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Development and Application of Development and Application of Geostatistical Methods to ModelingGeostatistical Methods to Modeling
Spatial Variation in Snowpack Properties,Spatial Variation in Snowpack Properties,Front Range, ColoradoFront Range, Colorado
Tyler Erickson and Mark WilliamsTyler Erickson and Mark Williams
Department of GeographyDepartment of Geography
Institute of Arctic and Alpine ResearchInstitute of Arctic and Alpine Research
University of Colorado, BoulderUniversity of Colorado, Boulder
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OutlineOutline
• Introduction
• Snow depth distribution (alpine valley)
• Meltwater discharge (forest meadow)
• Meltwater flowpaths (cubic meter)
• Conclusions / Future directions
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Mountain SnowpacksMountain Snowpacks
• Water source
• Recreation
• Habitat
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snowpack
infiltration
sublimationredistribution
snowmeltprecipitation
Physically-based Model
Empirical Model
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Snowpack DistributionSnowpack Distribution
• Physically-based models require spatially-distributed model inputs
• Snow properties are typically measured at only a few locations(1 site per 1650km2)
• How can we infer snow properties over large areas from limited measurements?
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Snowmelt ProcessSnowmelt Process
• Flow of meltwater through a snowpack is not uniform(meltwater flowpaths)– Allow for rapid movement of mass &
energy, even when snowpack is ‘cold’– Concentrate runoff at the base of the
snowpack– May be important for understanding
the “ionic pulse”
• How can we characterize the meltwater flowpaths?
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Spatial CorrelationSpatial Correlation• Measurements in close proximity to each
other generally exhibit less variability than measurements taken farther apart.
• Assuming independence, when the data are spatial-correlated may lead to:
1. Biased estimates of model parameters
2. Biased statistical testing of model parameters
• Spatial correlation can be accounted for by using geostatistical techniques
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OutlineOutline
• Introduction
• Snow depth distribution (alpine valley)
• Meltwater discharge (forest meadow)
• Meltwater flowpaths (cubic meter)
• Conclusions / Future directions
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Snow Depth in Green Lakes ValleySnow Depth in Green Lakes Valley
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ObjectivesObjectives
• Identify significant auxiliary variables for predicting snow depth in an alpine valley
• Estimate snow depth distributions at unsampled locations and/or times
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Methodology OverviewMethodology Overview
Geostatistics
- Spatial estimates
- Incorporates spatial correlation
Linear Regression
- Incorporates auxiliary variables
- Significance testing
Geostatistical with a Complex Mean
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Regionalized Variable ModelingRegionalized Variable Modeling
deterministiccomponent
stochasticcomponent
regionalizedvariable
z(x) = m(x) + (x)
linear model variogram model
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model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
model optimization model optimizationElder et al., 1991 discrete regions Bayesian classifier UNCORR (none)Hosang and Dettwiler, 1991 linear model OLS CORR (not specified)Carroll and Cressie, 1996 linear model OLS CORR WLSElder et al., 1998 discrete regions regression tree UNCORR (none)Balk and Elder, 2000 discrete regions regression tree CORR OLSErxleben et al., 2002
ordinary kriging constant kriging system CORR AICtrend surface linear model OLS UNCORR (none)modified residual kriging linear model OLS CORR AICbinary regression tree discrete regions regression tree UNCORR (none)binary regression tree + kriging discrete regions regression tree CORR AIC
Staehli et al., 2002 linear model OLS CORR visualWinstral et al., 2002 discrete regions regression tree UNCORR (none)
linear model kriging system CORR RMLCURRENT WORK
deterministic component stochastic componentReference
Spatial Modeling of SnowSpatial Modeling of Snow
z(x) = m(x) + (x)
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Auxiliary ParametersAuxiliary Parameters
• Elevation
• Slope
• Radiation
• Shelter
• Drift
z(x) = m(x) + (x)
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““Linear” ModelLinear” Model
Constant mean:
Linear trend:
Nonlinear trend:
# of base functions
base function coefficients
base functions
Base function coefficients (β) are optimized by solving a kriging system
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Kriging SystemKriging System
How do we determine the coefficients ()?
variogrammodel
trendmodel
measureddata
unknowns
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Variogram ModelVariogram Model• Used to describe spatial correlation
4
3
2
1
Variogram parameters (σ2 and L) are optimized by Restricted Maximum Likelihood
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Significance TestingSignificance Testing
Compact model:
Augmented model:
H0: β2 = 0
Is β2 significantly different from zero?
Is elevation a significant predictor of snow depth?
• Sampling snow depth– length = 1000m– spacing = 50m– # points = 21
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Example cont…Example cont…
H0 is TRUE
5% H0 rejected
5% H0 Rejected
5% H0 Rejected
H0 Rejected!
H0
Not Rejected
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Methodology FlowchartMethodology Flowchart
Measureddata
Auxiliarydata
Trendmodel
Variogramoptimization
(RML)
Base functionoptimization
(kriging)
Variogrammodel
Estimate orsimulation
maps
7 (annual surveys)
1 (exponential variogram)
3 (constant, linear, nonlinear)
5 (elevation, slope, radiation, wind shelter, wind drifting)
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Optimized CoefficientsOptimized CoefficientsBase Function Coeff. Units 1997 1998 1999 2000 2001 2002 2003 1998-2003
1 1 cm 251 229 221 199 182 111 216
Base Function Coeff. Units 1997 1998 1999 2000 2001 2002 2003 1998-2003
1 1 cm 251 222 196 182 174 145 222
ELEV' 2 cm/m -0.530 -0.18
SLOPE' 3 cm/deg 3.0
RAD' 4 cm/(W/m2) -0.591
SHELTER' 5 cm/deg 9.11 8.46 7.78 6.65 3.85 4.06 6.88
DRIFT' 6 cm 124 115 62.34
Base Function Coeff. Units 1997 1998 1999 2000 2001 2002 2003 1998-2003
1 1 cm 251 201 190 189 177 115 226
ELEV' 2 cm/m 0.32 -0.60 0.03 -0.15 -0.27 -0.17
SLOPE' 3 cm/deg 1.71 1.47 3.18 0.73
RAD' 4 cm/(W/m2) -0.588 -0.133 0.256 -0.420 -0.224 -0.498 -0.039 -0.195
SHELTER' 5 cm/deg 4.49 7.64 6.98 5.88 6.67 4.50 1.73 5.09
DRIFT' 6 cm 30 176 96 18 121 124 103
(SLOPE')2 7 cm/deg2 0.182 0.144 0.055
(SHELTER')2 8 cm/deg2 -0.206 0.178 0.096
ELEV' * SLOPE' 9 cm/(m deg) -0.021
ELEV' * RAD' 10 cm/(W/m) -0.0087 -0.0076 -0.0028
ELEV' * SHELTER' 11 cm/(m deg) -0.021 -0.039 -0.018
SLOPE' * RAD' 12 cm/deg/(W/m2) -0.040
SLOPE' * SHELTER' 13 cm/deg2 -0.189 0.107 -0.077
SLOPE' * DRIFT' 14 cm/deg 10.2
RAD' * SHELTER' 15 cm/deg/(W/m2) 0.063 0.043 0.060 0.034 0.023 0.047 0.032
RAD' * DRIFT' 16 cm/(W/m2) -1.28 -1.84 -1.03 -1.29 -1.58 -1.49 -1.43
Nonlinear Trend Model (variable mean model with nonlinear base functions)
Constant Trend Model
Linear Trend Model (variable mean model with linear base functions)
z(x) = m(x) + (x)
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Deterministic Snow Depth MapsDeterministic Snow Depth Maps
0 5 10
Snow depth [m]Constant
NonlinearLinear
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Model Error VariogramsModel Error Variograms
z(x) = m(x) + (x)
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Snow Depth MapsSnow Depth Maps
0 5 10
snow depth [m] model residual [m]
-5 5
1999 best estimate ofdeterministic component
1999 best estimate ofstochastic component
0
1999 conditionedbest estimate
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Correlation toCorrelation toSNOTEL SNOTEL β1 = 231cm
Remaining βs are obtained from multiyear modeling (’98, ’00, ’01, ’02, ’03)
564mm
2.4m2
111m
Developed from’98, ’00, ’01, ’02, ’03 data(excludes ’99)
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Comparison to Regression TreeComparison to Regression Tree(1999 Dataset)(1999 Dataset)
Regression Tree ModelWinstral et al. (2002)
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GLV SummaryGLV Summary
• Used a spatially continuous, nonlinear model of the mean snow depth
• Identified topographic parameters that are significant predictors of snow depth
• Used external data (SNOTEL) to make a prediction without snow depth sampling
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OutlineOutline
• Introduction
• Snow depth distribution (alpine valley)
• Meltwater discharge (forest meadow)
• Meltwater flowpaths (cubic meter)
• Conclusions / Future directions
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Characterizing MeltwaterCharacterizing Meltwater
1. Measure the basal meltwater discharge(snow lysimeters)
2. Measure the pathways directly(snow guillotine)
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Objectives – Snow LysimeterObjectives – Snow Lysimeter
• Determine the sampling area necessary to accurately estimate average meltwater discharge
• Determine whether snow depth is important in relating basal discharge to surface melt
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Soddie Lysimeter ArraySoddie Lysimeter Array
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Data CollectionData Collection
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Meltwater Discharge ProcessingMeltwater Discharge Processing
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Effect of Sample SizeEffect of Sample Size
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Discharge Variability vs. TimeDischarge Variability vs. Time
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Snow DepthSnow Depth
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Discharge vs. Snow DepthDischarge vs. Snow Depth
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Flow ConcentrationFlow Concentration
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Meltwater SummaryMeltwater Summary(field scale)(field scale)
• 30-40 lysimeters are needed to adequately estimate the mean snowmelt
• Variability decreases over time
• Correlation length appears to be between 3-9 meters
• Depth appears to be an important control on meltwater discharge for non-uniform snowpacks
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OutlineOutline
• Introduction
• Snow depth distribution (alpine valley)
• Meltwater discharge (forest meadow)
• Meltwater flowpaths (cubic meter)
• Conclusions / Future directions
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Meltwater Flowpaths OccurrenceMeltwater Flowpaths Occurrence
• Meltwater flowpaths occur at a much finer scale than that measured by the snow lysimeters
• Dye applied at the snow surface has been used to identify meltwater flowpaths
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Objectives – Snow GuillotineObjectives – Snow Guillotine
• Produce a 3-dimensional description of meltwater flowpath occurrence– validation for numerical models,
non-destructive sampling
• Relate statistics of meltwater flowpath occurrence to snowpack stratigraphy– non-spatial statistics
– geostatistics
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TheTheSnow Snow GuillotineGuillotine
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Image ProcessingImage Processing
• Original Image
• Georeferenced
• Band Ratio
• Data Cube
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3-Dimensional Data3-Dimensional Data
lowhigh
Relative dyeconcentration:
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RowRowResultsResults
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Meltwater SummaryMeltwater Summary(1m(1m33 scale) scale)
• The snow guillotine enables the collection of high-resolution 3-D datasets of meltwater flowpath occurrence
• The horizontal distribution of meltwater flowpaths is strongly affected by stratigraphic interfaces in the snowpack
• Well-defined vertical pathways are more prominent near the surface
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Future DirectionsFuture Directions
• Model snow depth distribution at other sites
• Incorporate remote sensing data– model scale changes– data assimilation
• Apply developed methodology to other environmental variables– soil moisture, precipitation, etc.
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AcknowledgmentsAcknowledgments
• Advisory committee:– Mark Willams, Konrad Steffen, Nel Caine,
Tissa Illangasekare, Gary McClelland
• Funding sources– Keck Foundation, CU Geography,
CU Graduate School, Sussman Grant, Beverly Sears Grant, LTER program
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AcknowledgmentsAcknowledgments
• CU Mountain Research Station / LTER– Andy O’Reilly, Mark Losleben, Kurt Chowanski,
Todd Ackerman, Tim Bardsley
• Green Lakes Valleysurvey participants
• Soddie snowpitteam and surveyers
• Snow guillotineexperiments
• Family and friends