Studies in sparsity constrained approaches to geophysical inversion

Abstract

Tomography is an indispensable tool in Earth sciences, providing essential insights into the planet's internal structures and the geological processes that shape them. However, traditional tomographic techniques often struggle to accurately represent complex geological features, especially those that vary across multiple scales. These methods typically produce images that are overly smoothed, blurring sharp boundaries that are crucial for detailed geological interpretation. Additionally, the need for intricate parameter optimisation in these traditional approaches can make the interpretation of large datasets challenging.

To address these challenges and fully harness the potential of tomographic imaging, this thesis introduces a pioneering methodology known as `overcomplete tomography'. This novel approach uses an expanded set of mathematical functions to represent the Earth's interior, allowing for high-resolution imaging even from relatively sparse datasets. By employing more functions than traditionally required, overcomplete tomography can capture a wider range of geological features with enhanced precision, leading to more accurate reconstructions of complex structures. This technique not only improves image quality but also simplifies the process of building models, making the inversion more intuitive and less dependent on arbitrary parameter settings.

A key aspect of our method is the use of a mathematical technique called L1 regularisation in the inversion process. This approach encourages simpler models by promoting sparsity, meaning that only the most significant features are represented in the final images. This ensures that the images are both precise and concise, making them easier to interpret and preserving critical structural details often lost in conventional tomographic reconstructions.

Furthermore, we extend the application of overcomplete tomography to include temporal analyses, thereby enhancing its utility for dynamic studies. This temporal dimension is critical for monitoring and understanding natural phenomena such as crustal deformation, seismic activity, and volcanic eruptions. Capturing changes over time not only enhances our ability to track and predict geological hazards but also provides deeper insights into the dynamic processes shaping the Earth's crust.

Overall, this research significantly advances the field of geophysical tomography. By overcoming the inherent limitations of traditional methods and introducing a high-resolution approach, overcomplete tomography promises to revolutionise our understanding of the Earth's interior and its dynamic behaviours. Show more