This article extends the existing finite-element boundary integral (FEBI) method to account for electromagnetic (EM) scattering and inverse scattering from both isotropic and anisotropic 2-D inhomogeneous objects straddling multiple planar layers. In the forward scattering computation, the inhomogeneous scatterers placed across several layers are enclosed by a 1-D smooth boundary within which the finite element method (FEM) is implemented to solve for the 2-D EM field distribution. The integral equation (IE) is formulated on the 1-D boundary to form the radiation boundary condition (RBC) to truncate the FEM domain. The 2-D layered-medium dyadic Green’s functions (DGFs) used in the boundary IE to solve for the equivalent current and used in the scattering data equations to compute the scattered fields at the receiver array are evaluated for both the transverse electric (TE) and transverse magnetic (TM) modes. In the full-wave inversion (FWI) to reconstruct isotropic scatterer dielectric parameters in the TE mode and anisotropic scatterer dielectric parameters in the TM mode, we compute the first-order derivatives of the system mass matrix with respect to isotropic model parameters and the first-order derivatives of the system stiffness matrix with respect to anisotropic model parameters to assemble the sensitivity matrices for the TE mode and TM mode, respectively. Finally, the Levenberg-Marquardt (LM) method is used to iteratively call the FEBI forward solver to fulfill the reconstruction. Numerical experiments are carried out to show the computation efficiency and correctness of the forward scattering and FWI solvers.

This letter presents the computation of electromagnetic (EM) forward scattering from and the full-wave inversion (FWI) of two-dimensional (2-D) inhomogeneous scatterers straddling multiple subsurface planar layers covered by a one-dimensional (1-D) locally rough surface. In the forward scattering, the electric field integral equation (EFIE) is formulated and the 2-D Green’s functions are evaluated by the transmission-line analogy method and the buried object approach (BOA) to account for both the reflection and transmission in the multiple planar layer boundaries and the random scattering from the locally rough surface. The computation accuracy of the forward solver is verified by comparing the simulation results with the finite element method (FEM) outcomes. Meanwhile, the additional computational cost caused by the rough surface is investigated in numerical experiments and also explained in theory. It is shown that, for the numerical case in this paper, the forward solver needs nearly 90 times computation time and consumes nearly 30 times memory when the rough surface is present compared with those when it is absent. Meanwhile, In FWI, the variational Born iterative method (VBIM) is adopted to reconstruct multiple scatterers straddling multiple subsurface planar layers when the rough surface is present or absent. Numerical experiments show that neglecting a rough surface with the root mean square height of 0.1 m almost causes the failure of the inversion.

This article studies the effect of transceiver array aperture size on the inversion ability of the linear integral equation-based solver for the electromagnetic (EM) reconstruction of 2-D scatterers embedded inside a planarly multilayered medium. The investigation is performed in three steps. First, we derive the analytical relationship between the spectra of the scattered electric fields at the receiver array and the reconstructable 2-D scatterer spectrum, which is composed of four different spectral components. This is completely different from the single spectral component for the 2-D scatterer directly placed in a homogeneous subsurface region. Second, the singular value decomposition (SVD) is adopted to compute the discretized integral operator’s right-singular function whose spectrum can also reflect the reconstructable 2-D scatterer spectrum but with the wave attenuation and evanescent mode contribution taken into account. The obtained spectrum shows a “bandstop” feature in the vertical direction with a decrease in the transceiver array aperture size, which is totally different from the “bandpass” feature for the 2-D scatterer embedded in the homogeneous subsurface region. Third, the features of the reconstructable spectrum of the 2-D scatterer embedded inside a planarly multilayered medium, especially the “bandstop” feature, are validated in a series of numerical experiments.

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.

Migration can reconstruct the geometric structure of a subsurface object from the ground penetrating radar (GPR) data. However, a GPR antenna is usually simplified as an ideal point/line source of normal migration algorithms, which ignore the influence of the antenna radiation pattern in subsurface soil. In this letter, the back-propagation algorithm is corrected with the analytical half-space far-field radiation pattern of an infinite line source. The superiority of this modified migration algorithm is verified through numerical, laboratory, and field experiments. The results show that the undesired diffractive artifacts at the target edges can be suppressed, while reserving the reflection amplitude in the migrated images with antenna pattern correction, compared with the conventional back-propagation and Kirchhoff algorithms.

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.

This paper presents an efficient and accurate three-dimensional (3-D) quantitative hybrid microwave imaging method. The linear sampling method (LSM) is first carried out to quickly find the approximate shapes and locations of the unknown objects in the imaging domain based on the scattered field data recorded by receivers which are placed in the far-field zone and wrap the domain. Then the full-wave inversion (FWI) is implemented in a downsized domain which tightly encloses the unknown objects instead of in the whole domain through the Born iterative method (BIM) to quantitatively retrieve the dielectric model parameters of the objects. Because the LSM fails to obtain the sufficiently accurate shapes of the unknown objects, a trained 3-D CNN U-Net is inserted between the LSM imager and the BIM solver to further refine the obtained shapes of LSM, which is expected to aid the following FWI. The proposed hybrid method is validated via the quantitative imaging of both inhomogeneous isotropic scatterers and multiple homogeneous anisotropic scatterers. It is shown that the hybrid method can achieve both higher reconstruction accuracy and lower computational cost compared with the direct BIM inversion. Meanwhile, its antinoise ability is also tested.

This article presents a novel hybrid electromagnetic inversion method. The traditional 3-D variational Born iterative method (VBIM) is combined with the unsupervised machine-learning expectation maximization (EM). In each iteration, VBIM first outputs the pseudo-randomly distributed model parameters in all discretized cells in the inversion domain. Then the EM algorithm is used to classify them and estimate the mean model parameter values of each homogeneous scatterer or subscatterer supposing that the reconstructed model parameters in all cells comply with the Gaussian mixture model (GMM). At last, partial cells in the inversion domain classified as “background” will be removed and the unknowns in the next VBIM iteration are reduced. This process is implemented iteratively until no “background” cell can be removed anymore and the data misfit between the measured scattered field and reconstructed field reaches the stop criterion. Finally, the mean value of the model parameter estimated by EM is mandatorily assigned for each homogeneous scatterer or subscatterer. Numerical examples show that the proposed hybrid method works efficiently for the reconstruction of isotropic, anisotropic, homogeneous, or inhomogeneous scatterers. It also has a certain antinoise ability.

This article presents a hybrid 3-D electromagnetic (EM) full-wave inversion (FWI) method for the reconstruction of subsurface objects illuminated by an antenna array with the limited aperture. The 3-D linear sampling method (LSM) is first used to qualitatively reconstruct the rough shapes and locations of the subsurface objects. Then, the 3-D convolutional neural network (CNN) U-Net is used to further refine the images of the unknown objects. Finally, the Born iterative method (BIM) is implemented to quantitatively invert for the dielectric parameters of subsurface inhomogeneous objects or multiple homogeneous objects in the restricted image regions. Numerical simulations show that, compared with the pure FWI method BIM, the proposed hybrid method can reconstruct subsurface 3-D objects from limited-aperture EM data with both higher accuracy and lower computational cost. In addition, the proposed hybrid method also shows a strong antinoise ability for the reconstruction of multiple subsurface objects.

Ground-penetrating radar (GPR) has been widely applied to the nondestructive inspection of concrete structures such as tunnel lining, bridge deck, and retaining wall, which are usually reinforced by steel bars. The scattering of electromagnetic (EM) waves caused by the dense steel rebar embedded in the concrete structures has a severe influence on the penetration capacity of GPR waves. In this letter, the scattering and penetration characteristics of EM waves propagating through rebar net are investigated via both numerical and laboratory experiments, with an aim to select the antenna nominal frequency for a different reinforcement density. The results show that the rebar, which is perpendicular to the polarization direction of GPR waves and has a very small diameter compared with the wavelength, is almost transparent to the impinged GPR waves. The scattering and interaction of GPR waves caused by the rebar that is parallel to the polarization direction result in a shielding effect, which is manifested as a blind band in the low-frequency range in the transmitted spectrum. This result violates the rule of thumb commonly used in the GPR community, i.e., the lower frequency has a deeper GPR penetration depth. In the end, a low cutoff frequency is recommended for selecting a GPR antenna with an appropriate nominal frequency when it is used in the detection of an anomaly inside and behind a reinforced concrete structure, in which the spacing of the rebar net is known.

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 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.

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.

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.

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.

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.

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.

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.

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.