DIRECT METHODS FOR INVERSE SCATTERING WITH TIME DEPENDENT AND REDUCED DATA. Jacob D. Rezac

Transcription

1 DIRECT METHODS FOR INVERSE SCATTERING WITH TIME DEPENDENT AND REDUCED DATA by Jacob D. Rezac A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics Summer 2017 c 2017 Jacob D. Rezac All Rights Reserved

2 DIRECT METHODS FOR INVERSE SCATTERING WITH TIME DEPENDENT AND REDUCED DATA by Jacob D. Rezac Approved: Louis Rossi, Ph.D. Chair of the Department of Mathematical Sciences Approved: George H. Watson, Ph.D. Dean of the College of Arts and Sciences Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education

3 I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Fioralba Cakoni, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: David Colton, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Houssem Haddar, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Peter Monk, Ph.D. Member of dissertation committee

4 ACKNOWLEDGEMENTS First and foremost, I would like to thank my Ph.D supervisor, Dr. Fioraba Cakoni. She has been a knowledgable and patient collaborator and mentor, and I am greatful for her help with this thesis. I have particularly appreciated her continued support, even after moving a few hours up the road. It was a true pleasure being her student. I would also like to acknowledge my thesis committee, Drs. David Colton, Houssem Haddar, and Peter Monk for sharing their extensive knowledge and experience with me, and for always providing interesting new ideas and a welcoming environment in which to study them. In particular, I would like to thank Dr. Haddar for hosting me for two productive and enjoyable extended visits to École Polytechnique. I would also like to thank Drs. Yehuda Braiman and Neena Iman, who hosted me at Oak Ridge National Laboratory, allowing me to work on applied problems which are unrelated to this thesis. The work done in this thesis was funded by NSF Grant DMS and INRIA DeFI team, whose support I gratefully acknowledge. The time I spent completing my thesis was significantly improved, both mathematically and personally, by many people. In particular, thanks to my math siblings, Isaac Harris, Shixu Meng, and Irene de Teresa Trueba, and my unrelated math brother Brennan Sprinkle. Thanks to the wings crew, Kevin Aiton, Zach Bailey, Thomas Brown, Matt Hassell, Allan Hungria, Lise-Marie Imbert-Gérard, Matt McGinnis, Tonatiuh Sanchez-Vizuet, and Francisco Sayas; to my friends at École Polytechnique, Simone Schavi, Nicolo Castro, Mohammed Lakhal, Helle Majander, Tobias Rienmuller, and Faisal Wahid; to Food Club members Amy Jannet and Madelyn Houser and to Brunch Club members Nick Kaufman and Frances Bothfeld; former office and house mates James Alexander and Michael Depersio; and to Mike Greco and Yolanda Lin. iv

5 Special thanks for my family, Siobhan, Rex, Ben, Aida, Kate, Mike, Maggie, Joe, Amelia, Violet, Hannah, and Charles. v

6 TABLE OF CONTENTS LIST OF TABLES ix LIST OF FIGURES x ABSTRACT xiii Chapter 1 FORWARD AND INVERSE WAVE SCATTERING PROBLEMS Wave Equations Scattering Problems Volume Integral Equations and the Born Approximation Qualitative Methods in Inverse Scattering Data Reduction and Primary Contributions of this Thesis QUASI-BACKSCATTERING IN THE FREQUENCY DOMAIN Introduction Direct Scattering Problem Forward Problem for Quasi-Backscattering Data Quasi-Backscattering Inverse Problem Inverse Problem for Small Obstacles A Second Range Test for Three-Dimensional Reconstructions Inverse Problem for Coplanar Small Obstacles Inverse Problem for Extended Obstacles Numerical Experiments Two-Dimensional Projections of Small Obstacles vi

7 2.4.2 Reconstruction of Third Coordinate Three-Dimensional Reconstructions DIRECT IMAGING OF SMALL SCATTERERS USING REDUCED TIME DEPENDENT DATA Introduction Forward Model and the Born Approximation Inverse Problem for the Time Domain Born Approximation Reconstruction of Point Scatterers from Time Domain Multistatic Data Reconstruction of Point Scatterers from Patches of Time Domain Multistatic Data Reconstruction of Point Scatterers from Time Domain Quasi-Backscattering Data Linear Sampling Method for Extended Objects Under the Time Domain Born Model Numerical Reconstructions MUSIC and LSM Reconstruction with Multistatic Data Quasi-Backscattering Reconstructions THE BORN TRANSMISSION EIGENVALUES PROBLEM Introduction Spherically Stratified Media Transmission Eigenvalue Free Regions Transmission Eigenvalues for General Shapes and Contrasts FAST METHODS FOR HELMHOLTZ EQUATION SIMULATION WITH APPLICATIONS TO TIME DOMAIN SCATTERING AND BAYESIAN INVERSE PROBLEMS Volume Integral Equations with Galerkin and FFT-Based Techniques Galerkin Approach Piecewise Constant Finite Element Discretization Numerical Results for Galerkin Approximation The Adaptive Integral Method (AIM) Interpolation Between Finite Element and Cartesian Grid vii

8 5.1.6 Numerical Results for the Time Harmonic Problem Convolution Quadrature for Time Domain Equations Approximation by Independent Helmholtz Equations and Error Analysis Implementation and Numerical Results Application I: CQ-AIM Error Analysis Numerical Results Application II: Bayesian Inverse Scattering Infinite Dimensional Bayesian Inverse Problems Numerical Implementation Simulations OUTLOOK AND OPEN PROBLEMS BIBLIOGRAPHY Appendix A SEPARATION OF VARIABLES SOLUTIONS B COPYRIGHT PERMISSIONS viii

9 LIST OF TABLES 1.1 Outline of algorithm used by many qualitative techniques. The curves Γ m and Γ i refer to the locations in R d 1 where receivers and transmitters are placed, respectively. We require that both curves contain D in their interior Simulation results for time-harmonic scattering from a ball with constant index of refraction Simulation results for time-harmonic scattering from a ball with non-constant index of refraction Simulation results for scattering from a non-convex shape with non-constant index of refraction Time domain simulation results for scattering from a ball with constant index of refraction Time and memory usage for calculation Tikhonov regularized solution of contrast ix

10 LIST OF FIGURES 2.1 A Comparison of multi-static data (left) and quasi-backscattering data (right). Red circles correspond to device locations. The thick blue line in the right figure denotes where the quasi-backscattering set-up is moved and where each transmitting device is located Symmetric addition of new objects, δ = π/50 (no noise) Symmetric addition of new objects, δ = π/50 (approximately 1% noise) Decreasing the quasi-parameter δ. Figures have δ = π (top-left), δ = π/50 (top-right), and δ = π/100 (bottom). Approximately 5% noise Two objects moving closer to each other, δ = π/100 (no noise). Thick bar at bottom corresponds to half of wavelength An L-Shaped geometry which requires 3 views to see all obstacles, δ = π/30 (approximately 1% noise) Results for co-planar obstacles, δ = π/60. Figures on left are noise-free and figures on right have approximately 1% noise When ĴΠ (z j ) is computed, peaks appear for i j. Red circles show exact location of L(z j ) Reconstructions lose accuracy in the presence of 0.1% noise. Red circles show exact location of L(z j ) Three-dimensional noise-free reconstructions of point obstacles based on multiple experiments. We take 95 incident directions, 95 observation points, and use δ = π/60. The top figure is noise-free while the bottom figure has approximately 1% noise. In both figures, we display isovalues of 0.6 times the maximum value of the imaging function x

11 3.1 Examples of limited aperture multistatic (left) and quasi-backscattering (right) measurements. In the limited aperture multistatic figure, the blue line represents the location of transmitters and the red line the location of receivers. In the quasi-backscattering set-up, Γ i is the large dashed circle, the thick solid line is Γ (y) m for a fixed y Γ i, and the circles on Γ i not located at y represent locations to which Γ (y) m will be moved Plots of (I LSM (z)) 1 (top) and (I multi (z)) 1 (bottom) for two different geometries Multistatic patch reconstructions of the same geometry of small circles, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color Multistatic patch reconstructions of the same geometry of medium-sized ellipses, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color Plots of (I quasi (z, τ)) 1 for four different geometries Plots of (Îquasi(z, τ)) 1 (top) and (I quasi (z, τ)) 1 (bottom), with a different number of transmitters in each row. On the left there are 5 transmitters, in the middle there are 10 transmitters, and the right there are 15 transmitters. Time harmonic data was computed with wavenumber k = Backscattering reconstructions using I 1 backscattering (z, τ) for two different geometries. In both figures, 30 transmitters are used and data is measured only at the location of the transmitter. Time domain data was simulated for 14 seconds with 480 time steps xi

12 3.8 Limited aperture reconstructions using multistatic data with I 1 multi (z, τ) (top), multistatic patch data with two patches (middle), and quasi-backscattering data with I 1 quasi (z, τ) (bottom). In both figures, 19 transmitters are used and in the case of quasi-backscattering data, 4 receivers were used Plot of the function d 0 (k) associated with m = r. When d 0 (k) crosses the real axis, k is a transmission eigenvalue Sample of the region in which transmission eigenvalues can appear for a real constant contrast Convergence rate for time harmonic scattering from a ball with constant contrast simulated with P 0 finite elements The shape of the non-convex object off which we simulate scattering. The finite element triangulation with N = 192 used is shown as well Convergence rate of CQ-Galerkin scheme for multiple meshes Tikhonov regularized solutions to f(m) = y obs. All images have the same color scale, and white lines indicate finite element mesh. Left column: Solutions corresponding to a Galerkin scheme h = (top) and h = (middle), and the exact solution projected onto a mesh of size h = (bottom). Right column: Solutions corresponding to an AIM scheme h = (top) and h = (middle), h = (bottom) Bayesian solution to inverse scattering problem. (Top right) A sample generated by the prior distribution on a mesh of size h = (Top left) A sample generated by the posterior distribution on a mesh of size h = (Bottom) The MAP solution on a mesh of size h = Note that the posterior sample and MAP solution use the same color scheme, which is different from the posterior color scheme. White lines indicate the mesh xii

13 ABSTRACT This thesis is focused on the motion of acoustic waves through penetrable media, and the use of such waves to reconstruct material properties of the fluid through which the waves are moving. The reconstruction methods developed in this thesis fall under the category of qualitative inversion methods and, as such, are fast and mathematically justified. Unlike in typical qualitative methods, however, these new methods require only small amounts of scattered field data to be collected. In particular, we demonstrate that with both far field time harmonic data and near field time dependent data, the location of weakly scattering point obstacles can be reconstructed with reduced data collection requirements compared to typical qualitative schemes. We give full mathematical justification for the time harmonic method and partial justification for the time dependent method. We also analyze the transmission eigenvalue problem for weakly scattering media, proving that, under this assumption, transmission eigenvalues are discrete and can sometimes have complex part which grows without bound. Finally, we introduce a fast method for simulating time harmonic and time dependent acoustic wave scattering and apply this method to optimization schemes for reconstructing penetrable media based on scattered field data. xiii

14 Chapter 1 FORWARD AND INVERSE WAVE SCATTERING PROBLEMS Much of applied mathematics research over the past 150 years has been dedicated to determining the behavior of a physical system with specific parameters. At the time, many of the mathematical and scientific breakthroughs of the era began to describe the motion of acoustic and electromagnetic waves and how they interact with their surroundings. Such was the focus on these problems that by the early 1900s, Jacques Hadamard had begun to describe the physical problems which could be solved by mathematics [55]. Indeed, this led to Hadamard s concept of a well-posed problem, Definition 1. A problem is said to be well-posed if 1. A solution exists; 2. the solution is unique; and 3. the solution depends continuously on its initial data. A problem which is not well-posed is called ill-posed. This focus on well-posed problems which describe a physical phenomenon given initial conditions - which we call here a forward problem or direct problem - led to many breakthroughs in theoretical and applied mathematics (as well as, for example, in the sciences, engineering, and economics). However, there is a huge group of physically important problems which are not well-posed. Often problems in which one is given data about a physical process and asked what initial conditions or model led to that data do not satisfy one (or any!) of the criteria in Definition 1. These ill-posed problems are called inverse problems, as a contrast with the forward problem. Although they are not well-posed, these problems are often vital in engineering and science and have become more mathematically tractable over the last 50 years. 1

15 This thesis will focus on one particular type of inverse problem, the inverse scattering problem. In an inverse scattering problem, we estimate the physical properties about hidden objects based on the way acoustic or electromagnetic waves interact with the objects. This estimation process is sometimes called, appropriately, parameter estimation or reconstruction. Although we rarely explicitly mention their applications in this thesis, these problems find use in many fields of science and engineering, including geophysical exploration, non-destructive testing of structures, and in medical devices used to scan the interior of bodies. In part due to this wide range of use, we are interested in developing methods which are fast, accurate, and mathematically justified. In this chapter, we introduce the basic theory for both forward and inverse wave scattering problems in acoustics. In Section 1.1, we derive the acoustic wave and Helmholtz equations from first principles. These will be the primary equations which we will study in later chapters. When designing methods for parameter estimation, the forward model of the physical situation represents an important piece of a priori information. As such, we will spend a significant amount of time describing the forward model of our problem. Once we understand the behavior of the physical system, we introduce the class of inversion algorithms which are the primary focus of this thesis: qualitative methods. We introduce a few qualitative methods for inverse scattering in Section 1.2. Qualitative methods are an important class of reconstruction techniques which only aim to estimate a few important parameters, rather than the entire parameter state. For example, we are often interested in estimating the size, shape, and location of a hidden object, rather than the precise value of each material property of the object at all points. Qualitative methods are significantly faster than, for example, non-linear optimization schemes, and do not require as much a priori information. However, they also do not return as much information. We will briefly introduce a nonlinear optimization-style reconstruction technique at the end of this thesis to compare with these qualitative schemes. 2

16 This discussion of qualitative methods is continued in Chapter 2 where we introduce a technique for reconstructing the location of small objects using less scattering data than is typically required by qualitative methods. In particular, we only collect scattered field data in a small region surrounding the location from which an incident field was transmitted, using an experimental set-up called quasi-backscattering. This is in contrast to usual qualitative methods which require scattered field data to be collected on a surface completely surrounding an object in order to reconstruct the location of the object. Chapter 3 continues the discussion of reconstruction algorithms using quasi-backscattering data. In contrast to Chapter 2 which makes use of time harmonic incident fields and scattered field data collected in the far field, Chapter 3 makes use of time dependent incident fields with scattered field data collected in the near field. Both Chapters 2 and 3 include reconstruction algorithms for small and weaklyscattering objects. One of the primary advantages of typical qualitative methods is that they do not require any such assumptions. However, as Chapters 2 and 3 demonstrate, a weak scattering assumption can lead to a less restrictive experimental set-up. As such, in Chapter 4, we discuss the justification of a qualitative method called the linear sampling method (LSM) for time dependent incident fields scattering from small and weak obstacles. In particular, we discuss the properties of an auxiliary eigenvalue problem, called the Born interior transmission problem, which would lead to the mathematical justification for the weak-scattering LSM. To end the thesis, in Chapter 5, we discuss the numerical simulation of forward acoustic scattering problems. In particular, we introduce an unconditionally stable method of optimal convergence order to simulate the scattering of both time harmonic and time domain acoustic waves. This volume integral equation method uses a finite element Galerkin approximation in space and convolution quadrature in time. Unfortunately, this method is slow and requires a large amount of memory. Hence, we also discuss a fast method to approximate some of the spatial operators in this slow method. We also apply this fast method to a different type of inversion scheme than 3

17 is discussed elsewhere in this thesis, a Bayesian inversion algorithm. In a Bayesian inverse problem, we look for a probability distribution to describe our reconstruction of material properties of the medium we are probing. These methods, unlike qualitative methods, require a large number of numerical simulations of acoustic wave scattering through many different fluid media. However, they result in significantly more information about the reconstructed object and in simulations do not have serious restrictions on the amount of scattered field data which is collected. Some of this thesis was originally presented in the following articles: 1. H. Haddar and J.D. Rezac, A quasi-backscattering problem for inverse acoustic scattering in the Born regime. Inverse Problems 31, (2015). 2. F. Cakoni, D. Colton and J.D. Rezac, The Born transmission eigenvalue problem. Inverse Problems (2016). 3. F. Cakoni and J.D. Rezac, Direct imaging of small scatterers using reduced time dependent data. Journal of Computational Physics 338, (2017). 1.1 Wave Equations We begin by deriving the acoustic wave equation, which describes the propagation of acoustic waves through media with changing material properties, such as sound speed or density. We follow [36, 43] in this derivation. Throughout the thesis, we focus on scattering from penetrable media - that is, media which allows wave to pass through it. This is opposed to obstacle scattering, which is a model which includes the assumption that scattered waves are completely reflected by the scattering media, completely absorbed, or only absorbed into a small amount of the object. Despite this distinction, we will often refer to scattering from penetrable media as scattering from the object defined by where material properties (such as the speed of sound) change from background properties. This abuse of notation will improve readability, and references to scattering from obstacles should not be interpreted as scattering from impenetrable media. Note that while this derivation is given in R 3, the resulting equations hold for R 2 as well. 4

18 Sound waves are propagating vibrations in a fluid medium. As such, we are interested in describing the way in which the properties, such as pressure and velocity, of a small volume of fluid change in response to an acoustic disturbance. Assume that, prior to its interaction with an acoustic wave, the fluid is in an equilibrium state with equilibrium pressure P 0 and density ρ 0 = ρ 0 (x), x R 3. Also assume that the medium of interest is inviscid and that there are no external forces on the fluid which affect sound propagation. Denote by Ω R 3 the small volume of fluid whose boundary is denoted by Ω with outward pointing normal ν(x). as Pressure and density are related through some equation of state which we denote P = f(ρ, S), (1.1) where S = S(x, t) is the specific entropy of the physical system. For example, the function f could describe an ideal gas law whose form is dictated by the system s underlying physical processes. The equilibrium state of the fluid is defined by the property that f t = f = 0. As a simplifying assumption, assume that acoustic disturbances occur quickly enough that heat diffusion does not affect sound propagation. This adiabatic approximation is valid for sound approximation in air, for example, and is expressed by S t where v = v(x, t) is the velocity of the fluid. + v S = 0, (1.2) When an acoustic wave interacts with the small volume of fluid, the volume will move and change shape, and hence its density will change. This change in density, in turn, affects the pressure of the volume, which then causes fluid motion due to a pressure gradient. Since pressure and density are related by f, the relationship between pressure and fluid motion can be described through a relationship between density and fluid motion. Absent external forces, the only way to change density is through the 5

19 change of mass into and out of Ω. In particular, [ ] ρ(x, t) dv (x) = ρ(x, t)v(x, t) ν(x) ds(x) t Ω Ω = (ρ(x, t)v(x, t)) dx. This results in the conservation of motion equation ρ t Ω + (ρv) = 0. (1.3) Finally, the way in which Ω moves due to changing pressure gradients can be described by conservation of momentum. Newton s Second Law gives that [ ] ρ(x, t)v(x, t) dv (x) = ρ(x, t)v(x, t) (v(x, t) ν(x)) + P (x, t) ν(x) ds(x) t Ω Ω = v(x, t) (ρ(x, t)v(x, t)) + P (x, t) dv (x). Ω Note that the change of momentum in Ω is due both to the change of momentum across Ω and due to the external pressure changes. Hence, we have the conservation of momentum equation (ρv) t + v (ρv) + P = 0. (1.4) In typical situations 1, equilibrium values of properties of the fluid are much larger than the changes caused by acoustic wave propagation. As such, we perform an asymptotic expansion of pressure, velocity, density, and entropy, and use equations (1.1)-(1.4) to describe the change in fluid properties based on an acoustic disturbance. Hence, for ɛ 1, let P (x, t) = P 0 + ɛp 1 (x, t) + O(ɛ 2 ) ρ(x, t) = ρ 0 (x) + ɛρ 1 (x, t) + O(ɛ 2 ) v(x, t) = ɛv 1 (x, t) + O(ɛ 2 ) S(x, t) = S 0 (x) + ɛs 1 (x, t) + O(ɛ 2 ). 1 An atypical situation might be an explosion or an airplane breaking the sound barrier. In these cases, nonlinear affects contribute significantly to sound propagation. 6

20 From the assumption, the linearized governing equations are S 1 t + v 1 S 0 = 0 ρ 1 t + ρ 0 v 1 = 0 f P 1 = ρ 1 ρ (ρ f 0, S 0 ) + S 1 S (ρ 0, S 0 ) v 1 t + 1 P 1 = 0. ρ 0 Using the assumption that f(ρ 0, S 0 ) is a stationary state, 0 = f = ρ 0 f ρ (ρ 0, S 0 ) + S 0 f S (ρ 0, S 0 ). Combining this equation with the linearized adiabatic approximation and the time derivative of the state equation yields P 1 t = c 2 (x) ( ) ρ1 t + v 1 ρ 0, (1.5) where c 2 (x) = f ρ (ρ 0(x), S 0 (x)) is the speed of sound in the medium. Finally, taking a time derivative of (1.5) and using the two as-yet unused linearized equations yields a wave equation for the change in pressure caused by an acoustic disturbance, ( ) 2 P 1 1 = c 2 (x)ρ t 2 0 (x) ρ 0 (x) P 1. This is the acoustic wave equation. For simplicity, we will always set ρ 0 to be constant so that 2 P 1 t 2 = c 2 (x) P 1. Hence, we are only considering acoustic propagation through a medium which changes only due to changes in entropy. gradient in the medium, for example. defined by This could correspond to a changing temperature The wave equation also governs the velocity potential, say u(x, t), which is Indeed, defined in this way, v 1 = 1 ρ 0 u and P 1 = u t. 2 u t 2 = c2 u. (1.6) Although the physical meaning of different variables will not be important in what follows, in the rest of this thesis we will discuss scattering problems through the velocity potential. 7

21 1.1.1 Scattering Problems Assume an acoustic incident wave, u i = u i (x, t; y), x y R d (d = 2 or 3), t > 0 was emitted from a point y and is traveling through a homogeneous medium which is large enough compared to the wavelength u i that we can approximate it as all of R d. Based on the discussion above, u i satisfies 2 u i t 2 = c2 0 x u i, x y R d, t > 0 (1.7) for some constant c 0 > 0. By rescaling variables, we always set c 0 1. The primary aim of scattering theory is to understand the way in which u i is affected by an inhomogeneity in the fluid medium. Hence, assume a variable speed of sound, c L (R d ) so that c(x) γ > 0 for some γ R. Define n(x) = c 2 (x) to be the square of the index of refraction and D = supp(1 n(x)) to be the location of the unknown scatterer. We also introduce the contrast function m(x) = n(x) 1, which will simplify notation below. The total acoustic field traveling through such a medium behaves according to (1.6). Separating the total field into the sum of an incident field, u i, and scattered field, u s yields the governing equation 2 u s t 2 us = m(x) 2 t 2 ( u i + u s), (x, t) R d R +. (1.8) We must ensure that at time t = 0, the incident field has not yet been affected by the inhomogeneity. As such, assume u i (x, t) = 0, x D, t 0, and u s (x, 0) = us (x, 0) x D. (1.9) t This assumption is referred to as a causal wave assumption. Combined, (1.7)-(1.9) constitute the time dependent wave scattering problem. Thus far we have not considered the form of the incident field. We will primarily consider two types of incident field: time harmonic and fully time dependent. A time harmonic incident field is of the form u i (x, t; y) = Re ( û i (x)e ikt) 1 t 0. 8

22 where 1 X is an indicator function on a set X and is only required to fit the causality assumption. Note that in the following, we will always use ˆ over an independent variable to indicate it is related to the time harmonic problem. Substituting this into (1.7) and assuming u s has the same form yields the time harmonic scattering problem û s + k 2 û s = k 2 m(x) ( û i + û s) û i + k 2 û i = 0. We also add the Sommerfeld radiation condition, ( ) û s lim r r (d 1)/2 r ikûs = 0, r = x (1.10) which ensures that waves are outgoing rather than incoming. The other type of incident field of interest is a time dependent one. Let χ C 2 (D) be a causal temporal pulse function (that is, χ and its derivatives vanish for t < 0). We define the incident field originating at a point y R d as the time convolution of χ with the fundamental solution Φ to the wave equation, H(t x ) 2π t 2 x, d = 2 2 Φ(x, t) = δ(t x ), d = 3, 4π x where H is the Heaviside function. For example, in R 3 u i (x, t; y) = χ(t x y ), (x, t) (R 3 \y) R +. 4π x y Volume Integral Equations and the Born Approximation (1.11) Solutions to the scattering problems, both in the time harmonic and time dependent, can be represented in many ways. In this thesis we will focus on their volume integral equation representation. For example, working formally, convolving both sides of (1.8) against Φ defined in (1.11) yields for x y R d and t > 0 that u s (x, t; y) = m(z)φ(x z, t τ) ( u s tt(z, τ) + u i tt(z, τ) ) dv (z) dτ. R D 9

23 With this form in mind, define the retarded volume potential operator V acting on f C0 (B R), with D B R d, by (V f)(x, t) := Φ(x z, t τ)f(τ, z) dv (z) dτ, (x, t) R d R. R D It is well-known [73] that v(x, t) = (V f)(x, t) satisfies v tt v = f in R d R. This leads to the time domain Lippmann-Schwinger equation for u s, u s (x, t; y) + (V [mu s ])(x, t; y) = (V [mu i ])(x, t; y) x y R d, t > 0. (1.12) Although the above has been formal, we will provide justification and mapping properties of V in Chapters 3 and 5. Note that in the case of time harmonic data, we arrive at a similar conclusion. In particular, û s satisfies û s (x; y) + ( ˆV [mû s ])(x; y) = k 2 ( ˆV [mû i ])(x; y), x y R d, (1.13) where for f C0 (B), D B, ( ˆV f)(x) := k 2 m(z)ˆφ k (x, z)f(z) dv (z), D x R d and where ˆΦ k (x, y) := i 4 H(1) is the time harmonic fundamental solution. 0 (k x y ) d = 2 exp (ik x y ) 4π x y d = 3. (1.14) Aside from providing a solution representation for the scattered field, the Lippmann- Schwinger equations provide a helpful form for computing asymptotic expansions of the scattered field. In this thesis, we will often make strong assumptions on the speed of sound in the medium which allow us to weaken assumptions on other aspects of the scattering problem. In particular, we will make a weak scattering Born approximation in which multiple scattering effects are minor and can be ignored. Assume that n(x) = 1 + ɛm B (x) for ɛ 1, m B = O(1), and that solutions to (1.6) take the 10

24 form u s (x, t) = u s 0(x, t) + ɛu s B (x, t). The function us B = us ɛ=0 is the first term in the well-known Born approximation. Indeed, if u i (x, t) is of the same order as u s 0(x, t) for x D, then separating into powers of ɛ yields u s 0 0 and that u s B satisfies 2 u s B t 2 u s B = m B 2 u i ɛ t 2 for (x, t) R d R + (1.15a) u s B(x, 0) = us B t (x, 0) = 0 for x Rd. (1.15b) Using the volume integral equation approach introduced above yields u s B(x, t; y) = (V [m B u i tt])(x, t; y), x y R d, t > 0. (1.16) Notice that the solution represented in this way does not require us to solve an integral equation to find the scattered field, but rather just to apply an integral operator. This approximation significantly simplifies calculations at the expense of applicability of results. The above is more typically done for time harmonic data, which we will discuss in depth in Chapter Qualitative Methods in Inverse Scattering A primary goal of inverse scattering theory is the reconstruction of information about unknown objects based on how acoustic or electromagnetic waves scatter off of them. Qualitative methods are able to quickly and accurately determine the shape and location of hidden objects, and require little a priori information about the objects. They are non-iterative in nature and do not require large scale wave simulations. We refer the reader to [23] for a comprehensive account of these methods for inhomogeneous media. Each qualitative method takes a similar form, as indicated by the general algorithm in Table 1.1 below. In particular, we define an indicator function, depending on z R d (and possibly on time τ [0, T ] for some T > 0), so that the function is large when z D and small otherwise. The bulk of research in qualitative methods is dedicated to deriving indicator functions and demonstrating that they are large when z D. 11

25 Algorithm Frequency (time) domain sampling methods for reconstruction of obstacles Step 1 Collect scattered field data at x Γ m, y Γ i (and t [0, T ]). Step 2 Select a set of sampling grid points Z (and τ [0, T ]). Step 3 Plot the indicator function for each z Z (and τ [0, T ]). Step 4 Post-process or regularize the indicator function to determine the collection of z D. Table 1.1: Outline of algorithm used by many qualitative techniques. The curves Γ m and Γ i refer to the locations in R d 1 where receivers and transmitters are placed, respectively. We require that both curves contain D in their interior. The primary idea behind each of these indicator functions is that a specified function, which depends on z, is in the range of an operator depending on u s if and only if z D. There are numerous specific examples of qualitative methods, such as the MUltipe SIgnal Classification (MUSIC) method [32, 42, 66, 69, 7, 58] and the factorization method [69]. The first to be developed, however, is known as the linear sampling method [35]. It is also the most thoroughly researched and hence there are linear sampling-type methods to find obstacles using time harmonic and time dependent data in the context of acoustic, electromagnetic, and elastic media (among other physical settings). We describe the linear sampling method here as a specific introduction to qualitative methods. To explain the LSM in more detail, consider far field scattering from a time harmonic incident field transmitted towards an unknown object D R d from directions ŷ S d 1, the surface of a ball in R d. Scattered field data is then collected on ˆx S d 1. The notation ˆ on a dependent variable indicates a normalized vector. This is an unfortunate conflict of notation with ˆ for time harmonic fields, though there should be no confusion between dependent variable fields and independent variable vectors. Here, far field scattering refers to the fact that transmitters and receivers are placed far (in terms of number of wavelengths) from D, which is a common assumption in the time-harmonic case. Under this assumption, it is neccessary to analyze the 12

26 behavior of the field far from the obstacle; in particular, it can be shown that û s (x) = eik x x (d 1)/2 û (ˆx) + O( x (d+1)/2 ), as x where the so-called far field pattern u is a function of observation directions ˆx S d 1. An important example of a far field pattern is that of the fundamental solution ˆΦ k : ˆΦ (x; ŷ) = γ d e ikx ŷ where γ d = e iπ/4 / 8πk for d = 2 and γ d = (4π) 1 for d = 3. We will use the far field formulation and the far field pattern of the fundamental solution in particular in Chapter 2. Also important when discussing far field problems is the Herglotz wave function, defined for functions g L 2 (S d 1 ) by v g (x) := e ikx ˆdg( ˆd) ds( ˆd) x R d. S d 1 This is a linear combination of incident fields emitted from directions on S d 1 and will prove helpful in the analysis of some inverse scattering problems below. In the case of far field scattering, we take û i (z; ŷ) = e ikz ŷ, z R d. The linear sampling method proceeds by exploiting properties of solutions to the far field equation, g z L 2 (S d 1 ), ( ˆF g z )(ˆx) = ˆΦ (ˆx, z), (1.17) where the far field operator ˆF is defined by ( ˆF g z )(ˆx) := û (ˆx; ŷ)g z (ŷ) ds(ŷ) S d 1 and where we take z Z for a set of points Z R d which contains D (or, in practice, which we think contains D). A key step in the justification of the linear sampling algorithm is to factor the far field operator so that ˆF = ĜĤ where the operators Ĝ and Ĥ have function-analytic properties which allow us to relate regularized solutions to (1.17) to D. More precisely, we define Ĥ : L2 (S 2 ) L 2 (D) by Ĥg := v g D, where v g is the Herglotz wave function 13

27 introduced above. To define Ĝ, introduce the function ŵ H1 loc (R2 ) to be the unique solution to the scattering problem ŵ + k 2 nŵ = k 2 m ˆϕ lim r r (d 1)/2 ( ŵ r ikŵ ) = 0, r = x for ˆϕ L 2 (D). Then the operator Ĝ : { ˆϕ L2 (D) : ˆϕ + k 2 ˆϕ = 0} L 2 (S d 1 ) is defined by Ĝv := ŵ. Note that if ˆϕ = û i, then ŵ = u s and so Ĝ maps to û. Indeed, it is this fact, along with the form of ˆF that leads to the factorization ˆF = ĜĤ. A key point in the justification of the LSM is that ˆΦ (, z) R(Ĝ) if and only if z D. Indeed, from this statement we see the relationship between D and ˆF which is vital to the LSM. While such a range test does not hold for ˆF, we can construct approximate solutions to the far field equation which serve as useful indicators to the location of D. Indeed, by construction, ˆF g is the far field corresponding to a linear combination of incident fields emitted from S d 1 weighted by g. As such, for z D, g z satisfies the homogeneous far field equation if and only if ˆv + k 2 nˆv = 0 v g + k 2 v g = 0 in D in D ˆv v g = ˆΦ k (, z) ˆv ν v g ν = ˆΦ k (, z) ν on D on D, where ˆv = ŵ + ˆϕ using the notation from above. This is only true, in general, for very specific choices of D, n, and k. However, theoretical conditions on D, n, and k exist under which we can find approximate solutions to the far field equation which allow us 14

28 to image D. Loosely speaking, these conditions follow from understanding the interior transmission problem, which consists of finding ˆv, ŵ, and nonzero k C which satisfy ŵ + k 2 nŵ = 0 in D ˆv + k 2ˆv = 0 in D (1.18) ŵ = ˆv, ŵ = ŵ ν ν on D. We call k a transmission eigenvalue if there exists a nontrivial solution to (1.18). If n and D are such that k is not a transmission eigenvalue, then one of the two following items holds [23]: If z D then there exists a sequence g α z L 2 (S d 1 ) such that or lim ˆF gz α ˆΦ (, z) 2 α 0 L 2 (S d 1 ) = 0 and lim α 0 Ĥgα z 2 L 2 (D) <, if z / D then for all g α z L 2 (S d 1 ) such that lim α 0 ˆF g α z ˆΦ (, z) 2 L 2 (S d 1 ) < ɛ, lim α 0 Ĥgα z 2 L 2 (D) =. This suggests a method for finding the shape of D by using the process described in Table 1.1: after collecting the scattered far field data, proceed by finding a regularized solution g α z L 2 (S d 1 ) to ( ˆF g α z )(ˆx) = ˆΦ (ˆx, z) for ˆx S d 1. By the result above, the indicator function ÎĤ,D,LSM (z) = Ĥg z 1 L 2 (D) otherwise. will be large when z D and small Note immediately that this method is extremely problematic in that ÎĤ,D,LSM (z) is a function of D which is unknown (and indeed, an operator H which cannot be computed from the data we have). In practice, the indicator function ÎLSM = g α z 1 L 2 (S d 1 provides satisfactory results. Another serious drawback of this method is that the above result does not indicate how to construct g α z. Typically, Tikhonov regularization is used and the far field equation is replaced by the regularized equation (αi + ˆF ˆF )g α z = ˆF ˆΦ (, z) for some sufficiently-small α. While this method seems to work in practice, it is not clear that the regularized solution satisfies the same blow-up properties as the theoretical solution to the far field equation. A more recently-developed technique, the 15

29 generalized linear sampling method (GLSM) combines aspects of both the linear sampling method and the factorization method and provides a theoretical and numerical technique for overcoming many of these downsides of the linear sampling method. Moreover, while the theoretical assumptions required for the application of the GLSM have partially limited its use, it allows for complete theoretical justification of a wide variety of scattering problems. The GLSM uses a second factorization of the far field operator as well as a more carefully constructed regularized solution, an indicator function similar to ÎLSM is fully justified method for locating D in the GLSM framework. More details are available in [10, 11, 23]. Note that all of this can also be done for near field problems where the far field equation (1.17) is replaced by the near field equation ( ˆNg z )(x) = ˆΦ k (x, z), z Z, (1.19) and where the near field operator ˆN is defined by ( ˆNg z )(x) := û s (x; y)g z (y) ds(y). Γ m In near field scattering, we take û i (x, y) = ˆΦ k (x, y) to be a time harmonic pulse and let x Γ i and Γ m simply be curves in the exterior of D. See, for example, [54], for details in the acoustic scattering case. As discussed above, indeed, in order for the linear sampling method to be theoretically justified, n and k must be so that (1.18) is uniquely solvable. We will discuss this problem in depth for the weakly-scattering case in Chapter 4. Note now that, until this year, a lack of understanding of the behavior of complex transmission eigenvalues has prevented full justification of the linear sampling method for penetrable media using time dependent data. However, [94] gives an optimal description of the growth transmission eigenvalues in the complex plane, leading to full justification of the technique under appropriate assumptions on n and D. Although this work is recent, we will discuss it more in the context of our own results in Chapter 4. 16

30 1.3 Data Reduction and Primary Contributions of this Thesis The above remarks have hardly considered the real-world usage of qualitative methods for inverse scattering problems. Technological advances have increased the reliability and affordability of sensors for detecting scattered field data. Nonetheless, qualitative methods have lagged behind in their requirement of large amounts of scattered field data, often requiring transmitters and receivers to completely surround an object of interest. On the other hand, reconstruction methods based on nonlinearoptimization schemes are often successful with less scattered field data, but require significant amounts of computing power, time, and a priori data in order to return satisfactory results. The main contributions of this thesis are related to addressing these problems. The linear sampling method discussed above was developed under the assumption of multistatic scattered field data - that is, every location from which an incident field is transmitted is also a location at which scattered field data is collected. Most examples in the literature have made a further restriction on the available data: multistatic data is available on a full aperture curve, completely surrounding the object of interest. This assumption requires access to an area surrounding the entire object of interest, which is infeasible for large objects or ones in difficult-to-reach places. It also increases the cost of imaging, as large devices consisting of transmitters and receivers must be constructed. The quasi-backscattering approach introduced in Chapters 2 and 3 of this thesis have been developed by the author and his colleagues in order to combat these problems. As will be described in detail in the relevant chapters, the quasi-backscattering approach makes use of a small device composed of transmitters and receivers which can be moved around the object. In this way, only a small region surrounding the object needs to be accessed at once. Note that, this improvement in experimental geometry requires the theoretical assumption of weakly-scattering objects. While this is a strong assumption, experimental evidence provided in Chapters 2 and 3 suggest the weak-scattering assumption is not completely in affect. 17

31 Another method for reducing spatial data collection requirements has been the use of time domain or multi-frequency data. For example, the time domain linear sampling method developed in [31, 52, 53] experimentally requires fewer spatial data collection points than the single-frequency linear sampling method in order to successfully reconstruct the location and shape of an object. The multi-frequency methods developed in [54] is strongly related to the time domain linear sampling method (without the requirement of causal data) and can be expected to have similar reductions in spatial data collection requirements. Inspired by these examples, we considered the time domain quasi-backscattering problem in Chapter 3. Indeed, as will be discussed in detail there, acceptable reconstructions are achievable with orders-of-magnitude fewer spatial data points than single frequency methods. We also contributed results on the theoretical justification of the time domain linear sampling method for penetrable media under the Born approximation; as shown in Chapter 3, the time domain linear sampling method for scattering in the Born regime produces accurate reconstructions with very few transmitters and receivers. However, its justification requires results for the Born transmission eigenvalue problem. As discussed above, in the case of non-weakly scattering data, these theoretical questions were not satisfactorily answered until very recently [94] where only non-absorbing media considered. To this end, Chapter 4 is only a first step in justifying the Born time domain linear sampling problem for penetrable, and possibly absorbing, media. Finally, in Chapter 5, we address the problem of time-consuming numerical simulations leading to slow object reconstructions from non-linear optimization schemes. In particular, the numerical scheme described in that chapter simulates acoustic wave scattering from penetrable media in an unbounded domain with computational complexity O(MN log N), where M is the number of desired time steps and N the number of points in a triangulation of the scattering object. Although this method is not highly accurate, it is one of the only fast methods for simulating time domain scattering through penetrable media on an unstructured mesh which exactly models the unbounded spatial domain inherent in scattering problems. 18

32 Chapter 2 QUASI-BACKSCATTERING IN THE FREQUENCY DOMAIN 2.1 Introduction In this chapter we propose a data collection geometry in which to frame the inverse scattering problem of locating unknown obstacles from far field measurements of time harmonic scattering data. The measurement geometry, which we call a quasibackscattering set-up, requires less data than traditional multistatic configurations. We demonstrate that the data collected can be used to locate inhomogeneities in problems in which the Born approximation applies. In particular, we are able to image a two-dimensional projection of the location of a small obstacle by checking if a test function, corresponding to a point in R 2, belongs to the range of a measurable operator. Combining several projections then allows us to identify the location of the small inclusions in R 3. We also show how this algorithm can be extended to the case of extended spherical inclusions. The quasi-backscattering inversion scheme we describe in this chapter makes use of a particular experimental set-up; one device acts as a transmitter and a line of receivers extends in one-dimension a small distance from the transmitter. Figure 2.1 demonstrates the difference between a usual multistatic set-up and the quasibackscattering geometry. As the figure demonstrates, the quasi-backscattering geometry requires significantly less data than the multi-static geometry, which can be beneficial in practical applications. In Section 2.2, the direct scattering problem is formulated and the quasi-backscattering data setting is explicated. In Section 2.3, we introduce and analyze the inversion procedure capable of identifying two-dimensional projections of small objects locations. We 19

33 Figure 2.1: A Comparison of multi-static data (left) and quasi-backscattering data (right). Red circles correspond to device locations. The thick blue line in the right figure denotes where the quasi-backscattering set-up is moved and where each transmitting device is located. then extend the algorithm to the case of extended spherical inclusions. In Section 2.4, extensive numerical experimentations are presented in order to show the performance of this new algorithm. We end by explaining how one can obtain three-dimensional locations from two-dimensional projections. 2.2 Direct Scattering Problem We begin by discussing the mathematical formulation for the problem of acoustic incident plane waves scattering against inhomogeneous media in three-dimensions which was discussed in Chapter 1. This problem has been studied extensively and more information about the related direct and inverse problems can be found in, e.g., [21, 36, 69]. Assume a plane wave incident field with a fixed wave number k is generated far from the area of an inhomogeneity. Such an incident field is described by û i (x, ˆd) = 20

34 e ik ˆd x for x R 3 and ˆd S 2. Recall from Chapter 1 that the total field û(x) satisfies û + k 2 n(x)û = 0 in R 3, (2.1a) û(x) = û i (x, ˆd) + û s (x), (2.1b) ( ) û s lim r r r ikûs = 0, (2.1c) where û s (x) is the scattered field, r = x is the Euclidean magnitude of x, n(x) is the bounded refractive index of the inhomogeneous medium, and (2.1c) is the Sommerfeld radiation condition which holds uniformly with respect to ˆx = x/ x. For wave numbers such that Im k 0 and compactly supported refractive indices in L, it is known that (2.1a) (2.1c) has a unique solution in H 1 loc (R3 ). Recall that the contrast function as m(x) = 1 n(x) is such that m(x) is nonzero only on a compact set D R 3 which contains the inhomogeneity. As discussed in Chapter 1, the Lippmann-Schwinger equation for time-harmonic scattering from plane waves is, û(x) = e ikx ˆd k 2 D m(y)ˆφ k (x, y)û(y) dy. (2.2) This gives an exact expression for the unique solution to (2.1a)-(2.1c) where, as in Chapter 1, ˆΦ k (x, y) = 1 e ik x y 4π x y, x y is the fundamental solution to the Helmholtz equation in R 3. Assuming [69] k 2 max m(y)ˆφ k (x, y) 1, y D D which ensures a Neumann series solution to (2.2) converges, the first term of this series gives the Born approximation û B (x) = e ikx ˆd k 2 D m(y)ˆφ k (x, y)e ik ˆd y dy. (2.3) Formally, this is Fourier transform of the time domain Born approximation introduced in Chapter 1, (1.16). The inverse problem in which we are interested is to find information about D given data about the asymptotic behavior of û s B (x), the Born approximation to the 21

35 scattered field. As in the full-scattering problem, we are able to explicitly characterize the asymptotic behavior of the scattered field because of the Sommerfeld radiation condition, (2.1c). Specifically, ( ) û s B(x) = eik x û B x (ˆx, ˆd) 1 + O, x x 2 where û B (ˆx, ˆd) is the Born approximation to what is known as the far field pattern. Using (2.3), and we conclude that û s B(x) = k 2 D m(y)ˆφ k (x, y)e ik ˆd y dy û B (ˆx, ˆd) = k2 e ik( ˆd ˆx) y m(y) dy, ˆx S 2. (2.4) 4π D Forward Problem for Quasi-Backscattering Data The above derivations have not fixed the measurement geometry. We now restrict ˆx and ˆd to the quasi-backscattering experimental set-up. In what follows, let ˆx = ˆd + ηê where η [ δ, δ] for a small constant δ and ê S 2 is a fixed unit vector which is orthogonal to ˆd. The traditional backscattering set-up corresponds with δ = 0. Using the orthogonality of ˆd with ê and the fact that both are unit vectors, a Taylor expansion about η = 0 yields ˆx = ˆd + ηê ˆd + ηê = ˆd + ηê 1 + η 2 = ˆd + ηê + O(η 2 ). As such, we choose ˆx in this way as an approximation to ˆx = ˆd+ηê ˆd+ηê up to O(η2 ). For this reason we continue to use the notation ˆx, although it is no longer normalized. Substituting this choice of ˆx into (2.4) gives û B ( ˆd + ηê, ˆd) = k2 e 2ik ˆd y e ikηê y m(y) dy, ˆd S 1 (ê) (2.5) 4π D where S 1 (ê) := { ˆd S 2 ; ˆd ê = 0}. Following the typical approach of sampling methods in inverse scattering problems, we introduce the quasi-backscattering far field 22

36 operator, ˆF : L 2 ([ δ, δ]) L 2 (S 1 (ê)) which we will use extensively in solving the inverse problem. In particular, ˆF is defined as ( ˆF g)( ˆd) = δ δ û B ( ˆd + ηê, ˆd)g(η) dη, ˆd S 1 (ê). (2.6) 2.3 Quasi-Backscattering Inverse Problem We now turn our attention to the inverse problem of reconstructing the location of inhomogeneities from the quasi-backscattering far field data. We first consider the case of obstacles which are small compared to the wavelength of the incident wave and which are sufficiently far from one another. In Section 2.3.4, we use the analysis for this case as the basis for finding the centers of extended spherical obstacles. The key result of this section is Theorem 1 which will allow us to locate obstacles by testing if a specific function is in the range of the quasi-backscattering far field operator Inverse Problem for Small Obstacles Assume there are M obstacles with supports described by D j R 3, j = 1,..., M, embedded in a homogeneous background. Let the contrast be defined by the weighted sum of characteristic functions m(x) = M j=1 m j1 Dj where m j are constants. If D j = z j + R j Ω j are small obstacles centered at a point z j R 3 with size and shape described by R j and Ω j respectively, then using (2.5) we obtain that up to O(max(R j ) 4 ) error terms, û B ( ˆd + ηê, ˆd) M τ j e 2ik ˆd z j e ikηe z j, ˆd S 1 (ê), η [ δ, δ]. (2.7) j=1 Here, τ j = k2 4π m j Ω j, where Ω j indicates the volume of Ω j, are constants related to the strength of each scatterer. Combining (2.6) and (2.7) reduces the quasibackscattering operator to ( ˆF g)( ˆd) = δ δ û B ( ˆd + ηê, ˆd)g(η) dη = M δ τ j e 2ik ˆd z j e ikηe z j g(η) dη. j=1 δ 23

37 To further simplify the far field operator, we write each obstacle s location in terms of its components parallel to ê and perpendicular to ê. For a fixed ê, we write z j = Π (z j ) + L(z j )ê, j = 1,..., M where Π maps onto the plane orthogonal to ê and where L isolates the component of a vector which is parallel to ê. For example, if ê = (0, 0, 1) and z 1 = (1, 2, 3), we would have Π (z 1 ) = (1, 2, 0) and L(z 1 ) = 3. Note that for the sake of notational conciseness we will sometimes treat Π (z) as a vector in R 2. Decomposing the locations of obstacles in this way, we can write the far field operator as ( ˆF g)( ˆd) = M j=1 δ τ j e 2ik ˆd Π (z j ) δ e ikl(z j)η g(η) dη. (2.8) Since ˆF can be computed from the measurable far field pattern data, we use it to solve the inverse scattering problem. Indeed, Theorem 1 gives conditions under which we can relate the range of ˆF to the location of a small obstacle. Such a characterization is typical for sampling-type methods such as the linear sampling or factorization schemes, as well as the MUSIC algorithm. Before stating this theorem, we prove two short lemmas which are required for its proof. Lemma 1. Assume ê S 2 is fixed and let z j R 3, j = 1,..., M be distinct points whose components in the direction of ê differ (i.e., L(z i ) L(z j ), i j). Then A = {η e ikl(z j)η, j = 1,..., M} is a linearly independent sequences of functions for η [ δ, δ]. Proof. We would like to show that the Wronskian matrix of A, denoted by W, is non-singular. A short calculation shows that det(w ) = c(η)det(v ) where c(η) = ( exp ikη ) M j=1 L(z j) is a function which never vanishes and V (i,j) = ω j 1 i for ω = ikl(z i ). Since V is a Vandermonde matrix, it has a non-zero determinant so long as ω i ω j for each i j, which is true by the assumption on L(z i ), i = 1,..., M. Lemma 2. Assume ê S 2 is fixed, let z j R 3, j = 1,..., M be distinct points, and let z R 3 be any point perpendicular to ê and distinct from each Π (z j ). Then 24

38 B = { ˆd e 2ik ˆd z, z = z, Π (z 1 ),..., Π (z M )} is a linearly independent sequences of functions for ˆd S 1 (ê). The proof of this lemma follows the idea of Theorem 4.1 in [69]. This theorem of Kirsch and Grinberg implies that the above can also be proven for a finite number of ˆd j, η j S 1 with a similar but more technical argument. Proof. To show that B is linearly independent, assume β 0 e 2ik ˆd z + M β j e 2ik ˆd Π (z j ) = 0 for ˆd S 1 (ê). j=1 The left-hand-side of the above equation is, up to a constant multiple, the far field pattern of the function where x β 0 ˆΦk (x, z ) + M β j ˆΦk (x, Π (z j )) j=1 ˆΦ k (x, z) = i 4 H(1) 0 (2k x z ), x z (2.9) is the (radiating) fundamental solution of the Helmholtz equation in R 2 with wave number 2k and H (1) 0 is a Hankel function. As such, since the far field pattern vanishes, Rellich s lemma and unique continuation show that β 0 ˆΦk (x, z ) + M β j ˆΦk (x, Π (z j )) = 0 for x / {z, Π (z 1 ),..., Π (z M )}. j=1 Taking the limit as x approaches each of z and Π (z j ), j = 1,..., M shows immediately that B is a linearly independent sequence of functions for each ˆd S 1 (ê). obstacles. With these lemmas in hand, we are ready to prove the key theorem for small Theorem 1. Assume ê S 2 is fixed and ˆd S 1 (ê). Let z j R 3 for j = 1,..., M and let z R 3 be orthogonal to ê. If the components of each z j parallel to ê are not equal (i.e., L(z i ) L(z j ), i j), then ˆφ z ( ˆd) 2ik ˆd z = e z {Π (z j ), j = 1,..., M}. R(F ) if and only if 25

39 Proof. Let z / {Π (z j ), j = 1,..., M} be orthogonal to ê. Assume by contradiction that there exists some g(η) L 2 ([ δ, δ]) such that ( ˆF g)( ˆd) = e 2ik ˆd z. From the definition of ˆF, this would imply e 2ik ˆd z = M c j e 2ik ˆd Π (z j ), j=1 δ where c j = τ j δ e ikl(zj)η g(η) dη are constants. However, this is a contradiction with the linear independence of { ˆd e 2ik ˆd Π (ζ), ζ = z, Π (z 1 ),..., Π (z M )}, which shows that if ˆφ z R( ˆF ) then z {Π (z j ), j = 1,..., M}. To prove the second half of the theorem, assume L(z i ) L(z j ), i j and z {Π (z j ), j = 1,..., M}. We will show that ˆφ z N ( ˆF ) = R( ˆF ) which gives the result since ˆF is a finite rank operator with closed range. A short calculation gives that ( ˆF h)(η) = M τ j e ikl(z j)η h( ˆd)e 2ik ˆd Π (z j ) ds( ˆd). j=1 S 1 (ê) If h N ( ˆF ), then M τ j e ikl(z j)η j=1 S 1 (ê) h( ˆd)e 2ik ˆd Π (z j ) ds( ˆd) = 0. The linear independence of {e ikl(z j)η, j = 1,..., M} proven in Lemma 1 gives that for each j = 1,..., M, 0 = S 1 (ê) h( ˆd)e 2ik ˆd Π (z j ) ds( ˆd) = ( h( ˆd), ˆφ Π (z j )( ˆd) ) L 2 (S 1 (ê)) where (, ) L 2 (S 1 (ê)) indicates the inner-product on L2 (S 1 (ê)). As such, ˆφ z N ( ˆF ) for each z {Π (z j ), j = 1,..., M}, which gives the result. The proof of Theorem 1 in fact implies a slightly stronger result. Corollary 1. With no restrictions on L(z j ) and the same hypotheses on ê as in Theorem 1, if ˆφ z R( ˆF ) then Π (z) {Π (z j ), j = 1,..., M}. 26

40 Another corollary to Theorem 1 is that, for the appropriate restrictions on Π (z j ), P ˆφ z = 0 if and only if z {Π (z j ), j = 1,..., M} where P : L 2 (S 1 (ê)) R( ˆF ) is the orthogonal projection onto the orthogonal complement of the range of ˆF. P This suggests that the function Î(z) = 1 ˆφz for each z perpendicular to ê within a region of interest will be large when z is near Π (z j ), j = 1,..., M and small otherwise. This is exactly the MUSIC-type algorithm which we will use to locate the centers of small objects. To construct the imaging function Î(z), let (u k, σ k, v k ), k = 1, 2,... be the singular system for ˆF where the left singular functions are u k L 2 (S 1 (ê)) and the right singular functions are v k L 2 ([ δ, δ]). Since R( ˆF ) is spanned by the left singular functions u k which correspond to singular values σ k = 0, we can write ( ( ) Î(z) = ˆφz, u k k=r+1 L 2 (S 1 (ê)) 2 ) 1, (2.10) where r is the number of non-zero singular values. Numerical results showing that Î(z) is large near obstacles are given in Section A Second Range Test for Three-Dimensional Reconstructions Assume that the range test described above has been performed so that {Π (z j ), j = 1,..., M} are known. From Theorem 1, ˆφ Π (z k )( ˆd) R( ˆF ) for a given k = 1,..., M. As such, there is a g Π (z k ) L 2 ([ δ, δ]) such that ( ˆF g Π (z k ))( ˆd) = M c j e 2ik ˆd Π (z j ) = e 2ik ˆd Π (z k ) j=1 δ where, as before, c j = τ j g δ Π (z k )(η)e ikl(zk)η dη. By linear independence, c j = τ j δ jk, where δ jk is the Kronecker delta function. This suggests a second indicator function which can be used to find L(z j ) when Π (z j ) are already known. Formally, ( δ 1 Ĵ Π (z k )(z) = g Π (z k )(η)e ikl(z)η dη ) (2.11) δ 27

41 is arbitrarily large when z = L(z j ), j k. This argument is formal and we have no guarantee that ĴΠ (z k )(z) is small away from z = z j, j k. Nevertheless, in the numerical examples in Section below, Ĵ Π (z k ) indicates the location of L(z j ), as expected when Π (z j ) are known accurately and g z (η) is calculated using Tikhonov regularization and the Morozov discrepancy principle. As the numerical simulations will demonstrate, however, calculating L(z j ) in this manner is not robust to noise Inverse Problem for Coplanar Small Obstacles Due to the hypotheses on Theorem 1, the algorithm outlined above does not necessarily locate an object in the case that L(z i ) = L(z j ) for some i j 1,..., M. This problem can be easily alleviated: in all proofs we have assumed a fixed ê S 2. Since L(z j ) is a function of ê, we can perform multiple quasi-backscattering experiments with different ê directions to solve the problem. Indeed, we recommend this for purely geometric reasons as well. Since the quasi-backscattering technique gives only twodimensional projections of the locations of scatterers, two obstacles which lie on top of each other with respect to ê (i.e., Π (z i ) = Π (z j ) but L(z i ) L(z j )) will appear as the same obstacle in the reconstruction. Multiple experiments corresponding to different ê directions helps to fix this problem as well. In Section 2.4.3, we outline a technique for using data from multiple experiments with different ê directions to reconstruct obstacles in three-dimensions. Before continuing, we note that the algorithm outlined above does not necessarily identify obstacles if L(z i ) = L(z j ) for all i j. In particular, under these conditions, we show below that there is no obvious reason which suggests that Î(z) will be arbitrarily large at z = Π (z j ), j = 1,..., M. Indeed, the numerical simulations in Section 2.4 indicate that the reconstruction of co-planar obstacles is sensitive to noise. Assume that L(z i ) = L(z j ) for each i, j = 1,..., M. Since we can shift the origin with a change of variables, we set each L(z i ) = 0 without loss of generality. In 28

42 this case, the far field operator becomes ( ˆF g)( ˆd) = M j=1 δ τ j e 2ik ˆd Π (z j ) δ g(η) dη, ˆd S 1 (ê), and Fubini s Theorem gives that for η [ δ, δ], ( ˆF h)(η) = M τ j j=1 S 1 (ê) e 2ik ˆd Π (z j ) h( ˆd) ds( ˆd) = ( h( ˆd), ) M τ j ˆφΠ (z j ) j=1 L 2 (S 1 (ê)) Let h( ˆd) = u k ( ˆd) for a fixed k where u k ( ˆd) is a left singular function of ˆF corresponding to a singular value σ k = 0. Since u k N ( ˆF ), ( ) u k ( ˆd), M τ j ˆφΠ (z j ) j=1 L 2 (S 1 (ê)) = ( ˆF u k )(η) = 0. However, we cannot conclude from the above equation that ( u k ( ˆd), ˆφ ) Π (z j ) = 0, j = 1,..., M. L 2 (S 1 (ê)) Inverse Problem for Extended Obstacles We now adapt the arguments given in the previous section to the problem of finding extended obstacles. We show that, for small δ, the arguments given in Theorem 1 apply directly to locating the center of extended spherical obstacles. While the spherical nature of the extended obstacles does not seem to be required, it is not clear that we can uncover more information than the location of the center of these obstacles. In this section, assume there are M obstacles D j again of the form D j = z j + R j Ω j, where z j are the obstacles center, Ω j their shape, and R j their size. Now, however, assume each Ω j = B(0; 1) is a ball centered at zero of radius one and that R j is of similar size as the wavelength or larger. Assume, as before, that the contrast is. 29

43 defined by m = M j=1 m j1 Dj where m j are constants. We will begin our discussion by calculating û B (ˆx, ˆd) under these assumptions. From (2.4) we have û B (ˆx, ˆd) = k2 4π = k2 4π M m j e D ik( ˆd ˆx) y dy j j=1 M m j e ik( ˆd ˆx) z j j=1 B(0;R j ) e ik( ˆd ˆx) y dy. To simplify this expression into a more useful one, we state the following lemma. Lemma 3. For a constant R > 0 and any two vectors x, y R 3, e ix y dy = 4π (sin(r x ) R x cos(r x )). x 3 B(0;R) Proof. Under the change of coordinates y rŷ where r = y, R e ix y dy = r S 2 e irx ŷ ds(ŷ) dr. (2.12) 2 B(0;R) 0 It is known that S 2 e irx ŷ ds(ŷ) = 4πj 0 (rx) where j 0 is the spherical Bessel function of order zero [77]. Since j 0 (x) = sin(x), an integration-by-parts gives the result. x We are interested in the above result for x = k( ˆd ˆx) and R = R j. With this choice of parameters, M ( û B (ˆx, ˆd) = k 2 m j e ik( ˆd ˆx) z j sin(kr j ˆd ˆx ) kr j ˆd ˆx cos(kr j ˆd ) ˆx ) (k ˆd. ˆx ) 3 j=1 Returning to the quasi-backscattering approach and letting ˆx = ˆd + êη, a Taylor expansion about η = 0 gives that for j = 1,..., M, sin(kr j ˆd ˆx ) kr j ˆd ˆx cos(kr j ˆd ˆx ) (k ˆd ˆx ) 3 = sin(2kr j) 2kR j cos(2kr j ) (2k) 3 + O(η 2 ). If we again define the quasi-backscattering far field operator as ( ˆF g)( ˆd) = δ δ ub ( ˆd, η)g(η) dη, we find ( ˆF g)( ˆd) = = M ( δ τj L e 2ik ˆd z j j=1 M j=1 δ ( δ τj L e 2ik ˆd Π (z j ) ) e ikηê z j g(η) dη + o(δ 2 ) δ ) e ikl(zj)η g(η) dη + o(δ 2 ) 30

44 where τj L = m j (sin(2kr j ) 2kR j cos(2kr j ))/8k. Here, the asymptotic analysis follows from an application of Cauchy-Schwarz. Up to constants and o(δ 2 ), the quasibackscattering operator for extended spheres is identical to (2.8), the quasi-backscattering operator for small obstacles. As such, using the same technique described in Section 2.3.1, we can find two-dimensional projections of the centers of extended spherical obstacles. Note, incidentally, that τj L = 1 3 k2 m j Rj 3 + O(Rj), 5 which matches the expression for τ j used in the case of small spheres. 2.4 Numerical Experiments In this section, we give numerical results demonstrating the effectiveness of the above technique. In all experiments, we approximate obstacles by spheres with a small radius. Specifically, the radius for each obstacle is 1/500 units. We will use simulated forward data which is corrupted by random noise. Using the formula given in (2.4) we simulate û (ˆx, ˆd) using numerical integration. Numerically integrating (2.6) gives a discrete representation of the far field matrix, ˆF ij, which is corrupted by ˆF ij (1 + γξ) where ξ is a uniform random variable in [ 1, 1] and γ is a constant related to the level of noise. To calculate the indicator function, we compute the singular value decomposition of ˆF = USV and use U to calculate a discrete version of (2.10). The approximate imaging function is regularized by computing with all but the first ten singular vectors (i.e., r = 9 in (2.10)). In all examples, we take k = 15 to be the wave number. Other parameters are given for each experiment. The experimental parameters discussed above merit a few comments. The first is related to our use of ten singular vectors in reconstructions. Typically when using a MUSIC-type algorithm, the number of singular vectors is related to the number of unknown obstacles, which is estimated by the numerical rank of the far field operator. However, in our numerical experiments we have found such a technique to be sensitive to added noise. As such, we took the number of singular vectors as an upper bound of the number of obstacles. The results do not change noticeably when using the same number of singular vectors as there are obstacles. The second comment is related to 31

45 the relatively-high wave number used in these experiments. In the case of extended obstacles, low wave numbers are used to ensure that the Born approximation of the far field is valid. However, because we assume our objects are very small (a radius of 1/500 units), we are justified in using a higher wave number. We present three types of numerical inversions. In Section 2.4.1, we show twodimensional projections of small obstacles. In Section 2.4.2, we generate the third unknown coordinate assuming the first two are known. Finally, we use multiple ê- directions to generate full three-dimensional reconstructions for small obstacles in Section Two-Dimensional Projections of Small Obstacles We give several numerical examples in this section which help demonstrate both the strengths and weaknesses of the quasi-backscattering technique. In all reconstructions, darker colors correspond to higher values of the imaging function which correspond with the predicted locations of the obstacles. A small red circle in each picture corresponds to the true location of each obstacle. Note that the size of the dark areas near obstacle locations do not correspond to an estimate of obstacle size, but are merely an artifact of the way in which reconstructions are displayed. In all reconstructions, we use 80 2 sampling points uniformly chosen in the unit-square. In all experiments we use 95 incident directions and for each incident direction we use 95 locations for ˆx between δ and δ. We call these points between δ and δ the observation points. This is a large number of both incident directions and observation points and, indeed, acceptable results are achievable with far fewer. However, we prefer to focus these experiments on the affect geometric and physical parameters have on reconstructions. The first example, given in Figures 2.2 and 2.3, shows the algorithm differentiating between multiple small obstacles, added one at a time. The obstacles are located at z 1 = ( 0.25, 0.25, 0.5), z 2 = (0.25, 0.25, 0.25), z 3 = (0.25, 0.25, 0.25), and z 4 = ( 0.25, 0.25, 0.5). 32

46 y y x x y y x x Figure 2.2: Symmetric addition of new objects, δ = π/50 (no noise). For the next example, we show the affect of δ on reconstructions. While the motivation for the quasi-backscattering set-up comes from a Taylor expansion about η = 0 (and hence small δ), the experiments in Figure 2.4 show that in the presence of noise, the reconstruction technique is not stable for too small of δ, in particular when many obstacles are present. All three figures have z 1 = (0, 0.5, 0.25), z 2 = (0, 0.5, 0.75), z 3 = (0.5, 0, 0.25), and z 4 = ( 0.5, 0, 0.75). 33

47 y y x x y y x x Figure 2.3: Symmetric addition of new objects, δ = π/50 (approximately 1% noise). y x y y x x Figure 2.4: Decreasing the quasi-parameter δ. Figures have δ = π (top-left), δ = π/50 (top-right), and δ = π/100 (bottom). Approximately 5% noise. In the experiment in Figure 2.5 we show the resolution achievable by the quasibackscattering technique. Often, inversion schemes based on the Born approximation or a Fourier transform are limited to a half-wavelength resolution. Though we do not 34

48 show this rigorously, the numerical example in Figure 2.5 suggests such a limitation for the quasi-backscattering technique. Indeed, we see that the method is unable to differentiate between obstacles once they are within half a wavelength of each other. In this case, the technique gives a large range of possible locations, containing the true centers of the obstacles. In this experiment, there is a constant 0.2 unit distance between the z-coordinate of the obstacles. y y x x y y x x Figure 2.5: Two objects moving closer to each other, δ = π/100 (no noise). Thick bar at bottom corresponds to half of wavelength. The final two experiments of this type show the need to take multiple experiments with different ê directions when the underlying geometry of the obstacles is complicated. Figure 2.6 shows reconstructions from three different ê directions of three small obstacles which would form an approximate L -shape if they were connected with straight lines. In particular, z 1 = ( 0.25, 0.25, 0.25), z 2 = (0.25, 0.24, 0.25), and z 3 = ( 0.25, 0.26, 0.25). Due to the geometry of the obstacles, taking ê = (0, 0, 1) or ê = (1, 0, 0) only gives reconstructions of two of the three obstacles. By taking ê = (0, 1, 0), however, we are able to find all three obstacles. 35

49 Finally, we apply the quasi-backscattering algorithm to the reconstruction of co-planar obstacles that is, obstacles which violate the assumptions in Theorem 1. As Figure 2.7 shows, in the absence of noise, reconstructions are acceptable. However, under the addition of noise, the reconstructions become less clean. Changing ê so that L(z i ) L(z j ) results in more acceptable reconstructions. The figures are located at z 1 = (0.75, 0.75, 0.25) and z 2 = ( 0.25, 0.25, 0.25). y x (a) XY -plane z y (b) Y Z-plane z x (c) XZ-plane Figure 2.6: An L-Shaped geometry which requires 3 views to see all obstacles, δ = π/30 (approximately 1% noise)

50 y z x (a) XY plane x (c) XZ plane y z x (b) XY plane x (d) XZ plane Figure 2.7: Results for co-planar obstacles, δ = π/60. Figures on left are noise-free and figures on right have approximately 1% noise Reconstruction of Third Coordinate We now show two reconstructions of the third coordinate of a small obstacle, assuming the other two coordinates are known. We use the indicator function given by (2.11) where g z (η) is calculated using Tikhonov regularization plus the Morozov discrepancy principle. In both reconstructions, we take δ = π/50 and 377 observation points and incident directions. Though this is an unrealistically-large number of observation points and incident directions, we will show that the indicator function is still sensitive to noise. In both reconstructions, ê = (0, 0, 1) so that we are generating the z-coordinate in a typical Cartesian plane. For this reason, we explore another technique for three-dimensional reconstructions in Section below. In Figure 2.8, let z 1 = ( 0.24, 0.24, 0.75), z 2 = (0.26, 0.24, 0), and z 3 = (0.26, 0.26, 0.75). Adding no noise and assuming the two-dimensional projections of 37

51 z j, j = 1, 2, 3 are known exactly, the figure demonstrates we are able to construct L(z j ) under ideal circumstances. J z z (a) Ĵζ(z), ζ = Π (z 1 ) 0.03 J z z (b) Ĵζ(z), ζ = Π (z 2 ) J z z (c) Ĵζ(z), ζ = Π (z 3 ) Figure 2.8: When ĴΠ (z j ) is computed, peaks appear for i j. Red circles show exact location of L(z j ). We consider a more realistic scenario in Figure 2.9. Let z 1 = ( 0.25, 0.25, 0.75), z 2 = (0.25, 0.25, 0), and z 3 = (0.25, 0.25, 0.75). However, we have added 0.1% noise and assume we guess Π (z 1 ) = ( 0.24, 0.24), Π (z 2 ) = (0.26, 0.24), and Π (z 3 ) = (0.26, 0.26). We see that even under small perturbations, the accuracy of the reconstructions is dramatically decreased. 38

52 J z 0.05 J z z (a) Ĵζ(z), ζ = Π (z 1 ) z (b) Ĵζ(z), ζ = Π (z 2 ) J z z (c) Ĵζ(z), ζ = Π (z 3 ) Figure 2.9: Reconstructions lose accuracy in the presence of 0.1% noise. Red circles show exact location of L(z j ) Three-Dimensional Reconstructions The inversion schemes described above do a good job locating obstacles within the two-dimensional plane perpendicular to the selected ê direction. Given data from multiple experiments with multiple ê directions, we are better able to find the full threedimensional coordinates of an obstacle or set of obstacles. As discussed above, there are many scenarios in which reconstructing obstacles with multiple ê is encouraged. In this section, we show that multiple ê directions can be used to calculate three-dimensional reconstructions of obstacle locations. The creation of three-dimensional images from a selection of two-dimensional projections has been thoroughly studied in the image processing literature and we do not attempt to use state-of-the-art techniques here. Instead, we perform multiple quasibackscattering experiments on the same obstacle set-up, interpolate the results from each experiment onto a fixed sampling grid, and average the results. We regularize 39

53 our results for each ê before computing the averaged result. In particular, we apply a total variation minimization algorithm (see [28, 29, 86]) which emphasizes changes in gradient and hence sharpens edges. We next locally normalize each two-dimensional projection over a 5 5 grid of sampling points to further sharpen edges. After these regularization steps are performed, we average on a sampling grid as described. We compute forward and inverse data for this section as we did in Section Here, however, we vary ê. Specifically, we take 30 values of ê from a circle in the XY -plane. The results are given as three-dimensional contour plots of the imaging function. The contour which is plotted is α max Î(z) where α is a value between 0 and 1. In Figure 2.10, we demonstrate the techniques described above to compute three-dimensional object reconstructions. In particular, we consider three small objects located at the points z 1 = ( 0.5, 0.5, 0, 5), z 2 = (0.5, 0.5, 0.5), and z 3 = ( 0.5, 0.5, 0.5). Notice that these are in a geometry which forms an L -shape, as in Figure 2.7. As demonstrated above, when we use two-dimensional projection techniques, we require at least three ê directions to locate all objects for such a geometry. By taking more ê directions, however, we are able to give a full three-dimensional image of the geometry. 40

54 Figure 2.10: Three-dimensional noise-free reconstructions of point obstacles based on multiple experiments. We take 95 incident directions, 95 observation points, and use δ = π/60. The top figure is noise-free while the bottom figure has approximately 1% noise. In both figures, we display isovalues of 0.6 times the maximum value of the imaging function. 41

55 Chapter 3 DIRECT IMAGING OF SMALL SCATTERERS USING REDUCED TIME DEPENDENT DATA In this chapter, we introduce qualitative methods for locating small objects using time dependent acoustic near field waves. These methods have reduced data collection requirements compared to typical qualitative imaging techniques. As in Chapter 2, we only collect scattered field data in a small region surrounding the location from which an incident field was transmitted. The new methods are partially theoretically justified and numerical simulations demonstrate their efficacy. We show that these reduced data techniques give comparable results to methods which require full multistatic data and that these time dependent methods require less scattered field data than their time harmonic analogs. 3.1 Introduction We propose two schemes in this chapter which significantly reduce the amount of data required for accurate reconstructions. In both schemes, we use a small array of transmitters and receivers constructed so that data is collected only in a small region. Incident waves are emitted from the transmitters, collected by the nearby receivers, and the entire device is moved to a new location where the experiment is repeated. In one scheme, we allow the device to contain many transmitters and receivers, collecting multistatic data only in patches with a small aperture. In the other scheme, a quasi-backscattering set-up, the array contains one transmitter and a small number of receivers in a small neighborhood of the transmitter. We must increase the amount of a priori information we assume about the object in order to justify these 42

56 methods theoretically. As in Chapter 2, we will assume objects are small and weakly scattering. The quasi-backscattering data collection scheme proposed here is somewhat similar to the time harmonic study initiated in Chapter 2 and studied further in [56], though the reconstruction method and applicability of the method here differs. Of particular importance here is that the algorithms described below directly use causal time dependent near field data and require no Fourier or Laplace transformation into frequency domain data. In many applications, ranging from medical imaging to nondestructive testing, time dependent data is readily obtained. Moreover, as our numerical examples will demonstrate, using time dependent data allows us to use significantly fewer transmitters and receivers than time harmonic data. Most previous studies of similar problems make use of time harmonic far field data with one or multiple frequencies. In some applications, far field data is required due to physical constraints on how near to an object sensors can be placed. Nonetheless, near field data is sometimes easier to obtain in practice, and typically results in higher resolution reconstructions. Furthermore, the type of data collection scheme suggested here is readily implementable in practice. For example, a device with transmitters and receivers concentrated in a small region was built in [45] to collect scattered field data for potential industrial applications. To make the above comments precise, assume scattering is caused by time dependent acoustic waves propagating through a medium with a variable speed of sound, c L (R d ) (d = 2 or 3) so that c(x) γ > 0 for some γ R. We assume a constant background speed of sound, c 0 = 1. Let u i (x, t; y) indicate the incident field emitted from a point y R d evaluated at a point x R d \{y} and time t R +. Recall from Chapter 1 that such an incident field satisfies the free space acoustic wave equation, u i tt x u i = 0 for x R d \{y}, t R +. 43

57 The resulting scattered field, u s (x, t; y), satisfies c 2 (x)u s tt u s = (c 2 (x) 1)u i tt (x, t) R d R + (3.1a) u s (x, 0) = u s t(x, 0) = 0 x R d. (3.1b) Define n(x) = c 2 (x) to be the index of refraction and D = supp(1 n(x)) to be the location of the unknown scatterer. We will be more precise about n and D below. Let χ C 2 (D) be a causal temporal pulse function (that is, χ and its derivatives vanish for t < 0). We define the incident field originating at a point y R d as the time convolution of χ with the fundamental solution Φ to the wave equation. Recall from Chapter 1 that H(t x ) 2π t 2 x, d = 2, 2 Φ(x, t) = δ(t x ), d = 3, 4π x where H is the Heaviside function. For example, in R 3 u i (x, t; y) = χ(t x y ), (x, t) (R 3 \y) R +. 4π x y (3.2) The inverse problem is to find D from u s (x, t; y) for x Γ m, y Γ i, t R + where the measurement and incident locations, Γ m and Γ i respectively, are sets in R d 1 which do not intersect with D. For example, in a full aperture multistatic set-up, Γ m = Γ i = B R (0), where B R (0) is the boundary of a ball of radius R > 0 centered at the origin where R is large enough that D B R (0). In the limited aperture case Γ m, Γ i B R (0) (possibly Γ m = Γ i ). See Figure 3.1 (left) for a sample of a limited aperture multistatic geometry. In this chapter, we will primarily use reduced data. First, we use a series of limited aperture multistatic arrays which are moved around the obstacles. For example, let Γ i = Γ m be patches with a small area in R 3. We collect multistatic data with this patch and then move the entire array to a new location and collect data again. The second type of reduced data is a quasi-backscattering set-up. To describe this data setup, let Γ i R d 1 be the curve on which we will place transmitting devices. We again 44

58 assume we can collect data only with a small device which moves around Γ i. Denote by δ > 0 a small constant. For each fixed y Γ i, data is collected on Γ (y) m := Γ i B δ (y), where B R (x) is the ball of radius R > 0 centered at x R d. See Figure 3.1 (right) for a sample set-up geometry in R 2. Note that this set-up requires more data than the related backscattering data, in which each transmitter has just one associated receiver, and both are located at the same point. u s D Γ m u i y Γ i D B δ (y) y Γ (y) m u i u s D D Γ i u s u s Figure 3.1: Examples of limited aperture multistatic (left) and quasi-backscattering (right) measurements. In the limited aperture multistatic figure, the blue line represents the location of transmitters and the red line the location of receivers. In the quasi-backscattering set-up, Γ i is the large dashed circle, the thick solid line is Γ (y) m for a fixed y Γ i, and the circles on Γ i not located at y represent locations to which Γ (y) m will be moved. There exist many qualitative methods for solving inverse scattering problems with multistatic time domain or multifrequency data. In [52, 53], a qualitative method known as the linear sampling method is used to approximate the shape of D using causal multistatic time-domain scattering data. In these papers, the theoretical justification of the method remains incomplete due to technical problems involving an associated problem called the interior transmission problem described in Chapter 1. However, a new result on transmission eigenvalues [94] alleviates this difficulty. This will be discussed in more detail in Section This is in contrast to the time domain linear sampling method for scattering from bounded objects with Dirichlet, Neumann, 45

59 or Robin boundary conditions whose theory is fully described in [31, 57]. The multifrequency linear sampling method, which can be seen as time dependent technique with non-causal waves, is studied in [54]. In [84, 91], it was shown that, under certain conditions, a potential function related to speed of sound can be calculated based on backscattered time domain data collected in the far field. While these require less data than we do, they solve a slightly different problem than we do here and do not provide a method for constructing the potential. Time reversal methods, described for example in the review article [44], are also popular for solving inverse scattering problems with time dependent data. The time harmonic backscattering problem for small and weak scatterers was studied in [50] using multiple frequencies. A number of recent reconstruction algorithms have been proposed, e.g. [1, 51, 80], which reduce data requirements by using only one incident source and scattered field data with receivers surrounding the objects. Such approaches result in fast data collection, since there is only one experiment required, but require that the objects can be simultaneously surrounded by receivers. We take a different approach to data reduction here, assuming there is enough time to perform many experiments, but that the objects cannot be completely surrounded by receivers at the same time. This approach is useful in the case of imaging large regions or in cases where it is costly to place many receivers at once. 3.2 Forward Model and the Born Approximation We begin by discussing the well-posedness of (3.1). This is well known, and to discuss it precisely we follow [13, 52, 73], introducing some space-time Sobolev spaces described through the Fourier-Laplace transform. This will allow us to introduce and state the well-posedness of a time domain weak scattering approximation and its frequency domain counterpart. This approximation is the Born approximation described in Chapter 1. These will be used in Section to validate a multistatic MUSIC-type algorithm in the time domain. 46

60 As discussed in the introduction to this chapter, we are able to reduce the amount of data required for reconstructions by making a priori assumptions on the contrast n and the scatterer D. In particular, we will make a weak scattering Born approximation in which multiple scattering effects are minor and can be ignored. Assume that n(x) = 1 + ɛm B (x) for ɛ 1, m B = O(1), and that solutions to (3.1) take the form u s (x, t) = u s 0(x, t) + ɛu s B (x, t). Recall from Chapter 1 that the function u s B = us ɛ ɛ=0 is the well-known Born approximation which satisfies 2 u s B t 2 u s B = m B 2 u i t 2 for (x, t) R d R + (3.3a) u s B(x, 0) = us B t (x, 0) = 0 for x Rd. (3.3b) This time domain Born approximation should be considered as a linearization of the scattered field with respect to the strength of scatterers, rather than as the first term of a series solution to (3.1) in the way that the time harmonic Born approximation sometimes is; as is discussed in Remark 4.5 of [70], terms associated with higher order terms in ɛ are not necessarily well-defined in any reasonable spaces. We follow the same process for solving (3.3) as in Chapter 1 and take a spacetime convolution of m B (x)u i tt with Φ(x, t). This results in a time domain Lippmann- Schwinger equation, u s B(x, t; y) + (V [m B u s B])(x, t; y) = (V [m B u i ])(x, t; y) x y R d, t > 0. Recall that the retarded volume potential operator V is defined by (V f)(x, t) := Φ(x z, t τ)f(τ, z) dv (z) dτ, (x, t) R d R R D where Φ(x, t) is given by (3.2). Later we will also use the related single layer potential, S Γ, defined by (S Γ f)(x, t) := Φ(x y, t τ)f(τ, y) ds(y) dτ, (x, t) (R d \Γ) R R Γ where Γ is some closed surface. 47

61 In order to make these equations precise, we recall the appropriate space-time Sobolev spaces, following [52, 73]. To this end, we first introduce the Fourier-Laplace transform. Let ω = η + iσ for η, σ R with σ > σ 0 > 0 for some σ 0 R. We use the notation C σ0 = {ω C : Im(ω) σ 0 > 0} to define this half-plane. Let X be a Hilbert space. The set of temporal, smooth, and compactly supported in [0, ) X-valued functions is denoted by D(R + ; X) = C0 (R; X). The associated X-valued distributions on the real line which vanish for time t < 0 are denoted by D (R + ; X) and the corresponding tempered distributions by S (R + ; X). Define L σ(r +, X) := {f D (R +, X) : e σt f S (R +, X), for some σ(f) < } to be the space of functions with well-defined Fourier-Laplace transforms. Indeed, the Fourier-Laplace transform of f = f(x, t) L σ(r +, X), denoted by ˆf(x, ω) is given by ˆf(x, ω) = 0 f(x, t) exp (iωt) dt, ω C σ0 for σ 0 = σ 0 (f) and x R d, t R. Note that ˆf(x, ω) = F(e σt f)(η) where F represents the typical Fourier transform on causal functions, so many properties of the Fourier transform will transfer to the Fourier-Laplace transform with little change. We can now define the Hilbert space for p N 0, σ R, H p σ(r +, X) := { f L σ(r + ; X) : R+iσ ω 2p ˆf(, ω) 2 By Parseval s theorem, the norm of this space is equivalent to f 2 Hσ(R p + ;X) = e 2σt p 2 f(, t) t p dt 0 X X } ds <. where we have used the fact that f and its derivatives vanishes for t < 0. For more details see e.g. [60]. With this notation in hand, we have the following result about the solvability of (3.3), where σ > σ 0 > 0 for a σ 0 depending on the specifics of the problem. 48

62 Theorem 2 ([73], Theorem 3.2). For r = 0, 1, 2 and p R, V : H p σ(r +, L 2 (D)) Hσ p+1 r (R +, H r (R d )) is a bounded linear operator. Moreover, if v = V (f) for some f H p σ(r +, L 2 (D)) then v(t) = 0 for t < 0 and v H p σ(r +, H 1 (R d )) satisfies v tt v = f in H p 1 σ (R +, L 2 (R d )). Theorem 2 allows us to write u s B(x, t; y) = (V mu i tt)(x, t) (3.4) = m(z)φ(x z, t τ)u i tt(z, τ; y) dv (z) dτ, (x, t) R d R +. R D For later, we introduce the bounded linear solution operator for (3.3), G : H p σ(r +, L 2 (D)) H p+1 r σ (R +, H r (R d )). (3.5) which takes u i to u s B with (3.4). Here σ, p, and r are as in Theorem 2. From the above, the solution of the Born wave equation satisfies u s B H p σ (R +,H 1 (R d )) C u i H p σ (R +,L 2 (D)). (3.6) See also [87] and Chapter 5 for a discussion of these properties in both R 2 and R 3. Taking the Fourier-Laplace transform of V gives an equivalent formulation in the frequency-domain. In particular, for f C0 (D) define the operator ˆV by ( ˆV f)(x; ω) = ˆΦ ω (x, z)f(z) dv (z), x R d D where ˆΦ ω (, ) is the fundamental solution of the Helmholtz equation with wavenumber ω C σ0. for some σ 0 > 0. It can be shown [73] that ˆV : L 2 (D) H 2 (R d ) and that if ˆv = ˆV f then ˆv satisfies ˆv + ω 2ˆv = f in R d. Hence, û s B + ω 2 û s B = ω 2 mû i (3.7) 49

63 has the solution û s B(x, ω; y) = ω 2 D m(z)ˆφ ω (x, z)û i (z; y) dv (z). (3.8) This is identical to the plane wave scattering case considered in Chapter 2, with a different incident wave. If Im(ω) = 0 then the Fourier-Laplace transform becomes the standard Fourier transform, (3.7) becomes the usual equation for a Born approximation to the time harmonic scattered field with wavenumber Re(ω), and (3.8) is the first term of the Born series. 3.3 Inverse Problem for the Time Domain Born Approximation We now discuss time dependent imaging algorithms, two for multistatic data and one for quasi-backscattering data. We first introduce a MUSIC-type method for imaging weak and small scatterers which is fully justified theoretically. As a specific case of this method, we describe a reconstruction algorithm using multistatic patch data. Next, we introduce a MUSIC-type method for quasi-backscattering data. As the numerical results in Section 3.4 suggest, both techniques can be used to find obstacles from time domain data. Finally, we discuss the linear sampling method for extended weak scatterers with multistatic data and why it lacks full justification. Each algorithm we develop below takes a similar form. In particular, we define an indicator function, depending on z R d (and possibly on time τ [0, T ] for some T > 0), so that the function is large when z D and small otherwise. As such, the bulk of this section is dedicated to deriving indicator functions and demonstrating that they are large when z D. The primary idea behind each of these indicator functions is that a specified function, which depends on z, is in the range of an operator depending on u s if and only if z D. For multistatic data, we are interested in the near field equation (N multi g z,τ )(y, t) = l ξ z,τ(y, t), (y, t) Γ i R + (3.9) 50

64 for each z Z, where the near field operator N multi : L 2 σ(r +, L 2 (Γ m )) L 2 σ(r +, L 2 (Γ i )), σ > 0 is defined by (N multi g)(y, t) = R Γ m u s B(x, t τ; y)g(x, τ) ds(x) dτ, (y, t) Γ i R +. (3.10) In Section below we will give more details about the mapping properties of N multi. Furthermore, l ξ z,τ(y, t) := is the convolution of a smooth compactly supported ξ C c R Φ(y z, t τ t 0 )ξ(t 0 ) dt 0, (3.11) (R) with the fundamental solution of the wave equation given by (3.2). For example, in R 3, l ξ z,τ(y, t) = ξ(t τ y z ). The idea for quasi-backscattering data is similar. 4π y z Reconstruction of Point Scatterers from Time Domain Multistatic Data Assume now that D is composed of M weak point scatterers located at the points z j R d, j = 1,..., M. Let the contrast m B be of the form m B (x) = M j=1 m j1 Dj (x) where m j are constant. In this section we collect multistatic data and introduce a MUSIC-type algorithm for locating small objects based on near field time domain data. Hence, let Γ i R d 1 be the curve from which incident fields are transmitted and let Γ m R d 1 be the curve on which the resulting scattered field is measured. We assume the curves do not intersect D and that they are either closed curves or open subsets of analytic curves. Below we will take Γ i = Γ m. In the above configuration, the near field operator (3.10) takes the form (N multi g)(y, t) = where M m j (Φ(z j y, ) j=1 ( ) ) S χ Γ m g) (z j, ) (t), (y, t) Γ i R +, (3.12) (S χ Γ m g)(x, t) = ( χ( ) (S Γm g)(x, )) (t) (3.13) 51

65 and indicates a time convolution. Hence, the point scattering near field equation (3.9) becomes M m j (Φ(z j y, ) j=1 ( ) ) S χ Γ m g) (z j, ) (t) = l ξ y,τ(y, t), (y, t) Γ i R +. The Fourier-Laplace transform N multi in this point scattering context yields the frequency domain weakly scattering near field operator, N multi : L 2 (Γ m ) L 2 (Γ i ) defined as ( N multig )(y, ω) = for some σ 0 > 0. M j=1 m j χ(ω)ˆφ ω (z j, y) ˆΦω (x, z j )ĝ(x, ω) ds(x), Γ m Similarly, the Fourier-Laplace transform of l ξ z,τ is l ξ z,τ(y, t)(y, ω) = ˆξ(ω)e iωτ ˆΦω (y, z), ω C σ0 (3.14) where ˆΦ ω is the fundamental solution for the Helmholtz equation. Thus the transformed point scattering near field equation reads M α j ˆΦω (y, z j ) = β ˆΦ ω (y, z), y Γ i, z R d (3.15) j=1 where α j = m j χ(ω) ˆΦω Γ m (x, z j )ĝ(x, ω) ds(x), β = ˆξ(ω)e iωτ are constants depending on ĝ, z j, τ, m j, Γ m, and ω. Point scattering approximations of the type derived here can also be derived through high frequency truncation of asymptotic expansions, as explained in [4]. For fixed ω, τ = 0, and ˆξ(ω) 1, (3.15) leads the MUSIC algorithm in the frequency domain; the following lemma allows us to characterize the range of ˆNmulti. Lemma 4. Let Γ i be a closed curve or an open subset of an analytic curve and let z j R d, j = 1,..., M be distinct points which do not lie on Γ i. Then the sets of functions {y ˆΦ ω (y, z j ) : j = 1,..., M, y Γ i } are linearly independent for any ω C σ0 with σ 0 = σ 0 (Φ) > 0. 52

66 Proof. Let a j C be constants so that M a j ˆΦω (y, z j ) = 0 y Γ i. (3.16) j=1 Since ˆΦ ω (y, z) are solutions to the Helmholtz equation, they are real analytic on y away from y = z. Without loss of generality, assume Γ i is a closed curve. Otherwise, we can analytically continue (3.16) to the analytic curve of which Γ i is a subset. Note that the left hand side of (3.16) is a radiating solution to the Helmholtz equation outside of Γ i. Hence, by unique continuation and uniqueness of the exterior Dirichlet problem with boundary Γ i, (3.16) is in fact true for all y R d \{z 1, z 2,..., z M }. Due to the singularity of ˆΦ ω (y, z) at y = z, letting y z j, we see that each a j must vanish identically. The above lemma enables us to characterize the range of the finite rank operator ˆN multi which in turn leads to a test to locate the point scatterers. Note that the proof of this theorem is very similar to the proof of the equivalent theorem for far field data given in [69]. We include this proof for completeness. Theorem 3. Assume ω C σ0 for some σ 0 > 0, D = {z 1, z 2, z M } and that Γ i and Γ m (not necessarily the same) are closed curves or open subsets of analytic curves which do not intersect D. Then ˆΦ ω (, z) Range( ˆN multi ) if and only if z = z j some j = 1,..., M. Proof. Assume by contradiction that z = z 0 / {z 1,..., z M }, and that there is some g L 2 (Γ m ) so that ( ˆN multi g)(y) = ˆΦ ω (y, z 0 ) for each y Γ i. By definition of ˆNmulti, ˆΦ ω (y, z 0 ) = M α j ˆΦω (y, z j ), y Γ i. j=1 This is a contradiction with the linear independence shown in Lemma 4. Hence, if ˆΦ ω (, z) Range( ˆN multi ) then z {z 1, z 2..., z M }. 53

67 Now assume z {z 1, z 2,..., z M }. We will show that ˆΦ ω (, z) Kern( ˆN multi ) = Range( ˆN multi ) (notice that ˆN multi is finite rank so Range( ˆN multi ) = Range( ˆN multi )). We explicitly calculate the adjoint of ˆN multi : L 2 (Γ i ) L 2 (Γ m ) as ( ˆN M multih)(x) = m j ˆ χ(ω)ˆφ ω (x, z j ) ˆΦω (y, z j )h(y) ds(y). Γ i As such, if h Kern( ˆN multi ) then M m j ˆ χ(ω)ˆφ ω (x, z j ) ˆΦω (y, z j )h(y) ds(y) = 0, Γ i j=1 j=1 for x Γ m and by the assumption on Γ m, the linear independence shown in Lemma 4 gives 0 = ˆΦω (y, z j )h(y) ds(y). Γ i Hence each ˆΦ ω (, z j ) Kern( ˆN multi ) = Range( ˆN multi ). While we have primarily proven Theorem 3 in order to prove a similar result for the time dependent case, it also gives a MUSIC-type inversion scheme for multistatic weakly scattering near field time harmonic data. In particular, both Lemma 4 and Theorem 3 follow in an identical way for real positive values of ω R, ω > 0. Then u s is time harmonic acoustic scattering data from a point source incident field. Let P ˆN multi : L 2 (Γ i ) R( ˆN multi ) be the orthogonal projection onto the orthogonal complement of the range of ˆNmuti. Theorem 3 gives that P ˆN ˆΦω (, z) = 0 if and only if z = z j. multi In a typical MUSIC application, the function I(z) = P ˆN ˆΦs (y, z) multi L 2 (Γ i ) serves as an indicator function to locate D: if D Z for some set of sampling points Z R d, then (I(z)) 1 is large for each z D and small otherwise. We change this slightly here and test the angle between Range( ˆN multi ) and ˆΦ ω (y, z) for each z Z. When the angle between these is very small, we assert that ˆΦ ω (y, z) Range( ˆN multi ) and hence that z = z j for j = 1,..., M. We find numerically that this results in a more stable reconstruction algorithm than the typical approach. To calculate this angle, introduce ) (ˆΦω (y, z), P ˆNmulti ˆΦ ω (y, z) Jˆ L multi (z) := 2 (Γ i ), ˆΦ ω (y, z) L 2 (Γ i ) P ˆNmulti ˆΦω (y, z) L 2 (Γ i ) 54

68 where P ˆNmulti is the projection operator onto the range of ˆN multi. Note that J ˆ multi 1 with equality if and only if ˆΦ ω (z) Range( ˆN multi ). Then the angle between the ˆΦ ω (z) ( )) and the range of the near field operator is Îmulti(z) = arccos Re( Jmulti ˆ (z). As ) 1 the numerical results demonstrate in Section 3.4, (Îmulti (z) is large if and only if z = z j. Note that ˆ Jmulti is similar in form to the indicator function introduced for time harmonic scattering in [64]. However, the two functions are derived in a very different fashion - the indicator function in [64] is not related to the range of the near field operator - and as far as the authors can tell, their similarity is only coincidental. The range test in the frequency domain formulated in Theorem 3 can now be used to obtain a range test for time domain scattering. Theorem 4. Assume D = {z 1, z 2, z M } and that Γ i and Γ m (not necessarily the same) are closed curves or open subsets of analytic curves which do not intersect D. Define l ξ z,τ(x, t) as in (3.11) with z R d and τ > 0 and ξ C 0 (R + ). Then l ξ z,τ Range(N multi ) if and only if z {z 1,..., z M }, where N multi is given by (3.12). Proof. Assume l ξ z,τ Range(N multi ). This is true if and only if there exists some g z,τ so that (N multi g z,τ )(y, t) = l ξ z,τ(y, t), which by Parseval s equality is true if and only if 0 = = 1 2π e 2σt (Nmulti g z,τ )(y, t) l ξ z,τ(y, t) 2 L 2 (Γ i ) dt, σ > 0 +iσ +iσ This holds true if and only if ( ) ˆNmulti ĝ z,τ (y, ω) l ξ z,τ(y, ω) 2 L 2 (Γ i ) ( ) ˆNmulti ĝ z,τ (y, ω) ˆl ξ z,τ(y, ω) 2 = 0, ω C σ. L 2 (Γ i ) dω. (3.17) Note that by analyticity ˆ χ(ω) = 0 and ˆξ(ω) = 0 only for a discrete set of ω C σ0 with σ > σ 0 > 0. Hence, recalling (3.11) and (3.14) we now have that l ξ z,τ Range(N multi ) if and only if M α j ˆΦω (y, z j ) = β ˆΦ ω (y, z), y Γ i, z R d, ω C σ0. j=1 55

69 where α j = m j χ(ω) ˆΦω Γ m (x, z j )ĝ z,τ (x, ω) ds(x) and β = ˆξ(ω)e iωτ But this is exactly the range test from Theorem 3, and so is true if and only if z {z 1,..., z M }. As in the frequency domain case, this leads to an inversion scheme for time dependent multistatic data. Indeed, to calculate the angle between Φ(y z, t τ) and Range(N multi ), introduce J multi (z, τ) := (Φ(y z, t τ), P Nmulti Φ(y z, t τ)) L 2 (Γ i R), Φ(y z, t τ) L 2 (Γ i R) P Nmulti Φ(y z, t τ) L 2 (Γ i R) where P Nmulti is the projection operator onto the range of N multi. Then the angle between the Φ(y z, t τ) and the range of the near field operator is I multi (z, τ) = arccos (J multi (z, τ)). Note that, unlike in the frequency domain case, we do not need to take a the real part of J multi since the time domain values are inherently real-valued. As seen in Section 3.4, the indicator function I multi (z, τ) := arccos(j multi (z, τ)) 0 if and only if z = z j. It is not completely clear how the sampling time τ affects reconstructions. However, numerical examples in Section 3.4 suggest that its choice is not very important for scattering from small and weak scatterers. On the other hand, numerical experiments with large obstacles, for which the above theory is not justified, show that a good choice of τ results in reconstruction of both the shape and location of an object. Poor choice of τ for large objects only allows the reconstruction of the location of the objects Reconstruction of Point Scatterers from Patches of Time Domain Multistatic Data A key point in the reduction of data collection requirements is that the above theorems make very weak assumptions about the geometry of Γ i and Γ m. As such, both Γ i and Γ m can be chosen as, e.g., sectors of a circle with a very small aperture. However, because of errors in data collection and limitations in measurement accuracy, this is not feasible in practice. Nonetheless, numerical simulations suggest that a patch 56

70 of multistatic data, in which Γ i = Γ m are, e.g., sectors of a circle with a very small aperture gives some indication of the hidden objects. These observations lead to a simple technique for limiting data collection requirements in obstacle reconstruction: collect multistatic data on a small patch array of transmitters and receivers, then repeatedly move the array around the obstacles and repeat the experiment. Once this data is collected, reconstruct the obstacles from each experiment and post-process these reconstructions to give one reconstruction incorporating each experiment. The simplest post-processing is to compute a weighted average of each reconstruction, though more complex processes may be applied. As is shown in Section 3.4, this patch data and post-processing step results in acceptable reconstructions. Indeed, in an error-free case it is theoretically justified. However, in practice it requires a possibly time consuming reconstruction process for each set of patch data. Each of these reconstructions may further require regularization and choice of regularization parameters. Furthermore, a simple average of each reconstruction does not take into account that reconstructions should be similar, as they come from the same objects. More sophisticated post-processing algorithms could certainly alleviate this problem, but we do not explore them here. In the next section, we propose a method which requires only one reconstruction using even less data. Unlike the multistatic patch method, however, we are unable to fully justify its theory Reconstruction of Point Scatterers from Time Domain Quasi-Backscattering Data For simplicity we assume that the transmitters are distributed on the boundary Γ i := S R of a large ball B R centered at the origin containing the scatterer D B R and for each transmitting point y Γ i the scattered field is measured at Γ (y) m := S R B δ (y), where B δ (y) is a small ball centered at y of radius δ. We will first consider briefly the full backscattering case when δ 0. As numerical results demonstrate below, the quasi-backscattering setting for δ > 0 produces better reconstructions than the full backscattering case. 57

71 R +, Consider the weak scattering near field backscattering operator for (y, t) Γ i (N backscattering g)(y, t) = = R u s (y, t τ; y)g(τ) dτ M j=1 m j χ(t τ 2 y z (4π y z j ) 2 j )g(τ) dτ. R Taking the Fourier-Laplace transform yields ( ˆN backscattering g)(y, ω) = ω 2 M j=1 m j (4π y z j ) 2 ˆχ(ω)ĝ(ω) exp (2iω y z j ) M = ω 2 m j ˆχ(ω)ĝ(ω)ˆΦ 2 ω(y, z j ) = j=1 M α j (ω)ˆφ 2 ω(y, z j ), j=1 where α j (ω) = ω 2 m j ˆχ(ω)ĝ(ω) and ω C σ0 for σ 0 > 0. Notice the similarities between N backscattering and N multi and define the sampling function where ξ C 0 ψ ξ z,τ(y, t) = ( ) F 1 [ˆΦ 2 ω(y, z)] ξ( τ) (3.18) and F 1 here denotes the inverse of the Fourier-Laplace transform defined in Section 3.2. Using the same arguments as in the proof of Theorem 4 for the multistatic case, the form of N backscattering suggests that ( ( ψ ξ z,τ (y, t), P Nbackscattering I backscattering (z, τ) := arccos ψξ z,τ(y, t) ) ) L 2 (Γ i R) ψz,τ(y, ξ t) L 2 (Γ i R) P Nbackscattering ψz,τ(y, ξ t) L 2 (Γ i R) acts as an indicator for the location of z j, where P Nbackscattering is the projection operator onto the range of N backscattering. As shown in Figure 3.7, reconstructions using I backscattering are quite recognizable, but less clear than their quasi-backscattering equivalents. Hence, we suggest collecting quasi-backscattering data with δ > 0. This differs from the results originally introduced for the quasi-backscattering context in Chapter 2, and in [56] in which a specific relationship between Γ i and Γ (y) m is required in order to reconstruct z j under 58

72 the assumption that x z j 1. Here we do not require any such relationship or that data is collected sufficiently-far from the point scatterers. However, unlike in [56, 59], we are unable to prove an exact characterization of which components of {z j } can be reconstructed. Indeed, other than an intuitive argument that more data leads to better reconstructions, we are unsure why the quasi-backscattering reconstructions are superior to the backscattering reconstructions. The quasi-backscattering time domain weakly scattering near field operator N quasi : H p σ(r +, L 2 (S R )) H p σ(r +, L 2 (S R )), σ > 0, p R is defined by (N quasi g)(y, t) = R Γ (y) m u s B(x, t τ; y)g(x, τ) ds(x) dτ, (y, t) Γ i R +. For a collection of point sources with strength m j centered at points z j, j = 1,..., M, N quasi takes the form (N quasi g)(y, t) = M ( m j (Φ(z j y, ) j=1 S χ Γ (y) m ) ) g) (z j, ) (t), (y, t) Γ i R + (3.19) and the Fourier-Laplace transform of the quasi-backscattering near field equation reads M j=1 m j χ(ω)ˆφ ω (y, z j ) ˆΦ ω (x, z j )g(x) ds(x) = ˆξ(ω)e iωτ ˆΦ2 ω (y, z) y Γ i, z Z, Γ m (y) (3.20) for ω C σ0, σ > σ 0 > 0. Unfortunately, as opposed to the backscattering or multistatic cases, in (3.20) the term J(y) := Γ (y) m ˆΦ ω (x, z j )g(x) ds(x) (3.21) is not proportional to ˆΦ ω (y, z j ). As such, the arguments used to justify Theorem 3 no longer apply. We would like to show that J(y) = cˆφ ω (y, z j ) + O(δ) where c is independent of y. To this end, for a small patch Γ (y) m on the sphere of radius R, say with diameter δ > 0, where δ is small compared to y z j for j = 1,..., M, up to order δ 2 we can replace Γ (y) m by the tangent plane in R 3 or line in R 2 at y; let us denote 59

73 it by T δ y. Hence x T δ y and we have that x = y + ηy where 0 < η < δ and y are the components of y parallel to the tangent plane. Simple asymptotic calculations, which for sake of the argument, we present here only in the R 3 case, give and hence, up to order O(δ 2 ) x z j = y z j + η (z j y ) y z j + O( δ 2) and x z j 1 = y z j 1 ( 1 η (z j y ) y z j 2 + O( δ 2) J(y) ˆΦ iωη(z j ( y ) y z ω (y, z j ) e j 1 η (z ) j y ) g(y + ηy )d(y + ηy ). ηy <δ y z j 2 Hence we obtain the desired result if (z j y ) = 0 for all y Γ i and j = 1,..., M. There is a special measurement configuration which works only in R 3 and is detailed in [56]. Here, the measurement geometry is specified so that the integral in the above equation is independent of y. Indeed, in this set-up, Γ i is set to be a line parallel to some direction ˆv and for each y Γ i, the corresponding measurements are taken on a line parallel to ˆv passing through y. In this set-up, ) M m j χ(ω)ˆφ 2 ω(y, z j ) = ˆξ(ω)e iωτ ˆΦ2 ω (y, z) j=1 y Γ i, z Z provides an exact range test for z j whose projections to ˆv differ. See Theorem 1 in [56] for more details. In the far field this setup becomes simpler and is studied in [59]. Nevertheless, for our data configuration in both R 2 and R 3, (3.20) shows that if z = z j for some j = 1,..., M, then we can find a g := g zj,y L 2 (Ty δ ) that solves exactly (3.20). Arguing heuristically, taking g as an approximating sequence of the Dirac delta function at y suggest that a range test as in the framework built up in the previous sections allowing us to introduce an indicator function for finding z j. As the numerical examples below demonstrate, these assumptions on the geometry of Γ m and Γ i do do not seem to be active. Indeed, based on the discussion above, 60

74 after taking the inverse Fourier-Laplace transform of (3.20), we introduce an indicator function to measure the angle between ψ z,τ and Range(N quasi ). Let J quasi (z, τ) := ( ψ ξ z,τ (y, t), P Nquasi ψξ z,τ(y, t) ) L 2 (Γ i R), ψz,τ(y, ξ t) L 2 (Γ i R) P Nquasi ψz,τ(y, ξ t) L 2 (Γ i R) where P Nquasi is the projection operator onto the range of N quasi. We demonstrate in Section 3.4 that the indicator function I quasi (z, τ) := arccos(j quasi (z, τ)) 0 if and only if z = z j. In exactly the same way, we can derive a time harmonic indicator function for quasi-backscattering data. In particular, time harmonic reconstructions are computed with ) (ˆΦ2 ω(y, z), P ˆNquasi ˆΦ2 ω (y, z) L Î quasi (z) = arccos Re 2 (Γ i ). ˆΦ 2 ω(y, z) L 2 (Γ i ) P ˆNquasi ˆΦ2 ω (y, z) L 2 (Γ i ) We will show below that time dependent reconstructions out perform time harmonic reconstructions when an incident field is transmitted from only few points Linear Sampling Method for Extended Objects Under the Time Domain Born Model Before showing numerical reconstructions of point objects, we discuss one technique, the linear sampling method, for extending the above multistatic results to extended obstacles which are weakly scattering. Full justification the linear sampling for the weak scattering case - as well as full justification under a strongly scattering model - requires new results about an associated interior transmission problem which we are unable to prove. Note that in the case of strongly scattering media, [94] provides the necessary results about the interior transmission problem for full justification of the linear sampling problem. We are unable to apply these results to the case of the Born regime. To introduce the linear sampling method for small objects, we follow [52], in which the linear sampling method for strongly scattering time domain data for inhomogeneous media is first investigated. As detailed in Chapter 1, the primary idea 61

75 in linear sampling is to find a regularized solution to the near field equation (3.9). Then I LSM (z, τ) = g z,τ 1 L 2 (D,R + ) is used as an indicator function. The values I LSM are then used to suggest where D is located; the z R d so that I LSM (z) is large are our reconstruction of D. By making use of the volume integral equation representation of u s B, we can factor N multi into the product of two well-studied operators. In particular, (N multi g)(y, t) = u s B(x, t τ; y)g(x, τ) ds(x) dτ R Γ m ( = m(z)φ(x z, t τ s) S χ Γ m is defined by (3.13). R Γ m R D ) u i tt(z, s, y) dv (z) ds g(x, τ) ds(x) dτ = m(z)u i tt(z, s; y) Φ(x z, t τ s) R D R Γ m = = This yields the following factorization D R g(x, τ) ds(x) dτ dv (z) ds m(z) (Φ(z y, ) χ( ) (S Γm g) (z, )) (t) dv (z) ( ) m(z)φ(z y, t τ) S χ Γ m g (z, τ) dv (z) dτ. D N multi g = γ Γi GS χ Γ m g, where G is the wave equation solution operator defined by (3.5), and γ Γi operator restricting the solution to Γ i. is the trace Using (3.6), and combining results for the multiple scattering linear sampling theory [52] with the Born transmission eigenvalue results in Chapter 4, it can be shown that the weakly scattering near field operator N multi : H p σ(r +, L 2 (Γ m )) H p σ(r +, H 1/2 (Γ i )), p R and σ > 0 is bounded, injective, and has dense range. Note that when p = 0 we have the mapping properties used above. By a contradiction argument it is also possible to show as in [52] that for z / D any approximate solution of (3.9) is such that g z,τ H p σ (R +,L 2 (Γ m)) 62

76 blows up as the regularization parameter in the equation goes to zero. However a complete justification of the linear sampling method, namely to describe the behavior of the approximate solution of (3.9) for z D, one need to find causal solution to the so-called interior transmission problem for the Born problem. This problem is to find w(x, t) and v(x, t) satisfying w tt (x, t) w(x, t) = m(x)v (x, t) D R + v tt (x, t) v(x, t) = 0 (x, t) D R + w(x, t) = l ξ z,τ(x, t) (x, t) D R + w(x,t) ν = lξ z,τ (x,t) ν (x, t) D R + w(x, 0) = w t (x, 0) = v(x, 0) = v t (x, 0) = 0 x D. The solution of this problem remains open both for the Born approximation model and the full multiple scattering model. Under the Born approximation, Fourier or Fourier- Laplace analysis fail to work for this problem since the transformed homogeneous problem (known as the transmission eigenvalue problem) is non-selfadjoint and its complex eigenvalues may have unbounded imaginary part, though in the full scattering case it is known from [94] the conditions under which transmission eigenvalues have bounded complex part. Moreover, there are either infinitely many real eigenvalues or a sequence of complex eigenvalues may approach the real axis (see Chapter 4 or [25]). Nonetheless, numerical implementation of the linear sampling method, i.e. finding a regularized solution of (3.9) shows that ( 1 g z,τ H p σ (R +,L 2 (Γ m))) remains finite inside D and is very small outside D. As such, the linear sampling method can be applied to image weak scatterers, as demonstrated in Figure Numerical Reconstructions To simulate forward scattering, we use the convolution quadrature-volume integral equation approach introduced in [73]. We use a Galerkin semi-discretization in space, as discussed in Chapter 5. With this, we simulate values of u s (x j, t k ; y i ) at some discrete values x j Γ m, y i Γ i, and at t k [0, T ], where T > 0 is some final time. We 63

77 choose T so that at least 99% of the energy of u s has left the computational domain. For simplicity, we always take n = 1 outside of D and n = 1.1 inside of D. Using these simulations we can calculate discrete approximations to N multi and N quasi, which we denote as N multi and N quasi, respectively. As we explain below, we are more interested in a partial singular value decomposition (SVD) of these near field operators than in the matrices themselves. Since the near field operator is a convolution, this decomposition can be calculated quickly and without explicitly forming N multi or N quasi by using a fast Fourier transform, as described in [57]. In order to avoid inverse crimes, we calculate the full multiple scattering data, not the Born approximation to this data. We further avoid inverse crimes by adding noise to reconstructions, replacing u s with (1 + ɛρ)u s where ɛ is what we refer to below as the noise level and ρ is a uniform random variable in [ 1, 1]. Our reconstructions use transmitters. This less than one-half of the amount of transmitter locations as are typically used in sampling-type schemes, though our reconstructions are of similar accuracy. This is likely both due to the increased amount of a priori data we assume about our obstacles and the fact that time domain data contains more information than the single frequency time harmonic data which is usually used. Unless otherwise noted in figure captions, obstacles are indicated by black lines. Red dots indicate location of transmitters and black dots the location of the receivers for each transmitter. Time domain data was simulated for 18 seconds with 600 time steps and with the impulse function χ(t) = sin(4t) exp( 1.6(t 3) 2 ). All figures have 5% added noise. All reconstructions are rescaled to [0, 1] for comparison purposes MUSIC and LSM Reconstruction with Multistatic Data We first use I multi and I LSM to find reconstructions using multistatic data. To this end, we compute N multi = USV, the SVD of N multi. Then, it is well-known that the projection operator can be written as P Nmulti = N multi (N multi N multi) 1 N multi = US (S S) 1 S U, so long as each of the inverse matrices exists. By using the singular value decomposition, we avoid the need to construct the possibly-large matrix N multi 64

78 and can easily regularize using a spectral cut-off method, by looking for a gap in the singular values on the diagonal of S and using only the large singular values in reconstruction. We calculate I multi (z, τ) in this way in order to reconstruct D. We calculate I LSM by solving (3.9) with a truncated SVD as in [52]. We calculate these multistatic indicator functions for two geometries in R 2. This is shown in Figure 3.2, where forward data is calculated for 10 incident points and 10 measurement points. For the point obstacles on the right, we use a sampling time of τ = 1, though the reconstructions seem acceptable for any 0 < τ < T. For the larger obstacle on the left, we set τ = 5.4. By choosing τ in this way, we reconstruct both the location and shape of D, while some values of τ only reconstruct the location. It is also an open problem to automatically chose τ for linear sampling methods under the full multiple scattering model. In these figures, large values indicate the reconstructed location of objects. We now restrict our figures to the averages of small multistatic patches. We demonstrate in Figure 3.3 that small objects and be reconstructed well if either the aperture of each patch is sufficiently large or if there are a large number of patches used. In Figure 3.4, we show the same experimental set-up used for larger objects, showing that reconstructions are worse, but still acceptable Quasi-Backscattering Reconstructions In Figures we use quasi-backscattering data to reconstruct the location of a number of objects. To construct I quasi, we again use the SVD of the discrete near field operator to calculate the angle between the test functions and their projection onto the range of N quasi. We again regularize with a spectral cut-off. In Figure 3.5 we demonstrate the feasibility of the proposed technique with different geometries. In this figure, we plot only the values of (I quasi (z)) 1 larger than a cut-off threshold chosen visually. In some ways, this figure then represents an idealized set of reconstructions. However, there are a number of algorithms which make this choice automatically, described for example in [9, 59]. For each figure, we use 20 65

79 Figure 3.2: Plots of (I LSM (z)) 1 (top) and (I multi (z)) 1 (bottom) for two different geometries. transmitters, each with 4 receivers, and set δ = π/100. For each image, we use the temporal sampling point τ = 1, though as before this choice does not seem to seriously affect reconstructions. In Figure 3.6 we demonstrate the dependence of reconstructions on the number of transmitter locations. We show both time harmonic and time dependent reconstructions. As expected, reconstructions become more accurate when more transmitter locations are used. Furthermore, time dependent reconstructions are more accurate than time harmonic reconstructions, until a sufficient number of transmitters are used. In the limiting case as δ 0, the quasi-backscattering set-up becomes a pure 66

80 Figure 3.3: Multistatic patch reconstructions of the same geometry of small circles, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color. backscattering set-up. In Figure 3.7 we show numerical examples of pure backscattering. There are a total of 30 transmitter locations, each with 1 receiver located at the same point. While reconstructions are not as sharp as, e.g., the full aperture multistatic reconstructions, there is a clear indication of object location. Finally, we compare limited aperture reconstructions from multistatic data to limited aperture reconstructions from patch and quasi-backscattering data. As expected, Figure 3.8 demonstrates that the multistatic reconstructions are superior to the patch and quasi-backscattering reconstructions, which are somewhat similar. Indeed, the quasi-backscattering algorithm results in reconstructions which are noisier and, the case of three point obstacles, only clearly reconstruct the two obstacles nearest the transmitter and receiver arrays, incorrectly indicating an extra obstacle in the 67

81 Figure 3.4: Multistatic patch reconstructions of the same geometry of medium-sized ellipses, indicated by black lines. (Top) Four patches are used with 5 transmitters and receivers each. From left-to-right, the aperture of each patch decreases from π/2 to π/4 to π/8. (Bottom) The same experiment as top but with 10 patches. Each set of circles indicates the location of transmitters and receivers in each patch. Transmitters and receivers in the same patch each have the same color. bottom right. Patch data reconstructions are also noisier than multistatic reconstructions, but do not have the same issues as quasi-backscattering reconstructions do with three point obstacles. The patch reconstructions do not separate the larger ellipses as effectively as the quasi-backscattering reconstructions. Note that both the multistatic and patch reconstructions required a careful selection of τ = 5 in order to produce optimal results, while the quasi-backscattering data did not need any such choice. 68

82 Figure 3.5: Plots of (I quasi (z, τ)) 1 for four different geometries. 69

83 Figure 3.6: Plots of (Îquasi(z, τ)) 1 (top) and (I quasi (z, τ)) 1 (bottom), with a different number of transmitters in each row. On the left there are 5 transmitters, in the middle there are 10 transmitters, and the right there are 15 transmitters. Time harmonic data was computed with wavenumber k = 3. 70

84 Figure 3.7: Backscattering reconstructions using I 1 backscattering (z, τ) for two different geometries. In both figures, 30 transmitters are used and data is measured only at the location of the transmitter. Time domain data was simulated for 14 seconds with 480 time steps. 71

85 Figure 3.8: Limited aperture reconstructions using multistatic data with I 1 multi (z, τ) (top), multistatic patch data with two patches (middle), and quasi-backscattering data with I 1 quasi (z, τ) (bottom). In both figures, 19 transmitters are used and in the case of quasi-backscattering data, 4 receivers were used. 72