Introduction
The widespread availability of precise Global Positioning
System (GPS) measurements has revolutionized the study of ice dynamics and
glacier mass balance (e.g., Gao and Liu, 2001). Continuously operating
dual-frequency GPS receivers provide high-frequency (1 Hz or less), highly
accurate (< 1–3 cm) measurements of position, which can be used to
derive surface velocity and elevation change. For applications involving ice
dynamics, these measurements offer important constraints for the mass
continuity equation, which equates surface elevation change with ice flux
divergence, surface mass balance (SMB), and basal mass balance (BMB). Here,
we explore a methodology to constrain each of these components directly from
GPS observables.
SMB processes include precipitation, sublimation, wind redistribution of
surface snow, and meltwater runoff. Regional climate models forced by
reanalysis output now provide daily estimates of Antarctic SMB on a
relatively coarse grid (∼ 5.5 to 27 km). In situ SMB measurements
are, however, still essential for model calibration and validation.
Traditionally, SMB is measured using stake networks, automated weather
stations (AWSs), near-surface radar surveys, and firn/ice cores, all of which
require substantial field operations in remote locations. These measurements
also tend to bias model calibration towards accessible locations, and recent
studies indicate that these biases can significantly affect mass balance
results, often resulting in overestimates of cumulative balance due to poor
sampling in dynamic areas (Andreassen et al., 2016).
Antarctic firn/ice core records indicate that SMB variability over most of
Antarctica during the last 800 years was statistically insignificant, but
accumulation increased more than 10 % for high-accumulation coastal
regions (e.g., the Amundsen Sea Embayment) since the 1960s
(Frezzotti et al., 2013). Historically, these areas have
been poorly sampled with traditional methods, providing limited data
available for validation of modeled SMB.
Accurate knowledge of firn compaction and its spatiotemporal variability is
essential for interpreting observed surface elevation change in remote
sensing data (e.g., satellite altimetry) and for partitioning this change
into components related to ice dynamics and SMB
(e.g.,
Shepherd et al., 2012; Wouters et al., 2015). Depth-dependent compaction
rates can be estimated from a number of different methods, including vertical
strain measurements (Arthern et al., 2010; Hamilton and Whillans, 1998),
borehole optical stratigraphy (Hawley and Waddington, 2011), repeat
phase-sensitive radio-echo sounding (pRES) measurements (e.g., Jenkins et
al., 2006) and ice-penetrating radar observations of internal layers over
time (e.g., Medley et al., 2014, 2015). In the absence of these measurements,
dynamic firn models forced by modeled SMB can provide estimates of compaction
rates throughout the firn column, which can be integrated to obtain estimates
for the contribution of firn compaction to surface elevation change over time
(e.g., Ligtenberg et al., 2011).
BMB for ice shelves (i.e., bottom melting, accretion) is driven by complex
ice–ocean interaction. State-of-the-art ice-shelf cavity ocean circulation
models offer some insight into sub-shelf ice–ocean interaction, but these
models lack validation, as in situ hydrographic observations are limited,
especially within the sub-shelf cavity and the ice–ocean boundary layer.
Some direct measurements are available from autonomous submersibles (e.g.,
Dutrieux et al., 2014) and instrumentation deployed through ice-shelf
boreholes (e.g., Stanton et al., 2013), but available data are limited to
short time periods and small spatial extents. Precise measurements of surface
elevation change from remote sensing observations (e.g., laser altimetry,
digital elevation models (DEMs)) can also be used to infer BMB (e.g.,
Dutrieux et al., 2013; Moholdt et al., 2014; Pritchard et al., 2012; Shean,
2016), but temporal resolution is limited, as time intervals between repeat
observations are typically several months to years.
Here, we use continuous GPS records from the Pine Island Glacier (PIG) ice
shelf to constrain local SMB, flux divergence, and BMB. We use changes in
observed GPS antenna elevation and reflectometry-derived surface elevation to
validate SMB and firn model output. Flux divergence is estimated from
observed strain rates between GPS stations. These estimates are then used to
isolate elevation change due to local BMB. This approach yields temporally
dense records of basal melt rates at spatially sparse GPS locations, which
are combined with high-resolution DEMs from the same time period to provide
spatial context. These complementary results for the PIG ice shelf provide
new information about the time-variable magnitude and spatial distribution of
basal melting, offering indirect observations of ice–ocean interaction and
BMB sensitivity to ocean heat content variability, with implications for
other rapidly evolving “warm-cavity” Antarctic ice shelves.
PIG background
Pine Island Glacier is one of the largest and most dynamic ice streams in
West Antarctica. Since the 1970s, PIG has experienced ∼ 30 km of
grounding-line retreat along its centerline (Rignot et al., 2014)
(∼ 8 km average retreat across full width of fast-flowing trunk;
Joughin et al., 2016), a ∼ 75 % increase in surface velocity
(Mouginot et al., 2014), and > 100 m of thinning (Bindschadler, 2002;
Pritchard et al., 2009), with accelerated retreat beginning in the 1990s.
These changes have been attributed to some combination of geometric
instability (i.e., marine ice-sheet instability) and external forcing (i.e.,
increased ocean heat content and/or changes in sub-shelf circulation) (Jacobs
et al., 2011; Joughin et al., 2010).
Present-day surface velocities are ∼ 4 km yr-1, with annual
discharge of ∼ 130–135 Gt (Medley et al., 2014; Mouginot et al.,
2014) and net mass loss estimates of 40 to 50 Gt yr-1 for the full PIG
catchment (Medley et al., 2014; Rignot, 2008). This mass loss is responsible
for ∼ 0.11 mm yr-1 global sea level rise (SLR), or approximately
40–45 % of the total ∼ 0.26 mm yr-1 Antarctic SLR
contribution (Church et al., 2013; Rietbroek et al., 2016; Shepherd et al.,
2012).
Context for Pine Island Glacier ice shelf with 2006–2016 median
surface velocity (Christianson et al., 2016; Joughin et al., 2010) over a
shaded relief map from October–December 2012 DEM mosaic. Black lines show
∼ 2-year paths between initial (green) and final (red) GPS station
locations. The yellow dot shows the Evans Knoll automated weather
station (AWS) and the blue squares show
RACMO grid-cell centers used during analysis. The purple triangles beyond
shelf front show locations of ocean mooring temperature data from
Christianson et al. (2016). The white line shows approximate 2011 grounding
line (Shean, 2016). Black rectangle shows location of Fig. 2a.
Figure 1 shows the fast-flowing portion of the PIG ice stream, which
terminates in an ice shelf (“main shelf”) that is ∼ 25 km wide,
∼ 100 km long, and ∼ 1–1.5 km thick across the grounding line.
Basal melting accounts for ∼ 60–75 % of mass loss from the ice
shelf, with estimated 2003–2008 melt rates of ∼ 95–101 Gt yr-1
(Depoorter et al., 2013; Rignot et al., 2013) and 2008–2015 melt rates of
∼ 80–90 Gt yr-1 (Shean, 2016).
The main shelf has complex surface topography, including kilometer-scale
surface ridges and troughs that correspond to basal keels and channels,
respectively (Bindschadler et al., 2011; Vaughan et al., 2012). A series of
longitudinal (along-flow) ridges and troughs are present along the shelf
centerline, with transverse (across-flow) ridges and troughs along the
lateral margins (Fig. 1). Local basal melt rates vary considerably across
these features (Dutrieux et al., 2013; Shean, 2016).
Hydrographic observations seaward of the PIG calving front in Pine Island Bay
suggest that basal melting declined by ∼ 50 % between 2010 and 2012
(Dutrieux et al., 2014). Long-term 2009–2015 mooring records seaward of the
southern calving front (Fig. 1) show a significant decrease in ocean
temperature (∼ 1–1.5 ∘C) over ∼ 450–770 m depths from
late 2011 to early 2012 and then again from mid-2012 to early 2013
(Christianson et al., 2016; Webber et al., 2017). These observations show
that the ocean heat content at the PIG ice-shelf front varies considerably
over monthly to interannual timescales.
PIG GPS sites
Several long-term GPS stations were installed on the PIG shelf as part of a
larger investigation of ice-sheet, ice-shelf, and ocean dynamics
(Bindschadler et al., 2011; Stanton et al.,
2013). During the early part of this effort, two GPS stations continuously
collected data from January 2008 to January 2010: one on the southern PIG
ice shelf (PIG2) and another on the fast-flowing, grounded ice upstream of
the grounding line (PIG1) (Fig. 1). In addition,
a ∼ 2×2 km array of five stations (SOW1–4, BOAR,
Fig. 2) was installed ∼ 50 km
downstream of the grounding line, near the center of the main shelf from
January 2012 to late December 2013.
The stations used dual-frequency Trimble NetRS GPS receivers (2008–2010
sites) and NetR9 receivers (2012–2014 sites), with Trimble Zephyr Geodetic
2 antennas mounted on 12 foot (3.66 m) poles with insulating pole-base
stoppers. The poles were driven into the snow by hand, with initial pole
bases set ∼ 0.5–1.0 m beneath the surface (M. Truffer,
personal communication, 2016).
(a) WorldView DEM from 11 November 2012 with 2012–2014 GPS
array positions overlaid. Note GPS positions relative to transverse
depressions and location of R1 rift associated with 2015 calving event (black
arrow). Ice-flow direction indicated by white arrow. White lines show
locations of profiles. (b) Smoothed surface elevation (0.5 km
window, approx. ∼ 1 ice thickness) and estimated freeboard thickness
for longitudinal profile X-X′ and (c) transverse profile Y-Y′.
Profile intersection is near BOAR (red point). Vertical exaggeration is
22×.
High-resolution optical imagery and DEM data (see Sect. 2.5) over the
2012–2014 sites show that SOW1, BOAR, and SOW3 were installed with
along-flow orientation in a longitudinal surface trough (Fig. 2) that
overlies a longitudinal basal channel. An ice-penetrating radar profile with
transverse orientation was collected upstream of the GPS array, providing ice
thickness estimates of ∼ 450–460 m near the apex of a longitudinal
channel and ∼ 540 m over adjacent keels
(Stanton et al., 2013).
Figure 2 shows estimated ice thickness for
longitudinal and transverse profiles across the GPS array.
A borehole was drilled through the ice shelf approximately 1.34 km upstream
of SOW1 (K. Riverman, personal communication, 2016), and an instrument
package with an upward-facing ice-bottom altimeter (acoustic ranger) was
deployed beneath the shelf from January to February 2012. Measurements from
this bottom altimeter and complementary pRES experiments provided basal melt
rate estimates of ∼ 14–25 m yr-1 within the longitudinal
channel (Christianson et al., 2016; Stanton et al., 2013).
The 2012–2014 GPS array was located near several transverse surface
depressions (Fig. 2), which are likely associated with transverse basal
channels and/or rifts. Local surface slopes were ∼ 0.6–0.9∘
within the largest of these depressions, immediately downstream of SOW3 and
SOW4. A notable linear surface depression located approximately 1 km
upstream of SOW1 (black arrow in Fig. 2) opened as a rift in ∼ 2014 (R1
in Jeong et al., 2016) and was subsequently the site of a large iceberg
calving event that occurred around July 2015. The placement of the 2012–2014
GPS array near these features complicates interpretation of GPS records but
also provides new constraints on the spatiotemporal evolution of strain rates
and rift formation for the PIG shelf.
Data and methods
GPS antenna position
As described in Christianson et al. (2016), GPS data were processed using
differential-carrier-phase positioning relative to bedrock GPS sites –
Backer Island (BACK; -74.26∘ N, -102.28∘ E;
∼ 60 km baseline) for 2012–2014 records and Howard Nunatak (HOWN;
-77.31∘ N, -8.65∘ E; ∼ 450 km baseline) for
2008–2010 records – with epoch-by-epoch zenith tropospheric delay
estimation. Daily-averaged positions of these base stations were calculated
using GAMIT and stabilized relative to a fixed circum-Antarctic reference
frame using a Kalman filter (GLOBK; Herring et al., 2015). Antenna positions
relative to the WGS84 ellipsoid were calculated every 30 s. We analyzed a
subset of these positions sampled at 10 min intervals and removed any
positions with uncertainty > 8 cm. The BOAR record was curtailed on
29 April 2013 (1.31-year duration), when an abrupt
∼ 2.0 m elevation decrease and corresponding horizontal offset
occurred, suggesting that the pole fell over.
We estimate initial antenna position accuracy of ∼ 1 cm. Positions
were converted to a local Cartesian horizontal coordinate system with final
antenna elevation values (zant) as orthometric height above the
EGM2008 geoid (Pavlis et al., 2012). Absolute geoid errors are poorly
constrained for coastal Antarctica, but relative geoid error for the
cumulative horizontal displacement of the GPS array (∼ 8 km over the
2-year period) should be < 1–2 cm. A constant offset of 3.71 m (3.66 m
pole length + 0.053 m phase center to bottom of antenna) was removed from
antenna elevation (zant) to estimate corresponding pole-base
elevation.
We estimated vertical tidal displacement for all GPS positions on the PIG ice
shelf using CATS2008A, an updated version of the model described by Padman et
al. (2002). We used mean sea level pressure values from the 0.75∘
grid-cell ERA-Interim reanalysis products (Dee et al., 2011) to correct for
vertical displacement due to the inverse barometer effect (IBE; e.g., Padman
et al., 2003). To do this, we removed the 2002–2016 median (985.21 hPa)
from 6 h sea level pressure and scaled the residuals by
∼ 1 cm hPa-1. Figure 3 shows that tidal amplitudes for the GPS
sites range from approximately -0.9 to +1.3 m and IBE amplitudes range
from -0.3 to +0.3 m. These signals were removed from the GPS antenna
elevation (zant), and residual high-frequency noise was removed
with a low-pass filter (1.5-day cutoff), yielding smoothed time series for
further analysis (Fig. 3). We conservatively estimate final zant
absolute accuracy of ∼ 0.1 m.
(a) Original GPS antenna elevation (light gray), after tide
correction (mid-gray), and after tide+IBE correction (black) for SOW4. The
red line shows smoothed time series and the yellow dashed line is linear fit
(-3.76 m yr-1). Sampled DEM elevations (cyan) show surface
elevation, which is offset from GPS antenna elevation by antenna–surface
distance (see Figs. 4 and 8). (b) High-frequency (< 1.5 days)
component of GPS record and CATS2008A tide model prediction, showing
excellent agreement. (c) Estimated inverse barometer effect (IBE)
magnitude from scaled sea level pressure.
Schematic of GPS station geometry. Surface elevation
(zsurf, dark blue line) is computed by removing antenna–surface
distance (hant-surf, dotted black line) from antenna elevation
(zant, black line). Pole-base elevation (red line) is computed
from pole length and antenna phase-center offset. At time t1 (right
panel), ongoing firn compaction resulted in decreased antenna and pole-base
elevation, while new snow accumulation offset surface lowering. The layer
within the firn column corresponding to the initial surface
(zsurf0) is represented by dotted blue line.
Antenna–surface distance
The GPS interferometric reflectometry (GPS-IR) method provides a precise
measurement of antenna phase-center height above a reflecting surface
(Larson, 2016). The reflecting surface for PIG is the interface
between the atmosphere and the snow/firn surface, and we define the antenna
height above this interface as the “antenna–surface distance”
(hant-surf). Figure 4 shows a schematic of this
GPS site geometry.
Assuming that the GPS pole base remains fixed within its original firn layer
(see Sect. 5.3 for further discussion), observed
decreases in the antenna–surface distance (hant-surf) can be attributed
to surface accumulation (e.g., snowfall, deposition of snow by wind).
Conversely, an increase in antenna–surface distance can be attributed to
surface ablation (e.g., melt, sublimation, removal of snow by wind) and
compaction of snow and/or firn above the pole base.
We computed mean daily antenna–surface distance for all sites using L1 C/A
code multipath surface reflections and the GPS interferometric reflectometry
methodology outlined in Larson et al. (2015). This method takes advantage of
the fact that the interference between the direct and reflected GPS signals
produces characteristic frequencies in signal-to-noise ratio data recorded by
the GPS receiver; these frequencies are directly related to the distance
between the GPS antenna phase center and the reflecting surface. Geodetic
antennas are designed to suppress multipath, so these interference patterns
are best resolved at low GPS satellite elevation angles. Reflector height
solutions were calculated for elevation angles of 5–25∘, which
sample the surface within a radial extent of ∼ 5–50 m. Local surface
slopes at each site are negligible, eliminating the need for an azimuthal
correction (e.g., Larson and Nievinski, 2013). Daily antenna–surface
distance (hant-surf) accuracy is estimated to be ∼ 1 cm
(Larson et al., 2015).
GPS-derived surface elevation
The antenna–surface distance (hant-surf) was subtracted from
antenna elevation (zant) to obtain daily records of surface
elevation zsurf (i.e., elevation of the air–snow interface above
the
EGM2008 geoid), with resulting
relative accuracy of ∼ 1–2 cm (absolute accuracy subject to the same
∼ 0.1 m zant uncertainty due to tidal, IBE, and geoid
corrections). The zsurf surface elevation values are directly
comparable with satellite or airborne laser altimetry data and stereo DEM
products. We use variable name zsurf rather than the more
traditional glaciological variable name h to limit potential confusion
between different “height” and “elevation” variables.
Continuous zsurf time series were generated for all seven PIG GPS
sites. The SOW3 record was curtailed on 22 August 2013, when antenna–surface
distance decreased below the minimum threshold of ∼ 0.5 m (Nievinski,
2013).
GPS velocity and strain rate
Horizontal velocities for each GPS station were computed from daily mean
antenna positions. We calculated principal strain rates for
eight different triangular sections
within the array (each defined by unique combination of three sites), using
the methods outlined by Savage et al. (2001). We tested multiple time
intervals for these strain rate calculations, from 2 to 120 days, and use
42 days as a compromise between temporal resolution and uncertainty (assuming
uncorrelated daily position error of ∼ 1 cm). We used observed
horizontal strain rates to estimate elevation change related to local flux
divergence.
Some component of observed GPS surface elevation change may also be related
to deformation due to local gradients in the driving stress and
surface-parallel flow due to advection over basal topography. The vertical
component of surface-parallel flow (V0 in Larson et al., 2015) can be
estimated using observed horizontal GPS paths and surface gradients from an
independent DEM. Advection over bed topography is irrelevant for a freely
floating ice shelf, and we attempt to estimate an upper bound for V0 due
to local deformation by considering local surface gradients and observed
relative horizontal displacements within the 2012–2014 GPS array.
High-resolution DEMs
In addition to the GPS elevation data, we generated WorldView/GeoEye stereo
DEMs (Shean et al., 2016) with 32 m posting over the PIG shelf (Shean, 2016)
to provide spatial context for the GPS time series. A total of seven
WorldView DEMs intersected the 2012–2014 GPS positions. We sampled DEM
surface elevation at corresponding GPS positions and compared with
GPS-derived surface elevation where possible.
High-resolution Lagrangian Dzsurf/ Dt maps (see methodology in
Shean, 2016; note we use D/Dt to indicate a Lagrangian differential
operator) were computed for the 2012–2014 GPS sites by forward-propagating
32 m DEM pixels from two initial DEM products (2 February 2012 and
23 October 2012) using interpolated, time-variable surface velocity maps from
Joughin et al. (2010) and Christianson et al. (2016). Lagrangian
Dzsurf/Dt maps were generated for all valid combinations of
these initial DEMs and ∼ 15 subsequent DEMs (∼ 0.5–2.5 years
later). Composite products were generated, with median
Dzsurf/ Dt values assigned to initial DEM pixel locations.
Surface mass balance
We analyzed estimates of 1979–2015 monthly and 2010–2013 daily SMB for
three 27 km grid cells over the PIG shelf from the Regional Atmospheric
Climate Model (RACMO) v2.3 (Ettema et al., 2009; Lenaerts et al., 2012; Van
Meijgaard et al., 2008; Van Wessem et al., 2014). The average 1979–2015 SMB
(a¯) is 0.91 m w.e. yr-1 for the grid cell closest to the
2012–2014 GPS array (-75.07∘ N, -100.80∘ E, Fig. 1).
The values for adjacent grid cells are 0.74 m w.e. yr-1 near the
grounding line of the main shelf (-75.15∘ N, -99.88∘ E)
and 0.84 m w.e. yr-1 over the south shelf (-75.30∘ N,
-101.14∘ E), providing some information on large-scale spatial
variability. These values are consistent with SMB estimates of
∼ 0.5–1.0 m w.e. yr-1 derived from CReSIS snow radar data
collected upstream of the PIG grounding line (Medley et al., 2014, 2015) and
SMB estimates of 0.99 and 1.06 m w.e. yr-1 for stake measurements
near 2006–2008 GPS sites over the upstream PIG trunk (Scott et al., 2009).
We conservatively estimate SMB uncertainty of 0.2 m w.e. yr-1.
AWS temperature data
To provide context for surface elevation change due to surface melt events,
we analyzed continuous 2011–2015 temperature data (3 h interval) from the
Evans Knoll AWS (-74.85∘ N,
-100.40∘ E; Lazzara et al., 2012), located at an
elevation of ∼ 178 m (height above EGM2008 geoid) on a bedrock outcrop
approximately 40 km north of the 2012–2014 GPS array (Fig. 1). We also
analyzed New York University (NYU) AWS
temperature data available near PIG2 from 9 January 2008 to 7 November 2009
and near BOAR from 19 January 2013 to 26 May 2015. Unfortunately, no AWS data
were collected on the PIG shelf during 2012. An analysis of overlapping time
periods for the Evans Knoll and 2013–2015 NYU AWS temperature records shows
a median offset of +1.24 ∘C (Evans Knoll warmer than NYU, with
normalized median absolute deviation (NMAD) of 2.76 ∘C), which is consistent with a dry adiabatic
lapse rate and local site conditions. This offset was removed from the Evans
Knoll temperature data to provide a continuous temperature estimate for the
GPS sites over the full 2012–2014 period.
To provide historical context, we extracted 2 m air temperature over the PIG
shelf from 0.75∘-resolution ERA-Interim reanalysis products (Dee et
al., 2011) for the 1979–2015 period with 6 h interval. The median offset
between the ERA-Interim temperature data and the 2013–2015 NYU AWS
temperature data was +0.10 ∘C (ERA-Interim warmer than NYU) with
NMAD 2.78 ∘C. This median offset was removed from the ERA-Interim
temperatures. We did not attempt to correct any seasonal bias in ERA-Interim
products (e.g., Jones et al., 2016).
Firn model
We used a dynamic firn model to simulate elevation change related to SMB and
firn processes. Model SMB output from RACMO2.3 (Sect. 2.6) was used to force
the semi-empirical 1-D IMAU-FDM dynamic firn model (Ligtenberg et al., 2011)
with 3 h timestep, and IMAU-FDM output was generated at 2-day intervals.
Velocity (vice) across the firn–ice transition (defined as the
layer with 917 kg m-3 density) was assumed to be in equilibrium with average 1979–2015 SMB
(a¯=0.91 m w.e. yr-1), so that vice=a¯a¯ρiρi. Vertical velocity components
for surface accumulation, surface sublimation, surface snow drift
erosion/deposition, surface melt, dry firn compaction, and a vertical
buoyancy correction (over floating ice-shelf grid cells) were computed for
the 2008–2010 and 2012–2014 periods (see Ligtenberg et al., 2011, for model
details). These components were combined to provide time series of simulated
surface elevation (zsurf̃) at each GPS station. In
addition, simulated elevations were computed over time for tracers
corresponding to the initial surface and pole base. We conservatively
estimate IMAU-FDM surface elevation uncertainty of ∼ 10 %, which
corresponds to ∼ 0.05 m for zsurf̃.
Derivation of basal mass balance
We combine the above observations and model output to estimate basal mass
balance for the PIG GPS sites. Mass conservation for a column with
ice-equivalent thickness Hice (after removing a thickness
correction d that accounts for total air content in the firn column)
relates Eulerian thickness change (dHice/dt) with dynamic
thinning or thickening due to flux divergence (∇⋅Hiceu, positive for extension), surface mass balance a˙ (meters ice
equivalent), and basal mass balance b˙ (meters ice equivalent, defined
as positive for melt):
∂Hice∂t=-∇⋅Hiceu+a˙-b˙.
The material derivative definition relates Eulerian (fixed reference frame)
and Lagrangian (reference frame moving with the ice column) thickness change:
DHiceDt=∂Hice∂t+u⋅(∇Hice).
Rearranging Eq. (2) and substituting into Eq. (1), we obtain the mass
conservation equation for Lagrangian thickness change:
DHiceDt=-Hice∇⋅u+a˙-b˙.
For a floating ice shelf in hydrostatic equilibrium, we can estimate
ice-equivalent thickness from air-column-corrected surface elevation
(zsurf-d), where zsurf is measured surface elevation
and d is total firn–air content:
Hice=(zsurf-d)ρwρw-ρi,
assuming a constant bulk density for ocean water (ρw=1026±1 kg m-3) and ice (ρi=917±5 kg m-3). We
substitute Eq. (4) into (3) and rearrange to estimate basal melt rate from
observed surface elevation change:
b˙=-DzsurfDt+(zsurf-d)∇⋅uρwρw-ρi+a˙.
Here, we assume that the total firn–air content remains constant for the period dt and drop the constant d from the material derivative term
(d≈12 m for the
PIG shelf, with uncertainty of ∼ 2 m; see Appendix in Shean,
2016). This
simplification is supported by the limited temporal variability (±0.3 m,
or ∼ 1–3 %) in modeled IMAU-FDM total firn air content for the three
PIG shelf grid cells during the relevant ∼ 2-year study periods.
The effects of processes that drive short-term surface-elevation change
(e.g., accumulation, melting) are largely absent below the upper few meters
of the firn column. Thus, elevation change at the GPS pole base (equivalent
to Dzant/Dt for constant pole length) displays less variability
than elevation change at the surface, as it is most sensitive to
(1) compaction rates within the underlying firn, (2) the long-term average
SMB (see Sect. 2.8), (3) BMB, and (4) flux divergence. For the pole-base
depths and time periods involved in this study, the first two terms display
limited to no variability, the flux divergence term is negligible, and
observed Dzant/Dt can capture basal melt rate variability that
might be obscured by surface accumulation and ablation signals in observed
Dzsurf/Dt.
Results
Horizontal velocity
Station velocities derived from daily mean positions for (a) 2008–2010 GPS sites and (b) 2012–2014 GPS sites. Note offset between SOW2
and SOW4, indicative of lateral shear across the ∼ 2 km wide
array, with greater extension near the center of the PIG shelf.
Figure 5a shows horizontal surface velocities of
the PIG1 and PIG2 stations. On the floating ice at PIG2, velocity increased
from ∼ 355 to ∼ 380 m yr-1 between 2008 and
2010 as the GPS moved downstream. Velocities for grounded ice at PIG1
increased at a relatively steady rate from ∼ 420 to
∼ 460 m yr-1 as the station moved toward the fast-flowing PIG
trunk (Fig. 1).
Figure 5b shows the 2012–2014 velocities for the
GPS array, which varied from ∼ 3830 to 4040 m yr-1
(Christianson et al., 2016). Velocities at SOW1, BOAR,
and SOW3 were similar, while SOW4 (closer to shelf centerline) consistently
moved ∼ 20 m yr-1 faster than these three sites, and SOW2
consistently moved ∼ 15 m yr-1 slower. Thus, there appears to be
∼ 30–40 m yr-1 dextral (right-handed) shear across the
∼ 2.4 km distance between the SOW4 and SOW2 sites. This
transverse velocity gradient is also apparent in velocity mosaics
(e.g., Christianson et al., 2016).
The velocity of all five stations varied by ∼ 2–4 % from
2012 to 2014, as described in detail by Christianson et al. (2016). In general, the stations displayed similar
relative velocity evolution, with several abrupt > 0.1–0.2 m day-1
velocity changes during the ∼ 2-year period
(Fig. 5).
Strain rate
Horizontal strain rates over 42-day periods for eight different
triangular sections within the GPS array. (a) First principal strain rate,
positive for extension. Error bars calculated for uncorrelated GPS position
error of 1 cm. (b) Second principal strain rate. (c) Velocity magnitude.
Diagrams in right column show color-coded triangular sections, with
∼ 2-year mean of principal strain rates plotted at centroids.
Figure 6 shows strain rate magnitude and direction
for 8 different strain triangles within the 2012–2014 GPS array. Mean
principal strain rates were +0.0018 yr-1 (extension approximately in
the along-flow direction) and -0.0001 yr-1 (compression approximately
in the across-flow/transverse direction). The array displayed a clockwise
rotation rate of ∼ 1∘ yr-1.
Spatial variations of strain rates within the array are small
(Fig. 6). Strain triangles including SOW1
experienced higher strain rates, while triangles including SOW3 experienced
lower strain rates, despite its location within the large transverse
depression (Fig. 2). Strain rate temporal
variability is also limited, but there do appear to be significant changes
correlated with shelf-wide velocity changes. In general, increased
(decreased) extensional strain rates were observed following an increase
(decrease) in absolute GPS array velocity.
Local surface slopes near SOW1, SOW2, and BOAR are negligible (< 0.2∘), so we assume no surface-parallel vertical motion for these
stations (i.e., V0=0). If all of the observed ∼ 3.4 m yr-1 relative displacement between SOW1 and SOW3 was attributed to flow down
∼ 0.6∘ local surface slopes at SOW3, then the
associated V0 magnitude would only be ∼ 0.03 m yr-1, which
is negligible compared to the observed ∼ 5.2 m yr-1
Dzsurf/Dt.
For estimated ice-equivalent thickness of ∼ 430–500 m, the observed
strain rates correspond to shelf thinning rates (DHice/Dt) of
∼ 0.5–0.9 m yr-1, with expected surface elevation change
(Dzsurf/Dt) of only ∼ 0.07–0.13 m yr-1. Based on
these estimates, we assume a value of -0.1±0.03 m yr-1 for the
divergence term in Eq. (5).
Antenna–surface distance
Initial antenna–surface distances (hant-surf) were ∼ 2.5
to 3.1 m, indicating that initial pole-base depths were ∼ 0.6
to 1.2 m below the initial surface (Figs. 7a,
8). Antenna–surface distance decreased over
time at all sites, with Dhant-surf/Dt rates of approximately -0.8 to -1.1 m yr-1 (Fig. 7a).
(a) Observed antenna–surface distance (hant-surf) for each
station. Legend lists original distance. (b) Observed antenna elevation
(zant) relative to initial absolute antenna elevation values listed in
legend. (c) Calculated surface elevation (zsurf) and simulated IMAU-FDM
surface elevation from SMB/firn (zsurf̃, thin black lines), both
relative to initial absolute surface elevation values listed in legend.
Time series of GPS antenna elevation (black), surface elevation
(thick blue), tracer for initial surface (dotted blue), and pole-base
elevation (red), all relative to initial absolute surface elevation. See
schematic in Fig. 4. Green points show sampled
WorldView DEM surface elevation. Note surface elevation decrease at all
sites but PIG2.
At both PIG1 and PIG2, there were periods of relatively rapid
antenna–surface distance decrease (e.g., from May to August 2008), followed
by a steady increase (e.g., August 2008 to February 2009). These changes are
consistent with periods of snow accumulation followed by several months of
ongoing firn compaction with limited snowfall. The 2012–2014 records show
similar periods of abrupt antenna–surface distance decrease and steady
increase, with more limited duration.
All GPS array records show an abrupt antenna–surface distance increase
(∼ 0.2–0.3 m) between December 2012 and January 2013, which
is consistent with surface melting and/or enhanced firn-compaction rates
above the pole base (i.e., upper few meters of the firn column).
GPS antenna and surface elevation change
Trends in observed antenna elevation change (Dzant/Dt) are
negative and highly linear (R2 0.98–1.00) for all PIG shelf sites, with
rates of -1.6 to -2.1 m yr-1 at SOW1, SOW2, and BOAR and higher
rates of -5.2 and -3.8 m yr-1 at SOW3 and SOW4, respectively
(Fig. 7b, Table 1). Observed Dzant/Dt over grounded ice at PIG1
is -7.6 m yr-1, with apparent concave-downward curvature. This is
consistent with V0 expected for surface-parallel flow (see Sect. 2.4)
and dynamic thinning over the PIG trunk associated with velocity increases in
2006–2008 GPS observations
(Scott et al., 2009) and satellite records
(Joughin et al., 2010; Mouginot et al., 2014).
GPS station data. Fields include surface elevation change relative
to tracer for initial surface D(zsurf-zsurf0′)/Dt, antenna elevation change Dzant/Dt
(equal to pole-base elevation change), surface elevation change
Dzsurf/Dt, and corresponding ice-equivalent basal melt rate
b˙. ∗ PIG1
values over grounded ice do not include correction to remove expected
Dzsurf/Dt due to advection along local surface slopes
(V0).
Site
Start date
End date
Days
D(zsurf-zsurf0′)/Dt
Dzant/Dt
Dzsurf/Dt
b˙
(m yr-1)
(m yr-1)
(m yr-1)
(m yr-1)
PIG1
2008-1-13
2009-9-4
601
0.93
-7.60∗
-6.76∗
–
PIG2
2008-1-10
2010-1-27
747
1.12
-1.12
-0.13
2.0 ± 0.9
SOW1
2012-2-10
2013-12-23
714
0.77
-1.81
-1.13
11.5 ± 1.1
SOW2
2012-2-10
2013-12-23
714
0.85
-2.08
-1.33
13.3 ± 1.2
BOAR
2012-2-10
2013-4-29
476
0.78
-1.58
-0.91
9.4 ± 1.1
SOW4
2012-2-10
2013-12-24
714
0.86
-3.76
-3.00
29.1 ± 1.7
SOW3
2012-2-9
2013-12-24
716
1.10
-5.23
-4.10
39.4 ± 2.1
The 2008–2010 surface elevation change (Dzsurf/Dt) at PIG2 is
limited (-0.13 m yr-1). By contrast, surface elevations decreased
significantly at all 2012–2014 GPS array sites, with rates of -0.9 to
-1.3 m yr-1 for SOW1, SOW2, and BOAR and rates of -4.1 and
-3.0 m yr-1 at SOW3 and SOW4, respectively.
Residuals about these linear fits (Fig. 9a and b) are small for PIG shelf
sites (root mean square error (RMSE) of 0.095 m for Dzant/Dt,
and RMSE of 0.143 m for Dzsurf/Dt), with some seasonal to
annual variability. We also note relatively abrupt (∼ days–weeks)
elevation changes that occurred across all stations in the 2012–2014 array
(e.g., -0.3 to +0.3 m anomaly during June 2012).
Surface mass balance
We consider surface elevation (zsurf) relative to a firn layer
tracer for the initial surface elevation (zsurf0′) to
estimate cumulative elevation change due to SMB after GPS installation.
Observed zsurf–zsurf0′ rates were ∼ 0.9–1.1 m yr-1 for
2008–2010 sites and ∼ 0.8–0.9 m yr-1 for 2012–2014 sites
(with SOW3 at ∼ 1.1 m yr-1) (Fig. 9c).
(a) Detrended GPS antenna elevation (zant, see
Fig. 7b for original records), with arbitrary y-axis offset. The legend
lists linear trends, which can be compared with Dzsurf/Dt trend
in panel (b) (see Eq. 6). (b) Detrended surface elevation
(zsurf, see Fig. 7c) and detrended IMAU-FDM simulated surface
elevation (zsurf̃), with arbitrary y-axis offset.
The legend lists linear trends. Note limited residual magnitude and dampened
seasonal signal of zant compared to zsurf. Unlike
zsurf, no significant change is observed in zant from
December 2012 to January 2013. (c) Surface elevation relative to
tracer for initial surface (zsurf-zsurf0′). As
annotated, positive slopes are indicative of new snow accumulation, shallow
negative slopes indicate ongoing compaction, and steep negative slopes likely
indicate surface melt. Note ∼ 0.2–0.3 m surface decrease from
December 2012 to January 2013. Legend values show linear fit at each site.
(d) Daily and monthly RACMO2.3 SMB. Note correlation of accumulation
events and increases in panel (c). (e) Scaled 2 m
temperature data from Evans Knoll AWS (black) and ERA-Interim (gray), with
above-zero AWS temperatures plotted in red. Note extended warm period from
mid-December 2012 to mid-January 2013, which corresponds to
∼ 0.2–0.3 m surface elevation decrease in panels (b) and
(c).
The average RACMO SMB over the central PIG shelf from 1979 to 2015 is
∼ 0.9 m w.e. yr-1. Monthly SMB climatology shows low
accumulation rates of ∼ 0.01–0.04 m w.e. month-1 over the PIG
shelf during the austral summer (November to February) and high accumulation
rates of ∼ 0.08–0.10 m w.e. month-1 during austral winter
(March to October) (Fig. 9d). Daily SMB products show periods of days to
weeks with increased accumulation (e.g., March 2013) that can be correlated
with abrupt decreases in antenna–surface distance.
The ∼ 3–4-week period between 24 December 2012 and 17 January 2013 was
relatively warm, with calibrated AWS temperatures of ∼ 1–5∘ C
for most days (Fig. 9e). We note that these are 2 m air temperatures and that once surface melting commenced, actual
surface temperatures would be lower but still above freezing. The daily RACMO
SMB data also show two accumulation events during the last week of
December 2012 (Fig. 9d), which involved rain on snow (M. Truffer, personal
communication, 2016). Surface elevations decreased by ∼ 0.2–0.3 m
across the entire GPS array during this warm and rainy period (Fig. 9b),
which is consistent with surface melting and/or enhanced firn-compaction
rates. No corresponding short-term changes were recorded by the antenna
elevations during the ∼ 3–4-week period (Fig. 9a), suggesting that the
processes responsible for the observed surface changes did not affect the
firn layers near the pole base (∼ 1.5 m depth). We note that there are
many warm periods between 1979 and 2015 with greater magnitude and duration
than the December 2012 to January 2013 period in the ERA-Interim 2 m air
temperatures over the PIG shelf.
Firn model
Figure 7c shows that the IMAU-FDM simulated surface elevation
(zsurf̃) ranges from -0.1 to +0.4 m from 2008 to
2010 and -0.2 to +0.2 m from 2012 to 2014. The observed
Dzsurf̃/Dt trend is +0.17 m yr-1 from 2008
to 2010, with no significant trend from 2012 to 2014. The magnitude and
timing of the simulated surface elevation variability is consistent with the
detrended observed surface elevation change (Fig. 9b). The observed
Dzsurf/Dt trends (-1 to -4 m yr-1), however, cannot
be explained by simulated elevation change due to SMB and firn processes
(Fig. 7c).
Basal melt rates
We computed basal melt rates from surface Dzsurf/Dt elevation
change using Eq. (5). The resulting melt rate estimates range from
∼ 2 m yr-1 at PIG2 to ∼ 39 m yr-1 at SOW3
(Table 1).
The 2012–2014 melt rate estimates show significant spatial variability. The
three upstream stations (SOW1, SOW2, and BOAR) experienced lower melt rates
of ∼ 9–13 m yr-1, while the downstream stations near the
transverse depression (SOW3 and SOW4) experienced higher rates of
∼ 29–39 m yr-1 for the same time period.
High-resolution DEMs
Figure 8 shows sampled DEM elevation compared with GPS surface elevation at
each site, with statistics provided in Table 2. In general, we observe good
agreement between the two datasets, with RMSE of 0.72 m and NMAD of 0.57 m
for the full sample (n=25). The DEMs display a slight bias (+0.43 m)
relative to the GPS surface elevation.
Statistics for WorldView DEM accuracy from comparisons with measured
GPS surface elevation data. Asterisks identify records with shorter time
interval and increased uncertainty.
Site
DEM
DEM dt
DEM Dzsurf/Dt
GPS-DEM RMSE
GPS-DEM mean
GPS-DEM SD
n
(days)
(m yr-1)
(m)
(m)
(m)
SOW1
5
302*
-2.30
0.69
-0.26
0.64
SOW2
5
619
-2.03
0.76
-0.46
0.60
BOAR
4
302*
-1.69
0.86
-0.55
0.66
SOW4
5
368*
-3.35
0.75
-0.61
0.44
SOW3
6
619
-4.32
0.54
-0.30
0.45
We observe good agreement between GPS-derived (Table 1) and DEM-derived
(Table 2) Dzsurf/Dt trends. The shorter DEM
Dzsurf/Dt intervals (e.g., ∼ 1 year for SOW1 and BOAR)
display larger errors than longer DEM intervals (∼ 2 years for SOW2 and
SOW3).
WorldView DEMs and composite Lagrangian Dzsurf/Dt products
generated using (a–b) initial DEM from 2 February 2012 and (c–d) initial DEM
from 23 October 2012. Note enhanced thinning observed within transverse
depressions and rift upstream of GPS array. The Dzsurf/Dt maps are used
to calculate basal melt rates (scaling factor of ∼ 9, e.g.,
Dzsurf/Dt of ∼ 1 m yr-1 corresponds to a basal melt rate
estimate of ∼ 9–10 m yr-1).
Figure 10 shows the composite DEM-derived Dzsurf/Dt maps, which
provide spatial context for the GPS-derived Dzsurf/Dt records.
Little or no elevation change was observed over longitudinal ridges, while
areas within and near transverse depressions experienced enhanced thinning.
This thinning was concentrated on the upstream side of the transverse
depressions. The Dzsurf/Dt products relative to the
23 October 2012 DEM (Fig. 10d) also show the spatial pattern of thinning
associated with the rift that opened upstream of SOW1 in ∼ 2014 (Jeong
et al., 2016).
Assumptions
The methods presented in Sect. 2 relied on
several simplifying assumptions. We now offer further discussion of these
assumptions and their potential influence on our results.
Hydrostatic equilibrium
In the absence of direct ice thickness measurements (e.g., radar profiles
near PIG GPS sites), we assume hydrostatic equilibrium and use surface
elevation to estimate freeboard ice thickness – a standard practice for
ice-shelf studies. While this assumption can lead to increased uncertainty
within a few ice thicknesses of the grounding line (Brunt et al., 2010;
Griggs and Bamber, 2011), it is reasonable for the mid-shelf location of the
GPS array, which in ∼ 2012, had been approaching hydrostatic
equilibrium for over 10–12 years since crossing the grounding line.
Previous studies using airborne ice-penetrating radar data have noted that
most of the PIG shelf is generally near hydrostatic equilibrium
(Bindschadler et al., 2011;
Dutrieux et al., 2013; Vaughan et al., 2012). Dense radar grids, however,
reveal narrow shelf-bottom channels, crevasses, and other features with
horizontal length scales of ∼ 10s–100s of meters that are not
apparent in ice-shelf surface topography
(Langley et al., 2014; Vaughan et al.,
2012). The thinner ice above these narrow features is partially supported by
lateral bridging stresses, so that the corresponding surface elevation will
appear higher than the expected freeboard surface elevation, providing
erroneously large ice thickness estimates using Eq. (4)
(Drews, 2015; Shabtaie and Bentley, 1982; Vaughan
et al., 2012).
Experiments with a high-resolution ice-flow model show that wider basal
channels tend to be near equilibrium, while increased bridging stresses
support ice over narrow basal channels (Drews, 2015). The PIG GPS array is
∼ 2 km across, which is > 4–5× the local ice thickness
(∼ 350–500 m). The ∼ 1–2 km length scale of nearby
longitudinal channels/keels is > 2–3× the local ice thickness,
with typical surface elevation difference between trough floors and adjacent
ridge crests of < 10 m. For the observed ice thickness, magnitude, and
length scale of surface variations, as well as the relatively long timescales
involved, we argue that the hydrostatic assumption is reasonable, and any
vertical elevation change due to evolving bridging stresses should be
negligible compared to the magnitude of observed Dzsurf/Dt and
our conservative error estimates.
SMB spatial variability
We used modeled SMB from a single RACMO2.3 grid cell to drive the IMAU-FDM
dynamic firn model, and applied the result to all GPS stations. We expect SMB
to vary spatially (e.g., Medley et al., 2015) due to local environmental
conditions (e.g., PIG2 elevation is > 400 m higher than SOW1–4 stations
on the shelf) and local surface topography (e.g., kilometer-scale
ridges/troughs), which will affect near-surface winds and snow
redistribution.
The larger zsurf-zsurf0′ values (a proxy for
surface accumulation) at SOW3 (Fig. 9c) indicate that greater local
accumulation occurred at this site within the transverse depression (Fig. 2),
potentially due to preferential deposition of wind-blown snow. However, we
also note that the accumulation histories of SOW4, which sits near a surface
ridge crest, and the three sites located on the floor of a broad, flat
surface trough (SOW1, SOW2, BOAR) appear similar (Fig. 9c).
The IMAU-FDM values do not account for horizontal advection of the firn
column through spatially variable RACMO fields (accumulation, surface
temperature, etc.) over time. The GPS sites over the PIG shelf are moving
∼ 4 km yr-1 (Figs. 1 and 5), which is nearly double the
observed PIG shelf velocities from the mid-1970s (Mouginot et
al., 2014). Thus, the local firn columns beneath the GPS sites likely
experienced variable SMB input over their ∼ 50–100 km
horizontal path during the corresponding 1979–2015 time period. This
suggests that the true firn column thickness and compaction rates may differ
from the IMAU-FDM estimates. For this reason, we use a constant firn air
content estimate (d≈12±2 m) derived from available airborne
ice-penetrating radar two-way travel time and altimetry surface elevation
measurements
(see
Appendix A of Shean, 2016).
Pole settling/tilting
We now consider whether some of the observed Dzsurf/Dt could be related to
settling, heating, or tilting of the GPS poles over time. We assume that the
poles froze in place shortly after installation, and the contact area
(∼ 1200 cm2 for a ∼ 1 m long cylinder
with ∼ 3.8 cm diameter) with surrounding firn should be
sufficient to counter the downward gravitational force. Thus, we expect that
antenna elevation change (Dzant/Dt) represents rates at the base of the
pole rather than rates within an overlying firn layer.
A related consideration involves heating of the exposed pole during summer,
which might lead to decoupling from the surrounding snow/firn and allow for
additional penetration of the pole base within the firn. The pole-base
stoppers should have prevented this penetration. In addition, we do not see
any indication of such settling from December 2012 to January 2013, when
surface elevations decreased by ∼ 0.2–0.3 m, but pole-base
elevations showed little change (Fig. 9a and b).
The lack of pole-base elevation change also suggests that surface meltwater
did not percolate more than ∼ 1–2 m below the surface.
Finally, we assume that the poles were installed with vertical orientation
and did not tilt over time. For an initially vertical pole with length of
3.71 m (including antenna phase-center offset), a 10∘ tilt would
introduce a -0.06 m vertical antenna elevation error
(-0.03 m yr-1 for a 2-year period), while a 20∘ tilt would
introduce a -0.22 m vertical error (-0.11 m yr-1). Thus, we
expect vertical error associated with any tilting to be negligible compared
to the large observed Dzant/Dt (-1.12 to
-7.60 m yr-1). These 10 and 20∘ tilts could, however,
introduce horizontal errors of up to 0.64 and 1.27 m, respectively, which
would affect intra-network displacement and strain rate
estimates. While it is possible that some minor tilting could have occurred
(especially during initial months), this was not noted during
servicing/removal, and the reflectometry results do not indicate any
systematic change in directional antenna–surface offset.
Strain rate length scales
We estimated ∼ 0.1 m yr-1 surface elevation change due to local
flux divergence, assuming that the observed relative horizontal displacements
are evenly distributed across the strain triangles, which have
∼ 1–2 km edges between
GPS stations. This assumption is supported by the ∼ 1 km spatial
extent of thinning signals within/near transverse depressions in the
Dzsurf/Dt maps (Fig. 10). Even if this strain is concentrated
over a shorter distance (e.g., ∼ 200 m), this contribution only
increases to ∼ 0.5 m yr-1, which is still small compared to
observed Dzsurf/Dt signals of ∼ 3–4 m yr-1. The
relatively large spatial variability in Dzsurf/Dt values
(∼ 1 to ∼ 4 m yr-1) and lack of spatial variability in
strain rates supports the assumption that the observed
Dzsurf/Dt is primarily caused by basal melt.
Discussion
SMB and firn compaction
The evolution of GPS-derived surface elevation relative to a tracer for the
initial surface (zsurf-zsurf0′) is consistent with
SMB estimates (a˙), providing qualitative validation for the RACMO SMB
and IMAU-FDM results. Based on these results, we suggest that it may be
possible to extract detailed SMB records for other sites using only observed
GPS antenna–surface distance and simple assumptions about firn densification
(e.g., Herron and Langway Jr., 1980). The problem is further
simplified for grounded ice with negligible BMB rates.
The limited variability in surface elevation at PIG2 (Fig. 7c) suggests that
the observed 2008–2010 SMB over the south PIG shelf was approximately equal
to basal melt during this period, assuming negligible velocity divergence for
this location. We observe large surface elevation trends for the 2012–2014
GPS sites with no significant simulated Dzsurf̃/Dt
trend, suggesting that SMB and firn compaction during this period were
consistent with average 1979–2015 SMB (a¯) values, and that the large
observed Dzsurf/Dt must be attributed to other processes,
specifically basal melting.
Residual elevation variability
The detrended surface (Fig. 9a) and antenna
(Fig. 9b) elevation residuals appear unrelated.
This suggests that seasonal surface processes (e.g., accumulation
influencing near-surface compaction rates) are not responsible for driving
antenna elevation variability. We considered several possible sources for the
observed subannual elevation variability, including ocean (e.g., currents,
sea surface height), atmospheric (e.g., pressure, temperature), and dynamic
processes (e.g., resistive stress from sea ice and/or mélange in shear
margins). Unfortunately, we were unable to definitively determine the
cause(s) for these variations in the ∼ 2-year GPS records.
Some of the short-term (days–weeks) variability (e.g., June 2012) observed
across all five 2012–2014 stations (Fig. 9b)
could be related to insufficient or incorrect IBE correction. The magnitude
and timing of these systematic anomalies, however, suggests that they are
likely related to grounding/ungrounding events (e.g.,
Joughin et al., 2016).
Strain rate history, rifting, and grounding evolution
The lateral shear across the GPS array is consistent with increased
longitudinal extension closer to the PIG centerline, potentially due to
locally enhanced ductile deformation (i.e., “necking”; Bassis and Ma, 2015)
across transverse depressions and/or expansion of basal/surface crevasses and
rifts. The SOW3 station, which lies within a large transverse depression
(Fig. 2), displays a slight acceleration in antenna elevation change
(Fig. 9a), potentially due to increased local extension within the
depression.
An upstream regrounding event would slow ice upstream of the GPS array,
initially resulting in increased extensional strain rates across the
transverse rifts/depressions, followed by a velocity decrease at the GPS
array. Conversely, an upstream ungrounding event would initially lead to
decreased extensional strain rates across the transverse rifts/depressions,
followed by an increase in observed GPS velocities. We suggest that an
upstream regrounding event (Joughin et al., 2016) in ∼ June 2012 could
be responsible for increased strain rates across the GPS array (Fig. 6).
Similarly, an ungrounding event in ∼ April 2013 followed by a grounding
event in ∼ November 2013 could explain the decrease and subsequent
increase in strain rates.
There is an abrupt ∼ 0.1–0.2 m antenna elevation (zant)
decrease at both SOW3 and SOW4 in late 2013, near the end of the records
(Fig. 9a). No surface elevation (zsurf) information is available
at SOW3 due to missing antenna–surface distance data for this period (see
Sect. 2.3), but a corresponding surface elevation decrease is observed at
SOW4 (Fig. 9b). These elevation decreases do not appear to be related to site
servicing. Rather, these observations are consistent with relatively abrupt
local extension within the transverse depression affecting SOW3 and SOW4 but
not the upstream GPS sites. The timing of this event corresponds with
observed lengthening of the large rift (R1) upstream of SOW1 (Jeong et al.,
2016), supporting the hypothesis that relatively rapid, localized extension
occurred across the transverse depressions and rifts during this period.
Comparison with in situ basal melt rate observations
The GPS-derived basal melt rate estimates (∼ 9–13 m yr-1 for
SOW1, SOW2, and BOAR sites) appear consistent with those from bottom
altimeter (∼ 14.7 m yr-1 from January to February 2012) and pRES
(∼ 15–25 m yr-1) measurements of Stanton et al. (2013). These
measurements provide some validation for the GPS results, as they are not
influenced by surface mass balance and firn processes. A direct comparison
may be imprudent, however, as the Stanton et al. (2013) borehole was
∼ 1.34 km upstream of SOW1 (near the R1 rift), which likely affected
local melt rates, and we observe considerable ∼ kilometer-scale spatial
variability in melt rates across GPS array. Furthermore, the bottom altimeter
sampled a ∼ 5 cm diameter spot with unknown upstream/downstream
orientation, approximately 30–40 cm from the edge of the 20 cm borehole.
Aside from local melt variability expected due to turbulent flow near the
altimeter pole or borehole edge, the altimeter provided a relatively small
spatial sample compared to the GPS results, which are sensitive to changes in
a column of ice with much larger footprint (100s to 1000s of
m2).
Basal melt rate spatial variability
The GPS records at SOW1, SOW2, and BOAR show similar Dzsurf/Dt
rates and residuals, which is consistent with their apparent orientation on
the same “block” between transverse rifts/depressions (Fig. 2) and supports
the hypothesis that they were exposed to similar sub-shelf circulation. The
DEM Dzsurf/Dt maps show enhanced surface elevation change
rates, and thus higher basal melt rates, on the upstream side of transverse
depressions (Fig. 10), which is consistent with increased
Dzsurf/Dt observed at the SOW3 and SOW4 sites.
This relationship is potentially related to enhanced buoyant flow and/or
turbulence over increased basal slopes (e.g., Jenkins, 2011) beneath
transverse surface depressions. We also suggest that these transverse basal
channels may offer conduits for meltwater flow between adjacent longitudinal
channels, potentially leading to increased circulation velocity and higher
melt rates within the transverse depressions.
Basal melt rate sensitivity to ocean temperature variability
Christianson et al. (2016) suggest that the subtle (∼ 2–4 %) changes
in 2012–2014 GPS velocity display a lagged correlation with observed
variations in ocean temperature records from moorings in Pine Island Bay (see
Fig. 1 for location), potentially implying causality. Our analysis supports
the alternative hypothesis of Christianson et al. (2016) that these velocity
variations are primarily related to upstream grounding evolution (Joughin et
al., 2016) and extension across a series of transverse depressions.
Rates of antenna and surface elevation change (Dzant/Dt and
Dzsurf/Dt) were essentially constant in time, with no significant variation
in inferred basal melt rates during this 2-year time period. If sub-shelf
melt rates beneath the GPS array covaried with observed ocean heat content
beyond the shelf front in Pine Island Bay
(Christianson et al., 2016; Webber et al.,
2017), a significant change in both Dzant/Dt and Dzsurf/Dt would be expected
during this period. The lack of any significant deviation suggests that melt
rates at these sites were not noticeably affected by observed ocean
temperature variability. This finding suggests that (1) these sites
are not representative of melt rates for the inner shelf (e.g., those near
the grounding line), (2) the oceanographic measurements near the PIG ice
front are not representative of water circulating beneath these ice-shelf
sites, and/or (3) local melt rates are less sensitive to the observed
oceanographic changes than previously assumed.
Future work
High-resolution velocity maps derived from submeter imagery could
potentially constrain local velocity divergence and length scales for
observed strain between GPS receivers. In addition, seismic data from
stations deployed near the GPS array and regional sites could help constrain
the timing and location of rift propagation and grounding/ungrounding
events.
High-resolution (< 1 km grid) SMB output and improved dynamic firn
model output would likely offer an improved understanding of local
variability across the GPS array. It may also be possible to further
constrain firn-compaction rates, and thus long-term SMB, using relative
layer thicknesses observed in CReSIS snow radar measurements
(e.g., Medley et al., 2015) or in situ pRES observations
(e.g., Jenkins et al., 2006). However, airborne radar data
over the PIG shelf suffer from clutter due to kilometer-scale surface/basal
topography and crevasses, while the available intermittent pRES records
(Stanton et al., 2013) likely lack the
sensitivity to detect small changes in firn layer thickness during the ∼ 3-week observation period.
These limitations highlight the current value of long-term GPS records to
constrain surface evolution where observations are sparse and model results
are poorly constrained. Expanding the scope of our study to include the full
archive of geodetic GPS data for the Antarctic and Greenland ice sheets
would offer a valuable new dataset for calibration/validation of models and
remote sensing data.
We offer the following recommendations to improve GPS-IR results for future GPS
deployments: (1) set the GPS elevation mask to 0∘ (default values are
typically ∼ 5–10∘), (2) track all possible signals (L2C, L5,
Galileo, and GLONASS), (3) ensure that antenna–surface distance will remain
> 0.5 m between servicing visits, and (4) document and photograph GPS
sites, noting antenna–surface distance and any pole tilt during install and
servicing.
Summary and conclusions
We analyzed GPS records for the PIG shelf for the 2008–2010 and 2012–2014
periods. We produced daily time series of antenna–surface distance
(hant-surf) and antenna elevation (zant, relative to EGM2008 geoid),
which were combined to accurately measure surface elevation (zsurf) at
each site. The surface elevation data can be directly compared with
remote-sensing measurements, providing independent validation for
high-resolution WorldView stereo DEM records (RMSE of ∼ 0.72 m, NMAD of ∼ 0.57 m).
The GPS-derived surface elevation data provide new information about local
SMB that can be compared with coarse-resolution model output and AWS data.
Surface elevation relative to a firn layer tracer for the initial surface (zsurf-zsurf0′) increased at rates of ∼ 0.8–1.1 m yr-1
for all GPS sites, which is consistent with modeled SMB of
∼ 0.7–0.9 m w.e. yr-1. An abrupt ∼ 0.2–0.3 m surface
elevation decrease, likely due to surface melt and/or enhanced firn
compaction, is observed across all GPS sites during a period of warmer
atmospheric temperatures from December 2012 to January 2013.
Trends in observed antenna (Dzant/Dt) and surface elevation change
(Dzsurf/Dt) were highly linear for all GPS sites on the PIG shelf. Observed
extensional strain rates were ∼ 0.001–0.002 yr-1 for the
2012–2014 GPS array, which corresponds to only ∼ 0.1 m yr-1
surface elevation change due to local flux divergence.
An alternative form of the mass conservation equation was used to estimate
BMB from observed Lagrangian surface elevation change, strain rates, and
SMB. Basal melt rates were ∼ 10 to ∼ 40 m yr-1
near the center of the fast-flowing PIG shelf, and ∼ 2 m yr-1
for the southern shelf. These melt rates are similar to those derived from
complementary in situ instrument records (Stanton et
al., 2013) and high-resolution stereo DEMs
(Shean,
2016).
Both GPS and DEM records show higher basal melt rates within and near
transverse surface depressions and rifts associated with longitudinal
extension. Basal melt rates for the 2012–2014 period show limited temporal
variability, despite substantial changes in ocean heat content at the ice
front and likely in the ice-shelf cavity. Residual elevation change
variability is likely related to upstream grounding/ungrounding events and
the local evolution of transverse depressions/rifts. Our results demonstrate
the value of long-term GPS records and interferometric reflectometry for
constraining ice-shelf mass balance estimates.