Extended Born Approximation Research Articles

Seismic imaging of subsurface geological structures by Kirchhoff’s migration based on extended Born approximation

Complex seismic signatures are generated because of the multifaceted nature of the subsurface. These features make the interpretation very complex. To understand the seismic behaviour, different numerical tools are available. In this present study, an attempt has been made to demonstrate both the modelling and imaging aspects of these complex subsurface features commonly encountered in petroleum exploration. The present work is an extended form of the Born approximation by using Green’s function based asymptotic ray theory. Subsequently, Kirchhoff’s depth migration has been applied to generate seismic shot gathers over structural as well as stratigraphic traps. From this analysis, it is observed that the technique is able to efficiently migrate both the structural and stratigraphic traps. The proposed technique also intends to handle strong velocity variation and amplitude restoration. However, some noise in terms of over-critical reflection has been observed in the depth migrated section corresponding to pinch-out and unconformity respectively.

Super-Resolution Ultrasound Imaging Based on a Fast Full-Waveform Solver

We propose a fast 2D full-waveform forward and inverse scattering solver to reconstruct ultrasonic speed and attenuation properties. The forward method is based on the volume integral equation, accelerated by the extended Born approximation and biconjugate-gradient fast Fourier transform method. The inverse method is based on the extended contrast source inversion. The research results show that our proposed method can fully unravel multiple scattering effects and achieve sub-wavelength resolutions. Numerical experiments indicate that the efficiency and robustness of the forward and inverse methods.

High-order generalized extended Born approximation algorithm for 3D electromagnetic responses modeling in anisotropic medium

In this paper, we present a high-order generalized extended Born approximation (Ho-GEBA) algorithm for modeling 3D electromagnetic responses of an arbitrary anisotropic body in transverse anisotropic background medium based on integral equation method. First, generalized series solutions of the integral equation are obtained by successive iterative technique, and a contraction operator is introduced for the anisotropic medium based on the iterative dissipation principle to guarantee the convergence of high-order series. Then, we derive the Ho-GEBA solutions of 3D electromagnetic responses in the anisotropic medium using the abnormal body domain decomposition method combining with the extended Born approximation. Analytical solutions of dyadic Green's functions in the transverse isotropic medium are used, which can improve the efficiency of the algorithm greatly. Numerical results show the validity of the algorithm by comparing it with the full integral equation method and the classical Born approximation.

Fast and accurate three‐dimensional controlled source electromagnetic modelling†

ABSTRACTWe present a fast approximate method for three‐dimensional low frequency controlled source electro‐magnetic modeling. We apply the method to a synthetic model in a typical marine controlled source electromagnetic scenario, where conductivity and permittivity are different from the known background medium. For 3D configurations, fast computational methods are relevant for both forward and inverse modelling studies. Since this problem involves a large number of unknowns, it has to be solved efficiently to obtain results in a timely manner, without compromising accuracy. For this reason, the Born approximation, extended Born approximation and iterative extended Born approximation are implemented and compared with the full solution of the conjugate gradient fast Fourier transformation method. These methods are based on an electric field domain integral equation formulation. It is shown here how well the iterative extended Born approximation method performs in terms of both accuracy and speed with different configurations and different source positions. The improved accuracy comes at virtually no additional computational cost. With the help of this method, it is now possible to perform sensitivity analysis using 3D modelling in a timely manner, which is vital for controlled source electromagnetic applications. For forward modeling the solution at the sea‐bottom is of interest, because that is where the receivers are usually located. For inverse modeling, the accuracy of the solution in the target zone is important to obtain reasonably accurate conductivity values from the inversion using this approximate solution method. Our modelling studies show that the iterative extended Born approximation method is fast and accurate for both forward and inverse modelling. Sensitivity analysis as a function of the source position and different reservoir sizes validate the accuracy of the iterative extended Born approximation.

THE CONTRAST SOURCE-EXTENDED BORN MODEL FOR 2D SUBSURFACE SCATTERING PROBLEMS

In this paper, we describe a new full-wave integral equation model to tackle electromagnetic scattering problems arising from objects buried in layered media. Such a model is a rewriting of the usually adopted Contrast Source integral equation and is named Contrast Source-Extended Born (CS-EB) owing to this circumstance and to the relationship existing among its linearization and the Extended Born approximation. By means of this alternative formulation, it is possible to modify the relationship among the scatterer permittivity and the fleld it scatters, thus possibly reducing the degree of non-linearity of this latter relationship. Accordingly, in these cases, the adoption of the CS-EB model may be convenient with respect to traditional ones in both forward and inverse scattering problems.

Low-frequency electromagnetic characterization of buried obstacles by differential evolution with strategy of communication between groups and multi-resolution

The characterization of conductive obstacles in a resistive subsoil in the induction regime is investigated. Validated by numerical experimentation, a simple model is proposed to calculate all electromagnetic quantities of interest, the interaction between obstacles being taken in account. It is based on the extended Born approximation, Lax-Foldy multiple diffraction theory, and introduction of equivalent ellipsoidal obstacles. Those are retrieved via a hybrid algorithm of Differential Evolution using a communication strategy between groups, which in particular enables separation of nearby coupled ones. The different steps of the algorithm are detailed. The resolution of the standard algorithm and the hybrid algorithm are compared. The effect of taking in account the interaction between obstacles in the model is analyzed.

Hybrid Differential Evolution and Retrieval of Buried Spheres in Subsoil

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The characterization of conductive obstacles in subsoil is investigated in the induction regime. Validated by numerical experimentation, a simple model is proposed to calculate the main electromagnetic quantities of interest, the interaction between the obstacles being taken into account. The model is based on the extended Born approximation, the Lax–Foldy multiple diffraction theory, and introduction of equivalent spherical obstacles. Those are retrieved via a hybrid algorithm of differential evolution using a communication strategy between groups, which in particular enables separation of coupled obstacles close to one another. </para>

Erratum: Iterative implementation of approximate solutions for electromagnetic scattering

AbstractOriginally published Microwave Opt Technol Lett 49: 2425–2429, 2007. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1716, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23399

An efficient 2.5D inversion of loop-loop electromagnetic data

We have developed an inversion algorithm for loop-loop electromagnetic (EM) data, based on the localised non-linear or extended Born approximation to the solution of the 2.5D integral equation describing an EM scattering problem. Source and receiver configuration may be horizontal co-planar (HCP) or vertical co-planar (VCP). Both multi-frequency and multi-separation data can be incorporated. Our inversion code runs on a PC platform without heavy computational load.For the sake of stable and high-resolution performance of the inversion, we implemented an algorithm determining an optimum spatially varying Lagrangian multiplier as a function of sensitivity distribution, through parameter resolution matrix and Backus-Gilbert spread function analysis. Considering that the different source-receiver orientation characteristics cause inconsistent sensitivities to the resistivity structure in simultaneous inversion of HCP and VCP data, which affects the stability and resolution of the inversion result, we adapted a weighting scheme based on the variances of misfits between the measured and calculated datasets.The accuracy of the modelling code that we have developed has been proven over the frequency, conductivity, and geometric ranges typically used in a loop-loop EM system through comparison with 2.5D finite-element modelling results. We first applied the inversion to synthetic data, from a model with resistive as well as conductive inhomogeneities embedded in a homogeneous half-space, to validate its performance. Applying the inversion to field data and comparing the result with that of dc resistivity data, we conclude that the newly developed algorithm provides a reasonable image of the subsurface.

Iterative implementation of approximate solutions for electromagnetic scattering

AbstractThis article introduces a recursive formula, in which any approximate method can, iteratively, be incorporated to solve electromagnetic scattering problems. The use of extended Born approximation and quasi‐analytical approximation in this formula results in two series (algorithms), whose convergence properties are discussed. Numerical examples are given to demonstrate merits and the range of applicability of the new series. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2425–2429, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22781

A Generalized Material Averaging Formulation for Modelling of the Electromagnetic Fields

In this paper, we derive a general electromagnetic material averaging formula for anisotropic inhomogeneous media. In deriving the homogenization formula for the effective medium, we employed the extended Born approximation applied to the Lipmann-Schwinger integral equation for multiple scattering. It is shown that even if the inhomogeneous medium to be averaged is isotropic, the resulting averaged medium can be anisotropic. We further show that, for some special cases, the homogenization formula reduces to well-known material averaging formulas such as the effective medium approximation for spherical inclusions and the Backus averaging formula for finely layered media. The derived averaging formula can be used as a tool to study rock heterogeneities, multiphase structures, as well as an upscaling scheme to assign proper material properties to finite-difference modeling grids when the model under consideration has features that are smaller than the grid size. The latter allows us to use nonconformal grids that will significantly reduce the simulation time of the forward modeling code.

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 CLtN 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

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.

Efficient low-frequency inversion of 3-D buried objects with large contrasts

An efficient inversion method is proposed using Cui et al.'s high-order extended Born approximations to reconstruct the conductivity object function of three-dimensional dielectric objects buried in a lossy Earth. High-order solutions of the object function are obtained, which have closed-form relations to the linear inverse-scattering solution. Because such relations can be evaluated quickly using the fast Fourier transform, the high-order solutions have similar simplicity as the linear inversion. When the contrasts of buried objects are large, the high-order solutions are much more accurate due to the approximate consideration of multiple-scattering effects within the objects. Hence, good-resolution images can be obtained for large-contrast objects using the new method by only solving a linear inverse problem. Numerical experiments have shown the validity and efficiency of the proposed method.

A New Approximation to Three-Dimensional Electromagnetic Scattering

We introduce a new source-dependent but diagonal scattering tensor to approximate the electromagnetic fields internal to a scatterer. The approximate analytical expressions for the three diagonal scattering components are derived. Numerical tests show that the new approximation has favorable accuracy and wider range of applicability than the existing approximations such as the widely used extended Born approximation and the quasi-analytical approximation.

A quasi‐analytical boundary condition for three‐dimensional finite difference electromagnetic modeling

Numerical modeling of the quasi‐static electromagnetic (EM) field in the frequency domain in a three‐dimensional (3‐D) inhomogeneous medium is a very challenging problem in computational physics. We present a new approach to the finite difference (FD) solution of this problem. The FD discretization of the EM field equation is based on the balance method. To compute the boundary values of the anomalous electric field we solve for, we suggest using the fast and accurate quasi‐analytical (QA) approximation, which is a special form of the extended Born approximation. We call this new condition a quasi‐analytical boundary condition (QA BC). This approach helps to reduce the size of the modeling domain without losing the accuracy of calculation. As a result, a larger number of grid cells can be used to describe the anomalous conductivity distribution within the modeling domain. The developed numerical technique allows application of a very fine discretization to the area with anomalous conductivity only because there is no need to move the boundaries too far from the inhomogeneous region, as required by the traditional Dirichlet or Neumann conditions for anomalous field with boundary values equal to zero. Therefore this approach increases the efficiency of FD modeling of the EM field in a medium with complex structure. We apply the QA BC and the traditional FD (with large grid and zero BC) methods to complicated models with high resistivity contrast. The numerical modeling demonstrates that the QA BC results in 5 times matrix size reduction and 2–3 times decrease in computational time.

Low-frequency detection of two-dimensional buried objects using high-order extended Born approximations

Linear inverse scattering method based on the Born approximation has found wide applications in the on-site detection of buried objects, such as in diffraction tomography. However, the Born approximation becomes invalid when the contrast of targets is high, in which the multiple-scattering effect within the target is more important. To avoid solving the nonlinear inverse scattering problem for large-contrast objects, quasi-linear and modified quasi-linear approximations are proposed based on the extended Born approximation. In this paper, new formulations are derived for the inversion of large-contrast dielectric objects buried in a lossy earth using the idea of backconditioner and lumped-mass approximation. We have shown that the zeroth-order form of such formulations is just the well-known Born approximation, and the first-order form is completely equivalent to the extended Born approximation-based quasi-linear approximation. Hence, high-order approximations are proposed in this paper, which are given in closed forms. Using such approximations, good resolution images can be obtained for large-contrast objects by only solving a linear inverse scattering integral equation. Reconstruction examples show the validity and efficiency of the proposed formulae.

Fast three-dimensional electromagnetic nonlinear inversion in layered media with a novel scattering approximation

This paper presents a fast three-dimensional (3D) electromagnetic nonlinear inversion method in a multilayered medium via a novel scattering approximation. Using the superposition principle, we introduce a new source-dependent but diagonal scattering tensor. The approximate analytical expressions for the three scattering diagonal components are derived. Numer- ical tests show that the new approximation has better accuracy and wider range of applicability than the existing approximations such as the extended Born approximation and the quasi-analytical approximation. The computational speed of the new scattering approximation is essentially the same as the Born approximation.With such a new approximation, we further develop an efficient linearized 3D electromagnetic inversion method in a layered medium. In this method, the inverse problem is cast into a weighted least-squares problem solved via a conjugate gradient scheme. In terms of eigenvalue analysis, we propose a choice of the data and model weighting matrices that can be constructed for a general, complex Fréchet sensitivity matrix. The suggested weights help the selection of the regularization factor within the specified range of [0, 1] for the inverse problem, and improves the condition of the original imaging system. Synthetic tomography experiments demonstrate the efficiency of the new scattering approximation and the fast 3D EM imaging technique.

New Tools and Series for Forward and Inverse Scattering Problems in Lossy Media

A convenient rewriting of the pertinent integral equation is exploited to introduce a new model for two-dimensional electromagnetic scattering by dielectric objects in lossy media. Exploiting this latter, a new series expansion is introduced to solve the forward problem accurately and effectively. The first term of such a series coincides, in particular situations, with the well-known extended Born approximation. Theoretical tools and results are given on the range of applicability and rate of convergence of the series, which favorably compares with the traditional Born one. These tools allow noticing that the new model exhibits a lower degree of nonlinearity with respect to parameters embedding dielectric characteristics as compared to the traditional model, thus suggesting its exploitation in the solution of the inverse problem. Numerical examples assessing effectiveness and convenience of the proposed models, tools and inversion methods are presented.

A Green function formulation of the Extended Born approximation for three-dimensional electromagnetic modelling

We derive an Extended Born approximation for electromagnetic scattering from three-dimensional scatterers with arbitrary shape in the diffusive frequency regime. In order to improve on the accuracy of the original Extended Born approximation we present an alternative formulation using a Green function approach. Unlike the original formulation, the source proximity effect is included in the scattering tensor of the new Extended Born approximation. The resulting equations, which still have an explicit dependency with respect to the material properties of the scatterers, are solved via a numerical procedure and its computational cost is still of the same order as the first-order Born approximation. However, its solution is far more accurate.