Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima

Journal of Oceanography Vol. 52, pp. 259 to 273. 1996 Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima NAOTO EBUCHI 1 and KIMIO HANAWA 2 1 Center for Atmospheric and Oceanic Studies, Faculty of Science, Tohoku University, Aoba, Sendai 980-77, Japan 2 Department of Geophysics, Graduate School of Science, Tohoku University, Aoba, Sendai 980-77, Japan (Received 29 May 1995; in revised form 1 November 1995; accepted 2 November 1995) The sea surface heights (SSHs) observed by the TOPEX altimeter are compared with tide gauge data at Chichijima in Ogasawara (Bonin) Islands and hydrographic data taken around the islands, in order to quantitatively verify the altimeter observations and oceanic tide corrections by three tide models proposed by Cartwright and Ray (1991), Ray et al. (1994), and Ma et al. (1994). First, performance of the new tide models is assessed by comparing tidal variations consisting of diurnal and semi-diurnal constituents with the tide gauge data at Chichijima. The tide model proposed by Ray et al. gives the smallest root-mean-squared (rms) difference of 2.61 cm. Errors in amplitude and phase in each tide model are evaluated by spectral analysis. The TOPEX SSHs corrected by the tide models are compared with sea level data at Chichijima. A long-term variation of a period of about 1 year is found in the residual between the SSHs and the Chichijima sea levels. This variation is also found in the difference between the dynamic height anomalies calculated from hydrographic data around the island and the Chichijima sea levels. By subtracting the long-term variation, the rms difference between the TOPEX SSHs and the Chichijima sea levels is reduced to about 4 cm and the slope of the regression line is improved to unity. The residual shows variations related to aliasing caused by incompleteness of the ocean tide correction with the repeat cycle of the altimeter observation. 1. Introduction TOPEX/POSEIDON is a joint U.S./France space mission to study the world ocean circulation. The satellite was launched in August 1992. The mission carries two altimeters, the National Aeronautics and Space Administration (NASA) dual-frequency TOPEX altimeter, and the advanced, experimental solid-state POSEIDON altimeter, designed by Centre National d Etudes Spatiales (CNES). The satellite is taking exact repeat orbits of 9.91 days period. The ground track spacing is 315 km at the equator and about 200 km at mid latitude. A scientific data set, called Geophysical Data Records (GDRs) has been distributed by the NASA Physical Oceanography Distributed Active Archive Center (PO-DAAC) at the Jet Propulsion Laboratory (JPL) of the California Institute of Technology. In order to obtain variations of sea surface topography related to geostrophic currents from altimeter-derived sea surface heights (SSHs), ocean tide should be subtracted from the observed SSHs. Two ocean tide models have been supplied in the TOPEX GDR: the Schwiderski model (Schwiderski, 1980a, b) and the Cartwright and Ray model (Cartwright and Ray, 1991). However, several verification studies (e.g., Molines et al., 1994; Schrama and Ray, 1994; Ma et

260 N. Ebuchi and K. Hanawa Fig. 1. Locations of Chichijima (C), the ground track, Pass 075 (P075), of the TOPEX altimeter ( and ) and stations of hydrographic observations by the R/V Koyo ( ). Grids of the TOPEX altimeter data used in the spatial interpolation to Chichijima are indicated by solid circles (see the text). al., 1994; Mitchum, 1994; Wagner et al., 1994; Ebuchi and Hanawa, 1995) have pointed out that the two models are not adequate in accuracy and that aliased waves caused by the errors in tidal constituents are not negligible. Based on the TOPEX/POSEIDON data, several improved models have recently been developed for applications (Ray et al., 1994; Ma et al., 1994; Schrama and Ray, 1994; Wagner et al., 1994). In the present study, performance of three new oceanic tide models such as the Cartwright and Ray (1991) tide model (hereafter CR), the Ray et al. (1994) model (RSC), and Ma et al. (1994) model (MSET), will be assessed by comparison with tide gauge data at Chichijima in Ogasawara (Bonin) Islands. The TOPEX SSHs corrected by using these models are also compared with sea level data at Chichijima. Effects of errors in tidal constituents of the models will be discussed. The tide gauge station at Chichijima is suitable for comparison with altimetry data since the island is located about 1000 km south of Honshu, the main island of Japan, and area of shallow sea around the island is not large (Fig. 1). The Chichijima Weather Station, where routine

Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima 261 meteorological observations have been made, is very close to the tide gauge station. Around the island, hydrographic observations have also been conducted monthly by the Ogasawara Fisheries Center using the R/V Koyo. Ebuchi and Hanawa (1995) compared variations of TOPEX SSH with sea level data at Chichijima, and found that aliasing caused by the M 2 tidal constituent error in the Cartwright and Ray (1991) tide model is significant. Sea level data observed by tide gauges at off-shore islands have often been used in quantitative verification of satellite altimeter data (e.g., Tai et al., 1989; Cheney and Miller, 1990; Mitchum, 1994). In comparisons of altimetry data with sea level data at islands, however, local bottom topography around tide gauge stations is expected to affect the observed sea level. In the present study, this effect will also be discussed by comparing the TOPEX SSHs, tide gauge data and hydrographic data taken around the island. 2. Data 2.1 TOPEX altimeter data We have analyzed the TOPEX GDRs for a period of the first 58 cycles, from 25 September 1992 to 21 April 1994. The repeat cycles 20, 31, 41 and 55, when the POSEIDON altimeter was in operation and no data were available for the TOPEX altimeter, are excluded from the present analysis. The data processing used in the present study is basically the so-called collinear analysis (Cheney et al., 1983). The satellite ground track, Pass 075, located closest to Chichijima is picked up. Figure 1 shows locations of the ground track and the island (27 05 N, 142 11 E), together with hydrographic stations around the island. The sampling interval of the data in the TOPEX GDRs is about one second, corresponding to 6.2 km interval along the satellite ground track. Flagged data indicating some errors in the measurements are discarded. Electromagnetic bias correction, ionospheric correction, dry and wet tropospheric corrections, inverse barometer correction and solid earth tide corrections are made to the observed SSHs according to the GDR Users Handbook (Callahan, 1993). Satellite radial orbit error is not corrected because the error is not considered to be significant (Fu et al., 1994). The three tide models, CR, RSC and MSET are examined for oceanic tide correction. The corrected SSHs are gridded by averaging in bins of 0.125 deg (about 15 km) in latitude along the ground track. The locations of grids are shown in Fig. 1. Mean SSH over the entire period of 58 repeat cycles and deviations from the mean for each cycle are calculated in each grid. In order to collocate the TOPEX SSHs with the tide gauge station at Chichijima, a Gaussian filter with an e-folding scale of 100 km, which is chosen by considering the distance between Chichijima and the TOPEX ground track, is adopted to the TOPEX SSH deviations. The SSH deviations at TOPEX grids within a distance of 100 km from Chichijima are averaged with a weighting function, Wr ( ) = exp( r 2 / R 2 ), R = 100 km, ( 1) where r is the distance from Chichijima. The TOPEX data grids used for the spatial interpolation are indicated by solid circles in Fig. 1. 2.2 Sea level data at Chichijima The tide gauge station at Chichijima in Ogasawara Islands has been operated by the Japan

262 N. Ebuchi and K. Hanawa Meteorological Agency since 1975. The hourly observed sea level data for a period from 1 September 1992 to 30 April 1994 are processed as follows. Tidal variation consisting of diurnal and semi-diurnal constituents is separated by using a tide-killer filter with cut-off period of 48 hours designed by Hanawa and Mitsudera (1985), and will be compared with hourly tides calculated by the tide models. Daily-averaged sea level is calculated from the residual. Inverse barometer correction is made by using daily-averaged sea level pressure observed at the Chichijima Weather Station. Anomalies from the mean over the whole period are calculated to be compared with the TOPEX SSHs. Running mean for a period of 11 days is operated to remove variations of time scale shorter than the TOPEX repeat cycle (9.91 days). The processed sea level data will be compared with the TOPEX SSHs corrected by the three tide models and hydrographic observation data around the island. 2.3 Hydrographic data observed by the R/V Koyo The Ogasawara Fisheries Center has monthly conducted hydrographic observations using the R/V Koyo at 18 stations around Chichijima since 1973. Locations of the stations are shown in Fig. 1. Since 1988, STD casts from the surface to a depth of 500 m have been made at these stations. Taneda (1994) made some corrections for the salinity data and calculated the surface dynamic height anomalies ( D) with a reference level of 500 db, where is the maximum depth of the casts as mentioned above. Please refer to the original article on details. Averages of D over the 18 stations for a period from September 1992 to April 1994 are used to represent sea surface topography around the island. Anomalies from the mean over the whole period are calculated to be compared with the TOPEX SSHs. 3. Result and Discussion 3.1 Comparison of tidal variations Figure 2 shows the comparison of hourly tidal variations of sea level data at Chichijima with those calculated by the three tide models, (a) CR, (b) RSC and (c) MSET, for a period of the first 324 days. The slopes of the regression line (not shown in Fig. 2) and root-mean-squared (rms) differences of the data scatter are summarized in Table 1. At Chichijima, these models show relatively good performance with rms differences less than 4 cm. The RSC model seems slightly better than the other models. However, the slope does not coincide with unity, indicating the existence of an error in amplitude. Also, the data points seem to distribute elliptically in each panel. This means that the model tide has phase difference with observed tide. In order to investigate errors in amplitude and phase of tidal constituents of the models, Table 1. Comparison of tidal variations in the sea level data at Chichijima with those calculated by the three tide models. (Number of data = 7776.) Tide model RMS difference (cm) Slope Correlation coefficient CR 3.35 1.02 0.993 RSC 2.61 0.94 0.997 MSET 3.81 0.92 0.992

Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima 263 Fig. 2. Comparison of hourly tidal variations consisting of diurnal and semi-diurnal constituents of the sea level data at Chichijima with those calculated by the three tide models; (a) CR, (b) RSC and (c) MSET.

264 N. Ebuchi and K. Hanawa Table 2. Amplitude ratio and phase difference between Chichijima tide and the three tide models for the eight main tidal constituents. Amplitude of the constituents are calculated by harmonic analysis of the sea level data at Chichijima. Period Amp. Aliasing period CR RSC MSET (h) (cm) (day) Amp. ratio Phase (deg) Amp. ratio Phase (deg) Amp. ratio Phase (deg) M 2 12.42 28.3 62.11 0.989 +1.50 0.932 +2.53 0.897 +5.59 S 2 12.00 12.9 58.74 1.015 +0.81 0.965 1.58 0.926 +6.54 N 2 12.66 5.2 49.53 1.146 +7.24 0.839 +1.58 0.933 +3.58 K 2 11.97 3.5 86.60 1.015 +0.81 0.965 1.58 0.926 +6.54 K 1 23.93 16.3 173.19 1.049 +3.15 0.940 +0.28 0.944 0.91 O 1 25.82 12.0 45.71 1.097 12.98 0.969 0.81 0.982 4.98 P 1 24.07 5.3 88.89 1.047 +1.81 0.938 0.79 0.942 1.94 Q 1 26.87 2.3 69.36 1.059 11.53 0.940 +1.63 0.985 9.07 power spectra of the sea level data at Chichijima and the model tides, and cross spectra between them are calculated. For the eight main constituents, phase differences are estimated from phase of the cross spectra, and amplitude ratios between the observed and model constituents are estimated from square root of ratio between the power spectral densities at the periods of constituents. The results are summarized in Table 2, together with the amplitudes of each constituents at Chichijima determined by harmonic analysis of the sea level data at Chichijima. Aliasing periods with the TOPEX repeat cycle are estimated by the following equation (Schlax and Chelton, 1994): T a = t / ( f t [ f t + 0.5] ), ( 2) where T a is the aliasing period, f is the frequency of the tidal constituent, t is the time interval of the TOPEX observation (=9.91 days), and [x] represents the greatest integer less than x. From the results in Table 2, amplitude of aliased waves can be estimated by a simple calculation of sinusoidal waves. Let r and φ be the amplitude ratio and phase difference between true and model tides, respectively. The residual between the model and true tide, δ(t), is expressed as, where δ( t) = ar sin( 2πft + φ) asin 2πft = a ( rcos φ 1) 2 + r 2 sin 2 φ { } 1/2 sin 2πft + α tan α = r sin φ / ( r cos φ 1), ( ), 3 ( )

Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima 265 Fig. 3. Estimated amplitudes of the aliased waves for the eight main constituents of the three tide models; (a) CR, (b) RSC and (c) MSET. and a is the true amplitude of the constituent. The amplitude of aliased waves estimated from the results in Table 2 using Eq. (3) for each constituent is shown in Fig. 3. In the CR model, errors in the O 1 constituent is significant rather than M 2. In the RSC and MSET model, the M 2 constituent is dominant in the aliasing. 3.2 Comparison of sea surface height variations (a) Influence of horizontal excursion of the TOPEX around track Before comparison of the TOPEX SSHs with Chichijima sea level data, influence of horizontal excursion of the TOPEX ground track is examined by using a geoid model composed by Fukuda (1995). Spatial resolution of the Fukuda geoid model is 5 in latitude and longitude. The upper panel of Fig. 4 shows the horizontal excursion of the ground track, Pass 075, in longitude at the latitude of Chichijima, 27 08 N. The location of the ground track varies about ±0.01 deg in longitude. Variation of geoid height caused by the excursion of the ground track is estimated by using the Fukuda geoid model, and plotted in the lower panel of Fig. 4. The estimated variation has an amplitude of 10 15 cm. It can be concluded that the influence of the horizontal excursion on the observed SSH is not negligible. (b) Intercomparison of the three tide models Correction for the horizontal excursion of the ground track is made for the raw one-second SSH data in the GDR by using three geoid models; Geoid (hereafter TP_GEO) and Mean Sea

266 N. Ebuchi and K. Hanawa Fig. 4. (a) Horizontal excursion of the ground track, Pass 075, in longitude at the latitude of Chichijima, 27 08 N and (b) variation of geoid heights caused by the excursion which is estimated by using the Fukuda (1995) geoid model. Table 3. Comparison of the rms differences and correlation coefficients between the TOPEX SSHs corrected by the three tide models and three geoid models and the Chichijima sea levels. (Number of data = 53.) Tide model Geoid model Not corrected TP_GEO TP_MSS FUKUDA RMS (cm) Corr. RMS (cm) Corr. RMS (cm) Corr. RMS (cm) Corr. CR 8.18 0.794 5.91 0.879 5.63 0.889 5.69 0.887 RSC 7.59 0.817 5.35 0.898 5.03 0.910 5.07 0.909 MSET 6.83 0.842 4.75 0.920 4.48 0.930 4.52 0.930

Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima 267 Fig. 5. Comparison of the TOPEX SSHs corrected by the RSC tide model and TP_MSS geoid model ( ) with the Chichijima sea level data ( ). The residual (TOPEX-Chichijima) is shown in the lower panel. Surface (TP_MSS) supplied in the GDR and the Fukuda model (FUKUDA). The corrected SSHs are processed as described in Section 2 and are compared with the sea level data at Chichijima. Figure 5 shows a result of comparisons for the SSHs processed using the RSC tide model and the TP_MSS geoid model. The TOPEX SSHs agree well with the Chichijima sea level with a rms difference of 5.03 cm. However, the residual shows oscillation with a period about 40 60 days, which is considered to be due to the aliasing caused by errors in the ocean tide correction with the sampling of 9.91-day repeat cycle. Radial orbit error and errors in corrections other than ocean tide are also included in the residual. In Table 3, rms differences are summarized for all cases of the tide and geoid models. The importance of the correction for horizontal excursion of the ground track using a geoid model is confirmed again. The correction reduces the rms difference by about 2 cm for the all geoid

268 N. Ebuchi and K. Hanawa Fig. 6. Comparison of the TOPEX SSHs corrected by the RSC tide model and TP_MSS geoid model ( ) with the Chichijima sea level data ( ) and the dynamic height anomalies D referred to 500 db which are calculated from hydrographic data observed by the R/V Koyo ( ). The residuals (TOPEX- Chichijima) and (Koyo-Chichijima) are shown in the lower panel by symbols of ( ) and ( ), respectively. Dashed line represents a long-term variation of the residual between the Koyo D and Chichijima sea level obtained by a 90-days running mean. models. Differences among the three geoid models are not significant. The MSET tide model shows the best agreement with Chichijima sea level, though it showed the largest rms difference in the comparison of tidal variations as listed in Table 1. The MSET tide model was developed using TOPEX altimeter data. Therefore, it might be considered that some portion of the radial orbit error and errors in the other corrections may also be involved in the tide model. This may result in the fact that the model tide shows worse agreement with tide gauge data but the corrected

Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima 269 Fig. 7. Residual of the TOPEX SSHs corrected for the long-term variation from the Chichijima sea level for the case of the RSC tide model and the TP_MSS geoid model. SSH agrees better. Further examination using TOPEX SSH data which were not used in the development of the tide model is necessary to solve this discrepancy. (c) Long-term variation of the residual In the lower panel of Fig. 5, the residual shows long-term variation other than the tidal aliasing. The residual is positive from January to June and negative from July to December. Period of the variation seems to be about one year. This long-term variation is discussed by comparing with hydrographic data around Chichijima. In Fig. 6, the dynamic height anomalies D referred to 500 db which are calculated from hydrographic data observed by the R/V Koyo are plotted together with the data in Fig. 5. In the upper panel, the D shows good agreement with the TOPEX SSH and Chichijima sea level. In the lower panel, residual of D from the Chichijima sea level shows a long-term variation similar to that of the TOPEX SSH. The dashed line represents the long-term variation of the D residual obtained by a 90-days running mean. It agrees well with the long-term variation of the TOPEX SSHs. From these results, the long-term variation of a period of about one year is considered to be caused by phenomena which are observed only by the tide gauge at coast and/or are enhanced in coastal shallow water. The cause of the long-term variation might qualitatively be explained as follows. Assuming that seasonal variation of sea surface height is dominated by steric change (Stammer and Wunsch, 1994), the steric effect might be enhanced in the shallow bay where the tide gauge station is located. This enhancement results in positive (negative) residual in winter (summer) as shown in Fig. 6. Positive (negative) values mean the Chichijima sea levels are lower (higher) than the TOPEX SSHs and the Ds in Fig. 6. In addition to this effect, it may be considered that setup of the sea surface at the coast of the island causes a seasonal variation in the residuals, since the setup is observed only by the tide gauge and may change seasonally. Futami Bay in Chichijima where the tide gauge is located opens west to the Pacific Ocean. The seasonal variation of the wind setup is considered to be caused by seasonal change of wind

270 N. Ebuchi and K. Hanawa Fig. 8. Scatter plot of the TOPEX SSHs before and after the correction for the long-term variation against the Chichijima sea level for the case of the RSC tide model (a) and the TP_MSS geoid model (b). Table 4. Comparison of the rms differences, correlation coefficients and slopes of regression line between the TOPEX SSHs before and after correction for the long-term variation and the Chichijima sea levels. The TP_MSS geoid model is used in the correction of horizontal excursion of the ground track. (Number of data = 53.) Tide model Before correction After correction RMS (cm) Corr. Slope RMS (cm) Corr. Slope CR 5.63 0.889 0.85 5.44 0.917 1.03 RSC 5.03 0.910 0.83 4.44 0.940 1.01 MSET 4.48 0.930 0.83 3.99 0.950 1.00 direction at the island due to the East Asian monsoon. Further study is necessary to discuss these processes quantitatively. By using the dashed line in Fig. 6, the long-term variation in the residual is removed from the TOPEX SSHs. The residual of the corrected TOPEX SSHs from the Chichijima sea level is plotted in Fig. 7. There still remains the variation of a period about 40 60 days corresponding to the tidal aliasing, though the long-term variation is removed. In Fig. 8, the TOPEX SSHs before and after the correction for long-term variation are plotted against the Chichijima sea level for the case using the TP_MSS geoid and RSC tide models. The rms difference is reduced from 5.03 cm to 4.44 cm, and the slope of regression line and correlation coefficient are improved from 0.83 and 0.910 to 1.01 and 0.940, respectively. The improvement of the slope is significant. In Table 4, changes of the rms difference, slope and correlation coefficient are listed for the cases using

Comparison of Sea Surface Heights Observed by TOPEX Altimeter with Sea Level Data at Chichijima 271 Fig. 9. Power spectra of time series of the residual for the three tide models; (a) CR, (b) RSC and (c) MSET. Aliasing periods due to the errors in the eight main tidal constituents are shown by arrows. the three tide models. For the case of the MSET tide model, the slope coincides with unity and the rms difference is reduced to less than 4 cm. (d) Spectral analysis of the residual In order to confirm that the variation in the residual as shown in Fig. 7 is caused by the tidal aliasing, power spectra of time series of the residual are calculated for the cases using three tide models with the TP_MSS geoid model. Gaps of the residual are filled by linear interpolation. A Hanning filter for three data points is adopted to smooth the power spectrum. The result is shown in Fig. 9. The aliasing period due to errors in the eight main tidal constituents (Table 2) are shown by arrows. Though the three power spectra show different feature with each other, most of the spectral peaks corresponds to the periods expected by the aliasing. As expected in Fig. 3, aliasing by the O 1 constituent is dominant for the case of the CR model, and that by the M 2 constituent is dominant for the case of the RSC model. In general, the features of power spectra of the residuals agree well with the results in Fig. 3. This supports the conclusion that the variation in the residual as shown in Fig. 7 is mainly caused by the tidal aliasing. For the case of the MSET model, however, a spectral peak is located at the alias period of the O 1 constituent, though the result in

272 N. Ebuchi and K. Hanawa Fig. 3 showed that the error in the M 2 is dominant. This result might be consistent with the discrepancy that the MSET model showed the largest rms difference in the comparison of tidal variation and the smallest rms difference in the comparison of TOPEX data with the Chichijima sea level (see Subsection 3.2(b)). 4. Conclusion In the present study, TOPEX SSHs, tide gauge data at Chichijima, and hydrographic data taken around the island were compared with each other to verify TOPEX observations and ocean tide corrections by tide models quantitatively. First, performance of the new oceanic tide models was assessed by comparison of tidal variation consisting of diurnal and semi-diurnal constituents with tide gauge data at Chichijima. The tide models proposed by Cartwright and Ray (1991, CR), Ray et al. (1994, RSC) and Ma et al. (1994, MSET) showed good agreement with rms differences less than 4 cm. The RSC model gave the smallest rms difference of 2.61 cm. Errors in amplitude and phase of each tide model were evaluated by spectral analysis. Then, period and amplitude of aliasing caused by these errors were estimated. The TOPEX SSHs corrected by using these models were compared with sea level data at Chichijima. It was confirmed that influence of horizontal excursion of the TOPEX ground track is not negligible in the sea near Chichijima, where change of geoid is very large. Correction for the influence using geoid models reduced the rms difference between the TOPEX SSHs and Chichijima sea levels by about 2 cm. A long-term variation of a period of about one year was found in the residual between the corrected SSHs and the Chichijima sea levels. This variation was also found in the difference between the dynamic height anomalies calculated from hydrographic data around the island and the Chichijima sea levels. The cause of the variation was discussed in relation to enhancement of the steric change by shallow water around the tide gauge station and setup of the sea surface by seasonally changing winds. This kind of differences between the sea level data observed at tide gauge stations on islands and the sea surface topography around the islands may affect results of comparisons of altimetry data with tide gauge data at islands. Finally, the rms difference between the TOPEX SSHs and the Chichijima sea levels was reduced to about 4 cm and the slope was improved to unity after the correction for the long-term variation. However, the residual showed variations related to the tidal aliasing. The spectra of the residual agreed well with the estimation from the results of comparison of tidal variations. Acknowledgements The Geophysical Data Records (GDRs) of the TOPEX altimeter were provided by the NASA Physical Oceanography Distributed Active Archive Center (PO-DAAC) at the Jet Propulsion Laboratory (JPL) of the California Institute of Technology. The sea level data at Chichijima were provided by the Oceanographic Division of the Marine Department, the Japan Meteorological Agency (JMA). The data of hydrographic observations by the R/V Koyo were provided by the Ogasawara Fisheries Center and were processed by Mr. Takeshi Taneda of Tohoku University. References Callahan, P. S. (1993): TOPEX POSEIDON GDR Users Handbook, Jet Propulsion Laboratory, D-9844. Cartwright, D. E. and R. D. Ray (1991): Energetics of global ocean tides from Geosat altimetry. J. Geophys. Res., 96, 16,897 16,912.

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