In this study, for the first time, an attempt is made to close the
sea level budget on a sub-basin scale in terms of trend and amplitude of the
annual cycle. We also compare the residual time series after removing the
trend, the semiannual and the annual signals. To obtain errors for altimetry
and Argo, full variance–covariance matrices are computed using correlation
functions and their errors are fully propagated. For altimetry, we apply a
geographically dependent intermission bias

If the sum of individual components is statistically equal to the total sea level variations, the budget is closed. Total sea level variations and its components are observed by in situ and satellite measurements, but can also be modelled. Several studies have attempted to close the sea level budget by using satellite altimetry, satellite gravimetry and observations or reanalyses of ocean temperature and salinity on a global scale. Closure of the budgets is required to get a consistent division between the mass component (MC) and steric-related sea level changes. This helps us to identify the contributors to present-day sea level changes. Contributors that affect the MC are glacier and ice sheet melt and land water storage, while heat fluxes between ocean and atmosphere contribute to steric changes. Ocean dynamics have an effect on both the MC and the steric change in sea level.

One of the first attempts to close the sea level budget compared time series
of total sea level from satellite altimetry with the sum of the MC from
satellite gravimetry and the steric component from Argo floats

While time series of satellite gravimetry and Argo observations became longer
and the processing of gravity fields improved, it became possible to look at
basin-scale budgets and patterns.

Some other studies focussed on sea level budgets in small basins.

Compared to previous studies, we improve the treatment of each dataset, in
particular with respect to an accurate description of the uncertainties. We
avoid using precomputed grids for Argo and altimetry, because no covariances
between grid cells are provided, and we use full variance–covariance matrices
of the GRACE gravity field solutions. Secondly, we address the effect of
several processing steps particularly on gravimetry data in terms of trend,
annual amplitude and (residual) time series. For altimetry, we briefly discuss
the effect of different averaging methods and analyse the effect on the
trends of having a latitude-dependent intermission bias

We apply our method to the North Atlantic basin, because the coverage of Argo is sufficient during the 2004–2014 period, which allows the construction of budgets over a 10-year time span. Secondly, for both steric sea level and the MC, different regimes are present in terms of trend, annual cycle and interannual variability, which allows us to investigate the performance of the method under various conditions. Additionally, we are able to address the effect of the glacial isostatic adjustment (GIA) on the trends, which is a large contributor in the northwest of the considered basin and therefore also a potentially large source of error.

This article will briefly describe the data used in Sect.

This section briefly discusses the data from the three observing systems that
are used to determine the sea level budgets. For the determination of the sum
of the steric and the mass components of sea
level, satellite altimetry data are used. The altimetry data are obtained from
the Radar Altimetry Database System (RADS)

Number of Argo floats within a 10

The steric component of sea level rise is determined using measurement
profiles of temperature and salinity from the Argo float network. Since the
first deployments of Argo floats in the year 1999, the number of Argo floats
has increased rapidly to approximately 3900 floats in the present day. Argo reached
maturity around the year 2007, when at least 3000 floats were in the water

Designations of filtered gravity field solutions.

The Earth's time-variable gravity field is measured since 2002 by GRACE. This mission measures changes in the
Earth's gravity field by low Earth orbit satellite-to-satellite tracking.
Traditionally the Earth's gravity field is expressed in spherical harmonics.
In this study, the release five monthly spherical harmonic solutions computed at
the Center for Space Research (CSR)

The data described in the previous section are processed such that they are
suited for establishing monthly regional sea level budgets. It implies that
the equation

As far as altimetry is concerned, after computing individual along-track sea
level anomalies, two important processing steps are described in this
section: a suitable averaging method to come to a time series of MSL for a
given area and a way to deal with geographical dependencies of the
intermission bias between the two Jason missions

To compute steric sea levels from Argo temperature and salinity measurements
the Thermodynamic Equation Of Seawater-2010 (TEOS-10) software is used

Monthly GRACE solutions of CSR and ITSG are provided with full
variance–covariance and normal matrices, which allows the use of an
anisotropic Wiener filter

Geographical differences between Jason-1 and Jason-2 sea level estimates averaged over the tandem period.

List of geophysical corrections applied in this study and for the MSLs of NOAA.

Individual sea level anomalies

In GMSL time series, an intermission bias correction is applied, which is
determined from the average GMSL difference between Jason-1 and Jason-2
during their tandem phase, in which the satellites orbit the same plane only
a minute apart

Due to the limited sampling of the Argo network and the relatively large
errors in the gravity field solutions it is necessary to integrate sea level
anomalies over extended areas. Previous studies producing GMSL time series
have used two different techniques

Therefore, it is suggested to average the sea level anomalies based on the
number of available measurements within a latitude band. The method connects
the weights assigned to the measurements to the number of measurements

For the error estimation, variance–covariance matrices are computed as
described in

Values for the parameters of the intermission difference correction.

Both the satellite altimetry mean sea level anomalies as well as the MC from
GRACE are affected by GIA. For the corrections
to GRACE and altimetry, we use the solution of

Because the CSR gravity fields are created on a monthly basis and the
altimetry measurements are averaged over a cycle of approximately 10 days,
the altimetry measurements are low-pass filtered. A low-pass filter

Using the Argo profile instead of a precomputed temperature/salinity (

First, steric sea levels are computed from the individual Argo

In the analysis, only profiles that reach at least 1000 m depth are included
and have at least a measurement above 30 m depth, which is the typical depth
of the mixed layer. A virtual measurement is created at 1 m depth,
assuming the same salinity and potential temperature values as the highest
real measurements, so that the top steric signal is not missed. Only
measurements that have error flag “1” (good) or “2” (probably good) are
used and the measurements are cleaned by moving a 5

To be able to average measurements monthly over a basin or a polygon, a grid
is constructed by statistical interpolation of the steric sea levels at the
profile locations based on the method described in

Consecutively, the background field vector

Using the covariances

To average the steric sea level anomalies, the values are weighted by the
cosine of the latitude, which results in

We use the full variance–covariance and normal matrices to filter the
spherical harmonic coefficients with an anisotropic non-symmetric (ANS)
filter

The filtered grids contain ringing effects around strong signals over
Greenland and the Amazon region, which can have substantial effects on the
estimated trends in the ocean, which is discussed in Sect.

Comparison between North Atlantic mean sea level time series of NOAA (green), the Wang and Rapp method (red), our method (blue) and our method using a geographically dependent intermission bias correction and the latest geophysical corrections (light blue).

Since the degree-1 coefficients are not measured by GRACE, we add those
of

The intersatellite accelerations of GRACE are de-aliased for high-frequency
ocean and atmosphere dynamics with the Atmospheric and Ocean De-aliasing
Level-1B (AOD1B) product. Monthly averages of the AOD1B are provided as the
GAD product for both CSR and ITSG, where the mass changes over land are set
to zero. To be able to combine the GRACE MC with inverse barometer-corrected
altimetry, the GAD products containing the modelled oceanic and atmosphere
mass are added back in the form of spherical harmonics. Because the ocean
model in the AOD1B product is made mass conserving by adding/removing a thin
uniform layer of water to or from the ocean, the degree zero is removed
before subtraction from the GAD product to compensate for the mean
atmospheric mass change over the ocean, which is not measured by inverse
barometer-corrected altimetry

To compute the MC at a specific grid cell, the

The averaging over an area is equal to that of the Argo grids. Suppose that

To correct the GRACE MC for the GIA trend, we first convert the GIA spherical
harmonic coefficients into EWH. The GRACE degree-2 coefficients are
different than those for the altimetry, due to changes in the Earth's
rotation axis

The mean MC anomaly

In this section, a comparison is made between existing products and the sea
levels from altimetry, gravimetry and Argo floats. First, we compare the MSL
time series over the North Atlantic Ocean with the existing time series
provided by the NOAA Laboratory for Satellite Altimetry

Figure

Differences in sea level trends computed with and without a latitude-dependent intermission bias.

As visible from the figure, hardly any differences are observed between all
four time series. The rms differences between all time series computed in
this study and NOAA are on the order of 3–4 mm, which is slightly larger
than differences found between the GMSL time series

The application of a latitude-dependent intermission bias has a substantial
effect on the trend. From the NOAA time series, a trend of 1.5 mm yr

In Fig.

Amplitudes of the annual signal (left) and trends (right) computed
with the Scripps grids

In terms of the amplitudes of the annual signal, all three methods provide similar results in terms of the large-scale features. Typically, large signals are found in the Gulf Stream region and close to the Amazon Basin, while the areas around Greenland and west of Africa have small amplitudes. The Glorys grid differs from the others primarily in the Labrador Sea and northwest of Ireland. Secondly, the grid computed in this study and the Glorys grid exhibit more short-wavelength spatial variability than the Scripps grid. As long as the regions over which budgets are made are large enough, the methods will not differ substantially in terms of annual amplitude.

The plots in the right column of Fig.

(Caption on previous page.) Amplitudes of the annual signal (left) and
trends (right) of the mass signal. The first to the third row show the
CSR96-DDK solutions

In Fig.

To determine how this effects sub-basin-scale MC time series, it is first
required to determine the minimum area over which the measurements have to be
integrated. GRACE gravity fields have a resolution of typically 250–300 km
half wavelength

To illustrate the effects of different filters and residual striping on
sub-basin-scale budgets, Fig.

The month-to-month noise of CSR60-W and CSR96-W time series is comparable for
all three polygons. The CSR60-DDK and CSR96-DDK time series become much noisier
for the meridionally oriented polygon, where month-to-month jumps of
10–20 mm occur. In addition, the DDK time series exhibit a substantially
different trend in the meridional polygon than the other time series, because
the orientation of the polygon is aligned with the residual stripes
(Fig.

Sub-basin-scale time series of the MC using various filters for
three polygons with different orientation: zonal

The first objective of this section is to reveal patterns of sea level
amplitudes and trends in the North Atlantic Ocean and how these resemble for
the two different methods: altimetry and the combined method of Argo and GRACE,
hereafter referred to as Argo

Amplitudes of the annual signal (left) and trends (right) computed
of the sum of the components (Argo

In Fig.

The grids and ground tracks shown in the left column show that large annual
signals are present in the Gulf Stream region and in a tongue stretching from
the Amazon to the Sahel. A region without any substantial annual signal is
located just west of Africa, which is clearly visible in both the Argo

The trends from altimetry in the right column of Fig.

The North Atlantic Ocean is split into 10 regions, divided in the middle by
the mid-Atlantic ridge, while trying not to cut through the major oceanographic
features in the latitudinal direction, like the salt water tongue in front
of the Mediterranean and the Gulf Stream, as shown in Fig.

Budgets for three representative regions, using the
Wiener-filtered MC solutions, are shown in Fig.

Time series of sea level components for regions B, D and I. Left: total sea level from altimetry in red, steric sea level in green and the ITSG90-W mass in blue. Right: total sea level from altimetry in red and the sum of steric sea level and mass in blue. In yellow and light blue: their 95 % confidence interval.

Trends of total sea level (mm yr

ASL

Amplitudes (mm) of the annual signal from total sea level from altimetry and the sum of steric and mass from Argo and GRACE for different filter solutions.

Trends computed from the time series of Fig.

In the northwest of the Atlantic, the choice of gravity field filter either
substantially influences the estimated trends (D and E), or they are just
outside of 2 standard deviations (C) for one or more solutions. Using the
CSR96-W solution, the budget is closed within 2 standard deviations for all
three polygons, whereas the other solutions do not close the budget. For
region C, the results of the all filters resemble one another quite well, but some
are just outside of 2 standard deviations from altimetry. For region D, the
CSR60-W results are far off, but the other results are close again. In this
region, sharp gradients occur not only in the MC with the presence of a
neighbouring continental shelf but also in the steric component. This might
lead to leakage of the continental shelf mass signal or problematic
interpolation of the Argo steric sea levels. In addition, for both of the
aforementioned regions, the GIA correction on the MC is relatively large.
Adding a GIA correction error of 10–20 %, which is smaller than
discussed in Sect.

Ultimately, only the budget in region H cannot be closed with any of the
solutions and there is no strong GIA signal present, which could be
responsible for a large bias. In addition, the sea level in this polygon does
not exhibit any strong gradients and the number of Argo floats is
substantial. This excludes interpolation or filtering problems. Therefore, we
argue that this can be explained by a deep-steric effect that could be
related to variations in the export of saline water from the Mediterranean

In conclusion, it is possible to close the sea level budget within two
standard deviations for 9 out of 10 regions using CSR96-W. If a
10–20 % GIA correction error is taken into account, the budget for
9 out of 10 polygons is also closed for CSR96-DDK and ITSG90-W. This also
suggests that the commonly assumed GIA correction error of 20–30 %

Time series of sea level components for polygons B, D and I after removing the trends and the annual and semiannual signals. Left: ITSG90-W mass in blue and steric sea level in green. Right: total sea level from altimetry in red and the sum of steric sea level and mass in blue.

We indicated that the seasonal cycles are primarily caused by steric
variations in sea level (Fig.

Table

Even though no error bars are computed for the CSR96-DDK, it is clear that
the results are far better in terms of budget closure. The results are
comparable to ITSG90-W, which closes 7 out of 10 budgets within 2
standard deviations. CSR DDK5

Using ITSG90-W, it is also possible to close the budget on the scale of the
whole North Atlantic Ocean (last row of Table

Fraction of explained variance,

Time series for the same regions as in Fig.

Using any of the filtered CSR or ITSG solutions, it is possible to detect the
interannual variability described, probably because most of the signal is of
steric origin. However, for the interannual signals that are less pronounced,
or for high-frequency behaviour of sea level, there are some differences
between the MC solutions. Table

The third column indicates that Argo in combination with CSR96-W does not
explain much of the residual variance, but mostly introduces additional
noise, which causes the negative values. Using the DDK5-filtered MC, the
explained variance increases, but the best performance is obtained with the
CSR60-W and especially the ITSG90-W gravity fields. The last column shows
that after reducing the trend, and the semiannual and annual signals,
between 24 and 53 % of the residual signal can be explained by the
combination of Argo and ITSG90-W. It is remarkable that for the whole North
Atlantic Ocean (last row), no variance is explained by the Argo

For the first time, it is shown that sea level budgets can be closed on a
sub-basin scale. With the current length of the time series, it is possible to
establish budgets over areas of approximately

To obtain proper averaged mass for sub-basin-scale polygons, the gravity
fields have to be filtered. The application of an anisotropic Wiener filter
on the CSR96 solutions leads to the best closure of the budget in terms of
sea level trends, with closure in 9 out of 10 regions. In the considered
regions, also the CSR96-DDK and the ITSG90-W solutions appear to close just as
many budgets when a 10–20 % GIA correction error is added. The results
of the CSR96-DDK filter, however, strongly depend on the orientation of
averaging area due to residual meridional striping. The strong resemblance
between trends also indicates that the errors on the GIA model are probably
smaller than the commonly assumed 20–30 %. Furthermore, a large
difference in trend between altimetry and Argo

The CSR60-W and CSR96-W solutions appear to underestimate the amplitude of the annual signal substantially. They also suffer from what appears to be leakage around the Amazon and Sahel, regions with a substantial annual hydrological cycle. Using the CSR96-DDK gravity fields and the ITSG90-W solutions, the sum of the steric and mass components becomes significantly closer to that of altimetry, with closure in 7 out of 10 regions. However, it must be noted that the altimetry signals tend to be slightly larger. This is likely due to partial destruction of the signal by filtering of the gravity fields or limited Argo coverage or, in some regions, deep-steric signals.

By removing the semiannual and annual signals and trends, interannual variability can be detected. Since most of the interannual variability in the North Atlantic Ocean is contained in the steric component, the type of filter on the gravity fields is not really important. However, if we look at differences on a month-to-month basis, high-frequency variations or small interannual fluctuations in mass, it is best to use the CSR60-W the ITSG90-W solutions, because the fraction of explained variance of the altimetric sea level time series by the sum of the components using these solutions is largest. Using the ITSG90-W solution, 24–53 % of the variability in the altimetry-derived sea level time series is explained. The CSR96-W solution only introduces noise and explains virtually no residual variability of the altimetry time series. Especially in the 4-day repeat orbits in 2004 and even the months around them, the Wiener-filtered solutions do not give proper estimates of the MC, which partly contributes to a lower explained variance.

To summarize, using the ITSG Wiener-filtered solution, the trend budgets close when an error of 10–20 % on the GIA correction is assumed. They perform, together with the standard DDK5-filtered CSR solution, best in terms of annual amplitude budget closure. Additionally, the combination of ITSG mass and Argo steric sea levels explains the largest fraction of variance in altimetry time series. Based on this, the best option to establish budgets, at scales considered in this paper, is the ITSG90-W solution. However, due to residual striping in the trend grids from the static background field that are added back after Wiener filtering, one must take care when averaging the MC over even smaller regions, or meridionally oriented polygons, which is a even a bigger problem for the standard CSR96-DDK solutions.

We added data in the supplement, so other people can reproduce trend plots and grids from the results section.

We would like to thank ITSG and CSR for providing their gravity fields including full normal and/or variance–covariance matrices. The Argo profile data are kindly provided on the website of the US National Oceanographic Data Center (NODC). We appreciate the service of the employees of Mercator for delivering the Glorys reanalysis product. We wish to express our gratitude to the RADS teams for constantly maintaining and updating their database. This study is funded by the Netherlands Organisation for Scientific Research (NWO) through VIDI grant 864.12.012 (Multi-Scale Sea Level (MuSSeL)). Edited by: M. Hoppema Reviewed by: two anonymous referees