We present a 1986 through 2017 estimate of Greenland Ice Sheet ice discharge.
Our data include all discharging ice that flows faster than 100 m yr-1
and are generated through an automatic and adaptable method, as opposed to
conventional hand-picked gates. We position gates near the present-year
termini and estimate problematic bed topography (ice thickness) values where
necessary. In addition to using annual time-varying ice thickness, our time
series uses velocity maps that begin with sparse spatial and temporal
coverage and end with near-complete spatial coverage and 6 d updates to
velocity. The 2010 through 2017 average ice discharge through the flux gates
is ∼488±49 Gt yr-1. The 10 % uncertainty stems primarily
from uncertain ice bed location (ice thickness). We attribute the ∼50 Gt yr-1 differences among our results and previous studies to our
use of updated bed topography from BedMachine v3. Discharge is approximately
steady from 1986 to 2000, increases sharply from 2000 to 2005, and then is
approximately steady again. However, regional and glacier variability is more
pronounced, with recent decreases at most major glaciers and in all but one
region offset by increases in the NW (northwestern) region. As part of the
journal's living archive option, all input data, code, and results from this
study will be updated when new input data are accessible and made freely
available at https://doi.org/10.22008/promice/data/ice_discharge.
Introduction
The mass of the Greenland Ice Sheet is decreasing (e.g.,
). Most ice-sheet mass loss – as iceberg discharge,
submarine melting, and meltwater runoff – enters the fjords and coastal
seas, and therefore ice-sheet mass loss directly contributes to sea-level
rise
.
Greenland's total ice loss can be estimated through a variety of independent
methods, for example “direct” mass change estimates from GRACE
or by using satellite altimetry to estimate surface
elevation change, which is then converted into mass change (using a firn
model, e.g., ). However, partitioning the mass
loss between ice discharge (D) and surface mass balance (SMB) remains
challenging (cf. ).
Correctly assessing mass loss, as well as the attribution of this loss (SMB
or D), is critical to understanding the process-level response of the
Greenland Ice Sheet to climate change and thus improving models of future
ice-sheet changes and associated sea-level rise .
The total mass of an ice sheet, or a drainage basin, changes if the mass gain
(SMB inputs, primarily snowfall) is not balanced by the mass loss (D and
SMB outputs, the latter generally meltwater runoff). This change is typically
termed ice-sheet mass balance (MB) and the formal expression for this rate of
change in mass is (e.g., )
dMdt=ρ∫AbdA-∫gQdg,
where ρ is the average density of ice, b is an area mass balance, and
Q is the discharge flux. The left-hand side of the equation is the rate of
change of mass, the first term on the right-hand side is the area A
integrated SMB, and the second term is the discharge D mass flow rate that
drains through gate g. Equation () is often simplified to
MB=SMB-D,
where MB is the mass balance and is referred to as the “input–output”
method (e.g., ). Virtually all studies agree on
the trend of Greenland mass balance, but large discrepancies persist in both
the magnitude and attribution. Magnitude discrepancies include, for example,
reporting a mass imbalance of -250±21 Gt yr-1 during 2003 to 2010, reporting
-181±28 Gt yr-1 during 2003 to 2008, and
reporting a mass imbalance of -265±19 Gt yr-1 during 2004 to
2008. Some of these differences may be due to different ice-sheet area masks
used in the studies. Attribution discrepancies include, for example,
attributing the majority (64 %) of mass
loss to changes in SMB during the 2005 to 2009 period, but
attributing the majority (85 %) of mass loss to
changes in D during the 2004 to 2008 period.
Discharge may be calculated through several methods, including mass flow rate
through gates (e.g., ), or
solving as a residual from independent mass balance terms (e.g.,
). The gate method that we
use in this study incorporates ice thickness and an estimated vertical
profile from the observed surface velocity to calculate the discharge. A
typical formulation of discharge across a gate Dg is
Dg=ρVHw,
where ρ is the average density of ice, V is depth-average
gate-perpendicular velocity, H is the ice thickness, and w is the gate
width. Uncertainties in V and H naturally influence the estimated
discharge. At fast-flowing outlet glaciers, V is typically assumed to be
equal at all ice depths, and observed surface velocities can be directly
translated into depth-averaged velocities (as in
). To minimize uncertainty
from SMB or basal mass balance corrections downstream of a flux gate, the
gate should be at the grounding line of the outlet glacier. Unfortunately,
uncertainty in bed elevation (translating to ice thickness uncertainty)
increases toward the grounding line.
Conventional methods of gate selection involve hand-picking gate locations,
generally as linear features (e.g., ), or
visually approximating ice-orthogonal gates at one point in time (e.g.,
). Manual gate definition is sub-optimal. For
example, the largest discharging glaciers draw from an upstream radially
diffusing region that may not easily be represented by a single linear gate.
Approximately flow-orthogonal curved gates may not be flow-orthogonal on the
multi-decade timescale due to changing flow directions. Manual gate selection
makes it difficult to update gate locations, corresponding to glacier termini
retreat or advance, in a systematic and reproducible fashion. We therefore
adopt an algorithmic approach to generate gates based on a range of criteria.
Here, we present a discharge dataset based on gates selected in a
reproducible fashion by a new algorithm. Relative to previous studies, we
employ ice velocity observation over a longer period with higher temporal
frequency and denser spatial coverage. We use ice velocity from 1986 through
2017, including 6 d velocities for the last ∼500 d of the time
series, and discharge at 200 m pixel resolution capturing all ice flowing
faster than 100 m yr-1 that crosses glacier termini into fjords.
Input data
Historically, discharge gates were selected along well-constrained flight
lines of airborne radar data . Recent advances
in ice thickness estimates through NASA Operation IceBridge
, NASA Oceans Melting Greenland (OMG;
), fjord bathymetry ,
and methods to estimate thickness from surface properties (e.g.,
) have been combined into
digital bed elevation models such as BedMachine v3
or released as independent
datasets . From these advances, digital bed
elevation models have become more robust at tidewater glacier termini and
grounding lines. The incorporation of flight-line ice thickness data into
higher-level products that include additional methods and data means gates
are no longer limited to flight lines (e.g., ).
Ice velocity data are available with increasing spatial and temporal
resolution (e.g., ). Until recently, ice
velocity mosaics were limited to once per year during winter
, and they are still temporally limited, often
to annual resolution, prior to 2000 (e.g.,
). Focusing on
recent times, ice-sheet-wide velocity mosaics from Sentinel 1A and 1B are now
available every 6 d (http://promice.org, last access: 23 May 2019). The increased availability of
satellite data has improved ice velocity maps both spatially and temporally,
thereby decreasing the need to rely on spatial and temporal interpolation of
velocities from annual/winter mosaics
.
The discharge gates in this study are generated using only surface speed and
an ice mask. We use the MEaSUREs Greenland Ice Sheet Velocity Map from InSAR
Data, Version 2 , hereafter termed
“MEaSUREs 0478” due to the National Snow and Ice Data Center (NSIDC) date
set ID number. We use the BedMachine v3
ice mask.
For ice thickness estimates, we use surface elevation from GIMP
(; NSIDC dataset ID 0715), adjusted
through time with surface elevation change from
and bed elevations from BedMachine v3 replaced by
where available. Ice sector and region
delineation is from . Ice velocity data are
obtained from a variety of products including Sentinel 1A and 1B derived by
PROMICE (see Appendix ), MEaSUREs 0478, MEaSUREs
0646 , , and
. Official glacier names come from
. Other glacier names come from
. See Table for an overview of
datasets used in this work.
This work uses 308 different velocity maps, biased toward the last 500 d of
the time series when 6 d ice velocities become available from the Sentinel-1
satellites. The temporal distribution is 1 to a few velocity maps per year
from 1986 to 2000, 9 to 13 velocity maps per year from 2000 through 2015, 24
in 2016, and 55 in 2017.
Summary of data sources used in this work.
PropertyName used in this paperReferenceBasal topographyBedMachineBasal topography for the southeastSurface elevationGIMP 0715Surface elevation changeSurface elevation changeBaseline velocityMEaSUREs 0478VelocitySentinelAppendix VelocityMEaSUREs 0646Velocitypre-2000Sectors and regionsSectors and regionsNames, MethodsTerminology
We use the following terminology, most displayed in Fig. .
“Pixels” are individual 200m×200m raster
discharge grid cells. We use the nearest neighbor when combining datasets
that have different grid properties.
“Gates” are contiguous (including diagonal) clusters of pixels.
“Sectors” are spatial areas that have 0, 1, or >1 gate(s) plus any
upstream source of ice that flows through the gate(s), and come from
.
“Regions” are groups of sectors, also from ,
and are labeled by approximate geographic region.
The “baseline” period is the average 2015, 2016, and 2017 winter
velocity from MEaSUREs 0478.
Overview showing fast-flowing ice (orange, greater than
100 m yr-1) and the gates for the top eight discharging glaciers
(Fig. ). Gates are shown as black lines in inset
images. Each inset is 30×30 km and all have the same color scaling,
but are different from the main map. Insets pair with the nearest label and
box. On the main map, regions from are
designated by thicker black lines and large bold labels. Sectors (same
source) are delineated with thinner gray lines, and the top discharging
glaciers are labeled with smaller font. H: Helheim Gletsjer; KB: Køge
Bugt; KG: Kangerlussuaq Gletsjer; KS: Kangilliup Sermia (Rink Isbræ); N:
Nioghalvfjerdsbræ; P: Petermann Gletsjer; SK: Sermeq Kujalleq (Jakobshavn
Isbræ); and Z: Zachariae Isstrøm. Basemap terrain (gray), ocean
bathymetry (blues), and ice mask (white) come from
BedMachine.
“Coverage” is the percentage of total, region, sector, or gate
discharge observed at any given time. By definition coverage is 100 %
during the baseline period. From the baseline data, the contribution to total
discharge of each pixel is calculated, and coverage is reported for all other
maps that have missing observations (Fig. ).
Total estimated discharge is always reported because missing pixels are
gap-filled (see Sect. below).
“Fast-flowing ice” is defined as ice that flows at more than
100 m yr-1.
Names are reported using the official Greenlandic names from
if a nearby name exists, and then
in parentheses.
Although we refer to solid ice discharge, and it is in the solid phase when
it passes the gates and eventually reaches the termini, submarine melting
does occur at the termini and some of the discharge enters the fjord as
liquid water .
Gate location
Gates are algorithmically generated for fast-flowing ice (greater than
100 m yr-1) close to the ice-sheet terminus determined by the
baseline-period data. We apply a 2-D inclusive mask to the baseline data for
all ice flowing faster than 100 m yr-1. We then select the mask edge
where it is near the BedMachine ice mask (not including ice shelves), which
effectively provides grounding line termini. We buffer the termini 5000 m in
all directions, creating ovals around the termini, and once again down-select
to fast-flowing ice pixels. This procedure results in gates 5000 m upstream
from the baseline terminus that bisect the baseline fast-flowing ice. We
manually mask some land- or lake-terminating glaciers which are initially
selected by the algorithm due to fast flow and mask issues.
We select a 100 m yr-1 speed cutoff because slower ice, taking longer
to reach the terminus, is more influenced by SMB (Fig. ).
Heatmap and table showing ice-sheet discharge as a function of gate
buffer distance and ice speed cutoff. The colors of the numbers change for
readability.
We select gates at 5000 m upstream from the baseline termini, which means
that gates are likely >5000 m from the termini further back in the
historical record . The
choice of a 5000 m buffer follows from the fact that it is near-terminus and
thus avoids the need for (minor) SMB corrections downstream, yet is not too
close to the terminus where discharge results are sensitive to the choice of
distance-to-terminus value (Fig. ), which may be indicative
of bed (ice thickness) errors.
Thickness
We derive thickness from surface and bed elevation. We use GIMP 0715 surface
elevations in all locations, and the BedMachine bed elevations in most
locations, except southeastern Greenland where we use the
bed. The GIMP 0715 surface elevations are
all time-stamped per pixel. We adjust the surface through time by linearly
interpolating elevation changes from , which covers
the period from 1995 to 2016. We use the average of the first and last
3 years for earlier and later times, respectively. Finally, from the fixed
bed and temporally varying surface, we calculate the time-dependent ice
thickness at each gate pixel.
2-D histogram of velocity and thickness at all gate pixels.
(a) Unadjusted (BedMachine and )
thickness. (b) Adjusted (as described in the text)
thickness.
Missing or invalid data
The baseline data provide velocity at all gate locations by definition, but
individual non-baseline velocity maps often have missing or invalid data.
Also, thickness provided by BedMachine is clearly incorrect in some places
(e.g., fast-flowing ice that is 1 m thick, Fig. a).
We define invalid data and fill in missing data as described below.
Invalid velocity
We flag invalid (outlier) velocities by treating each pixel as an individual
time series, applying a 30-point rolling window, flagging values more than 2
standard deviations outside the mean, and repeating this filter three times.
We also drop the 1972 to 1985 years from
because there is low coverage and extremely high variability when using our
algorithm.
This outlier detection method appears to correctly flag outliers (see
the
Appendix for unfiltered time-series graphs), but likely
also flags some true short-term velocity increases. The effect of this filter
is a ∼1 % reduction in discharge most years, but more in years with
high discharge – a reduction of 3.2 % in 2013, 4.3 % in 2003, and
more in the 1980s when the data are noisy. Any analysis using these data and
focusing on individual glaciers or short-term changes (or lack thereof)
should re-evaluate the upstream data sources.
Missing velocity
We generate an ice speed time series by assigning the PROMICE, MEaSUREs 0478,
MEaSUREs 0646, and pre-2000 products to their respective reported time stamps
(even though these are time-span products) or to the middle of their time
span when they cover a long period such as the annual maps from
. We ignore the fact
that any individual velocity map or pixel has a time span, not a time stamp.
Velocities are sampled only where there are gate pixels. Missing pixel
velocities are linearly interpolated in time, except for missing data at the
beginning of the time series which are back- and forward-filled with the
temporally nearest value for that pixel (Fig. ).
We do not spatially interpolate missing velocities because the spatial
changes around a missing data point are most likely larger than the temporal
changes. We visually represent the discharge contribution of directly
observed pixels, termed coverage (Fig. ), as
time-series graphs and opacity of dots and error bars in the figures.
Therefore, the gap-filled discharge contribution at any given time is equal
to 100 minus the coverage. Discharge is always reported as estimated total
discharge even when coverage is less than 100 %.
Invalid thickness
The thickness data appear to be incorrect in some locations. For example,
many locations have fast-flowing ice but report ice thicknesses of 10 m or
less (Fig. a). We accept all ice thicknesses greater
than 20 m and construct from this a thickness versus log10 speed
relationship. For all ice thicknesses less than or equal to 20 m, we adjust
thickness based on this relationship (Fig. b). We selected
the 20 m thickness cutoff after visually inspecting the velocity
distribution (Fig. a). This thickness adjustment adds
20 Gt yr-1 to our baseline-period discharge estimate with no
adjustment. In the
Appendix and Table we discuss
the discharge contribution of these adjusted pixels and a comparison among
this and other thickness adjustments.
Discharge
We calculate discharge per pixel using density (917 kg m-3), filtered
and filled ice speed, projection-corrected pixel width, and adjusted ice
thickness derived from time-varying surface elevation and a fixed bed
elevation (Eq. ). We assume that any change in surface elevation
corresponds to a change in ice thickness and thereby neglects basal uplift,
erosion, and melt, which combined are orders of magnitude less than surface
melting (e.g., ). We also assume
depth-averaged ice velocity is equal to the surface velocity.
We calculate discharge using the gate-orthogonal velocity at each pixel and
at each time stamp – all velocity estimates are gate-orthogonal at all
times, regardless of gate position, orientation, or changing glacier velocity
direction over time.
Annual averages are calculated by linearly interpolating to daily, and then
calculating the annual average. The difference
between this method and averaging only the observed samples is ∼3 %
median (5 % average, and a maximum of 10 % when examining the entire
ice sheet and all years in our data). It is occasionally larger at individual
glaciers when a year has few widely spaced samples of highly variable
velocity.
Discharge uncertainty
A longer discussion related to our and others' treatments of errors and
uncertainty is in the
Appendix , but here we describe how we estimate the
uncertainty related to the ice discharge following a simplistic approach.
This yields an uncertainty of the total ice discharge of approximately
10 % throughout the time series.
At each pixel we estimate the maximum discharge, Dmax, from
Dmax=ρV+σVH+σHW,
and minimum discharge, Dmin, from
Dmin=ρV-σVH-σHW,
where ρ is ice density, V is baseline velocity, σV is
baseline velocity error, H is ice thickness, σH is ice
thickness error, and W is the width at each pixel. Included in the
thickness term is surface elevation change through time
(dH/dt). When datasets do not come with error estimates,
we treat the error as 0.
We use ρ=917 kg m-3 because the gates are near the terminus in
the ablation zone and ice thickness estimates should not include snow or
firn, although regionally ice density may be <917 kg m-3 due to
crevasses. We ignore the velocity error σV because the
proportional thickness error (σH/H) is an order of magnitude
larger than the proportional velocity error (σV/V), yet both
contribute linearly to the discharge. W is location-dependent due to the
errors between our working map projection (EPSG 3413) and a more accurate
spheroid model of the earth's surface. We adjust linear gate width by up to
∼4 % in the north and ∼-2.5 % in the south of Greenland
(area errors are up to 8 %). On a pixel-by-pixel basis we used the
provided thickness uncertainty; except where we modified the thickness (H<20 m), we prescribe an uncertainty of 0.5 times the adjusted thickness.
Subsequently, the uncertainty on individual glacier-, sector-, region-, or
ice-sheet scales is obtained by summing, but not reducing by the square of
the sums, the uncertainty related to each pixel. We are conservative with our
thickness error estimates, by assuming the uncertainty range is from
Dmin to Dmax and not reducing by the sum of
squares of sectors or regions.
ResultsGates
Our gate placement algorithm generates 6002 pixels making up 276 gates,
assigned to 176 ice-sheet sectors from .
Previous similar studies have used 230 gates and
178 gates .
The widest gate (∼47 km) is Sermersuaq (Humboldt Gletsjer); the second
widest (∼34 km) is Sermeq Kujalleq (Jakobshavn Isbræ).
Twenty-three additional glaciers have gate lengths longer than 10 km. The
minimum gate width is 3 pixels (600 m) by definition in the algorithm.
The average unadjusted thickness of gates is 405 m with a
standard deviation of 260. The average thickness after adjustment is 439 m
with a standard deviation of 225. A histogram of unadjusted and adjusted
thicknesses at all gate locations is shown in Fig. .
(b) Time series of ice discharge from the Greenland Ice
Sheet. Dots represent when observations occurred. The orange stepped line is
the annual average. Coverage (percentage of total discharge observed at any
given time) is shown in (a) and also by the opacity of dot interior
and error bars in panel (b). When coverage is <100 %, total
discharge is estimated and shown.
(b) Time series of ice discharge by region. Same graphical
properties as Fig. . (a) The region with
highest coverage (CE), lowest coverage (NE), and coverage for the region with
highest discharge (SE) are shown. Coverage for other regions not shown to
reduce clutter.
(b) Time series of ice discharge showing the top eight
(mean of last year) discharging glaciers. Same graphical properties as
Fig. .
(a) Only an example high (Kangerlussuaq Gletsjer) and low
(Nioghalvfjerdsbræ) coverage shown to reduce
clutter.
Discharge
Our ice discharge dataset (Fig. ) reports a total
discharge of 438±43 Gt in 1986, has a minimum of 421±42 Gt in
1995, increases to 452±45 in 2000, and further to 504±49 Gt yr-1 in 2005, after which annual discharge remains
approximately steady at 484 to 503±∼50 Gt yr-1 during the
2005 to 2017 period. Annual maxima in ice discharged occurred in 2005
(504±49 Gt yr-1), 2011 (499±50 Gt yr-1), and 2014
(503±51 Gt yr-1).
At the regional scale, the SE glaciers (see Fig. for
regions) are responsible for 139 to 167 (±11 %) Gt yr-1 of
discharge (30 % to 34 % of ice-sheet-wide discharge) over the 1986 to
2017 period. By comparison, the predominantly land-terminating NO, NE and SW
together were also responsible for 131 to 168 of discharge (∼31 %
of ice-sheet-wide discharge) during this time
(Fig. ). The discharge from most regions has
been approximately steady or declining for the past decade. The northwest (NW) is the
only region exhibiting a persistent increase in discharge – from ∼89
to 113 Gt yr-1 (27 % increase) over the 1998 through 2017 period
(+∼1 Gt yr-1 or +∼1 % yr-1). This persistent
increase in NW discharge offsets declining discharge from other regions. The
largest contributing region, SE, contributed a high of 167±19 Gt in
2005 but dropped to 149 (155) ±18 Gt in 2016 (2017).
Focusing on the top eight contributors (mean of last year) at the individual
sector or glacier scale (Fig. ), Sermeq Kujalleq
(Jakobshavn Isbræ) has slowed down from an annual average high of ∼52 Gt yr-1 in 2012 to ∼45 Gt yr-1 in 2016 and ∼38 Gt yr-1 in 2017, likely due to ocean cooling
. We exclude Ikertivaq from the top eight
because that gate spans multiple sectors and outlets, while the other top
dischargers are each a single outlet. The 2013 to 2016 slowdown of Sermeq
Kujalleq (Fig. ) is compensated for by the many
glaciers that make up the NW region (Fig. ).
The large 2017 reduction in discharge at Sermeq Kujalleq is partially offset
by a large increase in the second largest contributor, Helheim Gletsjer
(Fig. ).
Discussion
Different ice discharge estimates among studies likely stem from three
categories: (1) changes in true discharge, (2) different input data (ice
thickness and velocity), and (3) different assumptions and methods used to
analyze data. Improved estimates of true discharge are the goal of this and
many other studies, but changes in true discharge (category 1) can happen
only when a work extends a time series into the future because historical
discharge is fixed. Thus, any inter-study discrepancies in historical
discharge must be due to category 2 (different data) or category 3 (different
methods). Most studies use both updated data and new or different methods,
but do not always provide sufficient information to disentangle the two. This
is inefficient. To more quantitatively discuss inter-study discrepancies, it
is imperative to explicitly consider all three potential causes of
discrepancy. Only when results are fully reproducible – meaning all
necessary data and code are available (cf.
) – can new works
confidently attribute discrepancies relative to old works. Therefore, in
addition to providing new discharge estimates, we attempt to examine
discrepancies among our estimates and other recent estimates. Without access
to code and data from previous studies, it is challenging to take this
examination beyond a qualitative discussion.
The algorithm-generated gates we present offer some advantages over
traditional hand-picked gates. Our gates are shared publicly, are generated
by code that can be audited by others, and are easily adjustable within the
algorithmic parameter space. This allows both sensitivity testing of gate
location (Fig. ) and gate positions to systematically evolve
with glacier termini (not done here). The total ice discharge we estimate is
∼10 % less than the total discharge of two previous estimates
and similar to that of
, who attribute their discrepancy with
to the latter using only summer velocities,
which have higher annual average values than seasonally comprehensive
velocity products. The gate locations also differ among studies, and glaciers
with baseline velocity less than 100 m yr-1 are not included in our
study due to our velocity cutoff threshold, but this should not lead to
substantially different discharge estimates (Fig. ).
Our gate selection algorithm also does not place gates in northeastern
Greenland at Storstrømmen, Bredebræ, or their confluence, because
during the baseline period that surge glacier was in a slow phase. We do not
manually add gates at these glaciers. The last surge ended in 1984
, prior to the beginning of our
time series, and these glaciers are therefore not likely to contribute
substantial discharge even in the early period of discharge estimates.
We instead attribute the majority of our discrepancy with
to the use of differing bed topography in
southeastern Greenland. When we compare our top 10 highest discharging
glaciers in 2000 with those reported by , we
find that the Køge Bugt discharge reported by
is ∼31 Gt, but our estimate is only
∼16 Gt (and ∼17 Gt in ). The
bed elevation dataset that likely uses the same bed
data employed by has a major depression in the
central Køge Bugt bed (Appendix ). This region of
enhanced ice thickness is not present in the BedMachine dataset that we and
employ (Fig. ). If the
Køge Bugt gates of are in this location,
then those gates overlie ice thicknesses that are
about twice those reported in BedMachine v3. With all other values held
constant, this results in roughly twice the discharge. Although we do not
know whether BedMachine or are more correct,
conservation of mass suggests that a substantial subglacial depression should
be evident as either depressed surface elevation or velocity
.
We are unable to attribute the remaining discrepancy between our discharge
estimates and those by . It is likely a
combination of differing seasonal velocity sampling
, our evolving surface elevation from
, or other previously unpublished algorithmic or
data differences, of which many possibilities exist.
Our ice discharge estimates agree well with the most recently published
discharge estimate , except that our discharge is
slightly less. We note that our uncertainty estimates include the
estimates, but the opposite does not appear to be
true. The minor differences are likely due to different methods.
use seasonally varying ice thicknesses, derived
from seasonally varying surface elevations, and a Monte Carlo method to
temporally interpolate missing velocity data to produce discharge estimates.
In comparison, we use linear interpolation of both yearly surface elevation
estimates and temporal data gaps. It is not clear whether linear or
higher-order statistical approaches are best-suited for interpolation as
annual cycles begin to shift, as is the case with Sermeq Kujalleq (Jakobshavn
Isbræ) after 2015. There are benefits and deficiencies with both methods.
Linear interpolation may overestimate
large changes if there are no other observations nearby in time. Statistical
models of past glacier behavior may not be appropriate when glacier behavior
changes.
It is unlikely that discharge estimates using gates that are only
approximately flow-orthogonal and time-invariant
have large errors due to this, because it is unlikely that glacier flow
direction changes significantly, but our gate-orthogonal treatment may be the
cause of some differences among our approach and other works. Discharge
calculated using non-orthogonal methodology would overestimate true
discharge.
Data availability
This work in its entirety is available at
https://doi.org/10.22008/promice/data/ice_discharge. The glacier-scale, sector, region, and
Greenland summed ice-sheet discharge dataset is available at
https://doi.org/10.22008/promice/data/ice_discharge/d/v0.0.1, where it will be updated as more velocity
data become available. The gates can be found at
https://doi.org/10.22008/promice/data/ice_discharge/gates/v0.0.1, the code at
https://doi.org/10.22008/promice/data/ice_discharge/code/v0.0.1, and the surface elevation change at
https://doi.org/10.22008/promice/data/DTU/surface_elevation_change/v1.0.0.
Conclusions
We have presented a novel dataset of flux gates and a 1986
through 2017 glacier-scale ice discharge estimate for the Greenland Ice
Sheet. These data are underpinned by an algorithm that both selects gates for
ice flux and then computes ice discharges.
Our results are similar to the most recent discharge estimate
, but begin in 1986 – although there is low
coverage and few samples prior to 2000. From our discharge estimate we show
that over the past ∼30 years, ice-sheet discharge was ∼430 Gt yr-1 prior to 2000, rose to over 500 Gt yr-1 from 2000
to 2005, and has remained roughly steady since 2005 at nearly
500 Gt yr-1. However, when viewed at a region or sector scale, the
system appears more dynamic, with spatial and temporal increases and
decreases canceling each other out to produce the more stable ice-sheet
discharge. We note that there does not appear to be any dynamic connection
among the regions, and any increase in one region that was offset by a
decrease in another has likely been due to chance. If in coming years when
changes occur the signals have matching signs, then ice-sheet discharge would
decrease or increase, rather than remain fairly steady.
The application of our flux-gate algorithm shows that ice-sheet-wide
discharge varies by ∼30 Gt yr-1 due only to gate position, or
∼40 Gt due to gate position and cutoff velocity
(Fig. ). This variance is approximately equal to the
uncertainty associated with ice-sheet-wide discharge estimates reported in
many studies (e.g.,
).
We highlight a major discrepancy with the ice discharge data of
, and we suspect this discharge discrepancy –
most pronounced in southeastern Greenland – is associated with the choice of
digital bed elevation model, specifically a deep hole in the bed at Køge
Bugt.
Transparency in data and methodology are critical to move beyond a focus of
estimating discharge quantities towards more operational mass loss products
with realistic errors and uncertainty estimates. The convention of devoting a
paragraph, or even page, to methods is insufficient given the complexity,
pace, and importance of Greenland Ice Sheet research. Therefore the flux
gates, discharge data, and algorithm used to generate the gates, discharge,
and all figures from this paper are freely available. We hope that the flux
gates, data, and code we provide here are a step toward helping others to
both improve their work and discover the errors in ours.
Errors and uncertainties
Here we describe our error and uncertainty treatments. We begin with a brief
philosophical discussion of common uncertainty treatments, our general
approach, and then the influence of various decisions made throughout our
analysis, such as gate location and treatments of unknown thicknesses.
Traditional and mathematically valid uncertainty treatments divide errors
into two classes: systematic (bias) and random. The primary distinction is
that systematic errors do not decrease with more samples, and random errors
decrease as the number of samples or measurements increases. The question is
then which errors are systematic and which are random. A common treatment is
to decide that errors within a region are systematic and that among regions
they are random. This approach has no physical basis – two glaciers a few
hundred meters apart but in different regions are assumed to have random
errors, but two glaciers thousands of kilometers apart but within the same
region are assumed to have systematic errors. It is more likely the case that
all glaciers less wide than some width or more deep than some depth have
systematic errors even if they are on opposite sides of the ice sheet, if ice
thickness is estimated with the same method (i.e., the systematic error is
likely caused by the sensor and airplane, not the location of the glacier).
The decision to have R random samples (where R is the number of regions,
usually ∼18 based on ) is also arbitrary.
Mathematical treatment of random errors means that even if the error is
50 %, 18 measurements reduce it to only 11.79 %.
This reduction is unlikely to be physically meaningful. Our 176 sectors, 276
gates, and 6002 pixels mean that even if errors were 100 % for each, we
could reduce them to 7.5, 6.0, or 1.3 %, respectively. We note that the
area error introduced by the common EPSG:3413 map projection is -5 % in
the north and +8 % in the south. While this error is mentioned in some
other works (e.g., ), it is often not
explicitly mentioned.
We do not have a solution for the issues brought up here, except to discuss
them explicitly and openly so that those, and our own, error treatments are
clearly presented and understood to likely contain errors themselves.
Invalid thickness
We assume ice thicknesses <20 m are incorrect where ice speed is >100 m yr-1. Of 6002 pixels, 5366 have valid thickness and 636
(12 %) have invalid thickness. However, the speeds at the locations of
the invalid thicknesses are generally much less (and therefore the assumed
thickness is less), and the influence on discharge is less than an average
pixel with valid thickness (Table ).
Statistics of pixels with and without valid thickness. Numbers
represent speed (m yr-1) except for the “count”
row.
Good pixelsBad pixelscount5366636mean821266SD1040235min10010125 %23012950 %48717175 %972281max10 0441423
When aggregating by gate, there are 276 gates. Of these, 187 (68 %) have
no bad pixels and 89 (32 %) have some bad pixels, 65 have >50 %
bad pixels, and 61 (22 %) are all bad pixels.
We adjust these thicknesses using a poor fit (correlation coefficient: 0.3)
of the log10 of the ice speed to thicknesses where the relationship is
known (thickness >20 m). We set errors equal to one half the thickness
(i.e., σH=±0.5H). We also test the sensitivity of
this treatment to simpler treatments, and have the following five categories.NoAdj
No adjustments made. Assume BedMachine thicknesses are all correct.
NoAdj+Millan
Same as NoAdj, but using the
bed where available.
300
If a gate has some valid pixel thicknesses, set the invalid
thicknesses to the minimum of the valid thicknesses. If a gate has no valid
thickness, set the thickness to 300 m.
400
Set all thickness <50 to 400 m.
Fit
Use the thickness versus speed relationship
described above.
Table shows the estimated baseline discharge to
these four treatments.
Effect of different thickness adjustments on baseline
discharge.
Finally, Fig. shows the geospatial locations,
concentration, and speed of gates with and without bad pixels.
Gate locations and thickness quality. (a) Locations of all
gates. Black dots represent gates with 100 % valid thickness pixels, blue
with partial, and red with none. (b) Percent of bad pixels in each
of the 276 gates, arranged by region. (c) Average speed of gates.
Color same as panel (a).
Schematic demonstrating coverage. Velocities are filled with linear
interpolation in time, and coverage is weighted by discharge. t columns
represent the same two gate pixels (A and B) at three time steps, where tn
are linearly spaced, but t2 is not observed anywhere on the ice sheet and
is therefore not included. Numbers in boxes represent example discharge
values. Gray parenthetical number is filled, not sampled, in pixel B at time
t3. Weighted filling computes the coverage as 9/11=0.81‾,
instead of 0.5 (half of the pixels at time t3 have
observations).
Missing velocity
We estimate discharge at all pixel locations for any time when there exists
any velocity product. Not every velocity product provides velocity estimates
at all locations, and we fill in where there are gaps by linearly
interpolating velocity at each pixel in time. We calculate coverage, the
discharge-weighted percent of observed velocity at any given time
(Fig. ), and display coverage as (1) line plots
over the time-series graphs, (2) opacity of the error
bars, and (3) opacity of the infilling of time-series dots. Linear
interpolation and discharge-weighted coverage are illustrated in
Fig. , where pixel A has a velocity value at all
three times but pixel B has a filled gap at time t3. The concentration of
valid pixels is 0.5, but the weighted concentration, or coverage, is 9/11
or ∼0.82. When displaying these three discharge values, t1 and t4
would have an opacity of 1 (black) and t3 would have an opacity of 0.82
(dark gray).
This treatment is applied at the pixel level and then weight-averaged to the
gate, sector, region, and ice-sheet results.
Filtered velocity
Here we show the same time series as in Figs. ,
, and , but
without any velocity filtering applied.
Same as Fig. but without the velocity filter.
Note the different y axis.
Same as Fig. but without the velocity
filter. Note the different
y axis.
Same as Fig. but without the velocity
filter. Note the different y axis.
Køge Bugt bed change between Bamber et al. (2013) and
Morlighem et al. (2017b)
Differences between BedMachine and
near Køge Bugt. (a) is baseline ice
speed, (b) BedMachine thickness,
(c) thickness, and (d) the
difference computed as BedMachine - Bamber. The curved line is the gate
used in this work.
Sentinel-1 ice velocity maps
We use ESA Sentinel-1 synthetic aperture radar (SAR) data to derive ice
velocity maps covering the Greenland Ice Sheet margin using offset tracking
assuming surface parallel flow using the digital
elevation model from the Greenland Ice Mapping Project (GIMP DEM, NSIDC 0645)
by and . The operational
interferometric post processing (IPP) chain
, developed at the Technical University
of Denmark (DTU) Space and upgraded with offset tracking for ESA's Climate
Change Initiative (CCI) Greenland project, was employed to derive the surface
movement. The Sentinel-1 satellites have a repeat cycle of 12 d, and due to
their constellation, each track has a 6 d repeat cycle. We produce a
Greenland-wide product that spans two repeat cycles of Sentinel-1 A. The
product is a mosaic of all the ice velocity maps based on 12 d pairs
produced from all the tracks from Sentinel-1 A and B covering Greenland
during those two cycles. The product thus has a total time span of 24 d.
Six-day pairs are also included in each mosaic from tracks 90, 112 and 142
covering the ice-sheet margin in the south as well as other tracks on an
irregular basis in order to increase the spatial resolution.
and have exploited
the high temporal resolution of the product to investigate dynamics of
glaciers. The maps are available from 13 September 2016 and onward, are
updated regularly, and are freely available from http://www.promice.org.
Software
This work was performed using only open-source software, primarily
GRASS GIS and Python, in particular the Jupyter, pandas,
numpy, statsmodel, x-array,
and Matplotlib packages. The entire
work was performed in Emacs using Org Mode. The parallel tool was used to speed up processing. We used
proj4 to compute the errors in the EPSG 3413
projection. All code used in this work is available in the Supplement.
Author contributions
KDM conceived of the algorithm approach and wrote the
code. KDM, WIC, and RSF iterated over the algorithm results and methods. ASO
provided the velocity data. SAK supplied the surface elevation change data.
All the authors contributed to the scientific discussion, writing, and
editing of the manuscript.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
We thank the reviewers for their constructive input that helped improve the
paper. Data from the Programme for Monitoring of the Greenland Ice Sheet
(PROMICE) were provided by the Geological Survey of Denmark and Greenland
(GEUS) at http://www.promice.org. Parts of this work were funded by the
INTAROS project under the European Union's Horizon 2020 research and
innovation program under grant agreement no. 727890.
Review statement
This paper was edited by Reinhard Drews and reviewed by
Ellyn Enderlin and one anonymous referee.
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