Introduction

Since lasers were invented in 1964, optical communications has been investigated for both military and commercial application because of its wavelength and spectrum availability advantages over radio frequency (RF) communications. Unfortunately, only fiber optic communications (FOC) systems have achieved wide implementation since then because of their ability to maximize power transfer from point to point while also minimizing negative channel effects. Recently, free-space optical communications (FSOC) has reemerged after three decades of dormancy due to the availability of new FOC technologies to the FSOC community, such as low-cost sensitive receivers and more power-efficient laser sources. Applications of FSOC, however, have been limited to local area (short range) networking because optical systems have been unable to effectively compensate for two atmospheric phenomena; cloud obscuration and atmospheric turbulence. Making a hybrid FSOC/RF communications system will compensate for cloud obscuration by using the RF capability when “clouds get in the way”. For this chapter, we will discuss how to mitigate the atmospheric turbulence for incoherent communications systems. In particular, we will discuss a new statistical link budget approach for characterizing FSOC link performance, and compare experimental results with statistical predictions. We also will comment at the end of the chapter on progress in coherent communications through turbulence. For those readers interested in science and modeling of laser propagation through atmospheric turbulence, we refer them to several excellent books on the topics for the details [1–5].

When a digital communications system experiences a very noisy/fading channel, the electrical signal-to-noise ratio may never be strong enough to obtain a low probability of detection error. Shannon proved in 1948 that a coding scheme could exist to provide error-free communications under those conditions. In particular, he showed it was possible to achieve reliable communications, i.e., error-free communications, over an unreliable, i.e., noisy, discrete memory-less channel (DMC), using block codes at a rate less than the channel capacity if the number of letters per code word, and consequently, the number of code words, are made arbitrarily large [1]. This result motivated a lot of research to find the optimum code, i.e., one that minimizes the number of letters and code words, to create reliable communications. The various approaches that have emerged from this research form a class of coding best known as forward error correction (FEC) coding (also called channel coding) [2–4]. Richard Hamming is credited with pioneering this field in the 1940s and invented the first FEC code, the Hamming (7,4) code, in 1950. He, and most others, basically has the originator systematically add generated redundant data to its message. These redundant data allows the receiver to detect and correct a limited number of errors occurring anywhere in the message stream without that person requesting the originator to resend part, or all, of the original message. In other words, the FEC provides a very effective means to correct errors without needing a “reverse channel” to request retransmission of data. This advantage is at the cost of a higher channel data rate if one wants to keep the information data rate the same. These techniques are typically applied in communications systems where retransmissions are relatively costly, or impossible, such as in mobile ad hoc networking when broadcasting to multiple receivers (multicast) [5–7], HF communications and optical communications. For the interested reader, Zhu and Kahn provide a detailed summary of FEC coding applied to the turbulence channel [8, Ch. 7, pp. 303–346].

Lidar (light detection and ranging) is an optical remote sensing technology that measures properties of scattered and reflected light to find range and/or other information about a distant target. The common method to determine distance to an object or surface is to use laser pulses, although as in radar it is possible to use more complex forms of modulation. Also as in radar technology, which uses radio waves instead of light, the range to an object is determined by measuring the two-way time delay between transmission of a pulse and detection of the reflected signal. In the military, the acronym Ladar (laser detection and ranging) is often used. Lidar has also expanded its utility to the detection of constituents of the atmosphere. It is not the intention of this book to expand on all the applications for which this technology has been applied, they are too numerous. Rather we will try to group applications by the technology needed. Thus, for example, a laser speed gun used for traffic monitoring uses the same basic technology as those used for surveying and mapping; i.e., bursts of nanosecond pulses. The wavelengths may vary due to eye safety requirements (>1.5 μm), and the pulse energies may vary because of the distances involved, but both use time of flight measurements. Also the scanning requirements can vary because of the areas to be covered, and the scanning equipment that is available can vary because of the pulse energies involved. For high-energy pulses reflective mirrors are needed, whereas for low-energy pulses electronic scanning with a CCD or CMOS shutter can be used. Most applications use incoherent optics. In some cases, where the additional cost warrants it, coherent (optical heterodyne) detection is used. Such an example is the measurement of clear air turbulence where coherent 10 μm laser ranging systems have been employed. Another class of incoherent applications use backscatter signatures to investigate chemical constituents, primarily in the atmosphere. Such systems use Raman, Brillouin, Rayleigh, and Mie scattering, as well as various types of fluorescence. As we discussed in Chapter 5, Mie theory covers most particulate scattering, ranging from the Rayleigh range ~1/λ4 for smaller particles (molecules) to the larger particles (aerosols). Raman and Brillouin scattering both induce wavelength shifts, requiring multiple wavelength generation and discrimination. The selection of interference filters, prisms, gratings or spectrascopic machines is determined by the resolution and costs associated with the specific application.

This paper serves as a survey of enclosure-type methods used to determine the obstacles or inclusions embedded in the background medium from the near-field measurements of propagating waves. A type of complex geometric optics waves that exhibits exponential decay with distance from some critical level surfaces (hyperplanes, spheres or other types of level sets of phase functions) are sent to probe the medium. One can easily manipulate the speed of decay such that the waves can only detect the material feature that is close enough to the level surfaces. As a result of sending such waves with level surfaces moving along each direction, one should be able to pick out those that enclose the inclusion.

Let (M, g) be a compact oriented Riemannian manifold with smooth boundary, and let SM = {(x, v) ∈ TM : |v| =1} be the unit tangent bundle with canonical projection π : SM → M. The geodesics going from ∂M into M can be parametrized by the set ∂+(SM) = {(x, v) 2 SM : x ∈ ∂M, (u, v) ≤ 0}, where is the outer unit normal vector to ∂M. For any (x, v) ∈ SM we let t →γ(t, x, v) be the geodesic starting from x in direction v. We assume that (M, g) is nontrapping, which means that the time τ( x; v) when the geodesic γ(t, x, v) exits M is finite for each (x, v) ∈ SM. The scattering relation

We develop a comprehensive theory and microlocal analysis of reverse-time imaging—also referred to as reverse-time migration or RTM—for the anisotropic elastic wave equation based on the single scattering approximation. We consider a configuration representative of the seismic inverse scattering problem. In this configuration, we have an interior (point) body-force source that generates elastic waves, which scatter off discontinuities in the properties of earth’s materials (anisotropic stiffness, density), and are observed at receivers on the earth’s surface. The receivers detect all the components of displacement. We introduce (i) an anisotropic elastic-wave RTM inverse scattering transform, and for the case of mode conversions (ii) a microlocally equivalent formulation avoiding knowledge of the source via the introduction of so-called array receiver functions. These allow a seamless integration of passive source and active source approaches to inverse scattering.

We develop a program and analysis for elastic wave-equation inverse scattering, based on the single scattering approximation, from two interrelated points of view, known in the seismic imaging literature as “receiver functions” (passive source) and “reverse-time migration” (active source).

We consider an interior (point) body-force source that generates elastic waves, which scatter off discontinuities in the properties of earth’s materials (anisotropic stiffness, density), and which are observed at receivers on the earth’s surface. The receivers detect all the components of displacement. We decompose the medium into a smooth background model and a singular contrast and assume the single scattering or Born approximation. The inverse scattering problem concerns the reconstruction of the contrast given a background model.

The global uniqueness for inverse boundary value problems of elliptic equations at fixed frequency in dimension n = 2 is quite particular and remained open for many years. Now these problems are well understood, with a variety of results appearing in the last 10 or 15 years, essentially all using the complex structure ℝ2 ⋍ℂ and ∂-techniques. This is therefore a good time to write a short survey on the subject. Although we tried to cover as much as we can, we do not pretend to be exhaustive and we apologize in advance for any forgotten reference, which is not a decision made on purpose but rather a sign of our ignorance. We have decided to give more details about the proofs of recent results based on Bukhgeim’s idea [2008], for there is already a survey by Uhlmann [2003] on the subject about older results. The results of Astala, Lassas, and Päivärinta using quasiconformal methods are the subject of a separate survey in this volume [Astala et al. 2013]. Finally, we do not discuss questions about stability and reconstruction, nor inverse scattering results

This text is an introduction to Calderón’s inverse conductivity problem on Rie-mannian manifolds. This problem arises as a model for electrical imaging in anisotropic media, and it is one of the most basic inverse problems in a geometric setting. The problem is still largely open, but we will discuss recent developments based on complex geometrical optics and the geodesic X-ray transform in the case where one restricts to a fixed conformal class of conductivities.

This work is based on lectures for courses given at the University of Helsinki in 2010 and at Universidad Autónoma de Madrid in 2011. It has therefore the feeling of a set of lecture notes for a graduate course on the topic, together with exercises and also some problems which are open at the time of writing this. The main focus is on manifolds of dimension three and higher, where one has to rely on real variable methods instead of using complex analysis. The text can be considered as an introduction to geometric inverse problems, but also as an introduction to the use of real analysis methods in the setting of Riemannian manifolds.

The success of most medical imaging modalities rests on their high, typically submillimeter, resolution. Computerized tomography (CT), magnetic resonance imaging (MRI), and ultrasound imaging (UI) are typical examples of such modalities. In some situations, these modalities fail to exhibit a sufficient contrast between different types of tissues, whereas other modalities, for example based on the optical, elastic, or electrical properties of these tissues, do display such high contrast. Unfortunately, the latter modalities, such as optical tomography (OT), electrical impedance tomography (EIT) and elastographic imaging (EI), involve a highly smoothing measurement operator and are thus typically low-resolution as stand-alone modalities.

In electrical impedance tomography one aims to determine the internal structure of a body from electrical measurements on its surface. To consider the precise mathematical formulation of the electrical impedance tomography problem, suppose that Ω ⊂ ℝn is a bounded domain with connected complement and let us start with the case whenσ: Ω → (0;∞) be a measurable function that is bounded away from zero and infinity.

The authors were supported by the Academy of Finland, the Finnish Centres of Excellence in Analysis and Dynamics and Inverse Problems, and the Mathematical Sciences Research Institute.

We survey recent results by the authors on multiwave methods where the high-resolution method is ultrasound. We consider the inverse problem of determining a source inside a medium from ultrasound measurements made on the boundary of the medium. Some multiwave medical imaging methods where this is considered are photoacoustic tomography, thermoacoustic tomography, ultrasound modulated tomography, transient elastography and magnetoacoustic tomography. In the case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann series expansion. We give almost necessary and sufficient conditions for uniqueness and stability when the measurements are taken on a part of the boundary. We study the case of a smooth speed and speeds having jump type of singularities. The latter models propagation of acoustic waves in the brain, where the skull has a much larger sound speed than the rest of the brain. In this paper we emphasize a microlocal viewpoint.

Multiwave imaging methods, also called hybrid methods, attempt to combine the high resolution of one imaging method with the high contrast capabilities of another through a physical principle. One important medical imaging application is breast cancer detection. Ultrasound provides high (submillimeter) resolution, but it suffers from low contrast. On the other hand, many tumors absorb much more energy from electromagnetic waves (in some specific energy bands) than healthy cells. Photoacoustic tomography (PAT) [Wang 2009] consists of exposing tissues to relatively harmless optical radiation that causes temperature increases in the millikelvin range, resulting in the generation of propagating ultrasound waves (the photoacoustic effect). Such ultrasonic waves are readily measurable. The inverse problem then consists of reconstructing the optical properties of the tissue. In thermoacoustic tomography (TAT)—see, e.g., [Kruger et al. 1999]— low frequency microwaves, with wavelengths on the order of 1 m, are sent into the medium.

We consider the inverse problem of electrical impedance tomography (EIT) in two dimensions [Borcea 2002]. It seeks the scalar valued positive and bounded conductivity σ(x), the coefficient in the elliptic partial differential equation for the potential u ∊ H1(Ω),

In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic, or conformally compact, spaces, which was introduced in [Vasy 2010a]. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of [Guillarmou 2005]), and gives high-energy estimates in strips. The key idea is an extension across the boundary for a problem obtained from the Laplacian shifted by the spectral parameter. The extended problem is nonelliptic—indeed, on the other side it is related to the Klein–Gordon equation on an asymptotically de Sitter space—but nonetheless it can be analyzed by methods of Fredholm theory. In [Vasy 2010a] these methods, with some additional ingredients, were used to analyze the wave equation on Kerr-de Sitter space-times; the present setting is described there as the simplest application of the tools introduced. The purpose of the present paper is to give a self-contained treatment of conformally compact spaces, without burdening the reader with the additional machinery required for the Kerr-de Sitter analysis.

In the past few years transmission eigenvalues have become an important area of research in inverse scattering theory with active research being undertaken in many parts of the world. Transmission eigenvalues appear in the study of scattering by inhomogeneous media and are closely related to non-scattering waves. Such eigenvalues provide information about material properties of the scattering media and can be determined from scattering data. Hence they can play an important role in a variety of inverse problems in target identification and nondestructive testing. The transmission eigenvalue problem is a non-selfadjoint and nonlinear eigenvalue problem that is not covered by the standard theory of eigenvalue problems for elliptic operators.

This article provides a comprehensive review of the state-of-the art theoretical results on the transmission eigenvalue problem including a discussion on fundamental questions such as existence and discreteness of transmission eigenvalues as well as Faber–Krahn type inequalities relating the first eigenvalue to material properties of inhomogeneous media. We begin our presentation by showing how the transmission eigenvalue problem appears in scattering theory and how transmission eigenvalues are determined from scattering data. Then we discuss the simple case of spherically stratified media where it is possible to obtain partial results on inverse spectral problems. In the case of more general inhomogeneous media we discuss the transmission eigenvalue problem for various types of media employing different mathematical techniques. We conclude our presentation with a list of open problems that in our opinion merit investigation.

Marc Kac [1966], in a famous paper, raised the following question: Let Ωℝ be a bounded domain and let be the eigenvalues of the nonnegative Euclidean Laplacian ΔΩ with either Dirichlet or Neumann boundary conditions. Is Ω determined up to isometries from the sequence λ0, λ1, . . .? We can ask the same question about bounded domains in Rn, and below we will discuss other generalizations as well. Physically, one motivation for this problem is identifying distant physical objects, such as stars or atoms, from the light or sound they emit. These inverse spectral problems, as some engineers have recently proposed in [Reuter 2007; Reuter et al. 2007; 2009; Peinecke et al. 2007], may also have interesting applications in shape-matching, copyright and medical shape analysis.