Understanding future impacts of sea-level rise at the local level is important
for mitigating its effects. In
particular, quantifying the range of sea-level rise outcomes in a probabilistic way enables coastal planners to
better adapt strategies, depending on cost, timing and risk tolerance. For a time horizon of 100 years,
frameworks have been developed that provide such projections by relying on sea-level fingerprints where
contributions from different processes are sampled at each individual time step and summed up to create
probability distributions of sea-level rise for each desired location. While advantageous, this method does not
readily allow for including new physics developed in forward models of each component. For example, couplings
and feedbacks between ice sheets, ocean circulation and solid-Earth uplift cannot easily be represented in
such frameworks. Indeed, the main impediment to inclusion of more forward model physics in probabilistic
sea-level frameworks is the availability of dynamically computed sea-level fingerprints that can be directly
linked to local mass changes. Here, we demonstrate such an approach within the Ice-sheet and Sea-level
System Model (ISSM), where we develop a probabilistic framework that can readily be coupled to forward process models
such as those for ice sheets, glacial isostatic adjustment, hydrology and ocean circulation, among others.
Through large-scale uncertainty quantification, we demonstrate how this approach enables inclusion of
incremental improvements in all forward models and provides fidelity to time-correlated processes. The
projection system may readily process input and output quantities that are geodetically consistent with space
and terrestrial measurement systems. The approach can also account for numerous improvements in our
understanding of sea-level processes.
Reliable projections of local sea-level change, together with robust uncertainties, are a key quantity for
stakeholders to shape adequate and cost-effective mitigation and adaptation measures to sea-level rise
. Most regional sea-level projections use a process-based approach, in which
all relevant processes are modeled separately and summed up together, including the individual estimates of error,
with their spatial signature
.
These projections are widely used by coastal planners and stakeholders, as is, for example,
demonstrated by the impact of on assessment reports across the United States
.
In their simplest form, these process based projections can be expressed straightforwardly. We generally refer to these as KOPP14 ) and write
RSLtotalθ,ϕ,t=∑i=1nFGRD,i(θ,ϕ)⋅Bi(t)+RSLsterodynamicθ,ϕ,t+RSLGIAθ,ϕ,t.
where RSLtotalθ,ϕ,t is the total projected relative sea-level (RSL) change at
time t, latitude θ and longitude ϕ. For all barystatic processes, or processes that change the total
ocean mass, the effects of gravity, rotation and deformation (GRD) on local sea level are computed by multiplying
the total barystatic contribution Bi(t) by the associated barystatic-GRD fingerprint (abbreviated by
“fingerprint” from here on), or FGRD,i(θ,ϕ), which is computed a priori. This procedure is
generally used to include the effects of glacier and ice-sheet mass loss, as well as for projected changes in
terrestrial water storage (TWS). Note here that our definition of GRD is not completely in line with
, as glacial isostatic adjustment (GIA) is considered as a separate contributor, and the GRD
contribution does contribute to
global-mean sea-level changes. It is rather in line with the definition of contemporary GRD in .
The effects of sterodynamic sea-level change RSLsterodynamicθ,ϕ,t, which is the
sum of global thermosteric expansion and local sea-level changes due to ocean dynamics, is generally included by
directly using estimates from Earth
system models (ESMs) and atmospheric–oceanic global circulation models (AOGCMs), such as the output of the Coupled
Model Intercomparison Project phase 5 CMIP5;. Finally, the GIA term
RSLGIAθ,ϕ,t is generally accounted for using output from a periodically updated global model.
To derive uncertainties for these local projections of sea-level change, the barystatic components Bi are often
sampled from a probability distribution found in published probabilistic projections, for example, from expert
elicitation projects e.g.,, or other ice-sheet models
. The sterodynamic contribution often uses the inter-model spread as a source of the
uncertainties. While the basis of each probabilistic projection is similar, each group adds additional components
and physics to Eq. (). For example, in and , a Gaussian
process regression model, based on tide-gauge observations, is used to account for the effect of non-climatic
vertical land motion. Or in and , the GRD effects of ocean dynamics
are explicitly taken into account, with computing
these effects over the entire projection time series.
One of the key strengths of this approach is its relative simplicity and transparency, as the process from probabilistic
estimates of the underlying processes into local sea-level changes is a simple multiplication operation with the
respective barystatic-GRD fingerprint. It provides a framework that outputs a probability density function (PDF)
for RSL at any desired location, from which the expected sea-level change and its confidence intervals can be derived.
This provides both efficient
calibration/validation quantities to projections and streamlines incrementally updated projections. In essence, each
modular input may be improved separately, so updates are unencumbered by the queueing up of new modules for
incorporation into more complex ESMs and AOGCMs.
Recently, a growing body of research indicates that additional processes should be considered in this process-based
approach. Inclusion of such processes is critical to improving the quantification of
uncertainties in local sea-level change predictions, but they are not directly feasible within the
framework of Eq. (). Below, we highlight some of the key contributors to
uncertainty that, until now, have not been considered together in large-scale estimates of sea-level
change.
First, in Eq. (), the multiplication of a barystatic mass contributor Bi(t) with a
fingerprint FGRD,i(θ,ϕ), assumes that the fingerprint is constant through time, which is not
always the case . Instead, a fingerprint results from feedbacks
between the geometry of sea-level components. For example, local sea level depends on the geometry of ice mass
loss, so temporal changes in ice geometry will directly translate into local sea-level changes e.g.,
. As a result, this temporal variability not only
affects the expected local sea-level changes but also its uncertainties, as the uncertainty of the input mass loss
also has a pronounced spatial pattern due to relative limitations in measurement and data interpretation. An
example of the inadequacy of temporally constant fingerprints is shown in Fig. for a
projection of Greenland's contribution to RSL at 2016 versus 2045 and 2075. Normalized RSL patterns are clearly different
between all three times, and the differences are not local to just Greenland but spill over into regions such as
North Europe, Alaska and the Canadian arctic.
Normalized fingerprints for Greenland in (a) 2016, (b) 2045 and (c) 2075,
based on the JPL ISSM experiment 5
simulation that contributed to ISMIP6 . The relative sea-level change patterns in the
North Atlantic and Arctic oceans vary significantly in both space and time, resulting in very different
contributions to local RSL in the years 2016, 2045 and 2075.
Second, covariance in time as well as the covariance between the individual processes is not
always small, though it is often considered to be or is approximated by a simple relationship
e.g.,.
Indeed, assuming so could cause a significant misrepresentation of the estimated uncertainties in local sea-level change.
For example, showed that most driving
factors of sea level are correlated with global-mean temperature changes, and ignoring this inter-process
covariance can underestimate uncertainty in local sea-level change. Note that in addition to covariance between processes,
the uncertainty in individual processes may also be correlated temporally. Propagating this full
spatiotemporal covariance into projections and its uncertainties promotes a better
understanding of the spatial and temporal coherence of uncertainties, which could, for example, allow
us to assess the likelihood of reaching specific sea levels by 2100 given observed sea-level change
during the next 20 years.
Thirdly, recent work on the Antarctic Ice Sheet (AIS) shows a strong coupling between GIA, elastic surface deformation and
ice mass loss . Such relationships between these processes
suggest that any uncertainties in computed ice-sheet histories and solid-Earth properties that
propagate into GIA projections can also feed back into ice-mass-loss
projections; thus, considering these processes as independent ignores these couplings. Here, the main problem is that
projection frameworks are articulated in terms of changes in mass, while most ice-sheet models, GIA models, TWS
evolution models and glacier models are explicitly described in terms of local mass change evolution (or thickness
changes, in meters per water equivalent per year; m w.e.yr-1). In order to be able to account for strong couplings, or to even be able to
ingest recent modeling results, one needs to propagate the local mass changes and the associated uncertainties into
regional sea-level projections. This is particularly relevant now given new modeling runs that have been carried
out within large modeling intercomparison projects (MIPs) such CMIP5 and CMIP6, as well as ISMIP6 or GlacierMIP2.
Contribution to uncertainty in 100-year extreme warming simulations of AIS and three
subregions of the AIS, tested for four different model variables independently. Each probability
distribution function represents an ensemble of 800 ISSM ice-sheet forward model runs, sampled using the DAKOTA uncertainty quantification framework .
Similarly, additional strong positive feedbacks between ice sheet and ocean dynamics have been evidenced in work
from and , among others. Specifically, these
studies suggest that strong coupling between sub-ice-shelf ocean circulation (in particular melt rates) and
ice-flow dynamics (in particular, grounding line dynamics and mass transport resulting in modifications of an
ice-shelf draft) results in significant retreat of ice streams such as Thwaites Glacier and Pine Island Glacier, as
Antarctica's warm circumpolar deepwater is advected close to their grounding line. Other high-frequency processes
(at the daily to monthly level) such as ocean tides, and in particular how tidal currents affect water mass
properties at ice-sheet marine margins , are critical in understanding how mass loss rates will
evolve. This will significantly impact how melt-rate parameterizations are developed to quantify melt rates,
especially in the West Antarctic Ice Sheet area . Significant work remains in calibrating such
melt-rate parameterizations to correctly account for all aforementioned effects. While more work is required in
terms of constraining such parameterizations, the impacts of such ice–ocean feedbacks have not been assessed in
probabilistic sea-level models (PSLMs).
Finally, in the past decade, extensive work has been carried out to probabilistically characterize components such
as
GIA or ice-sheet mass balance
. Substantial understanding of
the impact of rheological parameters and ice history on the distribution of bedrock uplift and rate of change in
geoid rates has been generated through modeling of GIA. Similarly, for ice-sheet models, significant knowledge
has been generated about how the mass balances of both the AIS and Greenland Ice Sheet (GIS) are impacted by surface
mass balance (SMB), ice-shelf basal melt, ice–bedrock friction, geothermal heat flux or ice rheology (see, e.g.,
Fig. a). All of these advances need to be fully integrated into new probabilistic projections of
sea-level change, and a new approach therefore needs to be envisioned that will allow for such new processes to be
accurately modeled.
Indeed, moving from strategies where continental-scale mass changes are sampled and multiplied with the
corresponding fingerprint to actually sampling upstream model inputs is important for improving the state of the
art. In particular, there is a strong need to fully account for spatial patterns of mass change and their
uncertainty (see Fig. b–d). This applies to, among others, SMB, basal friction or ice and
solid-Earth rheological properties.
For example, Eq. () relies on masses that are aggregated at the basin/continental level.
However, most ice-sheet models compute high-resolution thickness change patterns that are not aggregated. This
aggregation greatly reduces the complexity in representation of model physics and uncertainty propagated at the
interface between ice-sheet models and PSLMs. A more comprehensive approach that re-establishes interfaces between
forward models and PSLMs is therefore necessary, where model outputs are not aggregated or simplified.
Here, we propose a new framework for sea-level projections that is able to account for all terms in
Eq. (). We improve the existing process-based approach by using the Ice-sheet and Sea-level System
Model ISSM;, which allows for inclusion of forward model physics. It also improves the
modeling and sampling of covariance between input processes, both temporally and spatially through the computation of
high-resolution barystatic-GRD patterns. The latter feature builds the basis for a geodetically compliant projection
system where GRD patterns and their computation are done systematically and which does not introduce biases in the
projections.
MethodsTheory
Sterodynamic sea-level changes form a significant contributor to both global-mean sea-level rise and are
responsible for large parts of the regional deviations from the global-mean projected changes
e.g.,. Following the CMIP5
conventions, sterodynamic sea-level changes consist of a global-mean thermosteric contribution (variable name
) and a local dynamic contribution (variable name ) with a zero mean over the oceans.
Generally, an ensemble of model runs, either based on multiple models e.g., or on
large-ensemble experiments based on perturbing a single model for example,
, can be used to directly sample regional sea-level changes. An
alternative approach to generate more samples than model ensemble members is to determine common modes of
variability, for example, by extracting the largest empirical orthogonal functions from each model and perturbing
the associated principal components e.g.,Fig. 3.
While sterodynamic effects do not change the total ocean mass, ocean dynamics can be coupled to redistribution of
ocean mass, which manifests in ocean-bottom-pressure changes, particularly on shallow shelf seas
. These bottom-pressure changes load the solid Earth below and thus result
in GRD effects, which are often referred to as self-attraction and loading (SAL) effects
. These SAL effects could cause several centimeters of additional
sea-level rise above the sterodynamic signal in century-scale sea-level projections made by atmosphere–ocean
general circulation models (AOGCMs) . By adding the
ocean-bottom-pressure changes to the sea-level equation solver, this effect can be incorporated in regional sea-level
projections.
As depicted in Eq. (), in the classical approach, static sea-level fingerprints are computed a
priori for each individual process, which typically include glaciers (GLA), the GIS and AIS,
and TWS. These fingerprints are subsequently multiplied by the equivalent barystatic
contribution, which is often sampled from a PDF and added, together with the sterodynamic and GIA contribution, to
obtain local RSL changes and the associated confidence intervals. This method is both transparent and simple,
while maintaining computational efficiency due to the fact that the fingerprints do not have to
be computed for each sample or time step.
However, several issues arise from this approach, which can be mitigated using a different method. First, it is
assumed that the spatial pattern of mass loss is known a priori and does not vary over time. A common approach is
to assume that the mass loss is uniformly distributed over the ice sheet, or that it follows the spatial pattern
derived over the Gravity Recovery and Climate Experiment (GRACE) period. quantified
the errors induced by assuming a uniform mass loss
and found that this bias could be up to 1 and ≥ 10 cm for sites distant from and close to centers of mass
loss. Furthermore, the approximation of time-invariant fingerprints could lead to biases, when the spatial pattern
of mass loss varies over time.
Diagram of ISSM's Sea-Level Projection System (ISSM-SLPS) model. The system is driven by requirements from
Eq. (). ISSM-SESAW is the GRD core of the system (in pink). ISSM-SLPS is a combination of ISSM-SESAW
and a layer (in green) that handles STR, DSL and GIA inputs, as well as all uncertainty quantification aspects.
In our approach (Fig. ), ISSM Sea-Level Projection System (ISSM-SLPS) solves for RSL as follows:
RSL(θ,ϕ,t)=RSLSTR(t)+RSLDSL(θ,ϕ,t)+RSLGIA(θ,ϕ,t)+RSLGRD(θ,ϕ,t).
The first two terms on the right-hand side, i.e., RSLSTR(t)+RSLDSL(θ,ϕ,t),
together represent the
sterodynamic sea-level change. STR represents the global-mean thermosteric expansion and DSL local sea-level
changes due to ocean dynamics. These can be obtained from CMIP results. The GIA contribution to ongoing sea-level
change, RSLGIA, is given, for example, by . The last term, RSLGRD, refers to the
component of sea-level change due to mass-induced contemporary GRD response of the solid Earth (Gregory et al., 2019), excluding
the GIA processes. This implies that viscoelastic deformation is split between long-term timescales and short-term
fast rebound of the bedrock uplift, such as observed in West Antarctica , acting essentially
over timescales of 10–100 years. This includes mass transport between the land and the ocean, as well as that
due to dynamic ocean circulation. The latter field is provided by CMIP as the ocean-bottom-pressure (OBP) products.
Note that GRD associated with land–ocean mass transport is usually termed “sea level fingerprint”
e.g.,, while the GRD due to OBP variability is termed the “self-attraction and loading”
phenomenon e.g.,. As we shall see, we unify both of these elements of contemporary GRD sea level
in Eq. (). Note also that the global-mean OBP is removed from the ocean models, since ocean
dynamics do not add or remove any mass from/to the ocean. In fact, our projections rely on CMIP5 and CMIP6
“zos” (the sea-surface height change above geoid) and “zostoga” fields for which the Greatbatch correction has been
applied .
We compute RSLGRD using ISSM's Solid Earth and Sea-Level Adjustment module
ISSM-SESAW;. Assuming that all of land ice–water mass change directly modulates the ocean
mass, we define a global mass-conserving loading function, Mglobal(θ,ϕ,t), that describes the change
in mass per unit area on the solid-Earth surface as follows:
Mglobal(θ,ϕ,t)=Mland(θ,ϕ,t)[1-O(θ,ϕ)]+ρo[HOBP(θ,ϕ,t)+RSLGRD(θ,ϕ,t)]O(θ,ϕ),
where the land loading function (with dimensions of mass per unit area) Mland(θ,ϕ,t) is given by
:
Mland(θ,ϕ,t)=ρi[HAIS(θ,ϕ,t)+HGIS(θ,ϕ,t)+HGLA(θ,ϕ,t)]+ρwHTWS(θ,ϕ,t).
Here, ρi is the ice density, ρw is the freshwater density, ρo is the mean
density of ocean water,
and HOBP is the (ocean) water equivalent height of the ocean-bottom-pressure change. Similarly, HAIS,
HGIS and HGLA are the ice height change in the respective cryospheric domains, and HTWS is the
freshwater height change in the non-cryospheric land domain. Note that we invoke an ocean function
O(θ,ϕ) in Eq. () to ensure mass conservation in the system. This function is
equal to 1 over the oceans and 0 everywhere else.
The contemporary mass transport function Mglobal(θ,ϕ,t) loads the underlying solid Earth that is
self-gravitating, rotating and viscoelastically compressible. The induced spatial pattern of
RSLGRD(θ,ϕ,t) is dictated by the perturbation in Earth's gravitational and rotational potentials and
associated viscoelastic deformation of the solid Earth . In the
absence of dynamic sea level and meteorologically induced high-frequency signals, the sea-surface height mimics the
spatial pattern of the geoid . Therefore, we may write
RSLGRD(θ,ϕ,t)=C(t)+GGRD(θ,ϕ,t)-BGRD(θ,ϕ,t),
where GGRD(θ,ϕ,t) and BGRD(θ,ϕ,t) represent the change in geoid and bedrock
elevation induced by the loading of the solid Earth (Eq. ), respectively. Spatial invariant
C(t) is invoked to ensure mass conservation in the Earth system, and it may be readily derived by inserting
Eq. () into Eq. () and integrating it over the solid-Earth surface.
Both GGRD(θ,ϕ,t) and BGRD(θ,ϕ,t) appearing in Eq. () may be partitioned
into two components each: those related to gravitational potential and those to rotational potential. These
components can be computed by convolving Mglobal(θ,ϕ,t) with respective Green's functions. These may
be defined in terms of surface harmonics with loading Love numbers as coefficients. Given the structure and
viscoelastic properties of the solid Earth, these numbers characterize the axisymmetric deformational and
gravitational response of Earth to the applied unit surface load. The rotational components depend upon tesseral
second-degree loading and tidal Love numbers as well as on the perturbation in Earth's inertia tensor, which in turn depends on
Mglobal(θ,ϕ,t). In order to solve for RGRD(θ,ϕ,t), we require an a priori
knowledge of Mglobal(θ,ϕ,t), which in turn depends on RGRD(θ,ϕ,t) itself. The system of
Eq. () and () is therefore solved iteratively until a desired solution accuracy is
achieved. One key feature of this field is that as ice sheets lose mass, the near-field relative sea level drops,
and far-field sea level rises at a much larger rate than the barystatic term for the sake of mass conservation.
While theoretical/numerical treatments on the topic are found elsewhere e.g.,, version 1.0 of the SESAW algorithm where RSLGRD is solved for is
presented in .
A 3-D plot of the boundaries used to mesh each continental area of the Earth's surface. Regions include
South and North America, Australia, Eurasia and the Pacific, as well as Greenland and Antarctica. In
this particular scenario, Greenland has been subdivided into 18 regions along the boundaries defined in
.
Meshing
SESAW is a mesh-based convolution based on Eq. (2) in . As such, it relies on an anisotropic
unstructured mesh of the surface of the Earth which is refined according to specific metrics such as distance to
the nearest coastline, presence of loads (such as changes in ice thickness or TWS) and the complexity of the
coastline. Given the amount of inputs being sampled for in the SLPS system, a systematic approach to refining such
a mesh needs to be developed. The main tool for such a refinement is the ISSM implementation of the Bidimensional
Anisotropic Mesh Generator (BAMG) anisotropic mesh refiner . This is a 2-D-based
anisotropic mesher which can refine a mesh according to several constraints at the same time: a metric to specify
directions along which the mesh resolution needs to be improved, specific vertex or segment positions, in
particular vertex positions of the region outlines, and specified mesh resolutions for user-defined locations.
Combining these constraints, we develop an approach based on meshing of a set of 2-D continental areas of the Earth,
projection of such 2-D meshes onto the 3-D Earth surface and then stitching of the resulting meshes into one seamless
global 3-D mesh.
A 2-D adaptive mesh of South America using GRACE observations of ice and hydrological mass change
(in cm yr-1) from 2003 to
2016. Seismic effects are not removed in this rendering of Patagonian ice mass loss that
was directly taken from . The Global Self-consistent, Hierarchical, High-resolution Geography database (GHSSH) L1 coarse version coastline is shown in green. Segments of the triangular mesh are plotted
in black. The color bar for the thickness changes was saturated at [-1,1] cm yr-1 in order to improve
the contrast of the
figure given the high mesh resolution.
A plot of the 2-D regions is given in Fig. , which include South America, North America, Australia,
Eurasia and the Pacific. At the north and south, we have regions defined for Antarctica and Greenland.
Greenland itself has been further refined into 18 regions drawn along its main ice divides, following
Fig. 3. The approach facilitates a direct linkage of models from the existing
literature, or potentially from previous ISSM studies such as , without having to
remesh the entire Earth. This in turn allows for direct comparisons between uncertainty quantification projection
results where only one specific region is modified, hence allowing an approach where control runs can be compared
against specific variations of an uncertainty quantification projection run.
An example mesh of the South American continent is shown in Fig. . This mesh relies on
defined vertices for the outline, which match the outline vertices for the Pacific, Antarctica and Eurasian
meshes, so that the stitching within a larger 3-D mesh can be done without redundancy in vertices along continental
boundaries. In addition, GRACE ice mass trends from 2003 to 2016 are provided as a metric to
be used for refinement of the mesh, in particular around the Patagonian ice fields. The minimum mesh resolution
attained for this mesh is 500 m, and the largest is 1400 km. Finally, Global Self-consistent, Hierarchical, High-resolution Geography database (GHSSH) L1 coarse version was used as a vertex constraint, so that the final mesh perfectly
coincides with the coastline dataset (in black). This allows for the most optimum sea-level solution using the
SESAW solver.
Once each region has been meshed in 2-D using BAMG, it is projected onto latitude and longitude, and concatenated
together to create a 3-D mesh. This is possible because each 2-D mesh relies on the same set of boundaries as shown
in Fig. . The resulting mesh is shown in Fig. and comprises 38 944 surface
elements for 19 486 vertices. For comparison, an equiangular 1×1∘ grid would require 64 800 vertices, which
is 3 times as many as for a coarse grid resolution.
A 3-D Earth mesh stitched from 3-D projections of 2-D regional meshes of the following regions: South and
North America, Australia, Eurasia and the Pacific, as well as Antarctica and Greenland. GRACE observations
of ice mass change (in cm yr-1) from 2003 to 2016 are laid over the mesh. The left frame
azimuth is 30∘ with an elevation of 64∘. The right frame azimuth is 205∘ with an elevation of
23∘.
Sampling and partitioning
In order to sample variables at each time step, our approach is to use a geographical partitioning of the
unstructured mesh. An example is shown in Fig. , where a range of values from 1 to 5 has been
attributed for each vertex (and element) of the mesh, corresponding, respectively, to Antarctica (1), Greenland (2),
glaciers (3), the ocean (4) and land (5). For each partition and for each variable that is probabilistically
sampled, we define a probability density function (PDF). For normal distributions, for example, this will be done
through a mean and standard deviation.
Partition vector (values from 1 to 5: 1 for Antarctica, 2 for Greenland, 3 for glaciers around the
world, 4 for the ocean and 5 for ice-free land). The partition vector is used to sample probabilistic variables
in a geographically consistent way, with PDF distribution moments (mean and standard deviation) defined for
each partition area.
The algorithm for sampling through SLPS is explained in Algotithm 1, in the generic case where spatial covariance
is available between variables.
Here, is the time variable (ranging from the years 2019 to 2100, at 1-year intervals),
is the counter for each
sample, from 1 to nsamples (in our case, 10 000), is the sampled variable (from one of the SESAW inputs,
excluding RSLGIA which is deterministic in our framework), is a counter for all vertices in the mesh
, is the partition vector (for example, ranging from 1 to 5 in Fig. ),
is the
joint probability distribution of variables across all geographical locations, is the sample generator in
ISSM , is the jth sample matrix of scaling factors with size
(number of variables, number of partitions), the unmodified variable (stored in memory at the
beginning of the model run), and is the
sea-level solver, generating RSL for a specific sample of all the probabilistic variables. In this
algorithm, the PDF distribution is built by specifying its nature and parameters; e.g., the “type” argument can
indicate the choice of a multivariate Gaussian distribution and “pdfspec_arg” can specify the vector of means and
covariance matrix of alpha between each other and between partitions.
For this application, we assume that each variable and each partition is independent, and we set the mean of all
distributions to 1. This ensures that values of behave as scaling parameters. We use them to directly scale
a variable locally, according to which partition area this location geographically belongs. This method is
therefore significantly different from the approach in KOPP14, where the entire mass within a certain partition
(for example, GIS or AIS) is sampled. Here, the sampling is a scaling of a vectorial field, which therefore
preserves the local geographical distribution of a given variable. This is shown in Fig.
for a scenario where thinning rates of the GIS are sampled using one geographical partition (corresponding to
Fig. partition value of 2, in blue). We display the average thinning rates μ, μ+3σ
and μ-3σ (for an arbitrary value of the standard deviation σ=5 %). The structure of the thinning rate as it
is sampled is kept intact, implying that the spatial covariance of the variable being sampled across the mesh is
kept closely similar across samples and within any given partition.
Random sampling of thinning rates across Greenland. (a) GRACE-generated thinning rate pattern at 2005
(in m yr-1). (b) Thinning rate along the AB profile (from a) (in red, representing the
average of the PDF) and samples
generated at -3σ (blue) and +3σ (yellow) from the average.
ISSM-SLPS projections based on Intergovernmental Panel on Climate Change
Fifth Assessment Report (IPCC AR5) Representative Concentration Pathway 8.5 (RCP8.5). For each time step, we sample
(10 000 times with Latin hypercube sampling, or LHS) the following inputs: HGIS, HAIS, thermal
expansion of the ocean (STR), HTWS and glacier contributions HGLA (see AR5 WG1 chap. 13;
). Each input's PDF is calibrated using the AR5 5 %–95 % projection confidence
interval, similar to . The resulting global mean sea-level (GMSL) probability distribution function (PDF) distribution is shown in panels (a) (in
time) and (b) (at a subset of time steps). The 5 %–95 % confidence interval (likely range,
following AR5 definition) is plotted in black in panel (a), along with the temporal mean. Each time step is fully
decorrelated from the previous time steps, with this test being used to validate against existing an existing AR5
projection.
AR5 calibrated projection of RSL for nine cities around the world from 2007 to 2100. Sampling was
carried out for HAIS, HGIS, HGLA, STR and HTWS using mean and standard deviations
from AR5 . The patterns for ice thickness are from GRACE 2003–2016 trends
. DSL is fully deterministic, from the CMIP5 Norwegian Earth System Model (NorESM) runs .
RSLGIA was deterministically set to 0. Each time step was sampled for using 10 000 LHS samples.
Modularity
The advantage of the partition approach as implemented in SLPS is that various approaches to probabilistic
projections can be executed with the same framework. First, as we will show in the next section, the KOPP14
approaches are fully compatible with the SLPS framework. Indeed, fingerprint patterns can be recomputed using local
thickness change rate patterns that are spatially constant on the basis of only one partition, such as the entire
Greenland or Antarctic ice sheets (contrary to Fig. where Greenland is subdivided). Once
several partitions are adopted, however, the refinement in the fingerprint patterns significantly departs from the
KOPP14 approach. Second, existing probabilistic assessments for specific components (such as the impact of changes
in surface mass balance or basal friction in Antarctica – as shown in –
on ice thickness changes) can be used
directly, using model output (for example, for thickness change rates), or PDF distributions from such model
outputs. If the uncertainty quantification was done using a Bayesian framework, the model output statistics can be
reused directly (using some type of uniform discrete sampling of each model output), hence replicating a
Bayesian-type exploration approach of SLPS without incurring any additional computational cost (meaning, not having to rerun
the analysis carried out to compute such model outputs). Third, ISSM modules can be activated upstream of the SLPS
solver to push further the boundaries of the uncertainty assessment. For example, an analysis of the impact of
SMB variations in one specific region of Antarctica could be carried out using the ice-flow modeling core of ISSM,
capable of delivering ice thickness changes directly to the SLPS core. Fourth, these modules can be activated while
remaining coupled to other modules. For example, in , it was demonstrated that over centennial
timescales, coupling between the elastic uplift of the grounding line and ice-flow-related grounding line
migration are key to controlling the retreat of Thwaites Glacier in West Antarctica. Assessing the uncertainty
brought by such processes on sea-level rise (SLR) projections would require this coupling to be activated, which
could be done (assuming computational costs are still realistic) without modifications to the SLPS framework.
Finally, given how closely ISSM can be integrated within Web Server architectures using its native JavaScript
interface , SLPS is potentially fully compatible with open-source types of collaborative
approaches where inputs from the community could be provided directly to web servers running ISSM in the
background to generate model projections without significant investment in a computation core and/or an interface
to the latter.
Results and discussion
SLPS probabilistic projections were validated using model inputs from the Intergovernmental Panel on Climate Change
Fifth Assessment Report (IPCC AR5) . AR5 supplies several projection components in SLR
equivalents: the “expansion” term (STR), the “glacier” term (which can be converted into an average thickness
change rate for HGLA), “antnet” and “greennet” for net barystatic contribution from the Antarctica and
Greenland ice sheets, which can also be converted into an average change rate for HAIS and HGIS, and the
“landwater” term for TWS contribution to SLR (which can be converted into an average change rate for HTWS).
For each of these terms, AR5 supplies the mean projection and the 5 %–95 % confidence interval. We can
use this information to calibrate PDF distributions for thickness change rates at each time step, with the mean of
each PDF corresponding to the AR5 mean and the standard deviation calibrated from the 5 %–95 % interval
(corresponding to the -1.65σ to 1.65σ interval). Because AR5 does not supply spatial patterns, we rely
on GRACE 2003–2016 thickness change rate patterns from for HGLA, HAIS
and HGIS.
For HTWS, we assume a uniform spatial distribution over all the spatial partitions. STR is also considered
uniform over all the oceans. DSL is not sampled but rather deterministically set to the DSL term of the CMIP5
NorESM-ME runs . GIA is independently sampled (from ) and probabilistically
added as an independent PDF. The sampling is carried out on the partitions described in Fig.
with the notable exception that the GIS is further divided into 18 different basins as defined in
and as plotted in Fig. . For each year between 2007 and 2100, 10 000 sample runs of SLPS are
carried out (with full geodetic capabilities of the SESAW core). For each partition, samples for the corresponding
inputs are generated using a Latin hypercube sampling (LHS) algorithm. The runs were carried out on the Pleiades
cluster at the NASA Ames Research Center, on 20 Ivy nodes (20 cores per node for an equivalent 400 cores) over 7 h.
Figure shows projection results for GMSL computed at each time step between 2007 and 2100, and
histograms for several time snapshots. We match the mean and 5 %–95 % confidence intervals of AR5
(Fig. a) as expected. We also show the evolution of RSL in Fig. for nine cities
around the world. We provide the mean and standard deviation for each PDF and show how the sampling of
ice-related thickness changes impacts mean and standard deviation. In particular, as expected from the AR5 inputs,
we show a marked increase in PDF spreads as time evolves from 2007 to 2100.
Figure shows the impact of using existing statistics from to include into
SLPS. These statistics were evaluated using a Bayesian inversion method based on simulated annealing
, a variation of the Monte Carlo with Markov chains (MCMC) method
. They can be used directly in SLPS either during a standard probabilistic
projection run or a posteriori as is the case here. These statistics reflect the statistical fitness to a global
GIA dataset composed of paleo-RSL indicators and vertical GPS trends.
The impact of the migrating Laurentide isostatic bulge on
Norfolk, Virginia, is apparent in Fig. , with an offset of 16 cm in the average projection for
the city.
AR5 calibrated projection of RSL for Norfolk (in brown) versus same projection in which GIA statistics
from are used to account for GIA-induced RSL (in blue). Note that both PDF
distributions have standard deviations that are essentially identical within 0.4 % relative difference.
Figure shows results for a different experiment, in which we quantify the impact of refining
the amount of partitions used to sample the uncertainty in ice thickness change rates. For the area of Greenland,
we use either one partition (blue boundary) or 18 boundaries (brown basins) from the dataset.
Each basin is delimited by ice divides and thus represents a dynamically coherent area, expected to behave (short
of ice divides migrating actively) independently from one another. We rerun an SLPS projection using a similar AR5
setup and display the contribution of ice-related basins to SLR in New York and Hawaii for 1 and 18 partitions
respectively. As expected, the mean in PDF distributions are identical for both 1 and 18 partitions. However, the
tails are much larger for the one-basin scenario. The relative difference in standard deviations between 1 and 18
basins ranges from -23 % for New York to -34 % for Hawaii. This implies that current probabilistic RSL projections
are significantly overestimating (by 20 %–30 %) the width of the “likely” (5 %–95 %) range in
ice-melt contribution to RSL.
Impact of subsampling the GIS mass (from 1 basin for the whole ice sheet to 18 basins) on barystatic
sea-level rise in New York and Hawaii. The distributions are a result of SLPS, where HGIS, HAIS and
HGLA were sampled 10 000 times using an LHS algorithm. The means in PDF distributions for both scenarios
are identical; however, the tails are much larger for the one-basin scenario. The relative difference in standard
deviations between 1 and 18 basins ranges from -23 % for New York to -34 % for Hawaii. This implies that
current probabilistic RSL projections could significantly overestimate (20 %–30 %) the “likely”
(5 %–95 %) range in
ice-melt contribution from glaciers and ice sheets.
This is understandable because of the fact that in a one-partition scenario, variations of ice thickness are dictated
by scaling of the local ice thickness change rate mean by an identical scalar for the entire partition, which leads
to more extreme values for the contribution to RSL. With finer partitions, basins that have low thickness change
rates do not impact RSL as much, and in aggregate the total contribution range varies less. This can be visualized
better by taking the example of New York, where following contributions from South Greenland
are almost negligible. This implies that all the basins (and corresponding GRD patterns) in South Greenland will
contribute zero variance to the PDF for RSL in New York. This will therefore result in smaller tails for
projections that rely on more refined basins. A very similar conclusion was found in , where
Antarctica had to be subdivided in spatially coherent areas, which were not obvious initially and did not
mandatorily map into individual basins. The issue is that the error distribution in model inputs had a specific
spatial coherence that had to be respected. Assuming this coherence extended to the entire ice sheet led to
significantly larger and unrealistic uncertainty ranges in model outputs. Of course, given differing dynamics in
each geographical basin, we cannot assume that the input scaling should be similar (same standard deviation). This
will modify the results in Fig. . But our aim here is to point out the issue of
sub-partitioning as being essential in quantifying the right range of spread in modeled statistical outputs.
This analysis also shows that using SLPS, it is possible to efficiently address the question of how to sample
uncertainty in a manner that is consistent with the local behavior of separate basins, glaciers and ice sheets. In
, for example, it is shown that the impact of glacier ice thickness variations around the world
is significantly different and that relying on one fingerprint alone can lead to significant differences in the
projection of glacier contribution (up to several percent). Our approach in SLPS ensures that the GRD
contribution is systematically reassessed for each sample, at each time step, and the partitioning of our sampling
ensures that we correctly capture the specificity of each glacier/ice/hydrological area and their unique mass
change trends. It is to be noted that a similar approach is currently implemented in new instantiations of the
KOPP14 projection system based on sampling of glacier projections across the 19 Randolph Glacier Inventory (RGI)
areas used in the GlacierMIP results . However, these areas can be very large in spatial
extent (such as the low latitudes or north Asian areas) and should be broken down. Our approach scales for any
barystatic contributor at any spatial scale (for example, sub-basin or at the glacier level) required by the
structure of the error distribution of model inputs.
Conclusions
ISSM SLPS is a new sea-level probabilistic projection system which relies on a new partitioning approach to
sampling of boundary conditions, forcings and inputs. It is compatible with previous probabilistic frameworks but
allows for a more robust integration of state-of-the-art results in the modeling of ice flow in ice sheets and
glaciers, sterodynamic sea level, TWS evolution and GIA. It re-establishes temporal correlation in projections where
they were previously lacking and allows for better constraints on spatial and temporal covariance in the model
inputs. In particular, it is capable of systematically computing geodetically compliant patterns of sea level that
are consistent with space and terrestrial measurement systems. The system relies heavily on the use of
high-resolution anisotropic meshes and allows for a better interfacing with existing modeling frameworks which
operate at higher resolutions and consistently generate changes in mass density patterns around the globe.
SLPS has been validated against previous frameworks and is fully backwards compatible. Differences between SLPS and
previous approaches have also been shown both in terms of integration of GIA statistics and integration of new
high-resolution sampling of ice thickness change patterns in Greenland. This new approach offers a roadmap towards
further increasing the complexity and realism of sea-level probabilistic projection frameworks.
Code availability
The ISSM code and its SLPS components are available at http://issm.jpl.nasa.gov (last access: 29 September 2020). The instructions
for the compilation of ISSM and SLPS modules are available at http://issm.jpl.nasa.gov/download (last access: 29 September 2020). The public SVN repository
for the ISSM code can also be found directly at https://issm.ess.uci.edu/svn/issm/issm/trunk and downloaded using user name
“anon” and password “anon”. The version of the code for this study, corresponding to ISSM release 4.17, is SVN version
tag number 24683.
Data availability
All datasets used in the projections are freely available in the public domain and are referenced
in the text.
Author contributions
EL carried out all the simulations and implemented SLPS into ISSM. He wrote the bulk of
the manuscript. LC contributed computations for GIA. MM contributed enhancements to the adaptive
meshers in ISSM. All authors contributed to the manuscript in terms of text, figures and comments.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We would like to acknowledge the NASA High-End
Computing (HEC) Program through the NASA Advanced Supercomputing (NAS)
Division at Ames Research for all the high-performance computing and corresponding
support. We would also like to acknowledge Ed Zaron for his insightful comments on the
manuscript.
Financial support
This research has been supported by the Jet Propulsion Laboratory, California Institute of Technology (prime contract no. 80NMO0018B0004), the NASA Sea-level Change Team (N-SLCT, WBS no. 105393), NASA Cryospheric Sciences (WBS no. 105393), NASA Modeling Analysis and Prediction (MAP, WBS no. 105479) and NASA Earth Surface and Interior (ESI, WBS 105526) programs, as
well as the NASA GRACE and GRACE-FO science team programs (WBS nos. 32700 and 106749).
Review statement
This paper was edited by Steven Phipps and reviewed by three anonymous referees.
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