“The Earth is a laboratory, but one that is very different from those in experimental physics, where we are taught to carefully design an experiment so that we have full control.”

This quote, taken from Chap. 14 of this book, reflects one of the main challenges seismologists are faced with in their quest for imaging Earth’s interior using uncontrolled seismic waves observed by a sparse and irregular network of seismometers. Professor Guust Nolet, a leading scientist in the field of global seismology, has succeeded in writing a very readable treatise on the many aspects of imaging the interior of Earth and, albeit with less emphasis, the Sun. The book covers the underlying theory of seismic wave propagation and scattering, the practical aspects of seismic observations, and, last but not least, the theory and practice of modern seismic inversion and imaging methodologies. The existing approaches to imaging Earth’s interior range from normal mode analysis to reflection and transmission tomography. The emphasis in this treatise is on seismic transmission tomography, a methodology related to medical x-ray and ultrasonic tomography, ocean acoustic tomography, and cross-well tomographic imaging in exploration geophysics. Some of the specific challenges of seismic tomography are related to the fact that the interior of Earth is inhomogeneous at many scales and to the already mentioned aspect of the “uncontrolled experiment.” The author addresses these challenges with the authority of more than 25 years research experience in seismic tomography.

The book is written for seismic practitioners and students. The author has chosen for a somewhat informal style, with much more emphasis on physical explanation than on mathematical rigor. This is not to say that the mathematics is compromised, but only that the author has decided to limit the treatment of the underlying mathematics to what is needed for understanding the principles of seismic tomography. Readers interested in more extensive mathematical derivations are guided to the relevant literature. The physical explanations are often chosen from everyday life: once you have read the explanation why the sky is blue and clouds are white you will never forget the essential difference between Rayleigh and Mie scattering. What I consider very valuable in this book are the many exercises for students and, in particular, the references to software repositories, which help students improving their understanding of tomography and practitioners analyzing their data.

Let me further substantiate my appreciation of this book with a brief chapter-by-chapter discussion. The introductory chapter mentions a presentation by Keiti Aki at an AGU Conference in 1974, in which arrival times of seismic waves were interpreted in terms of an image, as the starting point of modern seismic tomography. This chapter further summarizes the early successes of seismic tomography with respect to geophysical discoveries, such as images of slabs subducting into the lower mantle and evidence of major low velocity regions near the core-mantle boundary, so-called superplumes. It concludes by mentioning that further improvement of the images requires a finite-frequency approach to tomography, accounting for the fact that arrival times are sensitive to perturbations within Fresnel zones around the rays.

Chapter 2 starts with the fundamentals of acoustic and elastodynamic wave theory and derives the differential equations for ray tracing in inhomogeneous isotropic Earth. It also contains a section on the corrections that are required for ray tracing in the Sun, where the flow of the medium plays a role. I like the fact that the author does not complicate things more than needed. For example, instead of giving a formal definition of correlation distance, he mentions in a footnote that “for the purpose of this book it is sufficient to define the correlation distance as the distance over which a parameter still looks smooth.” Chapter 3 introduces several approaches to ray tracing, i.e., solving the differential equations derived in Chap. 2, and discusses aspects such as convergence and stability over long distances. The author alternates theory with everyday life examples, e.g., by explaining the shortest path method as finding the shortest route between road signs in a town. Chapters 4 and 5 conclude the fundamentals of the theory for acoustic and elastodyamic body waves with a discussion of first order wave scattering (the Born approximation) and an overview of amplitude effects, such as geometrical spreading, anelastic damping, and scattering loss.

Chapter 6 discusses the various aspects of travel time observation, such as the determination of onset times, wavelet estimation, and determination of differential times via cross-correlation. It also includes a brief discussion on the sliding transition point between signal and noise and the recent developments of retrieving Green’s functions from the cross-correlation of ambient noise. The interpretation of travel times is discussed in Chap. 7. After an introduction on wave front healing due to finite frequencies, it is shown that travel times are affected by heterogeneities near the ray. The sensitivity to these heterogeneities is quantified by the so-called Fréchet kernel. Chapter 8 discusses the observation and interpretation of amplitudes along similar lines as the previous two chapters, including discussions on amplitude healing and amplitude Fréchet kernels.

Chapter 9 is dedicated to normal mode analysis. Since Earth is a finite body with a stress-free surface, the solution of the wave equation yields a discrete spectrum of eigenfrequencies. For very low frequencies, below 10 mHz, these frequencies are separated well enough to be measured and analyzed, leading to an estimate of the very long wavelength features of the Earth’s interior. The models obtained by transmission tomography should match these features at the low frequency end of their spectrum.

The treatment of surface waves, the most prominent arrivals on a seismogram, is complicated because surface waves do not have clearly defined wave fronts, like body waves, nor do they exhibit a discrete spectrum, like normal modes. In Chaps. 10 and 11, the author deals with this “ray-mode duality,” discusses how to measure the dispersion, and develops Fréchet kernels for finite frequency surface waves.

The following chapters address the various aspects related to the tomographic inversion itself. Chapter 12 deals with the art of model parametrization, i.e., finding the delicate balance between not imposing too much smoothness on the model, while allowing the data misfit to be sufficiently small and at the same time avoiding too much overparametrization. Parameters such as the ellipticity of Earth, surface topography, crustal thickness, and instrument response cannot be estimated from the data. Chapter 13 discusses how these parameters can be included as model corrections prior to inversion, assuming of course that a priori information on these parameters is available. Chapter 14 discusses the actual linear inversion. It reviews the basic aspects of maximum likelihood estimation, singular value decomposition, Bayesian inversion, and information theory. An inversion result only makes sense if one knows its limitations. Therefore, a chapter is dedicated to resolution and error analysis (Chap. 15).

In the first 15 chapters, Earth and the Sun were assumed to be isotropic. Chapter 16 discusses some of the modifications that are required to take anisotropy into account. In the final chapter (Chap. 17), the author looks ahead at promising new developments in imaging and observation, such as the analysis of multiple scattering, the application of non-linear inversion, and the strive toward global coverage of seismic sensors with programs like U.S. Array. The latter program involves a dense regular array of temporary seismic stations, moving across the continent of the United States from 2005 to 2015. This program constitutes an important step in the direction of “the Earth as a carefully designed experimental physics laboratory,” and will contribute to obtaining sharper and more reliable images of the interior of our planet.

Being an exploration geophysicist, I was only moderately familiar with the methodologies applied in global seismology. This very readable and inspiring book has brought my knowledge and understanding of this very interesting research field to a higher level. To my opinion, this book should be on the shelves of every seismology student and every practitioner of global seismic tomography. Moreover, it may be a very useful source for researchers in other disciplines employing tomography, such as ultrasonics, ocean acoustics, and exploration geophysics.