Complex Conductivity Distribution Research Articles

MTAINV3D: A Three-dimensional Adaptive Magnetotelluric Inversion Tool Using Unstructured Grids

Currently, 3D magnetotelluric inversion often uses a fixed inversion grid. Too sparse inversion grids cannot approximate the complex subsurface conductivity distribution, and may also lead to inversion results failing to converge. Too dense inversion grids will increase not only the computational cost, but also the non-uniqueness of the inversion. Aiming at the above problems, we have developed a three-dimensional magnetotelluric adaptive inversion strategy with unstructured finite element and limited memory BFGS (L-BFGS) algorithms. Firstly, based on the unstructured finite element method, the high accuracy forward responses of the 3D geo-electric model with arbitrary complex terrain and conductivity distributions can be obtained. Then, the forward and inversion meshes are decoupled by using the nested forward and inversion tetrahedral meshes. The Thikhonov regularization objective function with smooth constraints is established and minimized by L-BFGS optimization algorithm with inexact line search procedure, where the gradient of the objective function is solved by using the adjoint principle. Most importantly, an inversion grid adjustment strategy based on the dual constraints of sensitivity matrix and model gradient is proposed, which are used to guide the automatic refinement of inversion grid in the optimization process. Finally, a topographic model is used to validate and test the performance of the proposed adaptive inversion algorithm.

A fast approach to determine excitation eigenfrequencies for TD-EIT and FD-EIT

Electrical impedance tomography can reconstruct the complex conductivity distribution by injecting a current or voltage at a specific frequency into the target domain. The complex conductivity spectroscopy of numerous biological tissues is frequency-dependent. A suitable excitation frequency is vital to high-quality imaging over a wide frequency range. This paper investigates the relationship between the parameters of the biological tissue impedance model and the impedance spectroscopy. A frequency selection method based on the impedance spectroscopy is proposed, in which the impedance spectroscopy on a specific electrode at the domain boundary to be measured is first scanned. In TD-EIT, the difference between the target’s impedance spectroscopy and the null field is calculated, and the frequency corresponding to its extreme value is used as the excitation signal for TD-EIT. The excitation frequency in FD-EIT is the frequency corresponding to the extreme point in the target impedance spectroscopy, and we have also used this method to image the interior of the maize ear. This study provides a quick and efficient method for determining the excitation frequency for EIT, allowing researchers to find the best excitation frequency for high-quality imaging during actual measurements.

Multi-frequency complex conductivity sparse imaging of plant root zone based on space-frequency correlation

The root zone provides indispensable water and nutrients for plants and monitoring the dynamics of the root zone is highly important. However, due to the opacity and complexity of this zone, in situ characterization of the root zone poses many challenges. A sparse Bayesian learning complex model based on spatial frequency correlation (SF-SBLC) was proposed for fast nondestructive imaging of the growth trends in the root zone of plants, and multi-frequency electrical impedance tomography (mfEIT) was used to reconstruct the complex conductivity distribution of the root zone. To achieve the best sparse coefficients and enhance the quality of the reconstructed images, the SF-SBLC learned and utilized the spatial and frequency correlations of the complex conductivities at different frequencies. The reconstruction algorithm was further optimized using a concave-convex process method and a multi-measure vector model, which decreased its time complexity. When compared to conventional mfEIT reconstruction algorithms, the experimental findings demonstrated that the SF-SBLC could greatly reduce imaging artifacts, generate higher reconstruction performance (in terms of the image correlation coefficient, relative image error, resolution, position error, shape deformation, and ringing), and shorten imaging time. Moreover, the SF-SBLC was able to recreate the root zone's shape, size, and location in both a nutrient solution and a horticulture substrate. Finally, the effective volume of the root zone is defined and calculated based on the 3D EIT image. This technique offers a prospective tool that might be useful for monitoring the root growth trends and expansion rates of potted plants.

Investigation of Complex Permittivity/Conductivity Distribution by Electrical Tomography

The electrical tomography technology is a multi-phase flow detection technology. This technology has the advantages of non-radiation, non-invasive and visualization, can be widely used in traditional energy field and new energy field. The permittivity and conductivity, as the basic electrical parameters of material, are closely related to the distribution of material in multiphase flow. However, in the actual multiphase flow, due to the non-uniform distribution of permittivity and conductivity in time and space, the accuracy and validity of measuring instruments are challenged. The main target of this research is to investigate the application of electrical capacitance tomography (ECT), electrical resistance tomography with voltage excitation (ERTv) and electrical resistance tomography with current excitation (ERTc) on the multiphase flow with complex permittivity and conductivity distribution. The final objective based on the above research is to provide suitable tomography modual for different multiphase flow process. The experimental results show that ECT is suitable for measuring dry gas-solid fluids. ERTv is better for measuring high water content gas-solid fluids and low conductivity gas-liquid fluids, while ERTc is better for measuring continuous phase conductivity gas-liquid fluids.

A volume-surface integral approach for direct current resistivity problems with topography

We have developed an accurate volume-surface integral formula for 3D direct current (DC) resistivity forward modeling with heterogeneous conductivities and arbitrary homogeneous topography. First, a volume-surface integral formula is derived from its elliptic boundary value problem in terms of an artificial analytical function defined over the full space. That leads to a volume integral accounting for underground anomalous regions and a surface integral over the surface topography. Then, tetrahedral grids are used to discretize the volume anomalous bodies and triangular grids are adopted to approximate the complicated surface topography. The use of unstructured grids enables our volume-surface integral formula to deal with realistic earth models with complex geometries and conductivity distributions. Furthermore, linear shape functions are assumed in the tetrahedral and triangular elements to obtain the final system of linear equations. In the final system matrix, singularity-free analytical expressions are developed for entries arising from volume integrals over tetrahedral elements and Gaussian quadrature formulas are used to calculate surface integrals over triangular elements. To guarantee the accuracy of the final numerical solutions, direct solvers are used. At the end, three synthetic models are used to verify our newly developed volume-surface integral formula by comparison with published analytical solutions and finite-element solutions. Due to its high accuracy, solutions of our volume-surface integral approach can act as an efficient benchmark tool for other numerical solutions for complicated DC models with arbitrary homogeneous topographies.

Accelerated reconstruction of electrical impedance tomography images via patch based sparse representation.

Electrical impedance tomography (EIT) reconstruction is a nonlinear and ill-posed problem. Exact reconstruction of an EIT image inverts a high dimensional mathematical model to calculate the conductivity field, which causes significant problems regarding that the computational complexity will reduce the achievable frame rate, which is considered as a major advantage of EIT imaging. The single-step method, state estimation method, and projection method were always used to accelerate reconstruction process. The basic principle of these methods is to reduce computational complexity. However, maintaining high resolution in space together with not much cost is still challenging, especially for complex conductivity distribution. This study proposes an idea to accelerate image reconstruction of EIT based on compressive sensing (CS) theory, namely, CSEIT method. The novel CSEIT method reduces the sampling rate through minimizing redundancy in measurements, so that detailed information of reconstruction is not lost. In order to obtain sparse solution, which is the prior condition of signal recovery required by CS theory, a novel image reconstruction algorithm based on patch-based sparse representation is proposed. By applying the new framework of CSEIT, the data acquisition time, or the sampling rate, is reduced by more than two times, while the accuracy of reconstruction is significantly improved.

Ensemble-based Bayesian inversion of CSEM data for subsurface structure identification

A Bayesian inversion methodology for identification of large-scale subsurface structures (strata) from controlled source electromagnetic data is developed. The Bayesian inverse problem is solved by sampling from the posterior probability distribution, using the ensemble Kalman filter. Prior knowledge is incorporated in the methodology by utilizing structural prior information from, for example, interpreted seismic data. A model-based, composite parametrization of the electric conductivity distribution is applied to represent the subsurface structures. The model-based representation also allows for estimation of variation of electric conductivity within each stratum. To enhance stability and reduce model nonlinearity, a reduced representation of structure boundaries and conductivity variation within each stratum is applied. Numerical experiments on various test cases show that the methodology is able to identify fairly complex subsurface conductivity distributions reasonably well, with and without a hydrocarbon reservoir present. These experiments included strata with weak conductivity contrast and the application of various prior probability distributions. Furthermore, the methodology shows the ability to (almost completely) remove a reservoir present in the prior model that is not present in the true model (‘false positive’).

Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest.

A novel direct D-bar reconstruction algorithm is presented for reconstructing a complex conductivity distribution from 2-D EIT data. The method is applied to simulated data and archival human chest data. Permittivity reconstructions with the aforementioned method and conductivity reconstructions with the previously existing nonlinear D-bar method for real-valued conductivities depicting ventilation and perfusion in the human chest are presented. This constitutes the first fully nonlinear D-bar reconstructions of human chest data and the first D-bar permittivity reconstructions of experimental data. The results of the human chest data reconstructions are compared on a circular domain versus a chest-shaped domain.

Magnetoacoustic tomography with current injection

Magnetoacoustic tomography combines the advantages of electrical impedance tomography with sonography. It can reconstruct conductivity images of biological tissue with high spatial resolution and high contrast. In addition, magnetoacoustic tomography is a new nondestructive functional imaging technology, with which we can realize the early diagnosis of human diseases. In this paper, through theory analysis we deduced the analytical formula of acoustic pressure. Computer simulation and experimental studies on magnetoacoustic tomography with current injection (MAT-CI) were made, and a copper ring was used as the model for the simulation and experiment. Experimental setup and methods are introduced, and the acoustic signals and the image of a metal object by means of MAT-CI are presented. Results of the simulation and experiment were in a good agreement. The reconstructed image of an acoustic source can reflect the conductivity information of a sample. The conductivity distribution of a sample can be obtained by means of MAT-CI. The results provide foundations of study on a model with complex conductivity distribution and tissues for MAT-CI.

3D modelling of the human thorax for ventilation distribution measured through electrical impedance tomography

Thoracic electrical impedance tomography (EIT) aims to reconstruct a cross-sectional image of the internal spatial distribution of conductivity from electrical measurements made by injecting small alternating currents via an electrode array placed on the surface of the thorax. It is a non-invasive, radiation-free monitoring technique. In this paper, true 3D thorax models with conductivity distribution or complex conductivity distribution under different ARDS conditions are built up in comparison with the 2.5D ones, and EIT-derived numeric indices are also employed for evaluation of the lung ventilation. The purpose of this paper is to study different effects of different thorax models with either conductivity or complex conductivity on the reconstructed images and ventilation indices.

Factorization Method and Its Physical Justification in Frequency-Difference Electrical Impedance Tomography

Time-difference electrical impedance tomography (tdEIT) requires two data sets measured at two different times. The difference between them is utilized to produce images of time-dependent changes in a complex conductivity distribution inside the human body. Frequency-difference EIT (fdEIT) was proposed to image frequency-dependent changes of a complex conductivity distribution. It has potential applications in tumor and stroke imaging since it can visualize an anomaly without requiring any time-reference data obtained in the absence of an anomaly. In this paper, we provide a rigorous analysis for the detectability of an anomaly based on a constructive and quantitative physical correlation between a measured fdEIT data set and an anomaly. From this, we propose a new noniterative frequency-difference anomaly detection method called the factorization method (FM) and elaborate its physical justification. To demonstrate its practical applicability, we performed fdEIT phantom imaging experiments using a multifrequency EIT system. Applying the FM to measured frequency-difference boundary voltage data sets, we could quantitatively evaluate indicator functions inside the imaging domain, of which values at each position reveal presence or absence of an anomaly. We found that the FM successfully localizes anomalies inside an imaging domain with a frequency-dependent complex conductivity distribution. We propose the new FM as an anomaly detection algorithm in fdEIT for potential applications in tumor and stroke imaging.

Frequency-difference electrical impedance tomography (fdEIT): algorithm development and feasibility study

Frequency-difference electrical impedance tomography (fdEIT) has been proposed to deal with technical difficulties of a conventional static EIT imaging method caused by unknown boundary geometry, uncertainty in electrode positions and other systematic measurement artifacts. In fdEIT, we try to produce images showing changes of a complex conductivity distribution with respect to frequency. Simultaneously injecting currents with at least two frequencies, we find differences of measured boundary voltages between those frequencies. In most previous studies, real parts of frequency-difference voltage data were used to reconstruct conductivity changes and imaginary parts to reconstruct permittivity changes. This conventional approach neglects the interplay of conductivity and permittivity upon measured boundary voltage data. In this paper, we propose an improved fdEIT image reconstruction algorithm that properly handles the interaction. It uses weighted frequency differences of complex voltage data and a complex sensitivity matrix to reconstruct frequency-difference images of complex conductivity distributions. We found that there are two major sources of image contrast in fdEIT. The first is a contrast in complex conductivity values between an anomaly and background. The second is a frequency dependence of a complex conductivity distribution to be imaged. We note that even for the case where conductivity and permittivity do not change with frequency, the fdEIT algorithm may show a contrast in frequency-difference images of complex conductivity distributions. On the other hand, even if conductivity and permittivity values significantly change with frequency, there is an example where we cannot find any contrast. The performance of the proposed method is demonstrated by using computer simulations to validate its feasibility in future experimental studies.

A high-precision voltage source for EIT

Electrical impedance tomography (EIT) utilizes electrodes placed on the surface of a body to determine the complex conductivity distribution within the body. EIT can be performed by applying currents through the electrodes and measuring the electrode voltages or by applying electrode voltages and measuring the currents. Techniques have also been developed for applying the desired currents using voltage sources. This paper describes a voltage source for use in applied-voltage EIT that includes the capability of measuring both the applied voltage and applied current. A calibration circuit and calibration algorithm are described which enables all voltage sources in an EIT system to be calibrated to a common standard. The calibration minimizes the impact of stray shunt impedance, passive component variability and active component non-ideality. Simulation data obtained using PSpice are used to demonstrate the effectiveness of the circuits and calibration algorithm.

Reconstruction of the shape of conductivity spectra using differential multi-frequency magnetic induction tomography

Magnetic induction tomography (MIT) of biological tissue is used for the reconstruction of the complex conductivity distribution κ inside the object under investigation. It is based on the perturbation of an alternating magnetic field caused by the object and can be used in all applications of electrical impedance tomography (EIT) such as functional lung monitoring and assessment of tissue fluids. In contrast to EIT, MIT does not require electrodes and magnetic fields can also penetrate non-conducting barriers such as the skull. As in EIT, the reconstruction of absolute conductivity values is very difficult because of the method's sensitivity to numerical errors and noise. To overcome this problem, image reconstruction in EIT is often done differentially. Analogously, this concept has been adopted for MIT. Two different methods for differential imaging are applicable. The first one is state-differential, for example when the conductivity change between inspiration and expiration in the lung regions is being detected. The second one is frequency-differential, which is of high interest in motionless organs like the brain, where a state-differential method cannot be applied. An equation for frequency-differential MIT was derived taking into consideration the frequency dependence of the sensitivity matrix. This formula is valid if we can assume that only small conductivity changes occur. In this way, the non-linear inverse problem of MIT can be approximated by a linear one (depending only on the frequency), similar to in EIT. Keeping this limitation in mind, the conductivity changes between one or more reference frequencies and several measurement frequencies were reconstructed, yielding normalized conductivity spectra. Due to the differential character of the method, these spectra do not provide absolute conductivities but preserve the shape of the spectrum. The validity of the method was tested with artificial data generated with a spherical perturbation within a conducting cylinder as well as for real measurement data. The measurement data were obtained from a potato immersed in saline. The resulting spectra were compared with reference measurements and the preservation of the shape of the spectra was analyzed.

Reconstruction of the anisotropic complex conductivity distribution of a human thigh in three dimensions

In electric impedance tomography (EIT), a set of electrodes measures surface potentials resulting from known currents injected into the target region. Using a correct model of the target region, these potentials can be simulated very accurately, too. Since injected currents spread out in three dimensions, simulations in three dimensions are inevitable. In addition, the anisotropic nature of electric material properties of biological tissue should not be neglected to obtain appropriate results. The aim of this work is to reconstruct the anisotropic complex conductivity distribution of a human thigh. Pseudomeasurements making up for EIT data were available. A finite element model of a human thigh based on magnetic resonance images was generated. A classical Gauss-Newton algorithm with regularization was used to match the real material values and the ones fed into the simulation. Results for a simple, but unrealistic isotropic case as well as for a complicated anisotropic case are presented and discussed

Reconstruction of the anisotropic complex conductivity distribution in 3D

PurposeThe aim of the work is to reconstruct the anisotropic complex conductivity distribution with the common Gauss‐Newton algorithm.Design/methodology/approachA cubic region with anisotropic material properties is enclosed by a larger cube with isotropic material properties. Numerical simulations are done with tetrahedral nodal finite elements of second‐order.FindingsIt can be shown that it is possible to reconstruct anisotropic complex conductivity distribution if the starting values are chosen sufficiently close to the true values of the complex conductivity.Originality/valueIn this paper, the anisotropic electric conductivity and the anisotropic permittivity are reconstructed in 3D.

Solution of the inverse problem of magnetic induction tomography (MIT)

Magnetic induction tomography (MIT) of biological tissue is used to reconstruct the changes in the complex conductivity distribution inside an object under investigation. The measurement principle is based on determining the perturbation ΔB of a primary alternating magnetic field B0, which is coupled from an array of excitation coils to the object under investigation. The corresponding voltages ΔV and V0 induced in a receiver coil carry the information about the passive electrical properties (i.e. conductivity, permittivity and permeability). The reconstruction of the conductivity distribution requires the solution of a 3D inverse eddy current problem. As in EIT the inverse problem is ill-posed and on this account some regularization scheme has to be applied. We developed an inverse solver based on the Gauss–Newton-one-step method for differential imaging, and we implemented and tested four different regularization schemes: the first and second approaches employ a classical smoothness criterion using the unit matrix and a differential matrix of first order as the regularization matrix. The third method is based on variance uniformization, and the fourth method is based on the truncated singular value decomposition. Reconstructions were carried out with synthetic measurement data generated with a spherical perturbation at different locations within a conducting cylinder. Data were generated on a different mesh and 1% random noise was added. The model contained 16 excitation coils and 32 receiver coils which could be combined pairwise to give 16 planar gradiometers. With 32 receiver coils all regularization methods yield fairly good 3D-images of the modelled changes of the conductivity distribution, and prove the feasibility of difference imaging with MIT. The reconstructed perturbations appear at the right location, and their size is in the expected range. With 16 planar gradiometers an additional spurious feature appears mirrored with respect to the median plane with negative sign. This demonstrates that a symmetrical arrangement with one ring of planar gradiometers cannot distinguish between a positive conductivity change at the true location and a negative conductivity change at the mirrored location.

Accurate Reconstruction Algorithm of the Complex Conductivity Distribution in Three Dimensions

In electrical impedance tomography, an inverse problem has to be solved to reconstruct the complex conductivity distribution /spl kappa/=/spl sigma/+j/spl omega//spl epsiv/. The problem is ill posed, and therefore, a regularization has to be used. The aim is to reconstruct, as accurately as possible, both the electric conductivity /spl sigma/ and the electric permittivity /spl epsiv/ in three dimensions using finite elements of the second order for solving the forward problem. To this end, a new reconstruction algorithm based on a priori information has been developed.

Fast calculation of the sensitivity matrix in magnetic induction tomography by tetrahedral edge finite elements and the reciprocity theorem

Magnetic induction tomography of biological tissue is used to reconstruct the changes in the complex conductivity distribution by measuring the perturbation of an alternating primary magnetic field. To facilitate the sensitivity analysis and the solution of the inverse problem a fast calculation of the sensitivity matrix, i.e. the Jacobian matrix, which maps the changes of the conductivity distribution onto the changes of the voltage induced in a receiver coil, is needed. The use of finite differences to determine the entries of the sensitivity matrix does not represent a feasible solution because of the high computational costs of the basic eddy current problem. Therefore, the reciprocity theorem was exploited. The basic eddy current problem was simulated by the finite element method using symmetric tetrahedral edge elements of second order. To test the method various simulations were carried out and discussed.

Imaging the complex conductivity distribution in electrical impedance tomography

Electrical impedance tomography (EIT) uses current and voltage measurements made on the boundary of a body to compute the conductivity distribution within the body. The present reconstruction algorithm using sensitivity matrix images the complex conductivity distribution due to the permittivity distribution inside the body. A finite element model (FEM) with mesh refinement is used to solve the forward problem in order to improve image resolution. Computer simulations showed that the full information contained in the complex tissue conductivity (real and imaginary components) could be obtained by this method.