Contraction Integral Equation Research Articles

Study on the detectability of CSRMT in archaeological exploration: taking the ancient city wall remains as an example

Although electromagnetic methods have made significant advances in archaeological exploration, there remains a lack of a method that offers both high operating efficiency and high resolution. This study introduces a novel approach derived from the Controlled Source Radio Magnetotelluric (CSRMT) method. The CSRMT method combines the efficient configuration of the Controlled Source Audio-magnetotelluric (CSAMT) method with enhanced resolution capabilities. To evaluate the efficacy of the new method in archeological exploration, we designed a series of models based on existing references related to ancient city wall remains. Using the 3D contraction integral equation method, we numerically simulated the high-frequency component of the CSRMT method, and then on, the Laterally Constrained Inversion technique is used to invert the data. The effectiveness of the method was assessed qualitatively and quantitatively through analysing the electromagnetic field and apparent resistivity of models with varying widths, heights, top burial depths, and resistivity. Our analysis revealed that models with shallower burial depth and greater height of the underground city wall exhibited higher resolution. The findings demonstrate that the CSRMT method can efficiently and accurately identify buried remains, making it a highly promising tool for archaeological exploration and warranting further development.

Support Reconstruction of Dielectric and Metallic Targets via the Contraction Integral Equation

Support Reconstruction of Dielectric and Metallic Targets via the Contraction Integral Equation

3-D induction log modelling with integral equation method and domain decomposition pre-conditioning

SUMMARY The deployment of electromagnetic (EM) induction tools while drilling is one of the standard routines for assisting the geosteering decision-making process. The conductivity distribution obtained through the inversion of the EM induction log can provide important information about the geological structure around the borehole. To image the 3-D geological structure in the subsurface, 3-D inversion of the EM induction log is required. Because the inversion process is mainly dependent on forward modelling, the use of a fast and accurate forward modelling tool is essential. In this paper, we present an improved version of the integral equation (IE) based modelling technique for general anisotropic media with domain decomposition pre-conditioning. The discretized IE after domain decomposition equals a fixed-point equation that is solved iteratively with either the block Gauss–Seidel or Jacobi pre-conditioning. Within each iteration, the inverse of the block matrix is computed using a Krylov subspace method instead of a direct solver. An additional reduction in computational time is obtained by using an adaptive relative residual stopping criterion in the iterative solver. Using this domain decomposition scheme, numerical experiments show computation time reductions by factors of 1.97–2.84 compared to solving the full-domain IE with a GMRES solver and a contraction IE pre-conditioner. Additionally, the reduction of memory requirement for covering a large area of the induction tool sensitivity enables acceleration with limited GPU memory. Hence, we conclude that the domain decomposition method is improving the efficiency of the IE method by reducing the computation time and memory requirement.

Physically Unrolling Network Under Contraction Integral Equation for Limited-Aperture Inverse Scattering Problem

Physically Unrolling Network Under Contraction Integral Equation for Limited-Aperture Inverse Scattering Problem

Multiscaling Differential Contraction Integral Method for Inverse Scattering Problems With Inhomogeneous Media

Practical applications of microwave imaging often require the solution of inverse scattering problems with inhomogeneous backgrounds. Toward this end, a novel inversion strategy, which combines the multiscaling (MS) regularization scheme and the difference contraction integral equation (DCIE) formulation, is proposed. Such an integrated approach mitigates the nonlinearity and the ill-posedness of the problem to obtain reliable high-resolution reconstructions of the unknown scattering profiles. The arising algorithmic implementation, denoted as MS-DCIE, does not require the computation of Green’s function of the inhomogeneous background, and thus, it provides an efficient and effective way to deal with complex scenarios. The performance of the MS-DCIE is assessed by means of numerical and experimental tests in comparison with competitive state-of-the-art inversion strategies as well.

Learning-Assisted Inversion for Solving Nonlinear Inverse Scattering Problem

Solving inverse scattering problems (ISPs) is challenging because of its intrinsic ill-posedness and the nonlinearity. When dealing with highly nonlinear ISPs, i.e., those scatterers with high contrast and/or electrically large size, the traditional iterative nonlinear inversion methods converge slowly and take lots of computation time, even maybe trapped into local wrong solution. To alleviate the above challenges, a learning-assisted (LA) inversion approach termed as the LA inversion method (LAIM) with advanced generative adversarial network (GAN) in virtue of a new recently established contraction integral equation for inversion (CIE-I) is proposed to achieve a good balance between the computational efficiency and the accuracy of solving highly nonlinear ISPs. The preliminary profiles composed of only small amount of low-frequency components can be got efficiently by the Fourier bases expansion of CIE-I inversion (FBE-CIE-I). The physically exacted information can be taken as the input of the neural network to recover super-resolution image with more high-frequency components. A weighted loss function composed of the adversarial loss, mean absolute percentage error (MAPE), and structural similarity (SSIM) is used under the pix2pix GAN framework. In addition, the self-attention module is used at the end of the generator network to capture the physical distance information between two pixels and enhance the inversion accuracy of the feature scatterers. To further improve the inversion efficiency, the data-driven method (DDM) is used to achieve real-time imaging by cascading U-net and pix2pix GAN, where U-net is used to replace FBE-CIE-I in the LAIM. Compared with other LA inversion, both the synthetic and experimental examples have validated the merits of the proposed LAIM and DDM.

Fourier Bases-Expansion for Three-Dimensional Electromagnetic Inverse Scattering Problems

Three-dimensional Fourier bases-expansion contraction integral equation for inversion (FBE-CIE-I) is developed to solve electromagnetic inverse scattering problems (ISPs). Taking the shape of the scatterer into account, the size of the domain of interest (DoI) and thus the number of low-frequency Fourier bases are flexibly chosen in the proposed method. The accuracy and the stability of the proposed method are verified by the numerical tests on irregularly shaped targets. The results indicate that the accuracy and the convergence of the proposed 3-D FBE-CIE-I is insensitive to the parameters and the noises on the scattered fields.

A FORTRAN Program to Model Magnetic Gradient Tensor at High Susceptibility Using Contraction Integral Equation Method

The magnetic gradient tensor provides a powerful tool for detecting magnetic bodies because of its ability to emphasize detailed features of the magnetic anomalies. To interpret field measurements obtained by magnetic gradiometry, the forward calculation of magnetic gradient fields is always necessary. In this paper, we present a contraction integral equation method to simulate the gradient fields produced by 3-D magnetic bodies of arbitrary shapes and high susceptibilities. The method employs rectangular prisms to approximate the source region with the assumption that the magnetization in each element is homogeneous. The gradient fields are first solved in the Fourier domain and then transformed into the spatial domain by 2-D Gauss-FFT. This calculation is performed iteratively until the required accuracy is reached. The convergence of the iterative procedure is ensured by a contraction operator. To facilitate application, we introduce a FORTRAN program to implement the algorithm. This program is intended for users who show interests in 3D magnetic modeling at high susceptibility. The performance of the program, including its computational accuracy, efficiency and convergence behavior, is tested by several models. Numerical results show that the code is computationally accurate and efficient, and performs well at a wide range of magnetic susceptibilities from 0 SI to 1000 SI. This work, therefore, provides a significant tool for 3D forward modeling of magnetic gradient fields at high susceptibility.

Wavelet-Based Subspace Regularization for Solving Highly Nonlinear Inverse Scattering Problems with Contraction Integral Equation

A wavelet transform twofold subspace-based optimization method (WT-TSOM) is proposed to solve the highly nonlinear inverse scattering problems with contraction integral equation for inversion (CIE-I). While the CIE-I is able to suppress the multiple scattering effects within inversion (without compromising the accuracy of the physics), proper regularization is needed. In this paper, we investigate a new type subspace regularization technique based on wavelet expansions for the induced currents. We found that the bior3.5 wavelet is a good choice to stabilize the inversions with the CIE-I model and in the meanwhile it also can rectify the contrast profile. Numerical tests against both synthetic and experimental data show that WT-TSOM is a promising regularization technique for inversion with CIE-I.

Contraction Integral Equation for Modeling of EM Scattering by 3-D Bodies Buried in Multilayered Anisotropic Medium

Integral equation (IE) is regarded as an efficient method for 3-D modeling of electromagnetic (EM) responses. The main contribution of this article is to develop a contraction integral equation (CIE) algorithm with a high performance for modeling of EM scattering by 3-D bodies buried in multilayered anisotropic medium. The introducing of contraction operator can ensure that the discretized IE will be iteratively convergent in large conductivity contrast between abnormal bodies and background medium. Moreover, the contraction operator can also accelerate the iteration convergence of the IE. During the discretization process of CIE, direct waves are extracted from dyadic Green's functions (DGFs) of multilayered medium, and the singular integrals of direct waves are eliminated using equivalent sphere and ellipsoid methods. By splitting DGF into convolution terms and correlation terms, 3-D FFT can be used to accelerate the product evaluation of DGFs integral and scattering current encountered in each iteration. Validity of the proposed CIE is verified by comparing with the existing finite volume method. Modeling results show that the conventional IE will be iteratively divergent when the conductivity contrast between bodies and background is large, but the CIE is always iteratively convergent even under very large conductivity contrast.

Fourier Bases-Expansion Contraction Integral Equation for Inversion Highly Nonlinear Inverse Scattering Problem

The electromagnetic inverse scattering problems (ISPs) have counted severe multiple scattering effects, especially those for strong scatterers (the ones with high contrasts and/or electrically large dimensions). Iterative inversion methods based on the model of the Lippmann–Schwinger integral equation (LSIE) usually fail to reconstruct such highly nonlinear ISPs. In this article, in virtue of a new recently established contraction integral equation for inversion (CIE-I), we propose a Fourier bases-expansion with CIE-I (FBE-CIE-I) inversion method to handle highly nonlinear ISPs via the contrast source inversion (CSI) scheme. In the FBE-CIE-I, the unknown contrast source is spanned within the Fourier bases, and the multi-round optimization scheme is adopted by choosing the proper number of low-frequency components; not only the model of CIE-I can be stabilized well but also the computational cost can be further reduced compared with the traditional CSI. Besides, compared with the previous inversion method based on LSIE, the resolvability and robustness of the FBE-CIE-I could be significantly enhanced in handling highly nonlinear ISPs. Numerical and experimental tests verify the interests and efficiency of the proposed method.

A multi-resolution computational method to solve highly non-linear inverse scattering problems

An innovative computational method to solve inverse scattering problems is proposed for retrieving the electromagnetic properties of unknown targets. The proposed technique is based on the contraction integral equation for inversion (CIE-I) method to mitigate multiple scattering contributions when imaging strong scatterers. More specifically, the CIE-I is integrated in an effective multi-resolution (MR) scheme to reduce the ratio between unknowns and non-redundant data as well as to exploit iteratively acquired information on the scenario for yielding higher-resolution reconstructions. Some preliminary numerical results are reported to assess the capabilities of the proposed MR-CIE-I method.

A Multiresolution Contraction Integral Equation Method for Solving Highly Nonlinear Inverse Scattering Problems

The solution of highly nonlinear inverse scattering problems is addressed. An innovative multiresolution microwave imaging technique, which leverages on the profitable integration of the contraction integral equation for inversion (CIE-I) and the iterative multiscaling approach (IMSA), is proposed. Thanks to the joint exploitation of the features of the CIE-I formulation and the IMSA, the arising IMSA-CIE-I inversion technique provides faithful reconstructions of complex-shaped strong scatterers with enhanced resolution and computational efficiency with respect to the state-of-the-art competitive alternatives. Numerical and experimental test cases verify the effectiveness and limitations of the proposed method.

Solving Phaseless Highly Nonlinear Inverse Scattering Problems With Contraction Integral Equation for Inversion

At high frequencies, phase measurement is difficult to meet the accuracy required for imaging, and it is more susceptible to noise pollution. Consequently, it is important to develop inversion models and algorithms for electromagnetic inverse scattering problems with phaseless data (PD-ISPs). Compared to the ISPs with full-data (FD-ISPs), the PD-ISPs are more nonlinear due to the lack of the phase information. Aiming at solving highly nonlinear PD-ISPs, in this article we present a contraction integral equation for inversion (CIE-I) regularized with Fourier subspace via contrast source inversion method, denoted as PD-CSI-CIE-I. Under the model of CIE-I, the nonlinearity of PD-ISPs can be effectively alleviated. In order to stabilize CIE-I, a nested optimization scheme with multi-round procedure is adopted by choosing the proper number of low-frequency components from unknown induced current spanned in the Fourier subspace. By the Fourier bases expansion of induced current, the computational cost can be further reduced. Numerical and experimental examples validate the efficiency of the proposed inversion method. Futher, it is shown that the proposed PD-CSI-CIE-I has a stronger inversion capability than the one with the Lippmann-Schwinger integral equation (LSIE) when tackling high contrast scatterers with phaseless data.

Contraction Integral Equation for Three-Dimensional Electromagnetic Inverse Scattering Problems.

Inverse scattering problems (ISPs) stand at the center of many important imaging applications, such as geophysical explorations, industrial non-destructive testing, bio-medical imaging, etc. Recently, a new type of contraction integral equation for inversion (CIE-I) has been proposed to tackle the two-dimensional electromagnetic ISPs, in which the usually employed Lippmann–Schwinger integral equation (LSIE) is transformed into a new form with a modified medium contrast via a contraction mapping. With the CIE-I, the multiple scattering effects, i.e., the physical reason for the nonlinearity in the ISPs, is substantially suppressed in estimating the modified contrast, without compromising physical modeling. In this paper, we firstly propose to implement this new CIE-I for the three-dimensional ISPs. With the help of the FFT type twofold subspace-based optimization method (TSOM), when handling the highly nonlinear problems with strong scatterers, those with higher contrast and/or larger dimensions (in terms of wavelengths), the performance of the inversions with CIE-I is much better than the ones with the LSIE, wherein inversions usually converge to local minima that may be far away from the solution. In addition, when handling the moderate scatterers (those the LSIE modeling can still handle), the convergence speed of the proposed method with CIE-I is much faster than the one with the LSIE. Secondly, we propose to relax the contraction mapping condition, i.e., different contraction mappings are used in updating contrast sources and contrast, and we find that the convergence can be further accelerated. Several numerical tests illustrate the aforementioned interests.

On the modelling of M2 tidal magnetic signatures: effects of physical approximations and numerical resolution

The magnetic signatures of ocean hbox {M}_{2} tides have been successfully detected by the low-orbit satellite missions CHAMP and Swarm. They have been also used to constrain the electrical conductivity in the uppermost regions of the Earth’s mantle. Here, we concentrate on the problem of accurate numerical modelling of tidally induced magnetic field, using two different three-dimensional approaches: the contraction integral equation method and the spherical harmonic-finite element method. In particular, we discuss the effects of numerical resolution, self-induction, the galvanic and inductive coupling between the oceans and the underlying mantle. We also study the applicability of a simplified two-dimensional approximation, where the ocean is approximated by a single layer with vertically averaged conductivity and tidal forcing. We demonstrate that the two-dimensional approach is sufficient to predict the large-scale tidal signals observable on the satellite altitude. However, for accurate predictions of hbox {M}_{2} tidal signals in the areas with significant variations of bathymetry, and close to the coastlines, full three-dimensional calculations are required. The ocean–mantle electromagnetic coupling has to be treated in the full complexity, including the toroidal magnetic field generated by the vertical currents flowing from and into the mantle.

Large-scale seismically guided anisotropic inversion of towed-streamer EM data in the Barents Sea

This paper presents a fast and efficient large-scale anisotropic inversion technique for towed streamer electromagnetic (EM) data, which incorporates seismic constraints. The inversion algorithm is based on the 3D contraction integral equation method and utilizes a reweighted regularized conjugate gradient technique to minimize the objective functional (Zhdanov et al. 2014a, b). We have also introduced the concept of a moving sensitivity domain for seismically guided EM inversion, originally developed for airborne EM surveys (Zhdanov and Cox, 2013), which makes it possible to invert the entire large-scale towed streamer EM survey data while keeping the accuracy of the computation of the EM fields. The developed algorithm and software can take into account the constraints based on seismic and well-log data, and provide the inversion guided by these constraints. To demonstrate the practical effectiveness of this approach for large-scale inversion of marine EM data, as well as integration with seismic data, we apply the method to the inversion of about 2000 line-km of towed streamer EM data. The data form part of a larger survey in the Norwegian Barents Sea (McKay et al., 2016). We show that the technique produces a single resistivity model that is consistent with both the measured EM data and the main seismically defined structures. Thus, the resistivity model is ready to be interpreted and used in further quantitative interpretation studies.

Advanced 3D imaging of complex geoelectrical structures using towed streamer EM data over the Mariner field in the North Sea

Towed streamer electromagnetic (EM) data acquired at the Mariner heavy oilfield in the UK sector of the North Sea have been inverted using a fast and efficient 3D anisotropic inversion code. The towed streamer EM system was towed from a single vessel. The system consisted of a horizontal bipole source and electrode sensors housed in a streamer cable. This enabled a densely sampled grid of data over the subsurface volume of interest. The purpose was to estimate the resistivity structure in a volume including the Maureen and Heimdal reservoir structures in the Mariner complex. The 3D inversion algorithm is based on the contraction integral equation method and utilizes a re-weighted regularized conjugate gradient technique to minimize an objective functional (e.g., Zhdanov et al., 2014). A moving sensitivity domain approach is introduced to handle the large amount of data over the large area (Zhdanov, 2010; Zhdanov and Cox, 2012; Zhdanov et al., 2014; Cox and Zhdanov, 2014). This inversion method is proven to be fast and efficient and is here shown to be suitable for towed streamer EM data from complex geological environments such as the Mariner area. In this case, the final 3D resistivity cube after inversion and with a corresponding normalized misfit of 5.4% correlates well with the expected structure from seismic data and well logs. In particular, the 3D inversion was able to extract an anomaly with vertical and horizontal resistivities of 8-10 and 4-5 Ohm-m, respectively, corresponding to the Maureen and Heimdal reservoirs close to the resistive chalk and basement. The run time on a PC cluster was only seven hours for the full 3D inversion with data from all survey lines covering the area of interest.

Research on the Effect of 3D Body Between Transmitter and Receivers on CSAMT Response Using Integral Equation Method

AbstractIn the field work using the Control Source Audio‐frequency Magnetotelluric method, the space between the transmitter and the receivers is up to several kilometers. In order to study the effect of the anomalous body locating at between the transmitter and the receivers, we simulate the CSAMT response using 3D Contraction Integral Equation method. First we simulate the response of the target body in the survey area. We found that the center of the anomaly locates above the target at high frequency but shifts slowly away from the source with the frequency decreasing, therefore 3D interpretaion method should be considered if there is a 3D anomalous target body in the survey area. Then we simulate the effect of 3D anomalous body between the transmitter and receivers on the response of the target body in the exploration area. The 3D anomalous body could affect the response of the target only when the size of the anomalous body is large enough, otherwise the effect can be ignored. The effect is large when the anomalous body extends in source direction; it is medium when the body extends in horizontal direction and low when extending in vertical direction.

Three-dimensional numerical modeling of marine controlled-source electromagnetic responses in a layered anisotropic seabed using integral equation method

We perform a numerical simulation of marine controlled-source electromagnetic （MCSEM） responses in a layered anisotropic seabed using integral equation method. Firstly, we use the operator theory to deduce the contraction integral equation, which is always convergent regardless of any parameters, for it satisfies the contraction mapping condition. Then we introduce the subdomain multigrid and quasi-linear approximation to improve the computation efficiency. We also validate the accuracy and efficiency of the algorithm by synthetic numerical modelings. Finally, we simulate the three-dimensional MCSEM responses in the layered anisotropic seabed, and investigate the effects of anisotropy.