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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

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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)

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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

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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

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Regression Results (Continued)Regression Results (Continued)

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Model using Cordova MDDs

GASS B01Model using Bering

Glacier MDDs

Model using Yakutat MDDs

RSS: 637

RSS: 13,237

RSS: 17,384

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Regression Results (Continued)Regression Results (Continued)

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Bering Glaicer Average Daily Air Temperature

Cordova Average Daily Air Temperature

Yakutat Average Daily Air Temperature

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

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Snow Equilibrium LineB01

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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

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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