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Errors in recent ocean tide models: possible origin and cause
Prog. Oceanog. Vol. 40, pp. 325-336, 1997
Pergamon
PIh S0079-6611(98)00007-X
© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0079 6611/98 $32.00
Errors in recent ocean tide models: possible origin and cause A . J . E . SMITH 1
and O.B. ANDERSEN2
JDelfi Institute for Earth-Oriented Space Research, Kluyverweg 1, 2629 HS Delft, The Netherlands 2Kort-og Matrikelstyrelsen, Rentemestervej 8, 2400 Copenhagen NV, Denmark Abstract - Recently, the TOPEX/POSEIDON Science Working Team has recommended the FES95.2.1 and CSR3.0 ocean tide models for reprocessing the TOPEX/POSEIDON Geophysical Data Records. Without doubt, the performance of these models, especially in the deep oceans, is excellent. However, from a comparison of these hydrodynamically consistent models with the purely empirical DW3.2 and DEOS96.1 models, it appears that FES95.2.1 and CSR3.0 are affected by basin boundary related errors which are caused by the basin-wise solution procedure of the FES ocean tide model series. In their turn, the empirical DW3.2 and DEOS96.1 models seem to suffer from significant errors in the Antarctic seas due to the seasonal growth and decay of Antarctic sea ice. Also, bathymetry-induced differences were found between the hydrodynamically consistent models and the empirical models. Concerning these differences, TOPEX/POSEIDON and ERS-1 crossover statistics unfortunately do not provide conclusive results on which models are in error. © 1998 Elsevier Science Ltd. All rights reserved
I. INTRODUCTION Since the launch of G E O S A T in March 1985, the potential of using altimeter satellites as moving tide gauge stations has been widely recognized. This is clearly demonstrated by the numerous models that have used the altimetry from this satellite, e.g. CARTWRI6HT and RAY (1990, 1991) and WAGNER et al. (1994). More recently, the altimeter data of ERS-1 and T O P E X / P O S E I D O N have also found their applications in ocean tide modeling. Especially T O P E X / P O S E I D O N has led to a breakthrough in this field owing to its orbital accuracy in conjunction with its repeat period. The latter was carefully chosen as to avoid tidal aliasing to frequencies of ocean current variabilities (see F u et al., 1994) which led to alias periods typically less than 90 days for most of the largest tides. These small alias periods explain the huge number of recently published ocean tide models, some 15 in total, that somehow have incorporated the T O P E X / P O S E I D O N data, either as an empirical parameterization of the data itself or within data assimilation schemes incorporating dynamical constraints. After a detailed evaluation of 10 of these models by the T O P E X / P O S E I D O N Science Working Team ( S W T ) (see SHuM et al., 1996), the FES95.2.1 and CSR3.0 solutions were selected to replace the somewhat outdated models of SCHWlDERSKI (1983) and CARTWRICHX and RAY (1991) on the TOPEX/POSEIDON Geophysical Data Records ( G D R s ) for reprocessing. There should be no doubt about the great improvement in accuracy of these two models when compared with earlier models like those provided on the GDRs, i.e. the models of Schwiderski and Cartwright and Ray. Such is clearly demonstrated in SHUN et al. (1996) where all models included in the evaluation are estimated 325
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to have an accuracy of about 3 cm which unambiguously outclasses the 5 cm accuracy of the two earlier models. However, as the reprocessed GDRs will be used among a wide community of geodesists, geophysicists, and oceanographers, a detailed investigation of the FES95.2.1 and CSR3.0 models seems in place to identify possible errors that still may be present. Such an investigation is particularly useful for the diurnal and semi-diurnal tides because any error in these tides will alias in the TOPEX/POSEIDON altimetry to a signal of much longer period, e.g. 62 days for M2 and 173 days for K~, that may easily be mistaken for some oceanographic feature. The investigations in this paper mainly differ from those of SHUN et al. (1996) in the sense that the goal is not to obtain an accuracy assessment of the models by looking at the tide gauge fit or crossover statistics. Instead, the geographic pattern of the differences between the two models selected by the SWT and pure altimetric models are investigated which might indicate errors in any of the models. However, when possible, results are verified against independent data such as the altimeter crossover residuals of TOPEX/POSEIDON and ERS-I. Although a model comparison was also part of the investigations by SHUM et al. (1996), some interesting geographic structures have been found in FES95.2.1 and CSR3.0 that were not reported in that paper.
2. MODELS INCLUDED IN THE INVESTIGATIONS The investigations in this paper are based on four global ocean tide models. Two of these models account for hydrodynamics, i.e. FES95.2.1 and CSR3.0, while the other two, DW3.2 and DEOS96.1, are purely empirical, i.e. they fully rely on the tidal signal in the altimetry. The first three models were included in the evaluation by SHUM et al. (1996), while the DEOS96.1 model was developed later. Of these four models, only FES95,2.1 solves for the pure oceanic tide. The other three solve for the "elastic" ocean tide, i.e. the sum of the pure oceanic tide and the loading component of the sea floor. Because all comparisons in this paper are between elastic ocean tide models, the FES95.2.1 load tide was added to the FES95.2.1 ocean tidal solution. To provide some background information, a brief description of the models is given below. Details can be found in LE PROVOSTet al. (1994), SHUM et al. (1996), DESAI and WAHR (1995), and SMITH et al. (1996). The Grenoble FES95.2.1 model stems from the earlier purely hydrodynamic FES94.1 solution. FES95.2.1 is obtained by assimilating the earlier empirical CSR2.0 solution based on two years of TOPEX/POSEIDON altimetry into FES94.1 for ocean depths greater than 1 km. The last decimal in the FES designation indicates a change to FES95.2 after a small error in the assimilation scheme was discovered in October 1995 (P.S. Callahan, personal communication, 1996) which only affected the M2 component in the Arafura Sea north of Australia. The assimilation was performed over five basins separately, i.e. the North Atlantic, South Atlantic, Indian, North Pacific, and South Pacific oceans. Afterwards, the global solution is obtained by fitting the basinscale solutions through a set of tide gauge constants on the open boundaries. Essentially, the short-wavelength tidal features are controlled in the assimilation scheme by hydrodynamics whereas the longer wavelengths are controlled by altimetry. The model is harmonic and consists of 27 constituents. Eight constituents that define well the diurnal and semi-diurnal admittance are directly solved using the assimilation scheme while 19 minor tides have been deduced by admittance interpolation. The University of Texas/Center for Space Research CSR3.0 model is in essence a longwavelength correction to FES94.1 using two and a half years of TOPEX/POSEIDON altimetry.
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To convert the FES94.1 ocean tide model to elastic ocean tides, ocean loading as derived from CSR2.0 was added to the four dominant tides of FES94.1 in both the diurnal and semi-diurnal band. As it was recognized from a comparison with altimetry-derived tide models, e.g. that of SCnRAMA and RAY (1994), that FES94.1 contained some large-scale errors in M2 and S> the solutions of these constituents were taken from an adjusted version of FES94.1 by ANOERSEN et al. (1995). Next, a priori diurnal and semi-diurnal orthoweights were fit through the dominant tides in both bands. Corrections to the a priori orthoweights were then estimated from TOPEX/POSEIDON altimetry and added back to their a priori values to obtain the new model. The University of Colorado DW3.2 model is an empirical tide model based on two years of TOPEX/POSEIDON altimetry. The response formalism is applied to collinear track differences to solve for the long-period, diurnal, and semi-diurnal orthoweights. Of all four models, DW3.2 is the only one that solves for long-period tides instead of inferring them from equilibrium theory. However, care must be taken with applying the long-wavelength solutions as they seem to contain some instabilities (S.D. Desai, personal communication, 1996). The number of spectral lines that have been derived from the orthoweights and are provided by this model are 161 for the diurnal band, 116 for the semi-diurnal band, and 87 for the long-wavelength band. Most of these spectral lines are, however, quite small and if those with an equilibrium amplitude of e.g. 1 mm would be excluded only 41, 31, and 5 in the respective bands remain. Like DW3.2, the Delft Institute for Earth-Oriented Space Research DEOS96.1 model is also empirical. It is one of a series of models that are currently being developed to investigate the use of combining altimetry data from several satellites, i.e. TOPEX/POSEIDON, ERS-I, and GEOSAT, by applying both the harmonic and response analysis. The model discussed here is based on 3 years of TOPEX/POSEIDON sea surface height observations relative to the OSU91A geoid. It uses the harmonic analysis technique and solves for the major tides M2, S2, Ne, KI, and O~ together with geoid errors and the dynamic topography (modeled as a quadratic surface) and the harmonic constants of the annual and semi-annual seasonal cycles.
3. BATHYMETRY-RELATEDDIFFERENCES BETWEEN FES95.2.1/CSR3.0 AND DEOS96.1/DW3.2 In Fig. 1, the FES95.2.1, CSR3.0, and DEOS96.1 models are compared with the DW3.2 model for the M2 tide. The DW3.2 model was chosen as the reference for all model comparisons in Fig. 1 because it uses collinear tracks and thus is completely independent of errors in the surface to which the tides are referenced unlike the DEOS96.1 model which solves for the sum of the dynamic topography and geoid errors. Notice that all plots in Fig. 1 are illuminated from the northwest to accentuate the small-scale differences. Also notice that the differences in Fig. l(a,b) are smoother than those in Fig. l(c). Smoothness is enforced on the FES95.2.1 and CSR3.0 models by the hydrodynamic equations while DW3.2 uses a Gaussian smoothing procedure (DEsAt and WAnR, 1995). The DEOS96.1 model uses an internal smoothing procedure where a search area is defined around each grid point so that one altimeter observation may contribute to the normal matrices of several grid points (SMtxn et al., 1996). Apparently the internal smoothing of DEOS96.1 enforces less smoothness on the tidal solutions. In the deep oceans, the M2 vector differences are seen to be below 0.5 cm which indicates a remarkable agreement between the hydrodynamically consistent FES95.2.1 and CSR3.0 models and the empirical DEOS96.1 and DW3.2 models. Near the coast, the model differences are much larger and may occasionally exceed 50 cm. These large differences obviously result from the relatively poor resolution of the TOPEX/POSEIDON altimetry, i.e. a ground track spacing of 3° at the
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Errors in recent ocean tide models: possible origin and cause
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equator. As a consequence, the TOPEX/POSEIDON altimeter data fail to pick up small-scale tidal features in shallow waters which are due to a complex interaction of the tidal wave with the coastal geometry. Hence, because of differences in the data processing, smoothing of the solutions, and applied constraints, all models show relatively large disagreements (up to tens of centimeters) near the coast. If the differences are masked for an ocean depth less than 200 m (purple color in Fig. 1), the vector differences of FES95.2.1, CSR3.0, and DEOS96.1 with DW3.2 have a global mean of 1.4 cm, 1.2 cm, and 0.8 cm respectively. Clearly visible from Fig. l(a,b) is that the differences between the hydrodynamically consistent models with DW3.2 show a strong correlation with the topography of the ocean floor (Fig. 2). Obviously, the DW3.2 (and DEOS96.1 ) model will fail to capture most of the small-scale tidal signals which explains the topographic features in Fig. l(a,b) near the coast or small island groups. The limited potential of the TOPEX/POSEIDON spatial resolution to resolve these tidal signals has previously been noticed by SHUM et al. (1996). However, the illumination of the plots also makes visible some topographic features that can obviously be correlated with ridge structures like the Mid Atlantic Ridge, the Walvis Ridge (off the Namibian coast), and the Mariana Trench (south of Japan). As these features are in the deep oceans where the interaction of the tide with the ocean bottom should be negligible, the topographic features in these areas may just as well be due to FES95.2. I and CSR3.0 as to DW3.2 (or DEOS96.1). One possibility to explain these features is the ETOPO5 model that is used to specify the bathymetry in the hydrodynamic equations for the FES model series. Because errors in the
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ETOPO5 bathymetry model may be as large as several kilometers (NATIONALGEOPHYSICAL DATA CENTER, 1993), the bathymetry might have polluted the FES ocean tide model series from the very outset of FES94.1. Then, as the CSR2.0 model is based on TOPEX/POSEIDON altimetry, it cannot help FES95.2.1 to correct these small-scale bathymetry-induced errors. Likewise, this could explain why the bathymetry-induced errors also appear in CSR3.0 as they are simply copied from FES94.1 without being altered. Another possibility, of course, is that the interaction of the tidal wave with the bottom topography is still large enough to cause smallscale tidal features in the deep oceans. In that case, the coarse resolution of the TOPEX/POSEIDON altimetry implies that the DEOS96.1 and DW3.2 models would be in error. Notice that although both DEOS96. l and CSR3.0 solve for the mean sea surface (actually CSR3.0 solves for mean sea surface residuals), this procedure does not introduce topographic signals into the tidal solutions because no such signals can be seen in Fig. l(c). Of course this could be expected as there should be no correlation between the time-varying tidal signals and the constant bottom topography. The large-scale feature appearing in Fig. l(c) in the North Atlantic was found to be caused by a discrepancy in the M2 cosine between the DEOS96.1 solution and the other models. Hence, most of it reflects a small error in the M2 cosine of DEOS96.1. The fact that it partially seems to follow the Atlantic Ridge in the North Atlantic must be a coincidence and is unlikely to be caused by topographic errors in the DEOS96.1 model. Otherwise these errors would also have shown up in the other tidal components and at other locations of e.g. the Atlantic Ridge. Most likely, the bathymetry-induced differences were not earlier discovered in e.g. the model comparisons of ANDERSENet al. (1995), which included FES94. l, because this model contained some large-scale errors up to 5 cm magnitude (see ANDERSEN et al., 1995) which obscured these signals. From our comparisons, bathymetry-induced differences between FES95.2.1/CSR3.0 and DEOS96.1/DW3.2 were also discovered in $2 and N2. They occur mainly at the same locations as with M2, i.e. at the Atlantic Ridge, the Walvis Ridge, and at the Mariana Trench. However, compared to the l cm magnitude of these differences in M2, their significance is much less in S> about 0.5 cm, and quite negligible in N:, about 0.2 cm. In the diurnal tides, no bathymetryrelated differences could be detected in K~ and O~, most likely because the semi-diurnal tides are much stronger over the indicated areas of steep topography. Notice that these five tides are the only tides that can be resolved from altimetry. Any smaller tide will drown in the altimeter background noise with a harmonic solution. Even with the response method, results of the smaller tides can be doubted as they are based on the admittance of the stronger tides and thus are not independent. Hence, the comparisons in this paper are restricted to the five largest tides. To try to find out which of the models is responsible for the bathymetry-induced signals, i.e. FES95.2.1/CSR3.0 or DEOS96.1/DW3.2, a test has been performed in which the crossover statistics of TOPEX/POSEIDON and ERS-l were computed with each of the models over the Mid Atlantic Ridge. The crossover data of TOPEX/POSEIDON were derived from GDRs 125130 while for ERS- l, Ocean Products 1 l - 12 were used. The altimetry of neither of these crossover sets are used in the data reduction of any of the tidal models. Using tide gauge harmonic constants as an independent data type was considered less useful as there are few gauges over the Mid Atlantic Ridge, e.g. three in the tide gauge set provided by C. Le Provost (personal communication, 1994). Altimeter crossover residuals were computed applying the M2 tidal solution only in which the bathymetry signals were strongest. The ERS-1 crossovers were selected because of their much better spatial resolution (1 ° ground track spacing at equator) compared to TOPEX/POSEIDON. Unfortunately, the differences in crossover residual rms between the models are so small (less than a millimeter for both TOPEX/POSEIDON and ERS- l ) that they
Errors in recent ocean tide models: possible origin and cause
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do not allow any conclusion to be drawn on the origin of the bathymetric signals. Most likely, therefore, the bathymetry signals are due to both the hydrodynamically consistent and empirical models or they are not large enough to have significant effect on the crossover statistics. In that respect it makes no difference if one uses the FES94.1 model as a priori information, like CSR3.0, or not, like DEOS96.1 and DW3.2.
4. ERRORS IN FES95.2.1 AND CSR3.0 DUE TO BASIN BOUNDARYPROBLEMS IN FES94.1 The use of this Section is to show that FES95.2.1 and CSR3.0 contain errors that can be related to domain boundary problems in the older FES94.1 model. It is known that the basin-wise solution procedure of the FES models introduced some inconsistencies at the basin boundaries of FES94.1 (C. Le Provost, personal communication, 1996). A picture of the grid and the exact locations of the basin boundaries of the FES models can be found in LE PROVOST et al. (1994) and hence need not be reproduced here. As the FES94.1 solution is used as the basis for the assimilation of FES95.2.1 but also as the a priori model to CSR3.0, boundary errors appear in both FES95.2.1 and CSR3.0. The most obvious boundary error is the edge-like structure visible in the CSR3.0 - DW3.2 differences of M 2 in Fig. l(b). This feature, which is along the equator in the Pacific Ocean turning northeast south of Hawaii towards Mexico, has been noted at several occasions, e.g. ANDERSEN(1995) and SHUM et al. (1996). Other features in Fig. l(b) that cannot be explained by bathymetric differences are the straight lines between Africa and Brazil and south of South Africa. These lines respectively follow the FES boundary between the North Atlantic and South Atlantic oceans and the boundary between the South Atlantic and Indian oceans (see LE PROVOST et al., 1994), Features along these boundaries were not noticed by ANDERSEN (1995) and SHUM et al. (1996), probably because they did not apply shading to the differences. In Fig. l(b), it can be seen that the boundary errors in CSR3.0 are associated with errors of about 1.5 cm magnitude in M2 and that they are very local. In the case of FES95,2.1 (see Fig. l(a)), errors of 2 cm magnitude are seen to propagate into the South Atlantic and the Indian oceans due to a mismatch between the M: solutions at the boundary of these basins (C. Le Provost, personal communication, 1996). A close inspection of the ocean tide difference plots in ANDERSEN et al. (1995) reveals large discrepancies between all models and FES94.1 in M, and $2 in this region. Hence, it is likely that the errors in FES95.2.1 south of South Africa are due to an error in FES94.1 which has not been corrected by the assimilation scheme that produced FES95.2.1. The origin of the 1.5-2 cm signal in FES95.2.1 between Brazil and Afiica has not yet been confirmed but as it seems to follow the boundary in the Atlantic, which clearly appears in CSR3.0 (Fig. l(b)), a boundary problem might be a likely cause. It is interesting to note from Fig. l(a,b) that the boundary problem along the equator in the Pacific Ocean in FES94.1 has apparently been solved in FES95.2.1. The problems in the Atlantic Ocean and between the South Atlantic and Indian oceans, however, remain. Also interesting is that the solution procedure of CSR3.0, which basically is a long-wavelength adjustment to FES94.1, seems to detect most of the basin boundary signals which then become part of the adjustment. Hence, the large errors south of South Africa in FES95.2.1 have been removed by the adjustment procedure of CSR3.0 and thus are not present in Fig. l(b). The sharp features along the boundaries, of course, are of too short a wavelength to be detected by the 3 ° x 3° resolution o1" the TOPEX/POSEIDON tracks. Notice that the fact that CSR3.0 is a long-wavelength correction to FES94.1 provides an interesting check to verify that the signals in the Mid Atlantic and between
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the Atlantic and the Indian oceans are indeed due to boundary problems. Differencing CSR3.0 and FES95.2.1 will remove that part of the small-scale differences around the boundaries that both models have in common, i.e. the part that CSR3.0 cannot adjust, What remains are the sharp boundary features superimposed on the long-wavelength adjustments of CSR3.0. This is done in Fig. 3 which shows the M2 differences between FES95.2.1 and CSR3.0 which have a global mean of 0.9 cm for ocean depths larger than 200 m. Now the boundaries become visible as sharp lines which are due to CSR3.0 along the Equator and probably due to both models in the Mid Atlantic and between the Atlantic and the Indian oceans. Besides in M2, errors related to the FES94.1 boundary problems were also detected in some of the other constituents of FES95.2.1 and CSR3.0. In the case of CSR3.0, errors along the Equator and the Atlantic boundary were found of 1 cm in Sz, 0.5 cm in N2, and 0.2 cm in Kr. Notice, that the numbers for N2 and K1 are small compared to other differences in the constituents and that the only reason that they can so easily be detected is their characteristic shape. Hence, apart from M2 and $2, the boundary errors in CSR3.0 may be termed insignificant. In case of FES95.2.1, boundary errors of 1-1.5 cm in $2 and N2 could easily be detected along the boundary between the Atlantic and Indian oceans. In the Atlantic, only $2 showed a signal along the boundary which, however, may also be of a different cause. In the diurnal tides, no boundary-related signals were found. Hence, only boundary-related errors in FES95.2.1 of 2 cm in M2, and 1-1.5 cm in $2 and N2 along the boundary of the Atlantic and Indian oceans can be ascertained. A possible explanation why these errors are only found in the semi-diurnal tides of FES95.2.1 may be that these tides have a much larger amplitude south of South Africa than the diurnal tides. Hence, any discrepancy between the semi-diurnal basin solutions will show up more clearly. In Fig. 4 the difference of the crossover residual rms values of the DEOS96.1 and FES95.2.1 models is given for TOPEX/POSEIDON GDRs 125-130, i.e. DEOS96.1 - FES95.2.1. The crossover residual rms values were computed on a 4 ° × 4 ° grid and interpolated to pixel resolution. Clearly visible is the dark spot south of South Africa corresponding to a value of about 0.8 cm
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Errors in recent ocean tide models: possible origin and cause
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Fig. 4. Global crossover residual rms DEOS96.I-FES95.2.I in cm. Crossover data are from TOPEX/POSEIDON GDRs 125-130. which confirms the boundary error in FES95.2.1. Assuming that the difference in the crossover residual rms south of South Africa is mainly tidal error in FES95.2.1 and that this error can be equally divided over ascending and descending tracks, the value of about 0.8 cm indicates an rrns tidal error in FES95.2.1 of 0.6 cm (1/2 ~/2 × 0.8). It should be mentioned, however, that on a g l o b a l scale, none of the two models was found to perform better than the other. For instance, the global T O P E X / P O S E I D O N crossover residual rms values of the models were found to be 7.0 cm for DEOS96.1 and 7.1 cm for FES95.2.1 while in case of ERS-1 these values were 11.1 cm and 11.0 cm respectively.
5. ERRORS IN ALTIMETRY-DERIVED MODELS DUE TO SEASONAL CHANGE IN ANTARCTIC SEA ICE COVERAGE A detailed investigation of the differences displayed in Fig. 1 revealed values exceeding 2 cm in the Antarctic seas, i.e. from 80 ° west to 180 ° east longitude and below 60 ° south. Investigations into the formal errors of the DEOS96.1 model showed that the standard deviations of the tidal harmonic constants increased to about 10 cm in the Antarctic seas compared to a global average of 2 - 2 . 5 cm. For the seasonal cycles, however, the standard deviations were found to increase to over 30 cm compared to the same global average of 2 - 2 . 5 cm. Hence, the cause of the large differences in Fig. 1 is of seasonal nature and therefore must be the yearly growth and decay of Antarctic sea ice. As an example, Fig. 5 shows the number of TOPEX/POSEIDON altimeter normal points for each cycle in the 3 ° x 3 ° grid cell centered at 1.5 ° longitude and 64.5 ° latitude for the DEOS96.1 model. Clearly visible are the summer periods in the southern hemisphere, e.g. cycles 9 - 2 7 which correspond to the period 11 December 1992-17 June 1993, and the winter periods during which the growth of ice sheets may lead to a total absence of altimeter observations at Antarctic latitudes. Notice that cycles 102-117 coincide with the winter period in the southern hemisphere so that Fig. 5 does not go beyond cycle 101. As no altimeter observations over ice are used in the altimetry-derived models, the number of observations in
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Fig. 5. Number of TOPEX/POSEIDON altimeter normal points during cycles 9-117 in 3° x 3° grid cell centered at 1.5 ° longitude and - 64.5 ° latitude. grid points below 60 ° south varies with the yearly motion of the ice edge leading to an illconditioned system of normal equations in the Antarctic seas. As a consequence, all constituents of the altimetry-based models (DEOS96.1, DW3.2, and to a considerable extent also CSR3.0) show significant errors in the Antarctic seas which are reflected by the differences of e.g. DW3.2 with FES95.2.1 (Fig. l ( a ) ) which is less dependent on altimetry. The CSR3.0 solution in these seas is slightly constrained by the FES94.1 model. Hence, its differences with FES95.2.1 near the Antarctic are a bit smaller than those of DEOS96.1 and DW3.2. Especially the Weddell Sea in the South Atlantic is seen to be affected by the motion of the ice edge which reaches latitudes as high as 60 ° during winter in the southern hemisphere (GuzKOWSKA e t a l . , 1990). Differences of the altimetry-derived models with FES95.2.1 in the Weddell Sea were found to exceed 5 cm in amplitude. To quantify the r m s tidal errors in the DEOS96.1 and DW3.2 models in the Antarctic seas, Fig. 6 shows the difference of the crossover residual r m s of the DEOS96.1 and FES95.2.1 models, i.e. DEOS96.1 - FES95.2.1. Clearly visible is the larger r m s of the DEOS96.1 model around the Antarctic of about 1.5 cm. In the Weddell Sea these differences were found to exceed 3 cm. Assuming that the FES95.2.1 model correctly represents the tides in the Antarctic seas and that tidal errors can be equally divided over ascending and descending tracks, the seasonal changes in ice coverage lead to errors of about 1 cm magnitude in the DEOS96.1 (and DW3.2) model while errors are somewhat less in CSR3.0. In this respect, constraining the solutions of the empirical models by tide gauge constants available in these seas might be an interesting subject for further study. Notice, however, that for longitudes between 60 ° and 80 ° west, the
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Fig. 6. Crossover residual rms DEOS96. I-FES95.2.1 (cm) around Antarctic. Crossover data are from TOPEX/POSEIDON GDRs 125-130. crossover residual r m s of the FES95.2.1 model is larger than that of DEOS96.1. Hence, in that part of the Antarctic seas, the differences in Fig. l ( a ) may be caused by errors in the FES95.2.1 model due to difficulties with modeling ocean/ice sheet interaction (LE P~OVOST et al., 1994).
6. CONCLUSIONS The FES95.2.1/CSR3.0 and DEOS96.1/DW3.2 models show distinct differences in areas of steep topography. These differences have an amplitude of about 1 cm in M2 and 0.5 cm in S~. The cause is either related to the bathymetry model of the FES series for which ETOPO5 is used or to small-scale tidal features that are not picked up by the TOPEX/POSEIDON altimetry. Discontinuities along the basin boundaries of the earlier FES94.1 model cause some very distinct errors in FES95.2.1 and CSR3.0. Large differences are detected between the FES95.2. I model and the altimetry-derived DEOS96.1, DW3.2, and CSR3.0 models south of South Africa. These differences have a vector magnitude of about 2 cm in M2 and 1-1.5 cm in $2 and N~ and translate into a tidal error of approximately 0.6 cm in FES95.2.1 according to the T O P E X / P O S E I D O N crossover statistics. With CSR3.0, part of the inconsistencies along the basin boundaries have been removed by the adjustment procedure. Still, some remnants remain along the Equator in the Pacific and between Brazil and Africa of 1.5 cm in M2 and I cm in $2. These remnants are so local that they will have little effect on the global accuracy of CSR3.0. All altimetry-derived ocean tide models seem to contain errors of about 1 cm r m s near the Antarctic. The reason is the seasonal growth and decay of Antarctic sea ice, which locally leads to an absence of observations during periods as long as half a year. In the Weddell Sea, these errors can exceed 2 cm r m s . On a global scale, the models included in the investigations were found to perform equally well according to the crossover statistics of T O P E X / P O S E I D O N and E R S - I .
7. REFERENCES
ANDERSENO.B. (1995) Global ocean tides from ERS-I and TOPEX/POSEIDON altimetry. Journal (?f Geophysical Research 100(C12), 25249-25259.
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ANDERSEN O.B., WOODWORTHP.L. and FLATHER R.A. (1995) Intercomparison of recent ocean tide models. Journal of Geophysical Research 100(C12), 25261-25282. CARTWRIGHTD.E. and RAY R.D. (1990) Oceanic tides from geosat altimetry. Journal of Geophysical Research 95(C3), 3069-3090. CARTWRIGHT D.E. and RAY R.D. (1991) Energetics of global ocean tides from geosat altimetry. Journal of Geophysical Research 96(C9), 16897-16912. DESAI S.D. and WAHR J.M. (1995) Empirical ocean tide models estimated from TOPEX/POSEIDON altimetry. Journal of Geophysical Research 100(Cl 2), 25205-25228. Fu L.L., CHRISTENSENE.J., YAMARONEC.A., LEFEBVREM., MI~NARDY., DORRERM. and ESCUDIERP. (1994) TOPEX/POSEIDON mission overview. Journal of Geophysical Research 99(C12), 24369-24381. GUZKOWSKA M.A.J., RAPLEY C.G., RIDLEY J.K., CUDLW W., BIRKETT C.M. and SCOTT R.F. (1990) Developments in inland water and land altimetry. Final report, ESA contract CR-783988FFL, University College London, Mullard Space Science Laboratory, Dorking, Surrey, UK. LE PROVOSTC., GENCO M.L., LYARDF., VINCENT P. and CANCEIL P. (1994) Spectroscopy of the world ocean tides from a finite element hydrodynamic model. Journal of Geophysical Research 99(C12), 2477724797. NATIONAL GEOPHYSICALDATA CENTER (1993) Global Relief. CD-ROM, Boulder, Colorado. SCHRAMA E.J.O. and RAY R.D. (1994) A preliminary tidal analysis of TOPEX/POSEIDON altimetry. Journal of Geophysical Research 99(C 12), 24799-24808. SCHWIDERSKI E.W. (1983) Atlas of ocean tidal maps, part I: the semidiurnal principal lunar tide M 2. Marine Geodesy 6(3-4), 219-265. SHUN C.K., WOOOWORTHP.L., ANDERSENO.B., EGBERTG.D., FRANCISO., KING C., KLOSKOS.M., LE PROVOST C., LI X., MOLINESJ.-M., PARKEM.E., RAY R.D., SCHLAXM.G., STAMMERD., TEIRNEYC.C., VINCENT P. and WUNSCrt C.I. (1996) Accuracy assessment of recent ocean tide models. Journal of Geophysical Research 102(C 11 ), 25173-25194. SMITH A.J.E., AMBROSIUS B.A.C., WAKKER K.F., WOODWORTH P.L. and VASSIE J.M. (1996) Comparison between the harmonic and response methods of tidal analysis using TOPEX/POSEIDON altimetry. Journal of Geodesy 71, 695-703. WAGNER C.A., TAI C.K. and KUHN J.M. (1994) Improved M2 ocean tide from TOPEX/POSEIDON and Geosat altimetry. Journal of Geophysical Research 99(C 12), 24853-24865.
Pergamon
PIh S0079-6611(98)00007-X
© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0079 6611/98 $32.00
Errors in recent ocean tide models: possible origin and cause A . J . E . SMITH 1
and O.B. ANDERSEN2
JDelfi Institute for Earth-Oriented Space Research, Kluyverweg 1, 2629 HS Delft, The Netherlands 2Kort-og Matrikelstyrelsen, Rentemestervej 8, 2400 Copenhagen NV, Denmark Abstract - Recently, the TOPEX/POSEIDON Science Working Team has recommended the FES95.2.1 and CSR3.0 ocean tide models for reprocessing the TOPEX/POSEIDON Geophysical Data Records. Without doubt, the performance of these models, especially in the deep oceans, is excellent. However, from a comparison of these hydrodynamically consistent models with the purely empirical DW3.2 and DEOS96.1 models, it appears that FES95.2.1 and CSR3.0 are affected by basin boundary related errors which are caused by the basin-wise solution procedure of the FES ocean tide model series. In their turn, the empirical DW3.2 and DEOS96.1 models seem to suffer from significant errors in the Antarctic seas due to the seasonal growth and decay of Antarctic sea ice. Also, bathymetry-induced differences were found between the hydrodynamically consistent models and the empirical models. Concerning these differences, TOPEX/POSEIDON and ERS-1 crossover statistics unfortunately do not provide conclusive results on which models are in error. © 1998 Elsevier Science Ltd. All rights reserved
I. INTRODUCTION Since the launch of G E O S A T in March 1985, the potential of using altimeter satellites as moving tide gauge stations has been widely recognized. This is clearly demonstrated by the numerous models that have used the altimetry from this satellite, e.g. CARTWRI6HT and RAY (1990, 1991) and WAGNER et al. (1994). More recently, the altimeter data of ERS-1 and T O P E X / P O S E I D O N have also found their applications in ocean tide modeling. Especially T O P E X / P O S E I D O N has led to a breakthrough in this field owing to its orbital accuracy in conjunction with its repeat period. The latter was carefully chosen as to avoid tidal aliasing to frequencies of ocean current variabilities (see F u et al., 1994) which led to alias periods typically less than 90 days for most of the largest tides. These small alias periods explain the huge number of recently published ocean tide models, some 15 in total, that somehow have incorporated the T O P E X / P O S E I D O N data, either as an empirical parameterization of the data itself or within data assimilation schemes incorporating dynamical constraints. After a detailed evaluation of 10 of these models by the T O P E X / P O S E I D O N Science Working Team ( S W T ) (see SHuM et al., 1996), the FES95.2.1 and CSR3.0 solutions were selected to replace the somewhat outdated models of SCHWlDERSKI (1983) and CARTWRICHX and RAY (1991) on the TOPEX/POSEIDON Geophysical Data Records ( G D R s ) for reprocessing. There should be no doubt about the great improvement in accuracy of these two models when compared with earlier models like those provided on the GDRs, i.e. the models of Schwiderski and Cartwright and Ray. Such is clearly demonstrated in SHUN et al. (1996) where all models included in the evaluation are estimated 325
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to have an accuracy of about 3 cm which unambiguously outclasses the 5 cm accuracy of the two earlier models. However, as the reprocessed GDRs will be used among a wide community of geodesists, geophysicists, and oceanographers, a detailed investigation of the FES95.2.1 and CSR3.0 models seems in place to identify possible errors that still may be present. Such an investigation is particularly useful for the diurnal and semi-diurnal tides because any error in these tides will alias in the TOPEX/POSEIDON altimetry to a signal of much longer period, e.g. 62 days for M2 and 173 days for K~, that may easily be mistaken for some oceanographic feature. The investigations in this paper mainly differ from those of SHUN et al. (1996) in the sense that the goal is not to obtain an accuracy assessment of the models by looking at the tide gauge fit or crossover statistics. Instead, the geographic pattern of the differences between the two models selected by the SWT and pure altimetric models are investigated which might indicate errors in any of the models. However, when possible, results are verified against independent data such as the altimeter crossover residuals of TOPEX/POSEIDON and ERS-I. Although a model comparison was also part of the investigations by SHUM et al. (1996), some interesting geographic structures have been found in FES95.2.1 and CSR3.0 that were not reported in that paper.
2. MODELS INCLUDED IN THE INVESTIGATIONS The investigations in this paper are based on four global ocean tide models. Two of these models account for hydrodynamics, i.e. FES95.2.1 and CSR3.0, while the other two, DW3.2 and DEOS96.1, are purely empirical, i.e. they fully rely on the tidal signal in the altimetry. The first three models were included in the evaluation by SHUM et al. (1996), while the DEOS96.1 model was developed later. Of these four models, only FES95,2.1 solves for the pure oceanic tide. The other three solve for the "elastic" ocean tide, i.e. the sum of the pure oceanic tide and the loading component of the sea floor. Because all comparisons in this paper are between elastic ocean tide models, the FES95.2.1 load tide was added to the FES95.2.1 ocean tidal solution. To provide some background information, a brief description of the models is given below. Details can be found in LE PROVOSTet al. (1994), SHUM et al. (1996), DESAI and WAHR (1995), and SMITH et al. (1996). The Grenoble FES95.2.1 model stems from the earlier purely hydrodynamic FES94.1 solution. FES95.2.1 is obtained by assimilating the earlier empirical CSR2.0 solution based on two years of TOPEX/POSEIDON altimetry into FES94.1 for ocean depths greater than 1 km. The last decimal in the FES designation indicates a change to FES95.2 after a small error in the assimilation scheme was discovered in October 1995 (P.S. Callahan, personal communication, 1996) which only affected the M2 component in the Arafura Sea north of Australia. The assimilation was performed over five basins separately, i.e. the North Atlantic, South Atlantic, Indian, North Pacific, and South Pacific oceans. Afterwards, the global solution is obtained by fitting the basinscale solutions through a set of tide gauge constants on the open boundaries. Essentially, the short-wavelength tidal features are controlled in the assimilation scheme by hydrodynamics whereas the longer wavelengths are controlled by altimetry. The model is harmonic and consists of 27 constituents. Eight constituents that define well the diurnal and semi-diurnal admittance are directly solved using the assimilation scheme while 19 minor tides have been deduced by admittance interpolation. The University of Texas/Center for Space Research CSR3.0 model is in essence a longwavelength correction to FES94.1 using two and a half years of TOPEX/POSEIDON altimetry.
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To convert the FES94.1 ocean tide model to elastic ocean tides, ocean loading as derived from CSR2.0 was added to the four dominant tides of FES94.1 in both the diurnal and semi-diurnal band. As it was recognized from a comparison with altimetry-derived tide models, e.g. that of SCnRAMA and RAY (1994), that FES94.1 contained some large-scale errors in M2 and S> the solutions of these constituents were taken from an adjusted version of FES94.1 by ANOERSEN et al. (1995). Next, a priori diurnal and semi-diurnal orthoweights were fit through the dominant tides in both bands. Corrections to the a priori orthoweights were then estimated from TOPEX/POSEIDON altimetry and added back to their a priori values to obtain the new model. The University of Colorado DW3.2 model is an empirical tide model based on two years of TOPEX/POSEIDON altimetry. The response formalism is applied to collinear track differences to solve for the long-period, diurnal, and semi-diurnal orthoweights. Of all four models, DW3.2 is the only one that solves for long-period tides instead of inferring them from equilibrium theory. However, care must be taken with applying the long-wavelength solutions as they seem to contain some instabilities (S.D. Desai, personal communication, 1996). The number of spectral lines that have been derived from the orthoweights and are provided by this model are 161 for the diurnal band, 116 for the semi-diurnal band, and 87 for the long-wavelength band. Most of these spectral lines are, however, quite small and if those with an equilibrium amplitude of e.g. 1 mm would be excluded only 41, 31, and 5 in the respective bands remain. Like DW3.2, the Delft Institute for Earth-Oriented Space Research DEOS96.1 model is also empirical. It is one of a series of models that are currently being developed to investigate the use of combining altimetry data from several satellites, i.e. TOPEX/POSEIDON, ERS-I, and GEOSAT, by applying both the harmonic and response analysis. The model discussed here is based on 3 years of TOPEX/POSEIDON sea surface height observations relative to the OSU91A geoid. It uses the harmonic analysis technique and solves for the major tides M2, S2, Ne, KI, and O~ together with geoid errors and the dynamic topography (modeled as a quadratic surface) and the harmonic constants of the annual and semi-annual seasonal cycles.
3. BATHYMETRY-RELATEDDIFFERENCES BETWEEN FES95.2.1/CSR3.0 AND DEOS96.1/DW3.2 In Fig. 1, the FES95.2.1, CSR3.0, and DEOS96.1 models are compared with the DW3.2 model for the M2 tide. The DW3.2 model was chosen as the reference for all model comparisons in Fig. 1 because it uses collinear tracks and thus is completely independent of errors in the surface to which the tides are referenced unlike the DEOS96.1 model which solves for the sum of the dynamic topography and geoid errors. Notice that all plots in Fig. 1 are illuminated from the northwest to accentuate the small-scale differences. Also notice that the differences in Fig. l(a,b) are smoother than those in Fig. l(c). Smoothness is enforced on the FES95.2.1 and CSR3.0 models by the hydrodynamic equations while DW3.2 uses a Gaussian smoothing procedure (DEsAt and WAnR, 1995). The DEOS96.1 model uses an internal smoothing procedure where a search area is defined around each grid point so that one altimeter observation may contribute to the normal matrices of several grid points (SMtxn et al., 1996). Apparently the internal smoothing of DEOS96.1 enforces less smoothness on the tidal solutions. In the deep oceans, the M2 vector differences are seen to be below 0.5 cm which indicates a remarkable agreement between the hydrodynamically consistent FES95.2.1 and CSR3.0 models and the empirical DEOS96.1 and DW3.2 models. Near the coast, the model differences are much larger and may occasionally exceed 50 cm. These large differences obviously result from the relatively poor resolution of the TOPEX/POSEIDON altimetry, i.e. a ground track spacing of 3° at the
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Errors in recent ocean tide models: possible origin and cause
329
equator. As a consequence, the TOPEX/POSEIDON altimeter data fail to pick up small-scale tidal features in shallow waters which are due to a complex interaction of the tidal wave with the coastal geometry. Hence, because of differences in the data processing, smoothing of the solutions, and applied constraints, all models show relatively large disagreements (up to tens of centimeters) near the coast. If the differences are masked for an ocean depth less than 200 m (purple color in Fig. 1), the vector differences of FES95.2.1, CSR3.0, and DEOS96.1 with DW3.2 have a global mean of 1.4 cm, 1.2 cm, and 0.8 cm respectively. Clearly visible from Fig. l(a,b) is that the differences between the hydrodynamically consistent models with DW3.2 show a strong correlation with the topography of the ocean floor (Fig. 2). Obviously, the DW3.2 (and DEOS96.1 ) model will fail to capture most of the small-scale tidal signals which explains the topographic features in Fig. l(a,b) near the coast or small island groups. The limited potential of the TOPEX/POSEIDON spatial resolution to resolve these tidal signals has previously been noticed by SHUM et al. (1996). However, the illumination of the plots also makes visible some topographic features that can obviously be correlated with ridge structures like the Mid Atlantic Ridge, the Walvis Ridge (off the Namibian coast), and the Mariana Trench (south of Japan). As these features are in the deep oceans where the interaction of the tide with the ocean bottom should be negligible, the topographic features in these areas may just as well be due to FES95.2. I and CSR3.0 as to DW3.2 (or DEOS96.1). One possibility to explain these features is the ETOPO5 model that is used to specify the bathymetry in the hydrodynamic equations for the FES model series. Because errors in the
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ETOPO5 bathymetry model may be as large as several kilometers (NATIONALGEOPHYSICAL DATA CENTER, 1993), the bathymetry might have polluted the FES ocean tide model series from the very outset of FES94.1. Then, as the CSR2.0 model is based on TOPEX/POSEIDON altimetry, it cannot help FES95.2.1 to correct these small-scale bathymetry-induced errors. Likewise, this could explain why the bathymetry-induced errors also appear in CSR3.0 as they are simply copied from FES94.1 without being altered. Another possibility, of course, is that the interaction of the tidal wave with the bottom topography is still large enough to cause smallscale tidal features in the deep oceans. In that case, the coarse resolution of the TOPEX/POSEIDON altimetry implies that the DEOS96.1 and DW3.2 models would be in error. Notice that although both DEOS96. l and CSR3.0 solve for the mean sea surface (actually CSR3.0 solves for mean sea surface residuals), this procedure does not introduce topographic signals into the tidal solutions because no such signals can be seen in Fig. l(c). Of course this could be expected as there should be no correlation between the time-varying tidal signals and the constant bottom topography. The large-scale feature appearing in Fig. l(c) in the North Atlantic was found to be caused by a discrepancy in the M2 cosine between the DEOS96.1 solution and the other models. Hence, most of it reflects a small error in the M2 cosine of DEOS96.1. The fact that it partially seems to follow the Atlantic Ridge in the North Atlantic must be a coincidence and is unlikely to be caused by topographic errors in the DEOS96.1 model. Otherwise these errors would also have shown up in the other tidal components and at other locations of e.g. the Atlantic Ridge. Most likely, the bathymetry-induced differences were not earlier discovered in e.g. the model comparisons of ANDERSENet al. (1995), which included FES94. l, because this model contained some large-scale errors up to 5 cm magnitude (see ANDERSEN et al., 1995) which obscured these signals. From our comparisons, bathymetry-induced differences between FES95.2.1/CSR3.0 and DEOS96.1/DW3.2 were also discovered in $2 and N2. They occur mainly at the same locations as with M2, i.e. at the Atlantic Ridge, the Walvis Ridge, and at the Mariana Trench. However, compared to the l cm magnitude of these differences in M2, their significance is much less in S> about 0.5 cm, and quite negligible in N:, about 0.2 cm. In the diurnal tides, no bathymetryrelated differences could be detected in K~ and O~, most likely because the semi-diurnal tides are much stronger over the indicated areas of steep topography. Notice that these five tides are the only tides that can be resolved from altimetry. Any smaller tide will drown in the altimeter background noise with a harmonic solution. Even with the response method, results of the smaller tides can be doubted as they are based on the admittance of the stronger tides and thus are not independent. Hence, the comparisons in this paper are restricted to the five largest tides. To try to find out which of the models is responsible for the bathymetry-induced signals, i.e. FES95.2.1/CSR3.0 or DEOS96.1/DW3.2, a test has been performed in which the crossover statistics of TOPEX/POSEIDON and ERS-l were computed with each of the models over the Mid Atlantic Ridge. The crossover data of TOPEX/POSEIDON were derived from GDRs 125130 while for ERS- l, Ocean Products 1 l - 12 were used. The altimetry of neither of these crossover sets are used in the data reduction of any of the tidal models. Using tide gauge harmonic constants as an independent data type was considered less useful as there are few gauges over the Mid Atlantic Ridge, e.g. three in the tide gauge set provided by C. Le Provost (personal communication, 1994). Altimeter crossover residuals were computed applying the M2 tidal solution only in which the bathymetry signals were strongest. The ERS-1 crossovers were selected because of their much better spatial resolution (1 ° ground track spacing at equator) compared to TOPEX/POSEIDON. Unfortunately, the differences in crossover residual rms between the models are so small (less than a millimeter for both TOPEX/POSEIDON and ERS- l ) that they
Errors in recent ocean tide models: possible origin and cause
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do not allow any conclusion to be drawn on the origin of the bathymetric signals. Most likely, therefore, the bathymetry signals are due to both the hydrodynamically consistent and empirical models or they are not large enough to have significant effect on the crossover statistics. In that respect it makes no difference if one uses the FES94.1 model as a priori information, like CSR3.0, or not, like DEOS96.1 and DW3.2.
4. ERRORS IN FES95.2.1 AND CSR3.0 DUE TO BASIN BOUNDARYPROBLEMS IN FES94.1 The use of this Section is to show that FES95.2.1 and CSR3.0 contain errors that can be related to domain boundary problems in the older FES94.1 model. It is known that the basin-wise solution procedure of the FES models introduced some inconsistencies at the basin boundaries of FES94.1 (C. Le Provost, personal communication, 1996). A picture of the grid and the exact locations of the basin boundaries of the FES models can be found in LE PROVOST et al. (1994) and hence need not be reproduced here. As the FES94.1 solution is used as the basis for the assimilation of FES95.2.1 but also as the a priori model to CSR3.0, boundary errors appear in both FES95.2.1 and CSR3.0. The most obvious boundary error is the edge-like structure visible in the CSR3.0 - DW3.2 differences of M 2 in Fig. l(b). This feature, which is along the equator in the Pacific Ocean turning northeast south of Hawaii towards Mexico, has been noted at several occasions, e.g. ANDERSEN(1995) and SHUM et al. (1996). Other features in Fig. l(b) that cannot be explained by bathymetric differences are the straight lines between Africa and Brazil and south of South Africa. These lines respectively follow the FES boundary between the North Atlantic and South Atlantic oceans and the boundary between the South Atlantic and Indian oceans (see LE PROVOST et al., 1994), Features along these boundaries were not noticed by ANDERSEN (1995) and SHUM et al. (1996), probably because they did not apply shading to the differences. In Fig. l(b), it can be seen that the boundary errors in CSR3.0 are associated with errors of about 1.5 cm magnitude in M2 and that they are very local. In the case of FES95,2.1 (see Fig. l(a)), errors of 2 cm magnitude are seen to propagate into the South Atlantic and the Indian oceans due to a mismatch between the M: solutions at the boundary of these basins (C. Le Provost, personal communication, 1996). A close inspection of the ocean tide difference plots in ANDERSEN et al. (1995) reveals large discrepancies between all models and FES94.1 in M, and $2 in this region. Hence, it is likely that the errors in FES95.2.1 south of South Africa are due to an error in FES94.1 which has not been corrected by the assimilation scheme that produced FES95.2.1. The origin of the 1.5-2 cm signal in FES95.2.1 between Brazil and Afiica has not yet been confirmed but as it seems to follow the boundary in the Atlantic, which clearly appears in CSR3.0 (Fig. l(b)), a boundary problem might be a likely cause. It is interesting to note from Fig. l(a,b) that the boundary problem along the equator in the Pacific Ocean in FES94.1 has apparently been solved in FES95.2.1. The problems in the Atlantic Ocean and between the South Atlantic and Indian oceans, however, remain. Also interesting is that the solution procedure of CSR3.0, which basically is a long-wavelength adjustment to FES94.1, seems to detect most of the basin boundary signals which then become part of the adjustment. Hence, the large errors south of South Africa in FES95.2.1 have been removed by the adjustment procedure of CSR3.0 and thus are not present in Fig. l(b). The sharp features along the boundaries, of course, are of too short a wavelength to be detected by the 3 ° x 3° resolution o1" the TOPEX/POSEIDON tracks. Notice that the fact that CSR3.0 is a long-wavelength correction to FES94.1 provides an interesting check to verify that the signals in the Mid Atlantic and between
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the Atlantic and the Indian oceans are indeed due to boundary problems. Differencing CSR3.0 and FES95.2.1 will remove that part of the small-scale differences around the boundaries that both models have in common, i.e. the part that CSR3.0 cannot adjust, What remains are the sharp boundary features superimposed on the long-wavelength adjustments of CSR3.0. This is done in Fig. 3 which shows the M2 differences between FES95.2.1 and CSR3.0 which have a global mean of 0.9 cm for ocean depths larger than 200 m. Now the boundaries become visible as sharp lines which are due to CSR3.0 along the Equator and probably due to both models in the Mid Atlantic and between the Atlantic and the Indian oceans. Besides in M2, errors related to the FES94.1 boundary problems were also detected in some of the other constituents of FES95.2.1 and CSR3.0. In the case of CSR3.0, errors along the Equator and the Atlantic boundary were found of 1 cm in Sz, 0.5 cm in N2, and 0.2 cm in Kr. Notice, that the numbers for N2 and K1 are small compared to other differences in the constituents and that the only reason that they can so easily be detected is their characteristic shape. Hence, apart from M2 and $2, the boundary errors in CSR3.0 may be termed insignificant. In case of FES95.2.1, boundary errors of 1-1.5 cm in $2 and N2 could easily be detected along the boundary between the Atlantic and Indian oceans. In the Atlantic, only $2 showed a signal along the boundary which, however, may also be of a different cause. In the diurnal tides, no boundary-related signals were found. Hence, only boundary-related errors in FES95.2.1 of 2 cm in M2, and 1-1.5 cm in $2 and N2 along the boundary of the Atlantic and Indian oceans can be ascertained. A possible explanation why these errors are only found in the semi-diurnal tides of FES95.2.1 may be that these tides have a much larger amplitude south of South Africa than the diurnal tides. Hence, any discrepancy between the semi-diurnal basin solutions will show up more clearly. In Fig. 4 the difference of the crossover residual rms values of the DEOS96.1 and FES95.2.1 models is given for TOPEX/POSEIDON GDRs 125-130, i.e. DEOS96.1 - FES95.2.1. The crossover residual rms values were computed on a 4 ° × 4 ° grid and interpolated to pixel resolution. Clearly visible is the dark spot south of South Africa corresponding to a value of about 0.8 cm
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Errors in recent ocean tide models: possible origin and cause
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Fig. 4. Global crossover residual rms DEOS96.I-FES95.2.I in cm. Crossover data are from TOPEX/POSEIDON GDRs 125-130. which confirms the boundary error in FES95.2.1. Assuming that the difference in the crossover residual rms south of South Africa is mainly tidal error in FES95.2.1 and that this error can be equally divided over ascending and descending tracks, the value of about 0.8 cm indicates an rrns tidal error in FES95.2.1 of 0.6 cm (1/2 ~/2 × 0.8). It should be mentioned, however, that on a g l o b a l scale, none of the two models was found to perform better than the other. For instance, the global T O P E X / P O S E I D O N crossover residual rms values of the models were found to be 7.0 cm for DEOS96.1 and 7.1 cm for FES95.2.1 while in case of ERS-1 these values were 11.1 cm and 11.0 cm respectively.
5. ERRORS IN ALTIMETRY-DERIVED MODELS DUE TO SEASONAL CHANGE IN ANTARCTIC SEA ICE COVERAGE A detailed investigation of the differences displayed in Fig. 1 revealed values exceeding 2 cm in the Antarctic seas, i.e. from 80 ° west to 180 ° east longitude and below 60 ° south. Investigations into the formal errors of the DEOS96.1 model showed that the standard deviations of the tidal harmonic constants increased to about 10 cm in the Antarctic seas compared to a global average of 2 - 2 . 5 cm. For the seasonal cycles, however, the standard deviations were found to increase to over 30 cm compared to the same global average of 2 - 2 . 5 cm. Hence, the cause of the large differences in Fig. 1 is of seasonal nature and therefore must be the yearly growth and decay of Antarctic sea ice. As an example, Fig. 5 shows the number of TOPEX/POSEIDON altimeter normal points for each cycle in the 3 ° x 3 ° grid cell centered at 1.5 ° longitude and 64.5 ° latitude for the DEOS96.1 model. Clearly visible are the summer periods in the southern hemisphere, e.g. cycles 9 - 2 7 which correspond to the period 11 December 1992-17 June 1993, and the winter periods during which the growth of ice sheets may lead to a total absence of altimeter observations at Antarctic latitudes. Notice that cycles 102-117 coincide with the winter period in the southern hemisphere so that Fig. 5 does not go beyond cycle 101. As no altimeter observations over ice are used in the altimetry-derived models, the number of observations in
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A.J.E. SMITH and O.B. ANDERSEN
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Errors in recent ocean tide models: possible origin and cause
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6. CONCLUSIONS The FES95.2.1/CSR3.0 and DEOS96.1/DW3.2 models show distinct differences in areas of steep topography. These differences have an amplitude of about 1 cm in M2 and 0.5 cm in S~. The cause is either related to the bathymetry model of the FES series for which ETOPO5 is used or to small-scale tidal features that are not picked up by the TOPEX/POSEIDON altimetry. Discontinuities along the basin boundaries of the earlier FES94.1 model cause some very distinct errors in FES95.2.1 and CSR3.0. Large differences are detected between the FES95.2. I model and the altimetry-derived DEOS96.1, DW3.2, and CSR3.0 models south of South Africa. These differences have a vector magnitude of about 2 cm in M2 and 1-1.5 cm in $2 and N~ and translate into a tidal error of approximately 0.6 cm in FES95.2.1 according to the T O P E X / P O S E I D O N crossover statistics. With CSR3.0, part of the inconsistencies along the basin boundaries have been removed by the adjustment procedure. Still, some remnants remain along the Equator in the Pacific and between Brazil and Africa of 1.5 cm in M2 and I cm in $2. These remnants are so local that they will have little effect on the global accuracy of CSR3.0. All altimetry-derived ocean tide models seem to contain errors of about 1 cm r m s near the Antarctic. The reason is the seasonal growth and decay of Antarctic sea ice, which locally leads to an absence of observations during periods as long as half a year. In the Weddell Sea, these errors can exceed 2 cm r m s . On a global scale, the models included in the investigations were found to perform equally well according to the crossover statistics of T O P E X / P O S E I D O N and E R S - I .
7. REFERENCES
ANDERSENO.B. (1995) Global ocean tides from ERS-I and TOPEX/POSEIDON altimetry. Journal (?f Geophysical Research 100(C12), 25249-25259.
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ANDERSEN O.B., WOODWORTHP.L. and FLATHER R.A. (1995) Intercomparison of recent ocean tide models. Journal of Geophysical Research 100(C12), 25261-25282. CARTWRIGHTD.E. and RAY R.D. (1990) Oceanic tides from geosat altimetry. Journal of Geophysical Research 95(C3), 3069-3090. CARTWRIGHT D.E. and RAY R.D. (1991) Energetics of global ocean tides from geosat altimetry. Journal of Geophysical Research 96(C9), 16897-16912. DESAI S.D. and WAHR J.M. (1995) Empirical ocean tide models estimated from TOPEX/POSEIDON altimetry. Journal of Geophysical Research 100(Cl 2), 25205-25228. Fu L.L., CHRISTENSENE.J., YAMARONEC.A., LEFEBVREM., MI~NARDY., DORRERM. and ESCUDIERP. (1994) TOPEX/POSEIDON mission overview. Journal of Geophysical Research 99(C12), 24369-24381. GUZKOWSKA M.A.J., RAPLEY C.G., RIDLEY J.K., CUDLW W., BIRKETT C.M. and SCOTT R.F. (1990) Developments in inland water and land altimetry. Final report, ESA contract CR-783988FFL, University College London, Mullard Space Science Laboratory, Dorking, Surrey, UK. LE PROVOSTC., GENCO M.L., LYARDF., VINCENT P. and CANCEIL P. (1994) Spectroscopy of the world ocean tides from a finite element hydrodynamic model. Journal of Geophysical Research 99(C12), 2477724797. NATIONAL GEOPHYSICALDATA CENTER (1993) Global Relief. CD-ROM, Boulder, Colorado. SCHRAMA E.J.O. and RAY R.D. (1994) A preliminary tidal analysis of TOPEX/POSEIDON altimetry. Journal of Geophysical Research 99(C 12), 24799-24808. SCHWIDERSKI E.W. (1983) Atlas of ocean tidal maps, part I: the semidiurnal principal lunar tide M 2. Marine Geodesy 6(3-4), 219-265. SHUN C.K., WOOOWORTHP.L., ANDERSENO.B., EGBERTG.D., FRANCISO., KING C., KLOSKOS.M., LE PROVOST C., LI X., MOLINESJ.-M., PARKEM.E., RAY R.D., SCHLAXM.G., STAMMERD., TEIRNEYC.C., VINCENT P. and WUNSCrt C.I. (1996) Accuracy assessment of recent ocean tide models. Journal of Geophysical Research 102(C 11 ), 25173-25194. SMITH A.J.E., AMBROSIUS B.A.C., WAKKER K.F., WOODWORTH P.L. and VASSIE J.M. (1996) Comparison between the harmonic and response methods of tidal analysis using TOPEX/POSEIDON altimetry. Journal of Geodesy 71, 695-703. WAGNER C.A., TAI C.K. and KUHN J.M. (1994) Improved M2 ocean tide from TOPEX/POSEIDON and Geosat altimetry. Journal of Geophysical Research 99(C 12), 24853-24865.