Layer-stripping Algorithm Research Articles

Approximation of continuous EIT data from electrode measurements with Bayesian methods

The electrical impedance tomography (EIT) in its classical formulation seeks to estimate the electric conductivity distribution inside the body from the knowledge of the Dirichlet-to-Neumann (DtN) map of the conductivity equation at the boundary. Numerical methods for the solution of the EIT problem have been developed based on this formulation, most notably the d-bar method and the layer stripping algorithm. In practice, however, the EIT data (electrode data), collected by using a fixed number of contact electrodes, is tantamount to knowledge of the resistance matrix, a mapping between given current configuration and the corresponding vector of measured electrode voltages. Forward models corresponding to the DtN data and the electrode data differ in terms of the boundary values and no direct connection between them has been established. In this article, we analyze the relation between the two boundary data types, and propose to approximate the DtN data from the measured resistance matrix for solving the EIT inverse problem within the Bayesian framework, leveraging a sample based prior and a principal component model reduction.

Device for ultrasound imaging of standing trees

The aim of ARB'UST project is to develop an ultrasonic device for parametric imaging of standing trees. The device is designed to perform both transmission and reflection measurements, used for quantitative tomographic imaging. It allows various automatic acquisitions since the angular position of sensors can be precisely adjusted. The electronics and associated configuration enable particularly the measurement of velocity and attenuation of the ultrasonic waves during their propagation within the medium. Two tomography experiments were conducted on a plane tree sample (before and after drilling a hole) and tomograms were calculated by the “Layer Stripping” algorithm. Our first results show that the artificial defect is detected.

GPR data enhancement via multiple reflections' filtering

Based on a classification methodology for multiple reflections' (MR) identification recently published by the authors, this work proposes a filtering technique for GPR data to be integrated within the layer stripping (LS) approach. In LS, the reconstruction of the dielectric stack is obtained by recursively inverting the signals backscattered by each layer, from the top to the bottom. According to the state-of-the-art, to simplify the solution of the problem, MR are commonly neglected. The scope of this article is therefore to first discuss the implementation of the MR filtering procedure within a LS algorithm and evaluate from the quantitative point of view which improvements this will yield to the overall performances. © 2013 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2013.

Velocity-independent layer stripping of PP and PS reflection traveltimes

Building accurate interval velocity models is critically important for seismic imaging and AVO (amplitude variation with offset) analysis. Here, we adapt the [Formula: see text] method to develop an exact technique for constructing the interval traveltime-offset function in a target zone beneath a horizontally layered overburden. All layers in the model can be anisotropic, with an essential assumption that the overburden has a horizontal symmetry plane (i.e., up-down symmetry). Our layer-stripping algorithm is entirely data-driven and, in contrast to the generalized Dix equations, does not require knowledge of the velocity field anywhere in the medium. Important advantages of our approach compared to the Dix-style formalism also include the ability to handle mode-converted waves, long-offset data, and laterally heterogeneous target layers with multiple, curved reflectors. Numerical tests confirm the high accuracy of the algorithm in computing the interval traveltimes of both PP- and PS-waves in a dipping, transversely isotropic layer with a tilted symmetry axis (TTI medium) beneath an anisotropic overburden. In combination with the inversion techniques developed for homogeneous TTI models, the proposed layer stripping of PP and PS data can be used to estimate the interval parameters of TTI formations in such important exploration areas as the Canadian Foothills. Potential applications of this methodology also include the dip-moveout inversion for the P-wave time-processing parameter [Formula: see text] and stable computation of the interval long-spread (nonhyperbolic) moveout for purposes of anisotropic velocity analysis.

Localized Parametric Electromagnetic Inversion for Pavement Profiling with Ground Penetrating Radar

Ground penetrating radar (GPR) has proven to be an effective non-destructive tool for pavement profiling. Layer-stripping algorithm and parametric electromagnetic (EM) inversion algorithm (referred to as EMI algorithm) are two inverse scattering approaches to estimate multilayered media properties using monostatic GPR. In this paper, a localized parametric EM inversion approach (referred to as NHD-EMI algorithm) is proposed for analyzing pavements with spatial varying characteristics. A non-homogeneous detector (NHD) is first used to divide the whole data into multiple subsegments with similar media properties. Then EMI is applied to each segment with layer-stripping technique to provide the initial model parameter estimates needed by the EMI algorithm. Numerical results show that the proposed NHD-EMI algorithm exhibits better performance over the existing EMI algorithm for nonhomogeneous pavements.

3D description and inversion of reflection moveout of PS-waves in anisotropic media

Common-midpoint moveout of converted waves is generally asymmetric with respect to zero offset and cannot be described by the traveltime series t2(x2) conventionally used for pure modes. Here, we present concise parametric expressions for both common-midpoint (CMP) and common-conversion-point (CCP) gathers of PS-waves for arbitrary anisotropic, horizontally layered media above a plane dipping reflector. This analytic representation can be used to model 3D (multi-azimuth) CMP gathers without time-consuming two-point ray tracing and to compute attributes of PS moveout such as the slope of the traveltime surface at zero offset and the coordinates of the moveout minimum. In addition to providing an efficient tool for forward modelling, our formalism helps to carry out joint inversion of P and PS data for transverse isotropy with a vertical symmetry axis (VTI media). If the medium above the reflector is laterally homogeneous, P-wave reflection moveout cannot constrain the depth scale of the model needed for depth migration. Extending our previous results for a single VTI layer, we show that the interval vertical velocities of the P- and S-waves (VP0 and VS0) and the Thomsen parameters e and δ can be found from surface data alone by combining P-wave moveout with the traveltimes of the converted PS(PSV)-wave. If the data are acquired only on the dip line (i.e. in 2D), stable parameter estimation requires including the moveout of P- and PS-waves from both a horizontal and a dipping interface. At the first stage of the velocity-analysis procedure, we build an initial anisotropic model by applying a layer-stripping algorithm to CMP moveout of P- and PS-waves. To overcome the distorting influence of conversion-point dispersal on CMP gathers, the interval VTI parameters are refined by collecting the PS data into CCP gathers and repeating the inversion. For 3D surveys with a sufficiently wide range of source–receiver azimuths, it is possible to estimate all four relevant parameters (VP0, VS0, e and δ) using reflections from a single mildly dipping interface. In this case, the P-wave NMO ellipse determined by 3D (azimuthal) velocity analysis is combined with azimuthally dependent traveltimes of the PS-wave. On the whole, the joint inversion of P and PS data yields a VTI model suitable for depth migration of P-waves, as well as processing (e.g. transformation to zero offset) of converted waves.

Discrete layer-stripping algorithms and feasibilityconditions for the 2D inverse conductivity problem

We develop a discrete layer-stripping algorithm for the 2Dinverse conductivity problem. Unlike previous algorithms, thisalgorithm transforms the problem into a time-varying 1DSchrödinger equation inversescattering problem, discretizesthis problem and then solves the discrete problem exactly.This approach has three advantages: (i) the poor conditioninginherent in the problem is concentrated in the solution of alinear integral transform at the beginning of the problem, towhich standard regularization techniques may be applied and(ii) feasibility conditions on the transformed data are obtained,satisfaction of which ensures that (iii) the solution of thediscrete nonlinear inverse scattering problem is exact andstable. Other contributions include solution of discreteSchrödinger equation inverse potential problems withtime-varying potentials by both layer-stripping algorithms andsolution of nested systems of equations which amount to atime-varying discrete version of the Gel'fand-Levitan equation.An analytic and numerical example is supplied to demonstrate theoperation of the algorithm.

Layer Stripping for a Transversely Isotropic Elastic Medium

In this paper we develop a layer stripping algorithm for transversely isotropic elastic materials in three-dimensional space. We first prove that by making displacement and traction measurements at the boundary of these elastic materials one can determine the elastic tensor and all its derivatives at the boundary. The elastic measurements made at the boundary are encoded in the Dirichlet to Neumann (DN) map. Then we use the Riccati equation developed in [G. Nakamura and G. Uhlmann, A layer stripping algorithm in elastic impedance tomography, in Inverse Problems in Wave Propagation, IMA Vol. Math. Appl. 90, Springer-Verlag, New York, 1997, pp. 375-384] for the DN map in any anisotropic elastic medium to approximate the elastic tensor in the interior.

Electrical Impedance Tomography

t. This paper surveys some of the work our group has done in electrical impedance tomography.

Permittivity measurements of multilayered media with monostatic pulse radar

Electromagnetic (EM) inversion is a useful tool for quantitative analysis in short-range applications of pulse radars. To estimate multilayered media properties using monostatic radar, two inverse scattering approaches are discussed: (a) layer-stripping algorithm by exploiting amplitude and time delay of radar echoes after their detection, and (b) EM inverse problem of parameter optimization by minimizing the mean square error between measured and modeled data. Redundancy in the estimation of media properties is given by spatial continuous measurements of the investigated media. This property is exploited in both the approaches investigated. In the layer-stripping approach the medium within each layer is homogeneous and the interfaces are assumed laterally continuous. In the inverse problem permittivity is assumed to be laterally smooth, implicit smoothing being given in the model parameterization. It is implicit in both methods that the inversion accuracy is strictly related to the amplitude stability of the radar and plane wave approximation. Therefore, the system calibration and the compensation of some propagation effects (e.g., near field, losses due to conductivity and to scattering from particles distributed between layers and on interfaces, pulse distortion) become crucial aspects for each specific application.

Boundary determination of material parameters from electromagnetic boundary information

Consider the following inverse problem. From electromagnetic information obtainable at the boundary of a body, can one determine material parameters, and their normal derivatives, at the boundary of the body? In this paper we answer this question in two physically distinct situations. The first is when the relationship between the electromagnetic fields depends on the conductivity, the electric permittivity and the magnetic permeability of the body, and these parameters together with their normal derivatives are shown to be recoverable at the boundary. The second is when the constitutive relations for the fields are altered so as to further take into account the chirality of the body. We also show how a layer-stripping algorithm may be derived to estimate the unknown parameters near the boundary in both situations. The approach is to calculate an explicit asymptotic expansion for the symbol of a boundary operator which is assumed to be known (from boundary measurements); this expansion is shown in each case to determine the unknown parameters at the boundary.

Using the Hilbert uniqueness method in a reconstruction algorithm for electrical impedance tomography

Presents a new version of the layer stripping algorithm in the sense that it works essentially by repeatedly stripping away the outermost layer of the medium after having determined the conductivity value in this layer. In order to stabilize the ill posed boundary value problem related to each layer, the authors base their algorithm on the Hilbert uniqueness method and implement it with the boundary element method.

Layer stripping for time-harmonic Maxwell's equations with high frequency

The inverse boundary value problem for Maxwell's equations is considered. The objective is to estimate the electric permittivity and conductivity as well as the magnetic permeability within a body from stationary high-frequency field measurements on the boundary of the body. A layer-stripping algorithm for estimating the parameters in the body can be described as follows. First, the unknown parameters are estimated at the boundary of the body by applying highly oscillating field excitations. Then the surface data are propagated through the estimated surface layer by an invariant embedding equation. Repeating the process, one 'peels off' the body layer by layer. The aim of this article is to show that the necessary tools for the algorithm applied to Maxwell's equations exist. A propagation equation for the boundary data is derived and it is shown that measurements with high spatial variations give an estimate for the unknown material parameters at the boundary. Due to the energy dissipation, the method is expected to work near the boundary of the body.

Numerical performance of layer stripping algorithms for two-dimensional inverse scattering problems

Numerical results of implementing a two-dimensional layer stripping algorithm to solve the two-dimensional Schrodinger equation inverse potential problem are presented and discussed. This is the first exact (all multiple scattering and diffraction effects are included) numerical solution of a multi-dimensional Schrodinger equation inverse potential problem, excluding optimization-based approaches. The results are as follows: (1) the layer stripping algorithm successfully reconstructed the potential from scattering data measured on a plane (as it would be in many applications); (2) the algorithm avoids multiple scattering errors present in Born approximation reconstructions; and (3) the algorithm is insensitive to small amounts of noise in the scattering data. Simplifications of layer stripping and invariant imbedding algorithms under the Born approximation are also discussed.

Wave field splitting, invariant imbedding, and phase space analysis applied to ocean acoustic wave propagation modeling.

Due to the extremely large size of realistic three-dimensional ocean problems, there is an understandable desire to incorporate marching methods into the solution algorithm for the inherently elliptic (frequency-domain) problem. Recognizing that typical ocean propagation problems are essentially scattering problems in terms of a transition region and transversely inhomogeneous asymptotic half-spaces, wave field splitting, invariant imbedding, and phase space methods reformulate the problem in terms of an operator scattering matrix characteristic of the transition region. The subsequent equations for the reflection and transmission operators are first order in range, nonlinear (Riccati-like), and, in general, nonlocal. The system is well-posed, but stiff. The reflected and transmitted wave fields can be computed in a very efficient manner, while the wave field in the transition region (if desired) can be computed by essentially a layer-stripping algorithm. In principle, the transition region can be divided into subregions, allowing for parallel computations and subsequent recombination. Numerical examples will be discussed. [Work supported by NSF, AFOSR, ONR, ASEE, JEWC.]

System identification of lossless layered media from input-output data

Abstract In this paper we model the interior of the earth as a lossless layered medium, and we are concerned with the reconstruction of the parameters of such a medium from the input and the output (input-output data). In the first part of the paper we assume that the input-output data is noise free, and we derive a so-called layer stripping algorithm for the reconstruction of the parameters of the medium. The algorithm is just one of the many versions of layer stripping algorithms that are available in the literature. In the second and major part of the paper we assume that both the input and the output are corrupted by noise, and we derive two new methods for the estimation of the parameters of the medium using the noisy input-output data. The methods are based on the estimation of the parameters of an ARX-representation of a lossless layered medium. The distinction between the two methods is that in one method the ARX parameters are free, while in the other method the ARX parameters have to correspond to parameters of a lossless layered medium that are physically realistic. We illustrate the methods by means of a numerical experiments.

Stability analysis of the Yagle–Levy multidimensional inverse scattering algorithm

A layer-stripping algorithm suggested by Yagle and Levy for solving a multidimensional inverse scattering problem is analyzed. It is shown that the unfiltered version of the algorithm is unstable; for the filtered version, it is proven that the reconstruction depends continuously on the data.

A Layer Stripping Fast Algorithm for the Two-Dimensional Direct Current Inverse Resistivity Problem

The inverse problem of reconstructing the resistivity of the earth, varying both laterally and with depth, from direct current measurements is considered. The problem is formulated as a multidimensional inverse scattering problem and solved using a layer stripping algorithm. This algorithm recursively reconstructs the resistivity and electrical potential on horizontal planes of increasing depth by downward continuation. This is the first exact solution to the inverse resistivity problem for resistivity varying laterally as well as with depth. The algorithm is an extension of an algorithm proposed by Levy for resistivity varying in one dimension.

A fast layer stripping algorithm for reconstructing a lossy layered medium

A discrete layered absorbing acoustic medium is probed with an impulsive plane wave at normal incidence, and the reflection response of the medium is measured. The absorbing nature of the medium is specified by a complex wave speed, corresponding to a constant Q. A fast layer stripping algorithm for recovering the profile of the complex wave speed as a function of depth is obtained. The algorithm works by recursively computing the reflection response of that portion of the medium below the current depth, thus stripping away the effects of that portion of the medium at and above the current depth. The algorithm is, in principle, exact, includes the effects of multiple reflections, and admits a clear physical interpretation of its operation. Several computer simulations illustrate the performance of the algorithm on synthetic data.

A layer-stripping fast algorithm for reconstructing an elastic medium from its compressional wave reflection response

A fast algorithm for recovering profiles of density and compressional (P) and shear-vertical (SV) wave speeds as functions of depth for the inverse seismic problem in a continuous layered elastic medium is obtained. This algorithm differs from previous fast algorithms for this problem since it requires only the compressional wave P-P reflection response at the surface of the medium, for three different slownesses or angles of incidence. Previous algorithms have required shear stress data in the form of the P-SV and SV-SV reflection responses, making them unsuitable for an ocean environment. This algorithm is thus much more suitable for reconstructing the ocean floor from pressure data taken in the ocean. The algorithm is exact, and it includes the effects of multiple reflections and mode conversions. A computer run illustrates the performance of the algorithm on synthetic data.