Antarctic ice velocity maps describe the ice flow
dynamics of the ice sheet and are one of the primary components used to
estimate the Antarctic mass balance and contribution to global sea level
changes. In comparison to velocity maps derived from recent satellite images
of monthly to weekly time spans, historical maps, from before the 1990s,
generally cover longer time spans, e.g., over 10 years, due to the scarce
spatial and temporal coverage of earlier satellite image data. We found
velocity overestimations (OEs) in such long-span maps that can be mainly
attributed to velocity gradients and time span of the images used. In
general, they are less significant in slow-flowing grounded regions with low
spatial accelerations. Instead, they take effect in places of high ice
dynamics, for example, near grounding lines and often in ice shelf fronts.
Velocities in these areas are important for estimating ice sheet mass
balance and analyzing ice shelf instability. We propose an innovative
Lagrangian velocity-based method for OE correction without the use of field
observations or additional image data. The method is validated by using a
set of ground truth velocity maps for the Totten Glacier and Pine Island
Glacier which are produced from high-quality Landsat 8 images from 2013 to
2020. Subsequently, the validated method is applied to a historical velocity
map of the David Glacier region from images from 1972–1989 acquired during
Landsat 1, 4, and 5 satellite missions. It is demonstrated that velocity overestimations of up to 39 m a-1 for David Glacier and 195 m a-1 for Pine Island Glacier can be effectively corrected. Furthermore, temporal
acceleration information, e.g., on basal melting and calving activities, is
preserved in the corrected velocity maps and can be used for long-term ice
flow dynamics analysis. Our experiment results in the Pine Island Glacier (PIG) show that OEs of a
15-year span can reach up to 1300 m a-1 along the grounding line and
cause an overestimated grounding line (GL) flux of 11.5 Gt a-1 if not corrected. The
magnitudes of the OEs contained in both velocity and mass balance estimates
are significant. When used alongside recent velocity maps of 1990s–2010s,
they may lead to underestimated long-term changes for assessment and
forecast modeling of the global climate change impact on the Antarctic ice
sheet. Therefore, the OEs in the long-span historical maps must be seriously
examined and corrected. We recommend that overestimations of more than the
velocity mapping uncertainty (1σ) be corrected. This velocity
overestimation correction method can be applied to the production of
regional and ice-sheet-wide historical velocity maps from long-term
satellite images.
Introduction
Ice flow velocity fields on the Antarctic ice sheet (AIS) have been mapped
by using synthetic-aperture radar (SAR) and optical satellite images to study ice-sheet-wide ice flow
dynamics and AIS responses to global climate changes (Rignot et al.,
2011a; Gardner et al., 2018; Shen et al., 2018; Greene et al., 2020). An
important direct use of such velocity fields is to estimate the ice
discharge from the AIS to the Southern Ocean and perform mass balance
analyses (Shepherd et al., 2012, 2018; Gardner et al., 2018; Shen et al.,
2018; Rignot et al., 2019). Any errors in the reconstructed velocity fields
can cause uncertainties in the estimated AIS mass balance and associated
contribution to global sea level (GSL) change (Rignot et al., 2011b;
Church et al., 2013; DeConto and Pollard, 2016; IPCC, 2019).
The state of the reconstructed velocity fields can be represented by a
series of velocity maps that are derived from periodical satellite images.
The variations of the available satellite images in both spatial and
temporal coverage determine the extent and time span of the velocity maps.
While ice-sheet-wide annual velocity maps, e.g., in the Inter-mission Time
Series of Land Ice Velocity and Elevation (ITS_LIVE) project
(Gardner et al., 2018, 2019), have been produced from recent satellite
images (Storey et al., 2014; Lillesand, 2015; Li and Roy, 2017; Sabins
and James, 2020), regional velocity maps at a seasonal or monthly scale
have been generated from optical and SAR images (e.g., Landsat and Sentinel;
Frezzotti et al., 1998, 2000; Nakamura et al., 2010; Zhou et al., 2014;
Greene et al., 2017, 2018, 2020; Moon et al., 2021; Wulder et al., 2019). Furthermore, a weekly
ice velocity mapping scheme based on multi-mission satellite images was
proposed in Altena and Kääb (2017). However, owing to low image
quality, large geolocation errors, and low temporal and spatial coverage,
satellite images prior to 1990 are generally less available; appropriate
images for ice-sheet-wide or large regional velocity mapping with a shorter
time span (e.g., seasonal or annual), especially from 1960s and 1970s, are
scarce. Therefore, earlier velocity maps have been produced with a limited
extent and a longer time span. For example, combinations of first-generation
film-based ARGON images of 1963 (Ruffner, 1995; Kim, 2004), early
Landsat MSS images of 1970s, and TM images of 1980s (Chander et al., 2009)
have been used to create regional velocity maps with a time span ranging
from 1 to 23 years (Bindschadler and Scambos, 1991; Bindschadler et al.,
1996; Wang et al., 2016; Cheng et al., 2019; Rignot et al., 2019), although
the unique case of a 1963 velocity map of the Rayner Glacier from two ARGON
stereo image pairs has been presented (Li et al., 2017). Such a long time
span is problematic when we use the feature-matching technique for velocity
mapping. For example, at time1 a feature, with an initial velocity v0 at the
first location, is taken in the first image. The same feature is tracked in
the second image taken at time2 after traveling at the velocity v0 and an
acceleration a for a time span of Δt(time2–time1). Thus, the velocity v=v0+aΔt increases along with the time span Δt if acceleration a exists.
Given a constant acceleration, the velocity can be overestimated if the time
span is long or the velocity overestimation is proportional to the time
span. With a 10-year span, an overestimation of ∼ 70 m a-1 (or ∼ 10 %) was estimated for an area near the
grounding line of Totten Glacier (Fig. A1) (Chad A. Greene, personal communication, 2020). Berthier et
al. (2003) compensated for the overestimations in a Mertz Glacier mapping study
with an image span of 11 years by assigning the overestimated velocities to
middle points of traveled segments. This technique should have corrected a
large portion of the overestimation given the relatively weak spatial
velocity gradient along the main trunk of the glacier. Although it has not
been brought to further attention in publications, given its nature and
magnitude, this velocity overestimation issue should be fully understood, and
a comprehensive correction method should be developed so that corrected
historical velocity maps can be analyzed alongside modern maps to create a
long record of cohesive ice flow dynamics. Furthermore, this capability of
building a long record of AIS ice flow dynamics is important for estimation
of long-term AIS mass balance and prediction of the future GSL contribution
(Rignot et al., 2019).
This study is a part of our efforts to develop an ice velocity map of East
Antarctica for 1963 to 1989 (Li et al., 2017; Ye et al., 2017; Cheng et
al., 2019). In this paper we prove the existence of ice velocity
overestimation in long-term historical velocity maps and present an
innovative correction method. The proposed correction method is based on the
Lagrangian velocity that can be calculated from the overestimated velocity
map itself and thus does not require any field observations or additional
satellite images. We used a set of ground truth velocity maps with time
spans of 1 to 7 years derived from recent Landsat 8 images of 2013 to 2020
from the Totten Glacier (TG), East Antarctica, to validate the correction
method. We then applied the validated method to a historical velocity map in
the David Glacier region, East Antarctica, which was produced from images
from 1972 to 1989 acquired during satellite missions of Landsat 1, 4, and 5.
We show that the 17-year velocity overestimation can be successfully
corrected to within the uncertainty (1σ) of the 1-year map. Another
experiment at the Pine Island Glacier (PIG), one of the most dynamic
glaciers in Antarctica, was carried out to demonstrate that the
overestimation correction method can effectively adjust the overestimated
velocities as large as 195 m a-1, while preserving the velocity change
signature caused by temporal accelerations due to basal melting and calving
activities for long-term ice flow dynamics analysis.
MethodsThe velocity overestimation issue
We describe an acceleration-induced overestimation using a typical
scenario in AIS (Fig. 1a) where ice flow accelerates over a long slope from
several glaciers originating from the inland interior, running through the
main trunk, and discharging to the ocean (Bamber et al., 2000; Cuffey and
Paterson, 2010; Rignot et al., 2011a). In order to quantify the velocity
overestimation, for any point Po(xo,yo) on a glacier (Fig. 1a), we
define its velocity in two different frameworks, Eulerian and Lagrangian
(Chu and Fan, 2014; Chenillat et al., 2015; Altena and Kääb,
2017). First, the velocity in the Eulerian framework (E velocity) is defined
as VE=D/dt, the straight-line distance (D) divided by the time span (dt). Note that for
simplicity in equation derivation and discussion, we interchange a vector
with its scalar; thus, velocity and speed are not strictly distinguished in
this paper. Hence a velocity field described by a velocity map is also
defined in the Eulerian framework. In reality, the start point of D is
measured on the first image and end point on the second image; the two
images are taken with a time span of dt apart (Li, 1998; McGlone, 2013). The
reconstructed velocity map is stored as velocity components
(VxE,VyE) in the x and y directions, from which the velocity
field can be reconstructed. The velocity, represented by a velocity map, may
indicate an instantaneous or average velocity depending on the time span
dt (Cuffey and Paterson, 2010; Rignot et al., 2011a). Second, the velocity
in the Lagrangian framework (L velocity) is defined as VL=S/dt, the
curved distance traversed along the flow line (S) divided by the time span (dt; Altena and Kääb, 2017). Given a
time span and an initial point Po(xo,yo) (Fig. 1a), its
L velocity can be determined by tracking the point using a velocity map
(Chu and Fan, 2014; Chenillat et al., 2015). Operations in the Lagrangian
framework are often performed to estimate advected ice features (Altena and Kääb, 2017) or forecast future events in earth science
applications, e.g., mud and debris flow of landslides (Debella-Gilo and
Kääb, 2013; Feng et al., 2016), ocean currents at different depths
(Glenn et al., 2016; van Sebille et al., 2018), and storm center motion of
a typhoon or hurricane (Cram et al., 2007; Euler et al., 2019).
Illustration of acceleration-induced overestimation in a
velocity map derived from a satellite image pair with a long time span: (a)
schematic scene of accelerated ice flow in a glacier–ice-shelf system in
AIS, (b) calculation of E velocities at
Po(xo,yo) with an equal increment in
time span (1 year), and (c) increase in E velocity at the same point
Po(xo,yo) as the time span increases –
the cause of the acceleration-induced velocity overestimation.
Assume that we use a set of n+1 images (Imagei, i=0,1,…n) that are taken with a
time interval of 1 year to produce n velocity maps, V0-i, each derived
from an image pair (Image0, Imagei). The time span of the maps increases from
1 to n years. These maps are defined in the Eulerian framework (Fig. 1b). For
Po(xo,yo) its location after i years, Pi(xi,yi), is
determined in Imagei; its Eulerian distance D0-i is measured using both
Image0 and Imagei (i=1,2,…n). Consequently, the E velocity of a i-year span
(Δto-i) at Po(xo,yo) is
V0-iE=D0-iΔt0-i.
As the time span increases at a fixed rate of 1 year, the traversed
straight-line distance D0-i (red lines in Fig. 1b), correspondingly
E velocity V0-iE, increases rapidly because of the acceleration over
the traverse (Fig. 1c). In principle, every V0-iE (i=1,2,…n) value
represents the velocity at the same point Po(xo,yo) (Fig. 1a) in
these n velocity maps. In the cases where Imagei was not available and thus
the maps V0-i (i=1,…n-1) were not produced, we only had the map V0-n with
the longest span of n years. It is obvious that at Po(xo,yo) its
n-year velocity V0-nE is significantly larger than the 1-year
velocity V0-1E (Fig. 1c). In general, we define the velocity
overestimation of a i-year E velocity as
OE0-i=V0-iE-V0-1E.
Here we use a velocity map of a 1-year span as a baseline (overestimation-free) throughout the paper for simplicity, which can be changed for
glacier regions of different ice flow dynamics (spatial acceleration, mainly
caused by bed topography and slopes). For example the baseline span is 1 year for TG in Experiment 1 and 3 months for PIG in Experiment 3. We require that the
overestimation of the baseline map is negligible or smaller than σ
(velocity mapping uncertainty).
Overestimation correction based on Lagrangian velocity
We propose a method for correction of overestimation in a long-term velocity
map based on its Lagrangian velocities. The velocity field described by the
baseline map is used to calculate trajectories of ice mass and L velocities.
The objective is to calculate the overestimation OE0-i in V0-iE
(i=2,…n) so that these velocities of longer spans are corrected using Eq. (2) and describe the same velocity field as the 1-year V0-1E.
For point Po(xo,yo) in Fig. 1a, the E velocity V0-iE
(i=1,2,…n) is presented by individual bars in Fig. 2. Correspondingly its
L velocity of i-year span can be calculated as (Halliday et al., 2013)
V0-iL=S0-iLΔt0-i,
where S0-i is the Lagrangian trajectory (L trajectory) distance
(i=1,2,…n):
S0-iL=∫t0tiV0-1E(t)dt.
Within a baseline time span (e.g., 1 year or shorter), the difference
between the E distance and L distance may be considered within the mapping
uncertainty (1 σ): V0-1E≈V0-1L.
Beyond that span, the curved trajectory distance is longer than the straight
distance (S0-i≥D0-i, see Fig. 1b); thus, we have
V0-iL≥V0-iE (i=2,3,…n). An approximation of the increased
difference between V0-iL and V0-iE is presented by the
trends of L velocity (blue line) and E velocity (red line), which start from
the same velocity at V0-1L (or V0-1E) and reach the
maximum difference between V0-nL and V0-nE at the end (Fig. 2).
Derivation of equation for overestimation correction using
L velocity. Eulerian velocities
V0-iE (i=1,2,…n) are represented as bars. The red line is the average Eulerian velocity V‾0-iE of V0-1E
and V0-nE.
The blue line is the average Lagrangian velocity V‾0-iL of
V0-1L and
V0-nL derived from
V0-1E. The black line is the
average Lagrangian velocity V‾0-iL(n) of
V0-1L(n) and V0-nL(n) derived
from V0-nE.
Premise I. Within a baseline time span (e.g., 1 year or shorter) each segment (from Pi-1 to Pi in Fig. 1b) is
relatively short, and the E and L velocity difference is smaller than σ. Furthermore, over the map span of n years (e.g., 5–10 years or longer)
the accumulated E and L distances along the entire trajectory do not
deviate significantly from each other, so that the maximum velocity
difference in the Lagrangian and Eulerian frameworks (end points of blue and
red lines in Fig. 2) is limited within a threshold (V0-nL-V0-nE≤kσ), where k is a constant and σ is the
velocity mapping uncertainty.
In reality, the available historical images may only allow us to produce an
E velocity map with the longest time span, i.e., V0-n, which leads to the
maximum overestimation as defined in Eq. (2). Based on this map
V0-n of n-year span, the i-year span L velocity (black line in Fig. 2)
is defined as follow:
5V0-iL(n)=S0-iL(n)Δt0-iand6S0-iL(n)=∫t0tiV0-nE(ti)dt.
Consequently, V0-1L(n) (start point of black line in Fig. 2) is
set to be equal to V0-nE.
Premise II. Within the time span of n years (e.g.,
over 5–10 years), the velocity field described by V0-1E and
V0-nE does not change significantly, so that the line between
V0-nL(n) and V0-1L(n) (black line in Fig. 2) and that
between V0-nL and V0-1L (blue line in Fig. 2) are
approximately parallel to each other. Accordingly, the difference between
their simple averaged accelerations is within a threshold (a‾VL(n)-a‾VL≤k′σΔtn), where k′ is a constant and σ is
the velocity mapping uncertainty, and
a‾VL(n)=V0-nL(n)-V0-1L(n)Δtn,a‾VL=V0-nL-V0-1LΔtn.
Given the long-span velocity V0-nE we can calculate the n-year
L velocity V0-nL(n). Premise II measures the degree of spatial acceleration
invariance over the n-year span (parallelity between black and blue lines in
Fig. 2). Thus, it is a necessary condition for using the difference of V0-nL(n)-V0-1L(n), or ∼ (V0-nL(n)-V0-nE),
from the n-year-span map to substitute for the difference of
V0-nL-V0-1E from the 1-year-span map; furthermore, based on
Premise I the overestimation can be computed as
OE0-n=V0-nE-V0-1E=V0-nL-V0-1E=V0-nL(n)-V0-nE.
Consequently, using the map with the longest time span of n years, we can
effectively go back to V0-1E using a correction term defined as
correction =V0-nE-V0-nL(n).
In principle, Premise I presents a sufficient condition, V0-nL-V0-nE≤kσ. In case Premise I does not hold, for example, because of temporal
accelerations induced by calving and basal melting activities (Experiment 1 and Experiment 3), we
are still able to correct the OE portion induced by spatial acceleration and
preserve the uncorrected OE portion induced by temporal acceleration in the
residual ε for ice flow dynamics change analysis (Fig. A2).
Overestimation correction theorem. Assume that the
necessary condition in Premise II is met; spatial-acceleration-induced
overestimations in long-time-span velocities V0-nE can be corrected
or reduced using the following correction term, regardless of temporal acceleration:
9V0-1E=V0-nE+correction+ε,10correction=V0-nE-V0-nLn.
If Premise I holds, correction ≈-OE0-n; otherwise, correction<OE0-n, preserving the velocity increases
induced by temporal accelerations in the residual term ε (Fig. A2).
Additionally, given n discrete E velocities V0-iE (i=1,2,…n; Fig. 2), the
acceleration can be estimated using a linear regression model. Its least
squares (LS) estimation is (Montgomery et al., 2021)
a(VE)=n∑iV0-iE-∑i∑V0-iEn∑i2-∑i2.
On the other hand, this acceleration can also be approximated by an averaged
acceleration using the initial and end velocities:
a‾(VE)=V0-nE-V0-1EΔtn.
The two acceleration estimates can be compared to determine if the averaged
acceleration be used in the event that the intermediate velocities
V0-iE (i=2,…n-1) are not available. The red line is then used instead of
V0-iE (i=1,2,…n; Fig. 2).
Implementation aspects
Trajectory andLvelocity computation. The computation of the
L trajectory distances S0-iL and S0-iL(n) involves the
numerical implementation of the integral of the velocity field from
Po(xo,yo) to Pi(xi,yi) along a flow line on the maps of
V0-1 and V0-n using Eqs. (4) and (6), respectively. We select
an area of the velocity map of 2013 of Pine Island Glacier, West Antarctica
(Fig. 3a and b; Gardner et al., 2018), to show the details of integral
implementation. The velocity map has a time span of 1 year and a resolution
of 240 m. At each grid cell the velocity is stored according to its
components (VxE,VyE). We preprocess each component separately
by using a moving window smoothing filter to reduce noise. A 10-year
trajectory goes through an ice flow profile from P0 at 2560 m a-1
to P10 at 3935 m a-1 with an average acceleration of 138 m a-2. As shown in the enlarged area for the annual trajectory from P2
to P3 (Fig. 3c), the integral is carried out by accumulation of
sub-trajectories (between white dots) with a monthly increment. The sub-grid
positions of the monthly segments are interpolated for integration of the
overall L distance. Finally, we have the result of V0-1E= 2560 m a-1, S0-10L= 34 960 m, and V0-10L= 3496 m a-1,
compared to V0-10E= 3491 m a-1. The 10-year overestimation
(V0-10E-V0-1E) at P0 is approximately 931 m a-1
(36 %). Examples of overestimation correction in PIG are given in
Experiment 2 and the Discussion section.
Denoising and map quality. The quality of the reconstructed
velocity field is important for L trajectory calculation and subsequent
overestimation correction. Compared to the above Landsat 8 velocity map,
velocity maps derived from historical images may have higher uncertainties
due to the low image quality that is inherent in the satellite images of
early missions (Kim, 2004; Ye et al., 2017). In some cases, there may be
even gaps in the velocity map because image features may not be tracked by
using the optical flow or feature-matching techniques when ice features
disappear through glaciological processes or large advection motion in ice
shelf fronts (Scambos et al., 1992; Schenk, 1999; McGlone, 2013).
Therefore, preprocessing of the velocity maps should be performed to
eliminate outliers; then interpolation may be applied to fill small gaps so
that the integral of L trajectories and the calculation of L velocities can
be realized. Despite the sub-pixel accuracy of the orthorectification of
historical images and the sub-pixel precision of the cross-correlation-based
feature-matching method, the reconstructed velocity maps may still be
subject to a systematic bias called pixel locking, which results in matched
features moving close to integer pixel positions (Shimizu and Okutomi,
2005). This pixel level bias may introduce additional uncertainties to areas
of large velocity gradients. The pixel-locking effect can be reduced by
using a coarse-to-fine hierarchical feature-matching method as demonstrated
in the Antarctic and planetary environments (Debella-Gilo and
Kääb, 2011; Li et al., 2011; Heid and Kääb, 2012; Li et
al., 2017).
Implementation of numerical integral and L velocity
calculation in Pine Island Glacier, West Antarctica: (a) black rectangle
indicates the area of 10-year trajectory in panel (b), numbered yellow rectangles
with crosses are five selected areas in Experiment 3 (Fig. A3), and background is a 1-year velocity map (Gardner
et al., 2018); (b) enlarged trajectory area with annual points from
P0 to
P10; and (c) numerical integral of
the third-year sub-trajectory from
P2 to
P3 with a monthly increment.
ResultsExperiment 1: validation of the correction method at Totten
Glacier, East AntarcticaPreparation of validation velocity maps
This experiment is designed to validate the proposed velocity
overestimation correction method using the velocity map of Totten Glacier (TG) (Fig. 4a; Gardner et al., 2019),
which is one of the most active glaciers in East Antarctica and has been
experiencing significant mass loss since 1989 (Li et al., 2017; Gardner et
al., 2018; Shen et al., 2018). Figure 4b shows an ice flow velocity map of
TG from 2013 from ITS_LIVE (Gardner et al., 2019). This 240 m resolution velocity map was derived from the 15 m resolution Landsat 8
images with a 1-year span (January to December 2013). Under the primary
forces of gravity, margin shear, basal drag, and others (Bamber et al.,
2000; Cuffey and Paterson, 2010; Rignot et al., 2011a), the ice flow
accelerates over a path of 130 km along the centerline at a velocity of
∼ 600 m a-1 upstream of the grounding line to
∼ 2300 m a-1 at the ice shelf terminus.
Totten Glacier as an example for the validation of the
velocity overestimation correction method: (a) the Totten Glacier region and
five areas (white rectangles) selected in different parts of the ice shelf;
background is Landsat 8 image of 3 December 2013; black line is grounding
line from Rignot et al. (2011c); and (b) velocity map of
Totten Glacier of 2013 (Gardner et al., 2018); red and
yellow lines are the shelf front of 6 April and 9 September 2015,
respectively; light blue line is the PSI boundary (Fürst et al., 2016).
In order to validate the velocity overestimation correction method, we need
to determine if premises I and II are fulfilled. This further requires us to have E velocity
from the map series of V0-i (i=1,2,…n), which are not readily available
except the 1-year map of 2013 in Fig. 4b. To avoid lower-quality historical
velocity maps that may influence the effectiveness of the validation, we use
the earliest available high-quality Landsat 8 images from 2013 to 2020 to
produce velocity maps V2013–i (i= 2014, … 2020). The validation should also consider
velocities of areas with different ice flow dynamics. Thus, we select five
areas (white rectangles in Fig. 4) to produce smaller maps (∼ 10 km ×∼ 20 km), which contain 7-year trajectories
and represent different ice flow dynamic characteristics of the ice shelf,
i.e., Area 1 close to the grounding line, areas 2 and 4 along the main trunk, Area 5 on the tributary
flow, and Area 3 near a suture zone between the main and tributary glaciers.
Using the matching method adapted for ice surface features (Li et al.,
2011, 2017), we produced 35 smaller-sized velocity maps V2013–i
(i= 2014, … 2020) for seven time spans and five areas. Information about the Landsat 8
images used and the acquisition dates are given in Table A1. The OE-free
velocity maps of 1-year V2013–2014 of all five areas are illustrated in
panels 1a–5a of Fig. 5. The corresponding overestimated maps of 7-year span
V2013–2020 are shown in panels 1b–5b.
Velocities in five areas (rectangles in Fig. 4) of the
Totten Glacier are used to validate Premises I
and II. Panels (1a)–(5a) show the reconstructed
1-year velocity maps V2013–2014 in
the areas; matched points (red triangles) are used to map E velocity
V2013–iE (i= 2014, … ,
2020) (red lines in panels 1c–5c); points along the flow line
(blue dots) are tracked from the 1-year maps and used to calculate
L velocity V2013–iL (blue lines in panels 1c–5c). Similarly, panels (1b)–(5b)
illustrate the reconstructed 7-year velocity maps
V2013–2020 in the areas with the
matched points (red triangles) for E velocity
V2013–iE7; points along the flow line (black crosses) are tracked
from the 7-year-span velocity map and used to calculate L velocity
V2013–iL7 (black lines in panels 1c–5c). In
each area (panels 1c–5c) the difference between red and blue lines
increases with time but is limited within kσ at the end
(Premise I), except a large k in
Area 4 because of the effect of a calving event; the
black line is above the blue line due to the spatial-acceleration-induced OE
over 7-year span, but they are relatively parallel (Premise
II), and thus a correction can be estimated.
The uncertainty of the maps is estimated from two error sources: the
geolocation error of the satellite images σRef and the error
of feature matching σMatch:
σVelocity=1ΔtσRef2+σMatch2,
where σRef for Landsat 8 images is 18 m (Storey et al.,
2014) and σMatch is set to 8 m, ∼ 0.5 pixel (Li, 1998; Li et al., 2017). Thus, the uncertainty of the measured E distance is
20 m. Consequently, the velocity uncertainty σVelocity
becomes 20 m a-1 and 3 m a-1 for a 1-year map and a 7-year map, respectively.
In each area, we select an ice feature in the upper stream part in 2013. Its
locations in the following 7 years are determined from the satellite images
of 2014–2020 using the feature-matching method and shown as red triangles
in the maps (Fig. 5). Correspondingly, we track its annual trajectory
locations from the 1-year and 7-year maps using Eqs. (4) and (6) and
plotted them as blue dots and black crosses, respectively, on the maps.
Using these trajectories in the Eulerian and Lagrangian frameworks we
calculate V2013–iE, V2013–iL, and V2013–iL7 and plot them as red, blue, and black curves in panels 1c–5c,
respectively.
Validation of premises I and II
In general, the L velocities from the 1-year map (blue curves in Fig. 5) are
mostly faster than the corresponding E velocities (red curves). To validate
Premise I we further examine the L velocity V2013–2020L and E velocity
V2013–2020E of the 7-year span, i.e., differences between the end
points of blue and red curves in all five areas (panels 1c–5c of Fig. 5).
They are less than or equal to 8 m a-1 ((VL-VE)2013–2020 in
Table 1), except -31 m a-1 in Area 4, which was influenced by a calving event
in 2015 as mentioned in the Discussion section. The averaged difference of
all five areas is -4± 16 m a-1, that is, within 20 m a-1,
the uncertainty (1σ) of the 1-year velocity map (2013–2014) that is
considered OE-free. Consequently, the conditions in Premise I are met.
Velocity and acceleration in Eulerian and Lagrangian
frameworks used for validation of the overestimation correction method.
“Actual E velocity and OE” lists actually
mapped 1-year and 7-year E velocities and their differences as
overestimations in all five areas. Premise I
contains 7-year L velocities computed from the 1-year velocity map and
corresponding L and E velocity differences, which are used for validating Premise I. Premise
II illustrates averaged L accelerations computed from the 7-year
and 1-year velocity maps, respectively, as well as their differences, which
are used for validating Premise II.
“Overestimation correction” presents 7-year
L velocities computed from the 7-year map, overestimation corrections, and
E velocities and residuals (or errors) after correction.
AreaActual E velocity and OE Premise I Premise II Overestimation correction V2013–2014EV2013–20EOEActualV2013–20L(VL-VE)2013–20a‾VL(7)a‾VLa‾VL(7)-a‾VLV2013–20L(7)Corr.VCorr.Eε(m a-1)(m a-1)(m a-1)(m a-1)(m a-1)(m a-2)(m a-2)(m a-2)(m a-1)(m a-1)(m a-1)(m a-1)18148432985183.44.6-1.2861-188251121005100831007-11.60.41.21017-9999-631285131732132033.65.7-2.11337-20129712414141483691452-313.43.9-0.51502-1914645057547893579124.56.0-1.5819-307595Mean10541088341084-43.34.1-0.81107-19106914SD28930224290161.12.31.3300730421
For each area we calculated a least squares acceleration estimate
a(V2013–iL7) of the 7-year L velocity
(black curves in panels 1c–5c of Fig. 5) over all years (i= 2013, 2014, … 2020) using
Eqs. (11); then we calculated an average acceleration a‾V2013–2020L7
using Eq. (12) based only on the velocities of the beginning and end
years (2013 and 2020). In all five areas their differences (Table A2) are
less than 1.3 m a-2, which is less than σa (3 m a-2),
the acceleration uncertainty estimated from the velocity uncertainty of 20 m a-1 (1σ). That means that within the uncertainty of the 1-year
velocity map we can substitute the more rigid LS acceleration
a represented by a black curve in Fig. 5 with the average
acceleration a‾ estimated from the straight line
between the beginning and end velocities. Similarly, the differences between
accelerations a and a‾ of the 1-year
L velocity V2013–iL in all five areas (Table A2) are also less than
1.5 m a-2. Within the uncertainty of the 1-year velocity map we can
then substitute a (blue curve in Fig. 5) with a‾ (straight line) for the 1-year L velocity V2013–iL.
Validation of Premise II requires us to examine the accelerations represented by the
7-year L velocity V2013–iL7 (black curves in Fig. 5)
and the 1-year L velocity V2013–iL (blue curves). The relatively
well maintained parallelity of the black and blue curves in all five areas
shows that the velocity trend in the Lagrangian framework has not changed
significantly over the 7 years. This is further quantified by the
differences between the averaged 7-year and 1-year accelerations a‾VL(7)-a‾VL in Table 1. The differences are within
±2.1 m a-2 in all five areas, with an overall difference of -0.8±1.3 m a-2, which are all within σa (3 m a-2).
Thus, the conditions in Premise II are also fulfilled.
Overestimation correction
As shown in the first part of Table 1, we compare the overestimated
E velocity V2013–2020E in the 7-year map V2013–2020 with
V2013–2014E in the 1-year map V2013–2014. Their differences make the actual overestimation OEActual in five areas. The mean overestimation of 34 m a-1 with a range from 3 to 69 m a-1 is significant in comparison to σ (20 m a-1). Thus, the overestimation should be
corrected. It should be noted that a higher overestimation is expected for
historical velocity maps of longer time spans, e.g., 10–15 years.
Using our correction method in Eqs. (9)–(10), we estimated the
correction term (V2013–2020E-V2013–2020L(7)) and applied it to the overestimated V2013–2020E(Table 1). After the overestimation
correction, the corrected velocity VCorr.E from the 7-year span
V2013–2020E is agreeable to the 1-year span V2013–2014E
within an average difference of 14 m a-1 (ε, residual),
which is less than σ (20 m a-1). Overall, the overestimations
are effectively corrected.
Experiment 2: velocity overestimation correction in the David
Glacier region, East Antarctica
Experiment 2 in the David Glacier region on the Scott Coast, East Antarctica,
demonstrates the applicability of the introduced method to correct the
overestimation in historical multi-year span velocity maps derived from
long-term images from 1972 to 1989. Velocities in this region from 1988 to
1992 were mapped by using GPS and image-feature-tracking techniques
(Frezzotti et al., 1998, 2000). A new GPS campaign was carried out in the
region during 2005–2006 (Danesi et al., 2008). Velocity changes from 2016 to
2020 were detected using Sentinel-1A SAR images (Moon et al., 2021). In this
experiment we produced a velocity map of the region from 64 Landsat images
collected from 1972 to 1989 (Table A3). The image pairs used for velocity
mapping have mainly four time spans, namely 1, 15, 16, and 17 years, with
their footprints illustrated in Fig. 6a. The Landsat images were first
orthorectified using ground control points (GCPs), which were selected at
outcrops, blue ice, and ice rises (Ye et al., 2017). Then the velocity
field was reconstructed in three steps: (a) measurement of a set of manually
selected seed points for representing the initial structural velocity
information in stationary, low velocity, and dynamic flow areas; (b) velocity
point generation by the ice-surface-feature-based matching method controlled
by a triangular irregular network (TIN) model and initiated by the seed
points; and (c) dense grid matching for generation of the gridded ice
velocity field with a mixed time span from 1972 to 1989. This feature and
grid image matching method has been developed for surface mapping from
optical satellite images in challenging planetary and polar environments
and successfully applied to the reconstruction of the ice velocity fields in
the Rayner and Fimbul ice shelves in East Antarctica from historical optical
satellite images of 1963–1987 (Li et al., 2011, 2017). The produced
velocity map has a grid spacing of 500 m after a kriging interpolation from
all matched feature and grid points (Fig. 6b). The uncertainty (1σ)
of the velocity points is 4, 31, and 62 m a-1 for
time spans of 15–17 years, 1 year (TM), and 1 year (MSS), respectively.
Application of the overestimation correction method to a
historical velocity map of 1972–1989 of the David Glacier region, East
Antarctica. (a) Footprints and time spans of the Landsat image pairs used
for velocity mapping with a background of Landsat Image Mosaic of Antarctica
(Bindschadler et al., 2008), (b) produced velocity map
of 1972–1989 and selected points for examination of velocity overestimation
and correction, (c) overestimation values of all velocity points generated
using the image-feature-matching technique, and (d) velocity points with OE ≥ 1σ (red
points).
At all velocity points, including 8564 matched features from the feature-matching process and 59 783 grid points from the dense-grid-matching
process, we calculated the velocity overestimation values (Fig. 6c). The OEs
are generally low on the grounded ice, mostly within ±4 m a-1,
which are less than the mapping uncertainties (1σ) of different time
spans (Fig. 6d and Table 2). The higher OEs ranging from ∼ 4 to ∼ 50 m a-1 are on the ice tongue, ice
shelves, and glaciers. We selected a total of 16 points from four glaciers
and the inland region to examine detailed OEs in different areas of ice flow
dynamics (Fig. 6b and c). The OEs at three points in the inland region
(IL1–IL3) are negligible (Table 2). Points DG1–DG6 along the main trunk of
the David Glacier present an increasing trend of OEs from 1 to 4 m a-1
on the ice tongue up to 36 m a-1 at the ice shelf outlet, as the
average Lagrangian acceleration a‾VL(n) increases from 0.1 to 2.2 m a-2. However, DG7 is located in a relatively steady velocity area and
thus has negligible overestimation. Furthermore, the OE values at the
selected points on the smaller glaciers, Reeves Glacier (RG1 and RG2),
Priestley Glacier (PG1 and PG2), and Campbell Glacier (CG1 and CG2), range
from 0 to 12 m a-1 and are generally greater than those on
the grounded ice. At the velocity points where the OEs are greater than or
equal to the velocity mapping uncertainties (1σ), the corrected
velocities are calculated by applying the overestimation corrections.
Ice flow dynamics parameters, computed OEs, and
corrected velocities at 16 selected points.
Overall, for a historical velocity map of 1972–1989 with time spans from 1
to 17 years in the David Glacier region, the overestimations and corrections
mainly occur on the David Glacier and smaller glaciers where accelerations
in the spatial domain (not the temporal domain) exist, while those on the
grounded ice are non-significant. We recommend that the overestimations that
are greater than or equal to the mapping uncertainties (1σ) be
corrected.
Experiment 3: velocity overestimation correction at Pine Island
Glacier, West Antarctica
We further apply the OE correction method at the Pine Island Glacier, one of
the most dynamic glaciers in Antarctica. Four areas (1, 2, 3, and 4) along
the main trunk and one area (5) in the margin in PIG are selected (Fig. 3a).
The longest time span of 7 years (2013–2020) is determined, restricted by
the very high velocity (up to ∼ 4500 m a-1) on the ice
shelf, untraceable surface features over long spans, and availability of the
high-quality Landsat 8 images. Hence, most of the ice shelf cannot be
covered by a 7-year span velocity map, and the five selected areas are near
and upstream from the grounding line. The five areas were all mapped with a
baseline span of 3 months (panels 1a–5a in Fig. A3) but different longest
spans (n years, n=6, 7, 7, 7, and 3 for the five areas) (panels 1b–5b in
Fig. A3) because of difficulties in image feature tracking. Accordingly, in
each area one surface feature was selected and tracked for n years to
estimate E velocity V2013–iE; its L velocities of V2013–iL
and V2013–iL(n) were computed from the 3-month-span and n-year-span maps, respectively. Consequently, the resulting E velocity (red lines),
L velocity from the 3-month-span maps (blue lines), and L velocity from
the n-span maps (black lines) are illustrated in panels 1c–5c in Fig. A3.
The mapped ground truth data of 3-month- and 7-year-span velocities in
five selected areas showed actual OE values in PIG from 18 m a-1 in
Area 5 of the inland interior to 626 m a-1 in Area 2 near the grounding line at the glacier outlet (Table 3). Much of these OEs are attributed to the temporal
acceleration patterns in panels 1c–5c in Fig. A3 where the tracked
E velocities V2013–iE of red lines are, in the later years,
consistently above the L velocities V2013–iL of blue lines from the
3-month-span maps of 2013. These temporal accelerations were mainly
caused by the drastic calving activities in and after 2017 in PIG and
continuous basal melting in the Amundsen Sea sector during the study period
(Joughin et al., 2020, 2021). Furthermore, the black and blue lines are
approximately parallel (panels 1c–5c in Fig. A3). Hence, the proposed method
corrected on average 97 m a-1 of the spatial-acceleration-induced OE,
which is ∼ 40 % of the overall OE (245 m a-1) in five
areas (Table 3). The average residual of 148 m a-1, accounting for
another 60 % of the overall OE, represents the OE portion induced by the
temporal acceleration in Eq. (9).
Application of the OE correction method in PIG. “Actual
E velocity and OE” includes E velocities of 3-month and n-year spans and
their differences as OEs. “Overestimation correction” presents n-year
L velocities from n-year-span map, OE corrections, corrected E velocities,
and residuals (or errors) after correction.
Area ID (span)Actual E velocity and OE Overestimation correction V3 monthsEVnEOEActualVnL(n)CorrectionVCorr.Eε(m a-1)(m a-1)(m a-1)(m a-1)(m a-1)(m a-1)(m a-1)1 (6)393540541194090-364018832 (7)327839046264099-19537094313 (7)234125321912599-6724651244 (7)165119222712103-1811741905 (3)39241018416-640412Mean231925642452661-972467148SD138715062321538861477163
In summary, OEs exist in historical long-span velocity maps of Antarctica,
from smaller glaciers, such as David Glacier, to the fast-flowing glacier of
PIG, where spatial accelerations are produced by, e.g., bed topography and
slopes. In general, they are less significant in slow-flowing grounded
regions with low spatial accelerations. Instead, they take effect in places
of high ice dynamics, for example, near grounding lines and often in ice
shelf fronts. Velocities in these areas are important for estimating ice
sheet mass balance and contribution to global sea level changes, as well as for
analyzing ice shelf instability. For instance, in the David Glacier the
large OE corrections (up to 36 m a-1) occur on the ice shelf (Fig. 6).
The OEs of a 7-year span, up to 69 m a-1 (Table 1), would be about
∼ 50 % of the velocity increase detected during 1989–2015
in the Totten Glacier (Li et al., 2016). The PIG experiment showed an
extreme case in Antarctica where the OEs of a 7-year span can go as large as
626 m a-1 (∼ 20 %) near the grounding zone (Table 3);
furthermore, the OEs of a 15-year span can reach up to 1300 m a-1 along the grounding line and cause an overestimated GL flux of 11.5 Gt a-1 if not corrected (Fig. A4). Therefore, the magnitude of the OEs
contained in the long-span historical velocity maps is significant. When
overestimated historical maps of 1960s–1980s are used alongside recent
maps of 1990s–2010s for assessment of the long-term global climate change
impact on the Antarctic ice sheet and for forecast modeling, the
overestimated historical states may lead to underestimated long-term
changes. Furthermore, compromised forecasting may result. Thus, the OEs
in the long-span historical maps must be seriously examined and corrected.
Discussion
In this section we discuss the applicability of the proposed method in terms
of overestimation-free time span, influence of complex glacier geometry,
overestimation in fast flowing glaciers, and comparison with the
“midpoint” method.
Threshold of the overestimation-free time span for trajectory
segments
The choice of a short time span ΔtRef (e.g., a few months to
1 year) for the OE-free segments along a trajectory in Premise I makes sure that
the difference between the E and L velocities within the span is negligible,
or less than σ (velocity mapping uncertainty). It is also the
baseline time span of the initial OE-free velocity map. Determination of
this threshold has an implication on validation of Premises I and II, as well as the
integration period of the trajectory segments from Pi to Pi+1 (Fig. 3b). Estimation of ΔtRef can be performed in each glacier by a
linear regression between the E or L velocity V and time span Δt,
V=KΔt+b. Given b, K, and σ (20 m a-1), ΔtRef can be calculated as σΔtRef=σ-bK. Our experiment results show ΔtRef as 3.2 years and 3 months for TG and PIG, respectively. Thus, the estimated OE-free time spans appear to
be related to ice flow dynamics of the glaciers. In Experiment 1 and Experiment 3, we used 1 year for TG
and 3 months for PIG, respectively. We suggest that a linear regression for
ΔtRef estimation be performed before extensive historical
velocity mapping is carried out.
Glaciers with complex geometry
Within a long time span (e.g., over 5–10 years) in Premise I the difference between
L and E velocities accumulated over all segments along the entire
trajectory, i.e., the end-point deviation between the red and blue lines in
Fig. 2, is measured with a more tolerable threshold: (V0-nL–V0-nE)≤kσ. Although the OEs of the trajectory
segments are controlled by ΔtRef, ice mass moving along a
curved flow line over this long span may result in an accumulative
discrepancy. In Experiment 1 we showed that the average difference (V0-nL–V0-nE) in TG is within 1σ. Our further experiments in five smaller Antarctic glaciers with complex geometry, including the George VI,
Abbott, Dotson, Crosson, and Getz glaciers, resulted in (V0-nL–V0-nE) values that are negligible (smaller than 1σ) in all
five glaciers. We further investigated Area 2 of PIG (Fig. 3a), a section with a
high velocity of 3278 m a-1 and the highest curvature along the main
trunk (7 %, namely, a 1305 m cross flow deviation from the 19 720 m long
straight line). Based on the feature-tracking result using eight annual
Landsat 8 images, this curvature-induced E and L velocity difference is
calculated as 206 m a-1 over a 7-year span, among which 195 m a-1
(95 %) was corrected (correction for Area 2 in Table 2). The results in these glaciers suggest that the velocity overestimation in glaciers with complex geometry
can be mostly corrected and should not affect the applicability of the
proposed method.
OE correction in fast-flowing glaciers
The Totten Glacier and Pine Island Glacier are among the most dynamic
glaciers in Antarctica. The accelerated ice mass loss in these two
fast-flowing glaciers has been realized by ice shelf basal melting and front
calving (Li et al., 2015; Joughin et al., 2020, 2021). In Experiment 1 and Experiment 3 we show that such temporal accelerations do not alter the necessary condition in Premise II
significantly, so that a‾VL(n)-a‾VL≤k′σΔtn
holds with k′=1 and 3 on average for all experiment areas in TG (Table 1)
and PIG (Table A4), respectively. The spatial-acceleration-induced OEs
can be corrected. On the other hand, the impact of temporal accelerations is
represented in the sufficient condition in Premise I, (V0-nL-V0-nE) ≤kσ. For example, Area 4 of TG is located in a dynamic area of a
velocity of ∼ 1400 m a-1, about 30 km from the ice shelf
front (Fig. 4). Its E velocity (red curve in panel 4c in Fig. 5) exceeded
the L velocity (blue curve) in 2016 and showed a significant acceleration
thereafter. This resulted in the largest actual overestimation
OEActual of ∼ 69 m a-1 (Table 1), which may be
attributed to large shelf front calving occurring during the period between
6 April and 20 September 2015, with an area loss of 136 km2 (2 %;
Fig. 4b). The L velocities V2013–iL (blue curve in panel 4c of Fig. 5) are derived from the 1-year-span map V2013–2014 before 2015 and
thus do not reflect this temporal acceleration; furthermore, in
L velocities V2013–iL(7) (black curve) derived from the 7-year-span
map V2013–2020, the effect of the calving-induced acceleration is
averaged out over the 7-year span. Thus, the back and blue lines are
approximately parallel with an average acceleration difference of only -0.5 m a-2 (Table 1). Consequently, it appears that the calving event did
not significantly reduce the buttressing potential of the ice shelf because
the ice shelf front retreat occurred outside of the passive shelf ice (PSI)
boundary (light blue line in Fig. 4b; Fürst et al., 2016); the other
four areas are located farther away, inland, and away from the PSI boundary
and thus were not influenced as much as in Area 4. Therefore, the calving-induced temporal acceleration only occurred in Area 4 (k=∼ 1.5 in
Premise I) but not in the four other areas (k≤0.5). Hence, the calculated
correction of -19 m a-1 in Area 4 is at a similar level to that in the four
other areas and has only adjusted the OE caused by the spatial acceleration
along the trajectory. The velocity increase due to the temporal acceleration
by the calving event remains in the adjusted map as a large residual of
50 m a-1, which can serve as a signature to study the relationship
between calving activities and ice flow dynamics.
With k= 1, 2, 2, 10, and 5 in Premise I for the five selected areas from the inland interior to the grounding
zone in PIG (Table A4), the temporal accelerations caused by climate warming
impact are indicated by the high values of k (10 and 5) near the grounding line
at the glacier outlet. Overall, the proposed method was able to correct
∼ 40 % of the total OEs, leaving another ∼ 60 % in residuals as the signature of the velocity changes induced by the
temporal accelerations.
In addition, based on an annual E velocity map of 2013 in PIG from
ITS_LIVE (Fig. A4a), L velocities along the grounding line (GL;
Gardner et al., 2018) with time spans of 1 to 15 years (Fig. A4b) and the
associated ice discharge were computed. The actual flux gates were set with
nodes separated every 240 m, which were located up to 13 km upstream of the GL
to reduce the uncertainty of ice thickness data (Gardner et al., 2019) from
BedMachine (Morlighem et al., 2020). The results show that the velocity OEs
of the 15-year span can reach up to ∼ 1300 m a-1 in the
GL region. Such OEs in velocity can further cause an overestimation in ice
discharge across flux gates upstream the GL, which increases rapidly by
∼ 6.3 Gt a-1 within the first 4-year span (Fig. A4c);
thereafter, the flux overestimation slows down until a maximum of 11.5 Gt a-1 is reached at the 15-year span. This suggests that in fast-flowing
glaciers like PIG, OEs in velocity maps with a time span of greater than 0.5–2 years should be corrected. The fact that the OE effect in PIG appears
to level off after a couple of years is mainly because the velocity in the
majority of the ice shelf is approximately leveled at ∼ 4000 km a-1 and the acceleration is thus very small. This makes the average
L velocities along the GL increase in a much-reduced pace where an
integral of E velocities is performed in the leveled velocity area a few
years after crossing the GL (Fig. A4d). Consequently, the flux OE indicated
in Fig. A4c showed a similar trend. Extension of this exercise to the entire
continent involves a complex situation, including deceleration caused by ice
rises, mapping errors near ice shelf shear margins, GL data points that
would calve into the ocean within n-year span, and other potential factors.
It should be noted that given a velocity range in flux computation, the ice
flow angle between a flux gate and ice flow controls the flux magnitude. In
principle, at the same gate this angle changes with velocities of different
time spans and flow line patters, resulting in an overestimation or
underestimation in flux for individual glaciers. A systematic and in-depth
investigation should be carried out to handle the overestimation issue in GL
flux of all of Antarctica.
Comparison with the “midpoint” method
The “midpoint” method presented in Berthier et al. (2003) compensates for
overestimations by assigning the overestimated velocities to middle points
of the trajectories. We use the velocity measurements in Experiment 1 to compare the
performances of these two OE correction methods. Since Area 4 was affected by a
calving event during the time span, here we use the other four areas (areas 1, 2, 3, and 5, Fig. 4a). We estimated the E velocities of 7 years (2013–2020)
V2013–2020E and assigned them to the midpoints of the
trajectories in the four areas (Table A5). They were then compared to the
1-year-span velocity V2013–2014E at the midpoints to calculate the
bias εM. Similarly, the same overestimated
7-year-span E velocities V2013–2020E were corrected using the OE
correction method of this paper and assigned to the start points of the
trajectories as Vcorrected (Table A5). They were then compared to the
1-year-span V2013–2014E also, but at the start points to calculate
another set of bias εS. As shown in Table A5, the proposed OE correction method achieved a higher overall accuracy of
4 ± 10 m a-1 compared to 12 ± 14 m a-1 of the
midpoint method.
In summary, our validation and application experiment results demonstrated
that the proposed OE correction method can be applied to different types of
glaciers in Antarctica, regardless of their ice flow dynamics. While the OEs
are effectively corrected, temporal velocity increases caused by global-warming-induced activities can be preserved for long-term change studies.
The implication is that, when using newer velocity maps of short spans along
with historical maps of long spans produced in previous studies over the
past few decades, the overestimation of historical velocities could have
caused an underestimation in the long-term acceleration magnitude. On the
other hand, new efforts in historical velocity mapping at an ice-sheet-wide or a large regional scale should be made with a full consideration of OE
corrections. The applicability of the method to glaciers in Greenland should
be further investigated in future research.
Conclusions
Velocity overestimation exists in Antarctic ice flow velocity maps produced
from optical satellite images of long time spans. Such overestimations are
inevitable for historical velocity maps due to the poor availability of
earlier satellite images in Antarctica, especially before 1990. The results
in this study show that the overestimations can reach up to ∼ 69 m a-1 in the Totten Glacier, East Antarctica, over a 7-year span and
∼ 931 m a-1 in the Pine Island Glacier, West Antarctica,
over a 10-year span. The overestimated historical velocity maps should be
adjusted before they are combined with recent velocity maps to build a
long-term record for monitoring and forecasting the Antarctic ice flow
dynamics and impact of global climate changes on the ice sheet. We used a
set of ground truth velocity maps in the Totten Glacier produced from
Landsat 8 images of 2013 to 2020 to validate the proposed innovative method
for velocity overestimation correction. Based on the validated method, we
successfully corrected the overestimations of a velocity map of 1972–1989
with time spans from 1 to 17 years in the David Glacier region, East
Antarctica. Another experiment at the Pine Island Glacier (PIG) was carried
out to demonstrate that the method can effectively adjust overestimated
velocities as large as 195 m a-1, while preserving the velocity change
signature for long-term ice flow dynamics analysis. In summary we draw the
following conclusions.
The proposed Lagrangian velocity-based method is effective and easy to implement because the overestimation corrections are calculated by using the long-term velocity map itself only, without field observations or additional image data.
The premises of the correction method are validated by using a set of ground truth velocity maps developed from high-quality Landsat 8 images from 2013 to 2020 to show the rigidity of the method. The velocity overestimations of up to a 7-year span are proven to be corrected effectively to within the uncertainty (1σ) of the 1-year map.
The validated correction method is then successfully applied to correct overestimations in a historical velocity map of 1972–1989 with time spans from 1 to 17 years.
It is proven that ice velocity change information of temporal acceleration events, e.g., caused by shelf front calving and basal melting in the Totten Glacier and Pine Island Glacier, is preserved after the correction and can be used for long-term ice flow dynamics analysis.
The magnitude of the OEs contained in the long-span historical velocity maps
is significant. When overestimated historical maps are used alongside
recent maps for assessment of the long-term global climate change impact on
the Antarctic ice sheet and for forecast modeling, the underestimated
long-term changes and compromised forecasting may result. Thus, the OEs
in the long-span historical maps must be seriously examined and corrected.
It is recommended that OE values be computed for long-term historical
velocity maps and corrections be made when OEs are more than the velocity
uncertainty (1σ). This velocity overestimation correction method can
be applied to adjust velocity fields for the production of regional and ice-sheet-wide historical velocity maps from long-term satellite images before the
1990s.
Velocity map of the Totten Glacier from
ITS_LIVE (left) used to explain the concept of velocity
overestimation caused by acceleration (right): “Over 16
years, that parcel of ice travels about 13 km downstream (red path). It
begins at a velocity of about 720 m/yr, and in the first 8 months it travels
at an average rate very close to 720 m/yr. But then the ice picks up speed
as it moves downstream, so in the first 10 years it does not travel just
7200 m – it actually travels about 7900 m, or an average speed of 790 m/yr” (Chad A. Greene, personal communication, 2020).
Overestimation correction in case of temporal
accelerations.
Velocities in five areas (numbered yellow rectangles in
Fig. 3) of PIG are used to demonstrate application of the OE correction
method. Panels (1a)–(5a) show the reconstructed 3-month velocity maps V2013–2014 in the areas; matched
points (red triangles) are used to map E velocity
V2013–iE (i= 2014 (3 months), 2015, … , n) (red lines in panels 1c–5c); points along the flow line (blue
dots) are tracked from the 3-month maps and used to calculate L velocity
V2013–iL (blue lines in panels 1c–5c). Similarly, panels (1b)–(5b)
illustrate the reconstructed n-year velocity maps
V2013–n in the areas with the
matched points (red triangles) for E velocity
V2013–iEn; points along the flow line (black crosses) are tracked
from the n-year-span velocity map and used to calculate L velocity
V2013–iLn (black lines in panels 1c–5c). In
each area (panels 1c–5c) the difference between red and blue lines
increases with time; the black line is above the blue line due to the spatial-acceleration-induced OE over n-year span, but they are relatively parallel,
and thus a correction can be estimated. Background in panels (1a)–(5a) and (1b)–(5b) is
Landsat 8 image of 12 February 2014.
(a) One-year-span ice velocity map of PIG (2013) from
ITS_LIVE (Gardner et al., 2019), (b)L velocities of 1- to
15-year span
V0-iL (i= 1, 2, … 15) along GL calculated from the 2003
velocity map, (c) ice discharge across flux gates upstream the grounding line
estimated with time spans from 1 to 15 years, and (d)L velocities
V‾0-iL (averaged
over the GL portion across main trunk) vs. time span.
Information for the Landsat 8 images used for velocity
mapping in Experiment 1.
Acquisition date (UTC)Scene ID2013/12/03LC08_L1GT_102107_20131203_20170428_01_T2_B82014/11/13LC08_L1GT_101107_20141113_20170417_01_T2_B82015/12/25LC08_L1GT_102107_20151225_20170331_01_T2_B82016/11/25LC08_L1GT_102107_20161125_20170317_01_T2_B82017/11/05LC08_L1GT_101107_20171105_20171120_01_T2_B82018/12/17LC08_L1GT_102107_20181217_20181227_01_T2_B82019/12/29LC08_L1GT_101107_20191229_20200111_01_T2_B82020/10/19LC08_L1GT_102107_20201019_20201019_01_RT_B8
Acceleration estimates using least squares regression
a and simple average a‾
applied to L velocities of 7 years
VL(n) and 1 year
VL.
Information for the Landsat images used for velocity
mapping in Experiment 2.
SatelliteAcquisition date (UTC)Scene IDLandsat 1 (MSS)1972/11/28LM10641121972333AAA041972/11/28LM10641131972333AAA041972/11/28LM10651121972333FAK041972/12/1LM10671121972336AAA041972/12/1LM10671131972336AAA041972/12/2LM10681121972337AAA041972/12/5LM10711121972340AAA021972/12/11LM10781131972346FAK031973/1/2LM10641141973002AAA041973/1/3LM10831131973003AAA041973/1/16LM10611151973016AAA041973/1/17LM10781141973017XXX011973/1/25LM10681131973025FAK031973/1/31LM10741121973031FAK031973/1/31LM10741131973031FAK031973/1/31LM10741151973031FAK031973/2/6LM10801151973037AAA051973/2/8LM10641151973039AAA021973/2/9LM10831141973040XXX011973/2/11LM10671161973042FAK071973/2/12LM10861121973043FAK031973/2/14LM10701121973045AAA051973/2/14LM10701131973045AAA051973/2/20LM10761121973051FAK031973/2/28LM10661141973059AAA041973/2/28LM10661161973059AAA041973/3/6LM10721131973065XXX011973/10/29LM10751121973302AAA021973/10/29LM10751131973302AAA021973/11/4LM10811131973308FAK031973/11/4LM10811141973308FAK031973/11/7LM10841141973311FAK031973/11/21LM10801131973325FAK031973/11/24LM10641161973328AAA04
Continued.
SatelliteAcquisition date (UTC)Scene ID1973/12/20LM10731151973354FAK031973/12/25LM10601161973359AAA041973/12/31LM10661141973365AAA041973/12/31LM10661161973365FAK021973/12/31LM10671141973365FAK021973/12/31LM10671151973365FAK011974/1/5LM10711121974005AAA041974/1/5LM10711131974005AAA041974/1/12LM10781141974012AAA041974/1/12LM10781151974012AAA041974/1/14LM10621161974014AAA041974/1/16LM10641141974016AAA021974/1/18LM10661151974018AAA041974/1/20LM10861121974020AAA041974/1/25LM10731121974025AAA05Landsat 4 (TM)1988/12/15LT40601141988350XXX041988/12/15LT40601151988350XXX031989/1/29LT40551161989029XXX041989/3/24LT40651131989083XXX011989/3/24LT40651141989083XXX111989/3/24LT40651151989083XXX011989/11/12LT40641151989316XXX011989/11/14LT40621121989318XXX021989/11/14LT40621131989318XXX021989/11/26LT40661121989330XXX011989/11/30LT40621151989334XXX011989/12/5LT40651121989339XXX021989/12/16LT40621141989350XXX02Landsat 5 (TM)1986/1/4LT50561161986004XXX041986/12/13LT50571161986347XXX05
Application of the OE correction method in five selected
areas in PIG. Premise I contains n-year
tracked E velocities and L velocities computed from the 3-month velocity
map, as well as their differences. Premise II
illustrates averaged L accelerations computed from the n-year and 3-month
velocity maps, respectively, as well as their differences.
Area IDTime span (years)Premise I Premise II VnEVnL(VL-VE)na‾VL(n)a‾VLa‾VL(n)-a‾VL(m a-1)(m a-1)(m a-1)(m a-2)(m a-2)(m a-2)1640543954-1006.03.22.82739043699-20528.260.8-32.63725322453-799.515.9-6.44719221847-7525.627.8-2.253410402-82.23.6-1.4Mean625642471-9314.322.3-8.0*SD2150614477111.823.814.2
* With an average span of 6 years for all areas and σ=20 m a-1, the mean of a‾VL(n)-a‾VL in Premise II is 8 m a-2 (≤3σΔtn).
Comparison of the proposed OE correction method with the
midpoint method in Berthier et al. (2003).
Code is available from the corresponding author upon request.
Data availability
ITS_LIVE ice flow velocity maps used in this study are available at https://its-live.jpl.nasa.gov/ (last access: 26 February 2022) (Gardner et al., 2019).
Author contributions
RL led the study and developed the overestimation correction model. YC, MX, XY, ZL, and SL carried out velocity mapping. HC did the programming. RL, YC,
and GQ were involved in data analysis and presentation.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
We thank the editor and two reviewers for their constructive comments and
suggestions. We also thank the United States Geological Survey (USGS) for
the Landsat images.
Financial support
This research has been supported by the National Natural Science Foundation of China (grant no. 41730102), the National Key Research and Development Program of China (grant no. 2017YFA0603100), the Chinese Arctic and Antarctic Administration
(grant no. CXPT2020017), and the Fundamental Research Funds for the Central
Universities.
Review statement
This paper was edited by Etienne Berthier and reviewed by Chad Greene and one anonymous referee.
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