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 km

Smith, Pope, and Kohler Glaciers, three narrow (

The problem of projecting ice sheet behavior is challenging, in part due to
incomplete understanding of physical processes

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

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”

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

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.

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.

In

The Lagrange multipliers

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

Minimization of

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.

The time-dependent observations of velocity and surface elevation in
Eq. (

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

In addition to these time-dependent data sets, we use the BEDMAP2 bed
topography

The land ice model used in this study is that described in

The observations used in our transient calibration are those described in
Sect.

Our results in Sect.

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

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.

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

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.

Our snapshot calibration recovers MEaSUREs velocities to high accuracy. The RMSE with observations is reduced from 140 m a

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

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.

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

Aside from the boundary volume fluxes

This strengthening is offset by a decrease in backstress along the grounding
line (Fig.

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

Another noticeable feature is the “ribbed” pattern that appears in the

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.

Both snapshot and transient calibrations predict continued contribution to
sea level rise. The transiently calibrated model projects

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

Spatial patterns of projected grounding-line position for the transiently
calibrated run show significant retreat from 2011–2021 (Fig.

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

Finally, it is important to realize that these projections are

The projection of committed grounded volume loss of 21 km

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

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

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.,

To facilitate the discussion we define an

We display the estimated parameters for the linear-in-time boundary stresses
experiment in Fig.

We emphasize that the above results should be regarded with caution due to
the relatively small reduction in

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

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.

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

We briefly consider potential reasons for the discrepancy between our modeled
2011 grounding line and that of

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

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

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

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

Available data for the model include ICESat satellite altimetry data

All heights are relative to the WGS84 ellipsoid. BEDMAP2 bed elevations are adjusted for this geoid.

A general overview of the ice flow model used is given in

For the temperature-dependent ice stiffness parameter

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

In particular, let

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

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

In Fig.

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 10

The BEDMAP2 data set is available as supplementary material to the source
cited. The MEaSUREs data set can be downloaded from