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  • View in gallery

    Bathymetry of the study area. The numbered blue triangles represent the 25 tide gauges. The black lines are tracks of combined TOPEX, Jason-1, and Jason-2 altimeter satellites.

  • View in gallery

    Amplitude and phase of the M2 tide constituent estimated by HA, GOT4.7, TPXO7.2, and FES2012.

  • View in gallery

    Tide classification by CC of (a) HA, (b) GOT4.7, (c) TPXO7.2, and (e) FES2012. The red and blue asterisks represent points at seawater depth below 50 and between 50 and 3000 m, respectively.

  • View in gallery

    RMSmisfit between HA and models [(a-1)–(d-1)] GOT4.7, [(a-2)–(d-2)] TPXO7.2, and [(a-3)–(d-3)] FES2012 for the mean semidiurnal tide constituents.

  • View in gallery

    RMSmisfit between HA and models [(a-1)–(d-1)] GOT4.7, [(a-2)–(d-2)] TPXO7.2, and [(a-3)–(d-3)] FES2012 for the mean diurnal tide constituents.

  • View in gallery

    RSSmisfit of function of depth: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

  • View in gallery

    Percentage of contribution by each tide constituent in the RSSmisfit presented in Fig. 6: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

  • View in gallery

    RSSmisfit of function of latitude depth: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

  • View in gallery

    Percentage of contribution by each tide constituent in the RSSmisfit presented in Fig. 8: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

  • View in gallery

    RSSmisfit between each model analyzed and each tide gauge presented in Table 1. The legend represents the number of each tide gauge presented in Fig. 1 and Table 1.

  • View in gallery

    RMSmisfit between each model analyzed and each tide gauge.

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Extraction of Tide Constituents by Harmonic Analysis Using Altimetry Satellite Data in the Brazilian Coast

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  • 1 Centro de Hidrografia da Marinha, Diretoria de Hidrografia e Navegação, Rio de Janeiro, Brazil
  • | 2 Laboratório de Meteorologia Aplicada, Departamento de Meteorologia, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
  • | 3 Centro de Hidrografia da Marinha, Diretoria de Hidrografia e Navegação, Rio de Janeiro, Brazil
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Abstract

This paper analyzes the sea surface height dataset from the TOPEX, Jason-1, and Jason-2 satellites of a 19-yr time series in order to extract the tide harmonic constituents for the region limited by latitude 5°N–35°S and longitude 55°–20°W. The harmonic analysis results implemented here were compared with the tidal constituents estimated by three classical tidal models [i.e., TOPEX/Poseidon Global Inverse Solution 7.2 (TPXO7.2), Global Ocean Tide 4.7 (GOT4.7), and Finite Element Solution 2102 (FES2102)] and also with those extracted from in situ measurements. The Courtier criterion was used to define the tide regimes and regionally they are classified as semidiurnal between the latitude range from approximately 5°N to 22°S, semidiurnal with diurnal inequality from 22° to about 29°S, and mixed southward of latitude 22°S. The comparison results among all tide approaches were done by analyzing the root-sum-square misfit (RSSmisfit) value. Generally, the RSSmisfit difference values are not higher than 12 cm among them in deep-water regions. On the other hand, in shallow water, all models have presented quite similar performance, and the RSSmisfit values have presented higher variance than the previous region, as expected. The major discrepancy results were particularly noted for two tide gauges located in the latitude range from 5°N to 2°S. The latter was investigated and conclusions have mainly pointed to the influence of the mouth of the Amazon River and the considerable distance between tide measurements and the satellite reference point, which make it quite hard to compare those results. In summary, the results have showed that all models presently generate quite reliable results for deep water; however, further study should done in order to improve them in shallow-water regions too.

Corresponding author address: Victor Bastos Daher, Centro de Hidrografia da Marinha, Diretoria de Hidrografia e Navegação, Rua Barão de Jaceguay, s/n Ponta da Armação Niterói, 24048-900E Rio de Janeiro, Brazil. E-mail: victor@lma.ufrj.br; rosa@lma.ufrj.br; gutemberg@lma.ufrj.br; joalvarenga@uol.com.br; gregorio@dhn.mar.mil.br

Abstract

This paper analyzes the sea surface height dataset from the TOPEX, Jason-1, and Jason-2 satellites of a 19-yr time series in order to extract the tide harmonic constituents for the region limited by latitude 5°N–35°S and longitude 55°–20°W. The harmonic analysis results implemented here were compared with the tidal constituents estimated by three classical tidal models [i.e., TOPEX/Poseidon Global Inverse Solution 7.2 (TPXO7.2), Global Ocean Tide 4.7 (GOT4.7), and Finite Element Solution 2102 (FES2102)] and also with those extracted from in situ measurements. The Courtier criterion was used to define the tide regimes and regionally they are classified as semidiurnal between the latitude range from approximately 5°N to 22°S, semidiurnal with diurnal inequality from 22° to about 29°S, and mixed southward of latitude 22°S. The comparison results among all tide approaches were done by analyzing the root-sum-square misfit (RSSmisfit) value. Generally, the RSSmisfit difference values are not higher than 12 cm among them in deep-water regions. On the other hand, in shallow water, all models have presented quite similar performance, and the RSSmisfit values have presented higher variance than the previous region, as expected. The major discrepancy results were particularly noted for two tide gauges located in the latitude range from 5°N to 2°S. The latter was investigated and conclusions have mainly pointed to the influence of the mouth of the Amazon River and the considerable distance between tide measurements and the satellite reference point, which make it quite hard to compare those results. In summary, the results have showed that all models presently generate quite reliable results for deep water; however, further study should done in order to improve them in shallow-water regions too.

Corresponding author address: Victor Bastos Daher, Centro de Hidrografia da Marinha, Diretoria de Hidrografia e Navegação, Rua Barão de Jaceguay, s/n Ponta da Armação Niterói, 24048-900E Rio de Janeiro, Brazil. E-mail: victor@lma.ufrj.br; rosa@lma.ufrj.br; gutemberg@lma.ufrj.br; joalvarenga@uol.com.br; gregorio@dhn.mar.mil.br

1. Introduction

The Brazilian south tropical Atlantic Ocean region presents a complex system of tides due to the existence of regions with river mouths and freshwater discharges such as the Amazon River and Patos Lagoon, respectively, in conjunction with regions with different extensions of the continental shelf and a complex bathymetry. Vellozo and Alves (2004) classified the tides on the Brazilian coast by applying the criterion applied by Courtier (1938) using data measured by tide gauges. And their results show three different types of tide along the coast: semidiurnal for the latitude range from 5°N to approximately 22°S, semidiurnal with diurnal inequality from 22° to about 29°S, and a predominantly mixed tide at latitude greater than 29°S. In Mesquita et al. (1986), Mesquita and Leite (1986), Morettin et al. (1993), Mesquita et al. (1996), and Vellozo and Alves (2004), we can find an overview of the Brazilian coastal tide regime.

In fact, the coastal tidal processes significantly vary along the lengthy Brazilian coast; therefore, forecasting precisely and with high resolution the coastal tide is critical for several marine applications, in particular those related to oil offshore activities. By assimilating altimeter data, the current tide hydrodynamic numerical models are able to accurately predict with high resolution the tide behavior in deep waters. However, tide modeling in coastal and shallow shelf regions is harder. To overcome the latter, a tide model basically requires accurate information about the local bathymetry, the water balance at the mouth of the rivers, and a refined contour of the coastline. It is expected that a tide model with precise information about the mentioned physical parameters may accurately estimate, for example, the changes of a wave when it approaches the coast. Presently, there are no tidal models that include, in their contour conditions, precise information about bathymetry and the coastline of several regions and, the results from current tide models are still not very consistent near the coastal regions (Saraceno et al. 2010). Several works have investigated the aforementioned problem and other related problems, for example, Schrama and Ray (1994), Smith and Sandwell (1997), Pascual et al. (2006, 2007), Volkov et al. (2007), and Ray (2008). Alternatively, the technique of harmonic analysis can be used to estimate the tidal constituents using the measurement of sea surface height (SSH). Presently, the altimeter data available can be used to realize this analysis. The data used here are collected every 9.9156 days, which certainly does not fulfill Fourier theory conditions, that is, does not follow the Nyquist theorem, to extract the main tidal constituents by Fourier analysis. On the other hand, the Rayleigh criterion—that is, |F0F1| > 360°/NΔt, where F0, F1, N, and Δt are the aliased frequency of a tidal constituent, the other tidal constituent frequency, the number of observations, and the temporal satellite resolution (9.9156 days), respectively. The factor on the right-hand side in the above-mentioned inequality is the Rayleigh number. The latter is equal approximately to 0.002162° h−1 and all constituents have met the mentioned criterion for the 19-yr time series (N > 699) and thus all tidal constituents can be properly separated in this work (Franco and Alvarenga 2009).

Presently, according to literature, there is still a lack of tide studies of the Brazilian coast. Therefore, the main objective of this study is to investigate the quality of the tide constituents estimated by harmonic analysis (HA) of tides using remotely sensed altimeter data available in deep and shallow water and the tide constituents estimated from the three classical global tide models—that is, the TOPEX/Poseidon Global Inverse Solution 7.2 (TPXO7.2), the Global Ocean Tide 4.7 (GOT4.7), and the Finite Element Solution 2012 (FES2012)—in the region limited by the geographic coordinates of 5°N–35°S, 55°–20°W. In particular, in shallow water, the estimated tidal constituents are also compared with tidal constituents estimated from tide measurements of 25 tide gauges distributed alongside of the Brazilian coast. The eight main diurnal and semidiurnal tidal constituents (Vellozo and Alves 2004) analyzed are M2, S2, N2, K1, O1, K2, P1, and Q1. Figure 1 depicts the study area, its bathymetry, the locations of the tide gauges analyzed, and the references points along the satellite tracks.

Fig. 1.
Fig. 1.

Bathymetry of the study area. The numbered blue triangles represent the 25 tide gauges. The black lines are tracks of combined TOPEX, Jason-1, and Jason-2 altimeter satellites.

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

2. Material and methods

a. Data

1) Satellite data

The SSH data from multimission altimeter measurements from the TOPEX, Jason-1, and Jason-2 altimeter satellites from September 1992 to August 2011 are used. In particular, the used remotely sensed SSH data are commonly denominated as along-track corrected sea surface height (CorSSH). These altimeter products were produced and distributed by the Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO, http://www.aviso.altimetry.fr/), as part of the SSALTO ground processing segment. The data are presently available online (http://aviso.altimetry.fr/index.php?id=1267) and referred to in stationary or long-term geophysical and ocean phenomenon studies (http://www.aviso.oceanobs.com/fileadmin/documents/data/tools/hdbk_dt_corssh_dt_sla.pdf). Its spatial and temporal resolutions are approximately 6.2 km along track and 9.9156 days, respectively. In Fig. 1, the black lines represent the satellite main track locations where the SSH data were collected using remote sensing.

2) In situ data

Tide measurements collected by 25 tide gauges (with a sampling frequency of one measurement per hour) from the Directorate of Hydrography and Navigation of the Brazilian Navy were used to validate the HA and model results. The tide gauges were chosen based on two conditions: (i) their proximity compared to satellite tracks (Table 1) and (ii) the temporal time series with 30 days or more of measurements (Table 1) [i.e., according to Franco and Alvarenga (2009), this is the minimum period of measurements to extract or to infer the eight tide constituents analyzed in this work]. In this study, the tidal harmonic constants were extracted by HA using the computational code called Sistema de Marés do Centro de Hidrografia da Marinha (SISMARE), where it is currently implemented. The numbered dark blue triangles in Fig. 1 show the locations of the 25 tide gauges.

Table 1.

Comparison between tide gauge (TG), tide models, and HA.

Table 1.

3) Tide model data

The third data source are tide constituents estimated by tide models: TPXO7.2 (Egbert et al. 1994; Egbert and Erofeeva 2002), GOT4.7 (Ray 1999), and FES2012 (Carrère et al. 2012). It is a fact that the tide models have continually undergone tremendous improvements due to the use of orbital remotely sensed altimeter data in the last two to three decades or more. At the moment, most tide models have similar performances—that is, the tide output model differences are between a few centimeters and a maximum of 1 m in deep-water and shallow shelf regions, respectively (Zahran et al. 2006; Saraceno et al. 2010). The spatial resolution of GOT4.7 is ½° × ½°, TPXO7.2 is ¼° × ¼°, and FES2012 is ° × °. It is important to mention that only the TPXO7.2 and FES2012 have assimilated altimeter data.

b. Method of HA using satellite data

To achieve the proposed objective using HA, it is necessary to include the tide signal in the remotely sensed SSH data, which is done by Eq. (1) as follows:
e1
where SSHTide, CorSSH and OceanTide are SSH with tide signal, SSH without tide signal, and tide height provided with the CorSSH data file, respectively.

1) Harmonic analysis

By assuming that the amplitude R of any wave can be divided into and , and that ai and bi are coefficients to be determined, the remotely sensed SSHTide ζ(tk) can be written as
e2
where wi is the frequency of an ith tide component at kth time instant. Subtracting R0 in Eq. (2), we will be able to rewrite ζ(tk), at kth instant time, as a function of ai and bi. Thus, the problem of calculating the phase and amplitude tide constituents is reduced to solving a system of linear equations. It is possible to write the equation system as
e3
or
e4
In particular, the coefficient array is frequently singular or quasi singular or ill-conditioned, and due to of such characteristics, we have decided to use singular value decomposition (SVD) to solve Eq. (4). This solution assumes that , an array of dimension m × 2n, can be decomposed into
e5
where is an orthogonal matrix with the same rank of , is an m × 2n dimension array with null elements positive or diagonal (singular values), and T is the transpose of an orthogonal matrix of dimension n × n. Then, x is
e6
where (Franco and Alvarenga 2009).
Finally, the system solution is
e7
Knowing x, which represents ai and b, the amplitude R can be obtained. For example, the amplitude of harmonic constants Hi and phase Gi can be calculated using
e8
e9
where fi and ui are nodal correction factors for amplitude and phase, respectively; V0i is the astronomical argument that reflects the position of the sun or moon at the moment of harmonic analysis.

2) Validation

The amplitude and phase of the tide constituents estimated by HA, as described above, were compared with four data sources—that is, TPXO7.2, GOT4.7, FES2012, and tide gauge harmonic analysis. The only question was whether TPXO7.2 or FES2012, which already includes remotely sensed altimeter data in its data assimilation process, could be considered an independent database for comparison with the HA results. Glorioso (2000) and Saraceno et al. (2010) have discussed and justified the use of HA for comparison purposes based on the investigation of errors in the tide propagation scheme, which are particularly acute in areas of shallow water or complex bathymetry. Overall, their conclusions have pointed out that the errors are mainly due to low model skills over shelf and/or in some shallow water. On the other hand, the GOT4.7 model does not take remotely sensed altimeter data as input.

(i) Classification of tide
According to Feistel et al. (2008) and Franco and Alvarenga (2009), the tidal regime may be determined by the four most significant tide constituents—that is, K1, O1, M2, and S2—for a particular region. In this work the tidal regime was determined by Courtier coefficient (CC)—suggested by Courtier (1938)—which is represented by the following ratio:
e10

The CC ratio ranks the tide as semidiurnal, semidiurnal with diurnal inequality, mixed, and diurnal when 0 < CC ≤ 0.25, 0.25 < CC ≤ 1.5, 1.5 < CC ≤ 3.0, and CC > 3, respectively.

(ii) Root-mean-square misfit
The comparison between the amplitude and phase of the tide constituents estimated by HA and the other three data sources was done in terms of root-mean-square misfit (RMSmisfit), defined as
e11
where N is the number of points used, and H0 and G0 are amplitude and phase obtained from satellite, respectively. Similarly, Hm and Gm are amplitude and phase from other tide approaches used for comparison in this study, respectively. It is important to emphasize that the comparison was done two different ways: (i) interpolating the along-track satellites FES2012 and TPXO7.2 data to the GOT4.7 regular grid (0.5° × 0.5°) and (ii) interpolating the three models’ data to the along-track satellite points. The two interpolations are done by calculating the weighted average within a radius of 60 km around the reference point, taking as weight the inverse square of the distance between data.

In this work also is used the root-sum-square misfit (RSSmisfit), differing from RMSmisfit only in that there is no division by N and combining the sum of all the analyzed tide constituents results.

3. Results

a. Harmonic analysis versus tide models

To verify the capability of the proposed HA technique to estimate tide constituents, we have compared the eight diurnal and semidiurnal mean tide constituents (named M2, S2, N2, O1, K1, K2, P1, and Q1) that are estimated using the HA and those estimated by the three tide models previously described.

Figure 2 presents the amplitudes and phases of the M2 tide constituent estimated by HA, TPXO7.2, GOT4.7, and FES2012, respectively. The spatial pattern of M2 (amplitude and phase) obtained by the four approaches are similar for the entire region apart from the region influenced by the mouth of the Amazon River. Considering the latter region—for example, at the geographic coordinate at 0° (equator) and 50.5°W—the amplitudes and phases obtained are 88.64 cm and 33.95° by HA, 133 cm and 84.45° by GOT4.7, 13.75 cm and 300.4° by TPXO7.2, and 48.09 cm and 187.4° by FES2012, respectively. From the present results, it may not possible to affirm which model could properly estimate the tide constituents in this particular region. Perhaps, it could be attributed to the interpolation of HA along-track results to a regular grid of 0.5° and also due to remotely sensed data accessible only up to 30 km away from the coast. Besides, the tide at the mouth region of the Amazon River is considered to be the most complex tide due to substantial freshwater discharges of the entire study area. Further investigation showed that using the RMSmisfit values between HA (along track) and the interpolated models results can clarify the mentioned problem.

Fig. 2.
Fig. 2.

Amplitude and phase of the M2 tide constituent estimated by HA, GOT4.7, TPXO7.2, and FES2012.

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

Figures 3a–d show the tide classification by Courtier coefficient for HA, GOT4.7, TPXO7.2, and FES2012, respectively. The red and blue dots represent tide at seawater depth below than 50 m and higher than or equal to 50 m, respectively. The tide classification was regionally classified into one or more tide regimes according to the following four latitude intervals: 1) 5°N–22°S: the tides are classified as semidiurnal independent of model and sea depth; 2) 22°–25°S: the CC for all models has classified most tides as semidiurnal tide and semidiurnal tide with diurnal inequalities; 3) 25°–27° and 29°–31°S: the tide is semidiurnal tide with diurnal inequalities; and 4) 27°–28° and 31°–35°S: the tide can be classified as semidiurnal tide with diurnal inequalities and mixed tide.

Fig. 3.
Fig. 3.

Tide classification by CC of (a) HA, (b) GOT4.7, (c) TPXO7.2, and (e) FES2012. The red and blue asterisks represent points at seawater depth below 50 and between 50 and 3000 m, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

The difference is that the mixed tide was also produced by HA and two tide models—that is, HA (10 points), TPXO7.2 (10 points), and FES2012 (5 points) into the latitude range from 27° to 28°S, preferentially in regions with seawater depth below than 50 m. In general, the diurnal tide was not observed in the study region except by GOT4.7 results for only one point, which was discharged in this study for analyzing. Even though the results have also showed that there is no diurnal tide in the entire study region, because of declination and diurnal effects, semidiurnal tides with diurnal inequalities and mixed tide are generated in a latitude range higher than 22°S.

Figures 4 and 5 illustrate the RMSmisfit values between HA and three models (i.e., GOT4.7, TPXO7.2 FES 2012) for the mean semidiurnal and diurnal tide constituents analyzed here, respectively. The RMSmisfit values are very similar, with a few centimeters difference among them for all regions. The maximum values were found for M2 (RMSmisfit > 45 cm), as expected, due to its biggest amplitude. In general, the RMSmisfit values increased as the seawater depth decreased for the majority of constituents. In particular, for the constituents with the largest amplitude, such as M2, S2, and N2, the RMSmisfit values are even higher. We may easily blame the well-known limitations of the current global tide models in shallow waters; however, it is also important to mention that the remotely sensed altimeter measurements carry on a certain amount of possible error mainly regarding ionospheric state, sea state bias correction, dry and wet troposphere correction, and a geophysical correction (e.g., inverted barometer correction). Usually, a range of default corrections is applied to the altimeter data, but any local variability of these corrections can clearly produce an error—for example, in coastal regions, where the environmental conditions vary more than in offshore regions. Recently, Cheng and Andersen (2013) studied the impacts of altimeter correction observing significant error in the retrieval of altimeter data using default correction, and thus they applied an alternative correction around Taiwan. Their results showed, in particular, that dynamic atmosphere, wet atmosphere, and sea bias corrections play a significant role when precisely estimating local altimeter data.

Fig. 4.
Fig. 4.

RMSmisfit between HA and models [(a-1)–(d-1)] GOT4.7, [(a-2)–(d-2)] TPXO7.2, and [(a-3)–(d-3)] FES2012 for the mean semidiurnal tide constituents.

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

Fig. 5.
Fig. 5.

RMSmisfit between HA and models [(a-1)–(d-1)] GOT4.7, [(a-2)–(d-2)] TPXO7.2, and [(a-3)–(d-3)] FES2012 for the mean diurnal tide constituents.

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

Figure 6 represents the RSSmisfit of each tide constituent between HA and the three models versus sea depth. The models’ results were interpolated for the along-track point of altimetry satellites where the HA results were obtained. The RSSmisfit are increasing with the proximity of the shallow water. The RSSmisfit values are quite similar for the water depth higher than 200 m and their highest values are found at depths below that limit (i.e., in this region the maximum RSSmisfit values are 86.5 cm for TPXO7.2 vs HA and 58.27 cm for FES2012 vs HA). The FES2012 is the model that agrees with the HA even in shallow water.

Fig. 6.
Fig. 6.

RSSmisfit of function of depth: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

Figure 7 represents the percentage contribution of each constituent in the RSSmisfit values. By considering all three results of RSSmisfit [i.e., TPXO7.2 and HA (green dots), GOT4.7 and HA (red dots) and FES2012 and HA (black dots)], the M2 constituent could contribute up to about 50% and 77% of RMSmisfit value in deep (depth > 200 m) and shallow (depth ≤ 200 m) waters, respectively. The results also showed that the S2, N2, K2, K1, P1, O1, and Q1 RMSmisfit contributions to the RSSmisfit value can be up to 49% (TPXO7.2 and HA), 30% (GOT4.7 and HA), 31% (FES2012 and HA), 24% (FES2012 and HA), 39.5% (GOT4.7 and HA), 42.5% (TPXO7.2 and HA), and 24% (TPXO7.2 and HA), respectively, as in Fig. 7.

Fig. 7.
Fig. 7.

Percentage of contribution by each tide constituent in the RSSmisfit presented in Fig. 6: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

Figure 8 represents the RSSmisfit between HA and the three tide models, organized by function of the latitude. By studying the figure, it is possible to divide the results into three latitude intervals: 1) 5°N–2°S, where the maximum RSSmisfit is equal to 86.5 cm for TPXO7.2; 2) 17°S–20°S, where the maximum RSSmisfit is 22 cm for the GOT4.7; and 3) 33°S or higher latitude, where the maximum RSSmisfit is 40 cm for the TPXO7.2. Only for completeness of the text, the RSSmisfit results are much lower than the other two RSSmisfit values of GOT4.7 and TPXO7.2. The RSSmisfit values of FES2012 versus HA are not higher than 58.3, 16.4, and 18.5 cm for the previously mentioned latitude intervals, respectively. Again, by analyzing the contribution of each constituent RMSmisfit to RSSmisfit value versus latitude interval (Fig. 9), it is observed that S2 and N2 contribute slightly to the RSSmisfit value due to their low variations into latitude ranges. On the other hand, M2 and K2 constituents, which present a similar latitudinal distribution pattern as obtained via CC, have significantly contributed to the RSSmisfit value at the region corresponding roughly to the latitude lower than 22°S. Similarly, K1, P1, O1, and Q1 contribute significantly to the RSSmisfit value approximately from the latitude higher than the limit of 20°S. The highest RSSmisfit values of GOT4.7 versus HA are observed between 17° and 20°S and, perhaps, it is mainly due to the S2 contribution, as represented by red asterisks in Fig. 9.

Fig. 8.
Fig. 8.

RSSmisfit of function of latitude depth: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

Fig. 9.
Fig. 9.

Percentage of contribution by each tide constituent in the RSSmisfit presented in Fig. 8: TPXO7.2 × HA (green), GOT4.7 × HA (red), and FES2012 × HA (black).

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

b. Harmonic analysis, TPXO7.2, GOT4.7, and FES2012 versus tide constituents based on in situ measurements

In this section, the RMSmisfit and RSSmisfit values among the results from HA, TPXO7.2, GOT4.7, FES2012, and the tide constituents regularly estimated by the SISMARE are compared and analyzed for 25 tide gauges distributed along the Brazilian coast as in Fig. 1.

Table 1 and Fig. 10 depict the RSSmisfit values of HA and the models against the SISMARE results. Initially, in order to analyze the proposed comparison, an acceptable limit value for RSSmisfit should be defined. Therefore, considering that the studied tide models are quite reliable in deep-water regions, their RSSmisfit results were analyzed and it was noted that the values range from zero to 12 cm, which could reflect as a consistent model comparison. Thus, it assumes an RSSmisfit value ≤ 12 cm as the criterion to classify the model results as good (or acceptable) in shallow-water region too. Overall, it notes that all the models’ results are quite similar except for tide gauges 1 and 5, which their RSSmisfit values for models diverge significantly among them (see Fig. 10). Perhaps the greatest discrepancy found in tide gauge 1 analysis is due to this tide gauge being located about 150 km from the mouth of the Amazon River, where the mean RSSmisfit and the standard deviation from HA and the three tide models versus tide gauge 1 are 26.85 and 13.11 cm, respectively. In addition, the RSSmisfit of HA and FES2012 versus tide gauge 1 are 7.07 and 12.50 cm below the mean, respectively. For GOT4.7 and TPXO7.2, the RSSmisfit are 2.11 and 17.47 cm above the mean. For tide gauge 5, the discrepancy is certainly related to its position being about 94 km from the nearest satellite track point. The mean and standard deviation of the RSSmisfit of HA and all the models versus the tide gauge 5 results are 27.38 and 5.03 cm, respectively (again, the FES2012 has the better results with an RSSmisfit of 6.90 cm below the mean). Thus, to corroborate the latter discussion, it is clear by analyzing the RSSmisfit values for tide gauges 6–8 that they are much smaller than that obtained for tide gauge 5, since their locations are much closer (i.e., about 40, 37, and 12 km, respectively) to the same satellite reference point (see Table 1).

Fig. 10.
Fig. 10.

RSSmisfit between each model analyzed and each tide gauge presented in Table 1. The legend represents the number of each tide gauge presented in Fig. 1 and Table 1.

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

Figure 11 presents the RMSmisfit of HA and the three tide models versus all tide gauges for each tide constituent. By ranking the model result comparisons, the FES2012 showed the best performance, since its estimation of tide constituents is much closer from the tide gauge results than the other three approaches. The GOT4.7 result is quite similar to the FES2012 result, apart from the RMSmisfit, which is about 0.4 cm higher. The HA results comparison is slightly poor when compared to the other mentioned models’ results for K2 and the diurnal tide constituents.

Fig. 11.
Fig. 11.

RMSmisfit between each model analyzed and each tide gauge.

Citation: Journal of Atmospheric and Oceanic Technology 32, 3; 10.1175/JTECH-D-14-00091.1

4. Conclusions

This paper studied the estimation of tide constituents obtained by HA of altimetry satellite data and that obtained from three global tide models (i.e., TPXO7.2, GOT4.7, and FES2012) in the region corresponding to 5°N–35°S, 55°–20°W. Comparisons among tide constituents estimated by HA, tide models, and that calculated using in situ data from 25 tide gauges distributed along the Brazilian coast were done. For all results, the RMSmisfit values were systematically higher with decreasing seawater depth when the results of all the tide models were compared. As expected, the maximum RMSmisfit constituents’ values—that is, for M2 (>45 cm), S2 (>14 cm), and N2 (>12 cm)—have occurred in the region of influence of the mouth of the Amazon River, where the tide regime is semidiurnal.

Based on the RSSmisfit between HA and the three tide models, the highest difference values were found in shallow-water regions and in three particular coastal regions limited by the latitude ranges of 5°N–2°S, 17°–20S°, and 33°–35°S.

For deep water (depth > 200 m), the RSSmisfit between HA and the other three models are not higher than 12 cm, which means that the HA and all the tide models analyzed can produce reliable tide results for the mentioned region. By considering each constituent and its contribution to the RSSmisfit value, the M2 RMSmisfit is the most important and its contribution can be up to about 50% and 77% in deep and shallow waters, respectively.

Perhaps it is possible to state that HA results could be considered as steadier than the other three models in shallow water due to it being only a pure mathematics estimation rather than a physical estimation like those produced by the classical tide models, which take into the account variables associated with bathymetry and so on. On the other hand, it is important to say that HA and FES2012 and TPXO7.2 have as input altimetry satellite data, which carry uncertainties (or errors) associated with several previously mentioned sources. Thus, the errors associated with the altimetry data corrections may generally cause no reliable results by HA in the coastal regions.

By comparing the results from the four previously mentioned approaches (i.e., HA, TPXO7.2, GOT4.7, and FES2102) against tide in situ measurements, they have showed similar performances except for two particular tide gauges. The reasons for that were mainly because of the influence of the mouth of the Amazon River and the considerable distance between tide measurements and the satellite reference point, which make it harder to compare those results.

Finally, in order to overcome the present tide models’ difficulties in shallow-water regions, developments are required for ingesting precise information about local bathymetry and the contour of the coastline. In addition, for those tide models and HA that assimilate remotely sensed altimetry data, further investigation should be done to improve locally altimeter corrections.

Acknowledgments

The authors acknowledge the Oceanographic Modeling and Observation Network (a Brazilian initiative on operational oceanography) and Petróleo Brasileiro S.A. PETROBRAS for providing research funds and Agência Nacional do Petróleo, Gás Natural e Biocombustíveis (ANP) for their approval. The Brazilian Navy and Federal University of Rio de Janeiro also provided useful resources, such as the labor of specialized officers, a spacious building for researchers, and additional computational power.

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