The basal topography is largely unknown beneath most glaciers and
ice caps, and many attempts have been made to estimate a thickness field from
other more accessible information at the surface. Here, we present a two-step
reconstruction approach for ice thickness that solves mass conservation over
single or several connected drainage basins. The approach is applied to a
variety of test geometries with abundant thickness measurements including
marine- and land-terminating glaciers as well as a 2400 km

For the 210 000 glaciers and ice caps on this planet

In terms of input requirements, reconstruction approaches always need
information on the geometric setting. This normally comprises the glacier
outline and the surface topography. In the Ice Thickness Models
Intercomparison eXperiment

From an observational perspective, operational and regular satellite imagery
acquisition and processing has become an indispensable and continuously
growing source of information. Therefore, automated procedures have been
brought in place providing products such as glacier outlines

In light of the continuously growing body of information, it becomes easier
to gather the input fields for sophisticated thickness-reconstruction
approaches. In this regard, we present a two-step approach that provides a
physically based thickness field over entire glacier basins, ice fields or
ice caps. The first step has limited input requirements
(Sect.

The thickness-reconstruction approach is based on mass conservation and
largely originates from ideas presented in

Over the ice-covered domain

All source and sink terms are combined in the apparent mass
balance field

In a first step, the mass conservation (Eq.

With prior knowledge only on

By construction, the ice thickness is a priori unknown and so is the coupling
length scale (

To determine the flux magnitude

The direct flux solution to all input fields often shows widespread negative
values and high spatial variability. Therefore, we chose to iteratively
update the AMB-field

For the optimisation of the cost function, we rely on the “m1qn3”
module

Once a flux field is determined over the glacier domain, the ice thickness is
inferred in a post-processing step. Flux values are locally translated into
thickness values assuming the SIA

The ice-viscosity parameter

We apply a correction to the flux solution before computing the ice thickness
from Eq. (

Together with the thickness map, we present a formal error map. For this
purpose, the uncertainty of the input fields, i.e. the SMB and

Input uncertainties for the test geometries are presented in
Sect.

Another source of uncertainty relies on the fact that the mass-conservation
equation (Eq.

In a second step, the thickness map is updated in areas where reliable
surface velocity information is available by solving Eq. (

The ice-thickness solution is optimised as we cannot anticipate that input
fields are consistent in terms of the mass balance equation. Yet in this
step the optimisation makes use of three control variables. The AMB is
complemented by both horizontal velocity components

As above (Sect.

Errors are again estimated following the ideas presented in

Individual glacier outlines are first
partitioned into marine and land-terminating segments by searching whether the
surface elevation is 0 within 150 m of the outline point. Where the DEM
showed more advanced glacier fronts than the glacier inventory, a marine
termination is inferred within the same search radius but with 100 m as
surface elevation threshold. Subsequently, nunataks are automatically
accounted for in the mesh, if resolved by the target grid spacing. In
addition, we added grid points at each location where thickness measurements
were available. This was necessary to prescribe internal boundary conditions
on the error estimates. High-resolution thickness measurements were
subsampled a priori in accordance with the grid resolution. From the outline and
measurement locations, a 2-D mesh with triangular elements was generated using
the open-source finite-element grid generator Gmsh
(

In the first-step reconstruction, two external boundary conditions are
necessary around the glacier domain. At outflow boundaries along marine ice
fronts, no condition is imposed on the ice flux.
Where glaciers terminate on land, a zero-flux Dirichlet condition is imposed.
For the error estimation, internal boundary conditions are applied at
thickness measurement locations. There,

The two-step reconstruction approach is tested on three ice geometries on
Svalbard, where an abundant record of thickness observations was available
(Fig.

VIC is the second largest ice cap on the Svalbard archipelago

Overview map
of the Svalbard archipelago showing ice coverage (blue shading). The three
test sites are located in two areas on the archipelago (red shading and
rectangles). The test sites are Vestfonna ice cap (VIC) on Nordaustlandet,
the glacier complex comprising the marine-terminating Austre Torellbreen,
Hansbreen and Paierlbreen (THPB), and the land-terminating
Werenskioldbreen (WSB) in Wedel Jarlsberg Land. Background: grey-scale
hill-shaded topography based on a 50 m DEM from the Norwegian Polar
Institute (NPI;

Input fields to the ice-thickness reconstruction for
VIC

Austre Torellbreen is a marine-terminating glacier (Fig.

Glacier outline information is taken from the 2002–2010 glacier inventory
described in

Concerning the Svalbard surface elevation, we rely on a 50 m digital
elevation model from the 1990s (

VIC thickness measurements (Fig.

In the Hornsund area, Hansbreen is well studied and an ice-core drilling team
reached the bed at three locations in 1994

For the SMB information, we rely on the regional climate model MAR

To assess the sensitivity of the thickness reconstruction to the SMB input
(Appendix

Over VIC, 2003–2007 elevation changes (Fig.

Source code available at:

For Wedel Jarlsberg Land, elevation changes were calculated by differencing
the NPI 20 m DEM (

Using satellite imagery acquired between January 2015 and September 2016 by
the C-band synthetic aperture radar on-board Sentinel-1, we apply
intensity offset tracking to consecutive image pairs

Information on the fjord bathymetry is used to further constrain the
thickness reconstruction at marine ice fronts. The new International
Bathymetric Chart of the Arctic Ocean
(IBCAO version 3.0;

Ice-flux solution after cost optimisation for VIC

The target resolution for the meshing is set to 200 m for THPB and VIC and 100 m for WSB. Observations for all test geometries are very densely spaced and we decided to only keep measurements that are more than 50 m apart, which is half of the minimum grid spacing. The initial 20 792, 44 921 and 21 273 measurements collected on VIC, THPB and WSB were thus reduced to 4475, 5945 and 1189 points, respectively.

From the above presentation of the input fields available for the test
geometries, we define input uncertainties for the formal error propagation in
Sects.

This section covers the presentation and discussion of the ice-flux solution,
the reconstructed thickness and bedrock elevation fields as well as the error
estimates. In the error analysis, actual mismatch values from a fraction of
withheld measurements are compared to the formal error estimate
(Sect.

For Vestfonna ice cap, the ice-flux field is very instructive
(Fig.

First-step ice-thickness map for VIC

Bedrock topography associated to the thickness field in
Fig.

For WSB and THPB, the ice flux is small all along the land-terminating margin
and increases towards centrelines. For Austre Torellbreen, we find strong
flux convergence along Bøygisen and Løveisen. Further downstream,
ice-flux magnitudes remain constant as the AMB is close to zero. Unlike this
balanced situation, a pronounced surface subsidence over most of Paierlbreen is not
explained by the SMB and results in a positive AMB over the entire catchment
area. This imbalance is compensated by extensive downwasting implying a
gradual flux increase up to the marine ice front. The imbalance itself might
partially reflect the long-term geometric adjustment of Paierlbreen to the
surge in 1993–1999. Yet we cannot exclude that the SMB model underestimates
the magnitude of surface melting or that a bias is introduced by the DEM
differencing (Sect.

The first-step thickness map (Fig.

For the THPB and WSB systems in southern Spitsbergen (Fig.

For many glaciers, only few or even no thickness measurements are available,
and, therefore, we want to asses a lack of in situ measurements. For this
purpose, we recomputed all thickness fields relying on a random 1 % sample
of all thickness measurements (Fig.

Error-estimate map based on the error propagation presented in
Sect. (

Median values for the absolute thickness mismatches and the error estimates at measurement locations not included during the reconstruction. Marker colours indicate the respective fraction of all measurements withheld from the reconstruction. Dashed crosses span the interquartile range of all mismatch values (horizontal) and all formal error estimates (vertical). For orientation, the identity line was added in grey.

The following error analysis is two-fold: we first present and discuss
error-estimate maps from the formal error propagation of input uncertainties
as described in Sect.

Fraction of all validation measurements for which the absolute mismatch is less than the predicted error estimate. Values are given in percent.

Relying on a formal error propagation (Sect.

Considering only 1 % of all thickness measurements, the error estimates
become larger (Fig.

A pressing question is whether the magnitude of these error-estimate maps is reliable and falls into a realistic range. For this purpose, we withheld a random sample of all thickness measurements from the reconstruction and computed an absolute thickness mismatch for comparison. The sample size is defined as a fraction of all measurements and we investigated the range from 1 to 99 %.

In a first attempt, we directly compared the formal error estimates to the in
situ absolute mismatch values. Ideally, these two values would show a
positive correlation. Yet no clear dependence was discernible for any of the
sample sizes. Both data distributions, for mismatch values and error
estimates, are not normal. Being robust to outliers, we decided to quantify
these distributions in terms of medians and quartiles (Fig.

The above aggregate assessment suggested that the error estimates could serve
as upper constraints for the actual mismatch. However, it remains unclear how
reliable this interpretation is at individual measurement locations. We
therefore compute the data fraction of all withheld measurements for which
the actual mismatch is less than the predicted error estimate
(Table

We conclude that median error estimates overestimate the mismatch values and could therefore serve as an upper error constraint. Accepting this interpretation, we can provide a maximum error range for aggregate quantities. Mean thickness values for VIC, THPB and WSB fall into a range of 172–320, 141–217 and 46–508 m, respectively. For the area fraction of ice grounded below sea level, we find ranges of 7–23 % for VIC and 7–22 % for THPB. The maximum range on VIC is clearly exceeded by the 5% area fraction inferred by a direct interpolation of measurements. Despite this aggregate assessment, the spatial reliability of interpreting the error-estimate map in terms of an upper constraint for local measurements becomes increasingly difficult the fewer measurements are available.

The second step of this reconstruction is applied in one sub-domain for each
test geometry, where velocity measurements exceed 100 m yr

Ice thickness

Error-estimate map as in Fig.

On VIC, ice thickness is updated along eight fast-flowing outlet glaciers
(Fig.

In Wedel Jarlsberg Land, thickness fields are updated for the three
fast-moving frontal areas of the THPB complex. The wealth of thickness
observations implies that the first- and second-step reconstructions are very
similar (Fig.

The updated error-estimate map (Fig.

Median values for the absolute thickness mismatches and the error
estimates as in Fig. (

We repeat the aggregate error assessment from above (Fig.

In this section, we discuss the central assumptions and caveats of the
presented reconstruction approach. For the first step, sliding is neglected,
assuming that ice motion is an exclusive result from internal deformation. In
areas without thickness and velocity information, this assumption is likely
the dominant source of uncertainties and biases the results towards higher
thickness values. Other reconstruction approaches use an empirical scaling
relation

Another caveat in the first-step reconstruction is the assumption that the
ice flux follows a smoothed version of the surface-slope field
(Sect.

The provided error-estimate map is shown to be a practical measure for a
first error assessment. The underlying error analysis inherits all
assumptions made in the mass-conserving reconstruction and thereby accounts
for various input uncertainties. A fundamental assumption is that the error
estimate is the minimum value of two solutions
(Sect.

Concerning the sensitivity of the thickness map of VIC to changes in the
input SMB and the input DEM (Appendix

We present a two-step, mass-conserving reconstruction approach to infer glacier thickness maps with prior knowledge on source and sink terms in the mass budget. The two steps guarantee applicability in absence of velocity measurements. In the first step, a glacier-wide thickness field is inferred from a balance flux calculation on the basis of an apparent mass balance field. The second step requires velocity measurements, which are often not reliable over an entire glacier drainage basin. Therefore, the glacier thickness is only updated over a sub-domain. This updated field is consistent with the observed flow field and shows a seamless transition into the glacier-wide first-step map. In both steps, available thickness measurements are readily assimilated to constrain the reconstruction. Moreover, the inferred thickness field is provided together with an error-estimate map, based on a formal propagation of input uncertainties. Here, we present and apply this approach to various glacier geometries on Svalbard where an abundant thickness record was available.

The approach is found to be most beneficial in areas where thickness observations are sparse or unavailable. There, our reconstruction is informed by the glacier geometry and the AMB. Direct interpolations of thickness measurements often ignore this geometric and climatic information and fill a gaps according to distant measurements. The associated error map estimated from our reconstruction additionally highlights areas with least constrained ice thickness, namely away from observations and especially where ice flow is small or even stagnates. In an aggregate, glacier-wide sense, the actual thickness mismatch is shown to reach 25 % for glaciers with only few thickness measurements. In absence of such measurements, the aggregate mismatch freely scales with a priori choices for not-well-constrained parameters.

In light of the growing body of information on glacier changes with satellite
remote sensing, reconstruction approaches for mapping glacier ice thickness
are less and less limited from the input side. Therefore, 2-D approaches
become increasingly attractive and favourable because a final interpolation,
which fills gaps between reconstruction profiles, can be avoided. However,
the largest limitation on the applicability of 2-D approaches is the
availability of regional information on surface elevation changes and surface
mass balance. Elevation change maps from satellite remote sensing have
already been presented for several regions but further development is
necessary to reduce uncertainties associated with signal penetration and firn
properties. Concerning regional SMB fields, we can either rely on parametric
approaches or on regional climate models. In absence of both SMB and

The data can be requested from the corresponding author. In the near future, we will provide consistent Svalbard-wide fields for ice thickness and error estimates via one of the common online repositories.

To translate the ice-flux solution into an ice-thickness field, the
ice-viscosity parameter

The ice-viscosity parameter

For the THPB and WSB area, the

In summary, the interpretation of the viscosity field

In this section, we will briefly discuss how the AMB-field

Final apparent mass balance

Here, the sensitivity of the first-step reconstruction to the SMB input is
briefly discussed for VIC (Fig.

The sensitivity of the first-step thickness field to the DEM choice is
smaller as compared to the SMB sensitivity. The exchange of the 2010 DEM
(Sect.

On VIC and THPB, the area fraction with negative ice flux is 1.2 and 3.1 %,
respectively. On WSB, however, the flux solution over the main branch is
generally very small and shows many zero transitions. Consequently, the
area-fraction is higher at 4.1 %. The reason is that the AMB shows no
dominant source area in the upper glacier ranges. The zero transitions in the
flux solution would directly transmit into the ice-thickness field. To avoid
such transitions, we correct the flux as follows:

The flux correction applied during the first-step reconstruction
(Sect.

The first-step thickness solution is most sensitive to the flux correction in
small areas along divides and ridges (Fig.

Ice-thickness map for VIC as in Fig.

Ice-thickness map for VIC as in Fig.

In summary, the effect of the flux correction can lead to a considerable
difference in ice volume in the cases where no thickness measurements are
available and where small flux values prevail over a large area. Yet, where
measurements are available, a compensation is possible via the ice-viscosity
parameter

Ice thickness for VIC

Reconstruction sensitivity as quantified by the mean and maximum ice thickness, the ice volume and the area fraction grounded below sea level. The ‡symbol separates values stemming from a reconstruction using either all or only a 1 % fraction of the available thickness measurements.

JJF designed and implemented the reconstruction approach, applied it to the test cases and elaborated the details of the error estimation. The research aims and setup were developed in regular discussion with FG-C, TS, BS and MB. JJF led the writing of the manuscript, in which he received support from all authors. FG-C developed and provided the initial version of the optimisation routines. Input fields for the reconstruction are the Sentinel-1 surface velocities from TS; ice-thickness measurements from TJB, JAD, RP, FN and MG; DEMs from CN and BS; surface elevation changes from CN and GM; and surface mass balance fields from XF, CL and KA.

The authors declare that they have no conflict of interest.

This study received primary funding from the German Research Foundation (DFG) under grant number FU1032/1-1. Results presented in this publication are based on numerical simulations conducted at the high-performance computing centre of the Regionales Rechenzentrum Erlangen (RRZE) of the University of Erlangen-Nuremberg. The reconstruction approach also benefits from co-development work of the Elmer/Ice team at the CSC-IT Center for Science Ltd. (Finland). The velocity analysis on Svalbard was funded by DFG within the priority programme 1158 Antarctic Research with Comparable Investigations in Arctic Sea Ice Areas under contract number BR2105/9-1 and received financial support from the Helmholtz Association of the German Research Centres (HGF) Alliance on Remote Sensing and Earth System Dynamics. Thickness data collection in Wedel Jarlsberg Land was funded by the Spanish R & D projects C11093001 and C150954001, NCBiR/PolarCLIMATE-2009/2-2/2010 from the Polish National Centre for R&D, by IPY/269/2006 from the Polish Ministry of Science and Higher Education, by Polish-Norwegian funding through the AWAKE (PNRF-22-AI-1/07) project, by the EU FP7 ice2sea programme (grant number 226375) and by funds of the Leading National Research Centre (KNOW) received by the Centre for Polar Studies of the University of Silesia, Poland. The DEM generation in Wedel Jarlsberg Land received financial support from the European Research Council (grant 320816) and from ESA (project Glaciers CCI, 4000109873/14/I-NB). TanDEM-X data were provided under AO XTIGLAC6770. The WRF-SMB field was produced within the PERMANOR project funded by the Norwegian Research Council (255331). Edited by: Andreas Vieli Reviewed by: Fabien Maussion and Douglas Brinkerhoff