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.

The radio spectrum is one of the most important resources for communications. Traditionally, spectrum governance throughout the world has tended towards exclusivity of its use in large geographic areas, allocating frequency bands for specific applications and assigning licenses to specific users or service providers. This policy has generated a shortage of frequencies available for emerging wireless products and services, as most frequencies are now assigned. Moreover, exclusivity creates underutilization of the spectrum, as very rarely can all licensees make full use of the frequencies assigned to them. These facts have motivated the search for technologies able to alleviate the artificial scarcity of spectrum by adapting to changing environmental and network-usage conditions.

What is perhaps the most natural among these technologies involves opportunistic use of the spectrum, whereby secondary (unlicensed) users are able to occupy the portions of the spectrum left temporarily free by the licensed primary users. The stringent requirement here is that secondary users should not interfere with the primary users, which this paradigm of operation (later called interweaving) achieves using the simplest form of orthogonalization, one that only requires knowledge of the state of a frequency band, i.e., whether it is free or occupied. The fact that the spectrum can be shared by primary and secondary users, with the latter exploiting their cognition of the environment in which transmission is taking place, has led to the development of the concept of Cognitive Radio (CR), whose idea was first introduced in [1] in 1999.