1 Introduction

Changes in sea surface height (SSH) can be explained by a combination of physical processes: the addition or removal of freshwater (mass change), change in ocean water volume (steric change), a change in ocean bottom topography (bathymetry), and ocean water redistribution driven by changes in the geoid and ocean circulation (cf. Figure 1a). Typically, solid‐Earth changes are assumed to be driven primarily by glacial isostatic adjustment (GIA; herein, we only consider the viscous response of the solid Earth to deglaciation after the Last Glacial Maximum). Several GIA forward models are available and are used as a correction to SSH observations. Thus, conventional sea level budget (SLB) studies equate total SSH change to a sum of mass and steric sea level change (Bindoff et al., 2007; Leuliette & Willis, 2011). As a result, SLB studies are essential to understand the temporal evolution of different contributors to contemporary sea level rise.

Figure 1 Open in figure viewer PowerPoint 300‐km buffer for visualization purposes. The sea level budget including ocean bottom deformation: (a) schematic diagram of a column of the Earth's solid and fluid envelope. The initial (dashed) and final (solid) state of the ocean bottom (orange) and sea surface (blue); (b) SSH trend from ESA SLCCI altimetry product; (c) steric sea level trend from ensemble mean of four steric data products; (d) ocean mass sea level trend from JPL GRACE Release 06 data; and (e) the OBD trend calculated from our approach. All the maps have been filtered with a 1,000‐km half‐width Gaussian filter and masked over land300‐km buffer for visualization purposes.

The global‐mean SLB is reported to close within the uncertainties of the observation systems, when taking an ensemble mean of observational products available from different sources (WCRP, 2018). However, there can be large differences in the trends and spatial patterns from different sets of observational data products (Chambers et al., 2017; Dieng et al., 2015; WCRP, 2018), and the SLB at the ocean‐basin scale does not close (Dieng et al., 2015; Llovel et al., 2010; Purkey et al., 2014). A gap in the SLB can be attributed to inaccuracies or poor sampling in the observations and/or neglecting physical processes in the budget equation. Therefore, identifying individual processes and understanding how their magnitude is changing in time is important to improve sea level forecasting abilities. For example, there is increasing concern over the uncertainty in magnitude of the deep steric contribution to sea level change (that below 2,000 m not measured by the ARGO float network; Dieng et al., 2015; Purkey et al., 2014; WCRP, 2018), which motivated the community to invest in deep ARGO floats. However, elastic deformation of the ocean bottom due to changes in present‐day mass loading has not received as much attention.

It is known that the solid‐Earth responds instantaneously and elastically to changes in the surface mass load (Farrell, 1972). There is clear evidence that sea level rise, which was dominated by thermosteric change during the 20 Century, is now dominated by ocean‐land mass exchange (Bamber et al., 2018; Chen et al., 2017; WCRP, 2018). The small ocean mass increase during the last Century meant that elastic ocean bottom deformation (OBD) was significantly smaller than uncertainties and it could safely be ignored in the SLB equation. However, the contemporary acceleration in ocean mass makes OBD nonnegligible. Recently, it has been shown that the theoretical elastic OBD due to changes in mass load since 1993 contributes approximately 0.13 mm/year (or 3–4%) to global‐mean sea level change (Frederikse et al., 2017). This contribution is comparable in magnitude to the deep steric contribution, which has been identified as one of the top priorities in sea level research (Roemmich et al., 2019; WCRP, 2018). Due to ongoing global warming, the rate of ocean mass change is expected to continue to increase, which will in‐turn increase the elastic OBD in the near future.

In the SLB community, several different methods are employed to estimate mass sea level change from observations. Ocean mass changes may be directly observed over the oceans (as used by Chambers et al., 2004; Fenoglio‐Marc et al., 2006; Llovel et al., 2010) or may be indirectly determined invoking the conservation of mass for the whole Earth system, where ocean mass is the sum of contributions from land mass changes (e.g., Dieng et al., 2017; WCRP, 2018). To obtain a realistic spatial distribution of the indirect ocean mass, gravitation, rotation, and deformation effects are incorporated via the sea level equation (Farrell & Clark, 1976), resulting in “sea level fingerprints” (Adhikari et al., 2019; Frederikse et al., 2017; Riva et al., 2010). We note these latter sea level fingerprints intrinsically account for OBD as part of the gravitation, rotation, and deformation component, but not ocean mass redistribution (manometric changes) due to other processes such as wind stress and ocean circulation changes.

A small number of studies acknowledge the OBD component of sea level and recommend including it as a correction to the absolute SSH anomaly observations from satellite altimetry or intrinsically in sea level fingerprints derived from land mass change observations (Frederikse et al., 2017; Kuo et al., 2008; Ray et al., 2013). Yet OBD is still not included in SLB studies that use direct GRACE ocean mass change estimates or the simple sum of mass change estimates over land (WCRP, 2018). Since many studies use the SLB equation as a constraint to assess the quality of a time series or to estimate one component from the residual of the SSH minus the other budget components, updating the SLB equation, so that it accurately represents processes contributing to sea level change, is vital.

In this paper we first discuss the conventional SLB and then derive an updated SLB equation from a mass‐volume approach. The updated SLB equation has a dedicated term for OBD along with steric and mass terms. We show that the new SLB equation is equivalent to the conventional SLB equation under the assumption that the elastic deformation of the ocean floor due to ocean mass change is negligible. Our updated SLB equation is then applied using satellite altimetry SSH anomaly, GRACE ocean mass, in situ measured steric data, and an estimate of OBD in the updated SLB equation. We obtain our estimate of OBD from GRACE‐observed mass redistribution and elastic load Love numbers for a PREM solid‐Earth model (Dziewonski & Anderson, 1981). We discuss implications of this updated budget equation for SLB studies using various data products.