The practice of imaging has grown tremendously in the past few decades, both in sophistication and importance. There is a strong thread of commonality in the diverse quilt of applications of imaging in medicine, engineering, and the physical sciences. In particular, the same mathematical techniques, such as the use of trajectory-based and asymptotic methods, the central topic of this book, often serve as the underpinnings of each application. However, to the uninitiated, it might seem that each discipline has adopted a distinct formulation of the imaging problem. Thus, a sense of unity is lost in traversing the various applications. In addition, the development of imaging methods may be more extensive in one particular field compared to others. For example, in applied mathematics, trajectory-based imaging methods have been extended to a wide range of situations, such as diffusive and non-linear wave propagation. These advancements may not be appreciated or even known in other areas.

The goal of this book is to bring unity to the range of trajectory-based techniques for modeling fluid flow that may serve as the basis for efficient imaging algorithms. A secondary objective is to highlight the wide array of physical phenomena to which trajectory-based imaging methods lend themselves. It is widely known that a trajectory-based method, such as ray theory, is applicable to hyperbolic equations, typified by the wave equation. Less well known is the fact that trajectory-based methods may be used to study diffusive systems, governed by a parabolic equation. Similarly, ray methods for non-linear waves have been developed in applications such as gas dynamics and plasma physics, but are relatively unknown in such fields as hydrology. The fundamental techniques are then applied to important problems in the Earth sciences. Hopefully, after finishing this book the reader will glimpse the full range of trajectory-based imaging methods.

This book describes trajectory-based imaging from its mathematical formulation, through the formation and solution of the imaging equations, to the determination of the accuracy and resolution associated with the image. Our presentation is unique in that we cover a rather complete spectrum of physical phenomena. At the same time we have tried to focus on the practical side of imaging, emphasizing methods that are efficient and robust. Obtaining an image is not the end of the story, we need some measure of the reliability of our solution.