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

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

Evolution of GMSL time series (corrected for TOPEX-A drift using
the Ablain (2017) TOPEX-A correction) from six
different groups' (AVISO/CNES, CSIRO, University of Colorado,
SL_cci/ESA, NASA/GSFC, NOAA) products. The SL_cci/ESA covers the
period from January 1993 to December 2016, while all other
products cover the full 25-year period (January 1993 to December 2017).
Seasonal (annual and semi-annual) signals have been removed and a 6-month
smoothing has been applied. An averaged solution has been computed from the
six groups. GMSL time series have the same average on the 1993–2015 period
(common period) and the averaged solution starts at zero in 1993. The
averaged solution without TOPEX-A correction has also been represented. A
GIA correction of

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

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

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

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

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 (

Altimetry GMSL error budget given at

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:

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

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 (

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

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 (

Error variance–covariance matrix of altimeter GMSL on the 25-year period (January 1993 to December 2017).

The resulting shape of each individual

For the drifts, the

For residual time-correlated errors, the

All individual

We estimated the GMSL uncertainty envelope from the square root of the
diagonal terms of

Evolution in time of GMSL measurement uncertainty within a 90 %
confidence level (

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.

Evolution of the AVISO GMSL with different TOPEX-A corrections. On
the black, red and green curves, the TOPEX-A drift correction has been,
respectively, applied based on Ablain (2017), Watson et al. (2015), Dieng et
al. (2017) and Beckley et al. (2017). The uncertainty envelope, as well as
the trend and acceleration uncertainties, are given to a 90 % confidence
level (

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

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

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

GMSL trend uncertainties (mm yr

Based on the matrix

GMSL acceleration uncertainties (mm yr

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

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

Evolution of the GMSL trend uncertainties within a 90 %
confidence level (

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

GMSL acceleration using the AVISO GMSL time series corrected for the TOPEX-A drift using the correction proposed by Ablain (2017): the acceleration in the shaded areas is not significant (lower than the acceleration uncertainties at the 90 % confidence level). The length of the window (in years) is represented on the vertical axis and the central date of the used window (in years) is represented on the horizontal axis.

GMSL trends using the AVISO GMSL time series corrected for the
TOPEX-A drift using the correction proposed by
Ablain (2017). The length of the window (in
years) is represented on the vertical axis and the central date of the
window used (in years) is represented on the horizontal axis. A GIA
correction of

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

The global mean sea-level error variance–covariance matrix is available
online at

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

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.