What is waveform tomography?

Seismic tomography – in which we construct images of a body's interior using seismic waves – is an inverse problem; that is, our goal is to find a model that fits a set of existing data observations. This is much less straightforward than the reverse, forward problem (i.e., generating synthetic data from an existing model) due to the fact that multiple models can fit the same data or, in other words, the solution is non-unique. Furthermore, there may be parts of the model to which the data have no sensitivity, and small errors in the data can propagate into significant errors in the model (e.g., Trampert, 1998). In order to transform data space into model space, a seismic modeling algorithm is required that can generate synthetic data from an initial model, and then update the initial model to minimize the misfit between synthetic and observed data. There is therefore always a trade-off between the computational efficiency of the modeling algorithm and the accuracy or resolution with which it can represent the real seismic structure. Which kind of modeling algorithm is employed in a given situation depends very much on the nature of the structure being imaged, the quality of the data, and the available computational resources.

Traditionally, seismic tomography has used the travel times of wave phases between a seismic source and receiver to infer the sound-speed structure along the path between them. This so-called travel-time tomography, more typically referred to as “time–distance helioseismology” when applied to the Sun, is based upon ray theory, which assumes that waves travel with an infinitely high frequency, in much the same way as light rays propagating through a medium with smoothly varying refractive index, occasionally encountering a sharp interface. Under this approximation the seismic energy propagates along infinitesimally narrow geometric “ray paths.”

Introduction

There have only been three occasions in the past where seismic data from the surface of an extraterrestrial body were successfully acquired, although several unsuccessful attempts have been made, as discussed in Chapter 3. The first was during and following the Apollo landing missions on the Moon in 1969 through 1977, the second was during the Viking mission on Mars in 1976 through 1978, and the third was the Soviet Venera missions to Venus in 1982. During the Apollo project, three seismic experiments were conducted at various landing sites: Passive Seismic Experiment (PSE) at Apollo landing sites 11, 12, 14, 15, and 16; Active Seismic Experiment (ASE) at landing sites 14 and 16; and Lunar Seismic Profiling Experiment (LSPE) at the Apollo 17 landing site. These landing sites are indicated on Figure 7.1. During the Viking project, seismometers were deployed at two landing sites, Viking 1 and Viking 2, but the seismometer at the Viking 1 site failed to uncage, and only at the Viking 2 landing site were records of ground motion obtained. The Venera 13 and 14 measurements lasted only briefly, 127 and 57 minutes, respectively.

In this chapter, we briefly describe the Apollo and Viking experiments and summarize key findings of the contemporary analyses of the acquired data. Included are some historical accounts of events experienced at the time. Since the turn of the century, with vastly more powerful computers now available and with developments of new analysis techniques, these decades-old data are attracting renewed interest, and new results are emerging. These new results are described in Chapter 15. The results of the Venera experiment will not be discussed here because information available to the western world is scarce. We understand that only a single event of unknown origin was counted (e.g., Ksanfomaliti et al., 1982). The chapter will end with a brief account of remaining questions and suggested future directions in lunar and planetary seismology.

In this chapter, I discuss the first steps in asteroseismic analysis – following data through the analysis to the products that are what stellar modeling tries to explain. For asteroseismology, the time-varying signal is generally either the total brightness of the target (i.e., time-series photometry) or the (disk-integrated) radial velocity as measured via Doppler shifts of spectral lines. In either case, we measure a differential quantity (ΔY/Y) at regular intervals. More precisely, this quantity is integrated over a finite but short time (shorter than any expected signal from the star), producing an average value over the time interval of the integration. This produces the time series of the signal which I refer to as yn (tn) not in an instantaneous sense, but in a smoothed way.

Our time series represents an averaged, discretely sampled quantity as a function of time. From that time series, our goal is to determine the harmonic content of the signal – over what time scale(s) does the signal vary? Are there coherent (in phase and/or amplitude) sinusoidal variations? Are there preferred time scales for “noise” processes? Are there periodicities whose phase varies slowly over the entire interval? Are there “temporary” periodic signals? Answering these questions requires more and more sophisticated analysis techniques. In all cases, though, the sampling of the signal is critical in limiting the information content that we can derive.

There are many tools and techniques available for analysis of time-series data. In this chapter I will concentrate on Fourier analysis (i.e., power spectrum analysis), and also describe least-squares fits to amplitudes, phases, and frequencies. An equivalent analysis is via the Lomb–Scargle periodogram (Scargle, 1982; Appourchaux, 2014). Of course, more sophisticated tools are available but these have provided the dominant “one-two” punch for most asteroseismic studies. Various specific applications of asteroseismology employ subsequent analysis of the power spectra or frequency lists to derive oscillation parameters, but I leave those to subsequent chapters.

The beginning of the seismological inv estigation of the Moon dates back to the beginning of space flight: A working group implemented by NASA in 1959 (Hall, 1977) suggested the development of a seismometer for a hard landing on the Moon. This resulted in Ranger missions 3 to 5, which all unfortunately failed for technical reasons (Hall, 1977). The first measurement of elastic properties of lunar soil was conducted by the Surveyor landers a few years later (Christensen et al., 1968).

Besides these early attempts, seismological studies of the Moon divide into two phases: The first one saw the installation of a seismometer network on the Moon, starting with Apollo 11 on July 20, 1969 (Apollo 11 Mission Report, 1969), followed by the collection of continuous data until network shutdown on September 30, 1977 (Bates et al., 1979), and, in parallel and ongoing until the early 1990s, the analysis of the data. The second phase began in the late 1990s, when cheap computer power allowed for massive data processing on desktop workstations and the application of new methods.

This chapter aims to give a sketch of the Moon as it results from these two phases. The following sections will first describe the different types of seismic events observed on the Moon, and then detail the structure of the main layers of the lunar interior, i.e., the crust, mantle, and core. A summary section finally gives an overview of the present-day concept of the interior structure of the Moon.

Seismic sources and seismicity

Both endogenous and exogenous sources create seismic waves on the Moon. It is thus common to speak of “events” rather than “quakes,” unless the type of source has been identified. However, analysis of the spatial and chronological distribution of events, and of seismogram characteristics, leads to the identification of several classes of sources.

Introduction

Local helioseismology encompasses remote observations, data analysis, and theoretical modeling of solar oscillations to infer the three-dimensional structure within localized regions of the solar interior. What defines a region as “local” is relative, however, since targets of interest have included sunspots and convective elements with spatial scales ∼10−2R⊙ as well as large-scale plasma flows spanning much of a solar hemisphere. As a relatively new discipline first explored in the 1980s, local helioseismology has two main components: first, a research component to understand the interaction of solar oscillations (acoustic and surface gravity) with perturbations within the Sun and, second, the design and application of methods to infer the properties of the perturbations by modeling the measurements of those waves. Successful applications require a thorough understanding of the physics of the waves and their interaction with in homogeneities inside the Sun. The research component is particularly critical. For example, the types of perturbations found in the Sun can include magnetic fields for which the wave interactions can be quite complicated. Currently, the types of structures most amenable to modeling using local helioseismic measurements consist of isotropic wave-speed perturbations and the three components of plasma flows. Assessing the subsurface magnetic field directly is a challenging, but largely unrealized, goal of the field. While the status of the field is evolving, the determination of plasma flows in the first few tens of Mm below the solar surface remains one of the primary practical applications.

We outline in this chapter the practical applications of, and resulting measurements made with, common local helioseismic methods. Broadly speaking, local helioseismology can be roughly divided into Fourier methods (which operate in the frequency–wave number domain) and cross-covariance based methods (which operate in the space–time domain). The former (Section 6.3) can be considered in many ways as extensions of the analysis of global oscillations (Chapter 5) to localized regions of the Sun.

Introduction and overview

Sunquakes observed in the form of expanding wave ripples on the surface of the Sun during solar flares represent packets of acoustic waves excited by flare impacts and traveling through the solar interior. The excitation impacts strongly correlate with the impulsive flare phase, and are caused by the energy and momentum transported from the energy release sites. The flare energy is released in the form of energetic particles, waves, mass motions, and radiation. However, the exact mechanism of the localized hydrodynamic impacts that generate sunquakes is unknown. Solving the problem of the sunquake mechanism will substantially improve our understanding of the flare physics. In addition, sunquakes offer a unique opportunity for studying the interaction of acoustic waves with magnetic fields and flows in flaring active regions, and for developing new approaches to helioseismic acoustic tomography.

Solar flares represent a process of rapid transformation of the magnetic energy of active regions into the kinetic energy of charged particles, plasma flows, and heating of the solar atmosphere and corona. The primary energy release during the flares is believed to occur in the corona as a result of magnetic reconnection. It is generally believed that most of the energy released by the reconnection goes directly and indirectly (via plasma waves) to acceleration of electrons and protons which are injected into flaring magnetic loops (Figure 22.1a). Most of the observed radiation is produced either directly by these particles or indirectly through energization of the background plasma.

It was suggested long ago (Wolff, 1972) that flares may cause acoustic waves traveling through the solar interior, similar to the seismic waves in the Earth. Because the sound speed increases with depth, the waves are refracted in the deep layers of the Sun (Figure 22.1b), and then appear on the surface, forming expanding rings, similar to ripples on a water surface.

Introduction

Seismology of giant planets has long been considered as both a potentially powerful tool for probing their interiors and a natural extension of helioseismology. Giant planets are mostly fluid and convective, which makes their seismology much closer to that of solar-like stars than that of terrestrial planets. For this, we refer the reader to the introductory chapters about helio- and asteroseismology for basic concepts and vocabulary. By being the biggest and closest, Jupiter has attracted most of the efforts in this domain. Theoretical studies started in the late 1970s and the first observational attempts were undertaken in the late 1980s. So far, the two major results are a clear detection of acoustic oscillations of Jupiter (Gaulme et al., 2011), and the signature of Saturn f modes in the rings by the NASA Cassini spacecraft (Hedman and Nicholson, 2013).

This chapter first examines the theoretical motivation for developing seismology of giant planets, which mainly stands on an inaccurate knowledge of their interiors (Section 14.2). The next sections focus on two crucial points: why seismology can be done on giant planets (14.3), and what can it bring in terms of physics (14.4). We then present the observation techniques that have been used or envisioned to detect oscillations (14.5), and the main observational results (14.6).

Interior structure

Giant planets are planets massive enough to have retained the hydrogen and helium initially present in the circumstellar disk that led to the formation of the central star and its planets. The study of their composition is important in understanding both the mechanisms enabling their formation and the origins of planetary systems, in particular our own. Unfortunately, composition determination is complicated by the fact that their interiors are thought not to be homogeneous, so that spectroscopic determinations of atmospheric abundances are probably not representative of the planet as a whole.

Introduction

In this chapter we describe the recent field of diskoseismology: the study of extraterrestrial seismology in accretion disks. Accretion disks, formed by accumulation of matter falling toward a gravitating object, are ubiquitous in the universe, surrounding a wide variety of celestial objects.

They are a key ingredient in binary systems which host compact objects orbiting a so-called companion star, and attracting matter from it. In these objects, an accretion disk forms through accumulation of gas coming from the companion star, tidally attracted and passing through the inner L1 Lagrange point (see Figure 16.1). The compact object can be either a neutron star or a stellar-mass black hole (these systems are then called compact X-ray binaries, XRBs), or a white dwarf (systems called cataclysmic variables, CVs). Furthermore, they surround the supermassive black holes (SMBH) at the heart of active galaxies, we also detect them at cosmological distances in the extremely energetic gammaray bursts (GRBs), and finally, we find them around protostellar and young stellar objects (YSOs).

From the discovery of the first extra-solar X-ray source – Sco X-1, an XRB in the constellation of Scorpius – reported on December 1, 1962 by Giacconi et al. (1962), to the measurement of quasi-periodic oscillations and broad lines in accretion disks surrounding compact objects, our knowledge of accretion disks has greatly improved, ultimately leading to potential tests of general relativity, due to high gravitational forces naturally created by the presence of compact objects. Putting together these observations, simulations, and predictions, will eventually lead to unique results based on general relativity.

We will show in Section 16.2 that accretion disks are ubiquitous in the celestial objects of our universe, before reviewing in Section 16.3 the general characteristics of accretion disks. We will then introduce in Section 16.4 the field of diskoseismology, reporting the existing observations, numerical simulations, and theoretical predictions of the models, concerning the various celestial objects hosting accretion disks, before concluding in Section 16.5.

Introduction

The great success of helioseismology resides in the reearkable progress achieved in the understanding of the structure and dynamics of the solar interior. This success mainly relies on the ability to conceive, implement, and operate specific instrumentation with enough sensitivaty to detect and measure small fluctuations (in velocity and/or intensity) on the solar surface that are well below one meter per second or a few parts per million. Furthermore, the limitation of ground observations imposed by the day–night cycle (thus a periodic discontinuity in the observations) was overcome with the deployment of ground-based networks – properly placed at different longitudes all over the Earth – allowing longer and continuous observations of the Sun and consequently increasing their duty cycle.

In this chapter, we start with a short historical overview of helioseismology. Then we describe the different techniques used for helioseismic analyses along with a description of the main instrumental concepts. We particularly focus on the instruments that have been operating long enough to study the solar magnetic activity. Finally, we highlight the main results obtained with such high-duty cycle observations (>80%) over the past few decades.

A historical overview

The detection of solar oscillations goes back to more than 50 years ago when Leighton et al. (1962) discovered the five-minute oscillations in the solar photosphere. However, it was only with the observations of Deubner (1975) that their identity as standing acoustic waves, i.e., normal modes of the Sun of high spherical-harmonic degree, was established, confirming previous theoretical inferences by Ulrich (1970) and Leibacher and Stein (1971), thus setting the scene for the development of helioseismology. Coincidentally, a strong inspiration was the announcement in 1975 by the SCLERA (Santa Catalina Laboratory for Experimental Relativity by Astrometry) group of evidence for oscillations in the solar diameter (see Hill et al., 1976), of apparently truly global nature and hence containing information about the entire Sun.

Introduction

Variable stars have been observed since the discovery of the variability of Mira by David Fabricius at the end of the sixteenth century (Olbers, 1850) with a variation of 10 stellar magnitudes! The first known photometric measurement gf variability was done by Goodracke (1783) for Algol, comparing with stars differing by two magnitudes only. Using the same technique, Goodricke and Englefield (1785) discovered that δ Cephei was a variable star showang a periodical non-sinusoidal variation of about one magnitude.

Roberts (1889) suggested the use of photography for studying variable stars. The technique for detecting variable stars was quickly improved by photographically recording variations in the stellar spectra (Fleming, 1895), which would lead to visual identification of these variables (Reed, 1893).

The use of the selenium photometer by Stebbins and Brown (1907) and the development of the photoelectric cell, as anticipated by Stebbins (1915), introduced the electric determination of variability of stars (Stebbins, 1911). This was the start of an era: the detection of photons using the photoelectric effect. Improvement in the instrumentation led to lower detectable levels of variability that led to the discovery of rapidly oscillating Ap stars by Kurtz (1978). These stars oscillate with periodicity of about 6 to 10 minutes with an amplitude of about a t`ousandth of a magnitude (mmag). Kurtz and Shibahashi (1986) were even able to reach a noise limat of about 0.02 mmag on HR3831 Kurtz and Martinez, 2000).

From space, lower levels of variabilities had already been detected on the Sun using the ACRIM (Actave Cavity Radiometer Irradiance Monitor) instrument aboard the Solar Maximum Mission (Woodard and Hudson, 1983); the noise level was about one part per million (ppm, close to one μ mag). From the ground, clear detection of solar-like oscillations (pressure modes or p modes) in intensity and with spatial resolution was made possible using a property of the atmosphere combined with the significant angular diameter of the Sun (Appourchaux et al., 1995) (see also Chapter 2 of this book).