The Fractal Nature of Fault Slip and Its Incorporation into Earthquake Slip Inversions

In order to understand how earthquakes nucleate, propagate and terminate it is essential to understand the properties and stress conditions of the surfaces upon which earthquakes occur. Fault surfaces control frictional properties and by measuring exhumed faults we can better understand earthquake propagation and how this may be linked to fault structure. In order to forecast areas of a fault likely to be at risk from future failure it is necessary to accurately model the slip that occurs during each measured earthquake. In recent years many lines of evidence suggest that fault surfaces and earthquake slip show fractal properties. This includes high resolution scans of fault surfaces, observations of coseismic surface slip and analysis of published slip distributions. In this thesis I investigate how fault structure may affect the fractal properties of fault surface roughness, by investigating the along-strike changes in properties of the Campo Felice fault in the Italian Apennines. I then incorporate observations of fractal properties into earthquake slip inversions through a new form of regularisation, which I develop using Bayesian methods. Through this I aim to improve our understanding of the surfaces upon which earthquakes occur, how this links to fault structure and to improve our coseismic slip models, that provide the basis of stress models and hazard analysis. Fault surfaces displaying fractal properties mean that there is a power-law relationship between the topography of a fault and the wavelength of this topography: the magnitude of height fluctuations depends upon the scale at which they are observed. Whilst many studies have investigated fault roughness properties, here I present the first study of how fault roughness varies along the strike of a fault. I use terrestrial laser scans and laser profilometer scans at 14 locations along the length of the Campo Felice normal fault in the Italian Apennines, as well as a scan encompassing several hundred meters along the length of the fault. These scans show that the Campo Felice fault displays fractal properties over at least six orders of magnitude perpendicular to slip and at least three orders of magnitude parallel to slip. But, contrary to previous findings on other faults, I find that the Hurst parameter, which controls the fractal nature of the fault surface, changes considerably and unpredictably along the length of the fault, even between observations tens of metres apart. I suggest that this variability may be due to the variation of slip vector along the length of the fault, as is frequently observed in earthquakes. This variability could, additionally, be linked to fault asperities halting or impeding rupture, such that some areas of the fault experience more earthquakes, or experience different stress conditions during the same earthquake. I also find that the magnitude of topography displayed by Campo Felice fault is low compared to previous studies, suggesting it may be at risk of larger earthquakes. Observations of fractal fault surfaces suggest that earthquake slip should be fractal too. By using geodetic data taken at the surface before and after an earthquake we can perform slip inversions to give a model of how much slip occurred underground, on the fault surface. This is routinely performed for large, continental earthquakes. Due to noise and lack of data these inversions are frequently regularised to produce a stable solution, but the standard regularisation techniques have little physical basis. I incorporate fractal properties of earthquake slip into slip inversions by introducing a new regularisation technique: von Karman regularisation. I use a Bayesian method to fully explore parameter space and better understand uncertainties on the model parameters. From synthetic tests I find that this regularisation performs comparably, if not better, than other frequently used methods upon both fractal and Laplacian input slip distributions. Using InSAR (Interferometric Synthetic Aperture Radar) and GPS (Global Positioning System) data from the 2014 Mw 6.0 Napa Valley earthquake, I invert for slip using a two-segment fault model. I find that the choice of regularisation changes the location and magnitude of slip, which could have important implications for stress transfer and our understanding of the so-called shallow slip deficit. Through its incorporation of fractal properties, von Karman regularisation represents a more physical regularisation of earthquake slip along a fault plane. However, some bias can be introduced by incorrectly choosing the length and width of the fault plane. If a fault plane is too large, the regularisation can cause slip smearing, particularly at depth where the model is poorly constrained by the data, in order to improve the von Karman probability. To eliminate this bias I modify my Bayesian inversion scheme to solve for the size of the fault plane during the inversion, along with slip, rake and a hyperparameter controlling slip variance. This makes the inversion trans-dimensional, and aims to reduce the bias caused by an incorrect model. I apply it to the Mw 6.2 Central Tottori earthquake, Japan, using InSAR and GNSS (Global Navigation Satellite System) data. My model shows that the earthquake ruptured most of the seismogenic zone, in contrast to seismological studies. My results in this thesis further confirm that fault surface roughness shows fractal properties, and that fault structure may play an important role in the exact relationship between fault topography and the lengthscale of observation. Further investigation of exhumed fault surfaces can help inform earthquake models, including earthquake slip inversions, particularly if an earthquake were to occur on a fault upon which surface roughness measurements had already been taken. By incorporating observed fractal properties into earthquake slip inversions I aim to introduce less bias than other, less physical regularisations. With the European Space Agency's new satellites Sentinel-1a/b providing regular observations of the Earth's deforming regions, we are in a position to model earthquake slip better than ever before. I hope that by incorporating more realistic observations and using Bayesian methods to fully understand uncertainties, we can produce better, more realistic models. These models help our understanding of earthquakes, and, most importantly, earthquake hazard.