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Testing recent global ocean tide models with loading gravimetric data
Pergamon
P|I: S0079-6611 (98)00014-7
Prog. Oceanog. Vol. 40, pp. 369-383, 1997 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0079 6611/98 $32.00
Testing recent global ocean tide models with loading gravimetric data M. LLUBESand P. MAZZEGA UPS/CNRS/CNES, 18 av. E. Belin, 31401 Toulouse CEDEX 4, France Abstract - As soon as altimetric data from Topex/Poseidon satellite were available, several ocean tide models were able to be constructed by different teams in response to the scientific community request. Here we present a complement to a first comparison study made with ten tidal models (LLtmES and MAZZEGA, 1996). We use five more models of which some are updated versions computed using longer time series of altimetric data. The loading tide effects predicted by these models are compared to a data base of 228 gravimetric stations provided by the International Center for Earth Tides (MELCHIOR, 1994). Global statistical tests provide us with the relative performances of the models. Merging the five new models with the ten previous models allows us to make more general conclusions. CSR3.0 (EANES, 1994) remains the best of the fifteen models for the M2 constituent (standard deviation of 0.425/~gal). For the O~ constituent, Schwiderski (Scr~WIDERSrd, 1980) with a standard deviation of 0.298 p~gal, gives the best results. When considering only regional subsets of gravimetric data, no ocean tide model systematically performs better than any other. © 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION Over the last 15 years, the reference ocean tide model was that of Schwiderski built from hydrodynamics and assimilation of tide gauge data. However its tidal predictions, when compared to independent observations, have always shown significant discrepancies. The altimetric data from the Topex/Poseidon satellite mission were eagerly awaited and in 1994, new tide models became available. More than ten models were presented to the scientific community who need to choose the model best suited according to their own use, These new models can bring many benefits to geodesy, oceanography and geophysics (SHUM e t al., 1998). Tide gauge data collected at pelagic or island stations are commonly used to quantify the accuracy of the tide predictions provided by these models (ANDERSEN et al., 1995; SHUM et al., 1998). Another way is to use the harmonic constants deduced from relative gravimeters which, after various corrections, are empirical estimates of the gravimetric loading signal. A data base of 228 gravimetric stations, made available by the ICET (International Center for Earth Tides, Brussels) gives the constants of the semi-diurnal M2 and diurnal Ot constituents. Our study is based on statistical estimates of the differences between the measured gravimetric loading tides and the gravimetric loading tides as predicted from the competing models. These tests allow us to establish a ranking of the models. With regard to our previous study (LLUBES and MAZZEGA, 1996), updated models are tested using longer time-series of altimetric data. W e also present here the results for the "old" AG95.1 (Andersen,1995) and RSC94 (RAY et al., 369
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M. LLUBES AND P. MAZZEGA
1994) models, the new enhanced versions of the Ori-Nao (MATSUMOTOe t al., 1995) and TPXO.3 (EGBERT et al., 1994) models, and a updated version of ZAHEL (1995). 2. USING GRAVIMETRIC DATA A model evaluation may be made with tide gauge data, that is distributed world wide or is located in a limited area where the performance of the ocean tide models needs to be estimated. They bring a first point of view to the accuracy of the altimetric models. However, as some authors assimilate tide gauge data during their model computation, the previous test may be biased. Our comparisons rely on gravimetric records. These data are independent from all the models considered here, thus allowing some interesting confrontations to be made with conclusions drawn from other tests. The ICET data bank (MELCHIOR, 1994) contains the records from several types of gravimeters installed at land-based stations. The tidal harmonic constants are extracted from the observed temporal variations of gravity. Before using them for the computation of the loading tide prediction, the raw data must be corrected using standard models for the direct luni-solar attraction, the body tide, the atmospheric pressure and other environmental effects. The ICET paid attention to selecting only those stations with a sufficient quality level. They provide the name, latitude, longitude, height, distance to the nearest sea coast and observed vector already corrected (see MELCHIOR, 1994, for details) at 289 gravimetric stations for the M2 and O1 tidal constituents. Out of this set we keep the 228 data collected at stations that are more than 10 km from a coastline. Though much care was taken through dedicated experiments, the calibration of the gravimeters is the main source of errors in the data time series (BAKER et al., 1991, 1996; RYDELEK et al., 1991; MELCHIOR, 1995). Instrumental phase-lags and drifts also decrease the performances of the measurements. A modern spring gravimeter with electrostatic feedback has a 0.2 ~gal accuracy (FRANCISand MELCHIOR,1996) for one observed harmonic. MELCHIOR and FRANCIS (1996) completed a detailed study with the gravimetric data bank. They collected 11 ocean models and tested them using the gravimetric data in its original form, as provided by the modellers. We opted for a more original way of comparison, taking into account that the models do not have the same geographical coverages nor the same grid resolutions. MELCHIOR and FRANCIS (1996) can point out more clearly the influence of the Arctic ocean, the Weddel and Ross seas or detect imperfections in the raw version of the tide models. Choosing to compare them on the same coverage/resolution basis will filter out such model definition inhomogeneities. Hence we get a more robust ordering of the model performances. We can also single out the best loading correction, which may be not based on a single ocean model when considering different parts of the world ocean. AGNEW (1995) also tried to quantify the impact of the new altimetric models on the loading computations in the Southern latitudes. He used only one gravimeter station, located in Antarctica, giving a high quality data for loading observations. But he could only have a local view of the different performances of the models. Moreover the results are inherently limited by the accuracy of this single instrument. 3. THE MODELS The models considered for the test computations are purely oceanic tide models. Here we present a study based on five models. Some of them are updated ocean tide models including
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more altimetric data when compared to the previously released versions. These five models complement an earlier comparison study made with ten models (LLUBES and MAZZECA, 1996). We regroup all fifteen models together, in order to give general conclusions. For our computations, we only use the M 2 and Ol tidal constituents. But all the models offer more than these two constituents. The five models we recently consider are the following: AG95.1 (ANDERSEN, 1995) results from a long-wavelength adjustment of the FES94.1 (LE PROVOST et al., 1994) finite element models using altimetric data. They used the first two years of Topex/Poseidon data time series (70 cycles) to constrain the M2 and $2 solutions between 65°S and 65°N and keep FES94.1 unchanged for the outside latitudes. The other tidal constituents, including O~, are directly those of the FES94.1 model. ORI-NAO96 (MATSU~OTO et al., 1995) is a new computation of the previous ORI-NAO model, made with hydrodynamics and altimetric data assimilation. This version uses data of the Tope× cycles 9-120. It covers the whole Earth and has a 0.5 × 0.5 degree resolution. RSC94 (RAY et al., 1994) is based on the cycles 1 to 64 of Topex/Poseidon together with tide gauge data. The model is constructed on a regular 1° × 1° grid and covers the latitudes from 67°S to 67°N. Some semi-enclosed seas are lacking. TPXO. 3 (E~BERT et al., 1994) is an improved version of TPXO.2, now using crossover data from the Topex/Poseidon cycles with JGM-3 orbits. The resolution is always 0.58 ° by 0.70 ° on a band ranging from -80 ° and 70 ° in latitude. It is built with altimetric data assimilation into hydrodynamics. ZAHEL96 (ZAHEL, 1995) is a computation for the eight main tidal constituents, essentially using the previous model of Zahel. This version assimilates a restricted number of pelagic tide gauge data and also Topex/Poseidon data at more than 1400 sites, on a 1° × 1° grid. Table 1 lists the main characteristics of these models along with those of the ten previous models. They are referenced by the name and the letter quoted in the first two columns of the table. When an updated model has not been published, we give the previous paper as reference in the third column. The type, resolution and coverage are also presented in the table.
4. COMPUTATIONOF THE RESIDUALVECTORS In order to compare these models, we need to constrain them to have the same total coverage of the Earth. For this task, the FES94.1 model (LE PROVOST et al., 1994) is chosen as the reference model because of its largest ocean coverage. The zone of the world ocean where we need to use FES94.1 are located mainly in the high latitudes because of the relatively low inclination of the Topex/Poseidon satellite, in some marginal seas and near the coasts. We also redefine the grid of some models to put them on the same 0.5 ° × 0.5 ° regular grid. A bilinear interpolation algorithm is used for that purpose. The gravimetric data provided by the ICET includes 228 stations distributed mainly in Europe, but also in Africa, Asia and South America (Fig. 1). Unfortunately, very few stations are available in North America, Greenland, and Antarctica. The imbalance between the number of stations in the Northern and Southern hemispheres certainly biases our statistical tests. We shall complete the global results with some regional tests. To compute the loading gravity tide at each station, we apply the convolution method described by FARRELL (1972). First, the loading Love numbers are used to define the Green's functions of the response of the Earth when a point mass load is applied on the surface. These functions are tabulated according to the angular distance between the applied load and the obser-
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Table 1. List of the 15 global ocean tidal models used in the study Reference AG95.1 CSR3.0 DW95 FES94.1 FES95.2 OMP2 ORI---NAO O R I - - N A O 96 RSC94 SCHWIDERSKI SR95 TPXO.2 TPXO.3 ZAHEL Z A H E L 96
Authors
A B C D E F G H I J K L M N O
Type
Andersen HN + T/P Eanes T/P Desai and Wahr T/P Le Provost et al. HN Le Provost et al. HN + T/P Mazzega et al. T/P + TG Matsumoto et al. HN + T/P Matsumoto et al. HN + T/P Ray et al. T/P Schwiderski HN + TG Schrama and Ray T/P Egbert et al. HN + T/P Egbert et al. HN + T/P Zahel HN + TG Zahel HN + TG + T/P
Resolution
Coverage
0.5 ° x 0.5 ° Orthotides Orthotides 0.5 ° x 0.5 ° 0.5 ° × 0.5 ° 0.5 ° x 0.5 ° 1° x 1° 0.5 ° x 0.5 ° Orthotides 1° x 1° 0.5 ° x 0.5 ° 0.58 ° x 0.70 ° 0.58 ° x 0.70 ° 1° x 1° 1° x 1°
Med, A, An Med Med Med, A, An Med, A, An Med Med, A, An Med, A, An None A, An None Med, An Med, An A A, An
The models are quoted by their short name (column 1 ) and by a reference letter (column 2). The corresponding authors are listed in the next column. Then main characteristics of the models are given. Column HN is for hydrodynamical model, T/P or T/G for a model resulting from the analysis of Topex/Poseidon altimetry or tide gauge data, HN + TP (or equivalently HN + TG) for a model built by some kind of data assimilation into hydrodynamics. The grid resolution is given in the next column. The last column indicates whether Mediterranean sea (Med), Arctic (A) and peri-Antarctic (An) oceans are covered the original models. Those models lacking some oceanic areas are completed by the FES94.1 model
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Testing recent global ocean tide models with loading gravimetric data
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vation point. Then, the convolution integral between an ocean tide model and the adequate Green's function give the gravity loading prediction at any desired location, say our gravimetric stations. These calculations are done for each of the tide models, to a precision better than 2.5% (FRANCIS and MAZZErA, 1990). Note that during all the computations in this study, we used mass correction coefficients (see LLUBES and MAZZEGA, 1996, for details). Of the 228 stations selected (LLU~ES and MAZZErA, 1996), none are located closer than 10 km from a coast. Finally, subtracting the predicted vector from the observed vector given by the data bank provides us with the residual vector, at every gravimetric station. The observed vector contains two kind of information: the amplitude and the phase which are projected onto an in-phase and an in-quadrature component. The residual vectors are calculated for these two components separately. No correlation appears when they are plotted in a bivariate form. The results for the 228 residuals of the databank, obtained with the CSR3.0 loading predictions, are given for the M2 constituent and for the O1 constituent (Fig. 2).
5. STANDARD DEVIATIONS OF THE RESIDUALS 5.1. Test 1 We compute the standard deviations of the residual vectors for each ocean tide model, which provides us with a natural ranking of the models. In a first test we consider the all of the 228 stations. To illustrate the gaussian-like distribution of the data, histograms of the in-phase and in-quadrature components are drawn for the M2 constituent and for the Ot constituent (Fig. 3). The same plot shows the histogram of the raw data plus the histogram of the residual vectors which are obtained by correcting the data using the CSR3.0 model. The average and standard deviation for these histograms are also given on the diagram. It is apparent that there is a clear improvement in the repartition of the data after applying the loading correction. 5.2. Test 2 We will eliminate those stations whose residual vector is larger than three times the standard deviation obtained in the previous test. In Table 2 we give a list of those stations rejected for each tidal model by this criterion. In this way, eight stations are removed for the M2 constituent, and eight stations are removed for the Ot constituent, thus leaving 220 data for the computation of the standard deviations. 5.3. Standard deviations ( SD ) The results of test 2 is shown in Fig. 4 for the M2 constituent and for the O~ constituent. The fifteen models are presented in order of increasing SD. The results plotted here are averages of the in-phase and in-quadrature SD of the residual vectors. In order to give an easier overall intercomparison, we do not present separately the results for the five new models of the study. For the M 2 constituent the values of the SD range from 0.425 /xgals to 0.479 /xgals. The models seem to be distributed into three groups: one group with SD higher than 0.45 /xgals, one group with four models, including Schwiderski, with SD around 0.446p, gals and one group with the six better models, among those CSR3.0 performs best with a SD of 0.425 /xgals. We can remark that ORI-NAO96 gives values vastly improved compared to its previous version.
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For the O] constituent, the models are much closer to each other. Schwiderski gives the smallest SD (0.298 /xgals), but no model is over 0.32 /xgals. The accuracy reached in the determination of a tidal component is now about 0.2 /zgal for one gravimetric data (FRANCIS and MELCr[IOR, 1996). When gravimeters share the same cali-
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Table 2. Each station in the list presents residuals larger than 3 times the SD of the residual distribution obtained from the 228 gravimetric stations for those models indicated by their reference letter (e.g. Llanrwst has a residual > 3 SD for the FES94.1 (letter D) and RSC94 (letter I) models). Those stations preceded by the # symbol are rejected from the data base for test 2 Station name:
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M2 constituent 106 112 114 225 311 2551 2612 #2613
Llanrwst Taunton Redruth Veurne Grasse Penang Shangai Shangai T
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Memambetsu Nairobi
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Testing recent global ocean tide models with loading gravimetric data
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M2: Trait 2 CSR3.0 m SR95 m RSC94 m ORI_NAO g6 AG95.1 FES95.2 TPXO.2 SCHWIDERSKI TPXO.3
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Fig. 4. Standard deviations (SD) of the residuals, given in p~gals, for each of the 15 tidal models. They are computed for the M2 constituent with a subset of 220 stations (test 2). The same results are also plotted for OL. bration or instrumental operation, the observational data may be biased by a systematic error. The whole set of measurements can not be considered as independent data. However the data in the ICET bank is collected from several different models of gravimeters used by several different research teams. Even if the data are not strictly independent, they do have enough independence to greatly reduce the accuracy threshold, when the 220 data are used in a statistical study. For the M2 constituent, the results of our tests show clearly that the accuracy is good enough to discriminate between the various tidal models. When considering test 1 or test 2 the ordering of the ocean models is the same. Moreover, using tide gauge data, other authors (ANDERSEN et al., 1995; SHUM et al., 1998) concluded almost the same model ranking for M2. For the Ot constituent, the better performances of the Schwiderski's model would suggest that the introduction of hydrodynamics in the tidal modelling is needed for the "weak" diurnal Ot constituent. In the two statistical tests, the ordering is quite similar, tending to prove that we are above the significance level of the SD. But the ranking does not agree with the results of SHUM et al. (1998) and ANDERSEN et al. (1995) based on tide gauges records, which show
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Fig. 5. Standard deviations (SD) of the M2 load residuals from the 15 ocean models, calculated with 101 stations in the North Atlantic zone (test NA), 26 stations in the South Atlantic zone (test SA), 18 stations in the Indian Ocean zone (test IND), 26 stations in the West Pacific zone (test W P A C ) and 22 stations in the East Pacific zone (test EPAC). The results are shown in order of rank.
Testing recent global ocean tide models with loading gravimetric data
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conversely a clear improvement between the recent models and Schwiderski. The main differences in the tidal models are located in shallow water areas. Schwiderski assimilates a high number of coastal tide gauge data. The better fit along the coasts may be sufficient to make Schwiderski slightly better than the other models. Although the discrepancies are so tiny that interpretations must be given very carefully.
6. LOCAL TESTS This preliminary analysis is performed by computing SD from several subsets of the gravimetric stations. Compared to the previous study with only ten models, the zones for the local tests are changed (see Fig. 1). We try to regroup the stations according to their distribution around the main oceans. Five areas, including 194 among the 220 data of test 2 which represents a good coverage, are defined following this idea. One zone covers the North Atlantic (NA). It includes 100 gravimetric stations, located mainly
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01: T W I NA OMP2 SR95 ORI NAO 96 ORI NAO DW95 ZAHEL 96
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Fig. 6. Standard deviations (SD) of the Oj load residuals from the 15 ocean models, calculated with 99 stations in the North Atlantic zone (test NA), 27 stations in the South Atlantic zone (test SA), 18 stations in the Indian Ocean zone (test IND), 27 stations in the West Pacific zone (test WPAC) and 23 stations in the East Pacific zone (test EPAC). The results are shown in order of rank. in Europe, plus 9 in eastern North America, 1 in Greenland (eliminated for the O] constituent tests), 1 in Iceland and 1 in Africa. The zone over the South Atlantic (SA) includes 27 data distributed along the western side of Africa and the east of South America. The next zone circles the Indian ocean (IND). With only 18 stations this group has a very sparse coverage and most of the data are in East Africa. Around the West Pacific ocean ( W P A C ) we choose 27 stations, mostly in Japan, Korea and China. Very few are located South of this zone, Australia has only 2 to 3 stations, and there is one at Suva. One can see that five among the eight stations eliminated in test 2 are situated inside or just near the border of this area. This result may be due to either the systematic regional inaccuracy of most tidal models, or from less precision associated with these particular gravimetric stations.
Testing recent global ocean tide models with loading gravimetric data
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M. LLUBESAND P. MAZZEGA
Finally in the East Pacific zone ( E P A C ) 22 stations are selected in the western part of South America, central America and western North America. For the O~ constituent, the Uwekahuna station (Hawaii) is added. The results of the tests for the different zones are plotted in Fig. 5 for M2 and Fig. 6 for Ol. The Indian area gives the higher SD, but with only 18 data it is more sensitive to one bad residual. The model ranking of the West Pacific test is quite different from the test using the global statistics. This result is characteristic of larger discrepancies between the ocean tide models in this zone. Some further improvements in the Indian and West Pacific areas may be expected. Regarding the new versions of some models, we see that O R I - N A O 9 6 is improved as it is clearly better than the former version in four zones, for the M2 tidal constituent. This is generally not the case for the FES95.2, Zahe196 or TPXO.3 models. For the Oi constituent, the results are sometimes so close that they may not express any real tendency in the model ranking.
7. CONCLUSION In this paper, we present an additional group of models that complement a previous study that considered the performance of ten ocean tide models (LLUSES and MAZZEGA, 1996). Out of the 15 models used during the two studies CSR3.0 (EANES, 1994) remains the best model for the M2 constituent (with a SD of 0.425~gal) and it gives satisfactory residuals for O1 (with a SD of 0.305 /xgal). For the O1 constituent, the best result is given by the Schwiderski model (SD of 0.298 /xgal). The interpretation of the tests for the O1 constituent is less reliable due to the lower signal to noise ratio. W e are certainly very close to the significance threshold. The tests made on the local zones clearly show that there is no tidal model which is better than the rest in all o f the oceans. Regarding the results, one can still notice some differences for the loading effects. These may be attributed to the perfectible calibration of the gravimeters. W e can also expect further improvements in future ocean tide models.
8. ACKNOWLEDGEMENTS We wish to thank Prof. Melchior and Dr. O. Francis who kindly gave us the ICET gravimetric data. Most of the tidal models were available on ftp sites thanks to the modellers. We are grateful to Dr. O. Andersen, Dr. G. Egbert, Dr. K. Matsumoto and Prof. W. Zahel for their kindly and readily allowing us to use their models. The original manuscript has been significantly improved with the help of two anonymous referees.
9. REFERENCES AGNEW D.C. (1995) Ocean-load tides at the South Pole: a validation of recent ocean-tide models. Geophysical Research Letters 22, 3063-3066. ANDERSENO.B. (1995) Global ocean tides from ERS-1 and TOPEX/POSEIDON altimetry. Journal of Geophysical Research 100(C12), 25249-25259. ANDERSENO.B., WOODWORTHP. and FLATHERR.A. (1995) Intercomparison of recent ocean tide models. Journal of Geophysical Research 100(C12), 25261-25282. BAKERT.F., CURTISD.J. and DODSONA.H. (1996) A new test of Earth tide models in central Europe. Geophysical Research Letters 23, 3559-3562.
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BAKER T.F., EDGE R.J. and JEFFRIES G. (1991) Tidal gravity and ocean tide loading in Europe. Geophysical Journal International 107, 1-11. LANES R.J. (1994) Diurnal and semidiurnal tides from TOPEX/POSEIDON altimetry (abstract). LOS Transactions AGU 75(16), 1-108. EGBERT G.D., BENNETT A. and FOREMAN M. (1994) TOPEX/POSEIDON tides estimated using a global inverse model. Journal of Geophysical Research 99(C12), 24821-24852. FARRELL W.E. (1972) Deformation of the Earth by surface loads. Review of Geophysics 10, 761-797. FRANCIS O. and MAZZEGA P. (1990) Global charts of ocean tide loading effects. Journal of Geophysical Research 95(11), ll411-11424. FRANCIS O. and MELCHIOR P. (1996) Tidal loading in south western Europe: a test area. Geophysical Research Letters 23(17), 2251-2254. LE PRovost C., GENCO M.L., LYARD F., VINCENT P. and CANCEIL P. (1994) Spectroscopy of the world ocean tides from a finite element hydrodynamic model. Journal of Geophysical Research 99(CI2), 2477724797. LLUBES M. and MAZZEGA P. (1996) The ocean tide gravimetric loading reconsidered. Geophysical Research Letters 23(12), 1481-1484. MATSUMOTO K., OOE M., SATO T. and SEGAWA J. (1995) Ocean tide model obtained from TOPEX/POSEIDON altimetry data. Journal of Geophysical Research 100(C12), 25319-25330. MAZZEGA P., BERGE M. and FRANCIS O. (1994) TOPEX/POSEIDON tides: the OMP2 atlas (abstract). LOS Transactions AGU 75(44), 1-61. MELCHIOR P. (1994) A new data bank for tidal gravity measurements (DB 92). Physics of the Earth and Planetary Interiors 82, 125-155. MELCHIOR P. (1995) A continuing discussion about the correlation of tidal gravity anomalies and heat flow densities. Physics of the Earth and Planetary Interiors 88, 223-256. MELCHIOR P. and FRANCIS O. (1996) Comparison of recent ocean tide models using ground based tidal gravity measurements. Marine Geodesy 19, 291-330. RAY R.D., SANCHEZ B. and CARTWRIGHT D.E. (1994) Some extensions to the response method of tidal analysis applied to TOPEX/POSEIDON altimetry (abstract). LOS Transactions AGU 75(16), 1-108. RYDELEK P.A., ZORN W. and HINDERER J. (1991) On tidal gravity, heat flow and lateral heterogeneities. Physics of the Earth and Planetary Interiors 68, 215-229. SCHWIDERSKI E. (1980) On charting global ocean tides. Revue of Geophysics and Space Physics 18, 243-268. SHUM C.K., WOODWORTH P.L., ANDERSEN O.B., EGBERT G.D., FRANCIS O., KING C., KLOSKO S.M., LE PROVOST C., LI X., MOLINES J.-M., PARKE M.E., RAY R.D., SCHLAX M.G., STAMMERD., TEIRNEY C.C., VINCEN1~ P. and WUNSCH C.I. (1998) Accuracy assessment of recent ocean tide models. Journal c~fGeophysical Research 102(Cl l ), 25173-25194. ZAHEL W. (1995) Assimilating ocean tide determined data into global tidal models. Journal of Marine Systems 6. 3-13.
P|I: S0079-6611 (98)00014-7
Prog. Oceanog. Vol. 40, pp. 369-383, 1997 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0079 6611/98 $32.00
Testing recent global ocean tide models with loading gravimetric data M. LLUBESand P. MAZZEGA UPS/CNRS/CNES, 18 av. E. Belin, 31401 Toulouse CEDEX 4, France Abstract - As soon as altimetric data from Topex/Poseidon satellite were available, several ocean tide models were able to be constructed by different teams in response to the scientific community request. Here we present a complement to a first comparison study made with ten tidal models (LLtmES and MAZZEGA, 1996). We use five more models of which some are updated versions computed using longer time series of altimetric data. The loading tide effects predicted by these models are compared to a data base of 228 gravimetric stations provided by the International Center for Earth Tides (MELCHIOR, 1994). Global statistical tests provide us with the relative performances of the models. Merging the five new models with the ten previous models allows us to make more general conclusions. CSR3.0 (EANES, 1994) remains the best of the fifteen models for the M2 constituent (standard deviation of 0.425/~gal). For the O~ constituent, Schwiderski (Scr~WIDERSrd, 1980) with a standard deviation of 0.298 p~gal, gives the best results. When considering only regional subsets of gravimetric data, no ocean tide model systematically performs better than any other. © 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION Over the last 15 years, the reference ocean tide model was that of Schwiderski built from hydrodynamics and assimilation of tide gauge data. However its tidal predictions, when compared to independent observations, have always shown significant discrepancies. The altimetric data from the Topex/Poseidon satellite mission were eagerly awaited and in 1994, new tide models became available. More than ten models were presented to the scientific community who need to choose the model best suited according to their own use, These new models can bring many benefits to geodesy, oceanography and geophysics (SHUM e t al., 1998). Tide gauge data collected at pelagic or island stations are commonly used to quantify the accuracy of the tide predictions provided by these models (ANDERSEN et al., 1995; SHUM et al., 1998). Another way is to use the harmonic constants deduced from relative gravimeters which, after various corrections, are empirical estimates of the gravimetric loading signal. A data base of 228 gravimetric stations, made available by the ICET (International Center for Earth Tides, Brussels) gives the constants of the semi-diurnal M2 and diurnal Ot constituents. Our study is based on statistical estimates of the differences between the measured gravimetric loading tides and the gravimetric loading tides as predicted from the competing models. These tests allow us to establish a ranking of the models. With regard to our previous study (LLUBES and MAZZEGA, 1996), updated models are tested using longer time-series of altimetric data. W e also present here the results for the "old" AG95.1 (Andersen,1995) and RSC94 (RAY et al., 369
370
M. LLUBES AND P. MAZZEGA
1994) models, the new enhanced versions of the Ori-Nao (MATSUMOTOe t al., 1995) and TPXO.3 (EGBERT et al., 1994) models, and a updated version of ZAHEL (1995). 2. USING GRAVIMETRIC DATA A model evaluation may be made with tide gauge data, that is distributed world wide or is located in a limited area where the performance of the ocean tide models needs to be estimated. They bring a first point of view to the accuracy of the altimetric models. However, as some authors assimilate tide gauge data during their model computation, the previous test may be biased. Our comparisons rely on gravimetric records. These data are independent from all the models considered here, thus allowing some interesting confrontations to be made with conclusions drawn from other tests. The ICET data bank (MELCHIOR, 1994) contains the records from several types of gravimeters installed at land-based stations. The tidal harmonic constants are extracted from the observed temporal variations of gravity. Before using them for the computation of the loading tide prediction, the raw data must be corrected using standard models for the direct luni-solar attraction, the body tide, the atmospheric pressure and other environmental effects. The ICET paid attention to selecting only those stations with a sufficient quality level. They provide the name, latitude, longitude, height, distance to the nearest sea coast and observed vector already corrected (see MELCHIOR, 1994, for details) at 289 gravimetric stations for the M2 and O1 tidal constituents. Out of this set we keep the 228 data collected at stations that are more than 10 km from a coastline. Though much care was taken through dedicated experiments, the calibration of the gravimeters is the main source of errors in the data time series (BAKER et al., 1991, 1996; RYDELEK et al., 1991; MELCHIOR, 1995). Instrumental phase-lags and drifts also decrease the performances of the measurements. A modern spring gravimeter with electrostatic feedback has a 0.2 ~gal accuracy (FRANCISand MELCHIOR,1996) for one observed harmonic. MELCHIOR and FRANCIS (1996) completed a detailed study with the gravimetric data bank. They collected 11 ocean models and tested them using the gravimetric data in its original form, as provided by the modellers. We opted for a more original way of comparison, taking into account that the models do not have the same geographical coverages nor the same grid resolutions. MELCHIOR and FRANCIS (1996) can point out more clearly the influence of the Arctic ocean, the Weddel and Ross seas or detect imperfections in the raw version of the tide models. Choosing to compare them on the same coverage/resolution basis will filter out such model definition inhomogeneities. Hence we get a more robust ordering of the model performances. We can also single out the best loading correction, which may be not based on a single ocean model when considering different parts of the world ocean. AGNEW (1995) also tried to quantify the impact of the new altimetric models on the loading computations in the Southern latitudes. He used only one gravimeter station, located in Antarctica, giving a high quality data for loading observations. But he could only have a local view of the different performances of the models. Moreover the results are inherently limited by the accuracy of this single instrument. 3. THE MODELS The models considered for the test computations are purely oceanic tide models. Here we present a study based on five models. Some of them are updated ocean tide models including
Testing recent global ocean tide models with loading gravimetric data
371
more altimetric data when compared to the previously released versions. These five models complement an earlier comparison study made with ten models (LLUBES and MAZZECA, 1996). We regroup all fifteen models together, in order to give general conclusions. For our computations, we only use the M 2 and Ol tidal constituents. But all the models offer more than these two constituents. The five models we recently consider are the following: AG95.1 (ANDERSEN, 1995) results from a long-wavelength adjustment of the FES94.1 (LE PROVOST et al., 1994) finite element models using altimetric data. They used the first two years of Topex/Poseidon data time series (70 cycles) to constrain the M2 and $2 solutions between 65°S and 65°N and keep FES94.1 unchanged for the outside latitudes. The other tidal constituents, including O~, are directly those of the FES94.1 model. ORI-NAO96 (MATSU~OTO et al., 1995) is a new computation of the previous ORI-NAO model, made with hydrodynamics and altimetric data assimilation. This version uses data of the Tope× cycles 9-120. It covers the whole Earth and has a 0.5 × 0.5 degree resolution. RSC94 (RAY et al., 1994) is based on the cycles 1 to 64 of Topex/Poseidon together with tide gauge data. The model is constructed on a regular 1° × 1° grid and covers the latitudes from 67°S to 67°N. Some semi-enclosed seas are lacking. TPXO. 3 (E~BERT et al., 1994) is an improved version of TPXO.2, now using crossover data from the Topex/Poseidon cycles with JGM-3 orbits. The resolution is always 0.58 ° by 0.70 ° on a band ranging from -80 ° and 70 ° in latitude. It is built with altimetric data assimilation into hydrodynamics. ZAHEL96 (ZAHEL, 1995) is a computation for the eight main tidal constituents, essentially using the previous model of Zahel. This version assimilates a restricted number of pelagic tide gauge data and also Topex/Poseidon data at more than 1400 sites, on a 1° × 1° grid. Table 1 lists the main characteristics of these models along with those of the ten previous models. They are referenced by the name and the letter quoted in the first two columns of the table. When an updated model has not been published, we give the previous paper as reference in the third column. The type, resolution and coverage are also presented in the table.
4. COMPUTATIONOF THE RESIDUALVECTORS In order to compare these models, we need to constrain them to have the same total coverage of the Earth. For this task, the FES94.1 model (LE PROVOST et al., 1994) is chosen as the reference model because of its largest ocean coverage. The zone of the world ocean where we need to use FES94.1 are located mainly in the high latitudes because of the relatively low inclination of the Topex/Poseidon satellite, in some marginal seas and near the coasts. We also redefine the grid of some models to put them on the same 0.5 ° × 0.5 ° regular grid. A bilinear interpolation algorithm is used for that purpose. The gravimetric data provided by the ICET includes 228 stations distributed mainly in Europe, but also in Africa, Asia and South America (Fig. 1). Unfortunately, very few stations are available in North America, Greenland, and Antarctica. The imbalance between the number of stations in the Northern and Southern hemispheres certainly biases our statistical tests. We shall complete the global results with some regional tests. To compute the loading gravity tide at each station, we apply the convolution method described by FARRELL (1972). First, the loading Love numbers are used to define the Green's functions of the response of the Earth when a point mass load is applied on the surface. These functions are tabulated according to the angular distance between the applied load and the obser-
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Table 1. List of the 15 global ocean tidal models used in the study Reference AG95.1 CSR3.0 DW95 FES94.1 FES95.2 OMP2 ORI---NAO O R I - - N A O 96 RSC94 SCHWIDERSKI SR95 TPXO.2 TPXO.3 ZAHEL Z A H E L 96
Authors
A B C D E F G H I J K L M N O
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Andersen HN + T/P Eanes T/P Desai and Wahr T/P Le Provost et al. HN Le Provost et al. HN + T/P Mazzega et al. T/P + TG Matsumoto et al. HN + T/P Matsumoto et al. HN + T/P Ray et al. T/P Schwiderski HN + TG Schrama and Ray T/P Egbert et al. HN + T/P Egbert et al. HN + T/P Zahel HN + TG Zahel HN + TG + T/P
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0.5 ° x 0.5 ° Orthotides Orthotides 0.5 ° x 0.5 ° 0.5 ° × 0.5 ° 0.5 ° x 0.5 ° 1° x 1° 0.5 ° x 0.5 ° Orthotides 1° x 1° 0.5 ° x 0.5 ° 0.58 ° x 0.70 ° 0.58 ° x 0.70 ° 1° x 1° 1° x 1°
Med, A, An Med Med Med, A, An Med, A, An Med Med, A, An Med, A, An None A, An None Med, An Med, An A A, An
The models are quoted by their short name (column 1 ) and by a reference letter (column 2). The corresponding authors are listed in the next column. Then main characteristics of the models are given. Column HN is for hydrodynamical model, T/P or T/G for a model resulting from the analysis of Topex/Poseidon altimetry or tide gauge data, HN + TP (or equivalently HN + TG) for a model built by some kind of data assimilation into hydrodynamics. The grid resolution is given in the next column. The last column indicates whether Mediterranean sea (Med), Arctic (A) and peri-Antarctic (An) oceans are covered the original models. Those models lacking some oceanic areas are completed by the FES94.1 model
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vation point. Then, the convolution integral between an ocean tide model and the adequate Green's function give the gravity loading prediction at any desired location, say our gravimetric stations. These calculations are done for each of the tide models, to a precision better than 2.5% (FRANCIS and MAZZErA, 1990). Note that during all the computations in this study, we used mass correction coefficients (see LLUBES and MAZZEGA, 1996, for details). Of the 228 stations selected (LLU~ES and MAZZErA, 1996), none are located closer than 10 km from a coast. Finally, subtracting the predicted vector from the observed vector given by the data bank provides us with the residual vector, at every gravimetric station. The observed vector contains two kind of information: the amplitude and the phase which are projected onto an in-phase and an in-quadrature component. The residual vectors are calculated for these two components separately. No correlation appears when they are plotted in a bivariate form. The results for the 228 residuals of the databank, obtained with the CSR3.0 loading predictions, are given for the M2 constituent and for the O1 constituent (Fig. 2).
5. STANDARD DEVIATIONS OF THE RESIDUALS 5.1. Test 1 We compute the standard deviations of the residual vectors for each ocean tide model, which provides us with a natural ranking of the models. In a first test we consider the all of the 228 stations. To illustrate the gaussian-like distribution of the data, histograms of the in-phase and in-quadrature components are drawn for the M2 constituent and for the Ot constituent (Fig. 3). The same plot shows the histogram of the raw data plus the histogram of the residual vectors which are obtained by correcting the data using the CSR3.0 model. The average and standard deviation for these histograms are also given on the diagram. It is apparent that there is a clear improvement in the repartition of the data after applying the loading correction. 5.2. Test 2 We will eliminate those stations whose residual vector is larger than three times the standard deviation obtained in the previous test. In Table 2 we give a list of those stations rejected for each tidal model by this criterion. In this way, eight stations are removed for the M2 constituent, and eight stations are removed for the Ot constituent, thus leaving 220 data for the computation of the standard deviations. 5.3. Standard deviations ( SD ) The results of test 2 is shown in Fig. 4 for the M2 constituent and for the O~ constituent. The fifteen models are presented in order of increasing SD. The results plotted here are averages of the in-phase and in-quadrature SD of the residual vectors. In order to give an easier overall intercomparison, we do not present separately the results for the five new models of the study. For the M 2 constituent the values of the SD range from 0.425 /xgals to 0.479 /xgals. The models seem to be distributed into three groups: one group with SD higher than 0.45 /xgals, one group with four models, including Schwiderski, with SD around 0.446p, gals and one group with the six better models, among those CSR3.0 performs best with a SD of 0.425 /xgals. We can remark that ORI-NAO96 gives values vastly improved compared to its previous version.
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For the O] constituent, the models are much closer to each other. Schwiderski gives the smallest SD (0.298 /xgals), but no model is over 0.32 /xgals. The accuracy reached in the determination of a tidal component is now about 0.2 /zgal for one gravimetric data (FRANCIS and MELCr[IOR, 1996). When gravimeters share the same cali-
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Table 2. Each station in the list presents residuals larger than 3 times the SD of the residual distribution obtained from the 228 gravimetric stations for those models indicated by their reference letter (e.g. Llanrwst has a residual > 3 SD for the FES94.1 (letter D) and RSC94 (letter I) models). Those stations preceded by the # symbol are rejected from the data base for test 2 Station name:
Longitude
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M2 constituent 106 112 114 225 311 2551 2612 #2613
Llanrwst Taunton Redruth Veurne Grasse Penang Shangai Shangai T
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53.1 51.0 50.2 51.1 43.8 5.4 31.1 31.1
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Charters Towers Hamilton Taupo
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Testing recent global ocean tide models with loading gravimetric data
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M2: Trait 2 CSR3.0 m SR95 m RSC94 m ORI_NAO g6 AG95.1 FES95.2 TPXO.2 SCHWIDERSKI TPXO.3
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Fig. 4. Standard deviations (SD) of the residuals, given in p~gals, for each of the 15 tidal models. They are computed for the M2 constituent with a subset of 220 stations (test 2). The same results are also plotted for OL. bration or instrumental operation, the observational data may be biased by a systematic error. The whole set of measurements can not be considered as independent data. However the data in the ICET bank is collected from several different models of gravimeters used by several different research teams. Even if the data are not strictly independent, they do have enough independence to greatly reduce the accuracy threshold, when the 220 data are used in a statistical study. For the M2 constituent, the results of our tests show clearly that the accuracy is good enough to discriminate between the various tidal models. When considering test 1 or test 2 the ordering of the ocean models is the same. Moreover, using tide gauge data, other authors (ANDERSEN et al., 1995; SHUM et al., 1998) concluded almost the same model ranking for M2. For the Ot constituent, the better performances of the Schwiderski's model would suggest that the introduction of hydrodynamics in the tidal modelling is needed for the "weak" diurnal Ot constituent. In the two statistical tests, the ordering is quite similar, tending to prove that we are above the significance level of the SD. But the ranking does not agree with the results of SHUM et al. (1998) and ANDERSEN et al. (1995) based on tide gauges records, which show
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Fig. 5. Standard deviations (SD) of the M2 load residuals from the 15 ocean models, calculated with 101 stations in the North Atlantic zone (test NA), 26 stations in the South Atlantic zone (test SA), 18 stations in the Indian Ocean zone (test IND), 26 stations in the West Pacific zone (test W P A C ) and 22 stations in the East Pacific zone (test EPAC). The results are shown in order of rank.
Testing recent global ocean tide models with loading gravimetric data
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Fig. 5. Continued
conversely a clear improvement between the recent models and Schwiderski. The main differences in the tidal models are located in shallow water areas. Schwiderski assimilates a high number of coastal tide gauge data. The better fit along the coasts may be sufficient to make Schwiderski slightly better than the other models. Although the discrepancies are so tiny that interpretations must be given very carefully.
6. LOCAL TESTS This preliminary analysis is performed by computing SD from several subsets of the gravimetric stations. Compared to the previous study with only ten models, the zones for the local tests are changed (see Fig. 1). We try to regroup the stations according to their distribution around the main oceans. Five areas, including 194 among the 220 data of test 2 which represents a good coverage, are defined following this idea. One zone covers the North Atlantic (NA). It includes 100 gravimetric stations, located mainly
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Fig. 6. Standard deviations (SD) of the Oj load residuals from the 15 ocean models, calculated with 99 stations in the North Atlantic zone (test NA), 27 stations in the South Atlantic zone (test SA), 18 stations in the Indian Ocean zone (test IND), 27 stations in the West Pacific zone (test WPAC) and 23 stations in the East Pacific zone (test EPAC). The results are shown in order of rank. in Europe, plus 9 in eastern North America, 1 in Greenland (eliminated for the O] constituent tests), 1 in Iceland and 1 in Africa. The zone over the South Atlantic (SA) includes 27 data distributed along the western side of Africa and the east of South America. The next zone circles the Indian ocean (IND). With only 18 stations this group has a very sparse coverage and most of the data are in East Africa. Around the West Pacific ocean ( W P A C ) we choose 27 stations, mostly in Japan, Korea and China. Very few are located South of this zone, Australia has only 2 to 3 stations, and there is one at Suva. One can see that five among the eight stations eliminated in test 2 are situated inside or just near the border of this area. This result may be due to either the systematic regional inaccuracy of most tidal models, or from less precision associated with these particular gravimetric stations.
Testing recent global ocean tide models with loading gravimetric data
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Fig. 6. Continued
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Finally in the East Pacific zone ( E P A C ) 22 stations are selected in the western part of South America, central America and western North America. For the O~ constituent, the Uwekahuna station (Hawaii) is added. The results of the tests for the different zones are plotted in Fig. 5 for M2 and Fig. 6 for Ol. The Indian area gives the higher SD, but with only 18 data it is more sensitive to one bad residual. The model ranking of the West Pacific test is quite different from the test using the global statistics. This result is characteristic of larger discrepancies between the ocean tide models in this zone. Some further improvements in the Indian and West Pacific areas may be expected. Regarding the new versions of some models, we see that O R I - N A O 9 6 is improved as it is clearly better than the former version in four zones, for the M2 tidal constituent. This is generally not the case for the FES95.2, Zahe196 or TPXO.3 models. For the Oi constituent, the results are sometimes so close that they may not express any real tendency in the model ranking.
7. CONCLUSION In this paper, we present an additional group of models that complement a previous study that considered the performance of ten ocean tide models (LLUSES and MAZZEGA, 1996). Out of the 15 models used during the two studies CSR3.0 (EANES, 1994) remains the best model for the M2 constituent (with a SD of 0.425~gal) and it gives satisfactory residuals for O1 (with a SD of 0.305 /xgal). For the O1 constituent, the best result is given by the Schwiderski model (SD of 0.298 /xgal). The interpretation of the tests for the O1 constituent is less reliable due to the lower signal to noise ratio. W e are certainly very close to the significance threshold. The tests made on the local zones clearly show that there is no tidal model which is better than the rest in all o f the oceans. Regarding the results, one can still notice some differences for the loading effects. These may be attributed to the perfectible calibration of the gravimeters. W e can also expect further improvements in future ocean tide models.
8. ACKNOWLEDGEMENTS We wish to thank Prof. Melchior and Dr. O. Francis who kindly gave us the ICET gravimetric data. Most of the tidal models were available on ftp sites thanks to the modellers. We are grateful to Dr. O. Andersen, Dr. G. Egbert, Dr. K. Matsumoto and Prof. W. Zahel for their kindly and readily allowing us to use their models. The original manuscript has been significantly improved with the help of two anonymous referees.
9. REFERENCES AGNEW D.C. (1995) Ocean-load tides at the South Pole: a validation of recent ocean-tide models. Geophysical Research Letters 22, 3063-3066. ANDERSENO.B. (1995) Global ocean tides from ERS-1 and TOPEX/POSEIDON altimetry. Journal of Geophysical Research 100(C12), 25249-25259. ANDERSENO.B., WOODWORTHP. and FLATHERR.A. (1995) Intercomparison of recent ocean tide models. Journal of Geophysical Research 100(C12), 25261-25282. BAKERT.F., CURTISD.J. and DODSONA.H. (1996) A new test of Earth tide models in central Europe. Geophysical Research Letters 23, 3559-3562.
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