell discretization scheme is used to accurately describe arbitrarily undulant interfaces in a model. The ray tracing is carried out by forward–backward processing. In the forward step, the travel time in an irregular cell is expressed in terms of the bilinear interpolation of the known travel times on the cell's surfaces. Then, the wavefront is evolved from the source to the whole computational domain by using the newly developed travel-time solver and the fast wavefront expansion GMM. In the backward step, each ray path is traced from the receiver by finding the intersection points of potential ray propagation vectors with the surfaces of the relevant cells. The same travel-time solver is used to compute the candidate intersection points on all surfaces of each relevant cell, and the point with the minimum travel time is selected as a ray point from which the similar step is continued until the sources are found. Several numerical experiments are presented to demonstrate that the new algorithm is accurate, efficient, and robust.

Ground-penetrating radar (GPR) has been widely used for landmine detection due to its high signal-to-noise ratio (SNR) and superior ability to image nonmetallic landmines. Processing GPR data to obtain better target images and to assist further object detection has been an active research area. Phase-shift migration is a widely used method; however, its wavenumber space is nonuniformly sampled because of the nonlinear relationship between the uniform frequency samples and the wavenumbers. Conventional methods use linear interpolation to obtain uniform wavenumber samples and compute the fast Fourier transform (FFT). This paper develops two- and three-dimensional migration methods that process GPR data to obtain images close to the actual target geometries using a nonuniform fast Fourier transform (NUFFT) algorithm. The proposed method is first compared to the conventional migration approaches on simulated data and then applied to landmine field data sets. Results suggest that the NUFFT migration method is useful in focusing images, estimating landmine structure, and retaining relatively high signal-to-noise ratio in the migrated data. The processed data sets are then fed to the normalized energy and least-mean-square-based anomaly detectors. Receiver operating characteristic curves of data sets processed by different migration methods are compared. The NUFFT migration shows potential improvements on both classifiers with a reduced false alarm rate at most probabilities of detection.

In electrical impedance tomography (EIT), a forward solver capable of predicting the voltages on electrodes for a given conductivity distribution is essential for reconstruction. The EIT forward solver is normally based on the conventional finite element method (FEM). One of the major problems of three-dimensional (3-D) EIT is its high demand in computing power and memory since high precision is required for obtaining a small secondary field which is typical for a small anomaly. This accuracy requirement is also set by the level of noise in the real data; although currently the noise level is still an issue, future EIT systems should significantly reduce the noise level to be capable of detecting very small anomalies. To accurately simulate the forward solution with the FEM, a mesh with large number of nodes and elements is usually needed. To overcome this problem, we proposed the spectral element method (SEM) for EIT forward problem. With the introduction of SEM, a smaller number of nodes and hence less computational time and memory are needed to achieve the same or better accuracy in the forward solution than the FEM. Numerical results demonstrate the efficiency of the SEM in 3-D EIT simulation.

We introduce a new source-dependent but diagonal scattering tensor to approximate the electromagnetic fields internal to a scatterer. The approximate analytical expressions for the three diagonal scattering components are derived. Numerical tests show that the new approximation has favorable accuracy and wider range of applicability than the existing approximations such as the widely used extended Born approximation and the quasi-analytical approximation.

A three-dimensional (3-D) spectral-element method (SEM) based on Gauss-Lobatto-Legendre (GLL) polynomials is proposed to solve the Schro/spl uml/dinger equation in nanodevice simulation. Galerkin's method is employed to obtain the system equation. The high-order basis functions employed are orthogonal and the numerical quadrature points are the same as the GLL integration points, leading to a diagonal mass matrix and a more sparse stiffness matrix. Thus, the proposed method leads to a regular eigenvalue problem, rather than a generalized eigenvalue problem, greatly reducing the computer-memory requirement and central-processing-unit (CPU) time in comparison with the conventional finite-element method (FEM). Furthermore, the SEM is implemented for high geometrical orders, where curved structures can be modeled up to the accuracy comparable to the interpolation accuracy afforded by the basis functions. Numerical examples verify a spectral accuracy with the interpolation orders, and confirm that higher geometrical orders are essential for curved structures to achieve overall spectral accuracy. Examples of quantum dots in various structures, including a waveguide, are analyzed with mixed boundary conditions. Numerical results show that the SEM is an efficient alternative to conventional FEM and to the finite-difference method (FDM) for nanodevice simulation.

A two-dimensional nonlinear inverse scattering technique is developed for imaging objects in a multilayered medium that simulates the effects of building walls in the context of through-wall imaging (TWI). The effectiveness and capacity of the inversion algorithm and the feasibility of through-wall imaging is demonstrated via a number of numerical examples. It has been shown that using multifrequency data high-quality image reconstruction can be achieved with a limited array view.

We develop electromagnetic inverse scattering methods to reconstruct three-dimensional (3-D) objects buried in layered media. The nonlinear inverse problem is solved iteratively via the conjugate-gradient approach; within each iteration, the problem is linearized by Born and distorted Born approximations. The forward solution for layered media is provided by the stabilized biconjugate-gradient fast Fourier transform method. The inversion results from the Born and distorted Born iterative methods are presented, and the effects of the aperture size and noise on the inversion results are also investigated. Numerical results show that these methods are capable of reconstructing an arbitrary 3-D inhomogeneous object buried in a multilayered medium with high resolution.

A spectral-domain method is described for solving Schrodinger's equation based on the multidomain pseudospectral method and boundary patching. The computational domain is first divided into nonoverlapping subdomains. Using the Chebyshev polynomials to represent the unknown wave function in each subdomain, the spatial derivatives are calculated with a spectral accuracy at the Chebyshev collocation points. Boundary conditions at the subdomain interfaces are then enforced to ensure the global accuracy. Numerical results demonstrate that this spectral-domain method has an exponential accuracy and is flexible, and thus is an attractive method for large-scale problems. With only about four cells per wavelength, the results have an error less than 1% in our typical examples. For a typical quantum well, the method is about 51 and 295 times faster than the second-order finite-difference method for 1% and 0.1% accuracy, respectively. The spectral grid method has also been validated by results obtained by the finite-element method, semianalytical (Airy function) method, and the Numerov's method.

Near-surface electromagnetic characterization of objects buried in multilayered earth is important for the detection and identification of landmines, unexploded ordnance, and underground structures. However, so far little progress has been made in the development of fast algorithms for inhomogeneous objects in a layered medium. We report an iterative technique, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method, that simulates near-surface detection of three-dimensional, inhomogeneous objects buried in multilayered media. The CPU time and memory cost of the BCGS-FFT method is O(NlogN) and O(N), respectively, where N is the number of unknowns. This method is significantly more efficient than method of moments. It is capable of solving large-scale electromagnetic scattering problems with an arbitrary inhomogeneous object embedded in a layered medium with an arbitrary number of layers. Examples in subsurface detection of large buried objects are shown to demonstrate the efficacy of this method. At present, the object must be located completely within one single layer in this multilayer medium, but efforts are underway to remove this limitation.

Electromagnetic induction logging is one of the most important measurements in borehole characterization of an oil reservoir. With the ever increasing number of deviated and horizontal wells aiming for improved hydrocarbon production, simulation of induction well logging in realistic three-dimensional (3-D) environments has become an important subject of research. In this paper, we investigate a fast spectral-domain solver for the second-kind integral equation arising from Maxwell's equations for 3-D induction logging. We combine an iterative stabilized biconjugate-gradient (BiCGSTAB) technique with a fast Fourier transform (FFT) algorithm (BCGS-FFT method) to compute electromagnetic fields in 3-D inhomogeneous media at induction frequencies. The electric field integral equation is discretized through the use of the magnetic vector potential to weaken the singularity associated with the dyadic Green's function. The computational time and computer memory requirements of this BCGS-FFT method are O(NlogN) and O(N), respectively, where N is the total number of unknowns.

A multidomain pseudospectral time-domain (PSTD) method with a newly developed well-posed PML is introduced as an accurate and flexible tool for the modeling of electromagnetic scattering by 2-D objects buried in an inhomogeneous lossy medium. Compared with the previous single-domain Fourier PSTD method, this approach allows for an accurate treatment of curved geometries with subdomains, curvilinear mapping, and high-order Chebyshev polynomials. The effectiveness of the algorithm is confirmed by an excellent agreement between the numerical results and analytical solutions for perfectly conducting as well as permeable dielectric cylinders. The algorithm has been applied to model various ground-penetrating radar (GPR) applications involving curved objects in a lossy half space with an undulating surface. This multidomain PSTD algorithm is potentially a very useful tool for simulating antennas near complex objects and inhomogeneous media.

The authors propose the hybridization of the extended Born approximation (EBA) with the conjugate-gradient fast Fourier transform (CG-FFT) method to improve the efficiency of numerical solution of electromagnetic induction problems. This combination improves the solution efficiency in two ways. First, using the FFT in the extended Born approximation decreases the computational cost of the conventional EBA method from O(N/sup 2/) to O(N log/sub 2/ N) arithmetic operations, where N is the number of unknowns in the problem. This approach, referred to as the FFT-EBA method, applies to problems with a fairly large contrast. Secondly, using the EBA as a partial preconditioner for the CG-FFT method increases the convergence speed of the conventional CG-FFT method. This second approach, referred to as the EBA-CGFFT method, is in principle applicable to all problems with a homogeneous background, but is particularly efficient for problems with a higher contrast. Numerical experiments suggest that the combination of these two methods is more accurate and more efficient for electromagnetic induction problems.

Acoustic waves can be a viable tool for the detection and identification of land mines, unexplored ordnance (UXO), and other buried objects. Design of acoustic instruments and interpretation and processing of acoustic measurements call for accurate numerical models to simulate acoustic wave propagation in a heterogeneous soil with buried objects. Compared with the traditional seismic exploration, high attenuation is unfortunately ubiquitous for shallow surface acoustic measurements because of the loose soil and the fluid in its pore space. To adequately model such acoustic attenuation, we propose a comprehensive multidimensional finite-difference time-domain (FDTD) model to simulate the acoustic wave interactions with land mines and soils based on the Biot theory for poroelastic media. For the truncation of the computational domain, we use the perfectly matched layer (PML). The method is validated by comparison with analytical solutions. Unlike the pure elastic wave model, this efficient PML-FDTD model for poroelastic media incorporates the interactions of waves and the fluid-saturated pore space. Several typical land mine detection measurements are simulated to illustrate the application.

The authors develop a hybrid implementation of the extended Born approximation (EBA) with the conjugate-gradient fast Fourier Hankel transform (CG-FFHT) method to improve the efficiency of the numerical solution of borehole induction problems in axisymmetric media. First, they use the FFHT to accelerate the EBA as a nonlinear approximation to induction problems, resulting in an algorithm with O(N log/sub 2/ N) arithmetic operations, where N is the number of unknowns in the problem. This accelerated EBA is accurate for most formations encountered in practical applications. Then, for formations with extremely high contrasts, they utilize the accelerated EBA as a partial preconditioner in the CG-FFHT method to solve the problem accurately with few iterations. The seamless combination of these two approaches provides an automatic way toward a general efficient and accurate modeling algorithm for induction measurements in axisymmetric media.

The authors develop two nonlinear inverse methods to reconstruct the conductivity profile from electromagnetic induction (EMI) measurements: the improved two-step inverse method based on the extended Born approximation (EBA) and the combination of the EBA and the contrast source inversion (CSI) method. In the first method, the nonlinear problem is recast as a two-step linear inversion and is solved by using the extended Born approximation. The authors improve this method with the fast Fourier transform (FFT) algorithm and by a conjugate-gradient optimization procedure. For a problem with N unknown pixels and ill measurement data points, the computational time of this inverse EBA procedure is reduced to O(MN) in the first step (i.e., the underdetermined linear problem) and to O(Nlog/sub 2/N) in the second step (i.e., the well-determined linear problem) using the newly developed FFT-EBA method. Furthermore, the memory requirement is reduced to O(M/sub N/) (M/sub T/ is the number of transmitters). In the second inverse method, they apply the contrast source inversion (CST) method, but use the two-step linear inversion result as the initial solution. By using FFT, the CPU time costs O(C/sub 1/MN+C/sub 2/N log/sub 2/ N) in each iteration of the CSI procedure. Numerical examples illustrate the effectiveness of these methods, even up to a high contrast of 100:1.

We review our efforts to apply the nonuniform fast Fourier transform (NUFFT) and related fast transform algorithms to numerical solutions of Maxwell's equations in the time and frequency domains. The NUFFT is a fast algorithm to perform the discrete Fourier transform of data sampled nonuniformly (NUDFT). Through oversampling and fast interpolation, the forward and inverse NUFFTs can be achieved with O(N log/sub 2/ N) arithmetic operations, asymptotically the same as the regular fast Fourier transform (FFT) algorithms. Using the NUFFT scheme, we develop nonuniform fast cosine transform (NUFCT) and fast Hankel transform (NUFHT) algorithms. These algorithms provide an efficient tool for numerical differentiation and integration, the key in the solutions to differential equations and volume integral equations. We present sample applications of these nonuniform fast transform algorithms in the numerical solution to Maxwell's equations.

Many applications require time-domain solutions of Maxwell's equations in inhomogeneous, conductive media involving cylindrical geometries with both electrically small and large structures. The conventional finite-difference time-domain (FDTD) method with a uniform Cartesian grid will result in a staircasing error, and wastes many unnecessary cells in regions with large structures in order to accommodate the accurate geometrical representation in regions with small structures. In this work, an explicit FDTD method with a nonuniform cylindrical grid is developed for time-domain Maxwell's equations. A refined lattice is used near sharp edges and within fine geometrical details, while a larger lattice is used outside these regions. This provides an efficient use of limited computer memory and computation time. The authors use two absorbing boundary conditions to a nonuniform cylindrical grid: (1) the straightforward extension of Berenger's perfectly matched layer (PML) which is no longer perfectly matched for cylindrical interfaces, thus the name quasi-PML, (QPML); (2) the improved true PML based on complex coordinates. In practice, both PML schemes can provide a satisfactory absorbing boundary condition. Numerical results are shown to compare the two absorbing boundary conditions (ABCs) and to demonstrate the effectiveness of the nonuniform grid and the absorbing boundary conditions.

Recently, a pseudospectral time-domain (PSTD) algorithm was developed to simulate electromagnetic wave propagation. This technique uses the fast Fourier transform (FFT) algorithm for the spatial derivatives and uses the perfectly matched layer (PML) to eliminate the wraparound effect due to the spatial periodicity introduced by FFT. In this work, the author further analyzes this new method and compares it with the finite-difference time-domain (FDTD) and multiresolution time-domain (MRTD) methods for accuracy and efficiency. The PSTD algorithm is then applied to simulate large-scale problems for subsurface electromagnetic and acoustic measurements. For many problems encountered, since the spatial derivatives are obtained by the PSTD algorithm for continuous field components, this algorithm has a high order of accuracy in the spatial derivatives, and thus requires much fewer unknowns than the FDTD and MRTD methods. Numerical results confirm the efficacy of the PSTD method.

Recently an efficient pseudospectral time-domain (PSTD) algorithm has been developed to solve partial differential equations in computational electromagnetics and acoustics. It uses the fast Fourier transform (FFT) algorithm to approximate spatial derivatives, and the perfectly matched layer (PML) to eliminate the wraparound effect. Due to its high accuracy in the spatial derivatives, this method requires a significantly smaller number of unknowns than a conventional finite-difference time-domain (FDTD) method when solving large-scale problems. In this work, the authors further extend the PSTD algorithm to frequency-dependent media and apply the algorithm to simulate ground-penetrating radar (GPR) measurements in a dispersive Earth. The dispersion of the soil is treated by the recursive convolution approaches. The convergence property of the PSTD algorithm is investigated for the scattering of a dispersive cylinder. Multidimensional large-scale problems in GPR measurements are presented to demonstrate the efficiency of this frequency-dependent PSTD algorithm.

The inversion of induction tool measurements using the distorted Born iterative method (DBIM) and the conjugate gradient-fast Fourier-Hankel transform (CG-FFHT) is described. The inverse problem is formulated in terms of an integral equation of scattering where the unknown to be sought is the conductivity in the rock formation, when the measurements along a borehole axis are performed. The nonlinear problem is linearized at each stage using the distorted Born approximation. The inhomogeneous medium Green's function in the distorted Born approximation is found by solving a volume integral equation using the CG-FFHT method, which allows a rapid solution to a large problem with reduced computational complexity and memory requirement. In this manner, the inverse problem is solved with a computational complexity proportional to N/sub tl/N log N where N/sub tl/ is the number of transmitter locations used in the data collection and N is the total number of pixels used to model the unknown formation. The memory requirement is of order NN/sub tl/.>

Deals with the inversion of low-frequency electrode-type resistivity measurements for the conductivity distribution in a two-dimensional axisymmetric medium. It is well known that the inversion of such transverse magnetic measurements is much more nonlinear than that of transverse electric measurements. The distorted Born iterative method (DBIM) is applied to solve the nonlinear inverse problem. In each iteration of the DBIM, an efficient numerical mode-matching (NMM) method is used as a forward solver. In addition to its efficiency in solving for the predicted data, the NMM method gives a semianalytic expression for the partial derivatives of the Green's function required in the inversion. Several numerical results are presented to demonstrate the applications of the DBIM, and to address several practical issues related to the performance of the nonlinear inversion scheme. Because of the fast forward modeling and semianalytic Green's function available due to the NMM method, the inversion is fast and is practical for the interpretation of measurement data.>

The authors formulate and implement a numerical mode-matching (NMM) method to model electrode-type resistivity tools in invaded thin beds. The authors derive the low-frequency approximation of the Maxwell's equations to obtain the partial differential equation for the potential field. The new NMM program is validated by comparing the numerical results with those obtained from other dc programs. It is found that this new program is much faster than the program using the finite-element method (FEM), and hence is useful for routine interpretation of resistivity logs and for inversion.>

The author has used transverse electric measurements on the symmetry axis of an axisymmetric inhomogeneous medium to reconstruct its two-dimensional complex permittivity distribution. The distorted Born iterative method is applied to solve the nonlinear inversion problem, and the numerical mode-matching method is used to solve the forward problem in each iteration. It is shown through numerical examples that good imaging results can be obtained by using very simple measurements on the axis of symmetry.>

A symmetrical form of the solution for an electrical source in a multibed well-logging environment is derived. The method uses local reflection and transmission operators of a single-bed boundary and a general recursive algorithm to derive generalized reflection and transmission operators. Using this method, the computation time scales linearly as N, where N is the number of beds in the environment. A computer program was developed to implement the solution. The program is robust and generates accurate results from 20 kHz to 25 MHz.>