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Seismology is our primary method for inferring the structure and dynamics of Earth’s interior. A region of particular importance is the lowermost mantle near the core-mantle boundary. Detailed information about lowermost mantle seismic structures can be inferred by comparing synthetic seismic waveforms to those observed at Earth’s surface. In this thesis, a forward modelling method is presented which is based on first order single scattering approximations. These include the well-known Born approximation (which involves linearly perturbing the amplitude of the waveform), as well as the Rytov approximation (which involves linearly perturbing the phase of the waveform) and a whole family of intermediate approximations termed Marks approximations. The computation of these first order waveforms is facilitated by time and space dependent waveform sensitivity kernels. Kernels are computed for each source-receiver pair, but they are independent of the perturbed model. Once the kernel is calculated, synthetic seismograms for many models are generated very quickly (≈ 7 s per model). The applicabilities of the Born, Rytov and Marks approximations are tested by comparing to reference solutions computed with a fully numerical method. For small perturbations to the background model, an excellent agreement is found between the first order and reference solutions, providing validation of the method. The first order method breaks down with increasingly large and strong perturbations. The Rytov ap- proximation consistently performs better than Born (the worst) or Marks(intermediate). The method is valid for P, S and converted phases. The first order method is used to generate synthetic P/Pdiff waveforms under the Rytov approximation for 500 models representative of the large low velocity province under Africa. The database of synthetic data is probed in a statistical manner; sets of models which fit certain supposed observations are presented.

Machine learning (ML) has caused a fundamental shift in how we practice science, with many now placing learning from data at the focal point of their research. As the complexity of the scientific problems we want to study increases, and the amount of data generated by today's scientific experiments grows, ML is helping to automate, accelerate and enhance traditional workflows. Emerging at the forefront of this revolution is a field called scientific machine learning (SciML). The central goal of SciML is to more tightly combine existing scientific understanding with ML, generating powerful ML algorithms which are informed by our prior knowledge. A plethora of approaches exist for incorporating scientific principles into ML and expectations are rising for SciML to address some of the biggest challenges in science. However, the field is burgeoning and many questions are still arising. A major one is whether SciML approaches can scale to more complex, real-world problems. Much SciML research is at a proof-of-concept stage, where techniques are validated on simplified, toy problems. Yet, understanding how well they scale to more complex problems is essential for them to become widely applicable. This question is of central focus in this thesis. Firstly, multiple different physics-informed ML approaches are designed for three complex, real-world, domain-specific case studies taken from the fields of lunar science and geophysics, and their performance and scalability is assessed. Secondly, the scalability of physics-informed neural networks, a popular and general SciML approach, for solving differential equations with large domains and high frequency solutions is evaluated and improved. Common observations across these studies are discussed, and significant advantages and underlying limitations are identified, highlighting the importance of designing scalable SciML techniques.

This thesis aims to present the data measurement process, waveform modelling, full waveform sensitivity measurements, and inversion workflow, required to produce a P-wave tomography of the upper mantle, including the mantle transition zone, down to 1000km depth. Improved seismic velocity models are needed: many tomography models do not include regional travel-time datasets. This is because the first arrivals identified in travel-time datasbases are not enough on their own to seismically sample the mantle transition zone. Sharp velocity gradients at the top (410km) and bottom (660km) of the mantle transition zone induce triplications in body waves. This means that there is more than one possible path between source and receiver for a given waveform: these can range from a topside reflection off the 410km discontinuity, through to refraction at the base of the 660km discontinuity. It is therefore difficult to attribute the first arrival time in a body waveform to its true path in an automated manner, as needed for tomography. This is heightened by discrepancies between the assumed reference background model for data processing and inversion, with respect to the true background model: therefore, seismic travel-times must be measured relative to synthetic data based on full waveform modelling. A limited number of regional body wave velocity models based on the full seismic waveform exist, which can provide sufficient seismic illumination of the mantle transition zone, a known barrier for subducting oceanic lithosphere. In this study I present JEABOX, a P-wave tomography of the western Pacific upper mantle, using the joint inversion of existing teleseismic P-wave travel times based on multifrequency cross-correlation, with regional P-wave measurements between 14-30 degrees epicentral distance. I calculate Born sensitivity kernels for these latter triplicated data using the MC Kernel software based on the spectral element solver, Axisem. I showcase the integration of these novel triplicated data and their sensitivity kernels into a tried-and-tested finite-frequency tomography workflow, to perform joint inversion of teleseismic data and Dahlen kernels with regional travel-times and MC Kernels, to produce a hybridised P-wave velocity model based on Axisem, the first tomography of its kind. The resulting model corroborates several major seismic structures observed across western Pacific subduction zones in previous studies, including what appears to be slab stagnation beneath the Kuril and Japan trenches, and steep subduction under the Izu-Bonin trench. In the southern hemisphere, seismic signatures imply protracted subduction beneath the Sunda-Java trench under Borneo and surrounding area. Future work will tectonically interpret the JEABOX model in full.

The Anthropocene and the concomitant Great Acceleration – defined by super-exponential growth rates in a collection of measures of human activity – are having a tremendous impact on Earth and its ecosystems. To better understand the consequences of our activity, we must increase our knowledge of all the elements in the Geosystem, loosely defined as Earth, the oceans, atmosphere, and the Sun, and in constant evolution as human activity ties in new elements, e.g. the Moon or Mars. In this thesis, I am focusing on improving our knowledge of elements pertaining to geophysics, while also showcasing the diversity of thematics in the Geosystem. I also emphasize the common grounds and cross-cutting themes in our sundry of projects, in particular the importance of a data-centric approach, including the creation of new sensors, and the necessity of both methodological improvements to our numerical tools, and the adoption of novel methods from machine learning to help us exploit the newly available vast amount of data. I begin with wildlife conservation, where, with collaborators, we prototype an elephant monitoring system based on the classification of animal generated seismic signals with deep learning. We achieve promising results, with high accuracies in classification of elephants and their behaviour, and with further research I believe our method will bring about a seismic shift to wildlife monitoring. Next, I present a novel method for the computation of Frechet sensitivity kernels for full waveform inversion based on the discrete adjoint approach, resulting in a more advantageous frequency scaling of the computational cost than traditional methods. This discount is crucial to enable us to image the interior of the Earth at higher resolutions, which is necessary to unlock the answers to some of the biggest questions in Earth science, e.g. related to the mechanisms of mantle convection and plate tectonics. Finally, I introduce how, with my teammates, we developed a deep learning replacement to a now defunct Sun-monitoring spectrograph onboard a NASA satellite. The best performing model, a combined linear-CNN architecture, achieves 1.6% median error, virtually reviving the spectrograph and restoring a vital data source, for instance for satellite operators to model atmospheric dynamics and avoid catastrophic collisions.

Owing to continual improvements in computation power, we are getting closer to modelling seismic waves in ways that reflect the realistic complexity of the Earth. Nevertheless, modelling the true multi-scale structure of the Earth at large scale and/or high frequencies is currently still beyond our reach from a computational cost perspective. Hence, we rely on efficient modelling software to increase the scope of seismic simulations. Azimuthal Complexity Adaptation (ACA) is an innovative way of reducing the computational cost of seismic simulations by exploiting inherent sparsity of the wavefield. In this thesis, AxiSEM3D, an ACA-based modelling solver, is expanded to model wave propagation in local-scale settings. I find that the sparsity of wavefields persists in the presence of complex local-scale structures such as thrust faults and salt bodies and in models with full anisotropy. Further, I develop 3D wavefield scanning to locate complex areas of the wavefield and to investigate how they relate to structures in the subsurface model. The results suggest that high wavefield complexity is typically localised near abrupt wavespeed contrasts, on the low-wavespeed side thereof. I attempt to develop a new modelling solver which overcomes some of the limitations of AxiSEM3D. The resulting method is unstable, but it demonstrates the algorithmic complexity required to perform ACA in settings like coast lines. I conclude that a hybrid method which combines classic 3D modelling and ACA may be a simpler solution for increasing the scope of AxiSEM3D. Finally, I implement fault ruptures to simulate an earthquake scenario in the San Francisco Bay Area. Using ACA, I can include low-wavespeed bay muds which are usually neglected to save computational cost. Neglecting these muds leads to an underestimate of the shaking in the San Francisco Bay, even at low frequencies.

Seismology is one of the few geophysical probes which allows us to peer beneath the surface of a planet, moon, or star and map its structures and properties at a variety of scales and resolutions. This makes it a technique of enormous scientific power. Its application, however, suffers from problems of mathematical ill-posedness and poor conditioning. To alleviate some of these difficulties, we turn to computational seismic modelling. Seismic modelling is not without its own challenges, of cost and complexity, which have limited its applications. In this thesis, material will be presented which advances the frontiers of seismic modelling, especially in the domain of coupled solid-fluid media. Application of these improvements offers the opportunity to better understand the seismic behaviour, and hence the interior structures, of the worlds in question. In this thesis, we will cover advances in the seismic modelling of the Earth’s oceans, Mars’ atmosphere, the Sun, and other stars. The first five data chapters will cover the generation and analysis of synthetic data from the Earth, Mars (three times) and the Sun respectively. We will examine the influence that the oceans and their bathymetry have on surface wave propagation, followed by a presentation of results from the Oxford Seismology team’s participation in the blind test for the NASA InSight mission to Mars. Next, we will present modelling which was used to inform InSight’s attempts to detect the arrival of the Perseverance Rover in February 2021, followed by the results from this unique experiment. The penultimate chapter covers the extension of similar techniques to modelling seismic wave propagation in the Sun, and finally we will explore whether aster- oseismic techniques may be used to study the internal configuration of the λ Boötis class of stars.

This thesis presents P-wave tomographic images of mantle structures under South America and offers tectonic interpretations for the recovered features. To build this inversion, source time functions, a requirement for computing suitably realistic synthetic seismograms, were estimated from broadband waveforms of 3505 teleseismic earthquakes of magnitudes greater than 5.6, recorded between 1990 and 2016 by regional and global networks. Almost 5M traveltime anomalies were calculated using cross-correlation of synthetic and observed seismograms in eight frequency bands, with centre periods ranging from 30.0 to 2.7 seconds. Sensitivity kernels were calculated using finite-frequency theory. In order to better illuminate weakly sampled mantle regions, integrated traveltime picks from the International Seismological Centre's ISC-EHB catalogue were also included. Two main features are inferred from the tomographic images under South America. First, extensive lower-mantle fast anomalies under north-western region of South America may indicate ancient intra-oceanic subduction settings, which might have occurred to the west of the western margin of the continent. Secondly, high velocity provinces beneath western margin of South America, which appears only in the upper mantle, is interpreted to imply a younger subduction history. This fast anomalies distribution is associated to the Andean segmentation which has been archived in the geological records.

Earth structure and dynamics are largely inferred by seismology as the main tool for data-informed probing of Earth’s interior. Seismic wave propagation in physically realistic Earth models is one of the most funda- mental topics in seismology, not only essential to the analysis and interpre- tation of observed ground motions in terms of their source characteristics and structure-induced propagation effects, but also indispensable for seis- mic inverse techniques that deliver Earth models by minimising misfits between observations and theoretical predictions. Nevertheless, seismic wave propagation at a global scale still remains as one of the most chal- lenging problems in scientific computing, because of Earth’s multiscale constituents and the observable frequency band of global seismic data for their resolution. In this thesis, I present a new, computationally efficient numerical ap- proach to simulate global seismic wave propagation in realistic three- dimensional (3-D) Earth models. It is a hybrid method of spectral el- ement method (SEM) and pseudo-spectral method, that is, the azimuth dimension of a 3-D wavefield is characterised in terms of a global Fourier series parameterisation, such that the 3-D wave equation reduces to an al- gebraic system of coupled 2-D meridian equations, which is then solved by a 2-D SEM named AxiSEM3D. Computational efficiency of such a hybrid method stems from the inherent azimuthal smoothness of 3-D global wave- fields, resulted from the predominance of long-wavelength heterogeneity in Earth’s mantle and the axial singularity of a point seismic source. AxiSEM3D allows for material heterogeneities such as velocity, density, anisotropy and attenuation, as well as for finite undulations on radial discontinuities, both solid-solid and solid-fluid, and thereby a variety of aspherical Earth features such as ellipticity, topography and bathymetry, variable crustal thickness, and core-mantle boundary topography. Such interface undulations are equivalently interpreted as material perturba- tions of the contiguous media, based on the “particle relabelling transfor- mation”. Ocean is currently modelled as a hydrodynamic load. Benchmarked in reference to a discretised 3-D SEM for a variety of Earth models, including 3-D mantle and crustal structures, topography and bathymetry, and ellipticity, AxiSEM3D proves to be a convergent and ac- curate numerical method. Efficiency comparisons suggest that AxiSEM3D can be 1 to 3 orders of magnitude faster than a conventional 3-D SEM across a period range from 10 s down to 1 s, with its speedup increasing with simulated frequency and decreasing with model complexity. It is sufficiently comprehensive to cover all considered applications at global scale, but is maximally efficient for deep Earth studies with body wave data. The observable frequency range of global seismic data (up to 1 Hz) has been achieved for wavefield modelling upon a 3-D mantle model with moderate computing resources. The MPI-based high-performance C++ code scales up to more than 10,000 cores, available open-source at https://github.com/kuangdai/AxiSEM3D. Three applications are carried out, with different foci from Earth’s surface to the core-mantle boundary. First, surface waves are simulated within a state-of-the-art crustal model; the synthetics are compared to real seismic data to assess two different implementations of a 3-D crust. Second, wave scattering effects and cost impact of a localised small-scale structure with sharp and strong material contrast are investigated, considering a wide range of structure-wavelength combinations. And last, diffracted waves through an ultra-low velocity zone atop the core-mantle boundary are modelled at a 1 Hz frequency.