Analysis explanation

The following supporting information for the WOCE sea level tidal harmonic constants, confidence limits and power spectra has been provided by

Frederique Ponchant and Florent Lyard
groupe Nivmer
Laboratoire d'Etudes en Geophysique et Oceanographie Spatiales (LEGOS)
UMR 5566
Observatoire Midi-Pyrenees
18, Av. Edouard Belin
31 400 TOULOUSE Cedex

Harmonic constants

For each station, the full multi-year time series was analysed. From this analysis the harmonic tidal constants (amplitude and phase lag for each tidal constituent) were determined. Please note that the phase lag of the solar annual (SA) term refers to the astronomical argument of the astronomical annual variation, unlike in some analysis packages which use the 'radiational' astronomical argument. The files containing the tidal constants can be found under the 'Harmonic constants' heading.

Confidence limits

Following the computation of the Harmonic constants, the full time series were analysed year-by-year. These separate yearly analyses have been used to see how the constants fluctuate from year-to-year. From these yearly sub-analyses, a standard deviation 'sigma' has been computed, relative to the mean harmonic constant values from the full analysis. Also a 95% confidence interval range for the constants has also been computed. The results of this can be found in the files referenced under the 'Confidence limits' heading.

The columns in these files are as follows

1st column Station name
2nd column Wave name
For amplitude in cm
3rd column Lower bound of the confidence interval at 95%
4th column Mean value of the constant computed from the sub-analysis
5th column Upper bound of the confidence interval at 95%
6th column Relative error in percent ((5th column-4th column)/4th column)*100
The subsequent columns are the same for Phase Lag in degrees.

Power spectra

Following on from the computation of the confidence limits, the spectrum of the residuals of the tidal analysis was computed.

The aims of this are

1. To check that most of the tidal signal really has been removed. For some stations it was noticed that part of this signal still remains (i.e. presence of peaks at tidal frequencies). Nevertheless, these peaks are associated with very minor tidal waves of very low amplitude (not mentioned in Shureman tables).

2. To smooth the power spectra. For each frequency f, the values between f-(1/6months) and f+(1/6 months) values were averaged, and a smoothed power spectrum density (pink curve) obtained which gives an evaluation of the level of noise around the tidal frequencies.

From these averages a confidence interval at 95% (blue and green curves) can be determined, from which we have an estimate of the mean. By comparing the values given by the smoothed spectrum with the squared amplitudes we can decide if the harmonic constants can be taken with confidence or not. These smoothed spectra are available as PostScript files under the 'PostScript files of Power Spectra' heading.

For example, long period waves at Aburatsu station are completely absorbed into the noise. Therefore, the harmonics constants associated with these waves have to be considered with extreme care.

Energy transfer from tidal frequencies toward neighboring frequencies is responsible for the existence of tidal cusps which are clearly visible in the semi-diurnal band.

The noise level is about 0.1 mm**2 (quasi insignificant), reaching 1mm**2 around MSK2 frequency. Therefore, confidence in our estimation of harmonic constants for MSK2 is low.

Of course, M2 is completely above noise level (the noise level at this frequency is similar to the one taken for MSK2). Nevertheless, the interannual variability of the M2 amplitude is larger than 1 mm (i.e. +/- 3 mm, see nnnncfd1.arm files). This variability can be explained by the year to year evolution of the oceanic conditions (like the mean sea level, the polar ice cover) and/or by the occurrence of unusual meteorological events which can interact with the tides through non-linear processes.

Remarkably, the harmonic constants associated with the diurnal waves are pretty stable at Aburatsu station. The noise level is extremely low, implying that even the waves with low amplitude are estimated with a good confidence.

In conclusion, the smoothed power spectral density is a first estimator of the confidence we can have in harmonic constants. It indicates the level of potential distortion of the harmonic constants by the neighbouring oceanic signal. A complementary estimator is given by the variability of the individual annual sub-analyses which indicate the modifications of the tides by non tidal phenomena. In consequence, these two estimators should be used in conjunction to assess the best confidence intervals to be associated with the published constants derived from the complete multi-year data sets.