The thickness of ice shelves, a basic parameter for mass balance estimates,
is typically inferred using hydrostatic equilibrium, for which knowledge of
the depth-averaged density is essential. The densification from snow to ice
depends on a number of local factors (e.g., temperature and surface mass
balance) causing spatial and temporal variations in density–depth profiles.
However, direct measurements of firn density are sparse, requiring
substantial logistical effort. Here, we infer density from radio-wave
propagation speed using ground-based wide-angle radar data sets
(10

As a snow layer deposited at the ice-sheet surface is
progressively buried by subsequent snowfall, it transforms to higher-density
firn under the overburden pressure. The firn–ice transition, marked by the
depth at which air bubbles are isolated, occurs at a density of approximately
830

Knowledge of the depth–density profile and its spatial and temporal
variability is important for a number of applications: (i) to determine the
age difference of enclosed air bubbles and the surrounding ice in ice cores

Density profiles are most reliably retrieved from ice/firn cores either by
measuring discrete samples gravimetrically, or by using continuous dielectric
profiling

All of the aforementioned techniques, however, remain point measurements
requiring substantial logistics. A complementary approach is to exploit the
density dependence of radio-wave propagation speed. The principle underlying
the technique involves illuminating a reflector with different ray paths such
that both the reflector depth and the radio-wave propagation speed may be
calculated using methods such as the Dix inversion

A typical acquisition geometry is to position receiver and transmitter with
variable offsets so that the subsurface reflection point remains the same for
horizontal reflectors

Here, we investigate six WARR measurements collected in December 2013 on Roi
Baudouin Ice Shelf (RBIS), Dronning Maud Land, Antarctica. The WARR sites are
part of a larger geophysical survey imaging an ice-shelf pinning-point and a
number of ice-shelf channels which are about 2 km wide and can extend
longitudinally from the grounding line to the ice-shelf front

The basal mass balance inside ice-shelf channels can be mapped from remote
sensing assuming mass conservation

Herein, we calculate densities from WARR sites using traveltime inversion and
ray tracing (Sect.

We describe the propagation of the radar wave for each offset as a ray
traveling from the transmitter via the reflection boundary to the receiver
(Fig.

Similar to what has been done for wide-angle radar measurements in Greenland

The radar consists of resistively loaded dipole antennas (10

Location of the wide-angle (WARR) radar sites (red triangles) relative to the boreholes of 2010 and 2014 which were used for optical televiewing (OPTV). The depressed surfaces of ice-shelf channels appear as elongated lineations in the background image (Landsat 8, December 2013, provided by the US Geological Survey).

Wide-angle radar data showing air waves (AW, green lines) and
surface waves (SW, green dashed lines) with linearly increasing traveltime
with offset, while traveltime increases hyperbolically with offset for
internal (blue) and basal (red) reflectors. See Fig.

In multi-offset surveys, the traveltime of internal reflectors increases
hyperbolically with increasing offset

The traveltime

The traveltime is a non-linear function of the model parameters (and hence
the inversion results may be non-unique) because

Combining Eqs. (

Using Eq. (

The derivatives of Eq. (

An optimal set of model parameters

We apply the ray tracing model provided by

To solve the inverse problem we seek the set of parameters

We minimize

In order to compare different measurements at different locations, we
decompose the ice shelf into two layers of ice (

To test the inversion algorithm we use ray tracing with a prescribed
depth–density function and recording geometry (

We consider two ideal cases: a single reflector at 400 m depth, and two
reflectors at 30 and 400 m depth. Using the forward model, we simulated a new
set of reflectors with model parameters covering depth ranges of

Traveltime residuals (

Example for initial

Next, we consider effects of random and systematic errors and simulate four
ideal reflectors (

For each site, three internal reflectors were handpicked (

We first checked the consistency of the picked internal reflectors and
inverted for

In a second step, we inverted for all five remaining reflector combinations
containing three and four reflectors. We also considered a range for

Summary of the WARR results from sites 1–5 in terms of range of
offsets, number of offsets (

In general, the final results are more sensitive to the respective reflector
combination than to the initial guess of

Derived data summary of all sites (Site 3 is located in an ice-shelf
channel):

Depth profiles of density derived from WARR (dashed) and OPTV
(solid). WARR data are from Sites 1 and 3, closest to the OPTV sites. Site 3
and the 2014 borehole are both in the trough of an ice-shelf channel
(Fig.

Densities were evaluated independently from the radar analyses using OPTV
logs of two boreholes drilled in 2010 and 2014 (Fig.

Figure

The estimated 1 % error on the (depth-averaged) radio-wave propagation
speed translates into large error bars for the corresponding firn-air
contents (Fig.

To assess the derived depth–density profiles with an independent data set,
we compare Site 1 and Site 3 with the OPTV densities from the 2010 and 2014
boreholes, respectively (Fig.

A difference between the new study presented here and previous ones (e.g.,

A common problem when using the Dix inversion or semblance analysis is that
the applied normal moveout (NMO) approximation presupposes small reflection
angles (to linearize trigonometric functions) and small velocity contrasts

Data collection in a WARR survey is faster than a common-midpoint survey
because only the receiver (or transmitter) needs to be repositioned. A
common-midpoint survey, on the other hand, more easily facilitates the
corrections for dipping reflectors using dip-moveout

The main advantages of the method applied here are primarily linked to a more
robust inversion, which is less sensitive to reflector delineation because
reflectors are inverted simultaneously to constrain the density profile.
First, prescribing a global depth–density/velocity function for all internal
reflectors allows the coherency of the reflector picking to be checked by
investigating different subsets of reflector combination to single out
reflectors, which were picked with the wrong phase
(Sect.

Based on our synthetic examples, we found that the traveltime inversion used
here is unstable if all parameters (surface density, densification length,
reflector depths) are inverted for simultaneously. We therefore considered
the surface density to be laterally uniform, which is not supported by
empirical data. In principle, the surface density can be estimated from the
data by picking the linear moveout of the surface wave (green dashed lines in
Fig.

The WARR data presented here were collected with a 10 MHz radar. The
disadvantage of this low frequency is that fewer reflectors above the
firn–ice transition can be picked at this low resolution, relative to
higher-frequency data sets (cf.

We found velocity models for each site which adequately fit all reflector
combinations. There is no systematic deviation larger than the picking
uncertainty and hence there is no evidence that reflectors dip within
the interval between minimum and maximum offset (

The derived depth–density functions cluster into two groups: Sites 1, 4, and
5 have a mean firn-air content of

Even though uncertainties remain about what causes the density variations, we
have shown that traveltime inversion and ray tracing with a prescribed shape
for the depth–density function can produce results, which compare closely
with densities derived from OPTV (excluding small-scale variability due to
melt layers). The data presented here show that a lateral density variability
requires attention, particularly when using mass conservation to derive basal
melt rates in ice-shelf channels. Errors in the firn-air content propagate
approximately with a factor of 10 into the hydrostatic ice thickness, which
then substantially alters the magnitude of derived basal melt rates. Using
the same parameters as

We have collected six wide-angle radar
measurements on RBIS and used traveltime inversion in conjunction with ray
tracing to infer the local depth–density profiles. In the inversion, we
prescribed a physically motivated shape for the depth–density function,
which adequately takes curved ray paths and large reflection angles into
account and allows the simultaneous inversion of multiple reflectors. We find that this method
produces robust results, even with a comparatively low-frequency
(10

This paper forms a contribution to the Belgian Research Programme on the
Antarctic (Belgian Federal Science Policy Office), project SD/SA/06A
“Constraining ice mass changes in Antarctica” (IceCon), as well as the FNRS-FRFC
(Fonds de la Recherche Scientifique) project IDyRA. We thank the InBev
Baillet Latour Antarctica Fellowship for financing the RBIS fieldwork and the
International Polar Foundation for providing all required logistics in the
field. We thank in particular A. Hubert, K. Moerman, K. Soete, and L. Favier
for support in the field. Morgane Philippe is partially funded through a
grant from the “Fonds David et Alice Van Buuren”. The comments of two
reviewers and the editor R. Bingham have improved the initial version of this
manuscript. A version of the code is accessible on request and on GitHub
(