Abstract: In this chapter, I present an overview of waveform tomography, in the context of imaging of the Earth‘s whole mantle at the global scale. In this context, waveform tomography is defined utilising entire wide-band filtered records of the seismic wavefield, generated by natural earthquakes and observed at broadband receivers located at teleseismic distances. This is in contrast to imaging methodologies that first extract secondary observables, such as, most commonly, travel times of the most prominent energy arrivals (i.e. seismic phases), that can be easily identified and isolated in the records. Waveform tomography is a non-linear process that requires the ability to compute the predicted wavefield in a given three-dimensional Earth model and compare it to the observed wavefield. One of its main challenges, is the computational cost involved. I first review the history of methodological developments, specifically focusing on the global, whole mantle Earth imaging problem. I then discuss and contrast the two recent methodologies that have led to the development of the first three-dimensional elastic global shear velocity models that have been published to-date using numerical integration of the wave equation, specifically, using the spectral element method. I discuss how the forward problem is addressed, the data selection approaches, definitions of the misfit function, and computation of kernels for the inverse step of the imaging procedure, as well as the choice of the optimisation method. I also discuss model parametrisation, and, in particular, the important topic of how the strongly heterogeneous crust is modelled. In the final parts of this chapter, I discuss efforts towards resolving the difficult problem of model evaluation and present my views on promising directions and remaining challenges in this rapidly evolving field, aiming at further improving resolution of deep mantle elastic structure with the goal of informing our understanding of the dynamics of our planet.

Abstract: The continuously increasing quantity and quality of seismic waveform data carry the potential to provide images of the Earth’s internal structure with unprecedented detail. Harnessing this rapidly growing wealth of information, however, constitutes a formidable challenge. While the emergence of faster supercomputers helps to accelerate existing algorithms, the daunting scaling properties of seismic inverse problems still demand the development of more efficient solutions. The diversity of seismic inverse problems – in terms of scientific scope, spatial scale, nature of the data, and available resources – precludes the existence of a silver bullet. Instead, efficiency derives from problem adaptation. Within this context, this chapter describes a collection of methods that are smart in the sense of exploiting specific properties of seismic inverse problems, thereby increasing computational efficiency and usable data volumes, sometimes by orders of magnitude. These methods improve different aspects of a seismic inverse problem, for instance, by harnessing data redundancies, adapting numerical simulation meshes to prior knowledge of wavefield geometry, or permitting long-distance moves through model space for Monte Carlo sampling.

This chapter addresses a range of topics that hold considerable promise for future developments. We start by considering nested-inversions that allow definition of heterogeneity across a wide range of length scales from local through regional to global. This is followed by discussion of adaptive numerical gridding, exploitation of data redundancy, the development of efficient random sampling methods for inversion, and the use of HamiltonianMonte-Carlo techniques for efficient searching of high-dimensional spaces.

An important recent development has been the exploitation of the seismic noise field by the use of correlations between seismograms recorded at different positions. We discuss the nature of the ambient noise field, which is dominated by microseismic signals generated in the oceans. The dominant component of the correlation field comes from surface waves, and cross-correlation procedures can extract an empirical Green’s function representing propagation along the path between the stations being correlated.Such results are widely exploited in ambient noise tomography. In some circumstances body waves can also be extracted from the noise field.

The results of cross-correlation of seismic records depend on both the distribution of seismic sources and the structure in the vicinity of the path between the stations being correlated. The differences between the segments of the correlograms corresponding to opposite senses of propagation between the stations provide information on source excitation, while the properties of the dominant arrivals are mainly sensitive to structure.These properties can be exploited in inversion of the correlation wavefield, to extract both noise sources and Earth structure.

Descent methods of optimisation depend on calculating the derivatives of the composite function with respect to the large number of model parameters. Adjoint techniques allow the computation of derivatives in such complex models, with much less calculation than direct methods, and so enable practical non-linear inversion.Adjoint methods also allow effective computation of sensitivity kernels associated with the variation of critical parameters for both structure and sources. Sensitivity kernels can provide insight into the nature of Earth structure and the potential resolution of seismic tomography.

We here describe the process of waveform inversion for earthquake data, by reference to inversion for 3-D structure in seismic wavespeed and density in the eastern Mediterranean region using numerical simulation with the spectral element technique. Such waveform inversion needs to start from a good initial model, using low frequencies in the first stage. As the inversion proceeds higher frequencies and additional data can be to incorporated to achieve model refinement. We also examine the issues of practical resolution assessment, and validation of proposed models.

Current seismology depends on well-developed networks of instruments and a range of advances in both theoretical and computational developments. We provide a survey of the development of seismic recording, including the introduction of dense sets of portable instruments, so that major earthquakes are now captured by thousands of seismometers. We then discuss the way in which understanding of seismic waveforms has developed through the computation of synthetic seismograms and their exploitation in inversion. The last part of the chapter provides a description of the structure of the four Parts of the book