Abstract
Seismic free oscillations, or normal modes, provide a convenient tool to calculate low-frequency seismograms in heterogeneous Earth models. A procedure called ‘full mode coupling’ allows the seismic response of the Earth to be computed. However, in order to be theoretically exact, such calculations must involve an infinite set of modes. In practice, only a finite subset of modes can be used, introducing an error into the seismograms. By systematically increasing the number of modes beyond the highest frequency of interest in the seismograms, we investigate the convergence of full-coupling calculations. As a rule-of-thumb, it is necessary to couple modes 1–2 mHz above the highest frequency of interest, although results depend upon the details of the Earth model. This is significantly higher than has previously been assumed. Observations of free oscillations also provide important constraints on the heterogeneous structure of the Earth. Historically, this inference problem has been addressed by the measurement and interpretation of splitting functions. These can be seen as secondary data extracted from low frequency seismograms. The measurement step necessitates the calculation of synthetic seismograms, but current implementations rely on approximations referred to as self- or group-coupling and do not use fully accurate seismograms. We therefore also investigate whether a systematic error might be present in currently published splitting functions. We find no evidence for any systematic bias, but published uncertainties must be doubled to properly account for the errors due to theoretical omissions and regularization in the measurement process. Correspondingly, uncertainties in results derived from splitting functions must also be increased. As is well known, density has only a weak signal in low-frequency seismograms. Our results suggest this signal is of similar scale to the true uncertainties associated with currently published splitting functions. Thus, it seems that great care must be taken in any attempt to robustly infer details of Earth's density structure using current splitting functions.
Highlights
Our understanding of the Earth’s large-scale interior structure and dynamics draws heavily on observations of seismic free oscillations (‘normal modes’)
It is not clear how to define a meaningful prior for the splitting function inversion, and in any case such analysis would neglect the inherent non-linearity of the measurement process
By systematically increasing the number of modes in full-coupling calculations, we have demonstrated that accurate full-coupling requires inclusion of modes significantly beyond the highest frequency range of interest in the seismograms
Summary
Our understanding of the Earth’s large-scale interior structure and dynamics draws heavily on observations of seismic free oscillations (‘normal modes’). Al-Attar et al (2012) highlighted that use of either the self- or group-coupling approximations when computing synthetic spectra leads to errors of similar magnitude to the signal likely to be attributable to lateral variations in density within the Earth. Building on the existing body of work, the first aim of the present paper is to identify more precisely the conditions under which full-coupling can be regarded as ‘sufficiently accurate’ This information permits us to perform high-quality synthetic experiments to address the second question: can robust information about earth structure be inferred from measurements of ‘splitting functions’ (Woodhouse & Giardini 1985; Giardini et al 1987, 1988)? Repeating the experiment for models where density perturbations are set to zero, we conclude that splitting functions are contaminated by a significant approximation error, a similar regularization error and contain a density signal of the level of the total measurement uncertainty
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