MASS BALANCE OF GLACIERS IN THE WRANGELL RANGE, ALASKA
DETERMINED BY THE PTAA MODEL
Wendell Tangborn
Preliminary Report (June 2011)
Summary
The total glacierized area of the Wrangell Mountain Range in Alaska, approximately 4725 km2, is divided
into two separate regions of nearly equal area based on north and south orientations of the approximately 30
glaciers in this range. The North Wrangell glacier area is 2418 km2 and the South Wrangell area is 2307 km2.
Figure 1 is the area-altitude distribution for the South Wrangell glaciers, and Figure 2 the area-distribution
for the North Wrangells. Each of these glacierized regions is divided into 30.5 m (100 feet) elevation
intervals by ARC GIS.
The purpose of this project is to determine the historical mass balances of the Wrangell Mountain glaciers
using the PTAA model. To accomplish this, snow accumulation and snow and ice ablation at each of the
280 elevation intervals is calculated for each day of a 57 year period (1950-2006). These incremental
balances are then formed into annual mass balances for each region and for the full period of available
weather records.
Calibration of the PTAA mass balance model for each glacierized region is accomplished by calculating 9
balance parameters for each day of the summer season (approximately May 15 to September 30) throughout
the 1950-2006 period. Linear regressions are run between the calculated balance parameters for each day of
the summer season. Daily temperature and precipitation observations at two weather stations are converted
to snow accumulation and snow and ice ablation at each AA interval, then totaled for the glacierized areas.
(thus each iteration of the simplex optimization procedure requires 3-5 billion calculations). The first 15
iterations are derived from pre-set coefficients, the remaining are automatically computed in the simplex
optimizing subroutine. Usually 3-400 iterations are sufficient to close the simplex, which occurs when the
optimization function (ERRPC) has reached a designated minimum. It is emphasized that the determination
of the coefficients is not restricted in any way and can assume any value as long as the error produced by the
balance parameter regressions is minimized.
A detailed description of the PTAA model using South Cascade Glacier for its development is provided in
Tangborn (1999). Additional applications of the model in Alaska can be found in Tangborn (1997), Bhatt et
al (2007), Zang et al (2007a), Zang et al (2007b).
The optimization function, ERRPC, is equal to the average root-mean-square plus the complement of the R2
(1-R2) x 100, for the 19 regressions that relate the 9 balance parameters during each day of the summer
season (130-150 days), therefore, approximately 2700 linear regressions determine the value of ERRPC for
each iteration. Figure 3 demonstrates the optimization error (ERRPC) for each iteration when calibrating the
south Wrangell glaciers, and Figure 4 is the results for calibrating the north Wrangells. It should be noted
that once the 15 preset coefficients are applied to determine annual balances, the remaining errors are derived
solely by simplex optimization. The high value of the optimization error ERRPC (over 40 %) should not be
considered significant and is not related to the accuracy of the final balances.
The two weather stations used in this example are McKinley Park and Big Delta, located approximately 350
km from Mt. Wrangell. There are closer weather stations; however, no combination of four of them
produced a lower ERRPC than these two (that were also found to be optimum for Gulkana Glacier in an
earlier study). Calibration for the Wrangell Range glaciers was initiated using Gulkana coefficients, the
simplex optimization then produced a new set of coefficients for each of the north and south glacierized
areas (Tables 1 and 2).
Laser altimetry maps have been constructed for the Kennicott Glacier (1043 km2) in the South Wrangells
and the Nabesna Glacier (253 km2) in the North Wrangells. In addition to the 67 north and south facing
glaciers in the Wrangell Range, the PTAA model was also run individually on the Nabesna Glacier to represent
the north Wrangells and the Kennicott Glacier to represent the South Wrangells. This was done to
compare these data to aircraft laser altimetry data collected by the University of Alaska, Fairbanks. These model
runs also used the McKinley Park and Big Delta weather stations for the temperature and precipitation data
inputs. The Nabesna Glacier PTAA model run covered the time period 1957-2008 and had a total thickness
change of -33 meters of ice, or -0.57 m/yr for the 52 year period. The single glacier PTAA model runs for the
Nabesna and Kennicott Glaciers show the same patterns as the full area model runs, with the south oriented
Kennicott Glacier’s mean annual mass balance (-0.96 m.w.e) being some 40% greater than that for the north
facing Nabesna Glacier (-0.57 m.w.e). The mean annual balance for the 1950-2006 period of the North
Wrangell glaciers is –0.23 MWE and – 0.66 for the South Wrangell glaciers, corresponding total volume
losses of 556 km3 and 1523 km3, respectively.
These results predict that mass balances of south facing glaciers are nearly three times more negative (-0.66
to –0.23 mwe) than north facing glaciers, and that the ELA is 200 m higher than the north facing glaciers.
Glacier orientation is not included in the model for calculating ablation, but it is noteworthy that the PTAA
model apparently incorporates orientation in the calibration process as it produces realistic mass balances
using only weather observations from distant stations and a precise calculation of the area-altitude
distributions of these glacierized areas.
However, unquestionable verification of the model’s accuracy can only be determined using volume
changes found by DEM means during a sufficiently long segment of the 1950-2006 period.
FRACTION OF TOTAL AREA
SOUTH WRANGELL GLACIERS
AREA ALTITUDE DISTRIBUTION
TOTAL AREA = 2307 KM2
0.025
0.02
0.015
0.01
0.005
45
7
67
1
88
4
10
97
13
11
15
24
17
37
19
51
21
64
23
77
25
91
28
04
30
18
32
31
34
44
36
58
38
71
40
84
42
98
45
11
47
24
49
38
0
ELEVATION (METERS)
Figure 1. Area-Altitude distribution of the glaciers of the Southern region of the Wrangell Range,
determined for each 30.5 m interval by ARC GIS.
NORTH WRANGELL GLACIERS
AREA-ALTITUDE DISTRIBUTION
TOTAL AREA = 2418 KM2
FRACTION OF TOTAL AREA
0.035
0.03
0.025
0.02
0.015
0.01
0.005
91
4
10
97
12
80
14
63
16
46
18
29
20
12
21
95
23
77
25
60
27
43
29
26
31
09
32
92
34
75
36
58
38
41
40
23
42
06
43
89
45
73
47
55
49
38
0
ELEVATION (METERS)
Figure2. Area-Altitude distribution of the northern region of the Wrangell range, determined by ARC
GIS.
WRANGELL RANGE SOUTH
CALIBRATION ERROR VERSUS ITERATION
70
CALIBRATION ERROR - ERRPC (%)
1ST 15 ERRORS DETERMINED FROM PRE-SET COEFFICIENTS
65
60
55
50
45
757
736
715
694
673
652
631
610
589
568
547
526
505
484
463
442
421
400
379
358
337
316
295
274
253
232
211
190
169
148
127
85
106
64
43
22
1
40
ITERATION NUMBER
Figure 3. Calibration error (ERRPC) versus iteration of the south Wrangell Range. The first 15 errors
are determined by the pre-set coefficients shown in the input file OPT!IN.WRS.
WRANGELL RANGE NORTH
110
100
ERRPC (%)
90
80
70
60
50
701
676
651
626
601
576
551
526
501
476
451
426
401
376
351
326
301
276
251
226
201
176
151
126
101
76
51
26
1
40
ITERATION NUMBER
Figure 4..
Calibration error (ERRPC) versus iteration of the north Wrangell Range. The first 15 errors
are determined by the pre-set coefficients shown in the input file OPT1IN.WRN. The remaining
errors resulted from the simplex calibration process
WRANGELL RANGE NORTH
ANNUAL BALANCE VERSUS CALIBRATION ERROR
2
1.5
MEAN ANNUAL BALANCE (MWE)
MEAN BALANCE = -0.23
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
40
50
60
70
80
90
100
110
CALIBRATION ERROR (ERRPC) %
Figure 5. The mean annual balance of the north Wrangell glaciers, determined for the 1950-2006
period, versus the calibration error for each iteration. The minimum error is reached when the mean
balance is –0.23 MWE compared to a mean balance of -0.66 MWE for the south facing glaciers.
WRANGELL RANGE SOUTH
ANNUAL BALANCE VERSUS CALIBRATION ERROR
MEAN ANNUAL BALANCE (1950-2006) MWE
1.5
1
MEAN BALANCE = -0.66
0.5
0
-0.5
-1
-1.5
40
45
50
55
60
65
70
ERROR (ERRPC) %
Figure 6. The mean annual balance of the south Wrangell glaciers, determined for the 1950-2006
period, versus the calibration error for each iteration. The minimum calibration error is reached
when the mean balance is –0.66 MWE, compared to a mean balance of -0.23 MWE for north facing
glaciers.
WRANGELL RANGE NORTH
ANNUAL BALANCE
0.5
ANNUAL BALANCE (MWE)
0
-0.5
-1
-1.5
-2
04
02
00
98
96
94
92
90
88
86
84
82
80
78
76
74
72
70
68
66
64
62
60
58
56
54
52
06
20
20
20
20
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
50
-2.5
Figure 7. Annual balances for the north Wrangell glaciers for 1950-2006, calculated from the final
coefficients determined in the calibration. The mean annual balance for this period of these northerly
oriented glaciers is –0.23 mwe.
WRANGELL RANGE SOUTH
ANNUAL BALANCE
1
ANNUAL BALANCE (MWE)
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
20
04
20
01
19
98
19
95
19
92
19
89
19
86
19
83
19
80
19
77
19
74
19
71
19
68
19
65
19
62
19
59
19
56
19
53
19
50
-3.5
Figure 8. Annual balances for the south Wrangell glaciers for 1950-2006, calculated from the final
coefficients determined in the calibration. The mean annual balance for this period for these southerly
oriented glaciers is –0.66 mwe.
WRANGELL RANGE GLACIER ANNUAL BALANCES
1
SOUTH WRANGELL BALANCE (MWE)
0.5
0
-0.5
-1
y = 1.2987x - 0.2606
R2 = 0.8425
-1.5
-2
-2.5
-3
-3.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
NORTH WRANGELLS BALANCE (MWE)
Figure 9. Annual balance of glaciers with southern orientation versus annual balance of northern
glaciers
WRANGELL RANGE, ALASKA
CUMULATIVE ANNUAL BALANCES
NORTH AND SOUTH GLACIERS
CUMULATIVE ANNUAL BALANCE (MWE)
0
-5
-10
-15
-20
SOUTH WRANGELLS
NORTH WRANGELLS
-25
-30
20
04
20
01
19
98
19
95
19
92
19
89
19
86
19
83
19
80
19
77
19
74
19
71
19
68
19
65
19
62
19
59
19
56
19
53
19
50
-35
Figure 10. Cumulative annual balances of southerly oriented Wrangell glaciers and northerly oriented
glaciers.
SOUTH WRANGELL RANGE, ALASKA
MEAN ANNUAL, ACCUMULATION AND ABLATION BALANCE
VERSUS ELEVATION
MEAN BALANCE (1950-2006) MWE .
2
1
0
-1
-2
-3
-4
ANNUAL BALANCE
ACCUMULATION BALANCE
ABLATION BALANCE
-5
-6
-7
48
8
67
0
85
4
10
36
12
20
14
02
15
86
17
68
19
52
21
34
23
18
25
00
26
84
28
66
30
50
32
32
34
16
35
98
37
82
39
64
41
48
43
30
45
14
46
96
48
80
-8
ELEVATION (METERS)
Figure 11. Annual, accumulation and ablation balances of south oriented glaciers. Mean ELA is
approximately 2040 m elevation.
2
WRANGELL RANGE NORTH, ALASKA
MEAN ANNUAL, ACCUMULATION AND ABLATION BALANCE
VERSUS ELEVATION
1
BALANCE (MWE) .
0
-1
-2
-3
-4
ANNUAL BALANCE
ACCUMULATION BALANCE
ABLATION BALANCE
-5
94
4
10
97
12
50
14
02
15
54
17
07
18
60
20
12
21
64
23
17
24
70
26
22
27
74
29
27
30
80
32
32
33
84
35
37
36
90
38
42
39
94
41
47
43
00
44
52
46
04
47
57
49
10
-6
ELEVATION (METERS)
Figure 12. Annual, accumulation and ablation balances of north oriented glaciers. Mean ELA is
approximately 2230 m elevation
TABLE 1. SOUTH WRANGELL RANGE FINAL COEFFICIENTS
2.9231002330780029 1 CFF1 PRECIP MULTIPLIER AT MAX ALTITUDE
1.5325336456298828 2 CFF2 PRECIP MULTIPLIER AT TERMINUS
1.9415547847747803 3 CFF3 ALTITUDE OF MAX PRECIP (X 1000)
0.4155697822570801 4 CFF4 PRECIP MIXING FRACTION
0.4223370552062988 5 CFF5 LAPSE-RATE INTERCEPT (> AVG TEMP)
2.4088523387908936 6 CFF6 LAPSE-RATE LINE SLOPE (+2)
0.5889337658882141 7 CFF7 LAPSE-RATE INTERCEPT (< AVG TEMP)
2.1134562492370605 8 CFF8 LAPSE-RATE LINE SLOPE (+2)
13.4472980499267578 9 CFF9 ABLATION FROM TEMP (W/O PRECIP)
11.3437223434448242 10 CFF10 ABLATION FROM TEMP (WITH PRECIP)
-0.0121527016162872 11 CFF11 TEMPERATURE THRESHOLD
1.0233986377716064 12 CFF12 ABLATION FROM SOLAR RADIATION
8.0389041900634766 13 CFF13 SEASONAL SNOWLINE RISE MULTIPLIER
1.3465770483016968 14 CFF14 TEMPORARY SNOWLINE RISE MULTIPLIER
0.8375621438026428 15 CFF15 INTERNAL ACCUMULATION MULTIPLIER
TABLE 2. NORTH WRANGELL RANGE FINAL COEFFICIENTS
2.3887887001037598 1 CFF1 PRECIP MULTIPLIER AT MAX ALTITUDE
1.4978747367858887 2 CFF2 PRECIP MULTIPLIER AT TERMINUS
1.8936026096343994 3 CFF3 ALTITUDE OF MAX PRECIP (X 1000)
0.4049295783042908 4 CFF4 PRECIP MIXING FRACTION
0.3593062758445740 5 CFF5 LAPSE-RATE INTERCEPT (> AVG TEMP)
2.3861336708068848 6 CFF6 LAPSE-RATE LINE SLOPE (+2)
0.5852508544921875 7 CFF7 LAPSE-RATE INTERCEPT (< AVG TEMP)
2.1820788383483887 8 CFF8 LAPSE-RATE LINE SLOPE (+2)
13.8239889144897461 9 CFF9 ABLATION FROM TEMP (W/O PRECIP)
10.9378795623779297 10 CFF10 ABLATION FROM TEMP (WITH PRECIP)
0.0039875274524093 11 CFF11 TEMPERATURE THRESHOLD
0.8797739744186401 12 CFF12 ABLATION FROM SOLAR RADIATION
8.5338897705078125 13 CFF13 SEASONAL SNOWLINE RISE MULTIPLIER
1.3889465332031250 14 CFF14 TEMPORARY SNOWLINE RISE MULTIPLIER
0.8865205049514771 15 CFF15 INTERNAL ACCUMULATION MULTIPLIER
Conclusions
The PTAA model apparently generates realistic mass balances using only low-altitude weather observations
because the glacier has retained a memory of the past climate. Over the thousands of years of its existence,
erosion produced by the sliding glacier has carved its climatic history in the underlying bedrock. The surface
configuration of the glacier (its area-altitude distribution) reflects both the undulations of its bed and ice flow
of the glacier, thus by combining current weather observations with the current AA distribution, which is
embedded with the past climate, the current mass balance (snow accumulation minus snow and ice ablation)
can be calculated.
The surface configuration of these glaciers influence the timing, distribution and amount of snow
accumulation and ablation. Orientation of the surface strongly affects ablation, and to some extent
accumulation, and the Wrangell mass balances determined by the model show that south-facing glaciers have
3 times the ablation rate as north-facing glaciers. Therefore, the model appears to have sensed from the AA
profiles of the north and south facing glaciers that ablation would be much greater for the south-facing
glaciers Surface orientation is not an input to the model, also, the same weather stations were used for both
north and south glaciers. Theoretically, it should be possible to reconstruct the climate of this region for the
past several hundred years based only on how these glaciers respond to the current climate.
Acknowledgements
The area-altitude distributions were calculated by Indrani Das using ARC DIS. Funding for the Wrangell
project is provided by HyMet Inc
References
Zhang, J., U. S. Bhatt, W. V. Tangborn, and C. S. Lingle, 2007: Climate downscaling for estimating glacier
mass balances in northwestern North America: Validation with a USGS benchmark glacier, Geophysical
Research Letters, 34, L21505, doi:10.1029/2007GL031139. pdf version .
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with a Hierarchical Modeling Approach, Computing in Science and Engineering, 9 (2), 61-67. pdf version.
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America to Future Climate Change: an Atmosphere/Glacier Hierarchical Modeling Approach, Annals of
Glaciology, Vol. 46, 283 – 290. pdf version.
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the Area–Altitude Distribution of a Glacier , Geografiska Annaler: Series A, Physical Geography
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Columbia Glacier and relate it to calving and speed. Report of a Workshop, February 28 – March 2, 1997,
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Article first published online : 15 DEC 2003, DOI: 10.1111/1468-0459.00103