There is significant uncertainty over how ice sheets and glaciers will respond to rising global temperatures. Limited knowledge of the topography
and rheology of the ice–bed interface is a key cause of this uncertainty as models show that small changes in the bed can have a large influence on
predicted rates of ice loss. Most of our detailed knowledge of bed topography comes from airborne and ground-penetrating radar
observations. However, these direct observations are not spaced closely enough to meet the requirements of ice-sheet models, so interpolation and
inversion methods are used to fill in the gaps. Here we present the results of a new inversion of surface elevation and velocity data over Thwaites
Glacier, West Antarctica, for bed topography and slipperiness (i.e. the degree of basal slip for a given level of drag). The inversion is based on a
steady-state linear perturbation analysis of the shallow-ice-stream equations. The method works by identifying disturbances to surface flow which
are caused by obstacles or sticky patches in the bed and can therefore be applied wherever the shallow-ice-stream equations hold and where surface
data are available, even where the ice thickness is not well known. We assess the performance of the inversion for topography with the available
radar data. Although the topographic output from the inversion is less successful where the bed slopes steeply, it compares well with radar data
from the central trunk of the glacier for medium-wavelength features (5–50

Predicting the rate at which marine sectors of the West Antarctic Ice Sheet will retreat and contribute to globally rising sea levels is of increasing
importance due to persistent climate forcing across the region over the last few decades

Where ice-penetrating radar surveys have been undertaken with sub-ice-thickness line spacing (e.g. Rutford Ice Stream –

Bed conditions such as geology, hydrology and sediment distribution also play a role in controlling ice flow and behaviour

In this paper we exploit the relatively new availability of high-resolution surface elevation (REMA,

Following

Assuming that ice is a linear viscous medium (

We, however, are interested in the first-order momentum balance equations:

Also to the first order and importantly in the steady state, we have the following upper and lower kinematic boundary conditions:

Various points about the validity of the steady-state assumption for Thwaites Glacier are raised in the discussion (Sect.

All variables are then Fourier transformed with respect to the spatial variables

From depth integration of the Fourier-transformed incompressibility condition

Equations (

Starting once again with the shallow-ice-stream equations (Eqs.

This gives the following first-order momentum balance equations:

Fourier transforming with respect to the spatial variables

As there is no bed topography perturbation, the steady-state boundary conditions become

Equations (

Note that the transfer functions

These transfer functions can also be considered in a non-dimensional form, allowing us to make more general statements about the behaviour of the
system in terms of key variables, such as ice thickness as the characteristic length scale. For this purpose the same scalings as used in

The scale for slipperiness is given by

The non-dimensional form of the equations is obtained using the substitutions

The non-dimensional transfer functions (

For each set of wavenumbers in Fourier space (

Synthetic tests allow us to explore which bed features can and can not be resolved using this inversion method. First we create a synthetic bed
topography (

When running synthetic tests, several model parameters can be varied, in addition to the synthetic bed topography and slipperiness. Following

The effect of orientation to flow direction on how well landforms (top row; created synthetically) can be resolved by the inversion. These tests are presented on a 50

The effect of wavelength on how well landforms (top row; created synthetically) can be resolved by the inversion. These tests are presented on a 50

A two-dimensional Fourier transform decomposes an image into a weighted sum of two-dimensional sinusoidal basis functions. For this reason, all of our
synthetic tests used sinusoidal bed topographies and slipperiness as these are the most illustrative of the capabilities of the inversion. Sinusoidal
basis functions vary depending on three parameters: the wavenumbers in the

The effect of amplitude on how well landforms and slipperiness (rows 1 and 3, respectively; created synthetically) can be resolved by the inversion. These tests are presented on a 50

Figures

These synthetic tests show that in this simple least-squares inversion, the bed can be well resolved if the angle to the flow is greater
than 15

We now turn our attention to the methodology used to apply the synthetically tested inversion to real data, using the Thwaites Glacier catchment as our example.

Our base data for surface elevation and velocity were the REMA digital elevation model with 8

In the synthetic tests discussed above, the inversion was run over a single 50

When applied to real surface elevation and velocity data, this method generates four products: the mean and standard deviation of bed topography and
the mean and standard deviation of bed slipperiness. The standard deviation is

Inversion outputs across a 160

Figure

The bed topography product from the inversion is shown in Fig.

The standard deviation of the bed topography (Fig.

The bed slipperiness product from the inversion is shown in Fig.

In order to assess the success of the inversion, the bed topography output can be compared to existing radar data. Figure

We further compare the inverted bed topography with the bed topography sounded by swath radar along ice flow (Fig.

Our results demonstrate significant promise for being able to invert for bed topography across parts of Antarctica and other polar regions from
surface elevation and velocity datasets. Comparisons with existing radar data available from Thwaites Glacier suggest that within the central trunk
of the glacier, the bed features identified by the inversion are normally in the correct locations but are not always centred around the correct
depth. These average depth differences are primarily due to the mean ice thickness used in the inversion, which is a 50

The regions where the inverted bed deviates significantly from the topography picked from radar surveys are of particular interest in assessing the
potential of our inversion. Differences between inverted topography and radar lines are likely to be due to physical processes which are not
encapsulated by the shallow-ice-stream approximation. One place in which the shallow-ice-stream approximation is known to break down is where the mean
slope of the bedrock becomes too steep

A further consideration in comparing the inverted bed to real data is the steady-state assumption made when deriving the transfer functions. Without
repeat radar measurements for Thwaites Glacier we can not be sure of the stability of the bed. If the bed beneath Thwaites Glacier is changing
rapidly, as observed at Rutford Ice Stream

The steady-state assumption applies not only to the bed but also to the ice surface. Ice surface lowering due to glacier thinning would also affect
the steady-state assumption, but since generally the ice surface lowers in a relatively uniform way, this would not have a significant effect on the
first-order variations in the ice surface or the results of the model. More significant would be changes in the ice surface due to the filling and
draining of subglacial lakes, but these changes are normally fairly localised and would not propagate to the higher-wavelength Fourier
components. For Thwaites Glacier, the location of subglacial lakes is relatively well known

If the steady-state assumption is valid, then the age of the datasets used in the inversion is not important. However, input data from different years
or decades could also affect the steady-state assumption. The main surface expressions of known bed features appear to be fairly similar between REMA

As in any modelling study, it is important to explore the behaviour of the inversion when the parameters chosen are varied. In this inversion there
are just four fixed parameters which are not derived from the input data: the sliding-law exponent,

There is less certainty over what is the most suitable value for the non-dimensional mean slipperiness

This uncertainty in

The high-resolution swath radar grids

The effect of ice-thickness resolution on the results of the inversion in the Lower Thwaites region, as explored in Fig.

.

To explore the role that the 50

In Figs.

The results of the inversion (orange) compared to PASIN radar flight lines (grey) and BedMachine Antarctica (light blue) for an along-flow profile and an across-flow profile (locations shown in panels

A comparison of the two methods over an area where radar data have not yet been incorporated into BedMachine would allow an assessment of the
reliability of the two techniques and identification of any artificial bed features introduced by each. Since the two radar grids presented

We present the method and results of an inversion of ice surface elevation and velocity for bed topography and slipperiness in the Thwaites Glacier
region. Our method builds on the method used by

Starting from the shallow-ice-stream equations, linearising around a small perturbation in bed topography and taking the Fourier transform of the first-order equations, we have

Equations (

The determinant of the left-hand side of these equations is

Therefore we have

We then have

In the steady state, the kinematic boundary condition is

Substituting the expression from above gives

which can be rearranged as

In agreement with

Expanding the expression for

In agreement with

Expanding the expression for

In agreement with

Starting once again with the shallow-ice-stream equations

Equations (

The determinant of the left-hand side is

Therefore we have

which simplifies to

We then have

At a steady state,

In agreement with

Expanding the expression for

Expanding the expression for

The transfer functions (

Non-dimensionalised this gives

Since the system is over-determined, we can use a weighted least-squares inversion of Eqs. (

In matrix form we have the following forward model:

A least-squares inversion gives

For compactness we define the following:

The least-squares solution is then

This inversion is problematic where

To avoid this problem with an ill-conditioned inverse,

The filtered least-squares solution is then

The output data from the inversion are available on Zenodo at

The supplement related to this article is available online at:

RB, DG and AC developed the concept of the paper. DG advised on the use of the

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is an output of the NERC E4 Doctoral Training Partnership and the Geophysical Habitat of Subglacial Thwaites (GHOST) project, a component of the International Thwaites Glacier Collaboration (ITGC).

This research has been supported by the Natural Environment Research Council (grant nos. NERC NE/S007407, NERC NE/S006672, NERC NE/S006621/1, NERC NE/S006613/1, NERC NE/S006796/1) and the National Science Foundation (grant no. NSF PLR 1738934).

This paper was edited by Ginny Catania and reviewed by three anonymous referees.