Tyler Erickson and Mark Williams Department of Geography Institute of Arctic and Alpine Research

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

OutlineOutline

• Introduction

• Snow depth distribution (alpine valley)

• Meltwater discharge (forest meadow)

• Meltwater flowpaths (cubic meter)

• Conclusions / Future directions

Mountain SnowpacksMountain Snowpacks

• Water source

• Recreation

• Habitat

snowpack

infiltration

sublimationredistribution

snowmeltprecipitation

Physically-based Model

Empirical Model

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?

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?

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

OutlineOutline

• Introduction

• Snow depth distribution (alpine valley)

• Meltwater discharge (forest meadow)

• Meltwater flowpaths (cubic meter)

• Conclusions / Future directions

Snow Depth in Green Lakes ValleySnow Depth in Green Lakes Valley

ObjectivesObjectives

• Identify significant auxiliary variables for predicting snow depth in an alpine valley

• Estimate snow depth distributions at unsampled locations and/or times

Methodology OverviewMethodology Overview

Geostatistics

- Spatial estimates

- Incorporates spatial correlation

Linear Regression

- Incorporates auxiliary variables

- Significance testing

Geostatistical with a Complex Mean

Regionalized Variable ModelingRegionalized Variable Modeling

deterministiccomponent

stochasticcomponent

regionalizedvariable

z(x) = m(x) + (x)

linear model variogram model

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)

Auxiliary ParametersAuxiliary Parameters

• Elevation

• Slope

• Radiation

• Shelter

• Drift

z(x) = m(x) + (x)

““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

Kriging SystemKriging System

How do we determine the coefficients ()?

variogrammodel

trendmodel

measureddata

unknowns

Variogram ModelVariogram Model• Used to describe spatial correlation

4

3

2

1

Variogram parameters (σ2 and L) are optimized by Restricted Maximum Likelihood

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

Example cont…Example cont…

H0 is TRUE

5% H0 rejected

5% H0 Rejected

5% H0 Rejected

H0 Rejected!

H0

Not Rejected

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)

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)

Deterministic Snow Depth MapsDeterministic Snow Depth Maps

0 5 10

Snow depth [m]Constant

NonlinearLinear

Model Error VariogramsModel Error Variograms

z(x) = m(x) + (x)

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

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)

Comparison to Regression TreeComparison to Regression Tree(1999 Dataset)(1999 Dataset)

Regression Tree ModelWinstral et al. (2002)

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

OutlineOutline

• Introduction

• Snow depth distribution (alpine valley)

• Meltwater discharge (forest meadow)

• Meltwater flowpaths (cubic meter)

• Conclusions / Future directions

Characterizing MeltwaterCharacterizing Meltwater

1. Measure the basal meltwater discharge(snow lysimeters)

2. Measure the pathways directly(snow guillotine)

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

Soddie Lysimeter ArraySoddie Lysimeter Array

Data CollectionData Collection

Meltwater Discharge ProcessingMeltwater Discharge Processing

Effect of Sample SizeEffect of Sample Size

Discharge Variability vs. TimeDischarge Variability vs. Time

Snow DepthSnow Depth

Discharge vs. Snow DepthDischarge vs. Snow Depth

Flow ConcentrationFlow Concentration

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

OutlineOutline

• Introduction

• Snow depth distribution (alpine valley)

• Meltwater discharge (forest meadow)

• Meltwater flowpaths (cubic meter)

• Conclusions / Future directions

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

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

TheTheSnow Snow GuillotineGuillotine

Image ProcessingImage Processing

• Original Image

• Georeferenced

• Band Ratio

• Data Cube

3-Dimensional Data3-Dimensional Data

lowhigh

Relative dyeconcentration:

RowRowResultsResults

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

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.

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

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