To provide more accurate sea-level estimates in coastal regions, the ALES retracker operates in two stages. At first, it fits the leading edge of the waveform to have a rough estimate of the significant wave height (SWH). Then, depending on the SWH, the algorithm selects a portion of the waveform (known as subwaveform) and fits it to estimate the range (the distance between the satellite and the sea surface), the SWH, and the backscatter coefficient.

The dataset is freely available on the Open Altimetry Database website of the Technische Universität München (https://openadb.dgfi.tum.de/en/, last access: 22 July 2020). The European Space Agency (ESA) also provides, through the Sea Level Climate Change Initiative Programme, a coastal satellite altimetry dataset reprocessed with the ALES retracker. However, it only covers the northern latitudes up to 60∘ N and, therefore, only part of the region of interest in this study (Benveniste et al., 2020).

The dataset includes observations from the following altimetry missions: Envisat (version 3), Jason-1, Jason-1 extended mission, Jason-1 geodetic mission, Jason-2, Jason-2 extended mission, Jason-3, SARAL, and SARAL drifting phase. These are provided at a 1 Hz posting rate (equivalent to an along-track resolution of circa 7 km) and cover the period from June 2002 to April 2020, with the exception of one data gap between November 2010 (end of Envisat) and March 2013 (start of SARAL) to the north of 66∘ N. Data from different missions have been cross-calibrated so that there are no inter-mission biases.

Prior to distribution, several corrections have been applied to the satellite altimetry data. Among them, the geophysical corrections are of particular interest for the purpose of this study. Indeed, to validate the ALES-reprocessed altimetry against the Norwegian tide gauges, the same physical signal must be removed from both datasets. The geophysical corrections applied to the ALES-reprocessed altimetry data include the tidal and the dynamic atmospheric corrections (COSTA user manual, http://epic.awi.de/43972/1/User_Manual_COSTA_v1_0.pdf, last access: 22 July 2020). The correction for ocean and pole tides has been performed using the EOT11a tidal model. The solid-Earth-related tides have also been subtracted from the orbital altitude but, as it leaves the altimetry data in sync with the tide gauges (which are based on the solid Earth), this correction has no further interest for this study. The dynamic atmospheric correction (DAC), available at https://www.aviso.altimetry.fr/index.php?id=1278 (last access: 12 April 2021), removes both the wind and the pressure contribution to the sea-level variability at timescales shorter than 20 d and only the pressure contribution to the sea-level variability at longer timescales. The high-frequency component of the DAC is computed using the Mog2D-G high-resolution barotropic model (Carrère and Lyard, 2003), and it is removed because it would otherwise alias the altimetry data. The low-frequency component accounts for the static response of the sea level to changes in pressure, a phenomenon also known as the inverse barometer effect (IBE), according to which a 1 hPa increase or decrease in sea-level pressure corresponds to a 1 cm decrease or increase in sea level. To validate the ALES-reprocessed altimetry against the Norwegian tide gauges, the relevant physical signals at the relevant timescales must be removed from the tide-gauge data (Sect. 2.2).

The producers of ALES flag some of the data as unreliable. More precisely, they recommend excluding observations that fall within a distance of 3 km from the coast and whose sea-level anomaly (SLA), SWH, and standard deviation exceed 2.5, 11, and 0.2 m, respectively. We have followed these recommendations with one exception: we have lowered the threshold on the sea-level anomaly from 2.5 to 1.5 m because this choice leads to better agreement between the tide gauges and the ALES altimetry dataset between Måløy and Rørvik along the west coast of Norway (Fig. 1).

The Norwegian Mapping Authority (Kartverket) provides information on observed water levels at 24 permanent tide-gauge stations along the coast of Norway. Data are updated, referenced to a common datum, quality-checked, and freely distributed through a dedicated web API (http://api.sehavniva.no, last access: 28 April 2021).

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

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

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

The GIA results from the adjustment of the Earth to the melting of the Fennoscandian ice sheet since the Last Glacial Maximum, circa 20 000 years ago. The Earth's relaxation substantially affects the sea-level change relative to the Norwegian coast, with values ranging from approximately 1 up to 5 mmyr-1 (e.g. Breili et al., 2017). Along the Norwegian coast, the GIA affects the sea-level reading from the tide gauges because it induces vertical land movement (VLM) and, to a lesser extent, the sea level itself because it modifies the Earth's gravity field. The first effect has been corrected using both GNSS observations and levelling, whereas the second has not been corrected since the satellite altimetry data are also influenced by geoid changes (Simpson et al., 2017).

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

Since we have provided a description of the DAC in the previous section, here we only briefly describe how we have applied it to the tide-gauge data. At first, we have monthly-averaged the 6-hourly DAC dataset (available at the AVISO+ website, https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/dynamic-atmospheric-correction.html, last access: 12 April 2021). Then, for each tide gauge, we have computed the difference between the monthly mean sea level and DAC at the nearest grid point of the DAC product.

Over the time window covered by this study, the Institute of Marine Research (IMR) in Bergen, Norway, has maintained eight permanent hydrographic stations over the Norwegian continental shelf at a short distance from the coast (Fig. 1). Data are updated and available at http://www.imr.no/forskning/forskningsdata/stasjoner/index.html (last access: 11 November 2020).

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

We select the temperature and salinity profiles taken between January 2003 and December 2018 for them to overlap with the period covered by the ALES-reprocessed altimetry dataset. The data are irregularly sampled and are mostly collected once every 1 or 2 weeks. To allow a comparison with the satellite altimetry dataset, we have monthly-averaged the temperature and salinity profiles at each hydrographic station. We should note that the monthly averaged time series of temperature and salinity contain missing values (Fig. 2). Bud has the largest number of missing values with 76 gaps out of 192. It is followed by Indre Utsira and Ytre Utsira with 44 and 41 gaps, respectively. The remaining hydrographic stations have fewer than 16 gaps each.

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

Following an approach similar to the one found in previous papers (e.g. Cipollini et al., 2017; Breili et al., 2017), we use the Levenberg–Marquardt algorithm and fit the following function to sea-level records from remote sensing and in situ data: 1zt=a+b⋅t+c⋅sin⁡(2πt+d)+e⋅sin⁡(4πt+f), where a is the offset, b the linear trend, c and d the amplitude and the phase of the annual cycle, and e and f the amplitude and the phase of the semi-annual cycle. Then, we compare the linear trend, the amplitude, and the phase of the annual cycle and the detrended, deseasoned sea-level signals from remote sensing and in situ data. It is important to note that the use of this formula does not account for inter-annual variations of the seasonal cycle.

In this study, we present estimates of the sea-level trend from both satellite altimetry and tide gauges with corresponding 95 % confidence intervals (see below). Moreover, we assess how strongly the linear trends from altimetry depend on the time period considered and show those trends that are significant at a 0.05 significance level (see below). To compute the confidence intervals and the statistical significance, we account for the serial correlation in the time series. Indeed, successive values in the sea-level time series might be significantly correlated and, therefore, not drawn from a random sample. To account for this non-zero correlation, we compute the semi-variogram of the detrended and deseasoned SLA from satellite altimetry and the tide gauges and then determine the effective number of degrees of freedom, N∗, for each time series (Wackernagel, 2003), as described in Appendix A. To compute the 95 % confidence interval of the linear trends, we then use Eq. (A4) in Appendix A. Together with the semi-variogram, we also estimate the effective number of degrees of freedom using the formula N∗=N⋅(1-r1)/(1+r1), where N is the length of the time series and r1 is its lag-1 autocorrelation (Bartlett, 1935). However, in this paper, we opt for the more stringent approach and only present the confidence interval derived using the semi-variograms. Indeed, we find that the semi-variogram approach returns either the same or fewer effective degrees of freedom (not shown) when compared to the other method. This is not the case for the effective number of degrees of freedom of the detrended and deseasoned SLA difference between ALES and the tide gauges. However, we find that the choice of the approach does not alter our conclusions.

To compare the sea level from satellite altimetry and tide gauges, we first need to preprocess the altimetry observations since these are not colocated in space or in time with the tide gauges. The colocation consists of two steps. At first, we select the altimetry observations that are located near each tide gauge. Then, we average these observations both in space and in time to create, for each tide-gauge location, a single time series of monthly mean sea-level anomaly from altimetry.

During the process, we verify that the selected altimetry observations represent the sea-level variability at each tide-gauge location. More precisely, since tide gauges represent the sea-level variability along a stretch of the coast, we monthly-average all the altimetry observations within a certain distance d from the coast and a certain radius r from the tide gauge (Fig. 3). We try different combinations of d and r by allowing the first to range between 5 and 20 km, with steps of 2.5 km, and the second between 20 and 200 km, with steps of 15 km. Then, we pick the combination that maximizes the linear correlation coefficient between the detrended and deseasoned SLA measured by satellite altimetry and by the tide gauge (as, for example, in Cipollini et al., 2017). To set the maximum values of d and r at 20 and 200 km, respectively, we have first performed a sensitivity test and noted that larger values of d and r return slightly higher linear correlation coefficients (especially in northern Norway) but do not alter the main results of this study. At the same time, a maximum distance of 20 km from the coast and of 200 km from the tide gauge ensures that all the selected altimetry points are located over the continental shelf and that we can better capture the spatial-scale variability of the seasonal cycle of the sea level and of the sea-level trend.

We use the process described above to build a time series of monthly mean sea-level anomaly from altimetry at each tide-gauge location. The resulting sea-level time series have no missing values between Viker and Bodø. Instead, to the north of Bodø, they have 29 missing values which result from the lack of altimetry observations between November 2010 and March 2013.

We preprocess the altimetry observations to examine the steric contribution to the sea-level variability at each hydrographic station since the two datasets are not colocated in space or in time. More precisely, we select all the altimetry observations located within 20 km from the Norwegian coast and within 200 km from each hydrographic station. Then, for each station, we monthly-average the altimetry observations to build a sea-level anomaly time series from altimetry. The results in the previous subsection give confidence that the monthly mean sea level computed over such a large area is representative of the sea-level variability at each hydrographic station.

To compute the thermosteric and halosteric components of the sea-level variability at each hydrographic station, we first monthly-average the temperature and salinity profiles. Then, at each hydrographic station, we compute the monthly mean thermosteric and halosteric components of the sea level as in Richter et al. (2012): 2ηt=∫α(T∗,S∗)⋅T-T0dz,3ηs=-∫β(T∗,S∗)⋅S-S0dz, where α and β are the coefficients of thermal expansion and haline contraction, both computed at T∗=(T+T0)/2 and S∗=(S+S0)/2. For each hydrographic station, T0 and S0 are reference values and represent time-mean temperature and salinity averaged over the entire water column (Siegismund et al., 2007).

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

At each hydrographic station, we assess the contribution of temperature and salinity to the linear trend and the seasonal cycle of the SLA, as well as to the detrended and deseasoned SLA.

We do not use the harmonic analysis approach to estimate the linear trend and the seasonal cycle of the SLA and of the thermosteric, halosteric, and steric components of the sea level at each hydrographic station. Instead, we use simple linear regression to estimate the linear trend, and we compute the monthly climatology of each detrended time series to estimate the corresponding seasonal cycle. Indeed, the seasonal cycle of the SLA and of the thermosteric, halosteric, and steric sea level might depart from the linear combination of the annual and semi-annual cycles.

Before comparing the detrended and deseasoned SLA from altimetry and tide gauges, we briefly describe how the detrended and deseasoned SLA evolves along the Norwegian coast during the period under study. More precisely, we low-pass-filter the detrended and deseasoned SLAs with a 1-year running mean to identify their main features at each tide-gauge location. Figure 4 shows years when the detrended and deseasoned SLA variations are coherent along the whole Norwegian coast and years when the sea-level variability occurs at smaller spatial scales (between 100 and 1000 km). As an example, between mid-2009 and the beginning of 2011, the detrended and deseasoned SLA shows negative values of up to -6 cm along the entire Norwegian coast. On the contrary, between 2003 and mid-2009, we note a dipole pattern, with SLA with opposite sign in the south and in the north of Norway. Indeed, up to the beginning of circa 2006, the Norwegian coast experienced a negative SLA to the south of Hemsjø and a positive SLA to the north of Heimsjø. During the following 3 years, the opposite situation occurred. These results suggest that, although coherent sea-level variability occurs along the Norwegian coast as seen from tide gauges, there are periods when it does not: during these periods, the sea-level variability is likely driven by local changes.

Figure 5 shows very good agreement between the detrended and deseasoned monthly mean SLA from ALES and the tide gauges. The two datasets agree best along the west coast of Norway where, if we exclude Trondheim, the linear correlation coefficients exceed 0.90 and the root mean square differences (RMSDs) range between 1.5 and 2.5 cm. As expected, satellite altimetry performs better between Måløy and Rørvik than in southern and northern Norway because of the convergence of altimeter tracks in the region. We suspect that Trondheim is an exception because it is located in the Trondheim fjord, where satellite altimetry might not adequately capture local sea-level variations: the presence of land and patches of calm water affects the quality of the satellite altimetry measurements (Gómez-Enri et al., 2010; Abulaitijiang et al., 2015), and the complex bathymetry and coastline hamper geophysical corrections (Cipollini et al., 2010). Similar peculiarities of the coastline along the Norwegian Trench, in the Skagerrak, and in the Oslofjord are also likely to affect the agreement, causing the linear correlation coefficients to fall between 0.80 and 0.90 and the highest RMSDs to range between 2.5 and 4.5 cm. Instead, in northern Norway, where we find linear correlation coefficients between 0.80 and 0.90 (statistically significant at a 0.05 significance level) and RMSDs between 1.5 and 3 cm, the problem might result from the smaller number of altimetry observations in the region. Indeed, only the tracks of Envisat, SARAL, and SARAL drifting phase cover the Norwegian coast north of 66∘ N.

Figure 6 supports our previous conclusions on the relationship between satellite altimetry and the tide gauges at Trondheim, Oslo, and Oscarborg. In Fig. 6, we show, for each tide gauge, the standard deviation of the linear correlation coefficient and of the RMSDs over all the possible combinations of the distance from the coast and from the tide gauge to measure the geometrical uncertainty of the SLA estimates from satellite altimetry. We find that, at Trondheim, both the linear correlation coefficient and the RMSD depend more on the size of the selection window when compared to other regions of the Norwegian coast. Similarly, at Oslo and Oscarborg, we note an anomalously high standard deviation of the linear correlation coefficient. We expect anomalously high values of the standard deviation of the linear correlation coefficients and RMSDs because these three tide gauges are in sheltered areas (Trondheim is in the Trondheim fjord, whereas Oslo and Oscarborg are in the Oslofjord), which can favour the formation of patches of calm water and negatively affect the quality of the satellite altimetry observations.

Figures 7 and 8 show good agreement between the annual cycle estimated using the ALES altimetry dataset and the tide gauges. The difference between the amplitudes of the annual cycle from ALES and the tide gauges ranges between -1.2 and 1.8 cm. However, at most tide-gauge locations (15 out of 22), the differences are much smaller at between -1 and 1 cm, which is less than 10 % of the amplitude of the corresponding annual cycle (Fig. 7a). We note that the differences between the amplitudes are mostly negative along the southern and western coast of Norway and that, to the north of Rørvik, they become smaller and even change sign at some locations (Fig. 7b).

The difference between the phases of the annual cycle estimated using the ALES altimetry dataset and the tide gauges ranges between -10 and +10 d (Fig. 8b). Such a great similarity indicates that both radar altimetry and the tide gauges capture the phase lag of approximately 2 months between the annual cycle in the north and in the south of Norway. The annual cycle peaks during the second half of September in the Skagerrak and in the Oslofjord region, in October along the Norwegian Trench and in southwestern Norway, and mainly during the first week of November north of Kristiansund.

The differences between sea-level trend estimates obtained from the in situ and remotely sensed signals range between -0.85 and 1.15 mmyr-1 along the Norwegian coast (Fig. 9). Both datasets return a similar spatial dependence of the sea-level trend along the Norwegian coast, with the lowest values found in the Skagerrak and the Oslofjord (between 2 and 3 mmyr-1) and the highest to the north of Heimsjø (around 4 mmyr-1). Moreover, the two datasets return a similar uncertainty of the sea-level trend at each tide-gauge location.

Despite their similarities, we still find that the difference between the sea-level trend from altimetry and tide gauges is significantly different from zero at a 0.05 significance level at 3 out of 22 tide gauges. Following Benveniste et al. (2020), we assess the significance in terms of fractional differences (FDs). Fractional differences are defined as FD=|τ|/(t0.05/2⋅SE⋅N/N∗), where |τ| is the absolute value of the linear trend of the SLA difference between altimetry and each tide gauge, t0.05/2 is the critical value of the Student's t test distribution for a 95 % confidence level with N∗-2 degrees of freedom, SE is the standard error, and N/N∗ is the ratio between the total number of observations and the effective number of degrees of freedom. When FD>1, the difference between the two trends is statistically significant at a 0.05 significance level, a condition that occurs at Tregde, Måløy, and Bergen. Interestingly, none of these tide gauges are located north of 66∘ N despite only some of the altimetry missions considered in this study having an inclination exceeding 66∘ N (namely, Envisat, SARAL, SARAL drifting phase). Therefore, the fewer altimetry observations to the north of 66∘ N seem not to deteriorate the agreement between the ALES-reprocessed altimetry and the tide gauges.

Following Liebmann et al. (2010), we use the satellite altimetry data to assess how strongly the sea-level trend depends on the time length of the period considered. Each point in Fig. 10 shows the sea-level trend computed over the number of years on the y axis up to the year specified on the x axis. Between 2003 and circa 2013, we do not find a significant sea-level trend along the Norwegian coast. Indeed, with very few exceptions, the trends are not statistically different from zero at a 0.05 significance level. The exceptions consist of a small number of cases, each characterized by a sea-level trend lower than -4 mmyr-1.

On the contrary, with the exception of the three southernmost tide-gauge locations, we note a significant positive sea-level trend along the entire coast of Norway when the period considered for the calculation ends in 2015 or later. The linear trends decrease as the length of the period selected increases. When sea-level rates are computed over periods of a few years only, they even exceed 6 mmyr-1. Instead, over longer periods of time (e.g. more than 10 years), they mainly range between 3 and 5 mmyr-1. A visual inspection of the time series confirms that the sea level has increased since 2014.

The variability of the thermosteric and halosteric sea-level components along the Norwegian coast mainly occurs over two different spatial and temporal scales (Fig. 11). Notably, the seasonal cycle dominates the thermosteric sea-level variability at each hydrographic station and is responsible for the thermosteric sea level varying approximately uniformly along the coast of Norway. On the contrary, the halosteric component shows a variability at shorter spatial and temporal scales, possibly due to the contributions from local rivers. The main exceptions are, due to their proximity, the two sets of twin hydrographic stations, Indre Utsira–Ytre Utsira and Eggum–Skrova (Fig. 1).

Despite these differences, both the thermosteric and halosteric components of the sea level give a comparable contribution to the sea-level variability along the Norwegian coast (Fig. 11). This ranges approximately between -10 and 10 cm at each hydrographic station.

In the following sections, we investigate the spatial variability of these two components along the Norwegian coast, focusing on the linear trend, the seasonal cycle, and the residuals, as well as on their contribution to the sea-level variability in the region.

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

Over the period 2003–2018, we find that the linear trends of the thermosteric, halosteric, and steric components of the sea level approximately range between -1.0 and 2.5 mmyr-1. The steric contributions to coastal sea-level trends experience large spatial variability that is even negative at Sognesjøen and reaches a peak of approximately 55 % of the sea-level trend estimated from satellite altimetry at Lista and Ingøy. Moreover, when we compare the thermosteric and halosteric signals at these locations, we note that the latter contributes more than the former to the coastal sea-level trends (up to 50 % of the sea-level trend from altimetry). The width of the confidence intervals of the thermosteric, halosteric, and steric contributions ranges between 4.0 and circa 12.0 mmyr-1, with northern Norway exhibiting larger uncertainties (Fig. 12). This is a result of the high inter-annual variability of the thermosteric and halosteric components in the region (Figs. B1 and B2), which leads to fewer effective degrees of freedom and, therefore, to less accurate estimates of the linear trend.

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

With the exception of Lista, the choice of the dataset has a minimal influence on the estimates of the thermosteric, halosteric, and steric relative contributions to the sea-level trend along the coast of Norway. We reach this conclusion by visual inspection, but we also provide a more quantitative analysis based on the ratio between the linear trend of the SLA and of the thermosteric, halosteric, and steric components of the sea level. We find that, apart from Lista, the choice of the dataset modifies such a ratio by less than 13 %. At Lista, the change amounts to 59 % and results from the ALES-retracked satellite altimetry dataset returning a sea-level trend approximately 1.6 times larger than that provided by the tide gauge at Tregde (this is the tide gauge we use to compute the thermohaline contribution at Lista). Such a large variation is expected since, as we have already noticed, the sea-level rates obtained considering tide-gauge and satellite data at Tregde show less accurate agreement (Figs. 9 and C5).

In this section, we build on the results by Richter et al. (2012) and assess the thermosteric, halosteric, and steric contributions to the seasonal cycle of the sea level at each hydrographic station along the Norwegian coast.

We find that using the tide-gauge data, instead of satellite altimetry measurements, only minimally affects the estimate of the thermosteric, halosteric, and steric contributions to the seasonal cycle of SLA (Fig. 13), even though the tide gauges are not colocated in space with the hydrographic stations. Indeed, the seasonal cycle returned by satellite altimetry at each hydrographic station strongly resembles that returned by the nearby tide gauge (Fig. 13, fourth column). At the same time, the RMSD between the seasonal cycle of the SLA and steric sea level, scaled by the range (maximum minus minimum) of the seasonal cycle of SLA, minimally depends on the dataset used (Table 1, first and second columns).

We also note that density changes substantially contribute to the seasonal cycle of SLA along the Norwegian coast, as shown by Fig. 13 and Table 1. The seasonal cycle of SLA and steric sea level are 1 month out of phase along the southern and western coast of Norway up to Yndre Utsira and in phase over the remaining part of the Norwegian coast. Moreover, the ratio between the range of seasonal cycles of steric sea level and of SLA varies between 0.6 at Ytre Utsira and 0.9 at Bud (Table 1, third column).

Along the Norwegian coast, the seasonal cycle of steric sea level is more affected by variations in temperature than in salinity. We note that, with the exception of Bud and Skrova, the seasonal cycle of the steric component mostly resembles that of the thermosteric component in terms of both amplitude and phase. At the same time, we note a clear discrepancy between the seasonal cycle of the halosteric and steric components in both southern Norway, where they are in anti-phase, and at Bud, where the seasonal cycle of the halosteric sea level is dominated by the semi-annual cycle. A more quantitative analysis returns comparable results; the RMSD between the steric and halosteric seasonal cycles exceeds by a factor of 1.4 the RMSD between the steric and thermosteric seasonal cycles along the entire coast of Norway (with the exception of Skrova, where the ratio between the two RMSDs is 0.7).

The detrended and deseasoned thermosteric sea level along the Norwegian coast shows larger spatial variability compared to the detrended and deseasoned halosteric component (Fig. 14). The correlation matrix of the thermosteric sea level (Fig. 14a) shows larger values compared to the one obtained considering the halosteric sea-level signals (Fig. 14b). As an example, while the minimum linear correlation coefficient between two adjacent hydrographic stations in Fig. 14a is 0.52, it is only 0.19 in Fig. 14b. We briefly discuss the small spatial-scale variability of the halosteric sea level along the Norwegian coast in the “Discussion and conclusions” section of the paper.

From Fig. 14c, we also note that the values of the correlation matrix of the steric sea level fall between those of the thermosteric and halosteric components. This suggests that the thermosteric and halosteric components of the sea level give a similar contribution to the sea-level variability along the Norwegian coast.