Tide gauge (TG) records are affected by vertical land motion
(VLM), causing them to observe relative instead of geocentric sea level. VLM
can be estimated from global navigation satellite system (GNSS) time series,
but only a few TGs are equipped with a GNSS receiver. Hence, (multiple)
neighboring GNSS stations can be used to estimate VLM at the TG. This study
compares eight approaches to estimate VLM trends at 570 TG stations using
GNSS by taking into account all GNSS trends with an uncertainty smaller than
1

Tide gauges (TGs) measure local relative sea level, which means that they are
affected by geocentric sea level, but also by vertical land motion (VLM).
Knowing VLM at TGs is essential to convert the observed sea level into
a geocentric reference frame in which satellite altimeters
operate. TGs used in sea level reconstructions also require a correction for
VLM. The mean of VLM at TGs is not equal to that of the basin, and therefore
local VLM estimates are required to get an accurate estimate of ocean volume
change. The models for large-scale VLM processes, such as glacial isostatic
adjustment (GIA) and the elastic response of the Earth due to present-day
mass redistribution, are becoming more accurate. TGs are often only corrected
for the GIA signal, which typically reaches values of 10

One method to estimate VLM at TGs uses geodetic global positioning system
(GPS) receivers at fixed stations or Doppler Orbitography and
Radiopositioning Integrated by Satellite (DORIS) observations. Since many
other navigation satellites are currently providing range estimates as well,
we will refer to the GPS stations as global navigation satellite system
(GNSS) stations. Most studies compute GNSS VLM at TG stations from one of the
datasets by the University of La Rochelle (ULR)

A second way to observe VLM at TGs and to overcome the limitations of a sparsely
distributed GNSS network is differencing satellite altimetry and TG time
series, which we will refer to as ALT–TG time series from here on. Initially,
the ALT–TG time series were used to monitor the stability of satellite
altimeters for the global mean sea level (GMSL) record, which is currently
guaranteed up to 0.4

Recently, several studies have compared the GNSS trends to those of ALT–TG
globally

This study aims to further reduce the discrepancies between GNSS and ALT–TG
trends, while increasing the number of trend pairs. To do this, we will apply
several steps to improve the VLM estimates at tide gauges. First of all, the
number of reliable trend estimates is increased by using the GNSS trends
from the larger NGL database. Most TGs will neighbor multiple GNSS stations
for which several methods are applied to determine the best procedure.
Correlations between altimetry and TG time series are exploited to reduce
residual ocean variability, which is often present in ALT–TG time series

In this section, we describe the processing procedures for deriving GNSS and
ALT–TG VLM trends for comparison at TG locations. First, we will address the
estimation of GNSS trends at the TG locations. The estimation of ALT–TG
differenced trends is discussed in several steps. We briefly discuss the
selection of the tide gauges. After that we will discuss the altimetry
processing procedures. We briefly review the Hector software

The trend estimation at tide gauges primarily deals with two problems. First,
a trend is estimated from a GNSS time series, which contains an
autocorrelated noise signal and often undocumented jumps. We use
precomputed trends, of which the procedure is briefly reviewed in
Sect.

To obtain VLM trends at TGs, often the products of the
University of La Rochelle (ULR) are used. ULR versions 5 and 6 make use of
the Create and Analyze Time Series (CATS) software

In this study the results of NGL

Despite several recommendations to colocate GNSS
receivers with TGs, currently only a few have a record that ensures a trend
uncertainty of 1

Most studies simply average all neighboring TG trends or take the trend from
the closest station. However, many other and possibly better techniques are
possible. We compare trends from several approaches in
Sect.

Monthly TG data are obtained from the PSMSL database

List of geophysical corrections and orbits applied in this study.

Time series of ALT–TG differenced VLM at Winter Harbour. After averaging or weighting with the correlation a moving-average filter is applied to visualize the remaining interannual variability. In blue: without a threshold on the correlation and without correlation weighting. In red: with a threshold of 0.7 for the correlation and with correlation weighting. In the background are the time series without the moving-average filter applied.

VLM (

Range (

Statistics of trend differences between NGL and ULR5 at 70 stations for the eight approaches.

Number of TGs at which trends are estimated from differenced ALT–TG
time series. The “

To overcome the limitations of gridded products, we work with along-track
data and exploit the correlations between sea level at the satellite
measurement location and at the TG on interannual and decadal scales by using
a low-pass filter. We start by creating sea level time series every
6.2

Additionally, intermission biases between TP–J1 and J1–J2 are removed.

The Jason satellite series samples sea level every 10 days, and hence we average
three to four measurements in order to make a first set of time series that is
compatible with the monthly TG observations. As for the case of the TG
monthly solutions, observations more than 1

The monthly low-pass-filtered altimetry time series are kept if the
corresponding correlations from yearly low-pass-filtered time series are above
a certain threshold. We combine the remaining monthly altimetry time series
to get one averaged altimetry time series per TG. Alternatively, we also use
the correlations as weights to get one correlation-weighted altimetry time
series per tide gauge. In this case the monthly low-pass-filtered time series
are weighted by their corresponding correlation, then added together and
accordingly normalized so that the weights sum up to one. The resulting time
series are subtracted from the TG time series if there are at least
10 altimetry time series with a correlation above the threshold. The resulting
differenced ALT–TG time series with less than 15

Change in SD (mm) of the differenced time series using correlation thresholds and weighting. Note that a correlation threshold of 0.0 indicates positive correlations only.

Time series of ALT–TG differenced VLM at the Llandudno (UK) TG. A moving-average filter is applied to visualize the interannual variability. In blue: with a threshold of 0.0 for the correlation, but without correlation weighting. In red: with a threshold of 0.0 for the correlation and with correlation weighting. In the background are the time series without a moving-average filter applied.

ALT–TG trends (mm

RMS (

The ALT–TG time series have a monthly resolution, so they contain fewer
observations, and they exhibit substantial interannual variability. These
time series are therefore less suitable to be processed with the MIDAS
algorithm used to compute GNSS trends. For the computation of the ALT–TG
trends and the corresponding SD, we fit a power law in combination with
a white noise model by using the Hector software

The trends estimated from GNSS time series are computed over different time
spans than the ALT–TG trends and will be affected by nonlinear VLM induced
by elastic deformation due to present-day ice melt and changes in land
hydrology storage

This section first addresses the trends obtained from GNSS stations. The
averaging methods are discussed and the NGL trends are compared to those of
ULR5. Then the results of the correlation-weighted ALT–TG trends are
discussed. These are compared to those from

For 570 TGs at least one GNSS station is found within a 50

Even though the large-scale GIA process appears to be captured properly,
regional VLM has a large effect on the GNSS trends. In
Fig.

Using correlation thresholds, we try to minimize the residual ocean signal in
ALT–TG time series. Additionally, it will filter problematic stations when
no correlation between TG and altimetry observations is found. A higher
threshold therefore reduces the number of ALT–TG trends.
Table

In order to show that the method decreases the oceanic signal, we compare the
SD reduction by using correlation thresholds and weighting
(Fig.

Similar patterns of SD decrease, albeit reduced in magnitude, are observed
for the unweighted against the weighted VLM time series with a correlation
threshold of 0.0 (Fig. 4b), i.e., when only
positively correlated altimetry time series are taken into account. Instead
of 344 VLM trends, as for the comparison discussed above, 660 trends are
compared. The mean reduction of the SD is 1.4

Figure

Compared with the GNSS trends, the neighboring ALTG–TG trends show more
variation, which is especially true for the UK and Japan. It is difficult to
say whether this is a true VLM signal, but it is important to note that many
GNSS stations are placed on bedrock, which exhibits more stable trends than
the coastal locations of tide gauges. Secondly, the GNSS trends with an
uncertainty larger than 1

The results of applying correlation weighting and thresholding are shown
Fig.

Statistics of the differences between the median of the GNSS trends
(approach 2) and the ALT–TG trends for various correlation thresholds. The
“W” indicates that the altimetry time series are weighted by the
correlation. The row “W&M” shows the comparison with

Histogram of GNSS and ALT–TG trend differences. In blue are the results without any correlation threshold and in red with a correlation threshold of 0.7 and correlation weighting.

Trend differences (mm

In this section the VLM trends from GNSS using the eight approaches as
described in Sect.

Figure

In Table

Statistics of ALT–TG trend differences with the median GNSS approach for various correlation settings after applying a correction for nonlinear VLM.

On the right side of the table, we only included TGs for which all solutions
are available, which reduces the number from 155 to 137 because W&M
trends are also considered for comparison. The RMS of differences for 155
stations is only slightly larger as shown in
Table

Increasing the correlation threshold only slightly reduces the RMS between
GNSS and ALT–TG trends and the additional weighting has a neglectable effect
on the RMS. As mentioned before, the threshold increase and correlation
weighting generally reduced the SD (Fig.

The third column of Table

There is a nonlinear VLM signal due to present-day mass loss in both GNSS
and ALT–TG trends and since they cover different time spans this causes small
systematic differences between trends. Due to the inhomogeneous distribution
of the TGs and the spatial signal of nonlinear VLM, this affects not only
the mean, but also the skewness of the distribution. In
Fig.

We applied a correction for the effect of present-day mass loss to the trends
for the 155 stations for which a trend is found with all methods in
Table

We presented new ways to estimate VLM at TGs from GNSS and differenced ALT–TG
time series. A comparison is made between eight different methods to obtain
VLM at the TG from NGL GNSS trends. The range of the trends between the
approaches is at the same level as the SDs of the GNSS trends, with a mean of
0.92

For the ALT–TG trends we used along-track data from the Jason series of
altimeters. At every 6

The SD of the ALT–TG time series was reduced on average by approximately
10

The comparison with tide gauges also reveals that the trends from ALT–TG are
biased low (similar to

The trends in this publication (median GNSS and ALT–TG for all correlations)
are provided in the Supplement. The ALT–TG trends are accompanied by errors
bars computed using the Hector software. The provided uncertainties for the
GNSS use the MAD from the median of the trends within 50

The MIDAS GNSS trends are obtained from the Nevada Geodetic
Laboratory (NGL;

The latitude-dependent intermission biases are computed from

Values for the parameters of the latitudinal
intermission bias correction. These numbers are added to the sea surface
height anomalies of the respective satellites. “TP asc.” and “TP desc.” indicate
the function variables that should be added to the ascending and descending
tracks, respectively, of TOPEX/POSEIDON using Eq. (

The authors declare that they have no conflict of interest.

This study is funded by the Netherlands Organisation for Scientific Research (NWO) through VIDI grant 864.12.012 (Multi-Scale Sea Level: MuSSeL). We would like to thank Marta Marcos and Guy Wöppelmann for sharing their trend estimates. We thank Alvaro Santamaría-Gómez and an anonymous reviewer for their thorough reviews that helped to improve this article. Edited by: John M. Huthnance Reviewed by: Alvaro Santamaría-Gómez and one anonymous referee