This paper studies the initial sea surface displacement and its uncertainty after an earthquake based on tsunami waveforms. The spatial distribution is inferred with a Bayesian approach that provides probabilities that are interpreted as uncertainties of the displaced sea surface. The parameterization is nonlinear and treats apparent rupture velocity as unknown but assumes rise time to be fixed at 30 s. Importantly, the spatial complexity of the source is constrained by observations using a transdimensional algorithm based on a wavelet decomposition of the displacement field. In this approach, the number of wavelet coefficients is an unknown random variable that is also estimated as part of the inversion. The resulting parameterization is parsimonious in that it can adapt to the spatially varying source complexity while being consistent with the information in the tsunami waveforms. In this way, the resolution of displacement varies across the source region with more parameters introduced for parts of the source that are resolved well by the data and/or have significant complexity. The noise level (standard deviation) at each gauge is initially treated as unknown to estimate data covariance matrices. These matrices are applied in subsequent inversion and include unknown scaling which eliminates the requirement to assume station weights and accounts for temporally correlated waveform noise. The method is applied to waveforms recorded during the 2011 Japan Tsunami and results show high resolution (low uncertainty) in most parts of the source region and a previously unreported level of source detail. In particular, the main peak of the source is elongated trench parallel and shows a well-resolved bimodal finger-like feature in the northern source region that closely follows the trench.
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