Regularization Approach to Tidal Analysis

Since the times of Doodson it has been established that a record of length T is required to resolve tidal harmonics with a frequency separation 1/T. This rule, known as Rayleigh criterion, does not consider the actual resolution provided by the signal-to-noise ratio of the data. Available tidal analysis software, like Eterna, seek gravimetric parameters for a priori defined groups (sums) of harmonics that are assumed otherwise indistinguishable. The residual between the predicted tidal signal for groups and the recording is minimized with simple least squares (LS) fit.

We developed the new software, RATA, that abandons the concept of groups, so each tidal harmonic present in the catalogue receives its set of tidal parameters that are free to vary. The resulting ill-conditioned matrix is stabilized by Tikhonov regularization (ridge regression) in the LS objective function. To validate the results, we used the moving window analysis (MWA) technique for a priori groups, with the resulting local response model as the a priori model. Compared to the standard approach, which used the Wahr-Dehant-Zschau elastic analysis model, we clearly see that bias and beating patterns are significantly smaller or almost vanish. Hence, the local response model can capture the apparent temporal variations by appropriate tidal parameters within the MWA groups.

While the most information in each group is carried by the tidal wave with the largest amplitude, influence of other harmonics must be properly considered in estimated amplitudes and phases. Therefore, if amplification factor or phase from any other large amplitude harmonic in the group is significantly different from the expectation, the grouping parametrization might lead to an inaccurate (biased) estimate of tidal parameters. The trade-off parameter between data residuals and the model difference to the reference model is chosen at the corner of the misfit curve, indicating expected level of noise in the data. The resulting model parameters indicate “data-driven” groups to be inferred from significant harmonics in the inversion. To demonstrate the method and how it may be used to reveal system properties hidden by wave grouping, we analyzed 11.5 years gravity recordings from the superconducting gravimeter SG056 at the BFO (Black Forest Observatory, Schiltach). As a result, we distinguished 61 significant groups of harmonics for the local tidal response model, with no clear evidence that more groups are resolvable. Some of them highly violate Rayleigh criterion.