Introduction
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 mmyr-1
in Canada and Scandinavia . The elastic deformation due
to present-day mass redistribution is often ignored. However, elastic
deformation is becoming larger due to the increasing rate of Greenland's ice
mass loss and to a lesser extent other processes. Trends at TGs are also
affected by a large number of other local signals, including water storage,
post-seismic deformation and anthropogenic activities
. Since the local VLM processes cannot
be captured by models and the large-scale processes contain large
uncertainties, observations of VLM at TGs are essential.
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)
. Even though
ULR contains several GNSS solutions inland, its main focus is the coastal
zone. Currently, 754 GNSS stations are processed in the ULR6 database. A more
extensive database with approximately 14 000 GNSSs is processed by the Nevada
Geodetic Laboratory (NGL). They use a different processing procedure to
estimate trends from time series, which makes trends less vulnerable to jumps
. A statistical comparison between several GNSS solutions
was recently made by . They concluded that the number
of stations in the NGL database was larger, but that the differences between
neighboring stations was significantly larger than the Jet Propulsion
Laboratory (JPL) and ULR6 trend estimates. They also discussed systematic
errors due to differences in the origin of the reference frames, which were
on the order of 0.2 mmyr-1 globally. Furthermore, they found
that the local VLM uncertainty at the tide gauge was increased by 4×10-3 mmyr-1 per kilometer of distance between the TG and the
GNSS station . Most studies use the trends of either
colocated GNSS stations, the closest GNSS station or the mean of all GNSS
stations within a radius of several tens of kilometers
. Only involved
a more complex GNSS post-processing procedure using NGL trends based on
a combination of spatial filtering, Delaunay triangulation and median
weighting. One way to quantify the accuracy of GNSS-based VLM trends at TGs
is to compute the spread of individual geocentric sea level estimates or the
spread of geocentric sea level between regions . The
spread of regional trends reduced from 0.9 mmyr-1 in the ULR1
solution to 0.5 mmyr-1 in the ULR5
solution , which is approximately the
expected residual climatic signal. Any further improvements in the GNSS
trends therefore require another validation technique.
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 mmyr-1 . The
first study to infer VLM trends from ALT–TG time series was
. Based on the method of they
compared ALT–TG to DORIS at six stations. Later, several studies were
conducted on the regional and global scale of which an overview is given by
. The first study to estimate more than 100 VLM trends
obtained error bars for 60 of 114 TGs smaller than
2 mmyr-1. However, they noted that the TGs should be inspected
on a case-by-case basis to determine if the result was truly VLM.
increased the number of ALT–TG VLM trend estimates
sixfold to 641, but it included some outliers with trends above
20 mmyr-1. They also made a comparison between their study and
several earlier studies. The best agreement was found over a small set of 28
tide gauges, for which the results of differed from
by an RMS of 1.2 mmyr-1.
Recently, several studies have compared the GNSS trends to those of ALT–TG
globally . Several other
studies did an equivalent comparison with DORIS and ALT–TG for a limited
number of stations . While the older
studies primarily used along-track data from the Jason (TOPEX/POSEIDON: TP,
Jason-1: J1 and Jason-2: J2) series of satellite altimeters, the latest
studies used preprocessed grids, and made a comparison
between several gridded products and one along-track dataset. All recent
studies used ULR5 GNSS trends for comparison. The best results were obtained
with an interpolated altimetry grid provided by AVISO ,
yielding a median of differences of 0.25 mmyr-1 with an RMS of
1.47 mmyr-1 based on a comparison at 107 locations
. It is important to note that the time series for all
sites were visually inspected, primarily to remove those with nonlinear
behavior. Additionally, the corresponding correlations between altimetry and
TG time series were found to be highest for AVISO. did
not apply visual inspection and obtained a comparable result for 113 stations
(an RMS of 1.7 mmyr-1), while only incorporating GNSS trends
from stations within 10 km from the tide gauge.
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
. The reduction in ocean variability should lead to
more reliable ALT–TG VLM trends. Correlation thresholds additionally function
as a filter to remove time series that are uncorrelated due to differences
in ocean signals, possible (undocumented) jumps in the TG time series or
interannual VLM signals that cannot be separated from the ocean signal
. Additionally, we address the problem of contemporary
mass redistribution on trends over different time spans using
a fingerprinting method.
Data and methods
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
for the estimation of trends from differenced ALT–TG time series. Eventually,
trend corrections for contemporary mass redistribution using fingerprinting
methods are described.
GNSS trends
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. . Second, many GNSS stations are not directly colocated
to the TG station. Regular leveling campaigns to monitor the relative VLM
between the TG and the GNSS stations are often absent. Therefore, the
assumption is made that both locations are affected by the same VLM signal.
When multiple GNSS receivers are present in the vicinity of the tide gauge,
a method is required to estimate a single VLM trend from multiple GNSS
stations. This is discussed in Sect. .
GNSS trend estimation
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 ,
which is able to estimate trends and errors from time series by taking into
account temporally correlated noise. It has the advantage that it computes
a more realistic trend uncertainty. The software is also able to estimate and
detect discontinuities that occur due to earthquakes and equipment changes.
Even though a large proportion of the trend estimates have formal accuracies
better than 1 mmyr-1, undetected discontinuities might bias the
estimated trends .
In this study the results of NGL are used.
proposed the Median Interannual Difference Adjusted for
Skewness (MIDAS) approach, which is based on the Theil–Sen estimator. The
procedure estimates trends from couples of daily data points separated by
365 days. It then removes all estimates outside 2 SD, which are
computed by scaling the median of absolute deviation (MAD) by 1.4826
with respect to the median of the trend couples.
Afterwards, a new median is computed, which serves as the trend estimate.
demonstrated that MIDAS has a smaller equivalent step
detection size than methods that include step detection, such as those
computed by CATS and used by ULR5. Besides the advantage of detecting smaller
jumps, approximately 14 000 GNSS time series are processed, which is almost
20 times more than ULR6. Unlike , no manual screening
is applied to the time series or trends.
Trend estimation at tide gauges
Despite several recommendations to colocate GNSS
receivers with TGs, currently only a few have a record that ensures a trend
uncertainty of 1 mmyr-1 or better. Therefore we take all
stations into account that are within 50 km from a TG, provided that
the SD on the trend is lower than 1 mmyr-1 as estimated from the
MIDAS algorithm. The threshold on the SD ensures that most records containing
large nonlinear effects due to, for example, earthquakes and water storage
changes are removed from the analysis. Other studies used ranges from
10 km up to 100 km
. At 100 km the error due to relative VLM trends
increases substantially, on average more than 0.5 mmyr-1
for the NGL estimates, while taking a range of
10 km reduces the number of trends substantially. Therefore the range
is set to 50 km, but comparable results are found for 30 and
70 km, yielding a different number of trends (not shown).
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. and with the ALT–TG trends in
Sect. . In total eight different approaches are
considered. The first two involve all of the trends at neighboring GNSS
stations by computing their mean (1) and median (2). Method (1) is
applied by for regional sea level
reconstructions. One of the most frequently applied approaches uses the trend
at the closest station (3). It is used in two recent studies by
and . We also investigate inverse
distance weighting (4) in which the trend
dhTGdt is estimated as
dhTGdt=∑1didhidt∑1di,
where di and dhidt represent the
distance to the tide gauge station and the trend at GNSS station i. We also
use the GNSS trends based on the longest time series (5) and smallest error (6)
from stations within the 50 km radius. The seventh approach
involves weighting with the variances σi2 of the trends (7) such
that
dhTGdt=∑1σi2dhidt∑1σi2.
And the last method (8) takes into account spatial dependency and trend
uncertainty by combining methods (4) and (7), i.e., by weighting with the
variance and with the distance so that
dhTGdt=∑1σi2didhidt∑1σi2di.
Method (8) is a variant to the technique used in the altimeter calibration
study of . Note that the uncertainties range mostly
between 0.7 and 1 mmyr-1 and therefore method (8) is more sensitive
to the distance from the TG than to the variance of the GNSS trends. The
distance weights used in methods (4) and (8) quickly decrease with distance,
effectively reducing the number of GNSS trends involved in the estimate. In
several studies the method to estimate VLM trends at tide gauges from GNSS is
not documented.
Tide gauge time series
Monthly TG data are obtained from the PSMSL database . All
time series flagged after 1993 are removed. Any observations that are outside
of 1 m from the mean are considered outliers and removed from the
data. This number is similar to our altimetry sea level threshold and based
on the criterion used by NOAA for their global mean sea level estimates
. To be consistent with the altimetry observations, we
apply a dynamic atmosphere correction (DAC) consisting of a low-frequency
inverse barometer correction and short-term wind and pressure effects
. Initially, we consider all TGs with at least
10 years of valid data.
List of geophysical corrections and orbits applied in this study.
Satellite
TP
J1 and J2
Orbits
CCI
GDR-E
Ionosphere
Smoothed
dual-frequency
Wet troposphere
Radiometer
Dry troposphere
ECMWF
Ocean tide
GOT4.10
Loading tide
GOT4.10
Solid Earth tide
Cartwright
Sea state bias
CLS
Mean sea surface
DTU15
Dynamic atmosphere
MOG2D
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 (mmyr-1) at TGs using the median of the neighboring
trends.
Range (mmyr-1) of VLM estimates at TGs using eight
different approaches. The size of the symbols indicates the number of GNSS
trends available (with a maximum of 10).
Statistics of trend differences between NGL and ULR5 at 70 stations
for the eight approaches.
RMS
Mean
Median
Approach
Keyword
mm yr-1
mmyr-1
mmyr-1
1
Mean
1.11
0.07
0.05
2
Median
1.05
0.12
0.03
3
Closest
1.36
0.02
0.02
4
Dist. weight
1.21
0.00
0.03
5
Longest
1.29
0.32
0.20
6
Smallest error
1.15
0.24
0.17
7
Error weight
1.11
0.08
0.02
8
Dist. and error weight
1.23
0.01
0.05
Number of TGs at which trends are estimated from differenced ALT–TG
time series. The “-1.0” indicates that no correlation threshold is set.
Threshold
Number of TGs
-1.0
663
0.0
660
0.1
658
0.2
655
0.3
638
0.4
602
0.5
549
0.6
470
0.7
344
Differenced ALT–TG time series
obtained the smallest SD in the differenced time
series by averaging grid cells within 1∘ from the TG using the
AVISO interpolated product. The results obtained by taking the most
correlated grid point from AVISO within 4∘ around the TG
increased the SD. obtained lower correlations by
averaging Goddard Space Flight Center (GSFC) along-track altimetry
measurements within a radius of 1∘ from the TG. Note that the
AVISO grid is constructed using correlation radii of 50–300 km
and it includes measurements from all altimetry satellites,
not only the Jason series. The AVISO grid therefore effectively averages over
a much larger radius around the TG and it includes data from more satellites.
The larger uncorrelated noise using GSFC compared to AVISO, as shown by the
combination of the increased RMS and the spectral index
, is therefore likely an effect of the limited number
of GSFC altimetry measurements. However, using the large effective radius of
AVISO, data far away from the TG are included, which might not correlate with
the sea level signal at the TG.
This can result in a remaining ocean signal in ALT–TG time series, which contaminates the VLM trend estimates.
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 km along-track using the measurements from TP, J1 and J2 from the
RADS database between 1993 and 2015. In order to get
a consistent set of altimetry observations, the same geophysical corrections
are used for all satellites, as given in
Table . All time series within 250 km
from the TG are taken into account. This radius is larger than the open ocean
correlation distances used by and ,
except for the equatorial region where the correlation scales become much
larger. At distances larger than 250 km, one will still find some
highly correlated signals, but the trends caused by large-scale processes
like GIA and present-day mass redistribution will differ from those at the
TGs. It also ensures that at least one ground track of the altimeters is
within the range of the tide gauge at the Equator. Reducing the
250 km radius leads to a decreased number of trends.
Additionally, intermission biases between TP–J1 and J1–J2 are removed.
revealed a large dependence of the intermission biases on
the latitude. For the J1–J2 differences, a single polynomial is estimated
through the differences between the sea level observations of both
instrument such that the correction Δhsla,ib(λ) becomes
Δhsla,ib(λ)=c0+c1⋅λ+c2⋅λ2+c3⋅λ3+c4⋅λ4,
with λ as the latitude of the altimetry observations. For the TP-J1
differences, separate polynomials are estimated for four latitude regions and
the ascending and descending tracks . The values for the
parameters cn are given in Table . More details on
the computation procedure are found in Appendix .
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 m from the mean sea
surface are removed and the time series should have at least 10 years
of valid observations. Additionally, a second set of time series at each
satellite measurement location is created by applying a yearly
moving-average filter. This second set of altimetry time series is correlated
with a yearly low-pass-filtered version of the TG series in order to test
whether their signals match on interannual and longer timescales. The yearly
moving-average filter allows us to suppress the noise present in individual
altimetry measurements. The full pole tide from RADS (which contains a solid
Earth, loading and ocean tide as in ) is subtracted from
both time series before correlation, whereas for the TG time series we
restore the solid Earth pole tide as computed in . The
loading tide is at its maximum only a few millimeters, which has no
significant effect on the interannual correlation and is therefore not
restored. We also remove residual annual and semi-annual cycles and a linear
trend before correlation because the yearly moving-average filter has
side lobes, causing these seasonal signals to be partly retained. Other longer
filters are considered to reduce the side lobes, but they would introduce
larger transient zones. An iterative procedure removes sea surface heights
outside of 3 RMS up to a maximum of 10 % of the observations. The
outlier removal is primarily implemented to remove any spurious data present
in the RADS database. It is unlikely that more than 10 % of the
observations contain processing problems or outliers due to extreme events.
If more observations were discarded, high correlations might no longer
represent the corresponding ocean signal. The result is a set of
correlations that indicate which altimetry sea level time series resemble the
TG time series on interannual timescales and longer.
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 years of valid
observations are further discarded. This last requirement is due to the fact
that remaining ocean signals can still affect the estimated trends
significantly. An example of the reduction of variability due to correlation
thresholds and weighting is shown in Fig. . The
white noise in the unfiltered time series is reduced in the red curve;
however, the opposite might happen if the number of altimetry time series
decreases. It is most important to note that there is a strong reduction in the
variance of temporally correlated residuals, represented here by the low-pass-filtered time series. A correlated residual signal can strongly affect the
estimated trend, especially in areas with large variability due to
interannual events like ENSO. Note that for the differentiation of the time
series only the solid Earth part of the pole tide is added to the TGs,
as is done in the IERS 2010 conventions such that
the trends are consistent with those of the GNSS data. The main difference is
that the altimetry pole tide correction of is computed with
respect to a linearly drifting mean pole, while in the IERS conventions the
mean pole location is modeled as a third-order polynomial. If the pole tide
is not taken into account consistently, it can introduce biases of
0.1 mmyr-1 . Since the change rate of the
mean pole is nonlinear, this will introduce trend biases if the time spans
between GNSS and altimetry do not match. The drift of the mean pole is caused
by the redistribution of mass in the Earth system. This is corrected by using
the mass redistribution fingerprints discussed in Sect. , which are
computed using a model that includes elastic responses and rotation changes.
The drifting mean pole is primarily captured by the C21 and S21
spherical harmonic coefficients .
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 yr-1) estimated using no threshold (a),
with a correlation threshold and correlation weighting (b) and the
difference between them (c).
RMS (mmyr-1) of differences between GNSS and ALT–TG VLM trends.
The “W” indicates weighting by correlation. The “-1.0” indicates that no
correlation threshold is set. The numbers of the y axis refer to the
approaches used to combine the GNSS trends as described in
Sect. .
Differenced ALT–TG trends
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
spectrum of the white noise is flat, while the spectrum of power-law noise,
P(f), decays with frequency and is given by :
P(f)=1fs2σ2(2sin(πf/fs))2d,
where fs is the sampling frequency, σ the power-law noise scaling
factor and d links to the spectral index κ in
by κ=-2d. The value of d affects the effective number of
autoregressive parameters . This is required to capture the
temporal correlation in the ALT–TG time series as shown by Fig.
in which the low-pass-filtered time series give an idea of the memory in the
system. In order to handle several weakly nonstationary ALT–TG time series
we use the function “PowerlawApprox”, which uses a Toeplitz approximation
for power-law noise .
Contemporary mass redistribution
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 . To quantify those nonlinear VLM signals,
the response to mass redistribution is computed using a fingerprinting method
at yearly resolution. We take into account the loads of Greenland and
Antarctica,
glacier mass loss, the effects of dam retention and hydrological loads.
A detailed description of the input loads is given in .
To estimate the fingerprints of VLM, the sea level equation is solved,
including the rotational feedback . Since not
all load information for 2015 and 2016 is available yet, we will limit the
time series of ALT–TG up to 2015. Some GNSS trends are estimated from time
series that span beyond 2015. Therefore we linearly extrapolate the
fingerprint data, if necessary, to 2015 and 2016 based on the difference
between the years 2013 and 2014.
Results
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 . After
that, the GNSS and ALT–TG trends are compared and optimal settings are
discussed. For the comparison we take into account the fact that both trends are not
computed from time series covering the same period by correcting for
nonlinear VLM trends estimated from fingerprints.
Direct GNSS trends
For 570 TGs at least one GNSS station is found within a 50 km radius
with an uncertainty on the trend that is below 1 mmyr-1. The VLM
for these TGs is shown in Fig. using the median of the
surrounding GNSS stations in case there are multiple trends available. The
signature of GIA dominates the signal on large scales and is primarily
visible in Scandinavia and Canada. In Alaska there might be a significant
contribution of present-day ice mass loss. If GIA is removed the VLM signals
typically range between -3 and 3 mmyr-1
, with a few exceptions.
Even though the large-scale GIA process appears to be captured properly,
regional VLM has a large effect on the GNSS trends. In
Fig. the differences between the lowest and highest VLM
estimate from the eight methods discussed in Sect. are
shown. The extreme values primarily resulted from the “mean”, “median”
and “inverse distance” methods (not shown). The figure shows that the range
is generally higher when more GNSS trends are available. In particular the
seismically active zones like the US West Coast show a larger range. The
range of solutions, when considering all TGs with at least two GNSS trends,
has a mean of 0.92 mmyr-1 with 25th and 75th percentiles of 0.38 and
1.20 mmyr-1. In the case that at least three available GNSS trends are
considered, the mean of the differences rises to 1.09 mmyr-1 and
the 25th and 75th percentiles to 0.56 and 1.34 mmyr-1. Since we only
considered GNSS trends with a maximum SD of 1 mmyr-1, this
implies that a significant contribution of kilometer-scale VLM variations is
present along the West Coast of the US, where the difference between methods
is often larger than 1 mmyr-1. Note that the range of individual
GNSS trends is on average even larger than the range between methods.
estimated the global numbers for the impact of spatial
variations in VLM at 30 and 100 km of separation to be 0.2 and
0.5 mmyr-1. On the coasts of Europe and North America where most
tide gauges are located, these numbers are substantially larger; i.e., even
the range between methods is on average larger than 1 mmyr-1.
The differences between methods are often comparable in size to the VLM
signal, especially after the GIA is removed.
show that a comparison between their ALT–TG trends and
their GNSS trends yields an RMS of 1.47 mmyr-1. They use visual
inspection to remove tide gauges when clear nonlinear effects or
discontinuities were present. In Table a comparison is
made between the eight different approaches and the GNSS trends of
that were used in the aforementioned comparison with
ALT–TG trends at 70 locations. The values show that a substantial fraction of
the RMS between GNSS and ALT–TG trends can be explained by different
GNSS averaging and processing methods. Using the closest station (approach 3)
yields an RMS of 1.36 mmyr-1, which is comparable in magnitude to the
RMS between GNSS and ALT–TG trends found by . Note that
we remove all NGL GNSS trends with an uncertainty larger than
1 mmyr-1 and therefore colocated stations are sometimes
removed. The closest GNSS station in our selection is therefore not always
the same as the one used by . The best comparison is
found with the median (approach 2), even though the RMS of differences is
still above 1 mmyr-1. Since the closest station method depends
on a single station, there is a larger chance that some outliers are present, which
substantially increases the RMS of differences. For the closest station
method three trend differences larger than 3 mmyr-1 are found,
whereas only one is found for the median method.
Differenced ALT–TG trends
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 shows the reduction of the differenced VLM trends
when the correlation threshold increases. After a correlation threshold of
0.4, the number of observations drops substantially. At a threshold of 0.7,
the number of TGs for which a trend is computed is only half of that without
a threshold. The remaining trends are generally more reliable for two
reasons: VLM time series that exhibit relatively large residual ocean signals
are removed, and TG time series that contain large jumps due to
unidentified reasons (e.g., earthquakes or equipment changes) are removed.
In order to show that the method decreases the oceanic signal, we compare the
SD reduction by using correlation thresholds and weighting
(Fig. ). The plot in Fig. 4a shows the comparison
between the SD of the differenced time series using no correlation threshold
and the time series using a threshold of 0.7 together with a correlation
weighting. The mean reduction in SD is 3.9 mm, whereas the mean SD is
37 mm. The change in SDs at several locations are coherent, which is
expected because the sea level fluctuations along continental slopes are
coherent . Substantial reductions in SD are apparent on
both North American coasts, in Japan and in Northern Europe.
had already observed large discrepancies in
interannual ocean signals between TGs and altimetry in North America and in
Japan. This suggests that our technique is capable of reducing these ocean
signals, which is confirmed by the change in the median of the spectral
indices, κ, as discussed in Sect. . The median
of the spectral indices changes from -0.63 to -0.57, which indicates that
the autocorrelation in the residuals decreased. The Winter Harbour (Canada)
VLM time series (Fig. ) shows a typical example
in which the correlated noise is reduced. However, there are
several locations where the SD increases substantially. Most of them are
sporadic, but in a few locations, like in the UK and France, there is a coherent
increase.
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 mm, whereas the mean SD
is 38 mm. The strong reduction of the SD at the
southeast side of Australia is notable. In the UK and France an increase in SD is
present again. In most cases an increase in white noise, likely due to the
decreased effective number of altimetry measurements, is responsible for the
higher SD, as demonstrated in Fig. for a VLM
time series at Llandudno, UK. In most cases of an increasing SD, the
correlated ocean signals are still reduced or remain approximately equal.
Figure shows the VLM trends estimated from the ALT–TG
time series using no correlation threshold and a threshold of 0.7.
A comparison of Figs. and reveals that
the Indian Ocean and the southern Pacific Ocean are sampled better
using ALT–TG instead of GNSS trends. If the correlation threshold is set to
0.7, the number of trend estimates decreases, which particularly impacts
the number of trend estimates at TGs in South America and Africa.
Hence, for regional reconstructions, a careful choice should be made for the
correlation threshold.
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 mmyr-1 are removed, which reduces the
variability. Of the 663 ALT–TG trends, 293 (44 %) have a trend
uncertainty smaller than 1 mmyr-1. Therefore larger spatial
trend variability can also be induced by remaining ocean signals in the VLM
time series. In Fig. b showing the 0.7 threshold
trends, the number of trends is reduced due to the correlation threshold. It
removes most tide gauges in the highly variable regions previously mentioned and
the neighboring differences are therefore less erratic;
284 out of 344 trends (83 %) have a trend uncertainty smaller than 1 mmyr-1 using the 0.7 correlation threshold.
The results of applying correlation weighting and thresholding are shown
Fig. c. Two spots of coherent changes in the trends can
be clearly identified: in Norway the trends increased by approximately
1 mmyr-1, while on the East Coast of the US the opposite
happens. These spots exhibit longshore coherent sea level signals that are
not found in the open ocean . Note that both
locations also exhibit a strong reduction in standard deviation
(Fig. ). Coherent changes are also present around
Denmark. Other regions where substantial reductions in the SD are found do
not experience coherent changes in trends.
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
trends. The column “NoT” indicates the number of TGs
for which trend estimates are computed. On the left side of the table all
stations are taken into account, and on the right side only stations are taken
into account for which a solution exists for all correlation thresholds (including those from W&M).
All
Same
Correlation
RMS
Mean
Median
NoT
RMS
Mean
Median
NoT
mmyr-1
mmyr-1
mmyr-1
mmyr-1
mmyr-1
mmyr-1
-1.0
2.141
-0.241
-0.107
294
1.234
-0.167
-0.099
137
0.0
2.108
-0.248
-0.101
294
1.226
-0.175
-0.068
137
0.0 W
2.103
-0.250
-0.036
294
1.219
-0.172
-0.056
137
0.1
2.113
-0.258
-0.096
293
1.219
-0.174
-0.074
137
0.1 W
2.108
-0.260
-0.043
292
1.218
-0.170
-0.045
137
0.2
2.082
-0.233
-0.073
292
1.217
-0.163
-0.074
137
0.2 W
2.080
-0.234
-0.015
292
1.216
-0.168
-0.042
137
0.3
1.986
-0.152
0.047
283
1.221
-0.157
-0.066
137
0.3 W
1.991
-0.157
0.056
283
1.217
-0.165
-0.044
137
0.4
1.695
-0.106
0.065
264
1.223
-0.152
-0.050
137
0.4 W
1.696
-0.112
0.071
264
1.218
-0.158
-0.041
137
0.5
1.554
-0.086
0.044
239
1.220
-0.153
-0.058
137
0.5 W
1.552
-0.087
0.056
239
1.217
-0.155
-0.067
137
0.6
1.417
-0.093
-0.065
204
1.209
-0.155
-0.087
137
0.6 W
1.416
-0.093
-0.083
204
1.208
-0.156
-0.094
137
0.7
1.220
-0.142
-0.123
155
1.206
-0.140
-0.060
137
0.7 W
1.220
-0.144
-0.124
155
1.206
-0.142
-0.074
137
W&M
1.658
-0.177
-0.050
211
1.328
-0.101
0.020
137
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 yr-1) between the GNSS and ALT–TG time spans
induced by nonlinear VLM due to present-day mass redistribution.
GNSS vs. ALT–TG trends
In this section the VLM trends from GNSS using the eight approaches as
described in Sect. are compared with the differenced ALT–TG
VLM trends using various correlation thresholds. Based on the intercomparison
we determine the best solution for the GNSS approach and the correlation
thresholds for altimetry. Additionally, a comparison is made with
. We also investigate the effect of present-day mass
redistribution on the difference in trends due to varying time spans of the
GNSS and the ALT–TG methods.
Figure shows the RMS of trend differences between various
GNSS combination methods and correlation thresholds for ALT–TG. The RMS of
trend differences is computed at 155 TG stations for which all solutions are
available. The colors exhibit small differences horizontally and large
differences vertically, indicating that the GNSS method is more important in
reducing the RMS. The difference between the method with the lowest RMS of
differences, which is obtained by taking the median of the GNSS trends (2),
and the method with the highest RMS, which uses the closest GNSS station (3),
is approximately 0.12 mmyr-1. computed
VLM trends at TG locations by using a complex filtering procedure that also
implicitly takes into account the median of the GNSS trends. Next to taking
the median of the GNSS trends, taking the mean (1) within the 50 km
radius and using variance weighting (7) also yields substantially lower RMS
differences than the other five methods. However, the median method performs
slightly better. The median method is also less sensitive to large values
caused by GNSS trends with larger uncertainties (for which the mean method is
sensitive) and less sensitive to outliers caused by large local VLM differences
(for which the variance weighting method is sensitive).
In Table we analyze the results for different correlation
thresholds in more detail by comparing them to the GNSS trends based on the
median method. On the left side of the table the RMS, mean and median are
shown for all VLM estimates available for each correlation threshold. Setting
no correlation thresholds yields trend estimates at 294 TGs for comparison,
while setting a threshold at 0.7 leaves only 155. While the number of trends
decreases, the RMS decreases as well, indicating that the correlation
thresholds can serve as a selection procedure that filters out outliers.
This is confirmed by Fig. , in which we see the decrease in
the number of available trends, but also the removal of the outliers. If the
threshold is set to 0.7 only three discrepancies in trends larger than
3 mmyr-1 are found. Note that the reduction in RMS is not only
caused by the removal of problematic ALT–TG time series. Large earthquakes,
for example, might induce jumps or nonlinear behavior in both the TG and
GNSS time series, so the larger range in Fig. for no
correlation threshold may be partly attributed to problematic GNSS trends. In
the last row the trends are compared with our GNSS
trends. There is a similar RMS with the 0.4–0.5 correlation threshold trends,
but it is computed with a substantially smaller number of trends.
Statistics of ALT–TG trend differences with the median GNSS approach
for various correlation settings after applying a correction for nonlinear
VLM.
NoT: 155
NoT: 137
Correlation
RMS
Mean
Median
RMS
Mean
Median
mmyr-1
mmyr-1
mmyr-1
mmyr-1
mmyr-1
mmyr-1
-1.0
1.231
-0.102
-0.039
1.223
-0.100
0.030
0.0
1.225
-0.109
-0.027
1.215
-0.108
0.031
0.0
1.223
-0.106
0.016
1.209
-0.105
0.048
0.1
1.220
-0.107
-0.014
1.208
-0.107
0.034
0.1
1.222
-0.104
0.003
1.208
-0.104
0.072
0.2
1.220
-0.099
0.016
1.207
-0.096
0.027
0.2
1.221
-0.101
-0.001
1.206
-0.101
0.059
0.3
1.223
-0.091
0.011
1.211
-0.090
0.018
0.3
1.221
-0.098
-0.001
1.207
-0.098
0.036
0.4
1.226
-0.087
0.011
1.214
-0.085
0.021
0.4
1.223
-0.092
0.008
1.209
-0.091
0.037
0.5
1.225
-0.088
0.020
1.212
-0.086
0.042
0.5
1.222
-0.090
0.027
1.208
-0.088
0.045
0.6
1.222
-0.087
-0.007
1.202
-0.088
0.018
0.6
1.222
-0.087
-0.006
1.201
-0.089
0.028
0.7
1.220
-0.071
0.021
1.202
-0.073
0.037
0.7
1.219
-0.074
0.012
1.201
-0.075
0.036
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 . Note that the RMS of the
residuals using ALT–TG from W&M is 0.14 mmyr-1 lower
than those in the study of and about
0.4 mmyr-1 less than in , who incorporated
only 109 and 113 stations, respectively. This is a consequence of the
combined use of the median of the NGL trends and selection based on
correlation. Our altimetry solutions further decrease the RMS by another
0.1 mmyr-1 compared to W&M, even when no threshold on the
correlation is set. In the study of , the along-track
altimetry ALT–TG trends performed worse than the AVISO results. The reason
for this discrepancy could be the latitudinal intermission bias or the small
radius around the TG used in that study for including altimetry measurements.
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. ) of the ALT–TG
time series and Fig. shows coherent changes in trend.
Additionally, the NGL and ULR trends showed an RMS of differences and range
between the GNSS approaches of more than a millimeter. We argue that the
absence of a clear improvement or a change in RMS due to correlation
thresholds is a result of the relatively large noise in the GNSS trends. The
histogram in Fig. shows that for 155 stations, only three
discrepancies are larger than 3 mmyr-1. For these TGs (located
at Galveston and Eureka in the US and the Cocos Islands in Australia) we find
that the neighboring GNSS stations are located at the other side of lagoons
or on different islands. Therefore the likely cause of the largest
discrepancies is not the ALT–TG trend, but local VLM differences between the
GNSS stations and the TG.
The third column of Table shows that the mean is in all
cases negative; i.e., the GNSS trends are larger than those of ALT–TG. Trends
obtained with correlations of -1.0, 0.0, 0.1 and 0.2 are barely statistically
different from zero based on a 95 % confidence level, while the
others are not. The 95 % confidence level is taken as 2 times the
SD of the mean of the residual trends σnN, where N
is the number of trends and σn the SD of the residual trends). In the
right “mean” column for the 137 stations, the means are statistically
insignificantly different from zero at the 95 % confidence level,
whereas at a 90 % confidence level several are not.
The medians in both columns are closer to zero and deviate up to 0.2 mmyr-1 from the mean, which indicates a slightly skewed distribution.
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. the trend differences between the GNSS and ALT–TG
methods are visualized for all 294 stations. Most of the negative differences
in trends are observed in Europe and parts of North America, while positive
differences in trends are observed in Australia. In Europe there is an uplift
due to present-day mass loss, which increases over the last few years. Since
the GNSS time series are generally shorter, they measure a larger uplift
signal. By subtracting the present-day VLM that GNSS observes from altimetry
observations, we obtain negative signals in Europe.
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 . Similarly, this is done for the
137 stations so that the results are comparable with
Table . There is no significant reduction in RMS. The
maximal deviation of the median from zero is 0.06 mmyr-1 for the
155 stations and maximally 0.07 mmyr-1 for the 137 stations,
which is a reduction with respect to the values listed in
Table . The mean is also reduced to approximately
-0.1 mmyr-1, which is statistically equal to zero. This result
is at the level of the noise in the determination of the ITRF origin
and it is smaller than the 0.4 mmyr-1 to
which global mean sea level trends from altimetry are guaranteed
. Unless it is proven that the altimeters are more stable
and the uncertainties in the ITRF origin are reduced, a mean of trend
differences closer to zero cannot be expected.
Conclusions
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 mmyr-1 and a median of 0.71 mmyr-1.
A comparison with the estimates of ULR5 at 70 stations
yielded an RMS of at least 1.05 mmyr-1. A comparison with ALT–TG
showed that using the median of all neighboring GNSSs provided the best
results.
For the ALT–TG trends we used along-track data from the Jason series of
altimeters. At every 6 km along-track data were stacked to create
time series. The time series were low-pass filtered with a moving-average
filter of 1 year and correlated with low-pass-filtered TG time series. An
average or weighted monthly time series for altimetry was created by taking into
account only the time series corresponding to correlations above a threshold.
The TG time series were subtracted from the average of monthly low-pass-filtered altimetry time series to create a ALT–TG time series. Using the
Hector software between 344 and 663 trends were computed from the ALT–TG time
series, depending on the correlation threshold set.
The SD of the ALT–TG time series was reduced on average by approximately
10 % when a correlation threshold of 0.7 was used. Spatially
coherent differences in trends between various thresholds are observed on the
East Coast of the US and in Norway. We argue that residual interannual ocean
variability in ALT–TG time series can locally induce VLM trend biases,
especially when time series are short. For 155 stations globally distributed,
increasing the correlation threshold does not significantly affect the RMS of
differences between GNSS and ALT–TG trends. However, the correlation
threshold also works as a selection procedure. When considering 294 VLM
estimates from GNSS and ALT–TG at the same TGs for comparison, with no
threshold the RMS of differences was 2.14 mmyr-1, whereas an RMS
of 1.22 mmyr-1 was reached using 155 stations and a threshold of
0.7. This is a substantial improvement with respect to the
1.47 mmyr-1 RMS of at 109 TGs, the best
result so far. Note that increasing the threshold considerably reduces the
number of time series in the Southern Hemisphere and therefore other
thresholds might be better depending on the purpose.
The comparison with tide gauges also reveals that the trends from ALT–TG are
biased low (similar to ), even though this is barely
significant. Using mass redistribution fingerprints, a correction is applied
for trend differences caused by nonlinear behavior of present-day mass
changes. The RMS of differences is barely affected, but the mean of
differences is changed from about -0.2 to -0.1 mmyr-1, which
is now statistically insignificant.
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 km scaled by
1.4826 . If only a single
GNSS station is present, the MIDAS uncertainty is provided. If two GNSS
stations are present and both trends are statistically equal, it takes the
square root of the mean of the GNSS variances to avoid very small error bars.
When no correlation threshold is used 663 ALT–TG and 570 GNSS trends are
available at 939 different TGs. By setting the correlation threshold to 0.7,
the number of TGs for which a trend is estimated decreases to 759.
Depending on the application, the value of the threshold can be varied to
find an optimum between the reliability and the number of TGs for which
a trend is estimated. If both GNSS and ALT–TG trends are available, we
recommend using GNSS trends because of correlated residual ocean signals
between various ALT–TG time series. However, if a large discrepancy (>3 mmyr-1) is found between the GNSS and ALT–TG trends, we
recommend using the ALT–TG trend because the culprit is likely local VLM
differences between the TG and the GNSS stations. The GNSS–ALT–TG
histogram for no correlation threshold reveals large discrepancies between
the two methods of up to 10 mmyr-1. While the problems with ALT–TG
trends are mostly resolved by setting a higher threshold, the GNSS trends
might still require some inspection before they are used in sea level
studies. A faster practice is to use trend uncertainties that carry
information about the linearity of the trends, and when the MAD is used as
described above, also information about local VLM variability. However, when
only one GNSS station is present the information about local VLM variations
is absent.