Layer by Layer Reconstruction Methods for the Earth Resistivity from Direct Current Measurements

Several methods for reconstructing the resistivity profile of a layered laterally homogeneous earth from direct current measurements are described. These methods recover the resistivity of the earth layer by layer in a recursive way, and require a very small amount of computational effort. They are obtained by transforming the inverse resistivity problem into an equivalent inverse scattering problem, and by applying efficient signal processing algorithms such as the Schur, fast Cholesky or Levinson recursions to the transformed problem. These algorithms operate on a layer stripping or layer accumulation principle, and are shown to be related to previous reconstruction techniques of Pekeris, Koefoed, Kunetz and Rocroi, and others.     

A New Inversion Technique for Apparent Resistivity Measurements

A new algorithm for interpreting apparent earth resistivity data obtained from depth sounding measurements is presented. The method is based on fitting the data to an N layer earth model with equal layer heights h where h is equal to the required depth resolution. The spectral reflection coefficient at the earth's surface for the model used is derived as a ratio of two polynomials, the coefficients of which are a function of the resistivity ratios of adjacent layers. This provides the theoretical basis for inferring the model parameters. Measures for improving numerical stability are suggested and applied to a few examples. Results show that the present algorithm is most suited for continuously varying resistivity depth profiles.     

A Hybrid Method for Electromagnetic Propagated Resistivity Logging Data Inversion

Using an inverse technique for the array electromagnetic propagated resistivity logging (EPRL) data, a fine interpretation can be obtained about the resistivity distribution of an invaded profile. Generally, the Gauss-Newton algorithm (GN) is an efficient technique for the inverse problems; however, as a gradient-type optimization method, its accuracy and convergence depend strongly on the initial value. Even though this problem can be avoided by using a differential evolutionary algorithm (DE) as a global search optimization, it is computationally less efficient. In this paper, a hybrid inversion method of differential evolution has been developed to remove the strong dependence of the accuracy and convergence on the initial value. In this new method, an additional operation, which is designed with GN, is performed only to the best individual with a delay in the evolution processes of DE. Hence, the GN operation is used for the improvement of the convergence speed without leading to any decrease of the robustness of DE. The hybrid method is then extended to apply the inversion of EPRL data. Our results demonstrate its speed, steadiness, and efficiency of this hybrid method     

Permittivity measurements of multilayered media with monostaticpulse radar

Electromagnetic (EM) inversion is a useful tool for quantitative analysis in short-range applications of pulse radars. To estimate multilayered media properties using monostatic radar, two inverse scattering approaches are discussed: (a) layer-stripping algorithm by exploiting amplitude and time delay of radar echoes after their detection, and (b) EM inverse problem of parameter optimization by minimizing the mean square error between measured and modeled data. Redundancy in the estimation of media properties is given by spatial continuous measurements of the investigated media. This property is exploited in both the approaches investigated. In the layer-stripping approach the medium within each layer is homogeneous and the interfaces are assumed laterally continuous. In the inverse problem permittivity is assumed to be laterally smooth, implicit smoothing being given in the model parameterization. It is implicit in both methods that the inversion accuracy is strictly related to the amplitude stability of the radar and plane wave approximation. Therefore, the system calibration and the compensation of some propagation effects (e.g., near field, losses due to conductivity and to scattering from particles distributed between layers and on interfaces, pulse distortion) become crucial aspects for each specific application     

The Magnetotelluric Field for a Two-Dimensional Earth Modeled by a Nonuniform Buried Thin Sheet

Magnetotellurics (MT) is an electromagnetic sounding technique used in electrical geophysics to obtain information about the subsurface resistivity structure of the Earth. This information is then used to infer the presence of various natural resources such as oil accumulations and sources of geothermal energy. Extensive research has been done in the area of modeling and interpretation of the Earth in terms of one-dimensional Earth models. Difficulties in both modeling and interpretation occur when the conductivity structure of the Earth is allowed to vary laterally as well as vertically. This paper extends the analysis of a particular class of two-dimensional Earth models characterized by the "thin sheet" approximation. Integral equations are developed for the forward going problem of a plane wave normally incident on the surface of an Earth represented by a uniform half-space containing a thin sheet buried at an arbitrary depth and having an integrated conductivity that varies in one lateral direction. Both E perpendicular and E parallel polarization of the incident electric field are analyzed. The integral equations are evaluated numerically, and several example calculations are presented.     

Method for taking into account the shift parameter in the deconvolution of the depth composition distribution of semiconductor structures from SIMS depth profiles

A method for solving the direct and inverse problems of depth profiling in secondary-ion mass spectrometry is suggested. The advantages of solving the incorrect inverse problem in Fourier space with regularization by the Tikhonov method are discussed. Upon the reconstruction of element-concentration profiles, special attention is given to their shift as a feature of the SIMS resolution function. Consideration of the shift is achieved by joint solution of the direct and inverse problems of depth profiling. Examples of the operation of the deconvolution algorithm for both simulated and experimental profiles are given. It is shown that use of the proposed deconvolution algorithm makes it possible to increase the informativity and improve the depth resolution of the method. The suggested method for taking into account the shift enables us to avoid regular error in determining the position of thin layers near the surface.     

Electrical impedance tomography using the extended Kalman filter

In this paper, we propose an algorithm that, using the extended Kalman filter, solves the inverse problem of estimating the conductivity/resistivity distribution in electrical impedance tomography (EIT). The algorithm estimates conductivity/resistivity in a wide range. The purpose of this investigation is to provide information for setting and controlling air volume and pressure delivered to patients under artificial ventilation. We show that, when the standard deviation of the measurement noise level raises up to 5% of the maximal measured voltage, the conductivity estimates converge to the expected vector within 7% accuracy of the maximal conductivity value, under numerical simulations, with spatial a priori information. A two-phase identification procedure is proposed. A cylindrical phantom with saline solution is used for experimental evaluation. An abrupt modification on the resistivity distribution of this solution is caused by the immersion of a glass object. Estimates of electrode contact impedances and images of the glass object are presented.     

A Convex Optimization Approach for Depth Estimation Under Illumination Variation

Illumination changes cause serious problems in many computer vision applications. We present a new method for addressing robust depth estimation from a stereo pair under varying illumination conditions. First, a spatially varying multiplicative model is developed to account for brightness changes induced between left and right views. The depth estimation problem, based on this model, is then formulated as a constrained optimization problem in which an appropriate convex objective function is minimized under various convex constraints modelling prior knowledge and observed information. The resulting multiconstrained optimization problem is finally solved via a parallel block iterative algorithm which offers great flexibility in the incorporation of several constraints. Experimental results on both synthetic and real stereo pairs demonstrate the good performance of our method to efficiently recover depth and illumination variation fields, simultaneously.      

An adaptive multiscale inverse scattering approach to photothermal depth profilometry

Photothermal depth profilometry is formulated as a nonlinear inverse scattering problem. Starting with the one-dimensional heat diffusion equation, we derive a mathematical model relating arbitrary variation in the depth-dependent thermal conductivity to observed thermal wavefields at the surface of a material sample. The form of the model is particularly convenient for incorporation into a nonlinear optimization framework for is particularly convenient for incorporation into a nonlinear optimization framework for recovering the conductivity based on thermal wave data obtained at multiple frequencies. We develop an adaptive, multiscale algorithm for solving this highly ill-posed inverse problem. The algorithm is designed to produce an accurate, low-order representation of the thermal conductivity by automatically controlling the level of detail in the reconstruction. This control is designed to reflect both (1) the nature of the underlying physics, which says that scale should decrease with depth, and (2) the particular structure of the conductivity profile, which may require a sparse collection of fine-scale components to adequately represent significant features such as a layering structure. The approach is demonstrated in a variety of synthetic examples representative of nondestructive evaluation problems seen in the steel industry.The work of authors E. L. Miller and I. Yavuz was supported by a CAREER Award from the National Science Foundation MIP-9623721, an ODDR&E MURI under Air Force Office of Scientific Research contract F49620-96-1-0028, and the Army Research Office Demining MURI under grant DAAG55-97-1-0013. The work of authors L. Nicolaides and A. Mandelis was supported by a research contract from Material and Manufacturing Ontario (MMO).     

Optimal survey design using focused resistivity arrays

The first problem which needs to be solved when planning any geoelectrical survey is a choice of a particular electrode configuration that can give the maximal response from a target inhomogeneity. The authors formulate a problem of maximizing the response as an optimization problem for an applied current intensity distribution on the surface. The solution of this problem is the optimal intensity distribution of the current, which maximizes the response from the inclusion. This problem is solved numerically with singular value decomposition of an impedance matrix. The optimal current array is modeled as a current of varying optimal intensity injected at different electrodes. The problem does not need any information about the inclusion but its measured impedance matrix. Thus an optimal current array can be designed for every particular resistivity distribution. The optimal current patterns are found for a number of models of a conductive inclusion, and responses due to the optimal current are compared with responses due to conventional arrays. This method can be applied to any background and inclusion resistivity distribution     

Planetary exploration using a small electromagnetic sensor

A prototype broadband electromagnetic (EM) sensor, GEM-5, has been built and tested as a possible new probe for the future Mars rover to seek an ice-bonded layer at a given depth below the Martian surface. The sensor, with a vertical coaxial coil configuration, will measure the terrain resistivity and susceptibility to determine lateral variations in resistivity and magnetic susceptibility. The lateral variations will indicate regions of resistivity/susceptibility anomalies that may contain ice or water at depth. The forward solution for the sensor geometry over a layered formation and inverse algorithms to convert the EM data into the apparent susceptibility and resistivity are developed to investigate the ability of the sensor in detecting and resolving a buried (wet) ice layer in Mars-like geologic formations. Based on the simulated study, we find that the prototype sensor design should be able to resolve the lateral variations in resistivity/susceptibility under conditions of the Martian subsurface.     

Dependence of apparent resistance of four-electrode probes on insertion depth

The apparent resistance of a finite-thickness layer measured with a four-electrode plunge probe depends on the electrode insertion depth, electrode spacing, and layer thickness, as well as the resistivity ratio of an underlying layer. A physical model consisting of air, a saline solution layer, and an agar layer simulates the real situation of resistivity measurement. The saline layer represents the finite-thickness layer whose resistivity is to be measured by a plunge electrode probe, and the agar layer represents an underlying perturbing layer. A micropositioner controls the insertion depth of the four electrodes into the saline solution. With the apparent resistance measured on a semi-infinite-thickness layer of saline solution as standard, measurement results show decreasing apparent resistance and increasing error with increasing electrode insertion depth. This information is important for correct measurement of myocardial resistivity in vivo and in vitro.     

Absolute conductivity reconstruction in magnetic induction tomography using a nonlinear method

Magnetic induction tomography (MIT) attempts to image the electrical and magnetic characteristics of a target using impedance measurement data from pairs of excitation and detection coils. This inverse eddy current problem is nonlinear and also severely ill posed so regularization is required for a stable solution. A regularized Gauss-Newton algorithm has been implemented as a nonlinear, iterative inverse solver. In this algorithm, one needs to solve the forward problem and recalculate the Jacobian matrix for each iteration. The forward problem has been solved using an edge based finite element method for magnetic vector potential A and electrical scalar potential V, a so called A, A - V formulation. A theoretical study of the general inverse eddy current problem and a derivation, paying special attention to the boundary conditions, of an adjoint field formula for the Jacobian is given. This efficient formula calculates the change in measured induced voltage due to a small perturbation of the conductivity in a region. This has the advantage that it involves only the inner product of the electric fields when two different coils are excited, and these are convenient computationally. This paper also shows that the sensitivity maps change significantly when the conductivity distribution changes, demonstrating the necessity for a nonlinear reconstruction algorithm. The performance of the inverse solver has been examined and results presented from simulated data with added noise.     

3D-Object Space Reconstruction from Planar Recorded Data of 3D-Integral Images

The paper presents a novel algorithm for object space reconstruction from the planar (2D) recorded data set of a 3D-integral image. The integral imaging system is described and the associated point spread function is given. The space data extraction is formulated as an inverse problem, which proves ill-conditioned, and tackled by imposing additional conditions to the sought solution. An adaptive constrained 3D-reconstruction regularization algorithm based on the use of a sigmoid function is presented. A hierarchical multiresolution strategy which employes the adaptive constrained algorithm to obtain highly accurate intensity maps of the object space is described. The depth map of the object space is extracted from the intensity map using a weighted Durbin–Willshaw algorithm. Finally, illustrative simulation results are given.     

Infrared Thermography for Buried Landmine Detection: Inverse Problem Setting

This paper deals with an inverse problem arising in infrared (IR) thermography for buried landmine detection. It is aimed at using a thermal model and measured IR images to detect the presence of buried objects and characterize them in terms of thermal and geometrical properties. The inverse problem is mathematically stated as an optimization one using the well-known least-square approach. The main difficulty in solving this problem comes from the fact that it is severely ill posed due to lack of information in measured data. A two-step algorithm is proposed for solving it. The performance of the algorithm is illustrated using some simulated and real experimental data. The sensitivity of the proposed algorithm to various factors is analyzed. A data processing chain including anomaly detection and characterization is also introduced and discussed.      

改进的峰值检测反距离加权插值算法

采样频率限制和回波脉冲展宽是导致数字化激光脉冲测距峰值检测精度低的主要原因。传统的反距离加权插值算法只能解决低采样频率的问题,却无力解决回波脉冲展宽的问题。针对该问题,根据回波时间能量分布模型,从采样得到的峰值位置分别向上升沿和下降沿方向搜索,各自对搜索半径内的采样点按照距离远近赋予不同权重和插值,然后加权平均提取修正后的峰值时刻。该改进算法有效地解决了回波展宽的问题,减小了低采样频率和回波脉冲展宽带来的测量误差,通过实验论证了算法的有效性。     

The induced magnetic dipole of a polarized sphere

A homogeneous spherical body with a frequency dependent complex resistivity is placed in a uniform time varying magnetic field. It is shown that the induced magnetic dipole has a complex form that depends on the ratio of the diameter to the internal skin depth in a customary fashion. But the frequency dependence of the induced dipole is also influenced by the polarization properties of the material. A Cole-Cole form for the complex resistivity is employed to characterize the electrochemical polarization.     

J-substitution algorithm in magnetic resonance electrical impedance tomography (MREIT): phantom experiments for static resistivity images

Recently, a new static resistivity image reconstruction algorithm is proposed utilizing internal current density data obtained by magnetic resonance current density imaging technique. This new imaging method is called magnetic resonance electrical impedance tomography (MREIT). The derivation and performance of J-substitution algorithm in MREIT have been reported as a new accurate and high-resolution static impedance imaging technique via computer simulation methods. In this paper, we present experimental procedures, denoising techniques, and image reconstructions using a 0.3-tesla (T) experimental MREIT system and saline phantoms. MREIT using J-substitution algorithm effectively utilizes the internal current density information resolving the problem inherent in a conventional EIT, that is, the low sensitivity of boundary measurements to any changes of internal tissue resistivity values. Resistivity images of saline phantoms show an accuracy of 6.8%-47.2% and spatial resolution of 64 x 64. Both of them can be significantly improved by using an MRI system with a better signal-to-noise ratio.     

An automatization of Barnsley''s algorithm for the inverse problem of iterated function systems

We present an automatization of Barnsley's manual algorithm for the solution of the inverse problem of iterated function systems (IFSs). The problem is to retrieve the number of mappings and the parameters of an IFS from a digital binary image approximating the attractor induced by the IFS. M.F. Barnsley et al. described a way to solve manually the inverse problem by identifying the fragments of which the collage is composed, and then computing the parameters of the mappings (Barnsley et al., Proc. Nat. Acad. Sci. USA, vol.83, p.1975-7, 1986; Barnsley, "Fractals Everywhere", Academic, 1988; Barnsley and Hurd, L., "Fractal Image Compression", A.K. Peters, 1992). The automatic algorithm searches through a finite set of points in the parameter space determining a set of affine mappings. The algorithm uses the collage theorem and the Hausdorff metric. The inverse problem of IFSs is related to the image coding of binary images. If the number of mappings and the parameters of an IFS, with not too many mappings, could be obtained from a binary image, then this would give an efficient representation of the image. It is shown that the inverse problem solved by the automatic algorithm has a solution and some experiments show that the automatic algorithm is able to retrieve an IFS, including the number of mappings, from a digital binary image approximating the attractor induced by the IFS.     

Theoretical and computational aspects of 2-D inverse profiling

The authors discuss two techniques for solving two-dimensional (2D) inverse scattering problems by parameterizing the scattering configuration, and determining the optimum value of the parameters by minimizing a cost function involving the known scattered-field data. The computation of the fields in each estimated configuration is considered as an auxiliary problem. To improve the efficiency of these computations, the CGFFT iterative scheme is combined with a special extrapolation procedure that is valid for problems with a varying physical parameter such as frequency, angle of incidence, or contrast. Further, they analyze the dynamic range and the resolution of linearized schemes. To obtain an acceptable resolution for an object with a large contrast with respect to the surrounding medium, multiple-frequency information is used. Finally, the availability of a fast-forward solver was an incentive to consider nonlinear optimization. In particular, the authors use a quasi-Newton algorithm at only twice the computational cost of the distorted-wave Born iterative scheme