1
Supplementary Text for
2
Antarctic outlet glacier mass change resolved at basin scale
3
from satellite gravity gradiometry
4
J. Bouman, M. Fuchs, E. Ivins, W. van der Wal, E. Schrama, P. Visser, M. Horwath
5
6
Geophysical Research Letters, 2014
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Manuscript number MS# 2014GL060637
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S1.
GOCE gravity gradient data
9
GOCE measured the VXX, VYY, VZZ, VXY, VXZ and VYZ gravity gradients, with approximately X
10
along track, Y cross track and Z in radial direction. The gradients are calibrated and corrected for
11
temporal gravity field variations such as tides [Bouman et al., 2004, 2009]. The VXY and VYZ
12
gradients have low accuracy, whereas the other gradients are accurate, especially in the so-called
13
measurement band (MB), which was defined before launch to be between 5 and 100 mHz. In
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practice, the effective MB may differ from the pre-defined MB [Fuchs and Bouman, 2011].
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Figure S1 shows the spectral density of gradient differences GOCE – GOCO03S for two orbits
16
in November 2010. As the GOCO03S model includes and averages 18 months of GOCE
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gradient data as well as GRACE data for the long wavelengths, these differences predominantly
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show the errors in the GOCE data for the two orbital revolutions. The VYY differences increase
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for lower frequencies also in the MB, which is caused by systematic errors in this gradient,
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especially close to the magnetic poles. The VXX and VZZ differences are relatively flat in the MB,
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but we also see that somewhat below the MB the error increase is not severe. The upper bound of
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100 mHz roughly corresponds to L = 540 and slightly varying the upper bound will therefore
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hardly influence the results as temporal gravity field signal will be extremely small at these high
24
degrees.
25
In our analysis of the GOCE-only gravity gradients we used 3 mHz as the lower bound of the
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bandwidth. Even lower bounds would increase the influence of systematic errors in the data,
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whereas higher bounds filter out more gravity gradient signal. When analyzing GOCE-only data,
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a lower bound of 3 mHz seems to be a fair compromise between averaging out noise and keeping
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signal. In the combination with GRACE derived gravity gradients we used, as explained below,
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10 mHz as lower bound as this roughly corresponds to spherical harmonic degree L = 54, which
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is almost the maximum degree L = 60 of the GRACE CSR RL05 solutions.
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Fig. S1. Spectral density plots of differences between gradients observed by GOCE and
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predicted by GOCO03S. The GOCE MB is indicated by the vertical, dashed black lines, the
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vertical solid black line is at 3 mHz.
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S2.
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We have seen above that ice mass loss signal is visible in the GOCE gravity gradient data, but
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that these data suffer from errors at long wavelengths. On the other hand, GRACE is very
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accurate at long wavelengths, but suffers from stripes that become stronger for increasing
40
resolution (see Fig. 1B). We therefore combine GRACE and GOCE keeping as much as possible
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GRACE information for the long wavelengths and adding spatial detail from GOCE.
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Naturally GOCE and GRACE gradients can only be combined if data are available. Two major
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GOCE anomalies occurred in January - February 2010, and July – September 2010. In addition,
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no CSR RL05 monthly solutions are available for January, June, November and December 2011,
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and April and May 2012. The error reduction in the moving windows of 4 months primarily
46
depends on the amount of available data. An impression of the quality, of the derived gravity
Combination of GOCE and GRACE
2
47
fields for example, is obtained by looking at the available data relative to the maximum amount
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of data. Table S1 lists the percentage of number of days at which gravity gradient data are
49
available relative to the total number of days in a 4 month period. The percentage varies from 26
50
to 90%, where it has to be noted that our outlier flagging leads to a rejection of about 5% of all
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GOCE data for all 4 month windows.
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Table S1: Available data as a percentage of the number of days in 4 month periods.
Time Period
Data
Time Period
Data
Time Period
Data
2009/11 – 2010/02
80%
2010/11 – 2011/02
66%
2011/11 – 2012/02
47%
2009/12 – 2010/03
74%
2010/12 – 2011/03
67%
2011/12 – 2012/03
66%
2010/01 – 2010/04
74%
2011/01 – 2011/04
66%
2012/01 – 2012/04
66%
2010/02 – 2010/05
74%
2011/02 – 2011/05
90%
2012/02 – 2012/05
43%
2010/03 – 2010/06
87%
2011/03 – 2011/06
71%
2012/03 – 2012/06
27%
2010/04 – 2010/07
70%
2011/04 – 2011/07
70%
2010/05 – 2010/08
46%
2011/05 – 2011/08
71%
2010/06 – 2010/09
26%
2011/06 – 2011/09
67%
2010/07 – 2010/10
26%
2011/07 – 2011/10
90%
2010/08 – 2010/11
49%
2011/08 – 2011/11
66%
2010/09 – 2010/12
72%
2011/09 – 2011/12
43%
2010/10 – 2011/01
70%
2011/10 – 2012/01
46%
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S3.
Time-series of GRACE/GOCE gravity gradient change in West Antarctica
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It is illustrative to study the evolution in time of the GOCE along-track and cross-track gravity
55
gradients as this gives an indication of their directional sensitivity and shows the sensitivity for
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higher spatial resolutions. For large parts of the world the along-track direction of the GOCE
57
satellite was north-south. Because the inclination of the GOCE orbit was 96.7°, this is an
58
approximation. The maximum latitude is roughly 83° north and 83° south and for high latitudes
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the direction significantly changes. Close to the maximum latitude, for example, the along-track
60
direction is east-west and ascending and descending tracks are perpendicular around 80° latitude.
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We therefore analyzed ascending and descending tracks separately for the VXX and VYY as well
3
62
as their average. The advantage of the separate evaluation is that gradient signal with different
63
directions are not mixed, but the disadvantage is less error reduction.
64
In Fig. S2 and S3 the first row displays the differences for ascending tracks between
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GRACE/GOCE VXX and VYY and GOCO03S for two different time windows. The middle row
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shows the differences for descending tracks, whereas the last row is the average of ascending and
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descending tracks, which corresponds to north-south and east-west for VXX and VYY
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respectively, contaminated with part of the east-west and north-south gradient. Comparing the
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ascending and descending track differences for equal time-windows, one observes that up to
70
three patterns are visible. Differences occur that are caused by the different directional sensitivity
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for ascending and descending tracks, but also by errors that are not averaged out. The average of
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ascending and descending tracks has less amplitude and less error. In all cases the gradient
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differences increase with time. The along-track gradients have the tendency to be more sensitive
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to east-west oriented patterns, whereas the cross-track gradients tend to be more sensitive to
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north-south patterns such as the Thwaites Glacier. For the along-track gradient for descending
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tracks the glaciers feeding the Getz Ice Shelf, the Haynes/Smith/Kohler Glaciers and Pine Island
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Glacier appear to be overlain by gravity gradient maxima in July – October 2011 and there seems
78
to be sensitivity at relatively high spatial resolution.
79
We note that the differences between the snapshots are caused by temporal gravity field
80
variations, but also the residuals with respect to GOCO03S represent temporal variations.
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GOCO03S is a static global gravity field model that combines GRACE, GOCE and satellite laser
82
ranging (SLR) data with reference epoch 2005.0. Up to spherical harmonic degree and order 120
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the contribution of the GOCE gradients to the static gravity field is small, which means that the
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GOCO03S – GOCE gravity gradient residuals are representative of changes in the gravity field
85
relative to 2005.0.
4
86
Fig. S2. Snapshots of differences between GRACE/GOCE along-track gradients and GOCO03S
87
[mE]. The left column shows ascending & descending tracks and their average for July –
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October 2011, the right column for January – April 2012. Glaciers are delineated as in Fig. 1E.
5
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Fig. S3. Snapshots of differences between GRACE/GOCE cross-track gradients and GOCO03S
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[mE]. The left column shows ascending & descending tracks and their average for July –
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October 2011, the right column for January – April 2012. Glaciers are delineated as in Fig. 1E.
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S4.
Mass change derived from gravity gradient data
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The gridded GRACE/GOCE – GOCO03S vertical gravity gradient residuals at satellite altitude
95
are given in grids that cover the latitudes between -83° and -60° and all longitudes. The grids
96
were padded with zeroes to cover all latitudes from -90° to 90°, after which Stokes coefficients
97
were estimated for every 4 month window using global spherical harmonic analysis. The Stokes
98
coefficients could be used, e.g., to compute gravity anomalies at the Earth’s surface for different
99
maximum degrees. In addition, we used the gridded gravity gradient trace as a proxy for the
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vertical gradient errors (see Supplement S6) and applied spherical harmonic analysis to those
101
grids as well. The corresponding Stokes coefficients can be used to assess the errors.
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Gravity changes derived from the Stokes coefficients derived from the vertical gravity gradient
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represent the total mass effect. The secular change in local ice sheet mass is obtained correcting
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for glacial isostatic adjustment (GIA) and the solid earth elastic response to mass load change.
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We corrected for GIA using three recent, but rather different models, which are discussed below.
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A preponderance of evidence suggests West Antarctica has an ongoing, and likely large, change
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in the gravitational potential that is caused by GIA [Thomas, 1976; Wu and Peltier, 1983; James
108
and Ivins, 1998]. We assessed the GIA-effect in terms of EWH at the Earth’s surface using three
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distinct models: IJ05_R2 [Ivins et al., 2013], van der Wal [van der Wal et al., 2013] and the
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model of Whitehouse et al. [2012b]. The rationale of the first model is that it satisfies both the
111
most recent ice histories that may be assembled from in-situ rock and ice core records and the
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rates of GPS uplift trends now recorded on rock outcrops. The Whitehouse et al. [2012b] model
113
uses the W12 ice loading history [Whitehouse et al., 2012a] with modified Late Holocene ice
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history and mantle viscosities are fitted to Antarctic regional sea-level data. In contrast, the van
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der Wal model uses the ICE-5G ice loading history [Peltier, 2004] in combination with a 3D
116
composite rheology. In that rheology the viscosity is derived from global surface heat flow
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measurements and a seismic model, and two main deformation mechanisms for mantle rocks as
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found in laboratory measurements. The free parameters in the model are tuned to match uplift
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and gravity rates in Fennoscandia and North America. The three models give quite different
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EWH trends for some parts of West Antarctica, but the trend is small in the region where the
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main melting glaciers are located (see Fig. S4 that displays the EWH rate according to the
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IJ05_R2 and van der Wal models).
7
A
B
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Fig. S4. Effect of GIA on EWH rate at Earth’s surface [mm/yr]. (A) IJ05_R2: -1.2 – 24.1 mm/yr;
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(B) van der Wal: -6.8 – 41.3 mm/yr. Color scale has been set to -5 – 30 mm/yr.
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In view of recent GIA estimates based on geodetic observations, indicating significantly larger
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GIA uplift rates for the Amundsen Sea Sector [Groh et al., 2012; Gunter et al., 2013], present
127
GIA models may be revised in the future. However, errors of GIA models affect only the trends
128
themselves and not our main conclusions on reduced noise in high-resolution ice mass change
129
estimates.
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S5.
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The use of low-pass filtered Stokes coefficients from GRACE or GRACE/GOCE to determine
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mass changes in separate basins inevitably leads to leakage from one basin to another or from a
133
basin to the ocean [Horwath and Dietrich, 2009]. The amount of leakage depends on the
134
maximum spherical harmonic degree of the gravity field solutions, the type of low-pass filter,
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how large the mass change is in a basin and where it is located within the basin as well as on the
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shape of the basin. Mass trends for separate basins have been derived for Antarctica using
137
GRACE solutions to degree L = 60. King et al. [2012] simulated leakage effects based on
138
satellite radar and laser altimetry results and corrected those leakage effects from the purely
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GRACE-based estimates.
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We assess here how well basins can be separated in West Antarctica when purely satellite
141
gravimetry-based basin mass changes are computed. To that end, we performed simulations
142
based on realistic geographic patterns of height changes between 2003 and 2010, which were
Basin separation
8
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derived for the Pine Island, Thwaites and Haynes/Smith/Kohler Glaciers from ENVISAT radar
144
altimetry [Legrésy et al., 2006; Horwath et al., 2012]. The mass changes corresponding to those
145
patterns are -60.7 Gt/yr for the Thwaites and Haynes/Smith/Kohler Glaciers (basin 21) and -33.6
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Gt/yr for the Pine Island Glacier basin. We rescaled the PIG basin pattern with a factor of 1.8 to
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make it more compatible to satellite gravimetry results and accommodate for limitations of radar
148
altimetry to recover the full mass signal. The height changes were then expanded in spherical
149
harmonics with a maximum degree of 360 and converted to mass changes.
150
Different spherical harmonic degrees and Gaussian filters are applied to study the leakage
151
between these two basins as well as basin 20 (glaciers flowing into the Getz Ice Shelf) and all
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other basins in West Antarctica (1, 18, 19 and 23). We extended the integration masks for basins
153
20, 21, and 22 into the ocean to reduce leakage to the ocean (see Fig. S5). The leakage from the
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ocean to these basins is assumed to be small, because the mass change in the ocean is small
155
[Horwath and Dietrich, 2009]. Our analysis mainly focusses on basins 20, 21 and 22 in the
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Amundsen Sea Sector because they are known to be the main contributors to ice mass imbalance
157
in WAIS [King et al., 2012], and therefore for the other basins the original definition is adopted.
158
In all cases, mass changes from the Stokes coefficients are determined using a basin mask with 1
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for a particular basin and 0 outside the basin.
160
161
Fig. S5. Basins in West Antarctica used in this study. Definitions after [Zwally et al., 2012] with
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basin 20, 21 and 22 extended into the ocean.
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The amount of leakage between basins is first assessed for a mass loss of -60.7 Gt/yr in basin 21.
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Table S2 shows the recovered mass loss using the above basin definitions. Even for the
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maximum spherical harmonic degree of 360 there is about 10% leakage, especially to basin 22.
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We also see that filtered solutions with a Gaussian of 250 km half-width lead to significant
168
leakage. This is much reduced for a Gaussian of 90 km half-width, where there is a slight
169
reduction of leakage to basins 20 and 22 for increasing degree and a slight increase of leakage to
170
all other basins for increasing degree. For a Gaussian of 90 km half-width, the total recovered
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mass loss signal can be more than 100% of the input mass change. This is explained by the
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degree truncation, which leads to oscillations in the space domain (Gibbs effect).
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Table S2: Simulation results to assess leakage effects based on realistic mass change patterns for
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basin 21 with a total mass loss of -60.7 Gt/yr. Mass trends in WAIS as a function of spherical
175
harmonic degree and Gaussian filter half-width.
Gaussian filter
half-width 250 km half-width 90 km
Basin
L = 360
20 (GET)
Gaussian filter
L = 60
L = 90
L = 90
L = 120
-0.8
-6.3
-6.1
-3.2
-3.0
-54.6
-31.2
-31.3
-49.6
-48.7
22 (PIG)
-5.3
-10.7
-10.5
-8.2
-7.9
1 + 18 + 19 + 23
-0.1
-8.1
-8.3
-0.7
-1.1
WAIS
-60.8
-56.2
-56.1
-61.7
-60.7
% of total
100%
93%
92%
102%
100%
21 (THW + HSK)
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The above analysis was repeated for basin 22 with a simulated mass change of -60.5 Gt/yr. In
177
this case, the leakage is much less for L = 360 (see Table S3). Also here we see that a Gaussian
178
of 250 km half-width leads to significant leakage. When a Gaussian of 90 km half-width is being
179
used, there is a slight increase of leakage to basins 21 for increasing degree and a decrease of
180
leakage to all other basins for increasing degree. Overall we see that for increasing degree the
181
leakage reduction is more pronounced than for basin 21 (Table S2).
10
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Table S3: Simulation results to assess leakage effects based on realistic mass change patterns for
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basin 22 with a total mass loss of -60.5 Gt/yr. Mass trends in WAIS as a function of spherical
184
harmonic degree and Gaussian filter half-width.
Gaussian filter
half-width 250 km half-width 90 km
Basin
L = 360
20 (GET)
Gaussian filter
L = 60
L = 90
L = 90
L = 120
0.0
-0.1
-0.2
-0.2
0.1
-0.6
-13.2
-12.6
-3.0
-3.3
-59.8
-25.1
-25.9
-52.1
-53.9
-0.2
-14.9
-14.5
-5.3
-3.3
WAIS
-60.6
-53.1
-53.0
-60.6
-60.4
% of total
100%
88%
88%
100%
100%
21 (THW + HSK)
22 (PIG)
1 + 18 + 19 + 23
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Table S4 summarizes the summed effect of basin 21 and 22. The percentages in the third and
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fourth row are relative to the total mass change in the respective basins. If a Gaussian filter of 90
187
km half-width is used, then the total recoverable mass is 86% for basin 21 for L = 90 and L =
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120, whereas it is about 100% for basin 22 for L = 90 and L = 120. These percentages will be
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different for different actual mass changes and by taking into account the mass change in other
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basins. Nevertheless, it shows that the L = 90 and L = 120 cases have about the same mass trend
191
recovery capability, where the leakage to other basins is somewhat less for the L = 120 case. In
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addition, it seems to be more difficult to recover the correct unconstraint mass change for basin
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21 (Thwaites and Haynes/Smith/Kohler glaciers) than for basin 22 (Pine Island glacier). Finally
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we note that a separation of basins also should include an analysis of the error correlation
195
between basins, which is addressed in Section S8.
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11
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Table S4: Simulation results to assess leakage effects based on realistic mass change patterns for
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both basin 21 with a total mass loss of -60.7 Gt/yr and basin 22 with a total mass loss of -60.5
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Gt/yr (-121.2 Gt/yr altogether). Mass trends in WAIS as a function of spherical harmonic degree
200
and Gaussian filter half-width.
Gaussian filter
Gaussian filter
half-width 250 km half-width 90 km
Basin
L = 360
L = 60
L = 90
L = 90
L = 120
20 (GET)
-0.8
-6.3
-6.3
-3.4
-2.8
21 (THW + HSK)
-55.3
-44.4
-43.9
-52.5
-52.0
% of total
91%
73%
72%
86%
86%
22 (PIG)
-65.0
-35.8
-36.4
-60.3
-61.7
% of total
107%
59%
60%
100%
102%
-0.3
-23.0
-22.7
-6.1
-4.4
-121.4
-109.2
-109.0
-122.3
-121.1
1 + 18 + 19 + 23
WAIS
201
S6.
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We present here a method to assess the error of the vertical gravity gradients, and gravity field
203
products derived thereof, using the gravity gradient trace.
204
𝜖
𝜖
The measured or gridded diagonal gravity gradients can be written as 𝑉𝑋𝑋
= 𝑉𝑋𝑋 + 𝜖𝑋 , 𝑉𝑌𝑌
=
205
206
𝜖
𝑉𝑌𝑌 + 𝜖𝑌 , 𝑉𝑍𝑍
= 𝑉𝑍𝑍 + 𝜖𝑍 where 𝑉𝐼𝐼𝜖 are the gravity gradients with errors 𝜖𝐼 . Because of the
Laplace condition 𝑉𝑋𝑋 + 𝑉𝑌𝑌 + 𝑉𝑍𝑍 = 0, the sum (trace) of the measured gradients is
𝜖
𝜖
𝜖
𝑡𝑟𝑎𝑐𝑒 = 𝑉𝑋𝑋
+ 𝑉𝑌𝑌
+ 𝑉𝑍𝑍
= 𝜖𝑋 + 𝜖𝑌 + 𝜖𝑍 .
207
208
Error assessment using the gravity gradient trace
𝜖
Instead of the original 𝑉𝑍𝑍
, however, we use a combined vertical gravity gradient
209
𝜖
𝜖
𝜖
𝜖
𝑉𝐶,𝑍𝑍
= (−𝑉𝑋𝑋
− 𝑉𝑌𝑌
+ 𝑉𝑍𝑍
)/2 = (−𝑉𝑋𝑋 − 𝜖𝑋 − 𝑉𝑌𝑌 − 𝜖𝑌 + 𝑉𝑍𝑍 + 𝜖𝑍 )/2 =
210
1
1
1
1
(𝑉𝑍𝑍 + 𝑉𝑍𝑍 ) − 𝜖𝑋 − 𝜖𝑌 + 𝜖𝑍
2
2
2
2
211
= 𝑉𝑍𝑍 + 𝜖𝐶,𝑍 .
212
As the error standard deviation of VXX and VYY is approximately half the error standard deviation
213
of VZZ
12
214
215
216
217
218
𝜎𝑋 = 𝜎𝑌 = 𝜎,
𝜎𝑍 = 2𝜎
we have, when the errors are uncorrelated,
2
𝜎𝐶,𝑍
=
1 2
6
(𝜎 + 𝜎 2 + 4𝜎 2 ) = 𝜎 2
4
4
⇒ 𝜎𝐶,𝑍 ≈ 1.22 𝜎
or an error reduction of 40% compared with the original VZZ [Fuchs et al., 2013].
219
220
With the combined VZZ the trace becomes
221
1
1
1
1
1
1
𝜖
𝜖
𝜖
𝑡𝑟𝑎𝑐𝑒𝐶 = 𝑉𝑋𝑋
+ 𝑉𝑌𝑌
+ 𝑉𝐶,𝑍𝑍
= 𝜖𝑋 + 𝜖𝑌 − 𝜖𝑋 − 𝜖𝑌 + 𝜖𝑍 = 𝜖𝑋 + 𝜖𝑌 + 𝜖𝑍
2
2
2
2
2
2
222
2
2
and 𝜎𝑡𝑟𝑎𝑐𝑒,𝐶
= 𝜎𝐶,𝑍
. We therefore see that expected error standard deviation of the trace with the
223
combined VZZ equals that of the combined vertical gravity gradient.
224
𝜖
The error assessment of EWH trends (Supplement S7) is done as follows. 1) Grids of 𝑉𝐶,𝑍𝑍
at
225
satellite altitude are used to estimate Stokes coefficients with spherical harmonic analysis. 2) The
226
Stokes coefficients are used to derive EWHs on a grid at the Earth’s surface for each of the 26
227
windows between November 2009 and June 2012. 3) A bias and trend are estimated using
228
𝑥̂ = (𝐴𝑇 𝐴)−1 𝐴𝑇 𝑦
229
where 𝑥̂ is a vector with bias and trend, A is the design matrix and y are the 26 EWH values for
230
each grid point. Errors are obtained by repeating step 1) using grids of 𝑡𝑟𝑎𝑐𝑒𝐶 , which gives
231
Stokes coefficients of the errors. Next step 2) is repeated with the error Stokes coefficients,
232
which gives EWH errors e. The error matrix 𝑄𝑥 of the EWH bias and trend then is
233
234
𝑄𝑥 = 𝜎02 (𝐴𝑇 𝐴)−1
where 𝜎02 is the a priori variance factor and 𝜎0 is the standard deviation of the EWH errors e.
235
13
236
S7.
EWH trends
237
The Stokes coefficients derived from the GRACE/GOCE vertical gravity gradients were used to
238
derive equivalent water heights for each (overlapping) period of 4 months. As 3 of 29 windows
239
are affected by the GOCE data gap in July-August 2010, these windows are discarded, and a
240
trend was estimated using 26 EWH values. We validated these results using two time series of
241
GRACE monthly solutions: CSR RL05ext and GFZ RL05a. The latter are complete to spherical
242
degree 90, the former to degree 96. The GRACE/GOCE gradients are derived from the band-
243
pass filtered GOCE vertical gravity gradients and low-pass filtered gravity gradients from
244
GRACE CSR RL05, which is complete to SH degree 60. Thus, the contribution of CSR RL05 to
245
the GRACE/GOCE is 50% or larger below SH degree 54, decreases between SH degree 54 and
246
60 and has zero contribution starting from SH degree 61.
247
Figure S6 shows the EWH trends for November 2009 to June 2012 derived from the three
248
models evaluated to SH degree 90 and with 250 km Gaussian smoothing. To be consistent with
249
GRACE/GOCE, we averaged the available GRACE-only monthly solutions in a moving window
250
of 4 months and estimated an EWH trend from the 26 values. The spatial patterns as well as the
251
amplitudes of the EWH trends are all in good agreement between the models.
252
14
A
B
C
253
Fig. S6. EWH trends 11/2009 – 06/2012 for a maximum degree of L = 90 and 250 km Gaussian
254
smoothing [mm/yr]. (A) CSR RL05ext; (B) GFZ RL05a; (C) GRACE/GOCE. Glaciers are
255
delineated as in Fig. 1E.
256
GRACE/GOCE derived EWH trends are shown in Fig. S7A, S7C and S7E, for L = 90, 110 and
257
120, where a Gaussian of 90 km has been used. The Stokes coefficients derived from the gravity
258
gradient trace were used to assess the error in the EWH trends, which are shown in Fig. S7B,
259
S7D and S7F. We see that the EWH trends show increasing spatial detail for increasing degree
260
and the signal amplitude increases. The minima collocate with the glaciers feeding the Getz Ice
261
Shelf, and the Haynes/Smith/Kohler, Thwaites and Pine Island Glaciers. For the L = 90 case the
262
amplitude of the EWH trend agrees well with the CSR RL05ext and GFZ RL05a solutions (Fig.
263
4Cand 4D). The estimated errors of the EWH trends increase for increasing degree and are
264
generally smaller towards the south, which is attributed to the denser track distribution for high
265
latitudes.
15
A
B
C
D
E
F
266
Fig. S7. EWH trends and estimated errors November 2009 – June 2012 derived from a
267
combination of GOCE and GRACE [mm/yr]. Trends for maximum spherical harmonic degree
268
(A) L = 90, (C) L = 110, (E) L = 120. Estimated error for maximum spherical harmonic degree
269
(B) L = 90, (D) L = 110, (F) L = 120. Glaciers are delineated as in Fig. 1E.
270
S8.
271
The EWHs from the preceding section are equivalent to surface mass densities [Wahr et al.,
272
1998]. Applying a basin mask (basin = 1, other = 0) and subsequent integration gives the total
273
surface mass within a certain basin for each time window. Estimating a trend then gives the mass
274
change for the period November 2009 – June 2012. Table S5 summarizes these mass trends for
275
all basins in West Antarctica using the GRACE-only models CSR RL05ext and GFZ RL05a to
276
degree and order 90 where a Gaussian smoothing of 250 km and 90 km has been applied. Mass
Ice mass trends
16
277
trends for the GRACE/GOCE solutions were determined with the same settings and in addition
278
we used a maximum degree of 120 with Gaussian smoothing of 90 km.
279
In all cases, the time-variable part of the gravitational flattening coefficient 𝐶20 has been
280
replaced with values determined from SLR as this coefficient is less reliably estimated from
281
GRACE [Cheng et al., 2013]. The degree 1 Stokes coefficients cannot be determined well from
282
GRACE or GOCE as these coefficients are coupled with orbital parameters. Not accounting for
283
changes in these coefficients can have a large impact on estimated mass trends for Antarctica as
284
a whole [Ivins et al., 2013]. We re-estimated the mass trends in Table S5 using the trend values
285
for the degree 1 Stokes coefficients given by [Ivins et al., 2013] that are based on [Swenson et
286
al., 2008]. For basins 20 – 22 the trend became less negative by 0.3 – 0.4 Gt/yr, which is small
287
given the total effect and mass trend standard deviation, see Table S6.
288
The GIA effect on the mass trends was estimated for basin 20, 21 and 22 as well as for WAIS
289
using the three discussed GIA models. The W12a model gives values slightly larger than the
290
IJ05_R2 model, which in turn gives values slightly larger than the van der Wal model. For basins
291
20, 21 and 22 the GIA correction is small and at most -4.6 Gt/yr. The mass trend values in Table
292
1 include the GIA correction according to IJ05_R2.
293
The GRACE/GOCE uncertainties in Table S5 are 2σ values derived from the gravity gradient
294
trace and the GRACE-only models as follows. The gridded trace values are converted to Stokes
295
coefficients and these are used to determine surface mass density errors for each time window of
296
4 months. Integration then gives mass errors for each basin: a time series of 26 values for each
297
basin or sum of basins. These mass error time series are used to determine the variance factor
298
2
𝜎0,𝑡𝑟𝑎𝑐𝑒
for each basin or sum of basins, which is the same procedure that was used for the EWH
299
errors (see Supplement S6). This variance factor would then only include the GOCE part. To
300
assess the GRACE contribution to the errors to degree L = 60, we computed mass trend errors
301
from the difference between CSR RL05ext and GFZ RL05a. Both models are cosine tapered
302
between degree 40 and 60 to emulate the low-pass filter that was applied to GRACE derived
303
gravity gradients in their combination with GOCE. In addition, they are filtered using a Gaussian
304
of 250 km or 90 km. Also from the GRACE difference time series the variance factor is
305
determined for each basin or sum of basins. The standard deviation is multiplied with √2−1 to
306
obtain the standard deviation of a single GRACE solution. By summing the trace derived and
17
307
2
GRACE-only derived variance factors the GRACE/GOCE variance factor 𝜎0,𝑚𝑎𝑠𝑠
is obtained,
308
which is used to scale the error matrix (𝐴𝑇 𝐴)−1 of the estimated bias and trend.
309
The GRACE-only variance factors are obtained by computing mass trend errors from the
310
difference between CSR RL05ext and GFZ RL05a to degree L = 90. Their derived error standard
311
deviation is multiplied with √2−1 , assuming equal accuracy of the GRACE solutions and
312
uncorrelated errors. This assumption may lead to too optimistic error estimates as the GRACE-
313
only solutions are largely based on the same input data. Again the variance factors are used to
314
scale the error matrix (𝐴𝑇 𝐴)−1 of the estimated bias and trend. Note that in the determination of
315
the standard deviation for the sum of basins, one cannot simply add the variances of the single
316
basins as there is error correlation between basins. We therefore have, e.g., for basins 20 – 22:
317
√22 + 42 + 32 ≈ 5.4 Gt/yr ≠ 3 Gt/yr, for L = 90 with 250 km Gaussian smoothing.
318
The GRACE-only and GRACE/GOCE L = 90 derived trends are generally consistent for a
319
Gaussian of 250 km for basins 20, 21 and 22. The differences between GRACE/GOCE and CSR
320
RL05ext are 2 – 3 Gt/yr, and between GFZ RL05a and CSR RL05ext 3 – 8 Gt/yr. The
321
differences between GRACE/GOCE and GFZ RL05a are 0 – 11 Gt/yr, two of which are greater
322
than 3σ. The differences between the lumped sum of basins 20 – 22 are 2 – 6 Gt/yr, all less than
323
3σ, and therefore smaller than the differences between individual basins. This implies that the
324
estimated errors are too optimistic and/or that there are (small) systematic effects. The
325
differences between the GRACE/GOCE mass trends and GRACE-only mass trends for the
326
lumped sum of basins 1, 18, 19 and 23 are 18 – 22 Gt/yr. This is well above 3σ, where basin 1
327
and basin 18 extend below 83° S and are therefore partially not covered by GOCE data, which
328
may lead to unreliable trend estimates for these basins.
329
For a Gaussian smoother of 90 km half-width the GRACE-only solutions CSR RL05ext – GFZ
330
RL05a differ by +26 Gt/yr and -31 Gt/yr for basin 21 and 22 respectively, which is attributed to
331
systematic north-south stripes in the solutions that do not average out . The GRACE/GOCE mass
332
trend differences with CSR RL05ext are 14, 1, and 10 Gt/yr for basin 20, 21 and 22, whereas
333
they are 3, 27, and 21 Gt/yr for GFZ RL05a. Thus the GRACE/GOCE trends are closer to CSR
334
RL05ext, which is to be expected as CSR RL05 was used in the combination of GRACE/GOCE.
335
Nevertheless, for basin 20 the GRACE/GOCE trend is closer to GFZ RL05a than to CSR
336
RL05ext. For Basins 21 and 22, the differences between trends from GRACE/GOCE L = 90 and
18
337
L = 120 are small. This is consistent with simulations of the separation of signals in those two
338
basins, conducted based on realistic spatial patterns of glacier change with L = 90 and L = 120 in
339
an error-free scenario (Supplement S5).
340
Table S5: Mass trends West Antarctica [Gt/yr] (𝑪𝟐𝟎 from SLR), no GIA correction.
341
Uncertainties are 2σ values, Gaussian filter half-widths are indicated between brackets.
L = 90
L = 90
L = 120
(250 km)
(90 km)
(90 km)
Basin
GRACE +
CSR
GFZ
GRACE +
CSR
GFZ
GRACE +
GOCE
RL05ext
RL05a
GOCE
RL05ext
RL05a
GOCE
20
-44±2
-42±1
-38±1
-53±9
-67±6
-56±6
-57±10
21
-55±2
-58±3
-66±3
-61±11
-62±10
-88±10
-59±13
22
-38±2
-41±2
-38±2
-64±6
-74±14
-43±14
-63±9
20, 21, 22
-137±3
-141±2
-143±2
-179±14
-203±10
-188±10
-180±18
-39±4
-21±3
-17±3
-9±7
22±13
5±13
4±16
-176±5
-163±4
-160±4
-188±12
-181±6
-183±6
-175±11
1*, 18*,
19, 23
WAIS
342
*
343
The CSR RL05ext – GFZ RL05a difference for a Gaussian with 90 km half-width is -15 Gt/yr
344
for basins 20 – 22, whereas it is +17 Gt/yr for the sum of basins 1, 18, 19 and 23. The difference
345
between the GRACE-only solutions is just 2 Gt/yr for WAIS. All these differences, last three
346
rows in Table S5, are just above or smaller than 2σ, where we note that the error standard
347
deviation for the largest area, WAIS, is smaller than the error standard deviation of the subareas.
348
These are all indications that the GRACE-only solutions filtered with a Gaussian of 90 km half-
349
width suffer from systematic effects that average out for WAIS, but do not average out for
350
smaller areas.
Basins 1 and 18 extend below 83° S and are therefore not fully covered by GOCE data.
351
19
352
Table S6: GIA correction and degree one effect for basins in West Antarctica [Gt/yr].
353
Uncertainties are 2σ values.
Basin IJ05_R2 van der Wal W12a Mean GIA
degree l = 1
20
-2.0
-0.3
-2.8
-1.7 ± 1.8
0.4
21
-2.1
-0.6
-4.6
-2.4 ± 4.0
0.4
22
-3.3
-0.7
-4.4
-2.8 ± 3.8
0.3
WAIS
-22.9
-21.7
-34.3 -26.3 ± 14.0
2.5
354
355
The GRACE/GOCE mass trends are corrected for GIA and degree one effects. Table S7 shows
356
these mass trends for L = 90 and L = 120 with 90 km Gaussian filter half-width. The
357
uncertainties are 2σ values based on the GRACE/GOCE uncertainties from Table S5, the GIA
358
uncertainties from Table S6 and assuming that the 1-sigma error of the degree l = 1 term is as
359
large as the signal itself. More conservative GRACE/GOCE uncertainties would be obtained by
360
multiplying the uncertainties, e.g., by a factor of two as is done for GRACE-only, to account for
361
possible unmodeled temporal correlations [Horwath and Dietrich, 2009; King et al., 2012].
362
However, GOCE errors behave as white noise for higher frequencies and we are unaware of
363
analyses that show that there are temporal correlations. In addition, our estimates of mass
364
imbalance are consistent with independent estimates for basins 21 and 22, discussed below,
365
which gives reason to believe that our error estimates are reasonable.
366
The mass trends derived from CryoSat-2 are taken from Table 1 in McMillan et al. [2014], where
367
we summed the trend of basins 1, 18, 19 and 23 and obtained the associated uncertainty by error
368
propagation (McMillan et al. [2014] give 1σ values). For L = 90 the GRACE/GOCE trends differ
369
more from the CryoSat-2 trends for the sum of basins 1, 18, 19 and 23 than they do for L = 120.
370
Although the trend differences are 34 and 20 Gt/yr for L = 90 and L = 120 respectively, these
371
differences are not significant, since the GRACE/GOCE and CryoSat-2 uncertainties are large
372
for the sum of these four basins. Again we note that basins 1 and 18 extend below 83° S and are
373
therefore not fully covered by GOCE data. In addition, we did not apply offshore integration of
374
basins 1, 18, 19 and 23 because our study focusses on basins 20, 21 and 22. The GRACE/GOCE
20
375
trends do not significantly differ from the CryoSat-2 trends for basins 21 and 22, whereas they
376
do differ for basin 20 (discussed in the Main Text).
377
Medley et al. [2014] give mass budget estimates for five different catchments in ASS: Pine
378
Island, Wedge, Thwaites, East Thwaites and Haynes. Mass budgets are given for a number of
379
distinct periods with different surveys. The mass trends in Table S7, last column, are computed
380
from two time periods, 2009 – 2010, for which mass budgets are given for all five catchments
381
and for which the time period overlaps with the GRACE/GOCE data. The trends for the five
382
catchments are averaged per catchment and the trends for Pine Island and Wedge are added to
383
obtain the trend for basin 22, whereas the trends for Thwaites, East Thwaites and Haynes for
384
basin 21. Uncertainties are obtained from error propagation, where we note that Medley et al.
385
[2014] give 1σ values, while 2σ values are given here. The trends between brackets are obtained
386
assuming that they scale linearly with the difference in area size, which is about 12% larger for
387
both basins compared with the sum of the catchments. The scaled trends therefore represent an
388
upper bound.
389
Table S7: Mass trends West Antarctica [Gt/yr] with GIA correction from IJ05_R2 and
390
degree 1 correction. Uncertainties are 2σ values.
Basin
GRACE/GOCE
CryoSat-2
Mass budget
(11/2009 – 06/2012)
(2010 – 2013)
(2009 – 2010)
L = 90
L = 120
20
-55.0 ± 9.4
-58.4 ± 10.5
-23 ± 18
-
21
-63.1 ± 11.7 -61.1 ± 14.0
-64 ± 24
-52.5 (-58) ± 11.7
22
-67.0 ± 7.3
-66.4 ± 9.8
-56 ± 26
-47.4 (-53) ± 15.0
-23.3 ± 10.5
-9.7 ± 17.4
10 ± 36
-
-134 ± 54
-
1, 18,
19, 23
McMillan et al. [2014] Medley et al. [2014]
WAIS -208.5 ± 19.1 -195.6 ± 18.4
391
392
The error correlations in Table S8 between basins 20, 21 and 22 are derived from the
393
GRACE/GOCE mass error time series defined above. The error correlations are 0.5 or less,
21
394
which is not considered to be very large given the small number of samples (26 time windows).
395
The largest correlations are for L = 90 with 90 km Gaussian, which is possibly related to the
396
degree truncation and small filter-width that may lead to oscillations in the space domain.
397
Table S8: Error correlation between basins 20, 21 and 22.
Basin pair L = 90, 250 km L = 90, 90 km L = 120, 90 km
20, 21
-0.5
-0.4
-0.4
20, 22
0.1
0.5
0.2
21, 22
0.1
-0.1
0.2
398
399
Finally, we assess the amount of leakage between the basins based on the obtained mass trends
400
(Table S5) in combination with the simulation results (Table S2 and S3). The observed mass
401
trend 𝑚̇ in basin 20 – 22 is modelled as
402
𝑚̇20
𝑚̇21
𝑚̇22
=
=
=
𝛼1 𝑚̇20 + 𝛽1 𝑚̇21 + 𝛾1 𝑚̇22 + 𝛿1 𝑚̇𝑜𝑡ℎ𝑒𝑟 + 𝜀20
𝛼2 𝑚̇20 + 𝛽2 𝑚̇21 + 𝛾2 𝑚̇22 + 𝛿2 𝑚̇𝑜𝑡ℎ𝑒𝑟 + 𝜀21
𝛼3 𝑚̇20 + 𝛽3 𝑚̇21 + 𝛾3 𝑚̇22 + 𝛿3 𝑚̇𝑜𝑡ℎ𝑒𝑟 + 𝜀22
403
where 𝑚̇𝑖 is the true (unknown) mass trend in basin i, 𝛼1 , 𝛽1, etc. are coefficients that determine
404
the “leakage-out” and “leakage-in” effects, and 𝜀 are errors. The 𝛽 and 𝛾 coefficients can be
405
obtained from Table S2 and S3 and are, for L = 90 and a Gaussian filter of 90 km,
3.2
≈ 0.053
60.7
0.2
𝛾1 =
≈ 0.003
60.5
𝛽1 =
406
49.6
8.2
≈ 0.817 𝛽3 =
≈ 0.135
60.7
60.7
.
3.0
52.1
𝛾2 =
≈ 0.050 𝛾3 =
≈ 0.861
60.5
60.5
𝛽2 =
407
A rule-of-thumb estimate of the leakage of the other basins to basin 20 – 22 is obtained by taking
408
the maximum estimated mass trend of 22 Gt/yr (CSR RL05ext) for the sum of basins 1, 18, 19
409
and 23, and assuming that 20% of the signal leaks to other basins and the ocean. A value of 20%
410
is reasonable as we have leakage-out of 18% and 14% for basin 21 and basin 22 respectively. We
411
then have 4 Gt/yr that is redistributed over basins 20, 21, 22 and elsewhere. We therefore expect
412
that the summed leakage of basins 1, 18, 19 and 23 to basins 20, 21 and 22 is roughly zero,
413
which means that the 𝛿 coefficients are assumed to be zero.
22
414
The leakage of the basin 22 to the non-adjacent basin 20 is 0.3% and negligible. If we assume
415
that this is also true vice-versa, then 𝛼3 ≈ 0 and we have 𝑚
̅̇ 22 = 0.135 𝑚̇21 + 0.861 𝑚̇22 =
416
−63 Gt/yr, where 𝑚̇21 and 𝑚̇22 are GRACE/GOCE observed values. In other words, leakage-
417
out of basin 22 is largely compensated by leakage-in of basin 21 to 22 and the observed mass
418
trend for PIG seems to be a good approximation of the true mass trend although part of the trend
419
is caused by leakage from the Thwaites and Haynes/Smith/Kohler Glaciers. By setting 𝑚̇22 =
420
𝑚̇22 we have 𝛽3 𝑚̇21 = (1 − 𝛾3 )𝑚̇22 and 𝑚̇21 = −66 Gt/yr. With this value we obtain for the
421
leakage of basin 21 to 20 a value of 𝛽1 𝑚̇21= -3 Gt/yr and for the leakage of basin 20 to 21 we get
422
𝛼2 𝑚̇20 = -4 Gt/yr. Thus it appears that the leakage between basins 20 and 21 is about equal with
423
a net near-zero effect. For basin 20 we have -54 Gt/yr = 𝑚̇20 ≈ 𝛼1 𝑚̇20 + 𝛽1 𝑚̇21 . If we use the
424
above derived 𝑚̇21 = −66 Gt/yr and the given 𝛽1, then 𝑚̇20 ≈ −49.5/𝛼1 Gt/yr. The 𝛼1
425
coefficient was not derived in the simulation, but if we use the equivalent coefficient 𝛽2 or 𝛾3 as
426
approximate value, then 𝑚̇20 is -61 Gt/yr or -57 Gt/yr, respectively. Interestingly, this is close to
427
the mass trend for basin 20 that was estimated for the L = 120 case.
428
All-in-all one may therefore conclude that when GOCE gravity gradients are used to supplement
429
GRACE monthly fields, a clear separation of adjacent glacier drainage basin mass change in the
430
ASS is revealed, in contrast to GRACE-only monthly global solutions. Leakage effects between
431
basins 20, 21 and 22 are present but tend to be relatively small compared with the estimated
432
errors. In addition, leakage effects seem to compensate each other to a large extent and the
433
estimated trends are probably a good proxy for the true mass imbalance trends.
434
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