www.mtri.org Glacier Ablation Sensor System Glacier Ablation Sensor System 2008 Analysis and 2008 Analysis and Generalization Generalization Development of a melt model through multiple linear regression of remote sensing measurements Kevin Endsley B.S. Applied Geophysics candidate, Michigan Technological University Summer 2009 Research Intern, Michigan Tech Research Institute
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Glacier Ablation Sensor SystemGlacier Ablation Sensor System2008 Analysis and Generalization2008 Analysis and Generalization
Development of a melt model through multiple linear regression of remote sensing measurements
Kevin EndsleyB.S. Applied Geophysics candidate, Michigan Technological UniversitySummer 2009 Research Intern, Michigan Tech Research Institute
2
IntroductionIntroduction
Glacier Ablation Sensor System (GASS) measures hourly:
– Distance to glacier surface (ablation proxy)
– Air temperature– Irradiance (solar radiance)– Exitance (light emitted from
glacier)– Wind speed– Battery voltage– Latitude and longitude– Date and time
Photos Credit: Dr. Robert Shuchman
Observations from data:– Glacier migration– Total seasonal melt– Melt rate (mostly constant at 4
cm/day)
3
Model SpecificationModel Specification
All four driving factors measured (air temperature, irradiance, exitance, and wind speed) correlate with absolute melt at least in their time series aspect; less so in instantaneous measurements:
Irradiance known to have an effect; correlation is moderate (R²=0.47 at GASS B02) for instantaneous measurements
Temperature known to have an effect, correlation is moderate (R²=0.38 at GASS B02) for instantaneous measurements
Exitance only moderately correlated (R²=0.38 at GASS B02)
Wind speed not strongly correlated (R²=0.03 at GASS B02)
Problem is specification: instantaneous irradiance and temperature account for less than 50% (R²=0.50) of absolute melt variance, but explanatory power exceeds 90% (R²=0.90) when time series aspect included
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Model Specification (Continued)Model Specification (Continued)
Parameters were integrated (over gaps between ablation measurements) and correlated with melt:
(tmp_INT) (tlt_INT) (blt_INT) (wnd_INT)
Temperature Irradiance Exitance Wind Speed
B01 R² = 0.96 -- R² = 0.91 --
B02 -- R² = 0.92 R² = 0.89 --
B03 -- R² = 0.60 R² = 0.54 --
B06 -- -- R² = 0.75 R² = 0.07
Blanks indicate unlikely parameters (high p-values) in the full model. What is needed is a predictor variable that works for all sites
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Model Specification (Continued)Model Specification (Continued)
Goal is to define a predictor variable that has the time series aspect, is based on a physical parameter, and can be applied to any location on the glacier
Solution: Melt Degree Days (MDDs)
Calculation – ‘Cooling Degree Days’ (Dc) Formula:
If Tmax < Tbase, Dc = 0
If (Tmax + Tmin)/2 < Tbase, Dc = (Tmax – Tbase)/4
If Tmin ≤ Tbase, Dc = (Tmax – Tbase)/2 – (Tbase – Tmin)/4
If Tmin > Tbase, Dc = (Tmax + Tmin)/2 –Tbase
Where Tmin and Tmax are the minimum and maximum daily temperature, and Tbase is 0°C
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Model Specification (Continued)Model Specification (Continued)
Melt degree days (MDDs) are calculated from three different locations based on NWS¹ and AICC² air temperature data: Cordova, Yakutat and the Bering Glacier Field Camp
MDDs are added cumulatively from April 1st throughout the summer; first time minimum daily temperature at Bering Glacier broke 32°F in 2008 was April 4th
1, National Weather Service; 2, Alaska Interagency Coordination Center
YakutatYakutat
CordovaCordova
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Further Evidence for a Linear ModelFurther Evidence for a Linear Model
0
200
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600
800
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4/1/
04
4/15
/04
4/29
/04
5/13
/04
5/27
/04
6/10
/04
6/24
/04
7/8/
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7/22
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Ac
cu
mu
late
d M
elt
De
gre
e D
ay
s (
MD
Ds
)
Yakutat 2004
Cordova 2007
Yakutat 2008
Cordova 2008
Bering 2008
Cumulative melt degree days (MDDs) form straight lines through most of the summer melt season
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Model Properties and AssumptionsModel Properties and Assumptions
It is a proper model, and the best prediction will be determined by the method of least squares
Error determination: residual sum of squares (RSS)
Best model of absolute melt is a simple linear model (one predictor variable): absolute melt modeled by cumulative melt degree days (MDDs)
Residuals represent short-term deviations from a constant melt rate (which varies from 3.67 to 5.17 cm/day)
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Regression ResultsRegression Results
MDDs account for at least 97% of variation in response variable (variation in melt)
Standard deviation of regression coefficients from all GASS sites and with all air temperature measurements is 0.027 cm/MDD
Regression of Cordova MDDs yields best results
Model using Yakutat MDDs consistently overestimates absolute melt; model using Bering Glacier MDDs consistently underestimates melt
Model using Cordova MDDs as predictor is more accurate; coincidentally, Cordova represents an ‘average’ climate between Yakutat and the Bering Glacier Field Camp
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Defining the Ablation ModelDefining the Ablation Model
Can we create a model that applies to…– every GASS site? – a future GASS site at any latitude/elevation?– any potential location on the Bering Glacier?
R2 = 0.9994p-value: 0.01504
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 200 400 600 800 1000 1200 1400
Elevation
MD
Ds
Co
effi
cien
t
Snow Equilibrium LineB01
0.00
0.05
0.10
0.15
0.20
0.25
0.30
60 60.1 60.2 60.3 60.4 60.5 60.6 60.7 60.8
Latitude
MD
Ds
Co
effi
cien
t
B01
B02B02
B03B03 B06
B06
R² = 0.9994p-value: 0.01504
R² = 0.8818p-value: 0.15630
R² = 0.7058p-value: 0.10340
Coefficients vs Latitude Coefficients vs Elevation
(m)(°N)
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Defining the Ablation Model (Continued)Defining the Ablation Model (Continued)
There is only a 1.5% chance (p-value: 0.015) that there is no real correlation (null hypothesis is true) betweenelevation and the model coefficient
Compare to the 10-15% chanceof only a random correlationwith latitude
R2 = 0.9994p-value: 0.01504
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 200 400 600 800 1000 1200 1400
Elevation
MD
Ds
Co
effi
cien
t
Snow Equilibrium Line
0.24
0 @
226
m
0.21
5 @
412
m
0.18
9 @
628
m
Elevation does account for over 99% of the response variation
We define the model’s coefficient (c) as a linear function of elevation (h):
c(h) = 0.2392 – (5.206*10^-5)*h
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Defining the Ablation Model (Continued)Defining the Ablation Model (Continued)
The full model of ablation (M) as a non-linear function of melt degree days (d), accounting for elevation (h), assumed to be sufficiently general:
2nd Order Model (v1.0):
M(d,h) = [((3*10^-8)*h²) – (0.0001*h) + (0.2719)]*d – c
1st Order Model (v1.0):
M(d,h) = [0.2392 – (5.206*10^-5)*h]*d – c
COEFFICIENT TERM
COEFFICIENT TERM
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Evaluating Model PerformanceEvaluating Model Performance
Recall error definition: residual sum of squares (RSS)
The model with 1st-order coefficients and Bering Glacier MDDs is the model with the least residuals (smallest RSS)
For years when Bering Glacier data is unavailable (2007 and earlier), the Cordova 1st Order Model should be used
B01 B02 B03 B06 TOTALCordova 1st Order Model 10,477 2,665 7,136 15,571 35,849Cordova 2nd Order Model 9,752 48,031 22,851 41,151 121,78
5Yakutat 1st Order Model 44,918 14,311 618 15,571 75,418Yakutat 2nd Order Model 1,000 31,981 19,716 5,293 57,990Bering 1st Order Model 1,769 7,694 7,458 6,013 22,934Bering 2nd Order Model 51,299 140,103 44,760 26,516 262,67
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Evaluating Model Performance (Cont.)Evaluating Model Performance (Cont.)
How does the model perform for other years, elevations?
2007 Model without empirical data 2007 Model developed by regression of empirical data
Cordova, 1st Order RSS
B01 3,508 (17,466)
B03 2,521 (66,370)
B05 112,859 (70,017)
B06 15,099 (73,379)
Cordova, R Model RSS
B01 183 (225.3 m)
B03 218 (530.8 m)
B05 4,331 (989.7 m)
B06 1,674 (1222.9 m)Intercept Coefficient p
B01: -106.5 0.2058 2.2e-16
B03: -153.1 0.2668 2.2e-16
B05: -134.2 0.2017 2.2e-16
B06: -101.4 0.2156 2.2e-16
Here, the intercept was set to the difference between the first model result and the corresponding true melt measurement or, alternatively, the intercept calculated by regression.
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References, Citations, and LinksReferences, Citations, and Links
Alaska Interagency Coordination Center, Alaska Fire Service, “Predictive Services – Weather Database” – Bering Glacier Field Camp Weather, data accessed June, July, 2009 [http://fire.ak.blm.gov/wx/wxstart.php?disp=geog]
Degree Days Direct, “UK Monthly and Weekly Degree Day Figures” – How Degree Days Are Calculated, accessed June, July, 2009 [http://www.vesma.com/ddd/]
Hall, Myrna H. P., Daniel B. Fagre, “Modeled Climate-Induced Glacier Change in Glacier National Park, 1850-2100”, Vol. 53, No. 2, BioScience, February 2003
Josberger, Edward G., United States Geological Survey, personal communication, June 25, 2009
National Weather Service, “Alaska Interactive Climate Database Map” – Cordova and Yakutat Weather, data accessed June, July, 2009 [http://pajk.arh.noaa.gov/cliMap/climap.php]
Oerlemans, Johannes, “Simulation of Historic Glacier Variations with a Simple Climate-Glacier Model”, Vol. 34, No. 118, Journal of Glaciology, 1988
Thelen, Brian, Michigan Tech Research Institute, personal communication, June-July, 2009