Glacier Ablation Sensor System 2008 Analysis and 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

2 Introduction 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 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

4 Model Specification (Continued) Parameters were integrated (over gaps between ablation measurements) and correlated with melt: (tmp_INT)(tlt_INT)(blt_INT)(wnd_INT) TemperatureIrradianceExitanceWind Speed B01R² = R² = B02--R² = 0.92R² = B03--R² = 0.60R² = B06----R² = 0.75R² = 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

5 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’ (D c ) Formula: If T max < T base, D c = 0 If (T max + T min )/2 < T base, D c = (T max – T base )/4 If T min ≤ T base, D c = (T max – T base )/2 – (T base – T min )/4 If T min > T base, D c = (T max + T min )/2 –T base Where T min and T max are the minimum and maximum daily temperature, and T base is 0°C

6 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 1 st throughout the summer; first time minimum daily temperature at Bering Glacier broke 32°F in 2008 was April 4 th 1, National Weather Service; 2, Alaska Interagency Coordination Center Yakutat Cordova

7 Further Evidence for a Linear Model Cumulative melt degree days (MDDs) form straight lines through most of the summer melt season

8 Model 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)

9 Regression 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 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

10 Regression Results (Continued) Model using Cordova MDDs GASS B01 Model using Bering Glacier MDDs Model using Yakutat MDDs RSS: 637 RSS: 13,237 RSS: 17,384

11 Regression Results (Continued) Model using Cordova MDDs as predictor is more accurate; coincidentally, Cordova represents an ‘average’ climate between Yakutat and the Bering Glacier Field Camp

12 Defining 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? Snow Equilibrium Line B01 B02 B03 B06 R ² = p-value: R ² = p-value: R ² = p-value: Coefficients vs Latitude Coefficients vs Elevation (m) (°N)

13 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) between elevation and the model coefficient Compare to the 10-15% chance of only a random correlation with latitude Snow Equilibrium Line 226m 628m 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) = – (5.206*10^-5)*h

14 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: 2 nd Order Model (v1.0): M(d,h) = [((3*10^-8)*h²) – (0.0001*h) + (0.2719)]*d – c 1 st Order Model (v1.0): M(d,h) = [ – (5.206*10^-5)*h]*d – c COEFFICIENT TERM

15 Evaluating 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 1 st Order Model should be used B01B02B03B06TOTAL Cordova 1 st Order Model10,4772,6657,13615,571 35,849 Cordova 2 nd Order Model9,75248,03122,85141, ,785 Yakutat 1 st Order Model44,91814, ,571 75,418 Yakutat 2 nd Order Model1,00031,98119,7165,293 57,990 Bering 1 st Order Model1,7697,6947,4586,013 22,934 Bering 2 nd Order Model51,299140,10344,76026, ,678

16 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, 1 st 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 B (225.3 m) B (530.8 m) B05 4,331 (989.7 m) B06 1,674 ( m) InterceptCoefficientp B01: e-16 B03: e-16 B05: e-16 B06: e-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.

17 References, Citations, and Links Alaska Interagency Coordination Center, Alaska Fire Service, “Predictive Services – Weather Database” – Bering Glacier Field Camp Weather, data accessed June, July, 2009 [ Degree Days Direct, “UK Monthly and Weekly Degree Day Figures” – How Degree Days Are Calculated, accessed June, July, 2009 [ Hall, Myrna H. P., Daniel B. Fagre, “Modeled Climate-Induced Glacier Change in Glacier National Park, ”, 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 [ 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