Stabilized Biconjugate Gradient Fast Fourier Research Articles

Multiparametric Electromagnetic Inversion of 3-D Biaxial Anisotropic Objects Embedded in Layered Uniaxial Media Using VBIM Enhanced by Structural Consistency Constraint

This article presents the 3-D inversion of biaxial anisotropic objects embedded in layered uniaxial media by the variational Born iterative method (VBIM). Although previously only the isotropic complex permittivity profile was reconstructed, here we retrieve all the six diagonal anisotropic permittivity and conductivity parameters simultaneously. In the forward model, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is used to solve the volume electric field integral equation (EFIE). In the inversion model, VBIM is employed to minimize the cost function. In order to improve the inversion results, we propose a novel structural consistency constraint (SCC) applied to VBIM. The SCC is based on the Monte Carlo method and reduces the computational domain in the Born iteration. The major new contribution of this work is to combine the deterministic inversion algorithm VBIM with the stochastic Monte Carlo method to reconstruct all the diagonal anisotropic parameters of the scatterers. Several typical numerical models are simulated, and the results validate the efficiency, antinoise ability as well as the accuracy of VBIM-SCC for reconstructing 3-D biaxial anisotropic objects embedded in multilayered uniaxial media.

Retrieval of Composite Model Parameters for 3-D Microwave Imaging of Biaxial Objects by BCGS-FFT and PSO

This article presents the 3-D quantitative microwave imaging of homogeneous biaxial anisotropic objects with their optical axes being in arbitrary directions. Two sets of model parameters, the anisotropic dielectric parameters and the angles of optical axes with respect to the user-defined coordinate system, are retrieved iteratively. In the forward model, the total fields are solved from the discretized combined field volume integral equations (CFVIEs) by the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method. In the inversion model, the optimized solutions for two sets of model parameters are obtained by the particle swarm optimization (PSO) method. If the geometric models of the homogeneous anisotropic objects are unknown, a full-wave inversion method, the variational Born iterative method (VBIM), is combined with the multiparametric structural consistency constraint (SCC) to find the spatial boundaries of the homogeneous objects in advance. Two numerical examples are used to test the proposed method, and the results show that BCGS-FFT-PSO not only has strong adaptability for optical axes angles varying in wide ranges but also can retrieve composite model parameters of multiple homogeneous biaxial objects simultaneously even when the measured data are contaminated by noise.

3-D Electromagnetic Scattering and Inverse Scattering by Magnetodielectric Objects With Arbitrary Anisotropy in Layered Uniaxial Media

3-D electromagnetic (EM) scattering and inverse scattering of magnetodielectric objects with arbitrary anisotropy embedded in layered uniaxial media are studied. The stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is employed to solve the forward scattering problem formulated by the combined field volume integral equation (CFVIE). In the inversion, we use the variational Born iterative method (VBIM) enhanced by the structural consistency constraint (SCC) to reconstruct eighteen unknown dielectric parameters of the scatterers simultaneously. In order to further improve the accuracy of the reconstructed permittivity, permeability, and conductivity of the scatterers, we propose the structural continuity scanning (SCS) technique. Comparisons between the BCGS-FFT solution and simulated results from finite element method (FEM) are performed to verify the reliability and accuracy of the forward solver. Meanwhile, we reconstruct the multiple anisotropic parameters of several typical structures embedded in layered uniaxial media to show the feasibility and anti-noise ability of the inversion algorithm.

A Hybrid 3-D Electromagnetic Method for Induction Detection of Hydraulic Fractures Through a Tilted Cased Borehole in Planar Stratified Media

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.

Multifrequency 3-D Inversion of GREATEM Data by BCGS-FFT-BIM

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.

Fast Solution of Electromagnetic Scattering for 3-D Inhomogeneous Anisotropic Objects Embedded in Layered Uniaxial Media by the BCGS-FFT Method

In this paper, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is used to solve the electromagnetic scattering problem for 3-D inhomogeneous uniaxial anisotropic objects embedded in a layered uniaxial medium. The optical axis of the background medium is perpendicular to the layer boundaries. However, the optical axis of the scatterer can be rotated through any angles. The electric field integral equation (EFIE) is formulated in the layered uniaxial anisotropic background medium. The volumetric roof-top function is used for the basis function as well as the testing function. The dyadic Green’s functions (DGFs) for both the electric field and the magnetic vector potential are used to solve the EFIE, and the results as well as the performance are compared. Several numerical simulations are carried out to validate the accuracy and efficiency of the proposed method. The major new contribution of this paper is to extend the BCGS-FFT method to accommodate the uniaxial anisotropy of the layered background medium. Therefore, the DGFs for the layered uniaxial medium are evaluated before solving the EFIE. In addition, the DGF for the electric field is also used to solve the 3-D scattering problem through BCGS-FFT iterations, which is different from the previous work.

Electromagnetic Forward and Inverse Algorithms for 3-D Through-Casing Induction Mapping of Arbitrary Fractures

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.

Three-dimensional MR reconstruction of high-contrast magnetic susceptibility by the variational born iterative method based on the magnetic field volume integral equation.

To provide high-quality and high-contrast magnetic susceptibility mapping, a 3D MR reconstruction method for magnetic susceptibility based on the magnetic field volume integral equation with the variational Born iterative method (VBIM) is developed. Three-dimensional magnetic susceptibility is reconstructed from the positive rotating magnetic field component H1+ of the radiofrequency field acquired by B1 mapping. The stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is implemented in the forward problem to solve for the magnetic field, and the conjugate gradient fast Fourier transform method is implemented in the inverse problem to reconstruct the magnetic susceptibility distribution. Numerical results demonstrated that good effectiveness and high accuracy can be achieved for both the forward solver of the stabilized biconjugate gradient fast Fourier transform method and the inverse solver of the VBIM method. The method proved to be robust under noise contamination. Moreover, the magnetic susceptibilities with much higher contrasts than that of the non-full wave methods can also be efficiently reconstructed. The proposed method can reconstruct the magnetic susceptibility of not only human head, but also other human tissues or materials such as magnetic contrast agents with high magnetic susceptibilities. It has promising applications in high-contrast magnetic susceptibility mapping. Magn Reson Med 79:923-932, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Fast Simulation of Scattering Problem for Magnetodielectric Materials With General Anisotropy in Layered Media

In this paper, the mixed-order stabilized biconjugate gradient fast Fourier transform (mixed-order BCGS-FFT) method is presented to solve the scattering problem of magnetodielectric materials with general anisotropy in layered media. While the volumetric roof-top functions are used as the testing functions for the coupled field volume integral equation and the basis functions for flux densities, the second-order curl conforming basis functions are applied to expand the vector potentials with the aim of both preserving the continuity of their tangential components and avoiding the zero terms that might otherwise be caused by the divergence operator. The layered medium Green’s function (LMGF) is efficiently evaluated through the recursive matrix method along with an interpolation technique. Several numerical experiments are presented to demonstrate the high accuracy and efficiency of the method. Different from the previously published works that aim to solve the similar problem, the new contribution of this work is to extend the mixed-order BCGS-FFT method to accommodate the layered background medium. Therefore, the 3-D FFT acceleration for integral kernels associated with the LMGF as well as the interpolation technique has been implemented and combined with the mixed-order BCGS-FFT method.

The BCGS-FFT Method Combined With an Improved Discrete Complex Image Method for EM Scattering From Electrically Large Objects in Multilayered Media

This paper presents an efficient algorithm combining the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method with an improved discrete complex image method (DCIM) for electromagnetic scattering from electrically large objects in both lossless and lossy multilayered media. The required spatial Green's functions obtained by the improved DCIM are accurate both in the near- and far-field regions without any quasi-static and surface-wave extraction. Then, the scattering by buried objects is considered using the BCGS-FFT method combined with the improved DCIM. Numerical results show the improved DCIM can save tremendous CPU time in scattering involving buried objects.

A Full-Wave 2.5D Volume Integral Equation Solver for 3D Millimeter-Wave Scattering by Large Inhomogeneous 2D Objects

A two-and-a-half dimensional full-wave forward solver to compute the three-dimensional (3D) electromagnetic field scattered by an infinitely long inhomogeneous (lossy) dielectric cylinder with arbitrary cross-sectional shape under a given 3D time-harmonic illumination is presented. The relevant set of linear equations, obtained after performing a spatial Fourier transform of the fields along the axial direction and applying a method of moments discretization to the two-dimensional contrast source volume integral equation, is solved iteratively with a stabilized biconjugate gradient fast Fourier transform method. In this way, objects with cross-sectional dimensions of several to many wavelengths can be handled in a very fast way. Furthermore, a vectorial 3D Gaussian beam illumination, usually employed in active millimeter-wave (mm-wave) imaging systems, is implemented using a complex source Gaussian beam formulation. The validity of the method is proved by comparison with analytic results and with the results from a full-wave 3D solver. Finally, the method is applied to a millimeter-wave imaging example, showing the scattering from an object hidden under clothing on a human body.

Three-dimensional electromagnetic nonlinear inversion in layered media by a hybrid diagonal tensor approximation: Stabilized biconjugate gradient fast Fourier transform method

This paper presents an efficient three-dimensional nonlinear electromagnetic inversion method in a multilayered medium for radar applications where the object size is comparable to the wavelength. In the first step of this two-step inversion algorithm, the diagonal tensor approximation is used in the Born iterative method. The solution of this approximate inversion is used as an initial guess for the second step in which further inversion is carried out using a distorted Born iterative method. Since the aim of the second step is to improve the accuracy of the inversion, a full-wave solver, the stabilized biconjugate-gradient fast Fourier transform algorithm, is used for forward modelling. The conjugate-gradient method is applied at each inversion iteration to minimize the functional cost. The usage of an iterative solver based on the FFT algorithm and the developed recursive matrix method combined with an interpolation technique to evaluate the layered medium Green's functions rapidly, makes this method highly efficient. An inversion problem with 32 768 complex unknowns can be solved with 1% relative error by using a simple personal computer. Several numerical experiments for arbitrarily located source and receiver arrays are presented to show the high efficiency and accuracy of the proposed method.

An Improved Weak-Form BCGS-FFT Combined With DCIM for Analyzing Electromagnetic Scattering by 3-D Objects in Planarly Layered Media

Fast algorithms for electrically large objects buried in layered media are mainly hindered by two time-consuming processes. One is the table filling of Green's function, and the other is the solving of the impedance matrix equation. For the first, to accelerate the evaluation of the time-consuming Sommerfeld integral in the dyadic Green's function (DGF), the discrete complex image method (DCIM) is introduced to get a closed-form DGF. To further accelerate the calculation of DGF for the volume electric field integral equation (EFIE), DGF is split before applying DCIM. For the second, the iterative solver stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) is combined with DCIM for solving the matrix equations. Meanwhile, the closed-form DGF enables the spherical-mean Green's function, which eliminates the singularity of Green's function. Numerical results show that the weaker singularity results in a faster and steadier convergence rate for iterative solvers

A fast 2D volume integral-equation solver for scattering from inhomogeneous objects in layered media

The stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic and acoustic scattering from inhomogeneous objects embedded in a layered medium in two dimensions. Two-dimensional layered-media Green's functions are computed adaptively by using Gaussian quadratures after singularity subtraction. The Green's function is split into convolutional and correlational components in order to apply the FFT so as to solve the scattering problem efficiently. The CPU time and memory cost of this BCGS-FFT method is O(N log N) and O(N), respectively, where N is the number of unknowns, which is significantly more efficient than using the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic and acoustic scattering problems for inhomogeneous objects embedded in a layered medium with an arbitrary number of layers. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 47: 128–134, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21101

Simulation of Near-Surface Detection of Objects in Layered Media by the BCGS–FFT 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.

A fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered media

A newly developed iterative method, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium. In this fast solver, the weak-form formulation is applied to obtain a less singular discretization of the volume electric field integral equation. Several techniques are utilized to speed up the dyadic function evaluation. To accelerate the operation of the dyadic function on an induced current (i.e., the Green's operation), the function is split into convolutional and correlational components so that FFT can be applied. The CPU time and memory cost of this BCGS-FFT method is O(NlogN) and O(N), respectively, where N is the number of unknowns, significantly more efficient than the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic scattering problems in a planarly layered background. A large-scale scattering problem in a layered medium with more than three million unknowns has been solved on a Sun Ultra 60 workstation with 1.2 GBytes memory.

The BCGS-FFT method for electromagnetic scattering from inhomogeneous objects in a planarly layered medium

An iterative method, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method, combined with the FFT is developed for solving electromagnetic (EM) scattering from inhomogeneous objects embedded in a planarly layered medium. The CPU time and memory cost of this BCGS-FFT method is O(NlogN) and O(N), respectively, significantly more efficient than both direct and iterative solutions of the method of moment. As a result, this method is capable of solving large-scale EM scattering problems in a planarly layered background.