Investigation of bias in traditional grouping by means of regularization

<p>Harmonic tidal analysis bases on the presumption that since short records and close frequencies result in an ill-conditioned matrix equation, a record of length T is required to distinguish harmonics with a frequency separation of 1/T (Rayleigh criterion). To achieve stability of the solution, tidal harmonics are grouped. Nevertheless, if any additional information from different harmonics within the assumed groups is present in the data, it cannot be resolved. While the most information in each group is carried by the harmonic with the largest amplitude, time series from other harmonics is properly taken into account in estimated amplitudes and phases. However, if the signal from the next largest harmonic in a group is significantly different from the expectation, the grouping parametrization might lead to an inaccurate estimate of tidal parameters. That might be an issue since harmonics in a group do not have the same admittance factor, or if the assumed relationship between harmonics degree 2 and 3 is false.</p><p>The bias caused by grouping tidal harmonics can be investigated with methods used for stabilizing inverse problem solutions. In our study, we abandon the concept of groups. The resulting ill-posedness of the problem is reduced by constraining the model parameters (1) to reference values and (2) to the condition that admittance shall be a smooth function of frequency. The mentioned regularization terms are present in the least-squares objective function, and the trade-off parameter between the model misfit and data residuals is chosen by the L-curve criterion. We demonstrate how this method may be used to reveal system properties hidden by wave grouping in tidal analysis. We also suggest that forcing time series amplitude may be more relevant grouping criterion than solely frequency closeness of harmonics.</p>

Versatile Harmonic Tidal Analysis: Improvements and Applications

New computer software that permits more versatility in the harmonic analysis of tidal time series is described and tested. Specific improvements to traditional methods include the analysis of randomly sampled and/or multiyear data; more accurate nodal correction, inference, and astronomical argument adjustments through direct incorporation in the least squares matrix; multiconstituent inferences from a single reference constituent; correlation matrices and error estimates that facilitate decisions on the selection of constituents for the analysis; and a single program that analyzes one- or two-dimensional time series. This new methodology is evaluated through comparisons with results from old techniques and then applied to two problems that could not have been accurately solved with older software. They are (i) the analysis of ocean station temperature time series spanning 25 yr, and (ii) the analysis of satellite altimetry from a ground track whose proximity to land has led to significant data dropout. This new software is free as part of the Institute of Ocean Sciences (IOS) Tidal Package and can be downloaded, along with sample input data and an explanatory readme file.