Born Inversion Research Articles

Born-Inversion Procedure for Shape Reconstruction of Eccentric Defects in Cylindrical Components

In this paper, a cylindrical aluminum specimen with an eccentric circular hole is prepared and ultrasonic measurements are carried out by experimental means. The measurement area is restricted to the plane perpendicular to the axis of the cylindrical component. The measured wave data are fed into the approximate correction method formula—the Born inversion procedure—and cross-sectional images are obtained. Next, a 3D shape reconstruction of the defect in the aluminum specimen is performed by stacking the cross-sectional images. After correcting the defect’s echo amplitude, the defect reconstruction effect of the 2D section and 3D defect reconstruction effect improves remarkably.

Angle-domain generalized Radon transform for elastic multiparameter inverse scattering inversion

Linearized algorithms based on the Born approximation are well-known and popular techniques for quantitative seismic imaging and inversion. However, linearization methods usually suffer from some significant problems, such as the computational cost for the required number of iterations, requirement for background models, and uncertain and unstable multiparameter extraction, which make the methods difficult to implement in practical applications. To avoid these problems, we have developed an angle-domain generalized Radon transform (AD-GRT) inversion in 2D elastic isotropic media. This AD-GRT is an approximate transform between the seismic data and an angle-domain model, which acts as a scattering function, and the seismic data can be reconstructed accurately, even when the background models are incorrect. The density and Lamé moduli perturbation parameters can be extracted stably from the inverted angle-domain scattering function. Deconvolution of the source wavelet is taken into account to remove the effect of the wavelet and improve the resolution and accuracy of the inversion results. The derived AD-GRT inversion is noniterative and is as efficient as the traditional elastic GRT method. The additional dimension of the angle domain has little effect on the computational cost of the AD-GRT, as opposed to other extended-domain inversion/migration methods. Our method also can be used to solve nonlinear Born inversion problems using iteration, which can significantly improve their convergence rate. Three numerical examples illustrate that the angle-domain scattering function inversion, data reconstruction, and multiparameter extraction using the presented AD-GRT inversion are effective.

Lippmann–Schwinger–Lanczos algorithm for inverse scattering problems

Data-driven reduced order models (ROMs) are combined with the Lippmann–Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann–Schwinger equation, thus making further iterative updates unnecessary. We show numerical experiments for spectral domain domain data for which our inversion is far superior to the Born inversion and works as well as when the true internal solution is known.

A Bayesian-based approach to improving acoustic Born waveform inversion of seismic data for viscoelastic media

In seismic waveform inversion, the reconstruction of the subsurface properties is usually carried out using approximative wave propagation models to ensure computational efficiency. The viscoelastic nature of the subsurface is often unaccounted for, and two popular approximations—the acoustic and linearized Born inversion—are widely used. This leads to reconstruction errors since the approximations ignore realistic (physical) aspects of seismic wave propagation in the heterogeneous Earth. In this study, we show that the Bayesian approximation error approach can be used to partially recover from errors, addressing elastic and viscous effects in acoustic Born inversion for viscoelastic media. The results of numerical examples indicate that neglecting the modeling errors induced by the approximations results in very poor recovery of the subsurface velocity fields.

An alternative formula for approximate extended Born inversion

Various modifications of reverse time migration (RTM) provide asymptotic inverses to the subsurface offset extended Born modeling operator for constant-density acoustics. These approximate inverses have the same quality (asymptotic accuracy) as do generalized Radon transform pseudoinverses, but they can be computed without any ray tracing whatsoever. We have developed an approximate inverse of this type whose additional computational cost, above that of subsurface offset extended RTM, is negligible.

An approximate inverse to the extended Born modeling operator

Given a correct (data-consistent) velocity model, reverse time migration (RTM) correctly positions reflectors but generally with incorrect amplitudes and wavelets. Iterative least-squares migration (LSM) corrects the amplitude and wavelet by fitting data in the sense of Born modeling, that is, replacing migration by Born inversion. However, LSM also requires a correct velocity model, and it may require many migration/demigration cycles. We modified RTM in the subsurface offset domain to create an asymptotic (high-frequency) approximation to extended LSM. This extended Born inversion operator outputs extended reflectors (depending on the subsurface offset and position in the earth) with correct amplitude and phase, in the sense that similarly extended Born modeling reproduces the data to good accuracy. Although the theoretical justification of the inversion property relies on ray tracing and stationary phase, application of the weight operators does not require any computational ray tracing. The computational expense of the extended Born inversion operator is roughly the same as that of extended RTM, and the inversion (data-fit) property holds even when the velocity is substantially incorrect. The approximate inverse operator differes from extended RTM only in the application of data- and model-domain weight operators, and takes the form of an adjoint in the sense of weighted inner products in data and model space. Because the Born modeling operator is approximately unitary with respect to the weighted inner products, a weighted version of conjugate gradient iteration dramatically accelerates the convergence of extended LSM. An approximate LSM may be extracted from the approximate extended LSM by averaging over subsurface offset.

Shape reconstruction of mixed type void flaws using Born inversion method

We study an approximate technique for determining the shapes of mixed type void flaws in elastic media from knowledge of the ultrasonic scattering amplitudes. It is well known that the technique is highly effective under weak scattering conditions. In the paper, two cement paste cylindrical specimens with mixed type void flaws are prepared and ultrasonic measurements are carried out by experimental means. The measurement area is restricted in the plane perpendicular to the axis of cylindrical specimen. The measured wave data are fed into the approximate technique formula—the Born inversion method and cross-sectional image is obtained. We find that good results have been obtained for strong scattering void flaws such as mixed type void flaws in cement paste cylindrical specimens.

Improving the Performances of the Contrast Source Extended Born Inversion Method by Subspace Techniques

Subspace techniques have been introduced in the framework of contrast source (CS) extended born (CSEB) model, for improving its reconstruction capabilities. Two techniques are demonstrated. First, a scheme for generating a good initial guess of the scatterer profile is shown. Second, subspace-based optimization method is used for optimization. Using the suggested techniques, CSEB model can be applied for solving inverse electromagnetic scattering problem with an extended range of application with respect to previous contributions, particularly for very high contrast lossy scatterers.

Experimental Research on Shape Reconstruction of Eccentric Circular Cylindrical Flaw in Inhomogeneous Material

In this article, three dimensional Born inverse scattering method is modified to convenient form to reconstruct the shape of a three-dimensional flaw in an inhomogeneous cylindrical specimen. In this modified method, a measurement plane is restricted to the plane perpendicular to the axis of the inhomogeneous cylindrical specimen. Thus the cross-sectional image of the flaw can be obtained. By moving the measurement plane along the axis of the inhomogeneous cylindrical specimen, the cross-sectional image is obtained for each measurement plane. The three-dimensional flaw image is reconstructed by piling up the obtained cross-sectional images. Inhomogeneous cylindrical specimen with an eccentric circular cylindrical cavity model is prepared. The performance of the modified method to reconstruct the three-dimensional flaw is verified by using the experimentally measured waveforms. From the experimental research for the eccentric circular cylindrical defect, it is verified that the modified Born inversion works well for the volume type defect.

3D microwave imaging via preliminary support reconstruction: testing on the Fresnel 2008 database

In the solution of inverse scattering problems, support information is useful in order to reduce the degree of nonlinearity of the relationship among the data and the unknowns, as well as the overall computational burden. Moreover, by taking into account the spectral properties of the scattered fields, it can be helpful to reduce the effect of noise on data. Accordingly, a 3D inversion strategy in which one performs a preliminary estimation of the support of the targets is considered in this paper to process the experimental data provided by the Institut Fresnel of Marseille, France. In particular, the imaging problem is tackled in a step-wise fashion, in which one first pursues a morphologic characterization of the targets using a linear sampling method based procedure. Then, the retrieved support is exploited to ‘filter’ the data and, finally, in the quantitative characterization of the targets. This last step is performed by using the optimized contrast source extended Born inversion method wherein the estimated support is exploited in order to reduce both the investigated domain and the nonlinearity of the inverse problem.

Lineared Inverse Scattering Methods in Elasticity for Shape Reconstruction of Defects

Abstract Two linearized methods for the ultrasonic inversion are formulated for the shape reconstruction of defects in an elastic material. The methods are based on the elastodynamic inverse Born and Kirchhoff approximations. From the numerical analysis for two kinds of defects, it is verified that the Born inversion works well for volumetric defects and the Kirchhoff inversion for surface defects. Combination of two inversion methods leads to the separation of surface in a part from the volumetric part of the defect.

Noise propagation in linear and nonlinear inverse scattering

The propagation of noise from the data to the reconstructed speed of sound image by inverse scattering within the framework of the Lippmann-Schwinger integral equation of scattering is discussed. The inversion algorithm that was used consisted in minimizing a Tikhonov functional in the unknown speed of sound. The gradient of the objective functional was computed by the method of the adjoint fields. An analytical expression for the inverse scattering covariance matrix of the image noise was derived. It was shown that the covariance matrix in the linear x-ray computed tomography is a special case of the inverse scattering matrix derived in this paper. The matrix was also analyzed in the limit of the linearized Born approximation, and the results were found to be in qualitative agreement with those recently reported in the literature for Born inversion using filtered backpropagation algorithm. Finally, the applicability of the analysis reported here to the obstacle problem and the physical optics approximation was discussed.

Testing the contrast source extended Born inversion method against real data: the TM case

With reference to the case of TM data, in this paper we test the reconstruction capabilities of a recently proposed inverse scattering approach: the contrast source extended Born (CS-EB) inversion method. By relying on a suitable rewriting of the scattering equations, the CS-EB is claimed to be particularly effective whenever losses are present in the background medium and/or in the unknown scatterers. Here, by taking advantage of the Marseille data set concerning lossless inhomogeneous targets in free space, we show the effectiveness of the CS-EB inversion method also in the lossless case. In order to take advantage in a simple fashion of the available multifrequency information, data are processed within a frequency hopping scheme and suitable regularization schemes are exploited at each frequency. Moreover, in order to reduce the computational burden as well as the overall difficulty, the limited bandwidth properties of scattered fields are exploited by means of some simple processing to achieve a preliminary estimation of the region wherein the scatterers are located.

The diagonalized contrast source approach: an inversion method beyond the Born approximation

This paper deals with the imaging and inversion of the constitutive material properties of bounded objects embedded in a known background medium. The inversion utilizes measurements of the scattered field due to the illumination of the objects by a set of known single-frequency wave fields. We present two inverse scattering methods that approximately recast the full nonlinear inversion into a number of linear inversion steps. The two methods are as computationally efficient as the constrained Born inversion but provide reconstruction results that are far superior and are almost comparable in quality to the ones obtained using a full nonlinear iterative inversion. The linear inversion steps follow what is referred to as the source-type integral equation approach in which the contrast sources are inverted for in the first step from the linear data equation. In this step, we employ a local (diagonal) approximation of the operator that relates the contrast sources to the incident field. Once the contrast sources have been determined, the total internal wave fields follow from a direct application of the object equation. In the third and final step, the contrast function is estimated from either the constitutive relation (first method) or from solving the data equation again, this time in terms of the contrast profile (second method). The computational cost required by the first method is comparable to that of a constrained Born inversion. With only twice the computational cost, the second method invariably gives an almost perfect image. We will demonstrate the two methods for a number of representative synthetic examples, both in two and three dimensions.

超音波を用いた鉄筋形状および付着評価のための一解析法

The shape and location of reinforcing bars in the structural material are reconstructed from ultrasonic scattered waveforms in the low frequency. Two linearized inverse scattering methods are investigated for the imaging of reinforcing bars. The Born and Kirchhoff inversions reconstruct domains and boundaries of the bars, respectively. Although the dynamic interaction effect exists in the adjacent part of multiple bars, we can reconstruct the whole shape of bars with the Born inversion. On the other hand, the Kirchhoff inversion leads to the classification of the damaged area around a reinforcing bar. These performances of proposed imaging methods are demonstrated by numerical simulations using the boundary element method.

Forward Modeling and Inversion of 3‐D Cross‐hole Electromagnetic Fields

AbstractWe propose a method of modified local‐nonlinear iteration (MLNI) for volume integral equations, and use it to make calculations for forward and inverse problems of 3‐D cross‐hole electromagnetic fields. The large‐scale scatters are divided into two parts: the near region and the far region, whose location and size are determined by the position of the field point. The influence of the near region is computed by the method of local‐nonlinear approximation, while that of the far region is treated as an external exciting source and is calculated reiteratively. The method has the advantage of faster calculating speed, less computer memory requirements and more efficient convergency. The nonlinear inversion algorithm based on MLNI is applied during inversion, which can concentrate the imaging region not to the whole 3‐D domain but to a particular range. Since the algorithm considers the effect of the nonlinear term in each element of the Jacobi matrix, it is more accurate and stable comparing with the conventional iterative Born inversion method. The double cell decomposition method is applied during the first rough imaging process considering the limitation of the information and computer memory, then the imaging range is reduced and the second more accurate imaging process can be carried out. Numerical results indicate that MLNI is an efficient method to compute the scattering fields of large‐scale scatters and that the application of the method to inversion can give high‐resolution 3‐D cross‐hole conductivity images.

Elastodynamic inversion for shape reconstruction and type classification of flaws

Two linearized inverse scattering methods are investigated to reconstruct the shape of flaws in the elastic solid. One is based on the Born approximation and the other is based on the Kirchhoff approximation. The Born inversion is sensitive to the volumetric flaw but not to a crack-like flaw. On the other hand, the Kirchhoff inversion reacts to both boundaries of volumetric and crack-like flaws. The combined use of Born and Kirchhoff inversions leads to the classification method of flaw type. The performance of the proposed classification procedure is demonstrated by the numerical simulations and then by the experimental measurements for the two-dimensional models of flaws. An example for the three-dimensional shape reconstruction is also shown by using the numerically calculated backscattering amplitudes.

板内部の欠陥に対する逆散乱解析

A linearized inverse scattering method based on the Born approximation has been developed for the reconstruction of a defect shape in an elastic solid. In order to fully recover the shape of the defect, the Born inversion approach requires that the information must be measured at all frequencies and from all aspect-angles with respect to the scatterer. For a defect in a plate, however, the data cannot be measured at all directions and the complete information is seldom available.In this paper, the aspect-limited problem of the Born inversion is investigated for the inverse scattering of antiplane waves. After the formulation of the Born inverse method is described, the method is applied to numerically simulated data with limited aspect-angles for a defect in an infinite domain as well as for a defect in a plate. It is found that the illuminated portion of the defect can be reconstructed even using a limited amount of data. The defect shape in a plate can, therefore, be recovered very well by utilizing the reflected waves at the bottom surface.Pulse echo ultrasonic experiments using SH probes are conducted for a cylindrical cavity in a steel block. The Born inversion is applied to the data which include the direct back scatter by the cavity and to the data which include the back scatter due to reflection at the bottom of the steel block. It is found that the reconstructed result when both sets of data are utilized is far superior than those reconstructed by the individual data. It is also shown that even with the limited amount of data, the Born inversion can produce much better image of the defect than the aperture synthetic approach can.

Linearized Inverse Scattering Methods for Classification of Flaw Type(NDE3)

A classification method of the flaw type is discussed by utilizing the ultrasonic scattering amplitudes from flaws in an elastic body. The method is based on the combination of two linearized inverse scattering methods of Born and Kirchhoff inversions. The Born inversion is sensitive to the volumetric flaw but not to a crack-like flaw. The Kirchhoff inversion reacts to both surfaces of volumetric and crack-like flaws. The combined use of the Born and Kirchhoff inversions leads to the classification of flaw type. To show the versatility of the method, aluminum specimens that contain a circular cavity and artificial cracks are prepared, and the backscattered waveforms are acquired from the experimental measurement. The measured waveforms are processed and the waveforms are fed into the Born and Kirchhoff inversions. The results show that the crack-like flaw can be discriminated from the volumetric flaw. It is also shown that the scattered wave component in the low frequency range is important for the present classification method.

Ghosts in the Born images of a layer probed by acoustic waves

The Born inversion of time records of acoustic pulses reflected from a layer object gives rise to the images with ghost-like artifacts that make it difficult to pick out the exact thickness and composition of the object. The origin of these ghosts is explained herein and a means is proposed for their suppression.