## Seismic interferometry and non-linear tomography

##### Abstract

Seismic records contain information that allows geoscientists to make inferences about
the structure and properties of the Earth’s interior. Traditionally, seismic imaging and
tomography methods require wavefields to be generated and recorded by identifiable
sources and receivers, and use these directly-recorded signals to create models of the
Earth’s subsurface. However, in recent years the method of seismic interferometry has
revolutionised earthquake seismology by allowing unrecorded signals between pairs of
receivers, pairs of sources, and source-receiver pairs to be constructed as Green’s
functions using either cross-correlation, convolution or deconvolution of wavefields. In all
of these formulations, seismic energy is recorded and emitted by surrounding boundaries
of receivers and sources, which need not be active and impulsive but may even constitute
continuous, naturally-occurring seismic ambient noise.
In the first part of this thesis, I provide a comprehensive overview of seismic
interferometry, its background theory, and examples of its application. I then test the
theory and evaluate the effects of approximations that are commonly made when the
interferometric formulae are applied to real datasets. Since errors resulting from some
approximations can be subtle, these tests must be performed using almost error-free
synthetic data produced with an exact waveform modelling method. To make such tests
challenging the method and associated code must be applicable to multiply-scattering
media. I developed such a modelling code specifically for interferometric tests and
applications. Since virtually no errors are introduced into the results from modelling, any
difference between the true and interferometric waveforms can safely be attributed to
specific origins in interferometric theory. I show that this is not possible when using other,
previously available methods: for example, the errors introduced into waveforms
synthesised by finite-difference methods due to the modelling method itself, are larger
than the errors incurred due to some (still significant) interferometric approximations;
hence that modelling method can not be used to test these commonly-applied
approximations. I then discuss the ability of interferometry to redatum seismic energy in both space
and time, allowing virtual seismograms to be constructed at new locations where receivers
may not have been present at the time of occurrence of the associated seismic source. I
present the first successful application of this method to real datasets at multiple length
scales. Although the results are restricted to limited bandwidths, this study demonstrates
that the technique is a powerful tool in seismologists’ arsenal, paving the way for a new
type of ‘retrospective’ seismology where sensors may be installed at any desired location at
any time, and recordings of seismic events occurring at any other time can be constructed
retrospectively – even long after their energy has dissipated.
Within crustal seismology, a very common application of seismic interferometry is
ambient-noise tomography (ANT). ANT is an Earth imaging method which makes use of
inter-station Green’s functions constructed from cross-correlation of seismic ambient noise
records. It is particularly useful in seismically quiescent areas where traditional
tomography methods that rely on local earthquake sources would fail to produce
interpretable results due to the lack of available data. Once constructed, interferometric
Green’s functions can be analysed using standard waveform analysis techniques, and
inverted for subsurface structure using more or less traditional imaging methods.
In the second part of this thesis, I discuss the development and implementation of a
fully non-linear inversion method which I use to perform Love-wave ANT across the British
Isles. Full non-linearity is achieved by allowing both raypaths and model parametrisation
to vary freely during inversion in Bayesian, Markov chain Monte Carlo tomography, the
first time that this has been attempted. Since the inversion produces not only one, but
a large ensemble of models, all of which fit the data to within the noise level, statistical
moments of different order such as the mean or average model, or the standard deviation
of seismic velocity structures across the ensemble, may be calculated: while the ensemble
average map provides a smooth representation of the velocity field, a measure of model
uncertainty can be obtained from the standard deviation map.
In a number of real-data and synthetic examples, I show that the combination of
variable raypaths and model parametrisation is key to the emergence of
previously-unobserved, loop-like uncertainty topologies in the standard deviation maps.
These uncertainty loops surround low- or high-velocity anomalies. They indicate that,
while the velocity of each anomaly may be fairly well reconstructed, its exact location and
size tend to remain uncertain; loops parametrise this location uncertainty, and hence
constitute a fully non-linearised, Bayesian measure of spatial resolution. The uncertainty
in anomaly location is shown to be due mainly to the location of the raypaths that were
used to constrain the anomaly also only being known approximately. The emergence of
loops is therefore related to the variation in raypaths with velocity structure, and hence to
2nd and higher order wave-physics. Thus, loops can only be observed using non-linear
inversion methods such as the one described herein, explaining why these topologies have
never been observed previously. I then present the results of fully non-linearised Love-wave group-velocity tomography
of the British Isles in different frequency bands. At all of the analysed periods, the
group-velocity maps show a good correlation with known geology of the region, and also
robustly detect novel features. The shear-velocity structure with depth across the Irish Sea
sedimentary basin is then investigated by inverting the Love-wave group-velocity maps,
again fully non-linearly using Markov chain Monte Carlo inversion, showing an
approximate depth to basement of 5 km. Finally, I discuss the advantages and current
limitations of the fully non-linear tomography method implemented in this project, and
provide guidelines and suggestions for its improvement.