**Data description paper**
07 Aug 2019

**Data description paper** | 07 Aug 2019

# Uncertainty in satellite estimates of global mean sea-level changes, trend and acceleration

Michaël Ablain Benoît Meyssignac Lionel Zawadzki Rémi Jugier Aurélien Ribes Giorgio Spada Jerôme Benveniste Anny Cazenave and Nicolas Picot

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**Michaël Ablain et al.**Michaël Ablain Benoît Meyssignac Lionel Zawadzki Rémi Jugier Aurélien Ribes Giorgio Spada Jerôme Benveniste Anny Cazenave and Nicolas Picot

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^{1}MAGELLIUM, Ramonville Saint-Agne, 31520, France^{2}LEGOS, CNES, CNRS, IRD, Université Paul Sabatier, Toulouse, 31400, France^{3}Collecte Localisation Satellite (CLS), Ramonville Saint-Agne, 31520, France^{4}CNRM, Université Paul Sabatier, Météo France, CNRS, Toulouse, 31400, France^{5}Dipartimento di Scienze Pure e Applicate (DiSPeA), Urbino University “Carlo Bo”, 61029, Urbino, PU, Italy^{6}European Space Agency (ESA-ESRIN), Fracati, Italy^{7}Centre National d'Etudes Spatiales (CNES), Toulouse, 31400, France

^{1}MAGELLIUM, Ramonville Saint-Agne, 31520, France^{2}LEGOS, CNES, CNRS, IRD, Université Paul Sabatier, Toulouse, 31400, France^{3}Collecte Localisation Satellite (CLS), Ramonville Saint-Agne, 31520, France^{4}CNRM, Université Paul Sabatier, Météo France, CNRS, Toulouse, 31400, France^{5}Dipartimento di Scienze Pure e Applicate (DiSPeA), Urbino University “Carlo Bo”, 61029, Urbino, PU, Italy^{6}European Space Agency (ESA-ESRIN), Fracati, Italy^{7}Centre National d'Etudes Spatiales (CNES), Toulouse, 31400, France

**Correspondence**: Michaël Ablain (michael.ablain@magellium.fr)

**Correspondence**: Michaël Ablain (michael.ablain@magellium.fr)

Received: 14 Jan 2019 – Discussion started: 25 Jan 2019 – Revised: 12 Jul 2019 – Accepted: 16 Jul 2019 – Published: 07 Aug 2019

Satellite altimetry missions now provide more than 25 years of accurate, continuous and quasi-global measurements of sea level
along the reference ground track of TOPEX/Poseidon. These measurements are
used by different groups to build the Global Mean Sea Level (GMSL) record,
an essential climate change indicator. Estimating a realistic uncertainty in
the GMSL record is of crucial importance for climate studies, such as
assessing precisely the current rate and acceleration of sea level,
analysing the closure of the sea-level budget, understanding the causes of
sea-level rise, detecting and attributing the response of sea level to
anthropogenic activity, or calculating the Earth's energy imbalance.
Previous authors have estimated the uncertainty in the GMSL trend over the
period 1993–2014 by thoroughly analysing the error budget of the satellite
altimeters and have shown that it amounts to ±0.5 mm yr^{−1}
(90 %
confidence level). In this study, we extend our previous results, providing a
comprehensive description of the uncertainties in the satellite GMSL record.
We analysed 25 years of satellite altimetry data and provided for the first
time the error variance–covariance matrix for the GMSL record with a time
resolution of 10 days. Three types of errors have been modelled (drifts,
biases, noises) and combined together to derive a realistic estimate of the
GMSL error variance–covariance matrix. From the latter, we derived a 90 %
confidence envelope of the GMSL record on a 10 d basis. Then we used a
least squared approach and the error variance–covariance matrix to assess
the GMSL trend and acceleration uncertainties over any 5-year time periods
and longer in between October 1992 and December 2017. Over 1993–2017, we
have found a GMSL trend of 3.35±0.4 mm yr^{−1} within a 90 % confidence
level (CL) and a GMSL acceleration of 0.12±0.07 mm yr^{−2} (90 % CL). This is in agreement (within error bars)
with previous studies. The full GMSL error variance–covariance matrix is
freely available online: https://doi.org/10.17882/58344
(Ablain et al., 2018).

The sea-level change is a key indicator of global climate change, which
integrates changes in several components of the climatic system as a
response to climatic variability, both anthropogenic and natural. Since
October 1992, sea-level variations have been routinely measured by 12
high-precision altimeter satellites providing more than 25 years of
continuous measurements. The Global Mean Sea Level (GMSL)
altimeter indicator is calculated from the accurate and stable measurements of four
reference altimeter missions, namely TOPEX/Poseidon (T/P), Jason-1, Jason-2
and Jason-3. All four reference missions are flying (or have flown) over the
same historical ground track on a 10 d repeat cycle. They all have been
precisely inter-calibrated (Zawadzki and Ablain, 2016)
to ensure the long-term stability of the sea-level measurements. Six
research groups (AVISO/CNES, SL_cci/ESA, University of
Colorado, CSIRO, NASA/GSFC, NOAA) have processed the sea-level raw data
provided by satellite altimetry to provide the GMSL series on a 10 d basis
(Fig. 1). The six different estimates of the GMSL
record show small deviations between 1 and 2 mm at inter-annual timescales
(1- to 5-year timescales) and between ±0.15 mm yr^{−1} in terms of the trend
over the period 1993–2017. The spread across these estimates is due to the
use of various processing techniques, alternative versions of ancillary data
and different interpolation methods applied by the several groups
(Masters et al., 2012;
Henry et al., 2014). This spread is smaller than
the real uncertainty in the sea-level trend, because all the research groups
have used similar methods and corrections to process the raw data, and thus
several sources of systematic uncertainty are not accounted for in the
spread.

In a previous study, Ablain et al. (2009) have proposed a
realistic estimate of the uncertainty in the GMSL trend over
1993–2008, using an approach based on the error budget. They have identified
the radiometer wet tropospheric correction as one of the main sources of
error. They have also proposed the orbital determination, the
inter-calibration of altimeters and the estimate of the altimeter range,
sigma-0 and significant wave height (mainly on TOPEX/Poseidon) as
significant sources of error. When all the terms were accounted for, they
have found that the uncertainty in the trend over 1993–2008 was ±0.6 mm yr^{−1}
within a 90 % confidence level. This is larger than the uncertainty
of ±0.3 mm yr^{−1} over a 10-year period required by GCOS
(GCOS, 2011). In the framework of the ESA Sea Level Climate
Change Initiative (SL_cci), significant improvements have
been obtained estimating the sea level from space
(Ablain
et al., 2015; Quartly et al., 2017; Legeais et al., 2018) to get closer to
the GCOS requirements. New altimeter standards, including new wet troposphere
corrections, new orbit solutions, new atmospheric corrections and others,
were selected and applied in order to improve the sea-level estimation. The
GMSL trend uncertainties were then updated and estimated at different
temporal and spatial scales
(Ablain
et al., 2015; Legeais et al., 2018). During the second altimetry decade
(2002–2014), Ablain et al. (2015) have
estimated that the uncertainty in the GMSL trend was lower than ±0.5 mm yr^{−1} within a 90 % confidence level (CL) for periods longer than 10 years.

In previous studies, the uncertainty in GMSL has been assessed for long-term
trends (periods of 10 years or more, starting in 1993), inter-annual timescales (between 1 and 5 years) and annual timescales
(Ablain et al., 2009, 2015). This
estimation of the uncertainty at three timescales is a valuable first step,
but it is not enough, as it does not fully meet the needs of the scientific
community. In many climatic studies the GMSL uncertainty is required at
different timescales and spans within the 25-year altimetry record. In sea-level budget studies based on the evolution of GMSL components, these
estimates have been carried out at a monthly timescale. In this way, the GMSL
monthly changes have been interpreted in terms of changes in ocean mass
(Gravity recovery and climate experiment – GRACE – mission). This is also
the case for studies estimating the Earth's energy imbalance with the
sea-level budget approach (Meyssignac et al., 2019). In the
studies on the detection and the attribution of climate change
(e.g. Slangen et al., 2017), the uncertainty in the
trend estimates is needed, but over different time spans than those
addressed in Ablain et al. (2015,
2009) and in
Legeais
et al. (2018). The uncertainty in different metrics is often needed.
Dieng et al. (2017) and Nerem et
al. (2018) have recently estimated the acceleration in the GMSL over
1993–2017, finding a small acceleration (0.08 mm yr^{−2}) over
the 25-year long altimetry record.

In this paper we focus on the uncertainty in the GMSL record arising from instrumental errors in the satellite altimetry. The uncertainties of the measurements have been quantified in the GMSL record. This is important information for the studies in detection and attribution of the climatic changes, estimating the GMSL rise as a response to the anthropogenic activity. But this is not sufficient. In the detection–attribution studies the response of the GMSL to the anthropogenic activity needs to be separated from the response to the natural variability of the climate system, representing an additional source of uncertainty.

The objective of this paper is to estimate the error variance–covariance matrix of the GMSL (on a 10 d basis) from satellite altimetry measurements. This error variance–covariance matrix provides a comprehensive description of the uncertainties in the GMSL to users. It covers all timescales that are included in the 25-year long satellite altimetry record: from 10 d (the time resolution of the GMSL time series) to multidecadal timescales. It also enables us to estimate the uncertainty in any metric derived from GMSL measurements such as trend, acceleration or other moments of higher order in a consistent way.

We used an error budget approach to a global scale on a 10 d basis in order to calculate the error variance–covariance matrix. We considered all the major sources of uncertainty in the altimetry measurements, including the wet tropospheric correction, the orbital solutions, and the inter-calibration of satellites. We have also taken into account the time correlation between the different sources of uncertainty (Sect. 2). The errors have been separately characterized for each altimetry mission, since they have been affected by different sources of uncertainty (Sect. 2). On the basis of the error variance–covariance matrix we estimate the uncertainty in GMSL individual measurements on a 10 d basis (Sect. 3) and the uncertainty in trend and acceleration over all periods included in the 25-year satellite altimetry record (1993–2017) (Sect. 4). Note that in this article all uncertainties associated with the GMSL are reported with a 90 % CL unless stated otherwise.

The six main groups that provide satellite-altimetry-based GMSL estimates (AVISO/CNES, SL_cci/ESA, University of Colorado, CSIRO, NASA/GSFC, NOAA) use 1 Hz altimetry measurements from the T/P, Jason-1, Jason-2 and Jason-3 missions from 1993 to 2018 (1993–2015 for SL_cci/ESA). Each group processes the 1 Hz data with geophysical corrections to correct the altimetry measurements for various aliasing, biases and drifts caused by different atmospheric conditions, sea states, ocean tides and others (Ablain et al., 2009). They spatially average the data over each 10 d orbital cycle to provide GMSL time series on a 10 d basis. The differences among the GMSL estimates from several groups arise from data editing, from differences in the geophysical corrections and from differences in the used method to spatially average individual measurements during the orbital cycles (Masters et al., 2012; Henry et al., 2014).

Recently, the comparisons of the GMSL time series derived from satellite
altimetry with independent estimates are based on tide gauge records
(Valladeau et al., 2012; Watson et al., 2015)
or on the combination of the contribution to sea level from thermal
expansion, land ice melt and land water storage (Dieng et
al., 2017). They have shown that there was a drift in the GMSL record over
the period 1993–1998. This drift is caused by an erroneous onboard
calibration correction on TOPEX altimeter side-A (denoted TOPEX-A). TOPEX-A
was operated from launch in October 1992 to the end of January 1999. Then
the TOPEX side-B altimeter (denoted TOPEX-B) took over in February 1999
(Beckley et al., 2017). The impact on the GMSL changes
is −1.0 mm yr^{−1} between January 1993 and July 1995, and +3.0 mm yr^{−1} between
August 1995 and February 1999, with an uncertainty of ±1.7 mm yr^{−1}
(within a 90 % CL, Ablain, 2017).

Without taking into account the TOPEX-A drift correction, the differences
between all GMSL time series are small. The maximum trend difference between
all time series over 1993–2017 is lower than 0.15 mm yr^{−1}, representing less
than 5 % of the GMSL trend. The differences observed at interannual timescales are also small (<2 mm). By correcting the drift of TOPEX-A
using either of the available empirical corrections (WCRP Global
Sea Level Budget Group, 2018), the differences among solutions remain the
same (the difference between empirical corrections being smaller than the
difference between the raw GMSL time series). Therefore, the choice of one
or the other GMSL record is not decisive in this study, whose purpose is to
characterize the uncertainties. Hereafter, we use the GMSL AVISO record. The
corresponding altimeter standard corrections and the GMSL processing methods
are described on the AVISO website (https://www.aviso.altimetry.fr/msl/, last access: 1 August 2019).

This section describes the different errors that affect the altimetry GMSL
record. It builds on the GMSL error budget presented in
Ablain et al. (2009) and extends this work by taking into
account the new altimeter missions (Jason-2, Jason-3) and the recent
findings on altimetry error estimates. Three types of errors are considered:
(a) biases in GMSL between successive altimetry missions which are
characterized by bias uncertainties (±Δ) at a given time (*t*);
(b) drifts in GMSL characterized by a trend uncertainty (±*δ*);
and (c) other measurement errors which exhibit time correlation (so-called
residual time-correlated errors hereafter). The residual time-correlated
errors are characterized by their standard deviation (*σ*) and by the
correlation timescale (*λ*). All altimetry errors identified in this
study are summarized in Table 1 and are detailed
hereafter. Note that all uncertainties reported in Table 1 are Gaussians, and
they are given at the 1-*σ* level (i.e. we provide the standard deviation
of the Gaussian, denoted 1*σ* hereafter).

The biases can arise between the GMSL record of two successive satellite missions like between T/P and Jason-1 in May 2002, between Jason-1 and Jason-2 in October 2008 and between Jason-2 and Jason-3 in October 2016. These biases are estimated during dedicated 9-month inter-calibration phases when a satellite altimeter and its successor fly over the same track, 1 min apart. During the inter-calibration phases the bias is estimated and corrected for. Different missions show different biases, but the uncertainty in the bias correction is the same for all inter-calibration phases and amounts: ±0.5 mm (Zawadzki and Ablain, 2016). The situation is different for the switch from TOPEX-A to TOPEX-B in February 1999 because it was impossible to do any inter-calibration phase between the two sides of TOPEX (as both instruments were flying on the same spacecraft). For the switch, we assume that the uncertainty in GMSL is larger and is about 2 mm (Zawadzki and Ablain, 2016).

The drifts may occur in the GMSL record because of drifts in the TOPEX-A and
TOPEX-B radar instruments, because of drifts in the International
Terrestrial Reference Frame (ITRF) realization in which altimeter orbits are
determined or because of drifts in the glacial isostatic adjustment (GIA)
correction applied to the GMSL record. As explained before, the TOPEX-A
record shows a spurious drift due to an erroneous onboard calibration
correction of the altimeter (Beckley et al., 2017).
This drift has been corrected by using several empirical approaches
(Ablain, 2017;
Beckley et al., 2017; Dieng et al., 2017) that are all affected by a
significant uncertainty. We estimated this uncertainty to be ±0.7 mm yr^{−1} (1*σ* level) over the TOPEX-A period (1993–1998), with a comparison
against an independent GMSL estimate based on tide gauge records
(Ablain, 2017). For the TOPEX-B record, no
GMSL drift has been reported, but Ablain et al. (2012)
showed significant sigma-0 instabilities of the order of 0.1 dB, which
generate through the sea-state bias correction an uncertainty of ±0.1 mm yr^{−1} (1*σ* level) in the GMSL
record over the TOPEX-B period (February 1999–April 2002). Concerning the ITRF realization, Couhert
et al. (2015) have shown that the uncertainty in the ITRF realization drift
generates an uncertainty of ±0.1 mm yr^{−1} (1*σ* level) in the GMSL trend
over 1993–2015. We adopt this value here for the whole period 1993–2017. For
the uncertainty in the GIA correction applied to the GMSL, we use the value
of 0.05 mm yr^{−1} (1*σ* level) over the altimetry period from Spada (2017) (the value is taken from Table 1 in Spada, 2017). It has been
confirmed recently with an ensemble of 1000 GIA runs; see Melini
and Spada (2019). Combining the uncertainty in the GMSL trend over 1993–2017
from GIA and ITRF and assuming that they are not correlated yields an
uncertainty in the GMSL trend of ±0.12 mm yr^{−1} over 1993–2017 (1*σ*
level). In addition to the GIA correction and the TOPEX correction, we apply
an elastic correction to the GMSL record of +0.10 mm yr^{−1} to account for the
elastic deformations of the ocean bottom in response to modern melt of land
ice (Frederikse et al., 2017; Lickley et
al., 2018). The uncertainty in this correction arises from the uncertainty
associated with the computation of the elastic response of the solid Earth
(mainly from the uncertainty associated with the procedure to solve the sea-level equation, uncertainty in the choice of the Love numbers, and uncertainty
generated by the truncation degree of the spherical harmonics) and the
uncertainty in the mass redistributions that cause the elastic deformation.
Because the elastic response of the Earth is reasonably well defined
(Mitrovica et al., 2011), the uncertainty in the
elastic correction is largely dominated by the uncertainty in the mass
redistribution (Frederikse et al., 2017). The
uncertainty in the mass redistribution is about ±10 % on the
current ice mass loss (e.g. Blazquez et al., 2018).
It yields an uncertainty of ±10 % in the elastic correction
(because the elastic response of the Earth is linear). This uncertainty
amounts to ±0.01 mm yr^{−1}, which is very small. It is an order of magnitude
smaller than the uncertainty considered in this study (see Table 1). So we
neglect this source of uncertainty here.

The residual time-correlated errors are separated into two different groups,
depending on their correlation timescales. The first group gathers errors
with short correlation timescales, i.e. lower than 2 months and between
2 months and 1 year. The second group gathers errors with long
correlation timescales between 5 and 10 years. In the first group the
errors are mainly due to the geophysical corrections (ocean tides,
atmospheric corrections), to the altimeter corrections (sea-state bias
correction, altimeter ionospheric corrections), to the orbital calculation,
and to the potential altimeter instabilities (altimeter range and sigma-0
instabilities). At timescales below 1 year, the variability of the
corrections' time series is dominated by errors, such that the variance of
the error in each correction is estimated by the variance of the
correction's time series. For errors with correlation timescales lower than
2 months, we estimated the standard deviation (*σ*) of the error from the
correction's time series filtered with a 2-month high-pass filter. Since the
standard deviation of the errors depends on the different altimeter
missions, the standard deviation has been separately estimated for each
altimeter mission. We find *σ*=1.7 mm over the T/P period, *σ*=1.5 mm
over the Jason-1 period, and *σ*=1.2 mm over the Jason-2/-3 period. For
errors with a correlation timescale between 2 months and 1 year, we used the
same approach and filtered the correction time series with a band-pass
filter. In this case we find *σ*=1.3 mm over the T/P period, *σ*=1.2 mm over the Jason-1 period, and *σ*=1.0 mm over the Jason-2/-3 period.
Unsurprisingly, the highest errors are obtained for T/P and the lowest ones
for Jason-2/-3. This is because of (1) larger altimeter range instabilities
in T/P (Ablain et al., 2012; Beckley et
al., 2017), (2) the presence of a 59 d signal error in the altimeter range
of T/P (Zawadzki et al., 2018), and (3) the
deterioration in the performance of atmospheric corrections in the early
years of the altimetry era (Legeais et al., 2014). Note that
Jason-1 also shows higher errors than Jason-2 and Jason-3 at timescales
below 1 year (Couhert et al., 2015).

In the second group of residual time-correlated errors, errors are due to
the onboard microwave radiometer calibration, yielding instabilities in the
wet troposphere correction, and also due to the orbital calculation
(Couhert et al., 2015). Since these errors are
correlated at timescales longer than 5 years, they cannot be estimated
with the standard deviation of the correction time series, too short
(25-year long) to sample the time correlation. For this group of
residual time-correlated errors, we used simple models to represent the time
correlation of the errors. For the wet troposphere correction, several
studies (Legeais et al, 2018) have identified long-term differences among the
computed corrections from the different microwave radiometers and from the
different atmospheric reanalyses
(Dee et al., 2011). These studies report a difference in the wet tropospheric correction
for GMSL in the range of ±0.2–0.3 mm yr^{−1} for periods of 5 to 10 years.
Here, we adopt a conservative approach and we model the error in wet
tropospheric correction with a correlated error at 5 years with a standard
deviation of 1.2 mm (1*σ* level). The correlation is modelled with a
Gaussian attenuation based on the wavelength of the errors:
${e}^{\frac{-\mathrm{1}}{\mathrm{2}}{\left(\frac{t}{\mathit{\lambda}}\right)}^{\mathrm{2}}}$ with *λ*=5 years. In terms of trends, this residual time-correlated error generates an
uncertainty of ±0.2 mm yr^{−1} over 5-year periods. For the error in the
orbit calculation, comparisons of different orbit solutions showed
differences of ±0.05 mm yr^{−1} on 10-year timescales due to errors in
the modelling of the Earth's time-varying gravity field
(Couhert et al., 2015). We model this error with a
correlated error at a 10-year timescale with a standard deviation of 0.5 mm
(1*σ* level). The correlation is modelled by the same Gaussian distribution
as before with *λ*=10 years. In terms of trends, it corresponds to
an uncertainty of ±0.05 mm yr^{−1} over 10-year periods.

In the next section these different terms of the GMSL error budget are combined together to build the error variance–covariance matrix. Note that the different terms of the altimeter GMSL error budget described here are based on the current knowledge of altimetry measurement errors. As the altimetry record increases in length with new altimeter missions, the knowledge of the altimetry measurement also increases and the description of the errors improves. This implies that the error variance–covariance matrix is expected to improve and change in the future.

In this section we derived the error variance–covariance matrix (**Σ**) of the GMSL from the error budget described in Sect. 2. We
assumed that all error sources shown in Table 1 are
independent of each other. Thus the **Σ** matrix is the sum of the
individual variance–covariance matrix of each error source **Σ**_{i} in
the error budget (see Fig. 2). Each **Σ**_{i}
matrix is calculated from a large number of random draws (>1000)
of the simulated error signal using the model described in Sect. 2 (either a
bias, drift or time-correlated signal) fed with a standard normal
distribution.

The resulting shape of each individual **Σ**_{i} matrix depends on the
type of error (bias, drift or time-correlated signal; see
Fig. 2). For the bias, the **Σ**_{i} matrix
takes the shape of constant square blocks each side of the time occurrence
of the bias correction (see for example the square matrix for TOPEX-A and
TOPEX-B in the lower-left corner of Fig. 2 along
the diagonal). This square block shape means that the error in the bias
correction generates an error on the GMSL which is fully correlated along
time before and after the bias correction time, but which is not correlated
along time for dates that are apart from the bias correction time. This is
consistent with what we expect from a bias correction error. Note that in
this article (and in climate change studies in general) we are interested
only in GMSL changes, trends or acceleration but not in the mean time GMSL
(which is the absolute reference of GMSL). Thus, we have removed from the
GMSL time series the temporal mean over 1993–2017. The reference of the GMSL
is thus arbitrary and assumed to be perfectly known. This is
because the reference of the GMSL is not affected by the biases' correction
error here.

For the drifts, the **Σ**_{i} matrix takes the shape of a horse saddle.
This is because an error in the GMSL drift over a given period generates
errors in the GMSL time series which are correlated when they are close in
time and anti-correlated when they are on the opposite side of the drift period.

For residual time-correlated errors, the **Σ**_{i} matrix takes the shape
of a diagonal matrix with off-diagonal terms of smaller amplitude. The
further from the diagonal the off-diagonal terms are, the more attenuated
they are. The attenuation rate is a Gaussian attenuation based on the
wavelength of the time-correlated errors (${e}^{\frac{-\mathrm{1}}{\mathrm{2}}{\left(\frac{t}{\mathit{\lambda}}\right)}^{\mathrm{2}}}$), with various timescales *λ*.

All individual **Σ**_{i} matrices are summed up together to build the
total error variance–covariance matrix **Σ** of the altimetry-derived
continuous GMSL record over 1993–2018 (see Fig. 2). As expected, the dominant terms of the matrix are on the diagonal. They
are largely due to the different sources of errors with correlation timescales below 1 year (first group of errors in Sect. 2). The diagonal terms
are highest at the beginning of the altimetry period when T/P was at
work. This is because of larger altimeter range instabilities in T/P, the
presence of a 59 d signal error on the altimeter range of T/P and poorer
performance of atmospheric corrections in the early years of the altimetry
era (Legeais et al., 2014). The dominant off-diagonal terms
are also found during the T/P period (in the lower-left corner of the
matrix; see Fig. 2). The terms are induced by the
TOPEX-A trend error and the large bias correction uncertainty between
TOPEX-A and TOPEX-B (because of the absence of an inter-calibration phase
between TOPEX-A and TOPEX-B).

We estimated the GMSL uncertainty envelope from the square root of the
diagonal terms of **Σ** (see Fig. 3). As
expected, the GMSL time series shows a larger uncertainty during the T/P
period (5 to 8 mm) than during the Jason period (close to 4 mm). The bias
correction uncertainty between TOPEX-A and TOPEX-B in February 1999 is also
clearly visible, with a 1 mm drop in the uncertainty after the switch to
TOPEX side-B. Note that the uncertainty envelope has a parabolic shape and
shows smaller uncertainties during the beginning of the Jason-2 period (3.5 mm around 2008) than over the Jason-3 period (4.5 mm). This is not because
Jason-1 and Jason-2 errors are smaller than Jason-3's errors. Actually, Jason-2 and Jason-3 errors are slightly smaller than Jason-1 errors thanks to
better orbit determination. The uncertainties are smaller during the Jason-1
and Jason-2 periods because this period is in the center of the record. It
benefits from prior and posterior data that covariate and help in reducing
the uncertainty when they are combined together. In contrast, the Jason-3
period is located at the end of the record and does not benefit from
posterior data to help reduce the uncertainty.

In Fig. 4 we superimposed the GMSL time series (average of the GMSL time series in Fig. 1) and the associated uncertainty envelope. For the TOPEX-A period we tested three different curves with three different corrections based on the removal of the Cal-1 mode (Beckley et al., 2017), based on the comparison with tide gauges (Watson et al., 2015; Ablain, 2017), or based on a sea-level closure budget approach (Dieng et al., 2017). The uncertainty envelope is centred on the corrected record for TOPEX-A drift with the correction based on Ablain et al. (2017). As was expected, all the empirically corrected GMSL records are within the uncertainty envelope.

The variance–covariance matrix can be used to derive the uncertainty in any metric based on the GMSL time series. In this section we used the error variance–covariance matrix to estimate the uncertainty in the GMSL trend and acceleration over any period of 5 years and more within 1993–2017.

Recently, several studies
(Watson et al.,
2015; Dieng et al., 2017; Nerem et al., 2018; WCRP Global Sea Level Budget
Group, 2018) have found a significant acceleration in the GMSL record from
satellite altimetry (after correction for the TOPEX-A drift). The occurrence
of an acceleration in the record should not change the estimation of the
trend when calculated with a least squared approach. However, it can affect
the estimation of the uncertainty in the trend. To cope with this issue, we
address here at the same time both the estimation of the trend and
acceleration in the GMSL record. In order to obtain this objective, we used
a second-order polynomial as a predictor. Considering the GMSL record has *n*
observations, let *X* be an *n*×3 predictor where the first column
contains only ones (representing the constant term), the second column
contains the time vector (representing the linear term) and the third column
contains the square of the time vector (representing the squared term). Let
** y** be an

*n*×1 vector of independent observations of the GMSL. Let

**be an**

*ϵ**n*×1 vector of disturbances (GMSL non-linear and non-quadratic signals) and errors. Let

**be the 3×1 vector of unknown parameters that we want to estimate, namely the GMSL**

*β**y*intercept, the GMSL trend and the GMSL acceleration. Our linear regression model for the estimation of the GMSL trend and acceleration will thus be

with

where **Σ** is the variance–covariance matrix of the observation errors
(estimated in the previous section). **Σ** is different from the
identity because of the correlated noise (see Sect. 2).

The most common method to estimate the GMSL trend and acceleration is the
ordinary least squares (OLS) estimator in its classical form
(Cazenave
and Llovel, 2010; Masters et al., 2012; Dieng et al., 2015; Nerem et al.,
2018). This is also the most common method for estimating trends and
accelerations in other climate-essential variables
(Hartmann, et al., 2014, and references therein).
For these reasons, we turn here to the OLS to fit the linear regression
model. The estimator of ** β** with the OLS approach, denoted
$\widehat{\mathit{\beta}}$,
is

In most cases, ** ϵ** follows a

*N*(0,

*σ*

^{2}

*I*) distribution, which implies that $\widehat{\mathit{\beta}}$ follows a normal law:

The issue with this common framework is that the uncertainty in the trend and acceleration estimates does not take into account the correlated errors of the GMSL observations.

To address this issue, we used a more general formalism to integrate the
GMSL error into the trend uncertainty estimation, following
Ablain et al. (2009), Ribes et al. (2016) and IPCC AR5
(Hartmann, et al., 2014; see in particular Box 2.2 and the Supplement). The OLS estimator is left unchanged (and is
still unbiased), but its distribution is revised to account for **Σ**,
leading to

Note that this estimate is known to be less accurate than the general least
square estimate (GLS, which is the optimal estimator in the case where
**Σ**≠*I*) in terms of the mean square error, because its variance
is larger. A generalized least square estimate would probably help in
narrowing slightly the trend uncertainty, but the difference would likely be
small as the GMSL time series is almost linear in time. Important advantages
of using OLS here are that (i) OLS is consistent with previous estimators of GMSL
trends as well as estimators of trends in other essential climate variables
than GMSL (e.g. Hartmann, et al., 2014) and that (ii) the OLS best estimate does not depend on the estimated variance–covariance
matrix **Σ**.

Based on the matrix **Σ** defined in the previous section and the OLS
solution proposed before, we now estimate the GMSL trend (mm yr^{−1}) and
acceleration (mm yr^{−2}) uncertainties for any time span included in the
period 1993–2017. Results are synthetically displayed in
Fig. 5 for trends and in
Fig. 6 for accelerations. In
Fig. 5, the top of the triangle indicates that
the GMSL trend uncertainty over 1993–2017 is ±0.4 mm yr^{−1} (CL 90 %)
and that the GMSL acceleration uncertainty over the same period is ±0.07 mm yr^{−2} (CL 90 %, Fig. 6).
The GMSL acceleration uncertainty estimate is consistent with results of
Watson et al. (2015) on the January 1993 to June 2014 time
period, where they find an uncertainty of ±0.058 mm yr^{−2} at 1*σ*
which corresponds to ±0.096 mm yr^{−2} at the 90 %
confidence level. This is slightly larger than the
Nerem et al. (2018) estimate, which is
±0.025 mm yr^{−2} at the 1*σ* level on the full 25-year
altimetry era, which corresponds to ±0.041 mm yr^{−2} at
the 90 % confidence level. But the Nerem et
al. (2018) estimate is likely underestimated as they only consider omission
errors. The GMSL acceleration uncertainties have been calculated for all
periods of 10 years and more within 1993–2017
(Fig. 6). As expected, uncertainties tend to
increase when the period length decreases. At 10 years, the GMSL
acceleration uncertainties range from ±0.3 mm yr^{−2} over the T/P period to ±0.25 mm yr^{−2} over the Jason period. At 20 years they range between
±0.12 and ±0.08 mm yr^{−2}.

A cross-sectional analysis of the 10-year horizontal line in
Fig. 5 shows that the GMSL trend uncertainties
over 10-year periods decreased from 1.0 mm yr^{−1} over the first decade to 0.5 mm yr^{−1} over the last one. The larger uncertainty over the first decade is
mainly due to the TOPEX-A drift error, but also due to the large intermission
bias uncertainty between TOPEX-A and TOPEX-B, and, to a lesser extent, to the
improvement of GMSL accuracy with Jason-2 and Jason-3. Note that the current
GCOS requirement of 0.3 mm yr^{−1} uncertainty over 10 years (GCOS, 2011) is not
met at the 90 % confidence level. But the recent record over the last
decade based on the Jason series is close to meeting the GCOS requirement, with
a 90 % CL.

Figure 5 can also be analysed by following the
sides of the triangle. The results of this analysis are plotted in
Fig. 7. The plain line corresponds to the left
side, read from bottom left to the top of the triangle. The dashed line
corresponds to the right side, read from bottom right to the top of the
triangle. As expected, both curves show a reduction of the trend uncertainty
as the period over which trends are computed increases from 2 to 25 years.
The difference between the two lines shows the reduction of GMSL errors
thanks to the improvement of the measurement in the latest altimetry
missions. The lowest trend uncertainty is obtained with the last 20 years of
the GMSL record: 0.35 mm yr^{−1}.

The periods for which the acceleration in sea level is significant at the
90 % confidence level are shown in Fig. 8. The
acceleration is visible at the end of the record for periods of 10 years and
longer. The GMSL acceleration is 0.12 mm yr^{−2} with an
uncertainty of 0.07 mm yr^{−2} at the 90 % confidence level over
the 25-year altimetry era. This proves that the acceleration observed in the
GMSL evolution is statistically significant. It is worth noting that the
different empirical TOPEX-A corrections yield very similar results (0.126 mm yr^{−2}, Ablain, 2017; 0.120 mm yr^{−2}, Dieng et al., 2017; Watson
et al., 2015; 0.114 mm yr^{−2}, Beckley et
al., 2017). This acceleration at the end of the record is due to an
acceleration in the contribution to sea level from Greenland and from other
contributions, but to a lesser extent
(Chen et al., 2017;
Dieng et al., 2017; Nerem et al., 2018). A small acceleration is also
visible during the 1993–2005 period at the beginning of the record. This
acceleration is likely due to the recovery from the Mount Pinatubo eruption
in 1991 (Fasullo et al., 2016). Indeed,
Church et al. (2005) showed that the impact of large
volcanic eruptions on global ocean heat content is characterized by a rapid
reduction in global ocean heat content during the year following the
eruption followed by a period of recovery of a few years when global ocean
heat content increases faster than before the eruption (see also
Gregory et al., 2006, and Delworth
et al., 2005). The sea-level record starts in October 1992, which is 1.5 years after the eruption of Mount Pinatubo (15 June 1991). At that time
the global ocean heat content was starting to recover with an increasing
rate of rise (see Fasullo et al., 2016, their Fig. 2)
leading to an acceleration in sea level.

The period for which the trend in sea level is significant at the 90 % confidence level is shown in Fig. 9. In periods when the acceleration is not significant, the second-order polynomial that we used as a predictor to estimate the trend and the acceleration does not hold anymore in principle. For these periods, we should turn out a first-order polynomial. The use of a first-order polynomial does not affect the trend estimates, but only the trend uncertainty estimates. We checked for differences in trend uncertainty when using either second-order or first-order polynomial predictors. We found that these differences are negligible (not shown).

Figure 9 indicates that for periods of 5 years and longer, the trend in
GMSL is always significant at the 90 % CL over the whole record. At the end of
the record the trend tends to increase. This is consistent with the
acceleration plot in Fig. 6. Over the 25 years of satellite altimetry, we
find a sea-level rise of 3.35±0.4 mm yr^{−1} (90 % CL) after
correcting for the TOPEX-A GMSL drift. The differences due to the different
TOPEX-A corrections are negligible (<0.05 mm yr^{−1}).

The global mean sea-level error variance–covariance matrix is available online at https://doi.org/10.17882/58344 (Ablain et al., 2018).

In this study we have estimated the full GMSL error variance–covariance matrix over the satellite altimetry period. The matrix is available online (see Sect. 7). It provides users with a comprehensive description of the GMSL errors over the altimetry period. This matrix is based on the current knowledge of altimetry measurement errors. As the altimetry record increases in length with new altimeter missions, the knowledge of the altimetry measurement also increases and the description of the errors improves. Consequently, the error variance–covariance matrix is expected to change and improve in the future – hopefully with a reduction of measurement uncertainty in new products.

The uncertainty in the GMSL computed here shows the reliability of altimetry
measurements in order to accurately describe the evolution of the GMSL on
all timescales from 10 d to 25 years. It also shows the reliability of
altimetry measurements in order to estimate the trends and accelerations of
the sea level. Along the altimetry record, we find that the uncertainty in
each individual GMSL measurement decreases with time. It is smaller during
the Jason era (2002–2018) than during the T/P period (1993–2002). Over the
entire altimetry record, 1993–2017, we estimate the GMSL trend to 3.35±0.4 mm yr^{−1} (90 % CL, after correcting the TOPEX-A GMSL drift). We also
detect a significant GMSL acceleration over the 25-year period at 0.12±0.07 mm yr^{−2} (90 % CL).

In this study, several assumptions have been made that could be improved in the future. Firstly, the modelling of altimeter errors should be regularly revisited and improved to consider a better knowledge of errors (e.g. stability of wet troposphere corrections) and to consider future altimeter missions (e.g. Sentinel-3 and Sentinel-6 missions). Concealing the mathematical formalism, the OLS method has been applied because it is the most common approach used in the climate community to calculate trends in any climate data records. However, this is not the optimal linear estimator. The use of a generalized least square approach should involve some narrowing of trend or acceleration uncertainty. Another topic of concern is the consideration of the internal and forced variability of the GMSL. Here we only considered the uncertainty in the GMSL due to the satellite altimeter instrument. In a future study, it would be interesting to consider the partitioning of the GMSL into the forced response to anthropogenic forcing and the natural response to natural forcing and to the internal variability. Estimating the natural GMSL variability (e.g. using models) and considering it to be an additional residual time-correlated error would allow us to calculate the GMSL trend and acceleration representing the long-term evolution of GMSL in relation to anthropogenic climate change.

MA and BM led the study, developed the theory, and wrote the manuscript with input from all the authors. LZ and RJ contributed to the theory and performed the computations. AR contributed to the statistical methodology developed in this paper. GS supervised the part related to the GIA uncertainties. JB, AC, and NP discussed the results. All the authors contributed to the final manuscript.

The authors declare that they have no conflict of interest.

We would like to thank all contributors of the Sea Level CCI (SL_cci) project (Climate Change Initiative programme) and of the SALP (Service d'Altimétrie et de Localisation Précise), with special recognition to Thierry Guinle, SALP project manager at the CNES.

This research has been supported by the ESA in the framework of the Sea Level CCI (SL_cci) project (Climate Change Initiative programme) supported by the CNES in the framework of the SALP (Service d'Altimétrie et de Localisation Précise). Giorgio Spada is funded by a FFABR (Finanziamento delle Attività Base di Ricerca) grant of the MIUR (Ministero dell'Istruzione, dell'Università e della Ricerca) and by a DiSPeA (Dipartimento di Scienze Pure e Applicate of the Urbino 65 University) grant.

This paper was edited by Giuseppe M. R. Manzella and reviewed by two anonymous referees.

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- Abstract
- Introduction
- GMSL data series
- Altimetry GMSL error budget
- The GMSL error variance–covariance matrix
- GMSL uncertainty envelope
- Uncertainty in GMSL trend and acceleration
- Data availability
- Conclusions
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- GMSL data series
- Altimetry GMSL error budget
- The GMSL error variance–covariance matrix
- GMSL uncertainty envelope
- Uncertainty in GMSL trend and acceleration
- Data availability
- Conclusions
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References