Chapter 2 is a preparatory chapter for Chapters 3 and 4 and presents the main types and basic features of rolling element bearings, which are used in induction motors.

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. Theapplication 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 wavesand surface wave components emerge from the normal mode spectra of the Earth for radially stratified structure. Thistreatment 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

Chapter 4 presents an introduction to vibration spectrum analysis to diagnose faults in rolling element bearings in induction motors.This chapter is a precursor to the presentation of industrial case histories in Chapters 5 to 8 using conventional vibration spectrum analysis (VSA) to diagnose the onset of faults in rolling element bearings before actual bearing failures occur. It is emphasised that previously published books and papers that cover the theory and application of vibration monitoring to diagnose faults in rolling element bearings assume that vibration transducers such as accelerometers can be always placed on the bearing housings.In many cases access to bearing housings is not practically possible on induction motors and this a key fact which is demonstrated in this book.

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.

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.

The structure of the earth varies in 3-dimensions superimposed on the dominant trend of increase of seismic wavespeed with depth.We consider integral representations of the wavefield for media with horizontal variations, and from these develop the concepts of reflection and transmission operators that allow insight gained in the discussion of stratified media to be transferred to a more general 3-D environment.This operator approach also provides a convenient link between larger scale seismology and seismic reflection studies., and also helps in assessing approximations for propagation of both body waves and surface waves

Chapter 5 presents industrial case histories using VSA to diagnose cage faults in bearings of SCIMs.