Global sea-level budget and ocean-mass budget , with focus on advanced data 1 products and uncertainty characterisation

Müller Schmied (9, 10), Johnny A. Johannessen (11), Jan Even Øie Nilsen (11,12), Roshin P. Raj (11), 10 René Forsberg (13), Louise Sandberg Sørensen (13), Valentina R. Barletta (13), Sebastian B. Simonsen 11 (13), Per Knudsen (13), Ole Baltazar Andersen (13), Heidi Randall (13), Stine K. Rose (13), Christopher 12 J. Merchant (14), Claire R. Macintosh (14), Karina von Schuckmann (15), Kristin Novotny (2), Andreas 13 Groh (2), Marco Restano (16), Jérôme Benveniste (17). 14 15 (1) Technische Universität Dresden, Institut für Planetare Geodäsie, Dresden, D; 16 (2) LEGOS Toulouse, F; 17 (3) International Space Science Institute, Bern, Switzerland 18 (4) Centre for Climate Research Singapore, Meteorological Service Singapore, Singapore 19 (5) Institut of Geography and MARUM Center for Marine Environmental Sciences, University of 20 Bremen, Bremen, Germany 21 (6) University of Zurich, CH; 22 (7) University of Leeds, UK; 23 (8) Centre for Polar Observation and Modelling, University of Leeds, UK; 24 (9) Institute of Physical Geography, Goethe University Frankfurt, Frankfurt am Main, D; 25 (10) Senckenberg Leibniz Biodiversity and Climate Research Centre (SBiK-F), Frankfurt am Main, D 26 (11) Nansen Environmental and Remote Sensing Center, Bergen, NO; 27 (12) Institute of Marine Research, Bergen, NO. 28 (13) Technical University of Denmark, DK; 29 (14) University of Reading and National Centre for Earth Observation, UK; 30 (15) Mercator Ocean International, Toulouse, F; 31 (16) Serco/ESRIN, I 32 (17) ESA ESRIN, I 33 34 Correspondence 35 Martin Horwath, martin.horwath@tu-dresden.de 36 3


146
The three budget elements are spatial averages over a fixed ocean domain. We consider the global ocean 147 area in a first instance and we discuss restrictions to sub-areas further below.

148
More specifically, ΔSL(t) is the geocentric sea-level change from which effects of glacial isostatic 173 where the suffix "source" stands for Glaciers, Greenland, Antarctica, or LWS. 189 Here we assume 〈Δκ source 〉 Ocean = 〈Δκ source 〉 GlobalOcean , where the suffix "GlobalOcean" refers to  so that z(t) is a mean value over this period.

202
An alternative way of representing temporal changes is by the rates of change ( ), where t refers to a 203 time interval with length t (e.g. a month or a year) and z is the change of z during that interval.

204
Cumulation of ( ) over discrete time steps gives z(t): 205 206 We chose to primarily use the representation z(t) rather than ( ), that is, we use the evolution of state

213
where a1 is the constant part, a2 is referred to as the linear trend, or simply the trend, and 1 = 2 yr  .

214
The parameters a3, ..., a6 are co-estimated when considering time series that temporally resolve a 215 seasonal signal that has not been removed beforehand. We use the trend a2 as a descriptive statistic to 216 quantify the mean rate of change in a way that is well-defined and robust against noise. The trend a2 217 thus obtained for different budget elements is then evaluated in budget assessments according to Eq. (1),

219
We apply an unweighted regression in Eq. (10). While a weighted regression may better account for 220 uncertainties, it would imply that episodes of true interannual variation get different weights in the time 221 series of different budget elements, so that the trends a2 would be less comparable across budget 222 elements. As an exception, we apply a weighted regression in one case (the SLBC_cci steric product,

273
For the SLBC_cci project, the gridded sea-level anomalies were averaged over the 65°N-65°S latitude 274 range. The GMSL time series was corrected for GIA applying a value of -0.3 mm yr 1 (Peltier, 2004).

275
Annual and semi-annual signals were removed through a least square fit of 12-months and 6-months 276 period sinusoids.
277 Figure 1a shows the record of GMSL anomalies.

298
For each error source, the variance-covariance matrix over all months is calculated from a large number

333
Methods and product

334
The steric thickness anomaly for a layer, , of water with density is ℎ ′ = ℎ − ℎ , , where ℎ , is the 335 steric thickness of a layer with climatological temperature and salinity and ℎ = ( 1 − 1 0 ) 0 Δ 0 is the 336 "steric thickness" of the layer relative to a layer of reference density 0 and reference height ∆ 0 . ℎ ′ can 337 therefore be written in terms of layer density and climatological density for the layer , as . (11)

339
The monthly mean steric thickness anomaly for layer, , is found as the optimum combination of the

345
where is a column vector of 1s, is the vector of weights appropriate to a minimum error variance 346 average, and is the error covariance matrix of the steric thickness anomaly estimates.

347
The error covariance matrix, , is needed for the optimal calculation of the monthly average in Eq.

348
(12), as well as for the evaluation of uncertainty discussed below. To estimate this matrix, we need to

349
be clear about what "error" means here: it is the difference between the steric thickness anomaly for the 350 layer from a single profile (Argo, or climatological) and the (unknown) true cell-month mean. This 351 difference therefore has two components: the measurement error in the profile, characterised by an error 352 covariance ; and a representativeness error arising from variability within the cell-month . The 353 measurement error covariance is the smaller term and was modelled to be independent between profiles 354 within the cell (neglecting the fact that on occasion a single Argo float will contribute more than one

355
profile within a given cell in a month). The representativeness error covariance was modelled assuming 356 that this error has an exponential correlation form with a length scale of 2.5° and time scale of 10 days.

403
 Representativity errors are perfectly correlated vertically.

404
Having obtained cell-month mean SSLHA estimates and associated uncertainty, the global mean steric 405 sea-level height anomaly, ΔSLSteric, is the area-weighted average of the available gridded SSLHA results.

406
ΔSLSteric was calculated over the range 65°S to 65°N, consistent with other budget elements.

411
Uncertainty assessment

412
The uncertainties of the available cell-month mean SSLHA estimates were propagated to the ΔSLSteric.

413
In any given month, there are missing SSLHA cells, through lack of sufficient Argo profiles. Using

435
The use of a formal uncertainty framework allows separation of distinct uncertainty issues, namely, our 436 ability to parameterise and estimate the various uncertainty terms, our ability to estimate the error 437 covariance, and the model for propagation of error at each successive step.

465
The following GRACE monthly gravity field solutions series were considered:  rheology have been varied and tested against independent geodetic data to provide probabilistic 508 information. Table 1 demonstrates the sensitivity of GRACE OMC solutions to the GIA correction.

549
 Leakage errors arise from the vanishing sensitivity of GRACE to small spatial scales (high SH 550 degrees). In SLBC_cci, GRACE data were used up to a degree 60 (~333 km half-wavelength).

551
As a result, signal from the continents (e.g. ice-mass loss) leaks into the ocean domain.

552
Differences in methods to avoid (or repair) leakage effects can amount to several tenths of 553 kg m -2 yr 1 in regional OMC estimates (e.g., Kusche et al., 2016). Our buffering approach does 554 not fully avoid leakage. Moreover, the upscaling of the integrated mass changes to the full ocean 555 area is based on the assumption that the mean EWH change in the buffer is equal to the mean 556 EWH change in the buffered ocean integration kernel.

557
We adapted the uncertainty assessment approach used for GRACE-based products of the Antarctic Ice where t0 is the centre of the reference interval to which z(t) refers.

566
The noise was assessed from the GRACE OMC time series themselves as detailed by Groh et al. (2019a).

567
The de-trended and de-seasonalised time series were high-pass filtered in the temporal domain. The

568
variance of the filtered time series was assumed to be dominated by noise. This variance was scaled by

622
There are four global model parameters that need to be optimised: (i) the air temperature above which 623 melt of the ice surface is assumed to occur; (ii) the temperature threshold below which precipitation is 624 assumed to be solid; (iii) a vertical precipitation gradient used to capture local precipitation patterns not 625 resolved in the forcing datasets; and (iv) a precipitation multiplication factor to account for effects from 626 (among other processes) wind-blown snow and avalanching, which are not resolved in the forcing 627 dataset. For each of the eight forcing datasets cited above, we performed a multi-objective optimisation 628 for these four parameters, using a leave-one-glacier-out cross validation to measure the model's

740
The monthly grids were derived by applying a temporal window to aggregate the radar observations.

896
In the framework of this study, we used monthly globally-averaged (over 64432 0.5° by 0.5° grid cells)     Table 3.

978
All components exhibit a significant positive trend, i.e., water mass loss on land. Greenland ice masses 979 contribute 0.78 ± 0.02 mm yr 1 as assessed from GRACE or 0.89 ± 0.07 mm yr 1 as assessed from radar

992
In view of the systematic uncertainties inherent to several components of the mass budget, we stress that 993 any closure that is much better than the combined uncertainties does not indicate that the components     The phase difference between the red and the dark green vector corresponds to 7 days. We consider the two time periods P1 (altimetry era) and P2 (GRACE/Argo era) as introduced in Sect.    SLBC_cci Steric product (Fig. 3). Since the trend calculation accounts for these uncertainties (cf. Sect.

1138
The related monthly misclosure time series are shown in Fig. 11c. When using the SLBC_cci steric 1139 product, the monthly misclosure values are within the 1-sigma, 2-sigma, and 3-sigma range, 1140 respectively, for 89.6%, 100.0% and 100.0% of the months.

1150
The SLB misclosure varies interannually between roughly -6 mm and +4 mm (bold curves in Fig. 11b,   Table 1 is 1185 most accurate. In cases where the budget of trends is closed much better than the combined uncertainties 1186 (e.g., the second data columns in Table 3 and Table 4), this could just result from an incidental 1187 compensation of errors in the involved budget elements.

1188
The trends of the individual budget components assessed here for P1 and P2 agree within stated

1230
It is also important to mention that our 'global' mean sea level assessment as well as our assessment of 1231 the steric contribution, by its limitation to the 65°N-65°S latitude range, left out 6% of the global ocean 1232 in the Arctic and in the Southern Ocean. In the polar oceans, satellite altimetry has sampling limitations 1233 due to orbital geometry and sea-ice coverage. Likewise, Argo floats and other in-situ sensors have 1234 sampling limitations due to the presence of sea ice. Therefore, SLB assessments for the polar oceans 1235 (e.g., Raj et al., 2020) are even more challenging than for the 65°N-65°S latitude range focussed on in 1236 this paper. An assessment of the truly global mean sea-level and its contributions would involve higher 1237 uncertainties than quoted here for the 65°N-65°S range.

1277
For the GrIS contribution, we devised an empirical and effective way to convert the radar altimetry

Sea-level budget and ocean-mass budget 1297
As summarized in Table 3 and Table 4, the SLB and the OMB are closed within uncertainties for their 1298 evaluation periods P1 and P2 (SLB) and P2 (OMB). We may reformulate the budgets as follows. The

1299
GMSL linear trend over P1 and P2 is 3.05 mm yr 1 and 3.64 mm yr 1 , respectively. The larger trend 1300 over P2 than over P1 is due to an increased mass component, predominantly from Greenland but also 1301 from the other mass contributors. Over P1 (P2) the steric contribution is 38% (33%) of GMSL rise,

1306
Ranges quoted here arise from different options of assessing the contributions. Uncertainties given in 1307   Table 3 and 4 are not repeated here.

1308
We cannot attribute the statistically insignificant misclosure of linear trends. We tentatively attributed

1329
Limitations discussed in Sect. 7.2 call for further methodological developments. For example, the 1330 consideration of GIA as an own element in SLB analyses could help to enforce its consistent treatment.

1331
This will be particularly important for regional SLB studies, since GIA is a driver of regional sea-level

1336
While GMSL is an important global indicator, it is indispensable to monitor and understand the 1337 geographic patterns of sea-level change, that is, regional sea-level. Regional sea-level reflects the