We provide a discussion of the reflection wavefield in an exploration context, which allows links to be drawn to current developments for exploiting teleseismic arrivals. We provide a brief discussion of both land and marine seismic profiling, with emphasis on the multiple issues that arise in the case of a hard sea floor. We consider propagation issues using the operator development and show how this leads to understanding of migration procedures, with links to remapping of reflectivity as is currently being exploited in Marchenko techniques.

The theoretical foundations of the description of heterogeneity are established for both body waves and modal fields. We first consider perturbations of the wavefield using Born series and show how such concepts can be combined with the use of reflection and transmission operators to provide a flexible treatment of structures with varying heterogeneity in different zones of the model. Although the various modes of surface waves propagate independently in simple structure, the presence of heterogeneity induces cross-coupling that modifies the wavefield. Because the nature of heterogeneity differs in the various parts of the Earth, the interaction of different classes of heterogeneity has an important role in shaping the nature of the full seismic wavefield.

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 Hamiltonian Monte-Carlo techniques for efficient searching of high-dimensional spaces.

We examine the interaction of seismic waves with heterogeneity at all scales, with an emphasis on the influence of structure on multiple scales. Strong interactions occur when seismic wavelengths are comparable to the size of heterogeneity, producing complex scattering. Because seismic waves span a broad spectrum of frequencies and hence wavelengths, any heterogeneous structures will be perceived in different ways by the various aspects of the wavefield, with significant difference in behaviour between body waves and longer period surface waves. Such complications become most evident when a wide range of heterogeneity scales are present simultaneously, as in the lithosphere.

We here establish basic inversion framework in a Bayesian context, with introduction of measures of data fit and model suitability. We introduce Bayes’ theorem and identify the conditional probability with posterior probability distribution for model parameters through a composite misfit combining the match between observations and simulations and assumptions about the nature of acceptable models. We discuss Monte Carlo techniques and the assessment of model resolution, leading into the formulation of the non-linear inversion process in terms of optimisation of a measure of misfit.

We provide a broad survey of methods for inversion relevant to waveforms, showing how inversions can be performed for models described by very large numbers of parameters. As are needed for seismic tomography We concentrate on numerical optimisation of a composite misfit function using descent methods . We discuss steepest descent, conjugate gradient, quasi-Newton and subspace methods providing specific algorithms and illustrations.

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 large-scale structure of the Earth can be extracted with seismic tomography, but the finer scales of variation within the Earth lie beyond any capacity for direct imaging. Nevertheless, the scattered wavefield produced by small-scale heterogeneity contains important information on structure. We consider the representation of variations in Earth structure on scales from the global to the regional, and discuss ways in which numerical simulations and inversions can exploit data with differing station density to provide maximum resolution of structure. We contrast deterministic and stochastic (parametric) representations of heterogeneity, and examine the way in which ensemble results can be exploited for Earth structure that is time invariant. We also consider the way that effective media, with simpler structure, can be extracted from complex models by the process of wavespeed upscaling

We here examine the processes of scattering in the Earth, the various zones where it is important and the way that these different zones influence observed seismograms.Guided waves in heterogeneous structures play an important role at high frequencies, and can be described as a stochastic waveguide effect. Waves can be guided even within high wavespeed zones by elongate heterogeneity. We discuss examples from the propagation of deep earthquakes in subduction zone environments, and for the oceanic and continental lithosphere.

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.

The seismic signals from major events continue to propagate through the Earth for hours. The application of correlations between seismic stations to these long codas extracts steeply travelling body waves as the main contributors to the correlation wavefield. The properties of the coda correlation depend on the differences between the seismic phases that are being correlated. As a result, the correlation wavefield of the coda emphasises seismic phases that are difficult to detect in direct excitation by a source and so can provide new information on internal structure, e.g., an improved estimate of the shear wavespeed in the inner core.

The book covers propagation of seismic waves on all scales from the global to the local. We start by providing a discussion of the way that the higher frequency body waves and surface wave components emerge from the normal mode spectra of the Earth for radially stratified structure. This treatment provides a formulation that links global and local concepts, which is exploited in later chapters. We introduce the description of seismic wave propagation in terms of reflection and transmission through zones of Earth structure and show how this enables understanding of the physical processes that lead to observed seismic signals.

We address the nature of the correlation wavefield and its relation to the group of techniques collectively known as seismic interferometry. We establish a direct representation of the cross-correlation of the seismic signals between two stations and show how, with a suitable distribution of sources, this correlation can provide a virtual source-receiver pair whose phase properties arise from differencing. We then discuss the concept of generalised interferometry with an arbitrary distribution of sources, and illustrate the way in which processing procedures can affect the nature of correlated signals

We introduce the concepts of waveform correlations, and the way in which they can be exploited to extract information from seismograms. We show how correlation procedures can be used to determine time and phase delays. We then consider the closely related topic of transfer functions between aspects of the wavefield, and this leads into a discussion of the ways by which seismograms can be compared – a topic of importance in the comparison of observations and simulations. We also consider the nature of receiver functions and the correlation of teleseismic signals at a receiver to yield information on local structure.