Uncertainty quantification in seismic interferometry
It is a well-established principle that cross-correlating wavefield observations at different receiver locations yields new responses that, under certain conditions, provide a useful estimate of the Green's function between the receiver locations. This principle, known as wavefield interferometry, is a powerful technique that transforms previously discarded data, such as background noise or earthquake codas (the multiply scattered tails of earthquake seismograms), into useful signals that allow us to remotely illuminate subsurface Earth structures. The mathematical machinery that underlies wavefield interferometry assumes a number of ideal conditions that are not often found in practical settings. Furthermore, the original formulations are frequently simplified through a variety of approximations, in order to derive expressions that are more amenable to applications. These assumptions and approximations are frequently made in an ad-hoc fashion, without consideration of the errors thus introduced. This thesis centres on the study of errors introduced by violating two important assumptions of wavefield interferometry. Namely, that the noise sources are statistically uncorrelated, and that their energy contributions are isotropic. Violating these conditions makes the Green’s function and associated phases liable to estimation errors that so far have not been accounted for or corrected. We show that these errors are indeed significant for commonly used noise sources, and illustrate cases in which the errors completely obscure the phase one wishes to retrieve. Moreover, we consider the relevant case of the stationary phase approximation, widely invoked in interferometry theory and applications, and quantify mathematically the errors introduced both in an ideal setting and in the presence of correlated, anisotropic sources, applying and extending existing error quantification theory. Throughout these settings, this thesis implements an appropriate geostatistical correlation model to investigate the effect that smoothness and long-range correlations have on the interferometric estimate, particularly its phase. Analytical expressions are given for the first and second moments of these errors, as well as deterministic error bounds and probability bounds on the uncertainty of these approximations. These results are given in terms of statistical parameters that can be empirically estimated in practice, and numerically explored. Finally, this thesis contrasts the two main types of wavefield interferometry, active or controlled source interferometry and passive or ambient noise interferometry. The impact of violating the uncorrelatedness assumption is considered. This thesis proposes strategies to mitigate uncertainty in both settings, and in the case of ambient noise interferometry, the thesis presents a novel workflow that significantly mitigates errors introduced by the presence of statistical correlations in the sources. The methodology is general in the sense that it can be applied to noise with any degree of correlation, including completely uncorrelated sources. The methods are tested on synthetic data, illustrating significant improvement in the phase estimates in both settings. In all these cases we establish various bounds on the estimation error, and we analyse their significance and utility in real-life interferometric retrieval experiments.