Sea-level variability and change along the Norwegian coast between 1 2003 and 2018 from satellite altimetry , tide gauges and hydrography 2

Sea-level variations in coastal areas can differ significantly from those in the nearby open ocean. Monitoring 9 coastal sea-level variations is therefore crucial to understand how climate variability can affect the densely populated coastal 10 regions of the globe. In this paper, we study the sea-level variability along the coast of Norway by means of in situ records, 11 satellite altimetry data, and a network of eight hydrographic stations over a period spanning 16 years (from 2003 to 2018). At 12 first, we evaluate the performance of the ALES-reprocessed coastal altimetry dataset (1 Hz posting rate) by comparing it 13 with the sea-level anomaly from tide gauges over a range of timescales, which include the long-term trend, the annual cycle 14 and the detrended and deseasoned sea level anomaly. We find that coastal altimetry and outperforms conventional altimetry 15 products at most locations perform similarly along the Norwegian coast. However, the agreement with tide-gauges in terms 16 of trends are on average 10% better when we use the ALES coastal altimetry data. We later take advantage of the coastal 17 altimetry dataset to perform a sea level budget later assess the steric contribution to the sea-level along the Norwegian coast. 18 We find that the thermosteric and the halosteric signals give a comparable contribution to the sea-level trend along the 19 Norwegian coast, except for three, non-adjacent hydrographic stations, where salinity variations affect the sea-level trend 20 more than temperature variations. We also While longer time series are necessary to evaluate the steric contribution to the 21 sea-level trends, we find that the sea-level annual cycle is more affected by variations in temperature than in salinity, and that 22 both temperature and salinity give a comparable contribution to the detrended and deseasoned sea-level change along the 23 entire Norwegian coast. A conclusion from our study is that coastal regions poorly covered by tide gauges can benefit from 24 our satellite-based approach to study and monitor sea-level change. 25


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The Norwegian Mapping Authority (Kartverket) provides information on observed water levels at 24 permanent tide gauge 126 stations along the coast of Norway. Data are updated, referenced to a common datum, quality checked, and freely distributed 127 through a dedicated web API (api.sehavniva.no).

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Even though most tide gauges provide a few decades of sea-level measurements, in this study we only consider the period 130 between January 2003 and December 2018 because it overlaps with the time-window spanned by the ALES-altimetry 131 dataset. Moreover, we only select 22 of the 24 permanent tide gauges available: we exclude Mausund, since it has no 132 measurements available before November 2010, and Ny-Ålesund, because it is outside of our region of interest.

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Over the period considered, the only tide gauges with missing values are Heimsjø and Hammerfest, with a 1-month gap, and 135 Oslo, with a 2-month gap. We expect the Norwegian set of tide gauges to map the coastal sea-level with a spatial resolution 136 of circa 130 km as it corresponds to the mean distance between adjacent tide gauges. This estimate should be treated only as 137 a first order approximation of the spatial resolution since the distance between adjacent tide gauges varies along the 138 Norwegian coast and ranges from ~30 km, in southern Norway, to ~300 km, in western Norway (more precisely, between 139 Rørvik and Bodø).

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A number of geophysical corrections have been applied to the tide gauge data for them to be consistent with the sea-level 147 anomaly from altimetry. These include the effects of the glacial isostatic adjustment (GIA), the nodal tide and the DAC.

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The GIA results from the adjustment of the earth to the melting of the Fennoscandian ice sheet since the last glacial 150 maximum, circa 20 thousand years ago. The earth's relaxation affects substantially the sea-level change relative to the 151 Norwegian coast, with values ranging from approximately 1 up to 5 mm year -1 (e.g., Breili et al., 2017). The GIA affects the 152 sea-level because it induces a vertical land movement (VLM) and, to a lesser extent, because it modifies the earth's gravity 153 field. The first effect has been corrected using both GNSS observations and levelling, whereas the second has been corrected 154 using a GIA model (Simpson et al., 2017).

156
The low frequency constituents of ocean tide, derived from the EOT11a tidal model, are removed from the tide gauge data as 157 they are from the ALES-reprocessed altimetry dataset. Hammerfest, Honningsvåg and Vardø, the three northernmost tide 158 gauges ( Fig. 1), are located outside of the EOT11a model domain. Therefore, at these three locations, we remove the low 159 frequency constituents of ocean tide for Tromsø. The constituents in question are the solar semiannual, solar annual, and the 160 nodal tide. For Norway the solar annual astronomical tide is negligible, while the two latter constituents have amplitudes on 161 the order of 1 cm. The nodal tide has a period of approximately 18.61 years and results from the precession of the lunar 162 nodes around the ecliptic (Woodworth, 2012). As our time series are shorter than the nodal cycle, this constituent is not 163 negligible with regards to our trend analysis. None of the solid earth related tides needs to be removed from land-locked tide 164 gauge measurements to produce sea-level records comparable to altimetric sea surface height. Moreover, the ocean pole tide, 165 not provided by the EOT11a, has not been removed from the tide gauge data. However, it is negligible in our region.

167
Since we have provided a description of the DAC in the previous section, here we only briefly describe how we have applied 168 it to the tide gauge data. At first, we have monthly averaged the six hourly DAC dataset (available at the AVISO+ website, 169 https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/dynamic-atmospheric-correction.html). Then, for each 170 tide gauge, we have computed the difference between the monthly mean sea-level and DAC at the nearest grid point of the 171 DAC product.

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Over the time window covered by this study, the Institute of Marine Research (IMR) in Bergen, Norway, has maintained 175 eight permanent hydrographic stations over the Norwegian continental shelf, at a short distance from the coast (Fig. 1). Data 176 are updated and available at http://www.imr.no/forskning/forskningsdata/stasjoner/index.html.

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Along the Norwegian coast, the number of hydrographic stations is approximately one third the number of tide gauges.

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Therefore, compared to the tide gauges, the hydrographic stations provide a coarser spatial resolution of the physical 180 properties of the ocean. We find that the distance between adjacent hydrographic stations is approximately 250 km on 181 average. This distance is minimum between the twin stations Indre Utsira/Ytre Utsira and Eggum/Skrova, where it does not 182 exceed 30 km, whereas it is maximum in western Norway, between Bud and Skrova, where it is approximately 670 km.

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As for the tide gauges, we We select the temperature and salinity profiles taken between January 2003 and December 2018 185 for them to overlap with the period covered by the ALES-reprocessed altimetry dataset. The temperature and salinity profiles 186 at each hydrographic station are irregularly sampled and contain missing values (Fig. 2). Bud has the largest number of 187 missing values, with 76 gaps out of 192. It is followed by Indre Utsira and Ytre Utsira, with 44 and 41 gaps, respectively.

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The remaining hydrographic stations have less than 16 gaps each.

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The hydrographic data were used to obtain estimates of the thermosteric and the halosteric sea-level components over the 191 spatial domain considered in this study.

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In the present study, we present the estimates of the sea-level trend from both satellite altimetry and the tide gauges with the 215 corresponding 95% confidence intervals (Fig. 8). Moreover, we assess how strongly the linear trends from altimetry depends 216 on the time period considered and show those trends that are significant at a 0.05 significance level (Fig. 9). To compute the 217 confidence intervals and the statistical significance, we account for the serial correlation in the time series. Indeed,

218
successive values in the sea-level time series might be significantly correlated and, therefore, not drawn from a random 219 sample. To account for this non-zero correlation, we compute the variogram of the detrended and deseasoned SLA from 220 satellite altimetry and the tide gauges and, then, determine the effective number of degrees of freedom, * ,for each time 221 series (as described in Appendix A).

223
We compute the 95% confidence interval of the linear trend as follows: Where is the standard error of the linear trend, computed as if * = , the total number of observations in the time 226 series, and "."$/&, ) * *+ is the t-value computed using * − 6 degrees of freedom at a 0.05 significance level.

229
To compare the sea-level from satellite altimetry and tide gauges, we first need to preprocess the altimetry observations since 230 these are not colocated neither in space nor in time with the tide gauges. The colocation consists of two steps. At first, we 231 select the altimetry observations that are located nearby each tide gauge. Then, we average these observations both in space 232 and in time to create, for each tide gauge location, a single time series of monthly mean sea-level anomaly from altimetry.

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During the process, we verify that the selected altimetry observations represent the sea-level variability at each tide gauge

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We choose to maximize the linear correlation coefficient, instead of minimizing the root mean square differences (RMSDs),

249
since the former appears less sensitive in cases when there are few altimetry observations. There is one exception are three 250 exceptions: the Stavanger, Trondheim and Bodø tide gauges, where a very stringent colocation accidentally yields a high 251 correlation. Thus, for Bodø for these three stations, we select the second highest correlation, which corresponds to a distance 252 from the coast of 20 km and to a distance along the coast of 200 km.

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The results suggest that the spatial pattern associated with the detrended and deseasoned sea-level anomaly extends over

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We use the process described above to build a time series of monthly mean sea-level anomaly from altimetry at each tide

286
The steric component of the sea-level at each hydrographic station, -, , is simply the sum of the corresponding thermosteric 287 and halosteric components of the sea-level (Gill and Niller, 1973).

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At each hydrographic station, we assess the contribution of temperature and salinity to the linear trend and the seasonal cycle

311 312
Before comparing the detrended and deseasoned SLA from altimetry and tide gauges, we briefly describe how the detrended 313 and deseasoned SLA evolves along the Norwegian coast during the period under study. More precisely, we low-pass filter 314 the detrended and deseasoned SLAs with a one-year running mean to identify their main features at each tide gauge location.
315 Figure 3 shows years when the detrended and deseasoned SLA variations are coherent along the whole Norwegian coast, and 316 years when the sea-level variability occurs at smaller spatial scales (between 100 and 1000 km). As an example, between

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The complex geometry of the Norwegian coast can lead to small-scale variations in sea-level. This can partly explain the 352 difference between the sea-level estimates from tide gauges and from altimetry. Indeed, while the SLA time series measured 353 by the tide gauges are representative for particular locations, those from satellite altimetry, preprocessed as described above,

359
These results suggests that the detrended and deseasoned SLA in the south vary over smaller spatial scales compared to the 360 north. Indeed, both the linear correlation coefficient and the RMSD in southern Norway depend more on the size of the 361 selection window than in northern Norway. In Fig. 5a, we note that the standard deviation of the linear correlation 362 coefficients mainly ranges between 0.15 and 0.20 to the south of Trondheim, whereas it ranges between 0.10 and 0.15 to the 363 north of Trondheim. Likewise, the standard deviation of the RMSD follows a similar spatial pattern, with southern Norway

364
showing higher values compared to northern Norway.

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Figures 6 and 7 show a good agreement between the annual cycle estimated using the ALES altimetry dataset and the tide 374 gauges. The difference between the amplitudes of the annual cycle from ALES and the tide gauges ranges between -1.2 and 375 1.8 cm. However, at most tide gauge locations (16 out of 22), the differences are much smaller, between -1 and 1 cm, less 376 than 10 % of the amplitude of the corresponding annual cycle (Fig. 6a). We note that the differences between the amplitudes 377 are mostly negative along the southern and western coast of Norway and that, to the north of Rørvik, they become smaller, 378 and even change sign at some locations (Fig. 6b).

380
The difference between the phases of the annual cycle estimated using the ALES altimetry dataset and the tide gauges ranges 381 between -10 and +10 days (Fig. 7b). Such a great similarity indicates that both radar altimetry and the tide gauges capture the

400
The differences between sea-level trend estimate obtained from the in-situ and remote-sensed signals range between -0.8 and

406
Despite their similarities, we still find that the difference between the sea-level trend from altimetry and tide gauges is

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Interestingly, only one none of these tide gauges is located north of 66° N despite only some of the altimetry missions 415 considered in this study have an inclination exceeding 66° N (namely, Envisat, SARAL, SARAL drifting phase, Sentinel 3A 416 and 3B). Therefore, the fewer altimetry observations to the north of 66° N seem not to deteriorate the agreement between the 417 ALES-reprocessed altimetry and the tide gauges.

419
We can partly explain the discrepancy between the sea-level trend obtained from altimetry and the tide gauges by looking at 420 dependency on the distance from the coast. Indeed, from a visual inspection of Fig. 8   The variability of the thermosteric and the halosteric sea-level components along the Norwegian coast mainly occurs over 453 two different spatial and temporal scales (Fig. 10). Notably, the seasonal cycle dominates the thermosteric sea-level 454 variability at each hydrographic station and is responsible for the thermosteric sea-level to vary approximately uniformly 455 along the coast of Norway. On the contrary, the halosteric component shows a variability at shorter spatial-and temporal-456 scales, possibly due to the contributions from local rivers. The main exceptions are, due to their proximity, the two sets of 457 twin hydrographic stations, Indre Utsira-Ytre Utsira and Eggum-Skrova (Fig. 1).

459
Despite these differences, both the thermosteric and the halosteric components of the sea level give a comparable 460 contribution to the sea-level variability along the Norwegian coast (Fig. 10). This ranges approximately between -10 and 10 461 cm at each hydrographic station.

463
In the following sections, we investigate the spatial variability of these two components along the Norwegian coast, focusing

478
In this section, we perform a fit-for-purpose assessment of the Norwegian hydrographic station network to obtain estimates 479 of the steric sea-level trends from satellite altimetry and in-situ data.

481
We find that the linear trends of the thermosteric, halosteric and steric components of the sea-level approximately range 482 between -1.0 and 2.5 mm/year, the width of their confidence intervals ranges between 4.0 and 12.0 mm year -1 circa, with 483 northern Norway exhibiting larger uncertainties (Fig. 11). This is a result of the high inter-annual variability of the 484 thermosteric and the halosteric components in the region (Figs. B1 and B4), which leads to a fewer number of effective 485 degrees of freedom and, therefore, to less accurate estimates of the linear trend.

487
We also test if using tide gauges, instead of satellite altimetry, could alter our estimates of the relative contribution of these 488 components (thermosteric, halosteric and steric) to the sea-level trend along the coast of Norway. Such alteration may indeed 489 occur because the sea-level variations measured by the Norwegian tide gauges might not properly represent those occurring 490 in proximity of the hydrographic stations since the two sets of instruments are not colocated in space (Fig. 1).

492
With the exception of Lista, the choice of the dataset has minimal influence on the estimates of the thermosteric, halosteric 493 and steric relative contributions to the sea-level trend along the coast of Norway. We reach this conclusion by visual 494 inspection, but we also provide a more quantitative analysis based on the ratio between the linear-trend of the SLA and of the 495 thermosteric, halosteric and steric components of the sea-level. We find that, apart from Lista, the choice of the dataset 496 modifies such a ratio by less than 13%. At Lista, the change amounts to 59% and results from the ALES-retracked satellite 497 altimetry dataset returning a sea-level trend approximately 1.6 times larger than that provided by the tide gauge at Tregde 498 (this is the tide gauge we use to compute the thermohaline contribution at Lista). Such a large variation is expected since, as

502
In this section, we assess the steric contribution to the sea-level trends along the Norwegian coast, considering monthly 503 averaged coastal altimetry and hydrographic stations. Figure 11 shows the sea-level rates at each hydrographic station 504 considered in this study.

506
Over the period 2003-2018, we observe significant steric contributions to coastal sea-level trends, but mostly in the very 507 south and the very north of the Norwegian coast, at Lista and Ingøy, with the steric component explaining between 508 approximately 40-50 % of the sea-level trend estimates obtained from altimetry data. Moreover, when we compare the 509 thermosteric and the halosteric signals at these locations, we note that the latter contributes more than the former to the 510 coastal sea-level trends (up to 60 %).

512
At the other locations, the steric contribution to coastal sea-level is either more uncertain or considerably smaller. At Bud, 513 the steric component explains a large fraction of the sea-level trend comparable to the one found at Lista and Ingøy, but, 514 similarly to Lista and Ingøy, this mainly results from salinity changes. However, the uncertainty associated with these 515 estimates are larger at Bud than at the other two stations probably due to the large gaps in the temperature and salinity 516 recordings in the second half of the record. At the remaining five locations, the trends induced by the thermosteric, the 517 halosteric and the steric sea-level are considerably smaller than the altimetry rates. This suggests a larger influence of the 518 non-steric (mass induced) sea-level trend in these areas.

520
We note that the results in Fig. 11

541
To better understand what causes the spatial difference of the halosteric sea-level trend along the Norwegian coast, we 542 compute the linear trends at each hydrographic station as a function of depth level (Fig. 12). The results suggests that the We now assess the thermosteric, halosteric, and steric components of the sea-level annual cycle at each hydrographic station 556 along the Norwegian coast.

558
Contrary to what we observe for the sea-level trends, the steric sea-level gives a non-negligible contribution to the sea-level 559 annual cycle along the entire Norwegian coast (Table 1). Indeed, the steric signal explains more than 60 % of the sea-level 560 annual cycle at six out of eight hydrographic stations.

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In Table 1, we note that the annual cycle of steric sea-level is largely associated with ocean thermal expansion: except for  (Table 2). To the north of Ytre Utsira, the lag between the 571 thermosteric and the halosteric components of the sea-level decreases since the halosteric annual cycle peaks between

602
We find that using the tide gauge data, instead of satellite altimetry measurements, only little affects the estimate of the 603 thermosteric, halosteric and steric contributions to the seasonal cycle of SLA (Fig. 12), even though the tide gauges are not 604 colocated in space with the hydrographic stations. Indeed, the seasonal cycle returned by satellite altimetry at each 605 hydrographic station strongly resembles that returned by the nearby tide gauge (Fig. 12, fourth column). At the same time,

606
the RMSD between the seasonal cycle of the SLA and steric sea-level, scaled by the range (maximum minus minimum) of 607 the seasonal cycle of SLA, little depends on the dataset used (Table 1, first and second columns).

609
We also note that density changes contribute substantially to the seasonal cycle of SLA along the Norwegian coast, as shown 610 by Fig. 12

615
Along the Norwegian coast, the seasonal cycle of steric sea-level is more affected by variations in temperature than in

629
The detrended and deseasoned thermosteric sea-level along the Norwegian coast shows a larger spatial variability compared 630 to the detrended and deseasoned halosteric component (Fig. 13). The correlation matrix of the thermosteric sea-level (Fig.   631 13a) shows larger values compared to the one obtained considering the halosteric sea-level signals (Fig. 13b). As an 632 example, while the minimum linear correlation coefficient between two adjacent hydrographic stations in Fig. 13a

651
between Bodø and Tromsø, where the difference between the linear trend from ALES and the tide gauges are small (up to 652 0.7 mm year -1 ), compared to circa 1 to 3 mm year -1 obtained using a conventional altimetry dataset.

654
Along the Norwegian coast, the sea-level trend from the ALES-reprocessed satellite altimetry dataset is found to be

663
Regarding the comparison between the ALES-retracked and the along-track (L3) conventional altimetry datasets, we find 664 that the former shows, on average, a 10% improvement, despite it being well within the margins of error. This improvement

679
The results obtained from the ALES dataset also suggest that along the north-western coast of Norway, between Ålesund and

692
The ALES-retracked satellite altimetry dataset is found to underestimate the amplitude of the annual cycle along large 693 portions of the Norwegian coast (Fig. 6). Even though the difference between the two sets of estimates is not significant at a 694 95% significance level (the 95% confidence interval is approximately twice the standard error), we find this result interesting 695 because of its consistency. We do not expect such a consistency to depend on the ALES retracker since we find a 696 comparable result when we use the along-track (L3) conventional altimetry product (Fig. C3). We rather suspect a 697 dependence of the amplitude of the annual cycle on the bathymetry and, therefore, on the distance from the coast, as shown

709
We also assess the steric contribution to the seasonal cycle of SLA. Our results show that the steric variations and, in

727
The detrended and deseasoned sea-level varibility along the Norwegian shelf resembles the along-slope wind index proposed 728 by Chafik et al. (2019). We note that the similarities between the two are stronger along the western and the northern coast of 729 Norway than in the south. Indeed, from Olso to Ålesund, those SLA signals depart from the along-slope winds index 730 between 2003 and 2008, probably due to local effects, such as the Baltic outflow. We refer to local effects since Chafik et al.

731
(2019) attributed the interannual sea level variability over the northern European continental shelf to the along-slope winds, 732 which might regulate the exchange of water between the open ocean and the shelf through Ekman transport.

734
Because the detrended and deseasoned SLA pattern is coherent over large distances along the Norwegian coast (see also

743
The small-scale variability of the detrended and deseasoned sea-level halosteric component (Fig. 13)

764
Appendix A

765
To estimate the uncertainty associated with the sea level trends derived from tide gauges and the ALES-retracked satellite 766 altimetry dataset (Fig. 8), we need to account for the effective degrees of freedom in the sea-level anomaly time series.

767
Indeed, successive points in the SLA time series might be correlated and, therefore, not drawn from a random sample.

769
To determine the effective number of degrees of freedom, we produce the variograms of the detrended and deseasoned SLA

783
We use the fit to determine the lag at which each variogram reaches a plateau, since it indicates the decorrelation timescale

787
We find that the lag only little depends on the tide gauge location, and on whether we consider the detrended and deseasoned 788 SLA from the altimetry dataset or the tide gauges (Figs. A1 and A2). The variograms obtained from both altimetry and the 789 tide gauges return a lag of 2 months at each tide gauge location, with the exception of three stations in southern Norway

790
(Viker, Oscarborg and Helgeroa), where the SLA from the tide gauges is characterized by a 3-month lag.

792
We use the same approach to compute the uncertainty associated with the linear trend of the difference between the SLA 793 from satellite altimetry and the tide gauges, with only one exception. We noticed that the spheric model does not fit the 794 variogram for Trondheim. Therefore, for Trondheim, we opted for an exponential model: where h the fitting parameter, and a is the range parameter. An exponential function is preferred over the spherical function 799 when the time series shows a strong temporal correlation.

801
The serial correlation is negligible along the entire Norwegian coast with the exception of Viker, Oscarborg, Oslo and 802 Narvik, where the variograms return a 2-month lag (Fig. A3). At Trondheim, instead, we find a much larger lag

805
We use the effective number of degrees of freedom when we compute the confidence intervals of the sea-level rates in

838
Following the same argument as in the Appendix A of the Supplementary Material, to estimate the uncertainty associated 839 with the linear trends of the thermosteric, of the halosteric and of the steric components of the sea-level along the Norwegian 840 coast (Fig. 11), we need to account for the effective degrees of freedom in the corresponding time series.

842
As in Section A of the Supplementary Material, to determine the effective number of degrees of freedom, we first produce

847
The thermosteric sea-level (Fig. B1) shows the strongest serial correlation. The variogram of the thermosteric sea-level 848 returns lags ranging from 3 months, at Indre Utsira, to around 20 months at Skrova. In general, the thermosteric component 849 of the sea-level in northern Norway has fewer degrees of freedom than in the south.

856
Similarly to the Appendix A, we use the following formula to compute the 95% confidence interval of the linear trend of the 857 SLA and of the thermosteric, halosteric and steric components of the sea-level at each hydrographic station: 858 859

889
To compare the performance of the ALES-retracked and the conventional satellite altimetry dataset, we download the along-

893
We select the same satellite altimetry missions that have been reprocessed with the ALES-retracker. Moreover, we make 894 sure that both satellite altimetry datasets cover the same period.