From the raw altimeter measurement until the corrected SSH above a reference ellipsoid or the corresponding sea level anomaly (SLA) with respect to a known mean sea surface (MSS), the most common variable used in sea level studies, a number of corrections have to be applied to data, to account for the interactions with the atmosphere and with the sea surface.

The corrected sea surface height is computed as follows: (1) SSH = r − ( h + gcor )where h is the altimeter range corrected for instrumental effects, r is the satellite height above the TOPEX reference ellipsoid and gcor includes all geophysical corrections.

Using the NASA JGM3 orbit, [24] and the set of geophysical corrections summarised in Table 1 a reference data set, named DataSet1, has been computed. These corrections include: ionospheric correction, dry and wet tropospheric correction (wet_cor), sea state bias (SSB), earth and ocean tides, tidal loading, pole tide. When necessary, the corrections provided on the GDR-Ms were updated to introduce state-of-the-art models as described below.

The cycle dependent drift effect in the TOPEX Microwave Radiometer (TMR) wet tropospheric correction, including the yaw state correction to the TMR measured Brightness Temperatures (TBs), has been modelled according to the Scharroo model, [26].

The SSB correction was derived from the Chambers model [27]; a residual SSB correction of +2 mm was applied to cycles 236 and greater (TOPEX Side-B). This bias is very similar to the +3 mm value referred in [28]. The determination of this bias is explained later in section 4.1.

The TOPEX dual-frequency ionospheric correction has been smoothed using a second order Butterworth filter, [29] with a cut-off period of 20 seconds (λ≈120 km). The result of this filtering is similar to the procedure referred by [30], who used a 20-second averaging window, with the advantage that the effects of discontinuities introduced by data gaps or land regions are minimized.

The ocean tide model used was the NAO99b model, [31], which has been found to give smaller residuals in coastal regions, [32].

In the computation of DataSet1 no inverse barometer correction has been applied.

Sea level anomalies (SLA) have been derived from the corrected sea surface height values by removing the mean sea surface from the GSFC00.1 model, [33]. From the computed SLA data set only the TOPEX measurements for which the rain and the radiometer flag were not set have been selected.

For the purpose of analysing the effect of different models in the geophysical corrections and different orbit solutions other data sets have been derived using the same methodology. In each data set all geophysical corrections applied to DataSet1 have been kept except one, as described below. This information is summarised in Table 2.

DataSet1 - all corrections according to Table 1.

For each of the above described data sets, the along track sea level anomalies for each TOPEX cycle have been gridded using a 0.5° spacing. For each non empty 0.5° block the along-track SLA values are replaced by the median value at the centre of the block.

Time series of spatially aggregated sea level anomalies for the Tropical, Sub-Tropical and Sub-Arctic Atlantic regions are obtained through spatial averaging of SLA values for each cycle with equi-area weighting by a cosine function of latitude to account for the relative geographical area represented by each block. As mentioned before, only TOPEX measurements have been used. The 30 full Poseidon cycles were discarded and replaced by interpolated values; missing TOPEX cycles (118, 432, 433) were also replaced by interpolated values from adjacent cycles. Considering the 7 datasets and the three mentioned regions a total number of 21 time series of SLA values were generated. Figure 2 shows the sea level anomalies for DataSet1 in each of the three regions.

Flexible and robust non-parametric estimates of sea level inter-annual variability are obtained using STL, acronym for “Seasonal-Trend decomposition based on Loess”, [38]. Details concerning the non-parametric estimate of long-term sea level variability are given in [39]. STL decomposition is carried out for each time series of sea level anomalies using R implementation [40].

Sea level linear trend is finally estimated for each time series, using an ordinary least squares (OLS) adjustment to each interannual curve, [41]. For DataSet1, Figures 3a, 3b and 3c show the original SLA values, the interannual variations and the linear trend for the T, ST and SA regions respectively.

The development of SSB models for the various altimeter missions has been a subject of major research. The sea state bias correction is composed of several parts: the electromagnetic (EM), the skewness and the tracker biases. The EM is theoretically only a function of the frequency of the radar altimeter EM pulse. The skewness and the tracker bias are due to inaccurate tracking of the waveform by the on-board tracker and also depend on the sea state and on the altimeter frequency [1]. Since all components depend on the sea state and on the altimeter frequency, in practice the combined SSB effect for each altimeter is estimated as a function of the sea state parameters significant wave height (SWH) and wind speed (WS).

Most of the methods which have been used for determining the SSB correction fall in two main categories: parametric and non-parametric methods. The SSB is estimated by fitting an empirical model on altimeter-derived SSH differences, either at crossover points or along collinear tracks.

Parametric methods often assume an empirical relation between SSB, SWH and WS of the form (6), (6) SSH = SWH ( a 0 + a 1 SWH + a 2 W S + a 3 W S 2 )where SSB is the sea state bias in metres, SWH is the significant wave height in m and WS is the wind speed in m/s. The model parameters for each altimeter are estimated by a least squares fit of the model to sensor data for a given period. Gaspar [42,43] and Chambers [27] are examples of parametric SSB models developed for TOPEX using 4 parameters according to (6).

Non-parametric techniques have been developed for estimating the SSB from altimeter data themselves and applied to most satellite missions including TOPEX [44-46]. Each model is a look-up table expressing the two-dimensional SSB dependence on the SWH and WS fields. The non-parametric SSB model used in this study for the generation of DataSet2 is the most recent TOPEX solution obtained by Labroue et al. (Labroue, personal communication).

Scharroo et al. have been developing the so-called hybrib methods [47,48], consisting of an extension of the direct estimation method of the SSB from residual SSHs [49] through a parametric fitting followed by a successive smoothing of the residuals. The hybrid method essentially produces a non-parametric SSB model in the form of a smooth grid in the two-dimensional space determined by two sea state parameters.

The Chambers SSB [27] was developed as a correction model to the Gaspar BM4 model, [43] provided in the GDR-Ms which has been derived from TOPEX A data only and was found to be inadequate when applied to the entire TOPEX mission. Our results for the North Atlantic (not shown here) reveal that the difference between these two models has a strong annual latitude dependent signal, being almost an order of magnitude higher for TOPEX B than for TOPEX A. The effect on regional sea level trend of the difference between these models (with no TOPEX A/B bias applied) is 0.6 mm/yr in the Tropical region, 0.9 mm/yr in the Sub-Tropical region and 1.3 mm/yr in the Sub-Arctic region, which is in agreement with the Chambers [27] results, who reported a global increase on sea level rise of 1.1 mm/yr due to the difference between the two models.

Both the Chambers and the Labroue models used in this study have been developed as separate models for TOPEX A and TOPEX B, derived from data from each of the instruments, aiming to take into account the different response of the TOPEX A and B altimeters to the sea state. The two TOPEX A models and their differences are shown in Figures 11a to 11c; the corresponding models for TOPEX B are shown in Figures 12a to 12c. These figures evidence that there are major differences between the models. These differences are explained by the different computation procedures associated with each model: the non-parametric method allows a better fit to data than the parametric one. Very extreme values of the differences occur at sea state conditions that are less frequent, therefore with a worse data sampling in the adjustment procedure. The similarity between these models is higher for low values of SWH (up to ∼4 m), almost irrespective of the WS value.

Our first comparisons of time series of sea level anomalies derived with different SSB models, without applying any a priori TOPEX A/B bias, showed small but clear biases of few millimetres between the two altimeters. It has been pointed out by various authors [27,2] that the TOPEX A/B relative bias depends on the model adopted for the SSB correction. While different values have been reported for this bias when using the Chambers model, [28,2], no result has been published yet for the most recent Labroue model.

Motivated by these results we developed a simple method to determine the TOPEX A/B relative bias for the SSB models used in this study. Using the same corrections applied to DataSet1 and DataSet2 (apart from the TOPEX A/B bias) global mean sea level values were computed using an equal-area weighted average [2]. Differencing the means over the 10 cycles before and after the A/B switch produced a TOPEX A/B bias of 2 mm for both the Chambers and Labroue models. This value agrees, within 1mm range, with the values referred by [28] and [2], the last determined by calibration with tide gauges. The same method proved to be efficient in the determination of the TOPEX A/B relative bias for other SSB models (not shown here).

The derived biases were added to the TOPEX B SSB values in the generation of both DataSet1 and DataSet2. Equivalently, they could have been subtracted from the corresponding TOPEX B sea level anomalies. Note that, since all remaining datasets used in this study (DataSet3 to DataSet7) use the same SSB (Chambers) model, the same TOPEX A/B bias of 2 mm has been applied to all of them.

Our results show that, after applying the above mentioned biases, the pattern of the amplitude of the cycle mean differences between the Chambers and the Labroue models (Figure 5) is now very similar for TOPEX A and TOPEX B and the A/B transition is almost flat. The signal of the differences also shows a strong annual latitude dependence.

The seasonal and latitudinal dependence is explained by the fact that both SSB corrections are empirical models, functions of the significant wave height (SWH) and wind speed (WS), both measured by the altimeter. Both fields have a strong annual variation with intensity and variance increasing with latitude, particularly during the boreal winter months. This is clearly shown in Figures 13 and 14, representing the time evolution of the mean and standard deviation of the SWH and WS fields respectively, for each cycle, during the analysed period and for the three studied regions. Figures 15 and 16 illustrate the spatial patterns of the same fields for representative North Atlantic winter and summer conditions respectively. Figure 15 shows that during a typical winter cycle the amplitude of both SWH and WS fields have a strong latitudinal dependence with maximum values in the Sub-Artic region. On the contrary, Figure 16 shows that during a characteristic summer cycle the intensity of the same fields, although exhibiting some geographical patterns, does not possess a clear latitudinal dependence.

Figures 11c and 12c show that, for a large range of SWH and WS values the mean difference between the models is close to -0.02 m, in agreement with Figure 5. Typical values for winter conditions in the Sub-Arctic region are around 5-6 m for the SWH (Figure 13a) and 13-15 m/s for the WS (Figure 14a). For these mean extreme values in the study region the difference between the models lies in the region where the differences are closer to zero, in agreement with the pattern of the mean cycle differences shown in Figure 5.

The regional dependence of the SSB does not only influence sea level variations. Dong and coworkers [50] refer to the SSB influence in the estimation of the bias between different satellites. It is also known that the ionospheric correction derived from dual-frequency measurements at the Ku and C frequencies is influenced by any errors in the adopted Ku-band and C-band SSB models. However, while a SSB model derived from an “imperfect” ionospheric correction may not represent the “true” SSB, it will attempt to minimise the errors in the SSH (and in the ionospheric correction) related to the sea state [27].

According to Table 5 the difference between the two adopted SSB models induces a small change in sea level trend of 0.0 to 0.2 mm/yr in the considered regions. Apart from this linear trend, the interannual signal derived for both data sets reveals similar patterns within each region (see Figures 10a, 10b and 10c, green and blue curves). Note that the relative bias between TOPEX A and B when using the two SSB models (Figure 5) is of the order 1 mm, within the computation error, indicating that the two data sets seem to be consistent with respect to the TOPEX A/B transition. Further insight into the analysed models with respect to the continuity of TOPEX A and TOPEX B solutions can only be obtained through model comparisons at absolute calibration sites or with tide gauge data.

The GDR-Ms provide the radiometer wet tropospheric correction (wet_cor) as computed from the measured TBs of the 3 TMR channels (18 GHz, 21 GHz and 37 GHz). Several authors have reported drifts in the TBs of some of these channels and published correction models to account for these effects in the wet tropospheric path delay, [51,35,52,26]. The first two authors report a global effect on sea level change of about 1.0-1.5 mm/yr due to this instrument drift, between October 1992 and December 1996.

Ruf [35] and Callahan [52] have derived drift models for the 18 GHz TBs using a linear function of the TBs up to a certain date (late 1996 and early 1999 respectively) followed by a constant correction. The drift model used in this study for the generation of DataSet3 is a modified version of the Ruf model, as explained previously.

Apart from refining the 18 GHz model by using a piecewise linear function of the TBs, Scharroo et al. also derived drift models for the other two channels (21 GHz and 37 GHz). In addition, their model also takes into account the yaw state corrections to the TBs of all the three TMR channels.

Our results show that the effect on the North Atlantic sea level variation of using the two TMR drift models (Scharroo and Ruf) is of 0.1 mm/yr (Figure 6 and Table 5). Although non negligible this effect is much smaller than the effect of unmodelling the drift itself, which, considering Scharroo's model is of 0.4 mm/yr. Apart from the effect of using different drift functions, Figure 6 shows the effect of the T/P yaw state corrections, which range from 0 to 5 mm. These corrections were not applied to DataSet3 since they were reported after the development of the Ruf's model. In spite of the variability introduced in the derived SLAs, the yaw state corrections do not affect long-term sea level trend.

The effect of the TMR drift on sea level studies is highly dependent on the length of the analysed altimeter time series. Most TMR drift models, including the two analysed here, consider that the TMR drift after 1999-2000 is approximately constant. Therefore, the longer the time series the smaller the effect will be. In Figure 10 the effect on sea level interannual variation for the three different North Atlantic regions is shown by comparing the green (DataSet1) and red (DataSet3) curves. Although some latitudinal dependence could be expected related with the spatial variability of the various TBs, these figures and the results presented in Tables 3 and 5, show that the latitudinal dependency is not significant.

In spite of instrumental problems in the onboard microwave radiometers, affecting the wet tropospheric delay derived from the TBs measured by these instruments, provided these are properly modelled and accounted for, the radiometer derived wet_cor is still more accurate than the corresponding ECMWF model correction. For TOPEX, the TMR model provides a path delay correction with the accuracy better than 1.1 cm, even under the worst conditions of variable clouds and wind, [53]. Although the ECMWF model correction has been improving, several modifications introduced discontinuities in the model, making it unsuitable for use in sea level studies, [25,26].

Results show that the inverse barometer is the correction with the largest impact in the shape of the SLA time series for a given region (Figures 4, 7, 8 and 10). There is no consensus between the research community whether or not the IB correction should be applied to the altimeter data for sea level studies, [9,36,6], although this discussion was mainly related to the observation that an inverse barometer correction using a constant reference pressure creates unrealistic signals, since the mean correction is no longer zero, which is not consistent with ocean mass conservation [36].

In this study three different IB models have been compared, one that makes use of a constant reference atmospheric pressure (CRP) of 1013.3 mbar (DataSet4), one that uses a variable reference surface atmospheric pressure (VRP), interpolated from global average values obtained from the CNES/CLS (DataSet5) and the so-called combined-MOG2D model (DataSet6). In the computation of the VRP model, the values of P̅ have been smoothed using a (2-day)^-1 cut-off frequency. However it was found that the smoothing does not affect the mean sea level determination since the maximum difference in the values of P̅ is 0.7 mbar, which causes a maximum SLA change of 7 mm.

Figure 8a shows that the amplitude of the differences between the VRP and the CRP models can be as large as 2.5 cm and exhibits a strong seasonal signal. As expected, these differences are not region dependent, since the actual difference between the two models resides in the use of the constant value of 1013.3 in Equation (3) instead of the value of P̅ given by Equation (5) and the value of P̅ is global by definition, [36]. Both models have a similar impact on sea level trend with the VRP model giving slightly smaller trends by about 0.13 mm/yr (Tables 3, 4 and 5). Using only four years of T/P data [36] had found that globally the second model would give sea level variations 0.4 mm/yr smaller than the first one.

The pattern of the differences between the combined-MOG2D and the VRP models presented in Figure 8b is quite different from the previous two models. Now the seasonal signal has disappeared, as it could be expected since both models use the same modelling for the long wavelength atmospheric effects given by the inverse barometer response. This figure is now dominated by the high frequency effects modelled by the MOG2D barotropic model. Carrère et al. [37] found that the use of this model reduces the T/P crossovers variance by more than 15% with respect to the classical inverse barometer correction. The corresponding curves for the interannual signals shown in Figure 10 (dashed pink and dashed black curves) are almost coincident.

Figures 7a and 7b reveal the impact of the application of the IB correction from two of the mentioned models (CRP and VRP) on the shape of the corresponding SLA time series, showing the expected latitudinal dependence, related with the spatial variability of the pressure fields. Similar results were obtained for the combined-MOG2D model.

Figure 10 reveals that the shape of the interannual signal of an SLA time series without any IB correction (DataSet1) and the corresponding signals of the IB corrected ones (DataSet4, DataSet5 and DataSet6) are significantly different, particularly for high latitudes.

In global studies IB corrected SLA time series tend to give smaller sea level trends than the corresponding non IB-corrected, [25,36]. However, this work shows that in regional studies this is not always true. For the Tropical and Sub-Tropical regions the difference in sea level trend determined by a non IB-corrected series (DataSet1) and any of the IB-corrected ones (DataSet4, DataSet5, DataSet6) is in the range of 0.1 mm/yr to 0.3 mm/yr (Table 5). However, for the Sub-Arctic region it is significantly different, of the order of -0.8 mm/yr.

Atmospheric pressure variability exhibits a strong latitudinal dependence with the lowest values at tropical latitudes (∼2 mbar) and the highest values (∼15 mb) at polar latitudes, [54]. Thus the differences in the sea level interannual components related to whether the IB correction is applied or not to the altimetry measurements are most noticeable for the northern region.

The interannual components for the Subarctic region show a significantly different behaviour for IB-corrected and not corrected datasets. The sea level series for which the IB correction has not been applied departs significantly from the IB-corrected series for three distinct periods: near 1995, near 2000 and after mid-2003. These periods are associated with significant anomalies in the atmospheric patterns over the North Atlantic which influence the inverse barometer response of sea level.

The North Atlantic Oscillation (NAO) affects the sea surface height through a direct hydrostatic response to changes in the pressure field and indirectly through wind, ocean circulation and heat flux forcings [55-58]. The NAO index (Figure 17) switched from one of its highest positive values in 1995 to one of its most negative value for this century in 1996; a similar switch, but not as prominent, occurred in 2000-2001. We believe that the different shape of the IB-corrected and non IB-corrected curves shown in Figure 10 (particularly for the SA region, Figure 10c) are related to these NAO anomalies.

The period after mid-2003 shows a striking decrease in sea level in the non IB-corrected data sets and a significant departure from the IB-corrected values, but seems not to be directly related to the NAO. However, the northern mid latitudes experienced abnormal weather in the summer of 2003 which was not diagnosed by the NAO indices but was captured by the Northen Annular Mode (NAM) index, [59]. The abnormal summer of 2003 was associated with a horizontal anomaly in the structure of the 500 mbar geopotential height field. The pattern of positive anomalies over the high latitudes of the North Atlantic in association with the disappearance of the mean Icelandic Low persisted until 2005 affecting significantly the atmospheric circulation patterns. Ascertaining direct causes of sea level variability is very difficult due to the large number of interacting factors that affect sea level, but the coincidence of these climate anomalies with periods for which a departure from the IB response is observed is physically plausible.

Figure 9 shows the SLA differences between DataSet1 and DataSet7, computed using the NASA and the CNES orbits respectively. Most sea level studies using T/P altimeter data adopt the NASA solution. The CNES orbit globally increases the variance of T/P crossover differences by about 2.1 cm² relative to the NASA orbit [25], with a degradation of the CNES orbit particularly high from cycle 100 to cycle 246.

It has been reported [25] that large differences were found at hemispheric scale between the two orbits related mainly to changes in the ITRF reference frames. NASA and CNES have adopted different strategies in the way they handled the terrestrial reference frame adopted to refer the tracking stations. In the NASA processing, station horizontal velocities have been accounted for, allowing their position to evolve with time; in the CNES computation it was chosen to work on a fixed reference frame during a given period, based on the observation at the beginning of 1996 that the stations horizontal velocities were not known with sufficient accuracy. During this period, approximately from cycle 100 until cycle 246, the two orbits differ considerably. The jumps observed in cycle 247 and 320 are related with the CNES adoption of ITRF97 for this period and ITRF2000 from cycle 320 onwards, following the recommendation of the T/P and Jason-1 Science Working Team (SWT) in Miami, 2000. Cycle 320 also coincided with the introduction of the albedo model in the CNES orbit computation, accounting for the effect of surface radiation pressure. Following the SWT recommendation, the NASA orbit also adopted the ITRF2000 frame after cycle 360. Apart from a jump found at cycle 365, coincident with the T/P orbit transition, after cycle 360 the pattern of the differences between the two orbits decreases again to below the cm level as it was at the beginning of the mission.

ITRF effects on mean sea level have been discussed by [60,61]. The non-homogeneity of the adopted ITRF by the NASA and CNES groups seems to explain the trend detected in the difference between the two solutions, which can be approximately expressed by a z-axis coordinate drift between the two orbits. Due to the North–South asymmetric distribution of the oceans sampled by the altimeters, an error in the realization of the z-axis induces a systematic error in global mean sea level estimate. Morel and Willis [61] show that ITRF97 has an internal precision that would map to 3.0 mm uncertainty in the global mean sea level and 0.37 mm/yr in mean sea level rise.

Our results confirm that the pattern of the differences between the two GDR-M orbits for the North Atlantic has a regional dependence, with the amplitude of the differences increasing with latitude (Figure 9). These results concur with the known behaviour of the orbital error, which has a main period close to one orbital revolution and a geographically correlated pattern.

The orbit differences affect the shape of the interannual signal (Figures 10a, 10b and 10c, green and yellow curves) only at the middle of the series, in particular for the ST and SA regions. Due to the fact that the major differences occur in the middle of the sampled period and both solutions seem to be consistent at the beginning and end of the series, the difference in the linear sea level trend determined by both solutions is negligible, considering the errors of the estimated trends (Tables 3, 4 and 5).

Works mainly done within the scope of Jason-1 calibration/validation studies have pointed out differences between the two satellites, which seem to be related with some unidentified problems in TOPEX data. At the SWT meetings in November 2003 and November 2004 geographically correlated SSH differences between T/P and Jason-1 were reported, which seem correlated with the radial velocity. Clear hemispherical patterns were also found in TOPEX B SWH between ascending and descending passes. Further work needs to be performed in order to identify if the cause of these discrepancies is either on TOPEX or Jason-1 data and if it is related with orbital or tracker errors.

Recent orbit solutions based on GRACE derived gravity fields seem to substantially reduce the T/P orbital error as compared to the JGM3 orbits provided on the GDR-Ms. The availability of these improved orbits to users will certainly contribute to a further reduction of the overall error budget in altimeter data and will be an important contribution to long-term sea level variation studies.