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

1. Development and Application of Geostatistical Methods to ModelingSpatial Variation in Snowpack Properties,Front Range, Colorado Tyler Erickson and Mark Williams Department of Geography Institute of Arctic and Alpine Research University of Colorado, Boulder

2. Outline • Introduction • Snow depth distribution (alpine valley) • Meltwater discharge (forest meadow) • Meltwater flowpaths (cubic meter) • Conclusions / Future directions

3. Water source Recreation Habitat Mountain Snowpacks

4. Empirical Model Physically-based Model redistribution sublimation snowpack snowmelt precipitation infiltration

5. Snowpack 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?

6. Snowmelt 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?

7. Spatial 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: • Biased estimates of model parameters • Biased statistical testing of model parameters • Spatial correlation can be accounted for by using geostatistical techniques

8. Outline • Introduction • Snow depth distribution (alpine valley) • Meltwater discharge (forest meadow) • Meltwater flowpaths (cubic meter) • Conclusions / Future directions

9. Snow Depth in Green Lakes Valley

10. Objectives • Identify significant auxiliary variables for predicting snow depth in an alpine valley • Estimate snow depth distributions at unsampled locations and/or times

12. Methodology Overview Linear Regression - Incorporates auxiliary variables - Significance testing Geostatistical with a Complex Mean Geostatistics - Spatial estimates - Incorporates spatial correlation

13. Regionalized Variable Modeling regionalized variable deterministic component stochastic component z(x) = m(x) + e(x) linear model variogram model

14. Spatial Modeling of Snow z(x) = m(x) + e(x)

15. Auxiliary Parameters z(x) = m(x) + e(x) • Elevation • Slope • Radiation • Shelter • Drift

16. “Linear” Model # of base functions base functions base function coefficients Constant mean: Linear trend: Nonlinear trend: Base function coefficients (β) are optimized by solving a kriging system

17. Kriging System How do we determine the coefficients (b)? trend model unknowns variogram model measured data

18. Variogram Model • Used to describe spatial correlation 1 2 3 4 Variogram parameters (σ2 and L) are optimized by Restricted Maximum Likelihood

19. Significance 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

20. Example cont… 5% H0 Rejected H0 Rejected! H0 is TRUE 5% H0 Rejected H0Not Rejected 5% H0 rejected

21. Methodology Flowchart Measured data Variogram optimization (RML) 7 (annual surveys) Variogram model 1 (exponential variogram) Base function optimization (kriging) Estimate or simulation maps Trend model 3 (constant, linear, nonlinear) Auxiliary data 5 (elevation, slope, radiation, wind shelter, wind drifting)

22. Optimized Coefficients z(x) = m(x) + e(x)

23. Deterministic Snow Depth Maps Constant Snow depth [m] 0 5 10 Linear Nonlinear

24. Model Error Variograms z(x) =m(x)+ e(x)

25. Snow Depth Maps 1999 best estimate ofstochastic component 1999 conditionedbest estimate 1999 best estimate ofdeterministic component snow depth [m] model residual [m] 0 5 10 -5 0 5

26. Correlation toSNOTEL β1= 231cm Developed from ’98, ’00, ’01, ’02, ’03 data (excludes ’99) Remaining βsare obtained from multiyear modeling (’98, ’00, ’01, ’02, ’03) 111m 2.4m2 564mm

27. Comparison to Regression Tree(1999 Dataset) Regression Tree ModelWinstral et al. (2002)

28. GLV 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

29. Outline • Introduction • Snow depth distribution (alpine valley) • Meltwater discharge (forest meadow) • Meltwater flowpaths (cubic meter) • Conclusions / Future directions

30. Measure the basal meltwater discharge(snow lysimeters) Measure the pathways directly(snow guillotine) Characterizing Meltwater

31. Objectives – 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

32. Soddie Lysimeter Array

33. Data Collection

34. Meltwater Discharge Processing

35. Effect of Sample Size

36. Discharge Variability vs. Time

37. Snow Depth

38. Discharge vs. Snow Depth

39. Flow Concentration

40. Meltwater Summary(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

41. Outline • Introduction • Snow depth distribution (alpine valley) • Meltwater discharge (forest meadow) • Meltwater flowpaths (cubic meter) • Conclusions / Future directions

42. 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 Meltwater Flowpaths Occurrence

43. Objectives – 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

44. TheSnow Guillotine

45. Original Image Georeferenced Band Ratio Data Cube Image Processing

46. 3-Dimensional Data Relative dyeconcentration: low high

47. RowResults

48. Meltwater Summary(1m3 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

49. Future 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.

50. Acknowledgments • 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