Peter M. van den Berg

A contrast source inversion method

This paper describes a simple algorithm for reconstructing the complex index of refraction of a bounded object immersed in a known background from a knowledge of how the object scatters known incident radiation. The method described here is versatile accommodating both spatially and frequency varying incident fields and allowing a priori information about the scatterer to be introduced in a simple fashion. Numerical results show that this new algorithm outperforms the modified gradient approach which until now has been one of the most effective reconstruction algorithms available.

A Computational Model of the Electromagnetic Heating of Biological Tissue with Application to Hyperthermic Cancer Therapy

To investigate the potentialities of hyperthermia as a cancer therapy, computer simulations have been performed. This simulation consists of two tuccessive steps. First, the heat generated in a distribution of biological tissue when irradiated by a source of electromagnetic radiation is computed. The mathematical tool for determining the disbution of generated heat is the domain-integral-equation technique. This technique enables us to determine in a body with arbitrary distribution of permittivity and conductivity the electromagnetic field due to prescribed sources. The integral equation is solved numerically by an iterative minimization of the integrated square error. From the computed distribution of generated heat, the temperature distribution follows by solving numerically the pertaining heat transfer problem. The relevant differential equation together with initial and boundary conditions is solved numerically using a finite-element technique in space and a finite-difference technique in time. Numerical results pertaining to the temperature distribution in a model of the human pelvis are presented.

The contrast source inversion method for location and shape reconstructions

In this study we propose a multiplicative regularization scheme to deal with the problem of the detection and imaging of homogeneous dielectric objects (the so-called binary objects). By considering the binary regularizer as a multiplicative constraint for the contrast source inversion (CSI) method we are able to avoid the necessity of determining the regularization parameter before the inversion process has been started. We present some numerical results for some representative two-dimensional configurations, but we also show the three-dimensional reconstruction for a full vectorial electromagnetic problem. We conclude that the binary CSI method is able to obtain reasonable reconstruction results even when a wrong estimate of the material parameter is used. Moreover, generalization of the method allows us to handle inversion of more than one homogeneous scatterer having different material parameters.

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.

Application of the finite-difference contrast-source inversion algorithm to seismic full-waveform data

We have applied the finite-difference contrast-source inversion (FDCSI) method to seismic full-waveform inversion problems. The FDCSI method is an iterative nonlinear inversion algorithm. However, unlike the nonlinear conjugate gradient method and the Gauss-Newton method, FDCSI does not solve any full forward problem explicitly in each iterative step of the inversion process. This feature makes the method very efficient in solving large-scale computational problems. It is shown that FDCSI, with a significant lower computation cost, can produce inversion results comparable in quality to those produced by the Gauss-Newton method and better than those produced by the nonlinear conjugate gradient method. Another attractive feature of the FDCSI method is that it is capable of employing an inhomogeneous background medium without any extra or special effort. This feature is useful when dealing with time-lapse inversion problems where the objective is to reconstruct changes between the baseline and the monitor model. By using the baseline model as the background medium in crosswell seismic monitoring problems, high quality time-lapse inversion results are obtained.

Iterative Schemes Based on the Minimization of the Error in Field Problems

ABSTRACT The computation of electromagnetic fields In complex structures Is discussed. To handle the functional equation (e.g. integral equation) that results computationally from the analysis and at the same time have a measure for the accuracy attained, we introduce the global (i.e. Integrated over the domain of structure) root-mean-square error in the equality sign of the functional equation that has to be satisfied by the exact solution. This error criterion also enables us to develop an Iterative technique to solve the problem. In It, variational techniques are employed that enforce a monotonic decrease of the error in each iteration, and thus lead to an iterative improvement of the solution of the problem. Starting with an arbitrary Initial guess and a set of arbitrarily chosen varlational functions, some iteration schemes are derived. Suitable choices for the varlational functions are discussed. Some numerical results pertaining to a number of representative field problems illustrate the rate of co...

Three‐dimensional nonlinear inversion in cross‐well electrode logging

Electrode logging as known in the oil industry is a method for determining the electrical conductivity distribution around a borehole or between two boreholes from the static field (dc) measurements in the borehole. In this paper we discuss the reconstruction of the conductivity in a three-dimensional domain between two boreholes. The electric current density integral equation will be taken as point of departure to develop a nonlinear inversion scheme based on a new inversion method, the so-called contrast source inversion method (CSI). The CSI method considers the inverse scattering problem as an inverse source problem in which the unknown contrast sources in the object domain are reconstructed by minimizing the object and data error using a conjugate gradient step, after which the contrast is updated by minimizing only the error in the object. The robustness and the abilities of this full-vector three-dimensional inversion method have been demonstrated with a number of numerical examples where the synthetic “measured” data used were generated by solving a forward scattering problem using the conjugate gradient fast Fourier transform method.

Computation of electromagnetic fields inside strongly inhomogeneous objects by the weak-conjugate-gradient fast-Fourier-transform method

The computation of the electromagnetic field inside a strongly inhomogeneous dielectric object is formulated in terms of a domain-integral equation over the object. We discuss a weak form of the integral equation in which the spatial derivatives are integrated analytically. Doing so, we obtain an equation that is solved efficiently with the advantageous combination of a conjugate-gradient iterative method and a fast-Fourier-transform technique. Numerical computations are performed for a strongly inhomogeneous lossy sphere. For this case we compare the accuracy and the efficiency of the present method with the analytic solution based on the Mie series and the finite-difference time-domain approach. To show that the method is also capable of computing more-complex scattering problems, we assume the incident field to be generated by a (1/2)λ thin-wire dipole. In this case the absorbed power density is presented. All these test cases demonstrate that the weak form of the conjugate-gradient fast-Fourier-transform method can be considered as a comparatively simple and efficient tool for solving realistic electromagnetic wave-field problems.

Application of the multiplicative regularized contrast source inversion method on 3D experimental Fresnel data

This paper presents the results of inversion of multi-frequency electromagnetic scattered field data, measured by the Institute of Fresnel, Marseille, France, from three-dimensional (3D) homogeneous objects, both for co-polarization and cross-polarization antenna orientations. The reconstructions were obtained by using the multiplicative regularized contrast source inversion (MR-CSI) method. The inversion results indicated that the MR-CSI is able to robustly and efficiently invert these 3D data sets using minimal a priori information.

A Directional Borehole Radar System

In this paper we present the simulation and design of a directional borehole radar. In addition we discuss an imaging method for the radar system. The antenna system contains an electric dipole which is in one direction shielded by a cylindrical perfectly conducting reflector. The radiation pattern of the reflected wavefield is computed by first solving the integral equation. This equation combines the unknown electric surface current density on the reflector and the known incident field from the electric dipole. Once the electric surface current density is known, the radiation pattern of the system is computed using the integral representation over the reflector and the dipole. The radiation patterns for various configurations have been computed in order to find an optimal configuration. A prototype of the antenna system based on an optimal configuration has been built, and the directional radiation pattern has been measured in the plane perpendicular to the antenna system. The measurements were in good agreement with the computations. Subsequently a three dimensional imaging method for the borehole radar is presented. Here a deconvolution is carried out in the angular direction, making use of the computed radiation pattern. Some imaging results will be shown.