We consider imaging the reflectivity of scatterers from intensity-only data recorded by a single ... more We consider imaging the reflectivity of scatterers from intensity-only data recorded by a single moving transducer that both emits and receives signals, forming a synthetic aperture. By exploiting frequency illumination diversity, we obtain multiple intensity measurements at each location, from which we determine field cross-correlations using an appropriate phase controlled illumination strategy and the inner product polarization identity. The field cross-correlations obtained this way do not, however, provide all the missing phase information because they are determined up to a phase that depends on the receiver's location. The main result of this paper is an algorithm with which we recover the field cross-correlations up to a single phase that is common to all the data measured over the synthetic aperture, so all the data are synchronized. Thus, we can image coherently with data over all frequencies and measurement locations as if full phase information was recorded.
Tunable oscillatory modes of electric-field domains in doped semiconductor superlattices are repo... more Tunable oscillatory modes of electric-field domains in doped semiconductor superlattices are reported. The experimental investigations demonstrate the realization of tunable, GHz frequencies in GaAs-AlAs superlattices covering the temperature region from 5 to 300 K. The orgin of the tunable oscillatory modes is determined using an analytical and a numerical modeling of the dynamics of domain formation. Three different oscillatory modes are found. Their presence depends on the actual shape of the drift velocity curve, the doping density, the boundary condition, and the length of the superlattice. For most bias regions, the self-sustained oscillations are due to the formation, motion, and recycling of the domain boundary inside the superlattice. For some biases, the strengths of the low and high field domain change periodically in time with the domain boundary being pinned within a few quantum wells. The dependency of the frequency on the coupling leads to the prediction of a new type of tunable GHz oscillator based on semiconductor superlattices.
We propose a new strategy for narrow band, active array imaging of weak localized scatterers when... more We propose a new strategy for narrow band, active array imaging of weak localized scatterers when only the intensities are recorded and measured at the array. We consider a homogeneous medium so that wave propagation is fully coherent. We show that imaging with intensity-only measurements can be carried out using the time reversal operator of the imaging system, which can be obtained from intensity measurements using an appropriate illumination strategy and the polarization identity. Once the time reversal operator has been obtained, we show that the images can be formed using its singular value decomposition (SVD). We use two SVD-based methods to image the scatterers. The proposed approach is simple and efficient. It does not need prior information about the sought image, and guarantees exact recovery in the noise-free case. Furthermore, it is robust with respect to additive noise. Detailed numerical simulations illustrate the performance of the proposed imaging strategy when only the intensities are captured.
We study active array imaging of small but strong scatterers in homogeneous media when multiple s... more We study active array imaging of small but strong scatterers in homogeneous media when multiple scattering between them is important. We use the Foldy-Lax equations to model wave propagation with multiple scattering when the scatterers are small relative to the wavelength. In active array imaging we seek to locate the positions and reflectivities of the scatterers, that is, to determine the support of the reflectivity vector and the values of its nonzero elements from echoes recorded on the array. This is a nonlinear inverse problem because of the multiple scattering. We show in this paper how to avoid the nonlinearity and form images non-iteratively through a two-step process which involves ℓ 1 norm minimization. However, under certain illuminations imaging may be affected by screening, where some scatterers are obscured by multiple scattering. This problem can be mitigated by using multiple and diverse illuminations. In this case, we determine solution vectors that have a common support. The uniqueness and stability of the support of the reflectivity vector obtained with single or multiple illuminations are analyzed, showing that the errors are proportional to the amount of noise in the data with a proportionality factor dependent on the sparsity of the solution and the mutual coherence of the sensing matrix, which is determined by the geometry of the imaging array. Finally, to filter out noise and improve the resolution of the images, we propose an approach that combines optimal illuminations using the singular value decomposition of the response matrix together with sparsity promoting optimization jointly for all illuminations. This work is an extension of our previous paper [5] on imaging using optimization techniques where we now account for multiple scattering effects.
Summary We present a novel level set technique for shape reconstruction in history matching for n... more Summary We present a novel level set technique for shape reconstruction in history matching for non-conventional reservoirs. These reservoirs consist of several regions with dierent materials, e.g., shale or sand. The goal is to use the production data in order to estimate the unknown shapes and structure of these regions in the reservoir. Mathematically, we formulate this situation as an
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
ABSTRACT The problem of reconstructing images from measurements at the boundary of a domain belon... more ABSTRACT The problem of reconstructing images from measurements at the boundary of a domain belong to the class of inverse problems. In practice, these measurements are incomplete and inaccurate leading to ill-posed problems. This means that ‘exact’ reconstructions are usually not possible. In this Introduction the reader will find some applications in which the main ideas about stability and resolution in image reconstruction are discussed. We will see that although different applications or imaging modalities work under different physical principles and map different physical parameters, they all share the same mathematical foundations and the tools used to create the images have a great deal in common. Current imaging problems deal with understanding the trade off between data size, the quality of the image and the computational tools used to create the image. In many cases, these tools represent the performance bottleneck due to the high operational count and the memory cost.
For a beam impinging on a scattering medium the diffusion approximation to the radiative transpor... more For a beam impinging on a scattering medium the diffusion approximation to the radiative transport equation is not valid for analyzing the radiance near the source, especially if the medium scatters strongly with a sharp forward peak. To analyze the radiance, we use the Fokker-Planck approximation to the radiative transport equation. Numerical results show a backscattered ring appearing around the beam center. It also appears in Monte Carlo simulations of the radiative transport equation. This ring is manifested from successive near-forward scattering events, so it requires a directional description. Therefore the diffusion approximation cannot predict this ring.
Journal of Quantitative Spectroscopy and Radiative Transfer, 2001
We consider the matrix-valued radiative transfer equations for the Stokes parameters for the prop... more We consider the matrix-valued radiative transfer equations for the Stokes parameters for the propagation of light through turbulent atmospheres. A Monte Carlo method is introduced to solve the time dependent matrix-valued radiative transfer equations in 3D geometry. The Monte Carlo method is based on a probabilistic representation of the radiative transfer equations involving an augmented scalar transport equation where the polarization parameters are independent variables. The linear moments of the augmented transport equation with respect to the polarization parameters solve the matrix-valued radiative transfer equations. We show how polarization and depolarization e!ects develop in time for isotropic and unpolarized point sources, considered for concreteness in spherical and half-space geometries. We analyze in detail the creation of polarization by single-and multiple-scattering e!ects.
... Diego Álvarez Corresponding Author Contact Information , a , E-mail The Corresponding Author ... more ... Diego Álvarez Corresponding Author Contact Information , a , E-mail The Corresponding Author , Oliver Dorn a , E-mail The Corresponding Author , Natalia Irishina a , E-mail The Corresponding Author and Miguel Moscoso a , E-mail The Corresponding Author. ...
In this paper, we present a method for computing direct numerical simulations of narrow optical b... more In this paper, we present a method for computing direct numerical simulations of narrow optical beam waves propagating and scattering in a plane-parallel medium. For these computations, we use Fourier and Chebyshev spectral methods for three-dimensional ...
The local RBF is becoming increasingly popular as an alternative to the global version that suffe... more The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.
We study theoretically light backscattered by tissues using the radiative transport equation. In ... more We study theoretically light backscattered by tissues using the radiative transport equation. In particular we consider a twolayered medium in which a finite slab is situated on top of a half space. We solve the one-dimensional problem in which a plane wave is incident normally on the top layer and is the only source of light. The solution to this problem is obtained formally by imposing continuity between the solutions for the upper and lower layers. However, we are interested solely in probing the top layer. Assuming that the optical properties in the lower layer are known, we remove it from the problem yielding a finite slab problem by prescribing an alternate boundary condition. This boundary condition is derived using the theory of Green's functions and is exact. Hence, one needs only to solve the transport equation in a finite slab using this alternate boundary condition. We derive an asymptotic solution for the case when the slab is optically thin. We extend these results to the three-dimensional problem using Fourier transforms. These results are validated by comparisons with numerical solutions for the entire two-layered problem.
In this paper we propose and analyse a novel shape-reconstruction technique for the early detecti... more In this paper we propose and analyse a novel shape-reconstruction technique for the early detection of breast cancer from microwave data, which is based on a level-set technique. The shape-based approach offers several advantages compared to more traditional pixel-based approaches, as, for example, well-defined boundaries and the incorporation of an intrinsic regularization (in form of a-priori assumptions regarding the general anatomical structures present in the medium) that reduces the dimensionality of the inverse problem and thereby stabilizing the reconstruction. The level set strategy (which is an implicit representation of the shapes) frees us from topological restrictions during this reconstruction process. We present numerical results in 2D which demonstrate the performance of our scheme in various simulated realistic situations.
Our goal is to develop an inversion algorithm for reconstructing the shape of 3D breast tumors us... more Our goal is to develop an inversion algorithm for reconstructing the shape of 3D breast tumors using electromagnetic data. The method of moments (MoM) forward solver is used to calculate the electric and magnetic equivalent surface currents at the tumor interface and consequently the scattered electromagnetic fields. Using a so-called “adjoint scheme” for gradient calculation, the mismatch between calculated and
This paper describes the application of the adjoint method to the history matching problem in res... more This paper describes the application of the adjoint method to the history matching problem in reservoir engineering. The history matching problem consists in adjusting a set of parameters, in this case the permeability distribution, in order to match the data obtained with the simulator to the actual production data in the reservoir. Several numerical experiments are presented which show that
We consider imaging the reflectivity of scatterers from intensity-only data recorded by a single ... more We consider imaging the reflectivity of scatterers from intensity-only data recorded by a single moving transducer that both emits and receives signals, forming a synthetic aperture. By exploiting frequency illumination diversity, we obtain multiple intensity measurements at each location, from which we determine field cross-correlations using an appropriate phase controlled illumination strategy and the inner product polarization identity. The field cross-correlations obtained this way do not, however, provide all the missing phase information because they are determined up to a phase that depends on the receiver's location. The main result of this paper is an algorithm with which we recover the field cross-correlations up to a single phase that is common to all the data measured over the synthetic aperture, so all the data are synchronized. Thus, we can image coherently with data over all frequencies and measurement locations as if full phase information was recorded.
Tunable oscillatory modes of electric-field domains in doped semiconductor superlattices are repo... more Tunable oscillatory modes of electric-field domains in doped semiconductor superlattices are reported. The experimental investigations demonstrate the realization of tunable, GHz frequencies in GaAs-AlAs superlattices covering the temperature region from 5 to 300 K. The orgin of the tunable oscillatory modes is determined using an analytical and a numerical modeling of the dynamics of domain formation. Three different oscillatory modes are found. Their presence depends on the actual shape of the drift velocity curve, the doping density, the boundary condition, and the length of the superlattice. For most bias regions, the self-sustained oscillations are due to the formation, motion, and recycling of the domain boundary inside the superlattice. For some biases, the strengths of the low and high field domain change periodically in time with the domain boundary being pinned within a few quantum wells. The dependency of the frequency on the coupling leads to the prediction of a new type of tunable GHz oscillator based on semiconductor superlattices.
We propose a new strategy for narrow band, active array imaging of weak localized scatterers when... more We propose a new strategy for narrow band, active array imaging of weak localized scatterers when only the intensities are recorded and measured at the array. We consider a homogeneous medium so that wave propagation is fully coherent. We show that imaging with intensity-only measurements can be carried out using the time reversal operator of the imaging system, which can be obtained from intensity measurements using an appropriate illumination strategy and the polarization identity. Once the time reversal operator has been obtained, we show that the images can be formed using its singular value decomposition (SVD). We use two SVD-based methods to image the scatterers. The proposed approach is simple and efficient. It does not need prior information about the sought image, and guarantees exact recovery in the noise-free case. Furthermore, it is robust with respect to additive noise. Detailed numerical simulations illustrate the performance of the proposed imaging strategy when only the intensities are captured.
We study active array imaging of small but strong scatterers in homogeneous media when multiple s... more We study active array imaging of small but strong scatterers in homogeneous media when multiple scattering between them is important. We use the Foldy-Lax equations to model wave propagation with multiple scattering when the scatterers are small relative to the wavelength. In active array imaging we seek to locate the positions and reflectivities of the scatterers, that is, to determine the support of the reflectivity vector and the values of its nonzero elements from echoes recorded on the array. This is a nonlinear inverse problem because of the multiple scattering. We show in this paper how to avoid the nonlinearity and form images non-iteratively through a two-step process which involves ℓ 1 norm minimization. However, under certain illuminations imaging may be affected by screening, where some scatterers are obscured by multiple scattering. This problem can be mitigated by using multiple and diverse illuminations. In this case, we determine solution vectors that have a common support. The uniqueness and stability of the support of the reflectivity vector obtained with single or multiple illuminations are analyzed, showing that the errors are proportional to the amount of noise in the data with a proportionality factor dependent on the sparsity of the solution and the mutual coherence of the sensing matrix, which is determined by the geometry of the imaging array. Finally, to filter out noise and improve the resolution of the images, we propose an approach that combines optimal illuminations using the singular value decomposition of the response matrix together with sparsity promoting optimization jointly for all illuminations. This work is an extension of our previous paper [5] on imaging using optimization techniques where we now account for multiple scattering effects.
Summary We present a novel level set technique for shape reconstruction in history matching for n... more Summary We present a novel level set technique for shape reconstruction in history matching for non-conventional reservoirs. These reservoirs consist of several regions with dierent materials, e.g., shale or sand. The goal is to use the production data in order to estimate the unknown shapes and structure of these regions in the reservoir. Mathematically, we formulate this situation as an
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
ABSTRACT The problem of reconstructing images from measurements at the boundary of a domain belon... more ABSTRACT The problem of reconstructing images from measurements at the boundary of a domain belong to the class of inverse problems. In practice, these measurements are incomplete and inaccurate leading to ill-posed problems. This means that ‘exact’ reconstructions are usually not possible. In this Introduction the reader will find some applications in which the main ideas about stability and resolution in image reconstruction are discussed. We will see that although different applications or imaging modalities work under different physical principles and map different physical parameters, they all share the same mathematical foundations and the tools used to create the images have a great deal in common. Current imaging problems deal with understanding the trade off between data size, the quality of the image and the computational tools used to create the image. In many cases, these tools represent the performance bottleneck due to the high operational count and the memory cost.
For a beam impinging on a scattering medium the diffusion approximation to the radiative transpor... more For a beam impinging on a scattering medium the diffusion approximation to the radiative transport equation is not valid for analyzing the radiance near the source, especially if the medium scatters strongly with a sharp forward peak. To analyze the radiance, we use the Fokker-Planck approximation to the radiative transport equation. Numerical results show a backscattered ring appearing around the beam center. It also appears in Monte Carlo simulations of the radiative transport equation. This ring is manifested from successive near-forward scattering events, so it requires a directional description. Therefore the diffusion approximation cannot predict this ring.
Journal of Quantitative Spectroscopy and Radiative Transfer, 2001
We consider the matrix-valued radiative transfer equations for the Stokes parameters for the prop... more We consider the matrix-valued radiative transfer equations for the Stokes parameters for the propagation of light through turbulent atmospheres. A Monte Carlo method is introduced to solve the time dependent matrix-valued radiative transfer equations in 3D geometry. The Monte Carlo method is based on a probabilistic representation of the radiative transfer equations involving an augmented scalar transport equation where the polarization parameters are independent variables. The linear moments of the augmented transport equation with respect to the polarization parameters solve the matrix-valued radiative transfer equations. We show how polarization and depolarization e!ects develop in time for isotropic and unpolarized point sources, considered for concreteness in spherical and half-space geometries. We analyze in detail the creation of polarization by single-and multiple-scattering e!ects.
... Diego Álvarez Corresponding Author Contact Information , a , E-mail The Corresponding Author ... more ... Diego Álvarez Corresponding Author Contact Information , a , E-mail The Corresponding Author , Oliver Dorn a , E-mail The Corresponding Author , Natalia Irishina a , E-mail The Corresponding Author and Miguel Moscoso a , E-mail The Corresponding Author. ...
In this paper, we present a method for computing direct numerical simulations of narrow optical b... more In this paper, we present a method for computing direct numerical simulations of narrow optical beam waves propagating and scattering in a plane-parallel medium. For these computations, we use Fourier and Chebyshev spectral methods for three-dimensional ...
The local RBF is becoming increasingly popular as an alternative to the global version that suffe... more The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.
We study theoretically light backscattered by tissues using the radiative transport equation. In ... more We study theoretically light backscattered by tissues using the radiative transport equation. In particular we consider a twolayered medium in which a finite slab is situated on top of a half space. We solve the one-dimensional problem in which a plane wave is incident normally on the top layer and is the only source of light. The solution to this problem is obtained formally by imposing continuity between the solutions for the upper and lower layers. However, we are interested solely in probing the top layer. Assuming that the optical properties in the lower layer are known, we remove it from the problem yielding a finite slab problem by prescribing an alternate boundary condition. This boundary condition is derived using the theory of Green's functions and is exact. Hence, one needs only to solve the transport equation in a finite slab using this alternate boundary condition. We derive an asymptotic solution for the case when the slab is optically thin. We extend these results to the three-dimensional problem using Fourier transforms. These results are validated by comparisons with numerical solutions for the entire two-layered problem.
In this paper we propose and analyse a novel shape-reconstruction technique for the early detecti... more In this paper we propose and analyse a novel shape-reconstruction technique for the early detection of breast cancer from microwave data, which is based on a level-set technique. The shape-based approach offers several advantages compared to more traditional pixel-based approaches, as, for example, well-defined boundaries and the incorporation of an intrinsic regularization (in form of a-priori assumptions regarding the general anatomical structures present in the medium) that reduces the dimensionality of the inverse problem and thereby stabilizing the reconstruction. The level set strategy (which is an implicit representation of the shapes) frees us from topological restrictions during this reconstruction process. We present numerical results in 2D which demonstrate the performance of our scheme in various simulated realistic situations.
Our goal is to develop an inversion algorithm for reconstructing the shape of 3D breast tumors us... more Our goal is to develop an inversion algorithm for reconstructing the shape of 3D breast tumors using electromagnetic data. The method of moments (MoM) forward solver is used to calculate the electric and magnetic equivalent surface currents at the tumor interface and consequently the scattered electromagnetic fields. Using a so-called “adjoint scheme” for gradient calculation, the mismatch between calculated and
This paper describes the application of the adjoint method to the history matching problem in res... more This paper describes the application of the adjoint method to the history matching problem in reservoir engineering. The history matching problem consists in adjusting a set of parameters, in this case the permeability distribution, in order to match the data obtained with the simulator to the actual production data in the reservoir. Several numerical experiments are presented which show that
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