Tarek M. Habashy

Beyond the Born and Rytov approximations: A nonlinear approach to electromagnetic scattering

The Born and Rytov approximations, widely used for solving scattering problems, are of limited utility for low-frequency electromagnetic scattering in geophysical applications where conductivity can vary over many orders of magnitude. We present four new, relatively simple nonlinear estimators that can be used for rapid electromagnetic modeling. The first, termed the static localized nonlinear approximation, is designed specifically to correct the magnitude of the electric field internal to the scatterer. The second, termed the localized nonlinear approximation, improves the estimate of the phase of the scattered field and includes some of the cross-polarization effects due to full wave scattering. Two further new estimators, based on the Rytov transformation (the localized nonlinear Rytov and the static localized nonlinear Rytov approximations) are designed to further improve the estimation of the phase of the scattered field, especially at high frequency and for larger size scatterers. Although these approximations are nonlinear functions in conductivity, they are generally much faster to compute than the full forward problem, and are almost as efficient as the Born or Rytov approximations. Moreover, the enhanced accuracy of the new estimators has made us optimistic about their application to low-frequency three-dimensional inverse problems in electromagnetics. The approximations developed in this paper will also be applicable to fields such as quantum mechanics, optics, ultrasonics, and seismology.

A GENERAL FRAMEWORK FOR CONSTRAINT MINIMIZATION FOR THE INVERSION OF ELECTROMAGNETIC MEASUREMENTS

In this paper, we developed a general framework for the inversion of electromagnetic measurements in cases where parametrization of the unknown configuration is possible. Due to the ill-posed nature of this nonlinear inverse scattering problem, this parametrization approach is needed when the available measurement data are limited and measurements are only carried out from limited transmitter-receiver positions (limited data diversity). By carrying out this parametrization, the number of unknown model parameters that need to be inverted is manageable. Hence the Newton based approach can advantageously be used over gradient-based approaches. In order to guarantee an error reduction of the optimization process, the iterative step is adjusted using a line search algorithm. Further unlike the most available Newton-based approaches available in the literature, we enhanced the Newton based approaches presented in this paper by constraining the inverted model parameters with nonlinear transformation. This constrain forces the reconstruction of the unknown model parameters to lie within their physical bounds. In order to deal with cases where the measurements are redundant or lacking sensitivity to certain model parameters causing non-uniqueness of solution, the cost function to be minimized is regularized by adding a penalty term. One of the crucial aspects of this approach is how to determine the regularization parameter determining the relative importance of the misfit between the measured and predicted data and the penalty term. We reviewed different approaches to determine this parameter and proposed a robust and simple way of choosing this regularization parameter with aid of recently developed multiplicative regularization analysis. By combining all the techniques mentioned above we arrive at an effective and robust parametric algorithm. As numerical examples we present results of electromagnetic inversion at induction frequency in the deviated borehole configuration.

2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements

We present 2.5D fast and rigorous forward and inversion algorithms for deep electromagnetic (EM) applications that include crosswell and controlled-source EM measurements. The forward algorithm is based on a finite-difference approach in which a multifrontal LU decomposition algorithm simulates multisource experiments at nearly the cost of simulating one single-source experiment for each frequency of operation. When the size of the linear system of equations is large, the use of this noniterative solver is impractical. Hence, we use the optimal grid technique to limit the number of unknowns in the forward problem. The inversion algorithm employs a regularized Gauss-Newton minimization approach with a multiplicative cost function. By using this multiplicative cost function, we do not need a priori data to determine the so-called regularization parameter in the optimization process, making the algorithm fully automated. The algorithm is equipped with two regularization cost functions that allow us to reconstruct either a smooth or a sharp conductivity image. To increase the robustness of the algorithm, we also constrain the minimization and use a line-search approach to guarantee the reduction of the cost function after each iteration. To demonstrate the pros and cons of the algorithm, we present synthetic and field data inversion results for crosswell and controlled-source EM measurements.

An efficient finite-difference scheme for electromagnetic logging in 3D anisotropic inhomogeneous media

We consider a problem of computing the electromagnetic field in 3D anisotropic media for electromagnetic logging. The proposed finite-difference scheme for Maxwell equations has the following new features based on some recent and not so recent developments in numerical analysis: coercivity (i.e., the complete discrete analogy of all continuous equations in every grid cell, even for nondiagonal conductivity tensors), a special conductivity averaging that does not require the grid to be small compared to layering or fractures, and a spectrally optimal grid refinement minimizing the error at the receiver locations and optimizing the approximation of the boundary conditions at infinity. All of these features significantly reduce the grid size and accelerate the computation of electromagnetic logs in 3D geometries without sacrificing accuracy.

Rapid 2.5-dimensional forward modeling and inversion via a new nonlinear scattering approximation

We introduce a novel approximation to numerically simulate the electromagnetic response of point or line sources in the presence of arbitrarily heterogeneous conductive media. The approximation is nonlinear with respect to the spatial variations of electrical conductivity and is implemented with a source-independent scattering tensor. By projecting the background electric field(i.e., the electric field excited in the absence of conductivity variations) onto the scattering tensor, we obtain an approximation to the electric field internal to the region of anomalous conductivity. It is shown that the scattering tensor adjusts the background electric field by way of amplitude, phase, and cross-polarization corrections that result from frequency-dependent mutual coupling effects among scatterers. In general, these three corrections are not possible with the more popular first-order Born approximation. Numerical simulations and comparisons with a 2.5-dimensional finite difference code show that the new approximation accurately estimates the scattered fields over a wide range of conductivity contrasts and scatterer sizes and within the frequency band of a subsurface electromagnetic experiment. Furthermore, the approximation has the efficiency of a linear scheme such as the Born approximation. For inversion, we employ a Gauss-Newton search technique to minimize a quadratic cost function with penalty on a spatial functional of the sought conductivity model. We derive an approximate form of the Jacobian matrix directly from the nonlinear scattering approximation. A conductivity model is rendered by repeated linear inversion steps within range constraints that help reduce nonuniqueness in the minimization of the cost function. Synthetic examples of inversion demonstrate that the nonlinear approximation reduces considerably the execution time required to invert a large number of unknowns using a large number of electromagnetic data.

Simultaneous nonlinear reconstruction of two-dimensional permittivity and conductivity

A new inversion algorithm for the simultaneous reconstruction of permittivity and conductivity recasts the nonlinear inversion as the solution of a coupled set of linear equations. The algorithm is iterative and proceeds through the minimization of two cost functions. At the initial step the data are matched through the reconstruction of the radiating or minimum norm scattering currents; subsequent steps refine the nonradiating scattering currents and the material properties inside the scatterer. Two types of basis functions are constructed for the nonradiating currents: “invisible” (global) basis functions, which are appropriate for discrete measurements and nonradiating (local) basis functions, which are useful in studying the limit of continuous measurements. Reconstructions of square cylinders from multiple source receiver measurements at a single frequency show that the method can handle large contrasts in material properties.

Inversion algorithms for large-scale geophysical electromagnetic measurements

Low-frequency surface electromagnetic prospecting methods have been gaining a lot of interest because of their capabilities to directly detect hydrocarbon reservoirs and to compliment seismic measurements for geophysical exploration applications. There are two types of surface electromagnetic surveys. The first is an active measurement where we use an electric dipole source towed by a ship over an array of seafloor receivers. This measurement is called the controlled-source electromagnetic (CSEM) method. The second is the Magnetotelluric (MT) method driven by natural sources. This passive measurement also uses an array of seafloor receivers. Both surface electromagnetic methods measure electric and magnetic field vectors. In order to extract maximal information from these CSEM and MT data we employ a nonlinear inversion approach in their interpretation. We present two types of inversion approaches. The first approach is the so-called pixel-based inversion (PBI) algorithm. In this approach the investigation domain is subdivided into pixels, and by using an optimization process the conductivity distribution inside the domain is reconstructed. The optimization process uses the Gauss–Newton minimization scheme augmented with various forms of regularization. To automate the algorithm, the regularization term is incorporated using a multiplicative cost function. This PBI approach has demonstrated its ability to retrieve reasonably good conductivity images. However, the reconstructed boundaries and conductivity values of the imaged anomalies are usually not quantitatively resolved. Nevertheless, the PBI approach can provide useful information on the location, the shape and the conductivity of the hydrocarbon reservoir. The second method is the so-called model-based inversion (MBI) algorithm, which uses a priori information on the geometry to reduce the number of unknown parameters and to improve the quality of the reconstructed conductivity image. This MBI approach can also be used to refine the conductivity image obtained using the PBI approach. The MBI also adopts the multiplicative regularized Gauss–Newton method. The unknown parameters that govern the location and the shape of an anomaly are the locations of the user-defined nodes for the boundaries of the probed region, whereas the unknown parameter that describes the physical property is the conductivity. We will show some inversion results of synthetic and field data to demonstrate the advantages of both the PBI and MBI approaches. We further demonstrate that by combining both inversion algorithms we can arrive at a better interpretation of both CSEM and MT data.

Microwave Biomedical Data Inversion Using the Finite-Difference Contrast Source Inversion Method

We present a contrast source inversion (CSI) technique which is based on a finite-difference (FD) solver for use in microwave biomedical imaging. The algorithm is capable of inverting complex-permittivity biomedical data sets without the explicit use of a forward solver at each iteration. The FD solver is based in the frequency domain, utilizes perfectly matched layer (PML) boundary conditions, and the stiffness matrix is solved via an LU decomposition and Gaussian elimination. An important feature of the FD-CSI algorithm is that the stiffness matrix associated with the FD solver depends only upon the background medium and frequency, and thus the LU decomposition is only performed once, before the iterative inversion process. Unlike the usual integral equation (IE) based inversion techniques, the FD-CSI algorithm is readily capable of utilizing an arbitrary backarbitrary backgroundground medium for the inversion process.

A finite-difference contrast source inversion method

We present a contrast source inversion (CSI) algorithm using a finite-difference (FD) approach as its backbone for reconstructing the unknown material properties of inhomogeneous objects embedded in a known inhomogeneous background medium. Unlike the CSI method using the integral equation (IE) approach, the FD-CSI method can readily employ an arbitrary inhomogeneous medium as its background. The ability to use an inhomogeneous background medium has made this algorithm very suitable to be used in through-wall imaging and time-lapse inversion applications. Similar to the IE-CSI algorithm the unknown contrast sources and contrast function are updated alternately to reconstruct the unknown objects without requiring the solution of the full forward problem at each iteration step in the optimization process. The FD solver is formulated in the frequency domain and it is equipped with a perfectly matched layer (PML) absorbing boundary condition. The FD operator used in the FD-CSI method is only dependent on the background medium and the frequency of operation, thus it does not change throughout the inversion process. Therefore, at least for the two-dimensional (2D) configurations, where the size of the stiffness matrix is manageable, the FD stiffness matrix can be inverted using a non-iterative inversion matrix approach such as a Gauss elimination method for the sparse matrix. In this case, an LU decomposition needs to be done only once and can then be reused for multiple source positions and in successive iterations of the inversion. Numerical experiments show that this FD-CSI algorithm has an excellent performance for inverting inhomogeneous objects embedded in an inhomogeneous background medium.

A Finite Difference Scheme for Elliptic Equations with Rough Coefficients Using a Cartesian Grid Nonconforming to Interfaces

We consider the problem of calculating a potential function in a two-dimensional inhomogeneous medium which varies locally in only one direction. We propose a staggered finite difference scheme on a regular Cartesian grid with a special cell averaging. This averaging allows for the change in conductivity to be in any direction with respect to the grid and does not require the grid to be small compared to the layering. We give a convergence result and numerical experiments which suggest that the new averaging works as well as the standard homogenization with thin conductive nonconformal sheets and exhibits better accuracy for resistive sheets.