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«Harmonic tidal analysis methods on time and frequency domains: similarities and differences for the Gulf of Trieste, Italy, and Paranaguá Bay, ...»

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Harmonic tidal analysis methods on time and frequency domains: similarities and differences for

the Gulf of Trieste, Italy, and Paranaguá Bay, Brazil.

1,3 2 1

Eduardo Marone, Fabio Raicich & Renzo Mosetti

OGS, v. Auguste Piccard, 54, I-34151, S. Croce,Trieste, Italy;

CNR-ISMAR, Viale Romolo Gessi 2, I-34123, Trieste, Italy;

Contact: edmarone@ufpr.br


Two years of sea level data obtained at Trieste, Italy, and Paranaguá, Brazil, were used to compare the performances of two tidal analysis methodologies, one in the time domain (HMB) and other in the frequency domain (HMF). For each station the first year was analyzed to estimate the tidal constituents while the second year was used to compare observations against forecasted sea levels. Both methodsshowed equivalent performances but HMF is more user-friendly and offered better and more comprehensive results. The main reason seems to be linked to the amount of tidal constituents that HMF can estimate (more that 170) while HMB estimates around 110. Also, HMF showed better results for shallow water and long term components. However, the residuals showed that a significant amount of oscillating energy is left behind by both methods, suggesting that other deterministic signals not presentin the astronomic tidal frequencies have to be considered. We found that the semi-diurnal atmospherictide S2p is disturbing the sea constituent S2, as well as other frequencies, probably, in the diurnal specie, in the subtropical case of Paranaguá. These results are in agreement with the theoretical development of Chapman and Lindzen (1970) and the numerical simulations due to Arbic (2005). [Si mettono riferimenti anche nell‟abstract?] It is concluded that a better stochastic model for tidal analysis and forecast needs to be formulated in order to better represent the physics of sea level: while tidal forecast with the usual methods seems to work well in many practical cases, the high dependence of numerical models on initial and contour conditions suggests that sea level harmonic constituents estimation has to be improved.

Keywords: Tides, Analysis, Prediction, Harmonic Analysis, Trieste, Paranaguá CEM/UFPR (permanent) – P.O.Box 50002 – 83255-000 – Pontal do Paraná – PR – Brazil.


The astronomical tides has been studied in the last two centuries in different ways but the background of all the analysis and forecasting methodologies is based in the early development by Darwin (1898; 1907), th improved by Laplace, Lord Kelvin and other scientists, mostly in the 19 Century. This development is based on the principle that the sea level heights in a given place can be represented by a sum of N harmonic terms (from i= 0 to N-1), each one having an unique pair of amplitude (H i) and phase (Gi), oscillating at a particular frequency (i) defined by the astronomical tide generating potential and, when present, also by non-linear combination of the astronomical constituents (shallow water tidal components).

From the revolutionary work of Newton, the tides offered a great scientific challenge, but at the beginning th of the 20 Century, the great utility of a good acknowledge of the astronomical tides in a given location moved the interest to the applicability of the analysis and prediction techniques, which evolved from the use of complicated analysis tables to determine a restricted number of H,G pairs, passing through the analogue tide prediction machines, to the present digital computer based methodologies with almost no limitations, but the physics, for their determination. However, most if not all the past and present tidal analysis and forecasting methodologies kept the basic Darwin principle unchanged.

Nowadays, a giant number of different digital computer programs are widely used to analyze and predict the astronomical tides, using different algorithms and approaches. In a way, all analyze sea level data (evenly or unevenly time spaced), in the time or frequency domains, determining a set of H and G pairs for the given place, allowing the astronomical tide to be forecasted. All the actual methodologies work properly for prediction purposes but not necessarily when the H, G values are used for other purposes.

However, when we compare the sets of H and G obtained for a same place using the same sea level data but different analysis methodologies (Marone, 1991), we note that the tidal constituents differ in both their parameter values, but mainly in G, notwithstanding their use for forecasting purposes do not show substantial differences, which has not been yet well explained.

The extensive uses of numerical modelling in ocean sciences bring into play the tide as one of the most important dynamical forcing and, in most cases, there is still a need for improvement regarding the tidal constituents used to feed the models, which seems to work better if observed sea level data are used instead (Camargo & Harari, 2003; Harari et al., 2006; Hendershott, 1977; Lyard et al., 2006; Moreira et al., 2010), particularly, coastal and shallow waters numerical model results are still not fully satisfactory, particularly cause the reduced number of used tidal constituents, which covers less that 95% of the tidal energy (Padman et al., 2008).

If the tidal constants are not so constant, not only by physical reasons but also according to the used method of analysis, the efficiency of numerical models is compromised, as well as the interpretation of many ocean phenomena. It has been proved that the ocean micro-structure has high correlation with the turbulent dissipation and the tidal cycles (Polzin 1997; Ledwell et al. 2000). In coastal areas, tides contribute significantly with the vertical mixing, redistributing nutrients and oxygen, which impact marine life (Romero et al., 2006) and, also, with the flow of CO2 between sea and air (Bianchi et al., 2005).

The basic equations of tidal dynamics (Godin, 1972; Hendershott, 1972, 1977; Pugh, 1987; Munk and Wunsch, 1998; Pugh, 2004; Franco, 2009) are relatively simple and they are well known from Laplace times. However, in numerical modelling for instance, the need of higher precision of tidal constituents, particularly in coastal regions, is being indicated as a relevant scientific challenge (Lefevre et al., 2000;

Simionato et al., 2004; Lyard et al., 2006; Moreira et al., 2010).

In recent years, with the great evolution of satellite altimetry, the combination of satellite data assimilation into numerical modelling (Matsumoto et al., 1995) requires, for tidal fields to be well represented in a limited number of location inside the model domain, that constituents in such places have to be well and accurately known (Egbert & Erofeeva, 2002; Kantha, 1995).

It is then clear that the need for a better understanding of the principles and quality of the tidal constituents estimated by different methodologies have to be achieved. Also, when examining many of the actual tidal analysis methodologies, one faces other small but not less important problem: the names given to the tidal constituents are not yet fully standardized, and diverse methodologies identify the same i with different names, making the use of them, the comparisons, and the physical interpretation, more difficult. Also, the names of the different methodologies are so many and were created using such diverse approaches and styles that they help also to create confusion.

In this work we apply two methodologies, both based on the Harmonic Method, one solving the equations in the time domain (HMB) and the other in the frequency domain (HMF), to two sets of one year of 30minute sampled sea level data obtained in the Gulf of Trieste, Italy (Figure 1, top), and the Paranaguá Bay, Brazil (Figure 1, bottom), the results of the analysis were compared and used for prediction purposes. Having a subsequent year of sea level data for both places (Figure 2), the forecasted sea levels were compared with the observed data and the residuals were studied to clarify the causes of the differences, both physical and methodological, in the search for answers regarding the methodologies quality and, mainly, about the reasons of such differences. Such objective does not intend solely to solve some just formal issues, like the standardization of names or, worst, what methodology is the most accurate.

METHODS As already mentioned, we used here two analysis and prediction methodologies based in the Harmonic Method (HM) for our purposes, one solving the equations in the time domain (HMB) and other in the

–  –  –

and Foreman (1977) and it is actually a much evolved computer program (Bell et al, 1999), whose fundamentals will be explained below. Also, the HMF is an evolution of the work developed by Doodson (1921) due to Franco & Rock (1971) and its further updates (Franco, 1997, 2009), and it is shortly explained at the following section.

Harmonic analysis in the time domain - HMB The basic assumption for the application of the HMB method is that the tidal variations can be

represented by a finite number N of harmonic terms of the form:

–  –  –

conventionally expressed relative to the Greenwich Meridian. Usually, angular speeds are expressed in degrees per solar hour and phase lags in degrees. Each angular speed n = 2n / 360 (in radians per solar hour) is determined as a linear combination of the angular speeds 1,…,6 of the tidal components related to solar and lunar motions, namely mean lunar day (1), sidereal month (2), tropical year (3), Moon‟s perigee (4), regression of Moon‟s nodes (5) and perihelion (6). 0 is the angular speed related to the mean solar day. The linear combination coefficients are small integers. The most complete work on the subject as well as the determination of the coefficients mentioned above was performed by Doodson (1921). Details on the mathematical procedures can be found, e.g., in Pugh (1987).

In practice, in the harmonic analysis in the time domain we fit a tidal function:

–  –  –

where the unknown parameter are Z0, Hn and Gn (n = 1,…,N), to a time series of observed values O(t).

The terms fn are the nodal factors and un the nodal angles, while the terms Vn are the equilibrium phase angles for the constituents (for solar constituents f n = 1 and un = 0). Again, the convention is adopted to take Vn as for the Greenwich Meridian (Pugh, 1987). The fit is performed using the least-square

–  –  –

times of the observations.

Harmonic analysis in the frequency domain - HMF Analysis The harmonic method used here was originally developed by Franco and Rock (1971) and was permanently updated from their original routines in FORTRAN to modern languages capable to work in personal computers (Franco, 1997; 2009). It is the methodology used by the Brazilian Navy, which is the National tide authority, to analyse and forecast the astronomical tides all along the more than 8000 km of Brazilian coastline.

The HMF is based in the assumption that the observed sea level in a given place can be accurately represented by a stochastic model with a harmonic part (mostly the astronomical tide, as in Eq. 1) and a non deterministic residual (noise).

Departing from a deep acknowledge of the astronomical tidal potential, it uses the harmonic oscillation of

–  –  –

place. The data are then analysed in the frequency domain via, for instance, the Fast Fourier Transform or other algorithm, as the Watts method or the Direct Fourier Transform (Marone, 1991). Once the energy spectrum is obtained, the tricky approach of the HMF is the use, to separate very close but well known astronomic tidal constituents, using the angular frequency differences among neighbour tidal constituents instead of the Rayleigh principle (Pugh, 1987).

In a classical spectral analysis, a set of 2N harmonic equations is solved (in the time or frequency domains) in order to determine N pairs of H and G unknowns. In the HMF, the angular frequencies differences are used also in the array of equations to be solved, generating a redundant (more equations than unknowns) group which enhance the precision and increase the number of frequencies that can be solved for. This is possible because particular spectral peaks of sea level data are not “contaminated” by energy of the other tidal species (diurnal, semi-diurnal, etc.), allowing the method to treat each tidal species in separate sets with a redundant number of equations with respect to the unknowns. The split into sub-systems allows, for instance for the diurnal band (290n380 cycles per 8192 hours), to have 90 equations to solve just 20 tidal constituents, which can be easily solved using the least square method.

The HMF offers another advantage with respect to shallow water non-linear constituents, which are straightforwardly represented as combination of two or more astronomic tidal components in bi-linear or tri-linear arrays (Marone, 1991). Also, the method does not require sea level data series corresponding to exact multiples of half lunation (Franco and Rock, 1972). Finally, the HMF use a simple approach to estimate the quality of the results, calculating the signal-to-noise rate for each tidal species based in the stochasticity of the model and on the energy present in frequencies which do not correspond to astronomical tidal forcing (residual spectrum), and calculated from the variances of the residual energy present on each tidal band (species).


–  –  –

As much as N terms we have, by knowing the pairs H and G obtained with the HMB or HMF (or any other), the better will be the fit. The tidal prediction is then obtained solving the sea level equation (the harmonic development of Eq. 1) of the model for the period T of interest, reconstructing the astronomical

–  –  –

DATA SETS In order to help the readers with the geographical onset, Figure 3 depicts a couple of maps corresponding to the Gulf of Trieste, Italy, and the Paranaguá Bay, Brazil.

Gulf of Trieste The Gulf of Trieste lies in the northernmost part of the Adriatic Sea at about 45°40‟N latitude and 13°40‟E longitude. It is approximately a 20×20 km square, connected with the Adriatic on the SW side and surrounded by land on the three other sides. It is a shallow bay, with maximum depth of about 25 m.

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