遗传算法和多重网格在两步线性反演中的应用

基于扩展玻恩近似和电场积分方程,建立起非线性反演方程,然后应用两步线性反演方法进行迭代反演.在反演过程中,采用多重网格技术将待反演区域的离散网格从粗网格逐渐推演到细网格.在迭代前期,待反演区被离散为粗网格,利用实数编码的遗传算法对病态方程进行优化反演,并将其迭代反演结果作为迭代后期的初始值.在迭代后期,待反演区被离散成细网格,则采用共轭梯度(CG)法求解未知量较多的矩阵方程.二维轴对称非均匀介质剖面分布的反演结果表明,本文方法可反演高对比度的散射目标,具有较好的稳定性.     

A multiparameter iterative inversion of dual-laterolog inhorizontally layered media and its error analysis

The expression and algorithm of the Born approximation of dual-laterolog (DLL) are different from nonfocusing resistivity tools because the currents and potentials on the surfaces of the electrodes must be satisfied with the special focusing conditions. The Born approximation is described by a linear integral equation about the changes in current density, bucking currents, and potentials on the surfaces of all the electrodes, caused by small perturbations in the formation conductivity. Using the complex forward modeling of DLL, and the semi-analytic solution of the Green's function, we advance the fast algorithm of the Frechet derivative matrix containing the partial derivatives of both the apparent resistivities and bucking currents with respect to all components of the model vector. A normalization approach is used to transform the matrix into a dimensionless matrix. We then develop a multiparameter iterative inversion technique to simultaneously reconstruct all model parameters. Furthermore, we analyze the influence of both noises in data and errors of some model parameters on the inversion solution and give a method to choose the regularization factors. Finally, numerical result tests show the characteristics of the normalized Frechet matrix, the influence of errors of both bed thickness and flushed zone resistivity, etc, on the inversion solution when they are fixed, and the great improvement of the inversion quality when all model parameters are simultaneously reconstructed from both the apparent resistivities and bucking currents     

低频电磁场重建二维非均匀介质的有效方法

将玻恩迭代和变形玻恩迭代两种方法用于低频二维非均匀介质电导率的重建。利用上述两种方法求解非线性积分方程。用正则化方法对解加上约束克服重建问题中解的非唯一性和不稳定性。数值计算表明，两种迭代方法都可用于二维非均匀介质电导率的重建。同玻恩迭代法相比，变形玻恩迭代法显示出更快的收敛性，而同变形玻恩迭供法相比，玻恩迭代法的抗噪声能力更强。     

Efficient Numerical Simulation of Axisymmetric Electromagnetic Induction Measurements Using a High-Order Generalized Extended Born Approximation

This paper develops a high-order generalized extended Born approximation (Ho-GEBA) for the numerical simulation of electromagnetic scattering due to rock formations that exhibit axial symmetry around a wellbore. The resulting equations of Ho-GEBA are solved with a numerical procedure that is as efficient as the extended Born approximation (EBA). With the acceleration of a fast Fourier transform, the operation count is proportional to$O(CN)$, where$N$is the total number of spatial discretization cells and$Cll N$is a constant that depends on the number of discretization cells in the radial direction. Ho-GEBA remains accurate in the near-source scattering region and accounts for multiple scattering in the presence of large conductivity contrasts and relatively large frequencies. Numerical exercises conclusively indicate that the accuracy of Ho-GEBA is superior to that of EBA and the first-order Born approximation while maintaining the same level of algorithmic efficiency. These exercises are carried out on a variety of source–receiver configurations and frequencies typical of single-well borehole induction measurements.     

Electromagnetic conductivity imaging with an iterative Borninversion

A technique which uses low frequency electromagnetic fields to image the Earth's conductivity structure between two boreholes is developed. The scheme employs the Born approximation to linearize the integral equation formulation, and a regularized least squares method to invert the data for an initial estimate of the conductivity. A second Born approximation is then applied in a iterative manner to achieve better estimates of the structure. Numerical experiments show that this iterative method improves resolution when compared to normal Born inversion due to the fact that coupling between scattering currents is accounted for. The image quality is shown to be dependent on the operating frequency with the optimal frequency corresponding to the peak of a sensitivity function the authors have termed the Born kernel, which is the product of the primary electric field and the Green's function     

用变形玻恩迭代法反演电导率的二维非均匀分布

本文把变形玻恩迭代方法用于求解二维轴对称逆散射问题。该方法提供一种迭代收敛较快的反演成像算法。利用对称轴上２０ｋＨｚ电场的测量值在二维非均匀介质中对电导率分布进行反演。首先给出对于未知电导率分布的非线性积分方法，并用玻恩近似使积分方程线性化，然后用吉洪诺夫与正则化方法求出电导率分布。在迭代过程中，数值模式匹配法用于求解正演场．数值实例表明，用简单的对称轴上测量场能得到好的好的成像结果。     

The hybrid extended Born approximation and CG-FFT method forelectromagnetic induction problems

The authors propose the hybridization of the extended Born approximation (EBA) with the conjugate-gradient fast Fourier transform (CG-FFT) method to improve the efficiency of numerical solution of electromagnetic induction problems. This combination improves the solution efficiency in two ways. First, using the FFT in the extended Born approximation decreases the computational cost of the conventional EBA method from O(N²) to O(N log₂ N) arithmetic operations, where N is the number of unknowns in the problem. This approach, referred to as the FFT-EBA method, applies to problems with a fairly large contrast. Secondly, using the EBA as a partial preconditioner for the CG-FFT method increases the convergence speed of the conventional CG-FFT method. This second approach, referred to as the EBA-CGFFT method, is in principle applicable to all problems with a homogeneous background, but is particularly efficient for problems with a higher contrast. Numerical experiments suggest that the combination of these two methods is more accurate and more efficient for electromagnetic induction problems     

High-order generalized extended Born approximation for electromagnetic scattering

Previous work on the subject of electromagnetic scattering has shown that the extended Born approximation (EBA) is more accurate than the first-order Born approximation with approximately the same operation count. However, the accuracy of the EBA degrades in cases when the source is very close to the scatterer, or when the electric field exhibits significant spatial variations within the scatterer. This paper introduces a generalized extended Born approximation (GEBA) and its high-order variants (Ho-GEBA) to efficiently and accurately simulate electromagnetic scattering problems. We make use of a generalized series expansion of the internal electric field to construct high-order terms of the generalized extended Born approximation (Ho-GEBA). A salient feature of the Ho-GEBA is its enhanced accuracy over the Born approximation and the EBA, even when only the first-order term of the series expansion is considered in the approximation. This behavior is not conditioned by either the source location or the spatial distribution of the internal electric field. A unique feature of the Ho-GEBA is that it can be used to simulate electromagnetic scattering due to electrically anisotropic media. Such a feature is not possible with approximations of the internal electric field that are based on the behavior of the background electric field. Three-dimensional (3-D) models of electromagnetic scattering are used to benchmark the efficiency and accuracy of the Ho-GEBA, including comparisons against the first-order Born approximation and the EBA.     

Nonlinear inversion of electrode-type resistivity measurements

Deals with the inversion of low-frequency electrode-type resistivity measurements for the conductivity distribution in a two-dimensional axisymmetric medium. It is well known that the inversion of such transverse magnetic measurements is much more nonlinear than that of transverse electric measurements. The distorted Born iterative method (DBIM) is applied to solve the nonlinear inverse problem. In each iteration of the DBIM, an efficient numerical mode-matching (NMM) method is used as a forward solver. In addition to its efficiency in solving for the predicted data, the NMM method gives a semianalytic expression for the partial derivatives of the Green's function required in the inversion. Several numerical results are presented to demonstrate the applications of the DBIM, and to address several practical issues related to the performance of the nonlinear inversion scheme. Because of the fast forward modeling and semianalytic Green's function available due to the NMM method, the inversion is fast and is practical for the interpretation of measurement data     

轴对称二维非均匀介质的快速反演方法

首先采用变分原理和非线性积分方程推导出反演方程，然后利用玻恩迭代法对轴对称二维非均匀介质分布进行了反演。通过计算实例表明，该方法的收敛速度比玻恩迭代法快得多，在迭代前期与变形玻恩迭代法相当，而在迭代后期比变形玻恩迭代法的收敛速度更快；同时，每次迭代所需时间虽然多于玻恩迭代法，但明显少于变形玻恩迭代法。该方法同时具备了玻恩和变形玻恩迭代法两者的优点。     

基于阵列信息的二维轴对称电导率分布的有效反演方法

本文基于实际工程应用中的阵列感应测井仪（ＡＩＴ）的测量信息,利用变分坡昂迭代法（ＶＢＭ）,在非均匀背景介质中来重构和反演地层的电导率剖面。该方法基于非线性积分方程利用变分方法来建立反演方程,在反演迭代过程中,非均匀剪影介质中的格林函数无须更新,与变形玻昂迭代法（ＤＢＩＭ）相比其计算复杂性大大降低,文中仅利用沿井轴的ＡＩＴ响应对地层电导率进行反演,其胍演结果与起初地层电导率分布吻合的较好。     

基于积分方程的低数据量的快速反演

本文将变形玻昂迭代法（ＤＢＩＭ）并结合数值模式匹配法（ＮＭＭ）用于实际工程中的深感应测井仪（６ＦＦ４０）的低信息量的反演和剖面成像。在反演迭代过程中,用Ｔｉｋｈｏｎｏｖ正则化求解病态问题,利用ＮＭＭ解的ｚ向解析性质,避免了双重数值积分,同时提高了计算效率和反演质量。文中利用不完备的低信息量工程数据对地层电参数进行了反演,表明该方法实际工程应用的潜力。     

Effect of Nanoparticles on Electron and Thermoelectric Transport

Recent experimental results have shown that adding nanoparticles inside a bulk material can enhance the thermoelectric performance by reducing the thermal conductivity and increasing the Seebeck coefficient. In this paper we investigate electron scattering from nanoparticles using different models. We compare the results of the Born approximation to that of the partial-wave method for a single nanoparticle scattering. The partial-wave method is more accurate for particle sizes in the 1 nm to 5 nm range where the point scattering approximation is not valid. The two methods can have different predictions for the thermoelectric properties such as the electrical conductivity and the Seebeck coefficient. To include a random distribution of nanoparticles, we consider an effective medium for the electron scattering using the coherent potential approximation. We compare various theoretical results with the experimental data obtained with ErAs nanoparticles in an InGaAlAs matrix. Reasonably good agreement is found between the measured and theoretical electrical conductivity and Seebeck data in the 300 K to 850 K temperature range.     

Microwave Subsurface Imaging Using Direct Finite-Difference Frequency-Domain-Based Inversion

We have developed a new algorithm for electromagnetic inverse scattering problems in inhomogeneous media using finite-difference frequency-domain (FDFD) forward modeling, referred to as the FDFD-based inversion method. The key issue of this method is to build a linear expression for the inverse problem from an FDFD forward model by using Born approximation to neglect mutual coupling between scattered pixels and to then solve for the inverse coefficient matrix. An important advantage of this matrix-based method is that there is no need to specify a Green's function. As such, this inverse scattering algorithm is easily implemented and is robust to the heterogeneity in the background. We test the algorithm with a microwave subsurface object detection application using cross-well radar. The new method is compared with conventional inversion using Green's function-based Born approximation. Numerical experiments are presented for a 2-D borehole geometry for buried object detection in uniform soil and in multilayered soil backgrounds.      

Inversion of induction tool measurements using the distorted Borniterative method and CG-FFHT

The inversion of induction tool measurements using the distorted Born iterative method (DBIM) and the conjugate gradient-fast Fourier-Hankel transform (CG-FFHT) is described. The inverse problem is formulated in terms of an integral equation of scattering where the unknown to be sought is the conductivity in the rock formation, when the measurements along a borehole axis are performed. The nonlinear problem is linearized at each stage using the distorted Born approximation. The inhomogeneous medium Green's function in the distorted Born approximation is found by solving a volume integral equation using the CG-FFHT method, which allows a rapid solution to a large problem with reduced computational complexity and memory requirement. In this manner, the inverse problem is solved with a computational complexity proportional to N_tlN log N where N_tl is the number of transmitter locations used in the data collection and N is the total number of pixels used to model the unknown formation. The memory requirement is of order NN_tl     

WKB近似下的Fourier衍射成象方法

对于介质目标微波衍射成象，本文引入了ＷＫＢ近似来模拟目标内部总场。基于这种近似，我们导出了Ｆｏｕｒｉｅｒ衍射公式，并采用了广义滤波逆传播方法由目标空间谱实现目标特性的重建。计算机模拟结果表明采用ＷＫＢ近似重建目标特性较Ｂｏｒｎ近似有明显改善。     

A robust iterative method for Born inversion

We present a robust iterative method to solve the inverse scattering problem in cases where the Born approximation is valid. We formulate this linearized inverse problem in terms of the unknown material contrast and the unknown contrast sources and we solve the problem by minimizing a cost functional consisting of two terms. The first term represents the differences between the actual data and the modeled data, while the second term represents the misfit in the constitutive relations between the contrast sources and the incident fields. In each iteration, the contrast sources and the contrast are reconstructed alternatingly, using subsequently a conjugate gradient step for the contrast source updates and a direct inversion of a diagonal matrix for the contrast. A further regularization with a multiplicative regularization factor is discussed. In this regularization procedure the relative variation of the contrast is minimized as well. As a test case we consider the two-dimensional (2-D) transverse magnetic polarization problem. Synthetic numerical examples are presented in order to compare the presented algorithm to the traditional Born algorithm. Results with respect to the inversion of experimental data are presented as well. In addition, some inversion results for the subsurface sensing problem, both in two and three dimensions, are presented.     

A new approximation and a new measurable constraint for slabprofile inversion

A new approximation and a new measurable constraint for slab profile inversion is introduced. A comparison is given between the Born, the exact, and the new approximation. The new approximation and constraint can be utilized along with other methods such as the integral equation, the iterative or perturbed Born, and the alternating projection approaches. The latter is considered. The results obtained are superior to those obtained under the Born approximation     

Study of some practical issues in inversion with the Born iterativemethod using time-domain data

The performance of the Born iterative method of nonlinear two-dimensional profile inversion is examined for the reconstruction of large objects and in the presence of measurement noise. Time-domain data are used. It is shown that objects at least as large as about nine wavelengths can be inverted without any convergence problems. The algorithm is shown to perform well in the presence of 10% noise, or 20-dB signal-to-noise ratio. The simultaneous inversion of permittivity and conductivity profiles is formulated and solved using the Born iterative method. Objects with various loss tangents are reconstructed, and the limits of applicability of the algorithm are investigated     

Multiscale,Statistical Anomaly Detection Analysis and Algorithms for Linearized Inverse Scattering Problems

In this paper we explore the utility of multiscale and statistical techniques for detecting and characterizing the structure of localized anomalies in a medium based upon observations of scattered energy obtained at the boundaries of the region of interest. Wavelet transform techniques are used to provide an efficient and physically meaningful method for modeling the non-anomalous structure of the medium under investigation. We employ decision-theoretic methods both to analyze a variety of difficulties associated with the anomaly detection problem and as the basis for an algorithm to perform anomaly detection and estimation. These methods allow for a quantitative evaluation of the manner in which the performance of the algorithms is impacted by the amplitudes, spatial sizes, and positions of anomalous areas in the overall region of interest. Given the insight provided by this work, we formulate and analyze an algorithm for determining the number, location, and magnitudes associated with a set of anomaly structures. This approach is based upon the use of a Generalized, M-ary Likelihood Ratio Test to successively subdivide the region as a means of localizing anomalous areas in both space and scale. Examples of our multiscale inversion algorithm are presented using the Born approximation of an electrical conductivity problem formulated so as to illustrate many of the features associated with similar detection problems arising in fields such as geophysical prospecting, ultrasonic imaging, and medical imaging.