Vertical land motion (VLM) and changing sea levels result from a complex interplay of thermal expansion of ocean water, changing ice reservoirs, glacial isostatic adjustment (GIA), tectonic motion, and anthropogenic effects [1
]. VLM is a key for understanding long-term relative sea-level changes, and, in Fennoscandia, it is dominated by GIA. The present effect of GIA reaches values of up to 10 mm/year in the Gulf of Bothnia (center of former ice sheet) and decreases to nearly zero at the edges of the former ice sheet. Uplift signals from GIA can be computed by solving the sea-level equation (SLE):
. All three quantities of the SLE can be determined directly by time-series analysis of geodetic observations. Tide-gauge records constrain the relative sea-level change (
), which is the variation of the sea surface relative to the solid Earth [1
]. Tide gauges are attached to the Earth’s crust making their measurements affected by VLM. On the other hand, satellite altimetry and GPS (Global Positioning System) provide independent measurements of sea-level change
, respecively, with respect to a global geocentric reference frame. Based on the SLE, the combination of sea-surface measurements from altimetry and relative sea-level records from tide gauges can be used to isolate the VLM component
]. This method has been applied in previous studies, using point observations or gridded sea-level anomalies from conventional altimetry (e.g., Kuo et al. [1
], Nerem and Mitchum [2
], Pfeffer and Allemand [3
Nerem and Mitchum [2
] used a ∼7.5 years long time series of observations from TOPEX/POSEIDON (T/P) (with 10-days sampling interval) in combination with (daily averaged) measurements from 114 globally distributed tide gauges to compute VLM. The mean total error of the VLM estimates was 2.6 mm/year. Fenoglio-Marc et al. [4
] estimated VLM from T/P (1993–2001) and de-seasoned monthly tide-gauge data at 24 tide gauges in the Mediterranean Sea, which passed predefined selection criteria. Accounting for the serial correlation, the mean uncertainty of rates was 2.3 mm/year. Kuo et al. [1
] also combined T/P altimetry data from 1992 to 2001 with monthly averaged tide-gauge records at 25 sites in the Baltic Sea region. The estimated VLM rates ranged from −7.5 to 13.4 mm/year and had an average uncertainty of 4.6 mm/year. The average uncertainty of VLM rates was significantly reduced to 0.4 mm/year by applying a network adjustment algorithm. The algorithm exploited long-term (>40 years) tide-gauge records to link relative VLM between all the involved tide gauges. The application of this approach is only possible in areas with long tide-gauge records available. An improved algorithm was used by Kuo et al. [5
], which presented a VLM solution for Fennoscandia, with an average uncertainty of 0.5 mm/year. Ostanciaux et al. [6
] combined 16 years of T/P, Jason-1, and Jason-2 data with annual tide-gauge records at 641 globally distributed sites including 64 sites in Fennoscandia south of 66°N. At those sites, the VLM rates ranged from −12.8 to 11.3 mm/year, with an average VLM rate of 1.6 mm/year. Pfeffer and Allemand [3
] combined 20 years (1992–2013) of monthly averaged sea-level anomalies from a multi-satellite altimetry grid and monthly tide-gauge observations to evaluate VLM rates. Uncertainties of VLM rates ranged from 0.3 to 7.4 mm/year, with a mean uncertainty of 0.9 mm/year. Breili et al. [7
] compared sea-level rates from two sets of altimetry data (1993–2016) along the Norwegian coast with sea-level rates estimated from 19 tide-gauge records corrected for VLM. The coastal averages of the sea-level rates were within errors, indicating that no systematic errors are present in the observations nor in the applied corrections.
In this study, we explore the potential of using tide-gauge observations in combination with data from the European Space Agency (ESA) CryoSat-2 (CS2) [8
] to calculate VLM rates at 20 tide gauges along the Norwegian coast. The CS2 data have several advantages we will benefit from when combining it with tide-gauge observations. First, due to the orbit configuration of CS2, areas north of 66°N are covered by observations and its 369-days repeat orbit implies dense sampling of the ocean. Furthermore, CS2 has a synthetic aperture interferometric radar altimeter (SIRAL) that can be operated in three modes: low resolution, synthetic aperture radar (SAR), and interferometric SAR (SARIn). In SAR and SARIn modes, the footprint in the along-track direction is ∼300 m, in contrast to ∼8 km in the low-resolution mode. Smaller footprints reduce the risk of data contaminated by back-scattered energy from land features in the coastal zones, and allow the sea-surface height to be sampled closer to land. As a result of this, CS2 SARIn observations are available in a zone stretching out ∼40 km off the coast [9
] (see Figure 1
b), opposed to conventional altimetry that provides observation points, which are on average ∼54 km from the Norwegian tide gauges [10
] and need to be extrapolated towards the coast. Our results will be compared to independent VLM rates from a land-uplift model based on GPS, levelling, and geophysical modeling. We will test the existence of systematic errors in our results and investigate whether CS2 in combination with tide-gauge data is able to sample the spatial variation in VLM along the Norwegian coast. In addition to providing new estimates of VLM at Norwegian tide gauges, the present study demonstrates the coastal accuracy of CS2 data and its retracker.
describes data and methods used to determine VLM rates. Section 2.4
presents data used to validate our VLM rates, i.e., the absolute land-uplift model for the Nordic-Baltic region NKG2016LU_abs [11
]. Comparison results are shown in Section 3
. Finally, we discuss our results and give concluding remarks in Section 4
gives an overview of the amount of available 1 and 20 Hz CS2 SARIn observations within CS2 boxes as well as the mean difference and mean correlation between CS2 and tide-gauge time series computed over 20 Norwegian tide gauges. Generally, both 1 and 20 Hz CS2 time series agree well with the NMA tide-gauge time series. Averaged over all tide gauges, the 1 Hz CS2 time series show a standard deviation of differences and correlation of 11.9 cm and 0.82, respectively. The 20 Hz CS2 time series show a standard deviation of 12.1 cm and a correlation of 0.77. The agreement between 1 and 20 Hz CS2 time series with PSMSL is poorer. The 1 Hz CS2 time series show a standard deviation of 17.5 cm and a correlation of 0.53, while the 20 Hz time series show a standard deviation and correlation of 16.7 cm and 0.50, respectively. Hence, the tide-gauge time series from both NMA and PSMSL agree better with 1 Hz CS2 time series than with 20 Hz time series in terms of mean correlations.
Linear rates as well as associated uncertainties of VLM from the NKG uplift model and CS2 combined with tide-gauge records over the period 2010–2018 are illustrated in Figure 3
. The upper and lower panels show VLM rates based on 1 and 20 Hz CS2 time series, respectively. The NKG land-uplift model shows positive rates (from 1.3 mm/year at Måløy to 4.7 mm/year at Trondheim) for all 20 tide gauges. This is also confirmed by the majority of our estimated VLM rates. The rate uncertainties calculated by Equation (1
) take into account serial correlations of measurements. The uncertainties range from 3.1 to 27.1 mm/year when using 1 Hz CS2 sea-level anomalies and from 1.1 to 18.5 mm/year when using 20 Hz CS2 data. Largest uncertainties occur at tide gauges with few CS2 observations available, i.e., at Trondheim, Heimsjø, Oscarsborg, and Hammerfest (see Figure 2
lists key numbers from the comparison between our VLM estimates and rates from the NKG uplift model. VLM rates based on both 1 and 20 Hz CS2 data in combination with NMA tide-gauge measurements (VLM1HzNMA
) agree with the NKG uplift model within uncertainties for most of the sites. Their differences range from −13.9 to 8.1 mm/year. When using PSMSL tide-gauge data for the VLM estimation (VLM1HzPSMSL
), differences range between −23.2 and 16.3 mm/year. The standard deviations of differences between the NKG uplift model and rates of VLM based on PSMSL tide-gauge records are twice as large as the standard deviations of differences between the land-uplift model and rates from NMA tide-gauge data.
The last two columns of Table 2
list coastal averages of VLM and uncertainties defined in Equation (1
), where both quantities represent values averaged over tide gauges. We notice that the mean uncertainties when using 20 Hz CS2 observations are almost half of the mean uncertainties when using 1 Hz CS2 sea-level anomalies. The NKG uplift model has a coastal average of 2.8 mm/year over all 20 tide gauges. VLM1HzPSMSL
shows a coastal average of 4.4 mm/year and VLM1HzNMA
of 2.4 mm/year. VLM rates based on 20 Hz CS2 data show coastal averages of 5.5 mm/year (VLM20HzPSMSL
) and 3.4 mm/year (VLM20HzNMA
). We used the Welch’s unequal variances t
] (two-tailed, with
= 0.05) to check whether the coastal averages of estimated VLM rates are significantly different from the coastal average of the NKG uplift model rates. The average rates of our four VLM solutions along the Norwegian coast were not significantly different from the coastal average of the NKG uplift model at the 95% level. Though the coastal averages of VLM rates calculated with PSMSL data (VLM1HzPSMSL
) differ more from the coastal average of the NKG uplift model, they were not significantly different. This is most likely due to the high spread of the PSMSL-based VLM rates, and the fact that two most different estimates may pass the Welch’s unequal variances t
test if one of the two data sets has a large variance. The mean Spearman’s rank correlation coefficient between VLM estimates based on PSMSL data and the NKG uplift model is 0.53 when using 1 Hz CS2 data, and 0.46 when using the 20 Hz CS2 observations (Figure 4
a,c). Employing NMA tide-gauge records, the mean correlation between VLM rates and the NKG uplift model over all tide gauges is 0.58 and 0.43 for 1 and 20 Hz CS2 data, respectively (Figure 4
b,d). The mean correlations calculated for the Norwegian coast are lower than 0.77 obtained by Pfeffer and Allemand [3
] at 113 GPS sites. At the same time, they outperform the results by Nerem and Mitchum [2
] (0.35 at 33 nearby GPS and DORIS sites) and Ostanciaux et al. [6
] (0.40 at 57 GPS sites).
At Vardø, Harstad, Rørvik, Ålesund, Bergen, and Viker, for at least three out of four VLM solutions, differences to the NKG uplift model are maximum 5 mm/year or less. Although most of those tide gauges are inside fjords and land-confined (except Vardø and Viker), they show a good agreement to the NKG uplift model. At Honningsvåg, Hammerfest, Trondheim, Kristiansund, Måløy, Stavanger, and Tregde, the differences to the NKG uplift model range between 2 and 10 mm/year for three or four VLM solutions. Most of these tide gauges are land-confined, except Kristiansund and Tregde. In turn, high discrepancies between the NKG uplift model and VLM rates from CS2 and tide-gauge data are found at both tide gauges that are land-confined and located to the open ocean. For all four solutions, the largest misfits to the NKG uplift model are observed at Trondheim, Heimsjø, and Oscarsborg, all located deeply inside fjords. Leaving out these tide gauges, the coastal average of the NKG uplift model drops from 2.8 to 2.6 mm/year. Excluding Trondheim, Heimsjø, and Oscarsborg from the comparison reduces the minima and standard deviations of differences for all VLM solutions considerably (see lower part of Table 2
). A decrease in the coastal averages of estimated VLM rates is noted as well as reduced uncertainties. Unlike the mean values of differences calculated over all tide gauges, the mean values after excluding Trondheim, Heimsjø, and Oscarsborg are all positive, implying that all VLM solutions have consistently smaller rates than the observations in the NKG uplift model reflect. VLM1HzPSMSL
differ on average 1.4 and 1.5 mm/year from the NKG uplift model, respectively. VLM20HzPSMSL
differs on average 0.1 mm/year from the NKG uplift model and VLM20HzNMA
0.3 mm/year. The agreement of mean values between differences of estimated VLM rates and the NKG uplift model is also reflected in the agreement between their coastal averages. VLM rates based on 1 Hz CS2 data show coastal averages of 1.2 mm/year (VLM1HzPSMSL
) and 1.1 mm/year (VLM1HzNMA
), while estimates of VLM based on 20 Hz data have coastal averages of 2.5 mm/year (VLM20HzPSMSL
) and 2.3 mm/year (VLM20HzNMA
4. Summary and Discussion
We have assessed VLM at 20 tide gauges along the Norwegian coast by computing linear trends of differences between CS2 and tide-gauge time series for the period 2010–2018. Two tide-gauge data sets have been used: (i) monthly sea-level observations from PSMSL and (ii) 10-min sea-level measurements obtained from NMA. The agreement between the CS2 and tide-gauge time series is given in Table 1
. The resulting VLM rates with associated uncertainties are shown in Figure 3
and Supplementary Materials Table S1
, while the statistics of the comparison with the NKG uplift model is given in Table 2
and Figure 4
The 1 Hz CS2 data agree better with the tide-gauge time series than the 20 Hz data in terms of mean correlations. The agreement between the 20 Hz CS2 data and tide-gauge data from NMA (mean standard deviation of 12.1 cm and mean correlation of 0.77) represents encouraging improvements compared to the results in Idžanović et al. [22
]. For the same 20 tide gauges, they obtained a mean standard deviation of 14.7 cm and a mean correlation of 0.64 using standard CS2 corrections and the simple threshold retracker. Our results demonstrate that the SAMOSA 2 retracker has improved the coastal performance compared to the empirical retracker. Furthermore, the tide-gauge data with 10-min sampling interval agree significantly better with CS2 measurements than the monthly PSMSL time series. The smaller correlation and higher standard deviation between CS2 and the PSMSL time series (see Table 1
) can be explained by most different sampling rates. The 1 or 20 Hz sampling frequencies of CS2 imply that the altimetry observations include ocean signals, which are averaged to nearly zero in the monthly tide-gauge data. Consequently, differential ocean signals may be introduced in the series of differences between CS2 and the monthly tide-gauge data. This, in turn, may lead to less accurate VLM estimates and a possibly poorer fit to the NKG uplift model.
Our results are encouraging and suggest that CS2 in combination with the high-frequency NMA tide-gauge data can reflect the coastal average of VLM over 20 tide gauges along the Norwegian coast. At Honningsvåg, Kabelvåg, Rørvik, Mausund, and Viker, are differences between VLM rates based on NMA data and the NKG uplift model within the uncertainty of the land-uplift model. In addition, the agreement between the NKG uplift model and NMA-based VLM solutions indicates that there are no systematic errors in the Norwegian national sea-level observing system. Furthermore, the results obtained along the coast demonstrate that altimetry in combination with tide-gauge data can be used to determine VLM at tide gauges where there are no nearby GPS receivers nor rates available from a VLM model.
In general, the spread of rates from CS2 and tide gauges is larger than that of the NKG uplift model. This is especially notable for the PSMSL-based VLM rates, where the combination of CS2 with PSMSL tide-gauge records does not observe the VLM at some tide gauges. However, the NMA-based VLM solutions show a much smaller spread of differences to the NKG uplift model and a high spatial correlation. The comparison between estimated VLM rates and the NKG uplift model indicates that largest differences occur at tide gauges with an insufficient number of CS2 observations, i.e., at Trondheim, Heimsjø, and Oscarsborg. Excluding those tide gauges, the NKG uplift model shows a coastal average of 2.6 mm/year. Omitting Trondheim, Heimsjø, and Oscarsborg from the comparison, the standard deviations of differences decrease along with a significant drop in the mean uncertainties. In addition, eliminating Trondheim, Heimsjø, and Oscarsborg, the largest improvement is found for the PSMSL-based VLM solutions. Combining 1 and 20 Hz data with tide-gauge data provided by NMA gives coastal averages of 1.1 and 2.3 mm/year, respectively. VLM rates calculated combining tide-gauge measurements from PSMSL with 1 and 20 Hz CS2 data show coastal averages of 1.2 and 2.5 mm/year, respectively. In case of omitting the problematic tide gauges, we note a stronger dependence of VLM estimates to the CS2 sampling, and a better fit of VLM rates based on 20 Hz data to the NKG uplift model.
Different OT corrections applied to CS2 (FES2004) and tide-gauge measurements (OT corrections provided by NMA) are possible reasons for the misfit between the NKG uplift model and calculated VLM rates at some tide gauges. Particularly at Hammerfest, Trondheim, Heimsjø, and Kristiansund, discrepancies between signal standard deviations of FES2004 and local OT corrections within CS2 boxes are ranging from 6.7 to 28.5 cm. At these stations, we also find the largest differences to the NKG uplift model. In addition, the CS2 sea-surface observations come from multiple tracks within the CS2 boxes. Potential MSS errors will appear as SLA offsets between individual tracks, possibly introducing SLA errors [30
] and, in turn, affecting the VLM estimation. Especially at the coast, it becomes a problem where no observations are available, and the MSS is simply extrapolated. In general, the observed discrepancies between altimetric sea-level anomalies and tide-gauge sea level, and in turn the large spread of estimated VLM rates not seen in the NKG uplift model, might be due to an insufficient number of CS2 observations within CS2 boxes, instrument noise and complex ocean [22
] or other coastal processes (e.g., local subsidence not represented by the NKG uplift model).
The estimated errors of VLM rates are strongly dependent on the number of CS2 observations available in each CS2 box. Consequently, mean uncertainty estimates based on 20 Hz data (6.2 mm/year for VLM20HzPSMSL
and 3.2 mm/year for VLM20HzNMA
) are much smaller than those based on 1 Hz data (10.8 mm/year for VLM1HzPSMSL
and 6.2 mm/year for VLM1HzNMA
). Extension of the CS2 data span would improve the accuracy of the estimated VLM rates [5
]. A next step in the VLM estimation from CS2 and tide gauges could be a link of relative VLM between tide gauges, as presented in Kuo et al. [1
], using additional conditions and taking advantage of long-term tide-gauge records available in Fennoscandia. In addition, replacing the standard CS2 OT correction with a local one, as demonstrated in Idžanović et al. [22
], could possibly lead to a better agreement of estimated VLM rates with the NKG uplift model. Especially at tide gauges, where discrepancies between standard and local OT corrections are large. Furthermore, expanding the estimation of VLM rates using CS2 and tide-gauge measurements to the Baltic Sea region will be considered in the future, where the VLM signal reaches values up to ∼10 mm/year.