TCThe CryosphereTCThe Cryosphere1994-0424Copernicus GmbHGöttingen, Germany10.5194/tc-9-2429-2015Committed retreat of Smith, Pope, and Kohler Glaciers over the next 30 years inferred by transient model calibrationGoldbergD. N.dan.goldberg@ed.ac.ukhttps://orcid.org/0000-0001-9130-4461HeimbachP.JoughinI.SmithB.https://orcid.org/0000-0002-1118-7865Univ. of Edinburgh, School of GeoSciences, Edinburgh, UKUniversity of Texas, Institute for Computational Engineering and Sciences/Institute for Geophysics, Austin, Texas, USAApplied Physics Laboratory, University of Washington, Seattle, USAD. N. Goldberg (dan.goldberg@ed.ac.uk)21December2015962429244625July201525August20159November201530November2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/9/2429/2015/tc-9-2429-2015.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/9/2429/2015/tc-9-2429-2015.pdf
A glacial flow model of Smith, Pope and Kohler Glaciers is
calibrated by means of control methods against time varying, annually resolved
observations of ice height and velocities, covering the period 2002 to 2011.
The inversion – termed “transient calibration” – produces an optimal set
of time-mean, spatially varying parameters together with a time-evolving
state that accounts for the transient nature of observations and the model
dynamics. Serving as an optimal initial condition, the estimated state for
2011 is used, with no additional forcing, for predicting grounded ice volume
loss and grounding line retreat over the ensuing 30 years. The transiently
calibrated model predicts a near-steady loss of grounded ice volume of
approximately 21 km3 a-1 over this period, as well as loss of
33 km2 a-1 grounded area. We contrast this prediction with one
obtained following a commonly used “snapshot” or steady-state inversion,
which does not consider time dependence and assumes all observations to be
contemporaneous. Transient calibration is shown to achieve a better fit with
observations of thinning and grounding line retreat histories, and yields a
quantitatively different projection with respect to ice volume loss and
ungrounding. Sensitivity studies suggest large near-future levels of
unforced, i.e., committed sea level contribution from these ice streams under
reasonable assumptions regarding uncertainties of the unknown parameters.
Introduction
Smith, Pope, and Kohler Glaciers, three narrow (∼ 10 km wide),
interconnected West Antarctic ice streams, have exhibited substantial
thinning and speedup in recent years. As these ice streams are smaller than
neighboring Thwaites and Pine Island Glaciers – the contribution of Smith
Glacier to total Amundsen Embayment grounding-line flux is ∼ 7–8 times
smaller than that of Pine Island or Thwaites – focus is
often placed upon these larger ice streams, with regard to both modeling and
observations of the ice shelves and sub-shelf environments
e.g.,. However, high
thinning rates have been observed near the Smith terminus, even larger than
that of Pine Island and Thwaites .
Additionally, substantial retreat of the Smith grounding line has been
observed , suggesting that the ice stream may be subject to
the same instability thought to be underway on Thwaites .
As such, there is a need to develop a quantitative dynamical understanding of
the causes of this retreat; and if possible, to determine whether it will
continue at similar rates.
The problem of projecting ice sheet behavior is challenging, in part due to
incomplete understanding of physical processes , but also
due to difficulties in estimating the state of an ice sheet at any given
time. Unlike other components of the climate , ice sheet models
cannot be “spun up” to the present state, as the required historic forcing
fields are not available. Rather, the models must be initialized from
observations, which are mostly limited to surface properties such as surface
elevation and velocity. A widely used methodology is one to which we will
refer as “snapshot” calibration, first introduced by , and
which solves an inverse or optimal control problem. In this technique, an
optimal set of parameters relating to sliding stress (and possibly ice-shelf
viscosity) is found through a least-squares fit of the ice model's nonlinear
momentum balance to a given velocity field. Time dependence is not
considered, since the momentum balance (or rather stress balance) is
non-inertial. We choose the term “snapshot” because it applies to ice
velocity and geometry at a single instant, assumed to be the same for both data sets.
A number of studies have employed snapshot calibrations to make near-future
projections of the behavior of Pine Island and Thwaites Glaciers in response
to varying forcing scenarios . These
studies have deepened our understanding of the behavior of these ice streams.
However, the use of snapshot calibrations in ice sheet projections is
potentially problematic: any temporal inconsistencies among data sets can lead
to nonphysical transients which persist for decades, which is not ideal if
the goal is projection on a similar timescale. Inconsistency between the data
and model discretization can have a similar effect. For instance, co-located
gridded velocity and thickness data require interpolation for application to
a model whose discretization staggers these fields, potentially leading to
transient nonphysical artifacts . An oft-used approach is
to allow the model to adjust to these inconsistencies before conducting
experiments. The model may then have drifted to a state far from
contemporaneous observations, with potentially different sensitivities.
As the observational record grows, so does the availability of data for the
same geographic areas at multiple points in time. It is sensible, then, to
make use of this temporal resolution for the purpose of constraining the
time-evolving state of an ice stream, with the significant benefit of
producing initial conditions for forecasting from a realistic past
trajectory. Such an approach, which we term “transient calibration”, is
well developed in other areas of geophysics, e.g., in oceanography where it is
known as “state and parameter estimation” , or
reservoir modeling where it is known as “history matching”
. Here we present the results of such a calibration,
applied to Pope, Smith and Kohler Glaciers. Calibrated model parameters are
the result of an inversion in which a time-evolving model produces an optimal
fit to a 10-year time series of surface elevation and velocity observations.
The model is then run for an additional 30 years. In the transiently
calibrated run, rapid grounding-line retreat continues for another decade,
but then slows, while loss of grounded ice remains near constant at
∼ 21 km3 a-1 (or ∼ 0.06 mm a-1 sea level contribution). We show that the
predicted high levels of ice loss are relatively insensitive to any future
changes in forcing, and to any systematic errors in our calibration.
Transient calibration of a model of an Antarctic ice stream with
temporally resolved plan-view data has not previously been carried out,
though we point out that used methods similar to those used
in this study to infer surface mass balance over the Northeast Greenland Ice
Stream over a 6-year period from laser altimetry. No future projections were
made in their study.
We proceed with detailing what we mean by “snapshot” vs. “transient”
calibration of an ice flow model, and show how ice sheet observations are
used in this process (Sect. ). We then describe the
observational data (Sect. ), as well as the model and
the details of the calibration used in this study (Sect. ).
Results of the calibration and projection are presented in
Sect. , followed by an investigation of the sensitivity of
these results to plausible uncertainties in the parameter estimates (Sect. ).
Model calibrationSnapshot calibration
A widely used approach for single-time observations is to invert for
uncertain control variables, using a stress balance model, via the adjoint or
Lagrange multiplier method. applied such an optimal control
method, in which the misfit between model velocity,
u, and observed velocity, u*, is minimized
with respect to unknown (or uncertain) variables λ (often referred to
as a control variables), subject to the constraint that the velocity
satisfies the nonlinear stress balance, written in the generic form
L(u, λ)= 0. The misfit (or cost) function is expressed as
Jsnap=∑i=1N|ui-ui*|2ηui2,
where ui and ui* are at location i (grid
cell or node), and η(ui)2 the uncertainty of the
observation. The constrained optimization problem may be turned into an
unconstrained one by introducing Lagrange multipliers μi:
J′=Jsnap-∑i=1NμiLi,
where Li(u, β) = 0 is the discretized form of the stress
balance at node i. By finding a saddle point of J′ with respect to the
parameters and Lagrange multipliers, an extremal point of Jsnap is found
in parameter space with the stress balance enforced exactly. The coefficient β
of the linear sliding law
τb=β2u
is often used as the control variable λ. Jsnap is sometimes
extended with an additional “smoothing” term that penalizes small-scale
variations in the control parameters e.g.,. The ice
geometry (i.e., surface and bed elevation) is assumed to be known exactly.
In , the model considered is the shallow shelf approximation (SSA)
and the control variable is
β as above. Development of sophisticated glacial flow codes and the
consideration of ice-shelf physics have led to the use of alternative or
augmented control spaces e.g.,
and the use of higher-order stress balances e.g.,.
The Lagrange multipliers μi are then used to calculate the gradient of
Jsnap with respect to the control variables λ, which can in turn
be used to carry out the minimization of Jsnap via gradient descent or
quasi-Newton optimization methods. The μi are found by solving the
adjoint of L′, the linearization of the operator L. The adjoint
method is popular for snapshot calibrations in glaciology due to the fact
that L′ is self-adjoint, i.e., the adjoint operator, can be solved by the
same code used to solve L if the dependence of ice viscosity on strain
rates is ignored.
Transient calibration
When observations distributed in time are available together with a
time-evolving model, the “snapshot” calibration can be extended to what we
term “transient” calibration, which consists of optimizing agreement of the
model with observational data at multiple time levels, with both the
nonlinear stress balance and ice thickness evolution enforced as model
equations. This is equivalent to the following constrained cost function,
which should be compared against Jsnap:
Jtrans=ωu∑k=1T∑i=1Nχki(u)|ui(k)-ui(k)*|2ηui(k)2+ωs∑k=1T∑i=1Nχki(s)si(k)-si(k)*2ηsi(k)2,
where s is ice surface elevation, the superscript k is the time index,
and the asterisk indicates observational values. χki(u) and
χki(s) are equal to 1 if there is an observation at cell i and
time step k, 0 otherwise. ωu and ωs are weights to impose
relative importance of observations. The Lagrangian J′ now extends to one
with time-evolving Lagrange multipliers, i.e.,
J′=Jtrans-2∑k=1T∑i=1Nμi(k)xi(k)-Fi(x(k-1)),
where the model equations are written in generic form x(k+1)=F(x(k)),
and x represents the model state,
i.e., the minimal set of variables needed to step forward the model and to
evaluate Jtrans.
Minimization of Jtrans can be carried out in a similar manner, by use of
its gradient with respect to the control vector. However, gradient
calculation is more complicated, now requiring a time-dependent adjoint
model, which can be derived via the continuous-form adjoint of the model
equations, as has been done for simplified ocean models ,
or by means of algorithmic differentiation AD;. Used
extensively in ocean modeling e.g.,,
the use of AD tools in land ice modeling is becoming increasingly common
.
In this framework, the control parameters may now be chosen to be
time-dependent. However, doing so is meaningful only if physically justified
and if sufficient information is available to constrain the larger control
space. In the following, unless stated otherwise, time-independent parameters are used.
Observations
The time-dependent observations of velocity and surface elevation in
Eq. () come from two recently generated data sets. One
contains InSAR-derived surface velocities of the Smith Glacier region, binned
annually to a 500 m grid for the years 2006–2010
. Velocities are available for floating and
grounded ice. Coverage is not spatially uniform, but greater in later years.
The other data set is a series of annual surface digital elevation maps (DEMs)
from 2001 to 2011 on a 1 km grid. Coverage is consistent between
years, but data are not available seaward of the 1996 grounding line
, or on slow inter-stream ridges. Figure shows
the geographic region of study along with the acceleration and thinning
recorded by the transient data sets. The 2001 surface is not from 2001
measurements, but is simply an extrapolation backward in time from later
years. Further details of this data set are given in Appendix .
In addition to these time-dependent data sets, we use the BEDMAP2 bed
topography and the MEaSUREs (Making Earth System Data Records for Use in Research Environments) (450 m grid) data set
. We also use the accumulation
data set to estimate ice temperatures in the region, as explained in Appendix .
(a) Ice speed in the Pope/Smith/Kohler and Crosson/Dotson system.
The white contour is the grounding line as given by BEDMAP2, and the magenta
contour represents the limits of the transient surface elevation data set. The
rectangular box shows the subdomain used for our state estimate simulations – boundary
stresses are imposed along the black contour and boundary fluxes
are imposed along the light blue boundaries. (b) Norm of velocity change
between 2006 and 2010 within the model domain, excluding the areas of no
coverage in either 2006 or 2010. (c) Cumulative surface thinning, 2001–2011
in the surface elevation data set. The shaded region shows where data are
available. (d, e, f) Hövmoller plots of cumulative thinning along transects
in (c) in descending order.
Model and calibration setup
The land ice model used in this study is that described in
. The model's stress balance is depth-integrated,
similarly to the shallow shelf equations, but the effects of vertical
shearing are represented . Grounding-line migration is
implemented through a hydrostatic floatation condition. As described in
, the model has been successfully differentiated
using the AD software tool Transformations of Algorithms in Fortran
TAF;. We solve the land ice equations in the domain
shown in Fig. on the 500 m grid of the time-dependent velocity
set, and all other fields are interpolated to this grid. This allows for
resolution of the relatively narrow ice streams. However, the domain does not
include ice shelf seaward of the 1996 grounding line, as we explain below. We
do not account for the effects of firn on ice dynamics.
The observations used in our transient calibration are those described in
Sect. . The initial ice thickness in each model run is
from the 2001 DEM. Subsequent DEMs are applied to the cost function
Jtrans at the end of each model year, as are velocity constraints in the
years and locations available. For the snapshot calibration we use ice
geometry from the 2002 DEM; as we do not have 2002 velocities, MEaSUREs
velocities are used as constraints. As discussed below, in transient
calibrations the domain excludes ice shelves. We carry out snapshot
calibrations in the same domain to enable comparison, and the resulting
parameters become initial guesses in our transient calibration. Similar to
other ice model calibrations, the basal sliding parameter β2 is a
control parameter. Our other control parameters, less common in glaciological
inversions, arise from the nature of the transient calibration and the data
sets used, as explained below.
Our results in Sect. are generated assuming
time-invariant control parameters. In Sect. we allow for
time-dependent parameters, and consider the implications of the results.
Difference between (top panels) modeled and observed velocities in 2010
(the last year available) and (bottom panels) modeled and observed surface elevation
in 2011. Left panels: snapshot calibration. Right panels: transient calibration. The
magenta contours represent modeled grounding lines in 2011. In (d), the green
hatches give the 2011 grounding line position reported by
(digitized from the publication).
Boundary stresses as control parameters
Our transient surface observations only give values inland of the 1996
grounding line. Time-resolved annual velocity observations are provided for
the ice shelves, but only from 2007 to 2010. Including ice shelves in our
domain, then, would require estimation of transient ice-shelf thickness from
2001–2011. Such an estimate would be very poorly constrained (see
Sect. for a discussion on this topic). We overcome this
problem by formulating an open boundary estimation problem (see,
e.g., for an oceanographic analogue), with the 1996 grounding
line as the downstream boundary of the domain (see Fig. ).
Stresses at the grounding line, which would otherwise be part of the stress
balance solution, must now be imposed along this boundary. The action of the
membrane stress tensor along a horizontal boundary has
two components: normal membrane stress σ and shear membrane stress τ,
as explained in more detail in Appendix . In the
model, σ and τ can be defined along any horizontal boundary,
floating or grounded. These boundary stresses are not known a
priori, and we treat them as unknown spatially varying (along boundaries)
control parameters to be estimated via calibration, with two unknowns
(σj and τj) for each rectangular cell boundary j. Where the
domain borders a slow-moving ridge velocities are set to zero, and boundary
stresses are not applied.
Boundary volume flux as a control parameter
In our transient calibrations, the ice flux into the domain must be
estimated. This is due to the incomplete coverage of the time-dependent
velocities, which leaves the upstream regions poorly constrained, leading to
anomalously high thinning. To address this we consider boundary fluxesqx and qy as control parameters at x and y facing
boundaries, respectively. These boundary fluxes enter the model through the
continuity equation, which is solved via a finite-volume scheme, and are
treated as constant over a cell boundary. Boundary fluxes are not imposed
along the internal boundaries with slow-moving ridges, or where boundary
stresses are imposed. Note that qx and qy are only used in transient
calibration; for snapshot calibration, MEaSUREs velocities do not lead to
high thinning rates in these regions, despite no-flow conditions at the
upstream boundary.
Comparison of transient misfit of modeled surface elevation between
snapshot and transient calibration along different flow lines. From left to
right, panels correspond to flow lines in Fig. c in descending order.
Top row panels: snapshot calibration. Bottom row panels: transient calibration.
(a) Error |u-u*| of the “snapshot”
calibration to MEaSUREs velocities as in Eq. ()
(note color scale differs from that of Fig. a and b). (b) The pattern of sliding parameter β2 which
achieves the misfit in (a). (c) The adjustment of β2 in the transient
calibration relative to that of the snapshot calibration. (d) The pattern of
the buttressing inferred in the calibrations. Specifically, the profiles to
the left of the figure show -γσ(x) (cf. Eq. )
corresponding to points on the boundary at the same y position.
ResultsCalibration results
Our snapshot calibration recovers MEaSUREs velocities to high accuracy. The RMSE with observations is reduced from 140 m a-1 for the initial guess down to
50 m a-1 – but error is actually much lower any most areas outside the margins
of the narrow western branch of Kohler entering Dotson Shelf
(Fig. a). The control parameters adjusted in the snapshot
calibration, β, σ and τ, are then used in a transient (but
non-calibrated) run from 2002 to 2011. The degree to which this run agrees with
the transient observations is demonstrated in Fig. a, c, and
the top row of Fig. .
For velocities in the snapshot-calibrated run, the misfit for 2010 – the
last year in which velocity observations are available – is largest in
Kohler and Smith glaciers, and is up to ∼ 50 % or more of the observed
velocity. The misfit is largest at the boundary with the slow-moving ridge,
which may be because the no-flow condition imposed there by the model is not
accurate. By 2011, modeled surface elevation within 20–30 km of the grounding
line is ∼ 100 m higher than observed, a misfit that is larger than the
impact of the thinning signal itself over the period of integration. The
misfits grow with time, and so only the final years are shown at this level
of detail. Figure gives surface error along the flow transects
from Fig. c.
Relative to the time integration with initial state and parameters obtained
from the snapshot inversion, the transient calibration gives good agreement,
especially with respect to surface elevation (Fig. d). The 2011
surface elevation misfit field looks very different to the one inferred from
the snapshot calibration, with uniformly small misfits. Figure
(bottom row panels) shows the reduction in transient surface elevation misfit along
the transects. On Smith and Kohler, misfit in 2010 velocity has decreased,
though it is still substantial (Fig. b). The relatively low
decrease in velocity misfit between snapshot and transient calibration can be
explained by our choices of ωu and ωs, which favor surface elevation.
Also the grounding-line behavior is very different between the two
simulations. In the snapshot-calibrated run there is almost no retreat, while
in the transiently calibrated run the 2011 grounding line has retreated
considerably. The modeled 2011 grounding line is not completely coincident
with the observed grounding line of (digitized and plotted
for comparison), particularly in the western part of the Smith/Kohler
grounding region. The cause for this discrepancy is unclear; but in any
event, the ice in this region does unground in our simulation, it is simply
delayed by 5–10 years (see Sect. ).
Adjustment of control parameters
Aside from the boundary volume fluxes qx and qy, the control parameters
in the snapshot and transient calibrations have a one-to-one relationship.
Thus it is interesting to examine how the parameters are adjusted for
transient calibration. In both the snapshot and transient calibration, we
infer an area of very weak bed (basal stresses of ∼ 10 Pa or less) in the
fastest moving parts of the glaciers (Fig. b). The most
striking adjustment of basal stress parameters is a strengthening of the bed
under the trunks of Pope, Smith and Kohler Glaciers (Fig. c).
This strengthening is offset by a decrease in backstress along the grounding
line (Fig. d). It is possible that our snapshot calibration is
equifinal, i.e., that there is more than one combination of boundary
stresses and bed parameters to reproduce imposed velocity and elevation
observations. In this case our snapshot calibration does not correctly
estimate the dynamic state of the system, while the additional information
provided by the transient observations allows us to find a better balance
between boundary stresses and basal strength. Alternatively, it may be that
the temporal mismatch between velocity and altimetry in the snapshot
calibration demands a more extensive weak-bedded region than is realistic,
with additional buttressing required to match velocities at the grounding line.
A noticeable feature of the transiently calibrated solution is that of
“negative buttressing”; i.e., the normal membrane stress in some locations
is larger than what would be felt without any ice shelf. This could be for a
number of reasons. It is possible the model, and the fit to observations, is
insensitive to small-scale oscillations in the boundary stress field.
However, it could also be due to errors in the bed topography data: as
detailed in Appendix , boundary stresses are expressed as a
fraction of unconfined membrane stress, which depends on bed depth. Negative
buttressing could be compensating for an assumed bed that is too shallow.
Finally, the “negative buttressing” may be very real features of the ice
sheet. demonstrated that even in the absence of an embayed
ice shelf, alternating patterns of ridges and ice streams can lead to ice
shelf buttressing, whereby the fast-moving streams essentially “pull
forward” the ice on the slow-moving ridges. Such a situation could yield
negative buttressing at the ridges. Inspection of Fig. d shows
that the negative values occur at the slow-moving regions in between the
narrow, fast-flowing streams.
Another noticeable feature is the “ribbed” pattern that appears in the
β2 field, but not in the snapshot-calibrated field. The cause of this
discrepancy is uncertain. It is possible that the observed velocities could
be well-represented in the snapshot calibration without these features, but
that they are necessary to fit to surface observations. However, it may be
related to the “smoothing” term mentioned in Sect. .
In both models, a Tikhonov regularization term (i.e., the square-integrated
gradient of β) is added to the cost function – this is done because
ice model velocities are insensitive to high-wavenumber variations in the
basal sliding coefficient, and these scales are poorly constrained
. The term is multiplied by a weighting coefficient – and
this coefficient is chosen on the basis that β2 should not vary by
a considerable amount on scales smaller than the membrane stress scale
(∼ 5 km). Importantly, this weighting is the same for both transient and
snapshot calibrations. In making this choice, we may have implicitly
introduced a degree of subjectivity to our estimations .
Introducing prior information in a more objective manner is beyond the scope
of this study, but it is an important goal and should not be overlooked in
the future.
(a) Sea level contribution from the region and (b) total ungrounded
area in domain through 2041 based on snapshot and transient calibrations
(solid curves) and inferred from the DEM data, BEDMAP2, and Eq. ()
(red hatches).
Projected ice loss and behavior
The model state and parameters estimated via either snapshot or transient
calibration are used as initial conditions in two 40-year integrations out to
2041, i.e., extending into a 30-year prediction window 2011–2041. The results
are shown in Fig. a in terms of cumulative loss of volume above
floatation (VAF) from 2001. VAF does not include floating shelves or the
portion of a grounded column that would be supported by ocean pressure, and
thus is an indicator of sea level contribution. To calculate VAF from the
observational data, thickness hobs and height above floatation haf
must be inferred from surface and bed data as follows:
hobs=sobs-bobsbobs=maxR,-ρiρw-ρisobshaf=hobs+minρwρibobs,0,
where ρi= 918 kg m-3 and ρw= 1028 kg m-3 are ice and ocean
densities, respectively, sobs is surface elevation from the transient
DEM set, and R is BEDMAP2 bed elevation. haf is then integrated for VAF.
Both snapshot and transient calibrations predict continued contribution to
sea level rise. The transiently calibrated model projects ∼ 21 km3 a-1
grounded ice volume loss from 2011 to 2041 (∼ 0.06 mm sea level
equivalent), while the snapshot calibrated model suggests ∼ 25 % more.
Thus there is a quantitative impact of the initial state, and therefore of
the type of calibration used, on projected sea level contribution from the
region. There is an even more pronounced impact on projected grounding line
retreat: in the snapshot-calibrated run, almost no ungrounding takes place,
while in the transiently calibrated run ungrounding is significant (Fig. b).
Given the much closer fit of the transiently calibrated
simulation to surface observations in a least-square sense, we accept this
simulation as a better estimate of the dynamic state of the glaciers in the region.
Cumulative thinning since 2001 (shading) and grounding-line position
(red contours) in 40-year run from transient calibration. The 2021 and 2031
grounding lines are shown in successive plots with green and brown contours,
respectively.
Sensitivity of grounded volume (Volume above floatation, or VAF)
loss from the domain over the 40-year integration to (a) the sliding
parameter β2 and (b) bed topography R (see Sect.
above for explanation). (c) Annual cost functions
(cf. Eq. ) for various calibrated model runs.
(d)-γσ(x) (cf. Eq. ) at the
initial and final times in the “time-dep. boundary stress” estimation
referred to in (c). Values are plotted relative to the red curve in
Fig. c. (Note the difference in scale from Fig. c.)
Spatial patterns of projected grounding-line position for the transiently
calibrated run show significant retreat from 2011–2021 (Fig. ),
followed by a slight slowdown in retreat. In contrast, thinning rates remain
high throughout the 30-year integration. Grounding-line retreat does not
proceed down the deep troughs incised by Smith and Kohler Glaciers,
suggesting the retreat predicted by might not happen in the
near-term. We argue that this is because the troughs are quite narrow, and
lateral stresses from areas of shallower bed limit grounding-line retreat.
However, other studies suggest grounding line retreat in Amundsen and
Bellingshausen ice streams can be episodic rather than sustained due to
details of bed geometry . Furthermore,
while melting under parts of the ice shelf external to the model are
implicitly accounted for through boundary stresses and their sensitivities
(Sect. ), we do not apply melt rates to ice that goes
afloat within the domain. Such effects could lead to stronger retreat than
what is shown. Thus, we cannot discount further rapid grounding line retreat
in the future (i.e., beyond 2041), particularly since thinning rates remain
high throughout our simulation. Spatial patterns for the snapshot-calibrated
simulation actually show slight thickening in some areas downstream of the
observed 2011 grounding line, and otherwise show a more even pattern of
thinning (i.e., it is not skewed downstream).
The imposed mass fluxes at the inland boundary are not expected to influence
the results: the time scale (30 years) is less than the diffusive time scale
for grounding line changes to propagate across the domain
e.g.,, which we calculate to be ∼ 150 years based on a
nominal surface slope of 0.01, thickness of 1400 m, and velocity scale in the
upstream regions of 100 m a-1.
Finally, it is important to realize that these projections are
unforced: the estimated parameters and boundary conditions β,
σ and τ (and qx, qy where applicable) are held constant
over this time period, and no submarine melt is applied to any areas which
unground. This is the basis for referring to the projected grounded ice loss
as committed.
Uncertainties of estimated parametersUncertainty of sea level contribution projection
The projection of committed grounded volume loss of 21 km3 a-1 over the next 3 decades from 2011 onward is subject to uncertainty due its implicit
dependence on model parameters. The adjoint capabilities of the model allow
us to estimate reasonable bounds on this uncertainty through calculation of
sensitivities to these parameters, which can be integrated against parameter
field perturbations. For instance, Fig. a shows the adjoint
sensitivity of transiently calibrated VAF loss to the basal sliding
parameter β2. We refer to this quantity as δ*(β2), and it can be
interpreted as follows: assume the β2 is subject to a perturbation
P(x, y). Then the perturbation to VAF loss that follows from this parameter
perturbation is given by
δVAF=∫Dδ*β2Pdxdy,
where D is the model domain. δ*(R), the sensitivity of VAF loss to
topography, is plotted in Fig. b. Note that the influence of
β is sign-definite, i.e., decreasing β anywhere increases ice
loss, while lowering the bed only increases ice loss upstream of the
projected 2041 grounding line.
If we assume an error of 100 % for each basal sliding parameter – an
unlikely scenario, as this would affect the fit to observations – ice loss
projections would change by at most 57 %. Other parameters have lower
influence, assuming reasonable uncertainties. 100 % error in the boundary
stress parameters would change the ice loss projection by at most 1 %. The
influence of input fluxes qx and qy is similarly small. The full range
of bed elevation errors associated with the BEDMAP2 data set would change the
projection by at most 30 %. These values are based on linear sensitivities,
while our model is nonlinear – but the results are borne out by experiments
with finite perturbations. Of course, these fields would not vary
independently – but based on these relatively low sensitivities we
anticipate that the projected mass loss value is not overwhelmed by its
uncertainty. Thus, our conservative uncertainty analysis suggests a level of
committed sea level contribution from the region.
The above estimation of uncertainty bounds is tentative. Our inverted
parameters have no a priori estimates or uncertainties, and our minimization
does not provide a posteriori uncertainties or covariances. Thus we are
unable to provide accurate confidence intervals on ice loss based on
observational uncertainty. Estimation of a posteriori uncertainties based on
observational uncertainties may be possible e.g., through methods that infer
the Hessian of the cost function .
Enabling such calculations within our estimation framework is a future
research goal.
Time dependence of control parameters
Our adjoint-based calibration framework allows for the estimation/adjustment
of control parameters that vary not only in space, but also in time
(e.g., ). Justification for doing so derives
from the physical interpretation of these parameters, e.g., boundary stresses
representing far-field stresses in the ice shelves, which could change due to
crevassing or ocean melting. We investigate whether such time dependence can
be inferred from the observations. In our framework, parameters vary
piecewise-linearly over predefined time intervals of uniform length. For
instance, with intervals of 5 years, and over the interval from t= 5 years
to t= 10 years, σj (the normal stress at face j) takes on the values
σj(t)=σj(5)2-t5+σj(10)t5-1.
The parameters σj(5) and σj(10) (and σj(0))
are distinct for each cell face, and constitute additional parameters for the
system. Thus, the greater the temporal resolution, the more calibration
parameters are involved. Considering the increase in size of the parameter
space, the additional information is only meaningful if it improves the fit
of the calibration.
To facilitate the discussion we define an annual cost function,
i.e., a breakdown of Jtrans by year. That is, for each year k we define
J(k)=ωu∑i=1Nχki(u)|ui(k)-ui(k)*|2σ(ui(k))2+ωs∑i=1Nχki(s)si(k)-si(k)*2σsi(k)2.
In Fig. c this value is plotted by year for different
experiments. The annual cost functions resulting from the snapshot and
transient calibrations are plotted (although recall that the snapshot
calibration is not designed nor intended to explicitly reduce the transient
misfit reflected by Jtrans). Results from two additional calibrations
are shown as well. In the first, the β2 parameter is assumed
time-invariant, but boundary stresses are allowed to vary linearly over the
2001–2011 period as described above. In the second, boundary stresses are
constant while β2 is allowed to vary linearly in time. In each case,
the number of degrees of freedom which describe the time-variant control
doubles. The cost function Jtrans is reduced, but the reduction is
relatively small (∼ 20 %).
We display the estimated parameters for the linear-in-time boundary stresses
experiment in Fig. d, by plotting buttressing at the beginning
and the end of the simulation – or, more accurately,
-γσ(x,0) and -γσ(x,10), where
γσ determines buttressing level (cf. Eq. ) and the number in the superscript has the
same meaning as in Eq. () above. Results are displayed relative
to the time-independent parameters found above. The pattern corresponds to a
slight loss in buttressing from 2001–2011, albeit of a smaller magnitude than
its temporal average. (Note that the loss of basal stress due to
grounding-line retreat, found to be an important mechanism by
, is resolved by our model and therefore not implicit in
inferred boundary stresses.) Also, the pattern is slightly different at Smith
as compared to Kohler and Pope. The corresponding inferred pattern of
time-dependent β2 (not shown) is somewhat noisy but contains a clear
signal of bed weakening under fast-flowing regions just upstream of the 2011
grounding line.
We emphasize that the above results should be regarded with caution due to
the relatively small reduction in Jtrans resulting from additional
degrees of freedom. However, we are not aware of a quantitative measure to
determine whether the improvement is significant, i.e., whether the inferred
time-dependent adjustment of the parameters can be regarded as real, or just
“noise”. It is also possible that the small reduction of the cost function
is due to the shortness of the estimation period, over which the distinction
between time-varying vs. time-mean controls does not influence the
solution significantly. However, the pattern of temporal buttressing change
is at least plausible given observed submarine melt rates
and loss of ice rumples and pinning points
. Thus the information presented may be of use in future
studies of the region that include ice shelves, as it could be used to accept
or reject various ice shelf forcing scenarios on the basis of resulting
changes in buttressing. Questions regarding the level of temporal data
resolution required to constrain time-varying parameters, and of appropriate
criteria to identify overfitting of such parameters, are targets for future work.
Discussion
We do not hold our snapshot calibration to be the best possible in the sense
of reproducing spatiotemporally resolved observations. For this calibration
we used MEaSUREs velocities, which have a much later time stamp than the ice
geometry used. This choice was made because no 2002 velocity data were
available. Nevertheless, our results demonstrate that a snapshot calibration
with non-contemporaneous data, or data sets that might be inconsistent with
each other if used at face value in a dynamical framework, cannot be expected
to reproduce time-dependent behavior, whereas transient calibration can take
account of time-varying data in order to better reproduce observations,
thereby giving more confidence in near-future projections of ice sheet
behavior. The nonlinear least-squares framework ensures that mutually
incompatible data sets can be properly weighted, i.e., interpolated by the
model dynamics, instead of having to be simultaneously fulfilled exactly.
Importantly, within such a framework increased care must be taken to provide
useful error estimates for each observational element (the η entries in
Eq. ). This requires understanding of measurement
errors, potential systematic biases, and representation errors.
While transient calibration can potentially constrain time-varying behavior
of poorly known control parameters, care must be taken that the increase in
dimension of the parameter set yields an improved fit with observations.
Otherwise, the additional information provided (relative to time-invariant
parameters) may be of limited use. For our calibration, we see that allowing
for time-varying control parameters only provides a small improvement of fit,
and thus we do not reject the null hypothesis that far-field buttressing (and
bed strength) did not change from 2001–2011. While it is possible that
buttressing did decrease over this time, it is also possible that some
perturbation to the system occurred long before observations began, and the
2001–2011 retreat is just a continued response to this perturbation. More
investigation is needed regarding the details of how temporal observational
sampling is able to constrain temporal structure of poorly known parameters.
As explained in Sect. , the decision was made to remove the
ice shelves seaward of the 1996 grounding line from the domain in favor of
boundary stresses. It is worth briefly considering the implication of this
decision. BEDMAP2 draws ice shelf thickness data for the region from
, who give an effective timestamp of January 1995. It is
likely that the change in thickness from this date to 2001 was both
non-negligible and roughly on order with the change in thickness over the
2001–2011 window . Apart from BEDMAP2 our only available ice
shelf data are velocities in 2007–2010 (2006 had little ice shelf coverage);
we do not possess any data regarding ice shelf thickness change over time. In
order to model the evolution of the ice shelves, then, it would be required
to estimate 2001 ice shelf thickness, as well as the spatially and temporally
varying melt rates and effective Glen's Law ice stiffness parameter (A)
from 2001 to 2011. Data from grounded ice (such as velocities and surface
elevation) are not sufficient to infer such detailed information about ice
shelves, as modeling studies indicate that grounded ice evolution might be
insensitive to melt rates and ice stiffness over large parts of the ice
shelves . Thus estimates of the above parameters
would need to be made, with only velocity information at the end of the
decade as a basis of improving the estimates. Such a strategy would be
ill-posed owing not only to the limited temporal coverage, but also to the
fact that both ice shelf thickness and Glen's Law parameter determine
velocities. Thus the approach, while not impossible, would require very
careful quantification of a posteriori parameter uncertainty – which, as
stated previously, requires more sophisticated computational tools than those
used for this study. However, incorporation of ice shelf data and simulation
into transient calibration procedure is an important goal, and future efforts
should try to achieve this goal with the above limitations in mind.
In addition to the control parameters discussed above (boundary stresses,
upstream fluxes, and sliding parameters), two others were initially
investigated: adjustments to initial (2001) surface elevation, and
adjustments to bed elevation. These fields were considered to be potentially
important for observational agreement, as the 2001 DEM from which the initial
condition is derived is a backward-in-time extrapolation of later
measurements, and bed topography is considered to be a source of uncertainty for
ice flow . However, significant
adjustments were not found for either (the inversion adjusted initial surface
on the order of millimeters, and the bed on the order of meters), and their
inclusion did not improve the fit to observations. Thus these control
variables were not considered further. We point out these results may depend
somewhat on our prior assumptions of their variability, which is implicitly
imposed by the scaling of cost function gradients (see Appendix ),
and we stress the importance of choosing conservative and unbiased prior
information in future transient ice sheet calibrations.
We briefly consider potential reasons for the discrepancy between our modeled
2011 grounding line and that of . As mentioned in Sect. ,
we do not account for the effects of firn density in our
model, neither have our transient surface data been corrected for firn. As the
depth of the firn layer can affect the floatation condition
e.g,, it is reasonable to ask whether these omissions
can explain the disagreement between our modeled 2011 grounding line and
observations. Figure gives a detailed comparison between the
modeled and observed grounding lines, as well as the 2011 grounding line
inferred from the 2011 DEM and the BEDMAP2 data via Eq. ().
There is slight disagreement between the latter two grounding line estimates,
but it does not explain the erroneously grounded region in our model. Rather,
we suggest this region is anomalously thick (and therefore grounded) due to
buttressing from the small grounded “island” at the Smith Glacier grounding
line, which is not visible in the data. Furthermore, we
point out that grounding line agreement is not explicitly accounted for in
our transient cost function. Still, future studies should account for firn
effects in order to achieve better agreement with grounding line observations.
A detailed comparison of modeled grounding lines, the grounding line
implied by the data used in the modeling study, and directly observed
grounding-line position. The red shaded area represents the portion of the
domain which is ungrounded in 2011, inferred from floatation with the 2011
surface DEM and BEDMAP2, and assuming ice and ocean densities of 918 and
1028 kg m-3, respectively. The blue contour is the modeled 2011 grounding line,
and green hatches give the 2011 grounding line position from
. The thin black contour is the computational boundary, and
the thick black contour the 1996 grounding line. Note that the
data do not extend to Pope Glacier.
Conclusions
Generalizing optimal control methods based on steady-state adjoint models
well-known in glaciology to those using a transient forward and adjoint
model, enables us to perform model calibration based on simultaneous state
and parameter estimation through a nonlinear least-squares fit of a model to
time-resolved observations. We perform such a transient calibration for the
grounded portion of the Smith, Pope and Kohler Glacier region based on
velocity and surface observations covering the years 2001–2011. This
transient calibration is compared with a “snapshot” calibration of the same
region based on instantaneous (and assumed contemporaneous) observations. The
transient calibration agrees far better with spatially and temporally
resolved observations, giving increased confidence in near-future behavior
predicted by the model.
Extending the simulations beyond the 2001–2011 calibration period, both
snapshot- and transiently calibrated models are run in “predictive mode”
from 2011 to 2041, without any changes in boundary conditions or external
forcing. Both show a significant sea level contribution. That of the
transiently calibrated model is nearly 20 % smaller, but with significant
grounding line retreat and grounding line-concentrated thinning.
Sensitivity calculations suggest that, under reasonable assumptions regarding
parameter uncertainties, a committed grounded ice loss of
∼ 21 km3 yr-1 can be expected from the region, even in the absence of external forcing or
climate-induced feedbacks. Our sensitivity analysis does not replace a
comprehensive uncertainty quantification of projected ice volume loss, and a
more complete end-to-end uncertainty propagation chain is needed for
transient ice model calibration.
As the catchment of Smith, Pope and Kohler Glaciers is relatively small, the
potential for sea level contribution is not as large as that of Thwaites and
Pine Island . Nevertheless, the volume loss
from these glaciers is quite high given their size, and our projection shows
no indication of it slowing in the next few decades. Furthermore, significant
thinning of the region could affect flow of nearby ice streams by changing
surface gradients. The methodology of transient calibration introduced in
this study – which has not previously been applied to a marine-based
Antarctic ice stream – could be applied to other regions of Antarctica to
better constrain near-future behavior. To do this, better availability of
spatially and temporally resolved observations, for both grounded and
floating ice, along with credible error estimates for each observational
element will be essential.
Generation of surface elevation fields
The ice-sheet surface height used in the model is derived from a
least-squares fit of a time-varying surface model to laser-altimetry and
photogrammetric data. We represent the surface as a reference surface,
corresponding to 30 December 2010, and a set of elevation increments for
years between 2002 and 2012, each defined for the nodes of an irregular mesh.
The reference surface has a mesh resolution up to around 100 m, while each
elevation increment has a resolution of 2 km. The model's surface height as a
function of time is found through an iterative minimization of the sum of its
misfit to the data points and measures of its roughness and the roughness of
its temporal derivatives. The model fit is determined in part by the
numerical weight assigned to the roughness of the reference surface and the
elevation-change increments; we selected the weights to give expected
reference-surface errors due to random, uncorrelated data errors of around
0.06 m, and to give elevation-rate errors of around 0.03 m yr-1. It is likely,
however, that spatial correlation in data errors and irregular data
distribution resulted in considerably larger errors in some places.
Available data for the model include ICESat satellite altimetry data
, and airborne scanning laser altimetry data supplied
by NASA's Operation IceBridge program ,
and stereophotogrammetric data derived from the Worldview satellites, for
2011 and 2012. Each of these data sources is treated as a collection of
points with small, statistically independent errors for each point, and
larger, spatially uniform biases that are independent for each day on which
the data were collected. To ensure that all elevation-change estimates are
well constrained by data, we use only data for points that have a repeat
measurement within 1 km in at least 1 different year, and those
measurements acquired within 3 months of the reference date of 30 December 2010.
We fit the resulting data set with an initial elevation model, then
removed those data points whose residuals were larger than 3 times the
standard deviation of all model residuals, repeating this process until
either no further points were removed in an iteration, or until the
normalized standard deviation (equal to the standard deviation of the
residuals divided by their assumed errors) of the misfit reached unity.
All heights are relative to the WGS84 ellipsoid. BEDMAP2 bed elevations are
adjusted for this geoid.
Model description
A general overview of the ice flow model used is given in
. Here we discuss in detail features specific to,
or developed for, this study.
Temperature-dependent rheology
For the temperature-dependent ice stiffness parameter B in Glen's flow law,
we follow the approach of by stepping forward an
advection-diffusion equation for temperature to steady state, with velocity
and geometry held fixed. The upper surface temperature and kinematic boundary
conditions come from the parameterization of , and from
the accumulation data set of , respectively. A
constant geothermal flux of 100 mW m-2 out of the bed is assumed. From the
steady-state temperature field we calculate B, and use its depth-average in
all simulations, without adjustment.
Boundary stresses
Here, we describe in more detail how, in our experiments, the ice shelves are
omitted from the domain and replaced with a boundary condition that
represents the effect of the ice shelves on the grounded ice. Within the ice,
horizontal stresses are described by the membrane stress tensor
Se.g.,:
S=σh+TrσhI=ν‾4u‾x+2v‾yu‾y+v‾xu‾y+v‾x4v‾y+2u‾x,
where σh is the restriction of the Cauchy stress tensor
to the x and y directions. In this context, the stress balance solved by
the ice model for depth-average velocity can be written
∂jSij-τb,i=ρgH∂is,
where H is vertical thickness, and s is surface elevation, and summation
is over the j index. Along an arbitrary horizontal line ℓ within the
ice sheet or ice shelf, the force acting on the line, per unit length s and
in a depth-integrated sense, is
HS⋅n-Fn,
where n is the normal vector to ℓ, and F arises from
hydrostatic pressure. We henceforth refer to the two components of
S⋅n as σ, the component normal to
ℓ, and τ, the component parallel to ℓ (Fig. a).
Along a calving front, σ=σcf and τ=τcf are set by
local force balance:
σcf=ρg2HH2-ρwρzb2τcf=0,
where ρw is ocean density and zb is ice basal elevation
. Internally to the ice shelf and ice stream,
however, σ and τ depend on the nonlocal solution to Eq. ().
(a) Planform visualization of depth-integrated normal and shear
stress along a vertical front or grounding line. In the case of the ice
shelf, the stress balance must be solved within the glacier and ice shelf,
and stresses along the grounding line depend on this solution. If these
grounding-line stresses were imposed along the calving cliff, velocities in
the glacier would be the same in both cases. (b) Schematic of representation
of boundary stresses through parameters. Shaded cells represent computational
domain, and white cells represent area where an ice shelf would be, were it
included in the domain. Separate degrees of freedom describe normal and shear
stress at each cell face.
In particular, let ℓ coincide with the grounding line. For a given
solution to the stress balance, τ and σ will have a certain
dependence along the grounding line, and in general will vary with θ,
the distance along the grounding line. If the stress balance were again
solved, but only over the grounded part of the domain, with
S⋅n imposed to be equal to the same
(σ(θ), τ(θ)), then velocities and stresses within the ice
would be the same. (This is mathematically true for the depth-averaged
hydrostatic stress balance used in this study; while it does not hold for the
general Stokes balance, any nonhydrostatic effects will likely be limited to
the vicinity of the grounding line.) In other words, the effect of the ice
shelf on grounded velocities (and thickness evolution) is imposed solely
through σ(θ) and τ(θ).
Thus, in our runs, the boundary of the computational domain is internal to
the ice body (and initially coincides with the grounding line). As our model
has a rectangular grid, this boundary is not a continuous line but a
collection of cell faces, some directed in (i.e., normal to) the x direction
and some in the y direction (Fig. b). We implement σ
and τ as a set of parameters, with a separate value for each cell face.
Effectively, we implement a Neumann boundary condition, albeit one that does
not depend uniquely on the ice thickness and bed depth, as is the case for a
calving cliff. Rather, the boundary condition is a forcing that needs to be
estimated. These parameters are expressed not as stresses but as an excess
fraction of the unconstrained membrane stress. Thus,
σ=1+γσσcf,τ=γτσcf
and γσ, γτ are the actual parameters. Notice that in
this formulation σ and τ depend on bed depth at the cell face
according to the topographic data set (in this case BEDMAP2, ).
In some of our simulations, the boundary of the domain does not remain
coincident with the grounding line, as there is grounding-line retreat. The
grid cell faces along which stresses are imposed do not follow the grounding
line in this case; rather, they remain fixed and we effectively impose the
stresses on a portion of the shelf. However, they are still imposed far from
the calving front, and σ and τ(s) are still representative of
buttressing within the ice shelf.
In Fig. d we distinguish between σ(x) and
τ(x), boundary stresses along faces normal to the x direction (and
likewise γσ(x), τσ(x)), and σ(y) and
τ(y). Note than σ(x) and τ(y) enter into the
x momentum balance (and are therefore more relevant to flow predominantly
in the x direction).
Normalization of gradient information
When carrying out adjoint-based inversions or state estimations with
heterogeneous control fields, the units of the different control variables
must be accounted for. For instance, the boundary stress parameters as
described above nominally vary between 0 and 1 (dimensionless), while values
on the order 104 m2 a-1 were found for the input flux parameters. Thus
for a given stress parameter σi and a given flux parameter qj, one
might expect ∂Jtrans∂σi to be several
orders of magnitude larger than ∂Jtrans∂qj.
The gradient with respect to the parameter set, and thus the search direction
in parameter space, would be overwhelmed by the gradient with respect to
input fluxes. This issue is addressed by normalizing the cost function
gradient by nominal “unit” values, where the unit value corresponds to the
type of parameter. In our inversion, values of 0.1, 5 × 104 m2 a-1,
and 10 Pa (m a-1)-1 were used for boundary stresses, input fluxes, and
basal sliding parameters, respectively. Additionally, values of 1 and 10 m
were used for adjustments to the initial surface and the bed elevation,
respectively (see Sect. of main text). The normalization factor
for the initial condition was chosen since this value was in line with the
errors applied to the surface observations. The factor for the bed was chosen
due to the relatively small bed adjustments required by mass continuity
considerations for this region .
Acknowledgements
The BEDMAP2 data set is available as supplementary material to the source
cited. The MEaSUREs data set can be downloaded from
http://nsidc.org/data/NSIDC-0484. The time-dependent velocities and surface
DEMs can be obtained free of charge by e-mailing the corresponding author. All
materials for the ice model and optimization framework is available for
download from http://mitgcm.org, except for the automatic differentiation
software TAF, which is available for purchase from http://fastopt.com/. D. N. Goldberg
acknowledges funding through NERC grant NE/M003590/1. P. Heimbach was supported in
part by DOE grant SC0008060 (PISCEES) and NASA grant NNX14AJ51G (N-SLCT).
Edited by: G. H. Gudmundsson
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