The Gravity Recovery and Climate Experiment (GRACE) mission data have an
important, if not revolutionary, impact on how scientists quantify the water
transport on the Earth's surface. The transport phenomena include land
hydrology, physical oceanography, atmospheric moisture flux, and global
cryospheric mass balance. The mass transport observed by the satellite system
also includes solid Earth motions caused by, for example, great subduction
zone earthquakes and glacial isostatic adjustment (GIA) processes. When
coupled with altimetry, these space gravimetry data provide a powerful
framework for studying climate-related changes on decadal timescales, such as
ice mass loss and sea-level rise. As the changes in the latter are
significant over the past two decades, there is a concomitant self-attraction
and loading phenomenon generating ancillary changes in gravity, sea surface,
and solid Earth deformation. These generate a finite signal in GRACE and
ocean altimetry, and it may often be desirable to isolate and remove them for
the purpose of understanding, for example, ocean circulation changes and
post-seismic viscoelastic mantle flow, or GIA, occurring beneath the
seafloor. Here we perform a systematic calculation of sea-level fingerprints
of on-land water mass changes using monthly Release-06 GRACE Level-2 Stokes
coefficients for the span April 2002 to August 2016, which result in a set of
solutions for the time-varying geoid, sea-surface height, and vertical
bedrock motion. We provide both spherical harmonic coefficients and spatial
maps of these global field variables and uncertainties therein
(10.7910/DVN/8UC8IR; ). Solutions are provided
for three official GRACE data processing centers, namely the University of
Texas Austin's Center for Space Research (CSR), GeoForschungsZentrum Potsdam
(GFZ), and Jet Propulsion Laboratory (JPL), with and without rotational
feedback included and in both the center-of-mass and center-of-figure
reference frames. These data may be applied for either study of the fields
themselves or as fundamental filter components for the analysis of
ocean-circulation- and earthquake-related fields or for improving ocean tide
models.
Geodesists have long understood that the ocean mean sea surface follows the
shape of the Earth's geoid and that changes in on-land
water storage are a source of time-varying gravity . The
fundamental relationship of changes in land ice and water, solid Earth, and
sea-surface height is essential to the study of past and present relative
sea level e.g.,. Our recent
gain in confidence for monitoring the geographic locations and amplitudes of
both seasonal and supra-seasonal changes in global glacier and ice sheet mass,
dating to the beginning of the radar interferometry and altimetry era of the
early 1990s e.g.,, strengthens our ability
to effectively harness this information to construct informative models about
global sea-level variability associated with a self-attraction and loading
phenomenon . The mathematical
formalism relating changes in gravitational, rotational, and solid Earth
deformation responses to land ice and hydrological mass change has now niched
itself into contemporary studies of sea-level change: the prediction of
“sea-level fingerprints”. Sea-level fingerprints are a consequence of the
fact that the water elements being transported laterally between land and
oceans carry mass, gravitational attraction, and the ability to change the
radial stress at the solid Earth surface. These are characterized, for
example, as changes in relative sea level encircling areas of intense ice
mass loss such as Patagonia, coastal Alaska, the Amundsen Sea sector of West
Antarctica, and the Greenland Ice Sheet
e.g.,.
To date, space gravimetric measurements using Gravity Recovery and
Climate Experiment (GRACE) monthly gravity fields
and the subpolar ocean altimetry measurements from TOPEX/Poseidon and Jason
each have multiple geophysical signals and respective noise floors that are
generally high enough that clear detection of these contemporary land-mass-driven fingerprints in the oceans has remained elusive. However, it is
believed that these signals will eventually emerge in these data systems.
Such a belief springs, in part, from the fact that amplitudes of internal
ocean variability in intra- and interannual mass that GRACE observes are
relatively mute in comparison to on-land hydrology, two-way land-to-ocean
transport, and secular trends in land ice changes
. In fact,
have used ocean-bottom pressure data, in conjunction with
space geodetic data, to claim that fingerprints associated with decadal-scale
on-land mass changes are detected. Furthermore, have shown
that the fingerprints of Greenland ice mass loss have had measurable influences
on tide gauge records along the eastern coast of the US since the mid-1990s.
have noted that the influence of a land-mass-induced
fingerprint may be reflected in tide gauge records of relative sea level at
the northern Antarctic Peninsula, as there is a distinct change in trend at
about the year 2000 CE, possibly reflecting increased regional ice mass loss.
Each of these observations might be considered both intriguing and
preliminary in terms of providing the community with unambiguous detection of
sea-level fingerprints.
The effects of the fingerprints are nonetheless important to disentangle from
many geophysical and ocean circulation models and dataset. New insights into
the regional and global sea-level budgets are sought through explicitly
combining ocean altimetry with the space gravimetry information, and a key
part of this combination is to account for the details of sea-level
fingerprints e.g.,. Consideration of
land-ice- and water-driven fingerprints is also necessary when using geodetic
data to search for glacial isostatic adjustment (GIA) signals residing at or beneath the seafloor
e.g., or examining ice loss on land when ocean water
surrounds the region, such as in Graham Land of the Antarctic Peninsula
. Future applications of sea-level
fingerprints in geophysical geodesy should include the study of great
earthquakes (Mw≥8.0) at subduction zones and at
ocean rifts beneath the open ocean or adjacent to the
Antarctic Ice Sheet .
This paper describes a dataset of monthly changes in relative sea level,
geoid height, and vertical bedrock motion induced by mass redistribution from
land to ocean. These are derived from the Release-06 GRACE Level-2 monthly
Stokes coefficients for the period April 2002 to August 2016. The GRACE
mission data have been instrumental to the study of the Earth's climate system
and have helped us resolve numerous long-standing
questions in oceanography, hydrology, cryosphere, and geodesy. The GRACE
gravity solutions are now employed for providing new insights into changes in
ocean circulation .
The terrestrial water storage is now rigorously quantified for continents
as is the global cryospheric mass balance
. Land mass
change and its exchange with the global oceans, in fact, makes it possible to
successfully reconstruct subtle changes in the position of Earth's spin axis
on interannual timescales , thus providing a
confidence in the robustness of GRACE-based estimates of global surface mass
transport.
Key variables and deliverables
Relative sea level is defined as the height of the ocean water column bounded
by two surfaces: solid Earth surface and sea surface. Change in relative
sea level, ΔS, at a geographical location described by colatitude and
longitude (θ,ϕ) over the time interval Δt may be expressed
as follows:
ΔS(θ,ϕ,Δt)=ΔN(θ,ϕ,Δt)-ΔU(θ,ϕ,Δt),
where ΔN and ΔU are corresponding changes in sea-surface
height and bedrock elevation, respectively. Both of these variables are
usually expressed relative to the reference ellipsoid, which in turn is
defined relative to either the center-of-mass (CM) of the total Earth system
or the center-of-figure (CF) of the solid Earth surface. Tide gauges provide
direct measurements of ΔS, whereas satellite altimetry measures
ΔN in the CM reference frame.
Mass redistributed on Earth's surface provides a direct perturbation to the
Earth's gravitational and rotational potentials, causing a corresponding
perturbation in the geoid height. Since the geoid height on a realistic Earth
does not necessarily have to coincide with the sea-surface height, we write
ΔS(θ,ϕ,Δt)=1gΔΦ(θ,ϕ,Δt)-ΔU(θ,ϕ,Δt)2+ΔC(Δt),
where ΔΦ is the net perturbation in Earth's surface potential,
ΔC is a spatial invariant that explains the discrepancy between the
sea-surface height and geoid height , measured with
respect to the same reference ellipsoid, and g is the mean gravitational
acceleration at Earth's surface. As will be further discussed in Sect. 3,
ΔC is essential to conserve mass. Space-based gravity missions, such
as GRACE and GRACE Follow-On (GRACE-FO), provide direct measurements of the
non-rotational part of ΔΦ (to be defined explicitly in Sect. 3)
in the CM frame; the satellite system cannot measure the rotational part of ΔΦ because it retrieves data in an inertial reference frame.
In geodetic applications, global field variables are typically expanded in a
spherical harmonic (SH) domain. Most of the GRACE data processing centers –
including the University of Texas Austin's Center for Space Research (CSR),
GeoForschungsZentrum Potsdam (GFZ), and Jet Propulsion Laboratory (JPL) –
provide monthly solutions for normalized SH coefficients of the gravitational
potential termed “Stokes coefficients”. Stokes coefficient anomalies – the
values that deviate from the mean (static) field – can be used to readily
retrieve changes in on-land ice and water storage or ocean bottom pressure.
The goal of this paper is to provide Stokes coefficient anomalies (i.e., SH
coefficients of ΔΦ minus rotational centrifugal potential)
associated with the sea-level fingerprint of monthly changes in on-land ice
and water storage, which are derived from CSR, GFZ, and JPL GRACE Stokes
coefficients themselves. As we shall further clarify below, we provide these
new fingerprint coefficients and their corresponding spatial maps computed in
both CM and CF reference frames, with and without rotational feedback
included. We also provide solutions for change in relative sea level ΔS and bedrock elevation ΔU. Corresponding solutions for sea-surface
height, ΔN, may be retrieved using Eq. (). For
brevity, we hereafter drop the Δ symbol and assume that variables
imply “change” in respective fields – not the absolute fields – implicitly.
The sea-level equation
Here we briefly summarize the fundamental concept and a numerical technique
of solving the so-called “sea-level equation”. Much of the background and
supporting materials may be found, for example, in ,
, , and . Let
L(θ,ϕ,t) be the global, mass-conserving load function so that
L(θ,ϕ,t)=H(θ,ϕ,t)1-O(θ,ϕ)3+S(θ,ϕ,t)O(θ,ϕ),
where H(θ,ϕ,t) is the on-land change in water equivalent height
over the time period t, and S(θ,ϕ,t) is the corresponding change
in relative sea level on the oceanic domain O(θ,ϕ). By
definition, O=1 for the oceans and 0 otherwise. For ease of
discussion, we write F(θ,ϕ,t)=H(θ,ϕ,t)1-O(θ,ϕ) so that F(θ,ϕ,t) defines the
model “forcing” function.
The net change in on-land (water) mass directly affects the relative
sea level, hence conserving mass on a global scale. Such a redistribution of
mass on Earth's surface perturbs its gravitational and rotational potentials
and further redistributes the ocean mass. The net result of these
perturbations is the sea-level fingerprint: a unique spatial pattern of
relative sea level that is consistent with fundamental physical features of a
realistic Earth. For a self-gravitating elastically compressible rotating
Earth, we compute the sea-level fingerprint by satisfying the following sea-level
equation:
S(θ,ϕ,t)=E(t)+1gΦ(θ,ϕ,t)-U(θ,ϕ,t)4-1gΦ(θ,ϕ,t)-U(θ,ϕ,t).
The physical interpretation of the right-hand side terms is as follows:
The barystatic term, E(t), directly follows from the mass conservation principle. This spatial invariant describes S that would be
resulted in by distributing the net change in land water storage uniformly over the oceans.
Changes in the Earth's surface potential, Φ(θ,ϕ,t), and the solid Earth surface, U(θ,ϕ,t), may be partitioned as follows:Φ(θ,ϕ,t)U(θ,ϕ,t)=Φg(θ,ϕ,t)+Φr(θ,ϕ,t)Ug(θ,ϕ,t)+Ur(θ,ϕ,t),where Φg and Ug are the respective signals associated with the perturbation in gravitational potential. We may
compute Φg and Ug by convolving L (Eq. ) with the respective Green's functions. Similarly,
Φr and Ur are associated with the perturbation in rotational potential. The change in Earth orientation driven
by shift in the inertia tensor causes both solid Earth deformation and sea-level change . The net effects of the change
in orientation of Earth's spin axis thus provide a rotational feedback e.g.,. We may compute Φr and
Ur based on the perturbation in Earth's inertia tensor due to the global surface mass redistribution described by L
(Eq. ). We define all of the terms appearing in Eqs. () and () explicitly in Appendix A.
The last term in Eq. () represents the ocean-averaged value of (Φ/g-U). This spatial invariant is essential
to ensure that the global mean relative sea-level change is the same as the barystatic term.
To solve for the sea-level fingerprint in a conventional SH domain
e.g., and isolate useful SH coefficients noted in
Sect. 2, we express Eq. (), using Eq. (),
in the following form:
S(θ,ϕ,t)=X(θ,ϕ,t)+Y(θ,ϕ,t)+P(θ,ϕ,t)6+Q(θ,ϕ,t)+C(t),
where X=Φg/g, Y=Φr/g, P=-Ug, Q=-Ur, and C=E-<Φ/g-U>. By default, we account for the
rotational feedback, which when excluded, Eq. () takes a
reduced form with Y=0, Q=0, and C=E-<Φg/g-Ug>. We now multiply both sides of Eq. () by the
ocean function, O, to get the following:
S^(θ,ϕ,t)=X^(θ,ϕ,t)+Y^(θ,ϕ,t)+P^(θ,ϕ,t)7+Q^(θ,ϕ,t)+C^(t),
where S^=OS, X^=OX, and so on. In the
employed spectral methods (Appendix B), we find it more straightforward to
solve Eq. () rather than Eq. (). Since all of the
right-hand side terms appearing in Eq. () depend on
S^ itself (see Eqs. and and Appendix A), we solve the equation recursively until the desired solution convergence
is achieved (see Appendix B5). We consider the barystatic sea level (Eq. ) as the starting solution; i.e. S^=E^, where
E^=OE. Once Eq. () is solved for
S^, all of the terms appearing in Eq. () may be
retrieved easily.
We expand all of the terms appearing in Eqs. () and
() in the SH domain (as in Eq. ). Inserting these SH expansions into
Eq. () and equating the corresponding (degree l, order m) SH
coefficients, we find the following for any rth recursion:
S^lmr=X^lm(S^lmr-1)+Y^lm(S^lmr-1)+P^lm(S^lmr-1)+Q^lm(S^lmr-1)8+C^lm(S^lmr-1),
where r=1,2,…rmax is the recursion counter, and rmax is
the value of r for which the desired convergence is attained. Note that
dependence of right-hand side terms on S^lm itself is explicitly
stated. For r=1, we set S^lm0=E^lm. Since
E^lm does not depend on S^lm, it does not evolve during
the recursion. We define E^lm (Eq. ) and all of the other hatted coefficients appearing above (Eq. and so on)
in Appendix B. The hatted coefficients depend on corresponding non-hatted
coefficients, which are explicitly defined in Eqs. ()–() and ().
Once we obtain the final solution for S^lmr (after iteration
r=rmax), denoted for simplicity by S^lm, final solutions
for all of the non-hatted (degree p, order q) coefficients are obtained
as well. These non-hatted coefficients automatically satisfy the sea-level
equation itself (Eq. ) in the SH domain; i.e.,
Spq=Xpq(S^lm)+Ypq(S^lm)+Ppq(S^lm)+Qpq(S^lm)9+Cpq(S^lm).
Note that all of the SH coefficients appearing above are only a function of time
t. With final solutions achieved for all of the terms appearing in Eq. (), SH coefficients of geoid height change for a self-gravitating
Earth are given by Xpq(t) and those for a self-gravitating rotating
Earth by Xpq(t)+Ypq(t). Similarly, SH coefficients
of bedrock elevation change are given by -Ppq(t) for a self-gravitating
Earth and by -Ppq(t)+Qpq(t) for a self-gravitating
rotating Earth.
GRACE and sea-level fingerprints
Here we give a brief summary of the steps undertaken to develop sea-level
fingerprints and complementary data products. First, we note that the GRACE
processing centers, including CSR, GFZ, and JPL, have a variety of methods
employed to reduce noise, but the system has an inherent resolution limit of
about 300 to 400 km in radius at the Earth's surface. Hence, the Stokes
coefficients for the potential field provided by the official centers are
truncated at a varying degree and order, from 60 to 96. We employ a
truncation at degree and order 60, as many months may be much noisier than
others.
We use GRACE Level-2 Release-06 data products provided by all three premier
(and official) data processing centers (available at
ftp://podaac.jpl.nasa.gov/allData/grace/L2/, last access: 7 May 2019)
that are available for the spans April 2002 through August 2016 (CSR and JPL)
and January 2003 through November 2014 (GFZ). The Release-06 GSM files
represent the total gravity variability due to land surface hydrology,
cryospheric changes, episodic seismogenic processes, and GIA. We assume that
all mass transport information is contained within the post-processed GSM
files in which background models for the mass changes in atmosphere and
oceans having periodicities shorter than 1 month are removed
. GSM datasets are also corrected for solid Earth and
ocean tides by the processing centers
see. We also assume continuous transfer
of net mass to and from the oceans takes place on all timescales. This
includes a trend that supplies the mass term of sea-level rise. To do this
correctly, we derive degree 1 coefficients from JPL Release-06 data products
following the methods of . We replace degree 2 order 0
coefficients by those derived from satellite laser ranging analysis
that are compatible with Release-06 data products
(available at
ftp://podaac.jpl.nasa.gov/allData/grace/docs/TN-11_C20_SLR.txt, last
access: 7 May 2019). The physical origins motivating this replacement are
well known: there is far greater sensitivity to changes in degree 2 order 0
coefficients that can be retrieved from higher orbiting satellites tracked by
terrestrial laser stations than for GRACE . We apply GIA
correction coefficient by coefficient using the expected values from a
Bayesian analysis , available at
https://vesl.jpl.nasa.gov/solid-earth/gia/ (last access: 7 May 2019).
Finally, for all coefficients, we remove corresponding 11-year (January
2003–December 2013) mean values to retrieve Strokes coefficient anomalies.
By combining GSM Stokes coefficient anomalies with GIA and low-degree
coefficients as noted above, we may derive corresponding coefficients for
land water storage anomalies, Hlm(t), as follows :
Hlm(t)=aρe3ρw2l+11+kl′exp-14ln(2)lra210Clmgsm*(t)-Clmgia(t),
where ρw is the water density, ρe is the Earth's mean density,
kl′ are the load Love numbers of degree l, a is the Earth's mean
surface radius, r is the Gaussian smoothing window, and
Clmgsm and Clmgia are the GSM and GIA Stokes
coefficient anomalies, respectively. The term enclosed by braces is the
Gaussian smoothing filter. We consider r=300 km to comply with the so-called
gain factors that are used to restore the attenuated signals (detailed
below). An asterisk associated with GSM coefficients is meant to imply that
these solutions are corrected for more accurate low-degree Stokes
coefficients as noted above.
Barystatic sea-level time series. Our estimates
of trends and seasonal amplitudes for all three data centers are compared to
JPL mascon solutions . Results are plotted relative to the
time means over the period January 2003 through December 2013. Trend values
are provided for the period January 2005 through December 2015, except for
GFZ solutions (January 2005 to December 2013), for a comparison to the sum of
individual mass components (during January 2005 to December 2016) listed in
Table 13 of the report: 1.65±0.23 mm yr-1. As
an additional point of comparison, find GRACE-determined
mass changes for the barystatic sea-level trend at 2.04±0.08 mm yr-1 for January 2005–December 2013 from the mean of CSR,
GFZ, and JPL Level-2 Release-05 spherical harmonic solutions.
Monthly land water storage fields, H(θ,ϕ,t), may be generated by
assembling the coefficients (Eq. ) in an SH domain, as in
Eq. (). Gaussian smoothing aimed at removing the data noise also
attenuates the signals. An appropriate scaling of the fields is therefore
essential. For the ice sheet and peripheral glaciers in Greenland, three
non-overlapping sub-domains of Antarctica, and 15 regions of global glaciers
and ice caps, we compare our estimates of average rate of regional mass
change during February 2003 through June 2013 with those computed by
and derive the scaling factors – unique for CSR, GFZ,
and JPL data products – for each of these 19 cryospheric domains. As for the
non-cryospheric continental domains, analyzed monthly
land water storage signals obtained from the GRACE observations and the Noah
land surface model, simulated within the Global Land Data Assimilation System
(GLDAS-Noah), and derived global gridded gain factors. We combine these
factors to scale H(θ,ϕ,t) for the entire continents. Our estimates
of barystatic time series are comparable to, for example, JPL mascon solutions for both trends and seasonal amplitudes
(Fig. 1).
Effects of scaling on the select spherical harmonic coefficients.
(a) Scaling effects on the average rate of change in land water
storage, relative sea level and geoid height, during April 2002 through March
2016. These solutions are based on JPL Stokes coefficients and are computed
in the CM reference frame with the rotational feedback included.
(b) Distribution of energy, computed for a given degree l as a sum
of the squares of corresponding orders m, for unscaled and scaled solutions.
Note the log scale on the x axis of the right panel. Unlike the geoid height
change, the relative sea-level change has nonzero energy at l=0 with
magnitudes of 1.16 and 2.37 mm2 yr-2 for unscaled and scaled
solutions, respectively. Also note that solutions for geoid height change
employ a different scale (factor of 4) for appropriate visualization.
Land load function, sea-level fingerprint, and uncertainties
therein. Average rate of water equivalent height change in land water
storage (a, b) and associated change in relative sea
level (c, d) for the period April 2002 to March 2016 are shown with
their corresponding 1σ uncertainties. Maps for sea-level fingerprint
and uncertainty are produced by assembling the corresponding spherical
harmonic coefficients provided with this article. These solutions are based
on JPL Stokes coefficients and are computed in the CM reference frame with
the rotational feedback included. A zoomed-in map of the Mediterranean
Sea (e) is meant to highlight the local variability in sea-level
fingerprint. The fingerprint-predicted trends for tide gauges at TRIESTE
(1.26±0.18 mm yr-1) and CASCAIS (1.50±0.17 mm yr-1)
reflect differences that are comparable to those associated with interdecadal
atmospheric pressure trends (0.2±0.2 mm yr-1) and the
GIA-related fingerprint (≈0.2 mm yr-1) (see
, and , respectively). This
illustrates one example of the importance of contemporary sea-level
fingerprints for tide gauge data analysis and interpretation.
Panel (f) is meant to show that our solutions are comparable to
those obtained from a well-validated and higher-resolution ISSM sea-level
solver ; note that the solution discrepancy is within the
1σ uncertainties (compare with panel d).
A detailed description of scaling may be found in ,
who used the same recipe to post-process the CSR Release-05 GRACE Level-2 Stokes coefficients for robust
reconstruction of interannual variability in position of Earth's spin axis.
This gives us extraordinary confidence that the procedure for generating land
water storage fields and corresponding fingerprints are not only sound but
highly robust at long wavelengths. The effects of scaling on SH coefficients
of select fields are shown in Fig. 2. Our model solutions are also robust.
For example, our estimates of relative sea-level change (Fig. 3), vertical
bedrock motion (not shown), and geoid height change (not shown) are
consistent with the respective solutions computed using a well-validated
sea-level solver that operates on an unstructured global
mesh of the Ice and Sea-level System Model (ISSM;
https://issm.jpl.nasa.gov/, last access: 7 May 2019). We find that GFZ
solutions are slightly different from CSR and JPL solutions (Fig. 4),
although the difference in sea-level fingerprints is generally within
1σ uncertainties (compare Figs. 3d and 4d). We show the origin of
discrepancies by plotting the degree variance spectrum.
Comparison of data centers for select fields. JPL solutions are
subtracted from CSR and GFZ solutions for trend in land water storage
change (a, c) and relative sea-level change (b, d) during
January 2003–December 2013. The difference in the spectrum of energy
distribution is also shown (e, f). Results are computed in the CM
reference frame with the rotational feedback accounted for.
Based on CSR, GFZ, and JPL Stokes coefficients, we provide with this article
monthly SH coefficients of
model forcing function, Flm(t);
geoid height change, Xlm(t)+Ylm(t);
vertical bedrock motion, -Plm(t)+Qlm(t); and
relative sea-level change, Slm(t),
computed in both CM and CF reference frames with and without the rotational
feedback included. Effects of Earth's rotation and the reference frame origin
on select fields are shown in Fig. 5. The SH coefficients for sea-surface
height may be obtained by summing coefficients for bedrock motion and those
for relative sea-level change (see Eq. ). While one
may readily assemble these coefficients in an SH domain to retrieve the
corresponding monthly fields, we also supply 0.5∘×0.5∘ gridded solutions for user convenience.
We provide uncertainty associated with monthly
fields as well, both in terms of spatial maps and SH coefficients.
Quantification of the uncertainty is determined by the following recipe.
Based on the JPL Release-06 (GIA uncorrected) mascon solutions and associated
standard errors , we use a Monte Carlo approach
to generate 5000 ensemble members of monthly land water storage solutions. We
apply a unique GIA correction, computed by , to each of
these ensemble members. Next we solve the sea-level equation to derive an
equivalent number of solutions for S(θ,ϕ,t), U(θ,ϕ,t),
and other fields. Finally, we quantify the standard errors associated with
each field, weighted by the likelihood of each GIA model .
Figure 3 shows our estimates of standard errors associated with the trends in
land water storage and relative sea level.
Effects of (a) Earth's rotation and (b) reference
frame on model solutions. Rotational feedback on the relative sea-level
change exhibits a degree 2 spherical harmonic pattern and generally accounts
for ≈10 % of the total signal (compare with Fig. 3c). Degree 1 load Love numbers depend on the choice of
reference frame origin. The effect of reference frame – quantified here as
the model solution computed in the CM frame minus the solution computed in
the CF frame – is more pronounced in the vertical bedrock motion and the
geoid height change (not shown). These results are based on JPL Stokes
coefficients for the period April 2002 to March 2016.
Discussion
The utility of the data we provide
is that they may be used to rigorously remove those patterns that are
attributable to geoid height change and bedrock
motions caused by on-land mass changes from ocean altimetry, bottom-pressure,
and tide gauge studies. Such removal is essential for future studies of the
patterns of sea-level change owing to internal variability of the climate
system which drives changes in ocean density, fresh water fluxes, and
circulation e.g.,.
As we supply sea-level fingerprints and complementary data products with and
without rotational feedback, we owe the readers some additional words of
caution and recommendations. First, from the Eulerian equations of rotational
motion, we solve for the feedback consistently designed for periods longer
than 434 d (the period of the Chandler wobble). The rationale is that both
the solid Earth and ocean pole tide
are removed from the satellite solutions for GRACE
gravimetry and TOPEX/Poseidon and Jason altimetry on a routine basis
e.g.,. The improvements in the ocean
pole tide, in fact, have been accomplished by many years of assimilation of
the altimetric mission data. Hence, at periods near, or less than, 434 d,
the paths to unambiguously generating solutions to the sea-level equation
with centrifugal potential and loading changes from the pole tide are
unclear. We might assume that the relevant feedbacks are largely removed as a
processing step in rendering Level 1-b and Level 2 GRACE data products. We
keep, however, rotational feedback effects of an interannual nature in one
set of monthly solutions, and another set of solutions lack these effects.
The users of these data should understand the differences, as those employing
approaches to using the data to evaluate altimetric time series of order
10 years in length will certainly be interested in using the rotational
feedback version for the analysis of interannual trends and variability
adjacent to Greenland, for example , whereas users focusing
on seasonal timescale fingerprints are recommended to employ those
coefficients that lack the rotational feedback, as the altimetry and space
gravimetry products employed likely have the sea-surface height and gravity
effects of the annual polar motion, Chandler wobble, and associated pole
tides removed.
It is also worthwhile to note that on timescales of decades the mantle
primarily behaves elastically, perhaps with the exception at places where the
tectonic history has brought heat, volatiles, and changes in mineral
structure, such as water or reduced grain size, into the region, thus
reducing the effective viscosity to values below about 5×1018 Pa s e.g.,. At such low values of
viscosity in the upper mantle, stress relaxation can reduce both the
effective influence of gravitational loading and the amplitude of
fingerprints. While we acknowledge that this effect is quite difficult to
quantify, it should be a second-order effect.
Data availability
We presently store data in a public repository hosted by
Harvard Dataverse (10.7910/DVN/8UC8IR; ).
The first set of data we supply are SH coefficients of global field
variables. The zip file “SLFsh_coefficients.zip” contains a total of 1780
data files: 133×4 for GFZ and 156×4 each for CSR and JPL.
For the given data center, four files are provided for a particular GRACE
month: with and without Earth's rotational feedback included while solving
for sea-level fingerprints in both CM and CF reference frames. File names
follow the GRACE naming convention. Solutions that correspond to the GSM file
“GSM-2_2002095-2002120_GRAC_UTCSR_BA01_0600”, for example, are stored
in four files named “SLF-2_2002095-2002120_GRAC_UTCSR_BA01_0600” under
appropriate directories; we simply replace “GSM” by “SLF” to denote
“sea-level fingerprints”. The time stamp (in YYYYDoY-YYYYDoY format) and
the corresponding data center (five-character string containing CSR, GFZ, or
JPL) also appear in the file name. The example file considered above contains
sea-level fingerprint solutions for the period 95–120 d of year 2002 based
on the Stokes coefficients provided by CSR. Header lines 5–7 in each file
further clarify which data center the solutions correspond to, which
reference frame is considered, and whether or not Earth's rotational feedback
is accounted for. Each data file consists of a total of 18 columns: SH degree
l, SH order m, and SH coefficients for model forcing function Flm
(four columns), relative sea level Slm (four columns), geoid height
change [Xlm+Ylm] (four columns), and vertical bedrock motion
-[Plm+Qlm] (four columns). For each field, the first (last) two
columns store cosine (sine) coefficients for our predicted mean and 1σ
uncertainty, respectively. Users should note that the finite degree 0 order 0
harmonic in the monthly SLF files represents the finite mass changes for the
global oceans.
The second set of data we supply are gridded maps of global field variables. We provide a total of 12 NetCDF files:
four each for CSR, GFZ, and JPL. The file
“SLFgrids_GFZOP_CF_WITHrotation.nc”, for example, stores solutions based
on GFZ Stokes coefficients that are computed in the CF reference frame with
the rotational feedback accounted for.
Conclusions
In this paper we describe a data product that emerges from the Release-06
GRACE Level-2 Stokes coefficients, provided by CSR, GFZ, and JPL, which
contain the basic information necessary to create monthly sea-level
fingerprints, and these are general enough that they may be employed in
reconstructions of vertical bedrock motion, perturbed relative sea surface,
and geoid height change. We provide SH coefficients of each field and
uncertainty therein, computed in both CM and CF reference frames with and
without rotational feedback included. For user convenience, we also provide
spatial maps at 0.5∘×0.5∘ resolution.
A future space altimetry mission (Surface Water and Ocean Topography, or
SWOT) is aimed at providing real-time two-dimensional imaging of the
sea-surface height without the necessity of having to patch together
one-dimensional profiles e.g.,. In addition to
providing higher resolution, this will allow improved accuracy. When coupled
to GRACE-FO mapping of gravity changes, we should begin to see the emergence
of sea-level fingerprints. Perhaps more importantly, we may begin to more
confidently remove a part of the ocean altimetry signal that should not be
assimilated into dynamic ocean models: that which is associated with
self-gravitation and loading. Here we present the effects of land-based mass
transport and rotational effects, both together and separately. Recent
appreciation of the effects of solid Earth elastic and viscoelastic response
is now receiving increased scrutiny for the potential bias that may be
introduced into the altimetry trend record when not accounting for these
effects properly e.g.,. We have not treated
the influences of GIA on rotational deformation and/or the associated axial
displacement of the centrifugal potential, although we have employed the GIA
model of to analyze GRACE Level-2 for proper representation
of the monthly water height equivalent masses. As a consequence, users of the
data that we supply here should understand that folding the fingerprints into
analyses of any geodetic data, including tide gauges and ocean altimetry,
might want to carefully consider that the secular polar motion effects in the
Release-06 GRACE Level-2 products have been
removed and that the sea-surface height variability associated with polar
drift, annual, and Chandler wobble effect is currently removed from ocean
altimetry data in the manner described by . This fact allows
users to rather straightforwardly remove land-mass-change-related
fingerprints from either GRACE, ocean altimetry, tide gauge, or GPS-determined
vertical land motion data from April 2002 to August 2016 using the monthly
solutions we supply here as a data product.
Theory of the sea-level fingerprint
The fundamental theoretical concept of the so-called sea-level fingerprint is
summarized in Sect. 2. Here we provide explicit mathematical expressions
for all of the terms appearing in Eq. ().
The barystatic term is given byE(t)=-1AOa2∫H(θ,ϕ,t)1-O(θ,ϕ)dS,whereAO=a2∫O(θ,ϕ)dSis the ocean surface area, a is already defined in Eq. (10), and S is the surface domain of a unit sphere. The term enclosed by
brackets in Eq. () yields the net change in continental water volume.
Changes in gravitational potential, Φg, and associated changes in Earth's surface displacement, Ug, are obtained by
convolving the surface loading function (Eq. ) with respective Green's functions, GΦ and GU, as follows:Φg(θ,ϕ,t)Ug(θ,ϕ,t)=a2ρw∫GΦ(α)GU(α)L(θ′,ϕ′,t)dS′,where (θ′,ϕ′) are the variable coordinates. These variable coordinates at which the loading function is defined are
related to (θ,ϕ) at which Φg and Ug are evaluated via the great-circle distance, α, as follows:
cosα=cosθcosθ′+sinθsinθ′cos(ϕ′-ϕ). Green's functions are represented in the Legendre transform domain as follows:GΦ(α)GU(α)=34πa2ρe∑l=0∞g1+kl′hl′Pl(cosα),where Pl are Legendre polynomials (Eq. ), and kl′ and hl′ are the load Love numbers.
Changes in rotational potential, Φr, and associated changes in Earth's surface displacement, Ur, follow from the Eulerian theory of rotation :Φr(θ,ϕ,t)Ur(θ,ϕ,t)=10Λ00(t)Y00(θ,ϕ)A5+∑m=-221+k2h2/gΛ2m(t)Y2m(θ,ϕ),where Ylm are degree l order m spherical harmonics (Eq. ), Λlm are SH coefficients of perturbation in
rotational potential, and k2 and h2 are degree 2 tidal Love numbers. We may express changes in rotational potential in terms of changes in
Earth's rotation parameters, moment of inertia, and hence surface loading function. Considering leading-order terms only, we get the following nonzero coefficients
:Λ21(t)Λ2-1(t)=-115a2Ω2A6Ω(1+k2′)Aσ0-4π15ρwa4L21(t)L2-1(t),andΛ00(t)Λ20(t)=2/3-2/(35)a2Ω2A7-1+k2′C8π3ρwa4L00(t)-15L20(t),where Ω is the Earth's mean rotational velocity, A and C are the mean equatorial and polar moment of inertia, respectively,
σ0 is the so-called Chandler wobble frequency, and Llm are SH coefficients of the surface loading function (Eq. ). Note that the
terms inside brackets represent changes in Earth's moment of inertia: ΔI11 and ΔI22 (Eq. ) and
ΔI33 (Eq. ). Similarly, the terms enclosed by outer parentheses represent Earth's rotation parameters: polar
motion (m1,m2) (Eq. ) and change in length of day m3 (Eq. ).
The ocean-averaged term in Eq. (), denoted by 〈*〉, may be written as follows:1gΦ(θ,ϕ,t)-U(θ,ϕ,t)=1AO[a2∫{1gΦ(θ,ϕ,t)A8-U(θ,ϕ,t)}O(θ,ϕ)dS].When rotational feedback is excluded, Φ and U should be replaced by Φg and Ug, respectively.
Spectral methods for sea-level equationPrimer
Spherical harmonics.
For brevity, we define ω=(θ,ϕ) and drop explicit dependence of a function on time so that f(θ,ϕ,t)≡f(ω).
Any square-integrable function f(ω) can be expanded as the infinite sum of SHs as follows:
f(ω)=∑l=0∞∑m=-llflmYlm(ω)≡∑lmflmYlm(ω),
where flm are SH coefficients, and Ylm(ω) are (real) normalized SHs of degree l and order m. These SHs may be expressed in
terms of associated Legendre polynomials, Pl|m|, as follows:
Ylm(ω)=(2-δ0m)(2l+1)(l-|m|)!(l+|m|)!B2Pl|m|(cosθ)cos(mϕ)ifm≥0sin(|m|ϕ)ifm<0,
where δ0m is the Kronecker delta. For x∈-1,1 and m≥0, polynomials Plm(x) are given by
Plm(x)=(1-x2)m/2dmPl(x)dxm,
where
Pl(x)=12ll!dl(x2-1)ldxl
are the Legendre polynomials. This definition of SHs and their normalization
are consistent with those employed for GRACE data generation and processing
and can be evaluated
straightforwardly, for example, using MATLAB's
associated Legendre functions
(https://www.mathworks.com/help/matlab/ref/legendre.html, last access:
7 May 2019).
SH addition theorem.
It is useful to note here that the following relationship holds:
Pl(cosα)=12l+1∑m=-llYlm(ω)Ylm(ω′),
where α once again is the great-circle distance between coordinates ω and ω′.
Evaluation of SH coefficients.
For the chosen normalization, SHs obey the following orthogonality relationship
∫Ylm(ω)Yl′m′(ω)dS=4πδll′δmm′,
where δll′ and δmm′ are Kronecker deltas. Using this property, SH coefficients of f(ω) are obtained as follows:
flm=14π∫f(ω)Ylm(ω)dS.
Evaluation of surface integrals on a unit sphere.
We discretize the surface of a unit sphere using the so-called icosahedral pixelization method . It yields uniformly distributed
quadrature points with equal pixel area. This makes numerical integration fairly straightforward as follows:
∫f(ω)dS=4πNT∑j=1NTf(ωj),
where ωj is the centroid of the jth pixel, and NT is the total number of pixels. Note that the factor
4π/NT represents the area of each pixel on the surface of a unit sphere.
SH coefficients of some basic functions
Ocean function.
By definition, the ocean function is given by
O(ω)=1ifω∈SO0otherwise,
where SO is the ocean surface domain on a unit sphere. As in Eq. (), we may write
O(ω)=∑lmOlmYlm(ω).
Following Eq. () and using the definition of ocean function, we get
Olm=14π∫SOYlm(ω)dS,
where integration is performed only within the ocean surface domain. Following Eq. (), we obtain
Olm=1NT∑j∈SOYlm(ωj).
Model forcing function.
By definition, F(ω)=H(ω)1-O(ω).
We may write
Flm=14π∫SCH(ω)Ylm(ω)dS,
where SC is the continental domain on a unit sphere. We derive H(ω) from the GRACE Stokes coefficients as detailed in Sect. 3. Following Eq. (), we get
Flm=1NT∑j∈SCH(ωj)Ylm(ωj).
Global surface loading function. Since L=F+S^, we may write SH coefficients of L (Eq. ) as follows:
Llm=Flm+S^lm.
Some useful integrals and barystatic sea level
Ocean surface area on a unit sphere.
Since Y00(w)=1 (see Appendix B1), the SH coefficient of the ocean function (Eq. ) for l=0 and m=0 is given by
O00=NONT≡14π4πNTNO,
where NO is the number of pixels in the ocean surface domain SO. Since the term enclosed by parentheses represents the area
of each pixel, the total area of ocean surface (i.e., NO times the pixel area) on a unit sphere is given by
∫O(ω)dS=4πO00.
Continental water volume on a unit sphere.
The SH coefficient of the forcing function () for l=m=0 is given by
F00=1NT∑j∈SCH(ωj)B14≡14π∑j∈SC4πNTH(ωj).
Since the term enclosed by parentheses represents the area of each pixel, the sum of the bracketed term over SC
essentially yields the total continental water volume. Consequently, we may write
∫H(ω)1-O(ω)dS=4πF00.
Barystatic sea level. Using Eqs. (), (), (), and (), we get
E=-F00O00.
SH coefficients appearing in the sea-level equation
The coefficientE^lm. This coefficient is used as the first guess solution of S^lm (Eq. ) and
remains unchanged during the recursive process. Recalling that E^=OE and that E is a spatial invariant, we may write
E^lm=14πE∫O(ω)Ylm(ω)dS.
Noting that the integral is equivalent to 4πOlm (see Eq. ), and using Eq. (), we get
E^lm=-F00O00Olm.
Other hatted coefficients. All of the coefficients appearing in Eq. () may be evaluated in a similar manner.
Consider X^lm, for example. Recalling the definition that X^=OX, and following Eq. (), we may write
X^lm=14π∫O(ω)X(ω)Ylm(ω)dS.
Using the definition of ocean function, and expanding X(ω) as in Eq. (), we get
X^lm=14π∫SO∑pqXpqYpq(ω)Ylm(ω)dω.
Following Eq. (), we evaluate the integral as follows:
X^lm=1NT∑j∈SO∑pqXpqYpq(ωj)Ylm(ωj).
The coefficientXpq. By definition, X=Φg/g.
Using Eqs. () and (), we may write
X(ω)=3ρw4πρe∫∑p=0∞1+kp′Pp(cosα)L(ω′)dS′.
Using the SH addition theorem (Eq. ), and expanding L(ω′) as in Eq. (), we get
X(ω)=3ρw4πρe∫∑pq1+kp′2p+1Ypq(ω)Ypq(ω′)∑p′q′Lp′q′Yp′q′(ω′)dS′
or
X(ω)=3ρw4πρe∑pq∑p′q′1+kp′2p+1Lp′q′Ypq(ω)∫Ypq(ω′)Yp′q′(ω′)dS′.
Using the SH orthogonality relationship (Eq. ), we get
X(ω)=3ρwρe∑pq1+kp′2p+1LpqYpq(ω).
Using Eqs. () and (), SH coefficients Xpq are given by
Xpq=3ρw4πρe∫∑p′q′1+kp′′2p′+1Lp′q′Yp′q′(ω)Ypq(ω)dS.
Rearranging terms and applying the orthogonality relationship (Eq. ), we obtain
Xpq=3ρwρe1+kp′2p+1Lpq.
The coefficientYpq. By definition, Y=Φr/g. Using Eqs. () and (), SH coefficients Ypq are given by
Ypq=14πg∫Λ00Y00(ω)+∑2q′(1+k2)Λ2q′Y2q′(ω)Ypq(ω)dS.
Rearranging terms and applying the orthogonality relationship (Eq. ), we get
Ypq=Λ00gδp0δq0+(1+k2)Λ2qgδp2.
The coefficientPpq.
By definition, P=-Ug. Using Eqs. () and (), and following the procedure to derive Xpq (Eq. ), we obtain
Ppq=-3ρwρehp′2p+1Lpq.
The coefficientQpq.
By definition, Q=-Ur. Using Eq. (), and following the procedure to derive Ypq (Eq. ), we get
Qpq=-h2Λ2qgδp2.
The coefficientCpq.
By definition, C=E-Φ/g-U≡E-X+P+Y+Q. Using Eqs. () and () and similar equations for P, we may write
C=E-14πO00∫SO3ρwρe∑pq1+kp′-hp′2p+1LpqYpq(ω)+1gΛ00Y00(ω)+∑2q1+k2-h2Λ2qY2q(ω)dS.
Using Eqs. () and (), we get
C=-F00O00-3ρwρeO00∑pq1+kp′-hp′2p+1LpqOpqB24-Λ00g-1gO00∑2q(1+k2-h2)Λ2qO2q.
Note that C and all of the right-hand side terms are spatially invariant.
Using Eqs. () and (), we get
Cpq=-14πF00O00+3ρwρeO00∑p′q′1+kp′′-hp′′2p′+1Lp′q′Op′q′+Λ00g+1gO00∑2q′(1+k2-h2)Λ2q′O2q′∫Ypq(ω)dS.
Since Y00(ω)=1, we introduce a virtual expression Y00(ω) inside the integral and use Eq. () to find
Cpq=-δp0δq0F00O00+3ρwρeO00∑p′q′1+kp′′-hp′′2p′+1Lp′q′Op′q′B25+Λ00g+1gO00∑2q′(1+k2-h2)Λ2q′O2q′.
The terms inside the braces vanish when rotational feedback is not included. Note that the first term on the right-hand side of Eq. () and the first term inside the braces above
cancel out while solving Eq. (). It is, however, important to consider these terms explicitly for a clean isolation of SH coefficients of the desired fields.
Summary
Here we briefly outline the workflow of our computation.
Given the on-land change in water equivalent height H(θ,ϕ,t) over the time period t, we compute SH coefficients of the model forcing function Flm using Eq. ().
We compute SH coefficients of the “global loading function” Llm (Eq. ) by initializing S^lm, such that
S^lm≡S^lm0=E^lm, where E^lm is given by Eq. ().
Once S^lm, and hence Llm, are initialized, we solve the recursion equation for S^lmr (Eq. ) until
the solution is converged. The hatted SH coefficients appearing in Eq. () are expressed in terms of their non-hatted counterparts
as in, for example, Eq. (). For the chosen solid Earth model and the reference frame origin, we compute these
non-hatted coefficients using Eqs. ()–() and ().
The choice of the solid Earth model determines the load Love numbers kp′ and hp′ and the tidal Love numbers k2 and h2. In this
study, we consider the Preliminary Reference Earth Model .
The choice of the reference frame origin determines the degree 1 load Love numbers. In this study, we take the values from for the CM and CF reference frames.
Rotational feedback is accounted for via Eqs. (), (), and (); SH coefficients of perturbation in rotational potential Λpq appearing
in these equations are given by Eqs. () and (). To deactivate this feedback, we set Ypq=0 (Eq. ) and Qpq=0 (Eq. )
and remove the terms enclosed by braces in Eq. ().
As for the convergence criterion, we track the relative change in L-2 norm after each recursion and call the solution converged when it is less than 0.001 % of the L-2 norm of the solution
itself. This level of solution convergence is typically achieved after four to six iterations.
Unless stated otherwise, constants and parameters used in this study are taken directly from Table 1 of .
Once the solution is converged for S^lm, we appropriately combine corresponding non-hatted coefficients (i.e., Eqs. – and ) in order to retrieve
SH coefficients of relative sea level Spq (Eq. ), geoid height change [Xpq+Ypq], and vertical bedrock motion [-Ppq-Qpq]. We supply these
coefficients along with corresponding spatial maps.
Author contributions
SA and ERI conceived the research and wrote the first draft of the paper.
SA formulated the sea-level solver, with the help of ERI, and led the
calculations, with the help of FWL (in estimating degree 1 Stokes
coefficients) and TF and LC (in uncertainty quantification). All authors
contributed to the analysis of the results and to the writing and editing of
the paper.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
This research was carried out at the Jet Propulsion Laboratory (JPL),
California Institute of Technology, under a contract with National
Aeronautics and Space Administration (NASA). Constructive comments from the
reviewers greatly improved the quality of the paper.
Financial support
This research was supported by the JPL Research, Technology
& Development Program (grant no. 01-STCR-R.17.235.118; 2017–2019) and the
NASA Sea-Level Change Science Team (grant no. 16-SLCT16-0015; 2018–2020), as
well as the NASA MEaSUREs program (grant no. 17-MEASURES-0031; 2019–2023).
Review statement
This paper was edited by Giuseppe M. R. Manzella and
reviewed by Don Chambers, Xuebin Zhang, and Makan A. Karegar.
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