Beam-domain Data Research Articles

A Multi-Frequency Tomographic Inverse Scattering Using Beam Basis Functions.

We present an overview of a beam-based approach to ultra-wide band (UWB) tomographic inverse scattering, where beam-waves are used for local data-processing and local imaging, as an alternative to the conventional plane-wave and Green’s function approaches. Specifically, the method utilizes a phase–space set of iso-diffracting beam-waves that emerge from a discrete set of points and directions in the source domain. It is shown that with a proper choice of parameters, this set constitutes a frame (an overcomplete generalization of a basis), termed “beam frame”, over the entire propagation domain. An important feature of these beam frames is that they need to be calculated once and then used for all frequencies, hence the method can be implemented either in the multi-frequency domain (FD), or directly in the time domain (TD). The algorithm consists of two phases: in the processing phase, the scattering data is transformed to the beam domain using windowed phase–space transformations, while in the imaging phase, the beams are backpropagated to the target domain to form the image. The beam-domain data is not only localized and compressed, but it is also physically related to the local Radon transform (RT) of the scatterer via a local Snell’s reflection of the beam-waves. This expresses the imaging as an inverse local RT that can be applied to any local domain of interest (DoI). In previous publications, the emphasis has been set on TD data processing using a special class of localized space–time beam-waves (wave-packets). The goal of the present paper is to present the imaging scheme in the UWB FD, utilizing simpler Fourier-based data-processing tools in the space and time domains.

A compressed data approach for image-domain least-squares migration

We consider the problem of image-domain least-squares migration (LSM) based on efficiently constructing the Hessian matrix with sparse beam data. Specifically, we use the ultra-wide-band phase space beam summation method, in which beams are used as local basis functions to represent scattered data collected at the surface. The beam domain data are sparse. One can identify seismic events with significant contributions so that only beams with nonnegligible amplitudes need to be used to image the subsurface. In addition, due to the beams’ spectral localization, only beams that pass near an imaging point need to be taken into account. These two properties reduce the computational complexity of computing the Hessian matrix — an essential ingredient for LSM. As a result, we can efficiently construct the Hessian matrix based on analyzing the sparse beam domain data.

UWB Beam-Based Local Diffraction Tomography—Part II: The Inverse Problem

This two-part article is concerned with the medium reconstruction using the beam-based tomographic inverse scattering. This Part II is based on the results of Part I that has dealt with the preprocessing phase. Specifically, we defined there the beam-frame representation of the scattered field and the corresponding processing windows that transform the scattering data to the beam domain. We have also derived the “local time-domain diffraction tomography” relation according to which the beam-domain data are directly related to the local Radon transform (LRT) of the medium. This local transform can be inverted and is used here for the local reconstruction of the medium via beam-domain filtered backpropagation. In this article, we define the filtered backpropagated beam waves and then reconstruct the medium in any sub-domain of interest (DoI) by aggregating the contributions of the backpropagated beams that pass in or near that DoI. Specifically, we use the class of isodiffracting beam waves, namely, the isodiffracting Gaussian beams (ID-GBs) and the isodiffracting pulse beams (ID-PBs) for the frequency- and time-domain formulations, respectively. Explicit expressions for the filtered backpropagated reconstruction kernels are given. The efficacy of the beam-domain approach for local backpropagation and reconstruction is demonstrated via numerical examples of the synthetic noisy data.

UWB Beam-Based Local Diffraction Tomography—Part I: Phase-Space Processing and Physical Interpretation

In this two-part article, we study the ultrawideband beam-based schemes for tomographic inverse scattering. The theory utilizes discrete phase-space sets of beam waves, which constitute overcomplete frames everywhere in the propagation domain, and thus can be considered as a local alternative to the conventional plane-wave or Green’s function integrals used in conventional diffraction tomography. Specifically, we formulate two inversion schemes, a multi-frequency domain scheme, and a time domain scheme. The former utilizes isodiffracting Gaussian beams, while the latter utilizes isodiffracting pulsed beams. Both schemes consist of two phases: In the preprocessing phase, the scattering data are expanded as a sum of beams whose amplitudes, referred to as the “beam-domain data,” are extracted from the data using local beam-based transforms. In the imaging phase, the beam data are backpropagated and used for local reconstruction. In this Part I we discuss the preprocessing phase. We define the beam-based transforms, and then use the Born approximation to establish a cogent physical interpretation of the beam-domain data. Specifically, we show that these data are related to the local Radon transform of the medium, which is interpreted physically as a local Snell’s law. This relation will be used in Part II to reconstruct the medium.

Multifrequency beam-based migration in inhomogeneous media using windowed Fourier transform frames

SUMMARY We consider the problem of inhomogeneous subsurface imaging using beam waves. The formulation is based on the ultra-wide-band phase-space beam summation (UWB-PS-BS) method, which is structured upon windowed Fourier transform (WFT) expansions of surface fields and sources. In this approach, the radiated field is given as a superposition of beam propagators. Here, we use the beams first for expanding the surface sources and the scattered data, and then for imaging where we use the backpropagation and cross-correlation of beams. This formulation enables a target oriented imaging approach, where we take into account only pairs of source and receiver beams that pass near a region of interest, and thus extract only the relevant data arriving from this region. It also leads to a priori sparse representation of both the beam domain data and the beam propagators. A physical cogent for the beam domain data is obtained under the Born approximation. The beam domain data can be approximated as the local interaction between the beam propagators and the medium reflectivity. Thus, one may interpret the beam domain data as a local Snell’s law reflection in the direction defined by the vector summation of the incident beam and backpropagated beam ray parameters. We demonstrate a physical model for the beam domain data and the salient features of the proposed imaging algorithm using numerical examples.

MUSIC Imaging Using Phase-Space Gaussian-Beams Processing

In a recent paper we presented a beam-based backpropagation and correlation imaging scheme for targets in a homogeneous medium. This beam-based approach is extended here for MUSIC-imaging algorithms. In the beam approach, the fields of the physical arrays of point-sources and point receivers are expanded using special sets of collimated beam-sources and beam-receivers. This converts the physical scattering data into a beam-domain data, describing the scattering amplitudes seen (synthetically) by receiver beams due to excitation by source beams. The image is then formed by applying the MUSIC algorithm directly in the beam domain. We derive a closed form expression for the data transformation to the beam domain and then derive multi-experiments MUSIC-imaging algorithms that accommodate the measurement noise. The beam approach enables local imaging of any given sub-domain of interest by retaining only the subset of beams that pass through that domain, thus reducing the overall computation complexity relative to the conventional Green function approach, and filtering out of data and noise that arrive from non-relevant directions. Numerical investigations reveal that the beam approach also provides a better resolution under low S/N conditions.

Back-Propagation and Correlation Imaging Using Phase-Space Gaussian-Beams Processing

We present a beam-based imaging scheme for targets in homogeneous medium whereby the fields of the source and receiver arrays are expanded using special phase-space bases of collimated beam fields, thus converting the physical data into a beam-domain data describing the scattering amplitudes seen (synthetically) by the receiver beams due to excitation by the source beams. The image is then formed by correlating the backpropagated beam data with the incident beams. The formulation utilizes the ultra-wideband phase-space beam-summation method where the beam bases consist of Gaussian beams that emerge from a discrete set of points and directions in the source and the receiver domains. An important feature of this method is that the beam-sets are frequency independent and hence are calculated once and then used for all frequencies. A closed form expression for the data-transformation matrix from the physical domain to the beam domain is derived, leading to sparse beam-domain data. The beam approach enables local imaging of any sub-domain of interest by retaining only the subset of source and receiver beams that pass through that domain, thus reducing the overall computation complexity. The method properties are explored via numerical simulations in a noisy environment.

ESPRIT with multiple-angle subarray beamforming

Abstract This article presents a new approach of implementing signal direction-of-arrival estimation, in which subarray beamforming is applied prior to estimation of signal parameters via rotational invariance techniques (ESPRIT). Different from the previous approaches, the beam-domain data from multiple adjacent pointing angles are combined in a way that the displacement invariance structure required by ESPRIT is maintained. It is intended to further obtain a sub-beamwidth resolution for a conventional multi-beam system already having small beamwidths. Computer simulations show that for typical multi-beam system applications the new approach provides improved estimation mean-square errors over the original ESPRIT, on top of reduced requirements for signal-to-noise ratio, number of snapshots, and computational time.