Introduction
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 kgm-3 at depths typically ranging from 30 to 120 m
in polar regions Chapter 2. Densification continues
until air bubbles transform to clathrate hydrates and pure ice density is
reached (ρi≈ 917 kgm-3). The precise
nature of this densification depends on a number of local factors that also
vary temporally , including surface density and
stratification , surface mass balance and temperature
e.g.,, as well as dynamic recrystallization and the
strain regime. Recent studies also highlight the role of microstructure
and impurities
.
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
; (ii) to determine the depth and the cumulative mass above
radar reflectors in order to map surface mass balance with radar
; (iii) to interpret the seasonality of
surface elevation changes in terms of
surface mass balance, firn compaction, and dynamic thinning
e.g.,; and (iv) to infer ice-shelf thickness for mass
balance estimates from hydrostatic
equilibrium .
Density profiles are most reliably retrieved from ice/firn cores either by
measuring discrete samples gravimetrically, or by using continuous dielectric
profiling or X-ray tomography
. Techniques such as gamma-, neutron-,
laser-, or optical-scattering and references therein
circumnavigate the labor-intensive retrieval of an ice core and only require
a borehole, which can rapidly be drilled using hot water.
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 , semblance
analysis e.g.,, interferometry
, or traveltime inversion based on ray tracing
.
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 common-midpoint surveys, e.g..
Alternatively, only the receiver can be moved (Fig. ) resulting in
what is sometimes referred to as wide-angle reflection and refraction (WARR;
p. 165) geometry. In all cases, density can be
inferred from the radar-wave speed using density–permittivity relations
e.g.,.
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
. Ice inside the channels is thinner, sometimes more than
50 % , and the surface is depressed, causing the
elongated lineations visible in satellite imagery (Fig. ). Basal
melting inside channels can be significantly larger ,
correspondingly influencing ice-shelf stability .
Adjustment towards hydrostatic equilibrium resulting from basal melting can
weaken ice shelves through crevasse formation .
Channelized melting, on the other hand, can also prevent excessive area-wide
basal melting and hence stabilize ice shelves .
The basal mass balance inside ice-shelf channels can be mapped from remote
sensing assuming mass conservation e.g.,. This
approach calculates ice thickness from hydrostatic equilibrium which
engenders potentially two pitfalls. (i) Bridging stresses can prevent full
relaxation to hydrostatic equilibrium , and (ii) it may not
account for small-scale variations in material density. Evidence for
small-scale changes in density was suggested by and
, who found that the surface mass balance can be
elevated locally within the concave surface associated with ice-shelf
channels, which in turn may impact the local densification processes.
Atmospheric models typically operate with a horizontal gridding coarser than
5 km and cannot resolve such small-scale
variations in surface mass balance and density.
Herein, we calculate densities from WARR sites using traveltime inversion and
ray tracing (Sect. ). The data set is supplemented with
densities based on optical televiewing (OPTV) of two boreholes
(Fig. ; Sect. ). In Sects. and
, we compare both methods and discuss density anomalies
associated with the ice-shelf channels. We present our conclusions about the
derivation of density from radar in general, and the density anomalies in
ice-shelf channels in particular in Sect. , and discuss
consequences of our findings for estimating basal melt rates in ice-shelf
channels.
(a) Plain view of the wide-angle acquisition geometry:
transmitting (Tx) and receiving (Rx) antennas were aligned in parallel. While
the transmitter remained at a fixed location, the receiver was incrementally
moved farther away. A sketch of the corresponding ray paths is shown
in (b) with a synthetic velocity–depth function color coded. The
labels of example rays and their incidence angles are presented in
Eqs. ()–().
Development of a new algorithm to infer density from wide-angle radar
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. ). Using a coordinate system, where x is parallel to the
surface and z points vertically downwards, the ray paths are determined by
the spatially variable radio-wave propagation speed v(x,z) which is
primarily determined by density; unless v(x,z) is constant, ray paths are
not straight but bend following Fermat's principle of minimizing the
traveltime between transmitter and receiver. The geometry depicted in
Fig. is common in seismic investigations, and multiple techniques
exist for deriving the velocities from recorded traveltimes
.
Similar to what has been done for wide-angle radar measurements in Greenland
, we follow a variation of the approach delineated by
. measured common midpoint returns with a
100 MHz radar. They used a ray tracing forward model and inferred bulk
densities of individual intervals (hereafter interval densities) by inverting
reflector depths and interval velocities for single reflectors from top to
bottom (a.k.a. layer stripping). In this paper, we use a 10 MHz radar
providing improved depth penetration at the expense of lower spatial
resolution. In order to prevent small errors in interval densities and
velocities associated with shallow reflectors from being handed downwards, we
refine the method by parameterizing a monotonic depth–density function, and
by inverting simultaneously for a set of parameters specifying the density
and all reflector depths, described below.
Experimental setup
The radar consists of resistively loaded dipole antennas (10 MHz)
linked to a 4 kV pulser (Kentech) for transmitting, and to a digitizing
oscilloscope (National Instruments, USB-5133) for receiving
. Figure illustrates the acquisition geometry
in which the transmitter remained at a fixed location and the receiver was
moved incrementally farther away at 2 m intervals. The axis between
transmitter and receiver at Sites 1, 2, 4, 5, and 6 (locations,
Fig. ), was aligned across-flow (all antennas are parallel to the
flow) because we expect the ice thickness to vary little in the across-flow
direction and therefore internal reflectors are less likely to dip. For the
same reason, Site 3, which is located inside an ice-shelf channel, was aligned
parallel to the channel because in this particular area ice thickness varies
mostly in the across-flow direction. The transmitter–receiver distance was
determined with measuring tape, and recording was triggered by the direct air
wave. The latter is not ideal, and can be improved by using fiber-optic
cables. Processing of the radar data included horizontal alignment of the
first arrivals (a.k.a. t0 correction), dewow filtering, Ormsby bandpass
filtering, and the application of a depth-variable gain. Because triggering
was done with the direct air wave, a static time shift was added to each
trace to account for the delayed arrival of the air wave for increasing
offsets.
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. for locations
of Sites 1–6. Site 6 was excluded from further analysis because the basal
reflection is ambiguous (probably due to off-angle reflectors in the
vicinity).
In multi-offset surveys, the traveltime of internal reflectors increases
hyperbolically with increasing offset e.g.,, while the
surface wave (traveling in the firn column directly from transmitter to
receiver) has a linear moveout. The maximum amplitude of the basal reflector
was detected automatically and shifted with a constant offset to the first
break. Internal reflectors were handpicked. Figure shows
radargrams collected at all sites with the picked reflectors that were used
for the analysis. The maximum offset for each site was chosen to equal
approximately the local ice thickness. At Site 6, basal and internal
reflectors are overlaid with signals from off-angle reflectors and cannot be
picked unambiguously. We present the data here to exemplify a case for which
WARR does not yield reliable results and exclude this site from further
analysis.
Model parameterization and linearization
The traveltime tNr,No of a ray reflected from a
reflector Nr (r∈[1,R]) at depth Dr measured at
offset No (o∈[1,0]) is given by a line integral over the
inverse of the radio-wave velocity v along the ray path L (extending from the transmitter to the receiver
via the reflection boundary).
tNr,No=∫Lmv,Dr1vmvdl.
Figure illustrates the notation. For each site, we pick a number
of reflectors at different depths mD=D1,…,DRT, and we parameterize the velocity function as a
function of density using the model parameters mv. We use
an inverse method to reconstruct both the reflector depths and the velocity
profile from the measured traveltimes.
The traveltime is a non-linear function of the model parameters (and hence
the inversion results may be non-unique) because L depends both on the
initially unknown radio-wave propagation speed and the reflector depth. The
velocity between two radar reflectors is often represented as piecewise
constant or piecewise linear , making the model parameters
mv either the interval velocities or the interval velocity
gradients, respectively. Here, we introduce additional constraints from
who fit a depth profile of density of the form:
ρ=910-Ae-rz
to density measurements of the borehole recovered at RBIS in 2010. The
parameters A and r are tuning parameters for the surface density and the
densification length, respectively. We relate density to the radio-wave
propagation speed v using the complex refractive index method (CRIM) equation
:
ρ=cv-1-1cvi-1-1ρi,
where vi=168 mµs-1 is the radio-wave
propagation speed in pure ice and c is the speed of light in a vacuum.
Combining Eqs. () and () leads to
v(A,r)=ckρ(A,r)+1,
with k=1ρicvi-1 and
mv=(A,r)T. We use Eq. () and assume that
(i) radio-wave propagation speed v depends only on density (i.e., excluding
ice anisotropy); (ii) density is horizontally homogeneous over the maximum
lateral offset of the receiver (≤ 404 m) but varies with depth
so that v only varies with depth in that interval; and (iii) within this
interval, internal reflectors are horizontal. We aim to detect lateral
variations of the velocity profiles on larger scales (i.e., between Sites 1
and 5) by finding optimal sets of parameters m=mD,mv=A,r,D1,…,DRT∈RNm describing
the data at each site. The number of model parameters Nm=R+2
depends on the number of reflectors.
Using Eq. () and approximating the integral through a summation over
Nz depth intervals, Eq. () reads
tNr,Nom≈1c∑i=1Nzlzimkρmv+1.
The problem is linearized using an initial guess (marked with superscript 0)
and a first-order Taylor expansion:
tNr,Nom≈tNr,No0+∑j=1Nm∂tNr,No∂mj|mj0mj-mj0.
An equation of type () holds for all O offsets of all R
reflectors and can be summarized in matrix notation:
ε=SΔm,
where we define ε=tmod-tobs∈RNp as a vector composed of the
residuals between the observed (tobs) and the modeled
(tmod) traveltimes. Np is the total number of
picked datapoints for all reflectors (not all reflectors can be picked to the
maximum offset O), S∈RNp×Nm is a matrix containing all partial derivatives, and
Δm∈RNm is the model update vector.
One synthesized reflector is composed of more than 50 independent
measurements and at each site R = 4 reflectors (including the basal
reflector) were picked. There are therefore six model parameters
(Nm=4+2 for four reflector depths and two parameters A and
r describing the depth–density function) and the number of measurements
(Np) is typically larger than 200, turning Eq. ()
into an overdetermined system of equations.
The derivatives of Eq. () with respect to A and r are
∂tNr,No∂A=-kc∑i=1Nzlzie-rzi∂tNr,No∂r=Akc∑i=1Nzzilzie-rzi,
and ∂tNr,No∂Dn
(n∈[1,R]) follows from geometric considerations :
∂tNr,No∂Dn=2cosΘNr,Nov(Dn)δnr,
where ΘDn,No is the incidence angle of ray
No at the reflector boundary Nr=n
(Fig. b); δnr=1 for r=n and 0 otherwise.
An optimal set of model parameters m is found as follows.
(i) Starting with an initial estimate for the reflector depths
mD0 and the velocity model mv0, a ray tracing forward
model (Sect. ) calculates the expected traveltimes
tNr,No0 for a given set of transmitter–receiver
offsets; the difference between modeled and observed traveltimes results in
the misfit vector ε in Eq. ().
(ii) The overdetermined system is inverted for the unknown
parameter-correction vector Δm (Sect. ), and
(iii) the parameter set is updated with m1=m0+Δm and serves as new input for the forward model. These steps are repeated
iteratively until the parameter updates are negligible.
Ray tracing forward model
We apply the ray tracing model provided by only to
reflected (and not to refracted) rays. For a given set of reflectors in a
v(z) medium, no analytical solution exists which directly provides a
ray path from the transmitter to a given offset via a reflection boundary. The
problem is solved iteratively by calculating fans of rays with varying
take-off angles until one ray endpoint emerges within a given minimum
distance (≤ 0.5 m) to the receiver. For some v(z)
configurations no such ray can be found, indicating that the prescribed
v(z) medium does not adequately reproduce the observations.
Inversion
To solve the inverse problem we seek the set of parameters m that
minimizes the cost function J:
J=12εTCt-1ε+12λm-m0TCm-1m-m0,
in which the first term is the ℓ2 norm of the traveltime residual
vector weighted with Ct=diag{σi2}, where σi is the
uncertainty of the traveltime picks. The second term is a regularization
(weighted with Cm=diag{σj2}, where
σj is the estimated uncertainty of the model parameters) penalizing
solutions which are far from the initial guess. Regularization with the
Lagrange multiplier λ is needed because outliers in the data are
weighted disproportionally in a least-squares sense, which can lead to
overfitting the data.
We minimize J by updating m iteratively according to the
Gauss–Newton method:
mi+1=mi-STCt-1S+λCm-1-1∇J,
with ∇J=Ct-1Sε+λCm-1m-m0. High values of λ result in a
final model vector remaining close to the initial guess; lower values of
λ allow for larger changes in the parameter updates. We stop
iterating when changes in J are below an arbitrarily small threshold.
Sensitivity of the firn-air content
In order to compare different measurements at different locations, we
decompose the ice shelf into two layers of ice (Hi) and air
(HA) so that ρ¯H=ρiHi+ρaHA and Hi+Ha=H (i.e.,
HA=ρ¯-ρiρa-ρiH). The firn-air content HA (with air density
ρa) is a quantity independent of the local ice thickness (as
long as the depth-averaged radio-wave speed is determined below the firn–ice
transition) and changes thereof indicate changes in the depth-averaged
density due to a changing firn-layer thickness. The firn-air content in
Antarctica can vary from HA= 0 m in blue ice areas up to
HA= 45 m for cold firn on the Antarctic plateau
. Using the CRIM equation to determine HA
results in
HA=cHρi1v¯-1viρa-ρicvi-1.
We consider errors in HA from uncertainties in the depth-averaged
radio-wave propagation speed (v¯), and uncertainties in ice thickness
(H):
δHA2≈cρiv2ρa-ρicvi-1Hδv¯2+cρi1v¯-1viρa-ρicvi-1δH2.
Assuming δv¯≈ 1 %, and δH≈ 10 %
renders the first term of Eq. () about 8 times larger than
the second for the parameter ranges considered here, and we therefore neglect
errors in ice thickness for the error propagation. Equation ()
shows that the uncertainty of HA scales with the local ice
thickness so that small errors in the depth-averaged velocities
(< 1 %) result in significant errors in terms of HA. We
use HA as a sensitive metric, both for comparing sites laterally
and illustrating uncertainties of the radar method. In the following, we use
synthetic data to choose optimal parameters for the inversion, and to
investigate how errors in the data propagate into the final depth–density
estimates.
Testing with synthetic examples
To test the inversion algorithm we use ray tracing with a prescribed
depth–density function and recording geometry (A=460 kgm-3,
r=0.033 m-1; transmitter–receiver offsets between 30 and
300 m with 2 m spacing) to create a synthetic traveltime data set
with multiple reflectors. We first investigate whether the solution is well
constrained for ideal cases, and then we discuss effects of systematic and
random errors in the data.
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
±5 m from the ideal depths and depth–density functions defined
by r=0.01-0.1 m-1 (A was fixed). This density range
corresponds to firn-air contents from HA=5 to 50 m. The
root-mean-square differences (Δtrms) between the perturbed
and the ideal reflector are equivalent to the first term of the objective
function J (Eq. ) and indicate how well constrained the solution
is. Figure a illustrates that for a single reflector the solution
is not well constrained, meaning that different sets of model parameters give
similar results to the ideal solution (i.e., dense firn/shallower reflector or
less dense firn/deeper reflector). For example, positioning the reflector at
392 m depth with r=0.063 m-1 results in a firn-air content of
∼ 11 m, whereas positioning the reflector at 410 m depth with
r= 0.014 m-1 corresponds to a firn-air content of approximately
40 m. Both cases have a small model–data discrepancy and are barely
distinguishable from the ideal solution. Using two reflectors simultaneously
better constrains the solution, particularly if the shallower reflector is
above the firn–ice transition (Fig. b). We conclude from these
simple test cases that using the basal reflector alone is inadequate.
Instead, multiple reflectors should be considered and inverted for
simultaneously. Using this type of testing, we also find (i) that treating
A as a free parameter introduces significant tradeoffs with r even for
small noise levels. We therefore keep A fixed and assume in the following
that the surface density is laterally uniform; (ii) plotting both terms of
the objective function J (Eq. ) versus each other for different
λ (a.k.a. L-curve) helps to choose an optimal λ. We find that
λ≈ 0.1 marks approximately the kink point between too large a
model–data discrepancy on the one hand and overfitting on the other hand. We
keep λ=0.1 from hereon to prevent overfitting, but note that results
are largely independent of λ for λ≪ 0.1.
Traveltime residuals (Δtrms) calculated with
ray tracing between ideal reflector in a fixed depth–density profile
(A= 460 kgm-3; r= 0.033 m-1) with reflectors
perturbed in terms of depths and density. Ideal solutions are marked with red
crosses: (a) traveltime residuals for an ideal reflector at 400 m
depth; (b) volumetric slice plot of traveltime residuals for two
idealized reflectors at 30 and 400 m depth.
Example for initial (a) and final (b) fit between
the ray tracing forward model and the reflectors at Site 2. In this case,
three reflectors (black dots) were used for the inversion and one reflector
was kept for control. The forward model corresponds to the red dashed curves
and the control reflectors to the blue dashed curves. Initial estimates shown
here were r0= 0.05 m-1, D1= 68.2 m,
D3= 112.9 m, D4= 291.2 m; the best fit resulted in
r= 0.027 m-1, D1= 67.7 m, D3= 111.2 m,
and D4= 293.3 m. The traveltime residual between the model and data
for initial (x) and final fit (o) is shown in (c).
Next, we consider effects of random and systematic errors and simulate four
ideal reflectors (D1= 100 m, D2= 150 m, D4= 200 m, D4= 400 m) to which we add normally
distributed noise (i.e., simulating picking errors and variability in
aligning the direct waves used for triggering) and linear trends (i.e.,
simulating accumulated errors in positioning, unaccounted reflector dipping,
etc.). We then test the
robustness of the inversion for different initial guesses, and different
magnitudes of noise and systematic errors. We find that the limiting factor
for the initial depth guess is the forward model which does not find ray
paths for all offsets if the initial guess is further than
∼ 15 m from the true solution. For all initial guesses
deviating less than that, the inversion recovers the true depths robustly
within decimeters, even for noise levels with a mean amplitude of 5 times the
sampling interval (0.01 µs). However, the inversion is most
sensitive to trends in the data. For example, if reflectors deviate
systematically from 0.04 to -0.04 µs for large offsets,
reflector depths are reconstructed with an error of 2–3 m. The
corresponding densities deviate in terms of firn-air content more than
5 m from the ideal solutions. We conclude from these test cases that
reflectors need to be picked accurately (i.e keeping the same phase within
the individual wavelets); if systematic differences between the forward model
and data occur (e.g., the modeled reflector is tilted with respect to the
observations), then results should be interpreted with care.
Inversion of field data
For each site, three internal reflectors were handpicked (D1–D3) to
complement the automatically detected basal reflector (D4,
Fig. ). Initial guesses for reflector depths are based on standard
linear regression in the traveltime2–offset2 diagrams ;
r0= 0.033 m-1 and A= 460 kgm-3 stem from the
2010 OPTV density profile .
We first checked the consistency of the picked internal reflectors and
inverted for r and the depths of one internal reflector together with the
basal reflector. The remaining two internal reflectors were not used for the
inversion, but to validate the results. We did this for all three
combinations (D1–D4, D2–D4, D3–D4) in order to check
whether internal reflectors had been picked with the correct phase. Results
were considered consistent if the model–data discrepancy for each reflector
was within ±0.02 µs (cf. radar sampling interval is
0.01 µs). Picking a wrong phase typically causes inconsistent
results for one of the combinations. In such a case the corresponding
reflector was repicked.
In a second step, we inverted for all five remaining reflector combinations
containing three and four reflectors. We also considered a range for r0
between 0.021 and 0.056 m-1, corresponding to a firn-air content of
24 and 9 m, respectively. Figure illustrates an example
where three reflectors were used for the inversion and one was left for
validation; the model–data discrepancy is large for the initial guess. After
the inversion, the model–data discrepancy is smaller for all reflectors
including the reflector that was used for control only.
Summary of the WARR results from sites 1–5 in terms of range of
offsets, number of offsets (O), ice thickness (H), depth-averaged density
(ρ‾), depth-averaged radio-wave propagation speed
(v‾), firn-air content (HA), the decay length (r)
parameterizing the depth–density function, and the deviation from
hydrostatic equilibrium (ΔH). The ranges correspond to the lower and
upper limits of five reflector combinations at each site (four reflector
combinations contain three reflectors, and one combination contains all four
reflectors).
No.
Offset
O
H
ρ‾
v‾
HA
r
ΔH
(m)
(m)
(kgm-3)
(mµs-1)
(m)
(m-1)
(m)
1
26–308
141
280.2–281.3
847–855
173.0–173.8
16.8–19.3
0.026–0.030
15
2
30–318
144
266.1–266.6
864–867
171.9–172.2
12.4–13.2
0.039–0.041
19
3
20–222
101
156.7–157.0
828–832
175.2–175.5
13.3–14.0
0.036–0.038
-4
4
25–366
170
292.9–293.4
850–859
172.6–173.4
16.1–19.0
0.027–0.032
5
5
20–404
142
395.0–396.1
872–874
171.2–171.5
15.2–16.4
0.031–0.036
-13
In general, the final results are more sensitive to the respective reflector
combination than to the initial guess of r0. For the latter we chose the
one resulting in the smallest model–data discrepancy
(r0= 0.033 m-1). Differences between the final five parameter
sets give a lower boundary for an error estimate.
Derived data summary of all sites (Site 3 is located in an ice-shelf
channel): (a) depth–density profiles inverted from four reflectors,
(b) ice thickness, (c) depth-averaged density, and
(d) firn-air content. Black crosses in (b–d) represent the
outcomes for five combinations containing three or more reflectors. Error
bars assume a 1 % error in depth-averaged radio-wave propagation speed.
The blue crosses correspond to depth-averaged solutions using normal moveout
of the basal reflector only .
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. ). The envelopes of the radar-derived densities correspond to
the lower and upper limit of five reflector combinations used for the
inversion. The OPTV logs were smoothed with a 0.5 m running
mean.
Results
Figure and Table summarize the derived depth–density
functions, ice thicknesses, radio-wave propagation speeds, depth-averaged
densities, and the firn-air contents of the five WARR sites. The reconstructed
thicknesses vary between 157 and 396 m (86 % percentage
difference), the depth-averaged densities vary between 828 and
874 kgm-3 (∼ 5 % percentage difference), and
corresponding firn-air contents vary from 13.2 to 19.3 m (38 %
percentage difference). For the five different reflector combinations at each
site, the inverted ice thicknesses differ by less than 1.5 m
(< 1 % percentage difference), the inverted depth-averaged densities
differ by less than 10 kgm-3 (< 1 % percentage
difference), and the final firn-air contents differ by less than 3 m
(< 17 % percentage difference; Fig. b–d). This indicates
that the results are numerically robust to the combination of reflectors
used, and that the local ice thickness and depth-averaged density can be
determined with high confidence. However, we cannot derive rigorous error
estimates from the inversion itself. We found that picking the internal
reflectors is the most sensitive step and, similar to , we
estimate that the depth-averaged velocity can be determined within
±1 %. We used this value to calculate errors for the depth-averaged
densities and the equivalent firn-air content. These errors roughly take into
account the assumptions of non-dipping reflectors, ice isotropy, and
uncertainties of the density–permittivity model.
The estimated 1 % error on the (depth-averaged) radio-wave propagation
speed translates into large error bars for the corresponding firn-air
contents (Fig. d), impeding the comparison between sites.
Nevertheless, Sites 2 and 3 show lower firn-air contents
(∼ 13 m) than the other sites (∼ 17 m).
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. ). Site 3 is located inside an
ice-shelf channel, about 10 km north of the 2014 borehole located in the
same channel. Site 1 is about 6 km south of the 2010 borehole
(Fig. ). Both radar WARR measurements and the OPTV logs show a
depth–density profile that is denser inside than outside the ice-shelf
channel. This increases our confidence that the WARR method developed here
indeed picks up significant differences in firn-air content on small spatial
scales.
Discussion
Benefits of traveltime inversion using ray tracing
A difference between the new study presented here and previous ones (e.g.,
) is how the radio-wave propagation speed is
parameterized. Previous studies used piecewise linear or uniform speed
between individual reflectors, while we parameterize the speed as a
continuous function of depth (Eq. ). Here, we examine the benefit of
this approach for interpreting the radar results
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
. In our case reflection angles can be large
(< 45∘), particularly near the maximum offsets; contrary to NMO,
ray tracing is not adversely influenced by wide incidence angles. NMO
presupposes small velocity contrasts, because ray paths are approximated as
oblique lines neglecting raybending from a gradually changing background
medium. Traveltime inversion with ray tracing equally relies on this
approximation as long as interval velocities are assumed. In this study, we
prescribe a realistic shape of a depth–density/velocity function, which
changes gradually with depth, and raybending is taken into account adequately
during the ray tracing. We have tested both the small angle and the small
velocity contrast limitations quantitatively by using the OPTV-based
depth–density/velocity function and ray tracing in order to simulate
synthetic traveltimes of reflectors at various depths (50–500 m) and
horizontal offsets (50–500 m). We then used the synthetic
traveltimes for calculating the reflector depths and the depth-averaged
velocities (averaged from the surface to the reflector depths) subject to the
NMO equations. Differences in depth-averaged velocities were smaller than
0.5 %, and differences in reflector depths were smaller than
0.5 m. Similar to the findings of , this confirms
that in our case the NMO approximation essentially holds, even for
comparatively large horizontal offsets and a continuously changing
depth–velocity function. This must not always be the case and ray tracing
easily allows the NMO approximation to be checked for each specific setting.
For the examples considered here, solutions based on the Dix inversion, using
only the basal reflector, typically result in thicker ice and higher
depth-averaged densities (and correspondingly lower firn-air contents,
Fig. c–d).
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
choice for the acquisition geometry thus depends on the time available in the
field and on the glaciological setting (i.e., whether dipping reflectors are
to be expected). Traveltime inversion can cope with both types of acquisition
geometries. If reflector dips are important, the routine presented here can
be adapted to include one dip angle per reflector in the inversion. However,
given that including the surface density as an additional free parameter is
difficult if all parameters are inverted simultaneously, an iterative
approach may be required to find one depth–density function for all
reflectors while solving for the reflector dips individually (layer
stripping; ).
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. ). This step is important, particularly when
using lower frequencies as was the case here (10 MHz). At this stage
the basal reflector is useful, because it can be identified unambiguously.
Once more than two shallow internal reflectors are reliably picked, we found
that the inversion results were largely independent of the inclusion of the
basal reflector. Second, by inverting for reflectors simultaneously, it is
less likely that deeper reflectors inherit uncertainties from shallower
reflectors. This can happen when solving for reflectors individually where
tradeoffs between interval velocities and reflector depths are subsequently
handed downwards. Third, when using interval velocities, the parameter set
describing the depth–density/velocity function is larger than is the case
here. For example, for four reflectors eight parameters are required when
using interval velocities (four velocities and reflector depths,
respectively), and this compares with only the five parameters that we
required for the method applied here (r and four reflector depths). Simpler
models with fewer model parameters are preferable when using inversion.
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. , cf. ). However, in our 10 MHz data set
the surface wave cannot be identified unambiguously, resulting in a large
range of possible surface densities. We addressed this point with a
sensitivity analysis including a range of surface densities (300≤A≤ 500 kgm-3). The smallest model–data discrepancies are
found with A≈ 400 kgm-3, but in all cases the final
results do not deviate more than the error bars provided in Fig. .
This means that the ill-constrained surface density is essentially corrected
for during the inversion by adapting the densification length.
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. who derived an 8 %
velocity error with a 25 MHz radar versus a 2 % error with 200 MHz
radar). We found that the method applied here can cope with the picking
uncertainties at 10 MHz, whereas using Dix inversion frequently
resulted in interval densities much larger than the pure ice density. The
advantage of using a 10 MHz radar is that the entire ice column is
illuminated, including the unambiguous basal reflector. This opens up the
possibility for more sophisticated radar-wave velocity models including ice
anisotropy originating from aligned crystal orientation fabric below the
firn–ice transition . The radar data set is
also suited for other glaciological applications, for example, using the
basal reflections for deriving ice temperature (via radar attenuation rates)
from an amplitude versus offset analysis and
constraining the alignment of ice crystals using multistatic radar as a
large-scale Rigsby stage .
Radar- and OPTV-inferred densities
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 (≤ 404 m). The
results are numerically robust for different reflector combinations,
indicating equal validity for all results based on three reflectors or more
(Sect. ).
The derived depth–density functions cluster into two groups: Sites 1, 4, and
5 have a mean firn-air content of ∼ 17 m, whereas Sites 2 and 3
have lower values of ∼ 13 m. While these differences are minor
from a radar point of view, they are quite significant from an
atmospheric-modeling point of view. For example, propose
that the firn-air content around the entire Antarctic grounding line is bound
between 13 m (for the Dronning Maud Land area) and 19 m (for
ice shelves in West Antarctica). Including transient effects, such as surface
melt, the variability increases but typically stays within 5–20 m
. Because the aforementioned models run on 27 km grids
(approximately the size of our research area) they may overlook effects
acting on smaller scales. However, with the estimated uncertainty of the
depth-averaged wave speed (±1 %) the radar-derived variability in
firn-air content is barely significant (Fig. d); notwithstanding,
we find that Site 1 (which is closest to the 2010 borehole) agrees closely
with the OPTV of 2010, and a similarly good fit is found between Site 3 and
the 2014 OPTV (both located inside the same ice-shelf channel,
Fig. ). The implications are twofold: first, the correspondence
between the OPTV-derived density variations and those derived from the WARR
method provides independent validation of the latter technique. Second, the
fact that both techniques show increased density within the surface channel
indicates that the effect is real and should be accounted for by
investigations based on hydrostatic equilibrium. However, given that Site 2
also shows a comparatively low firn-air content, we cannot unambiguously
conclude from the data alone that firn density is elevated in ice-shelf
channels in general. One potential mechanism for such a behavior is the
collection of meltwater in the channel's surface depressions. At RBIS,
surface melt can be abundant in the (austral) summer months, particularly in
an about 20 km wide blue ice belt near the grounding line. The most recent
Belgian Antarctic Research Expedition (January 2016) observed frequent melt
ponding and refreezing in this area, mostly in the vicinity of ice-shelf
channels where meltwater preferentially collects in the small-scale surface
depressions. If this holds true, the increased density observed in the WARR
data close to the ice-shelf front is an inherited feature from farther
upstream. The channel's surface depressions likely also cause a locally
increased surface mass balance , and in general ice-shelf
channels can have a particular strain regime . Both of these
factors may also influence the firn densification rate, but given our limited
data coverage we refrain from an in-depth analysis here. More work is
required to determine whether firn in ice-shelf channels is systematically
denser.
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 compare the WARR-derived ice
thickness with the hydrostatic ice thickness for each site. We find a maximum
deviation of 19 m for Site 2, and a minimum deviation of 4 m
for Site 3 (Table ). Assuming the absence of marine ice, those
deviations are comparatively small given the uncertainties of the geoid and
the mean dynamic topography, both of which are required parameters for the
hydrostatic inversion.