1. Development and Application of Geostatistical Methods to Modeling Spatial 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. Mountain Snowpacks Water source Recreation Habitat

4. snowpack infiltration sublimationredistribution snowmelt precipitation Physically-based Model Empirical Model

5. Snowpack Distribution Physically-based models require spatially-distributed model inputs Snow properties are typically measured at only a few locations (1 site per 1650km 2 ) 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: 1.Biased estimates of model parameters 2.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 Geostatistics - Spatial estimates - Incorporates spatial correlation Linear Regression - Incorporates auxiliary variables - Significance testing Geostatistical with a Complex Mean

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

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

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

16. “Linear” 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

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

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

19. Significance Testing Compact model: Augmented model: H 0 : β 2 = 0 Is β 2 significantly different from zero? Is elevation a significant predictor of snow depth? Sampling snow depth –length = 1000m –spacing = 50m –# points = 21

20. Example cont… H 0 is TRUE 5% H 0 rejected 5% H 0 Rejected H 0 Rejected! H 0 Not Rejected

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

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

23. Deterministic Snow Depth Maps 0 510 Snow depth [m] Constant NonlinearLinear

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

25. Snow Depth Maps 0 510 snow depth [m]model residual [m] -55 1999 best estimate of deterministic component 1999 best estimate of stochastic component 0 1999 conditioned best estimate

26. Correlation to SNOTEL β 1 = 231cm Remaining βs are obtained from multiyear modeling (’98, ’00, ’01, ’02, ’03) 564mm 2.4m 2 111m Developed from ’98, ’00, ’01, ’02, ’03 data (excludes ’99)

27. Comparison to Regression Tree (1999 Dataset) Regression Tree Model Winstral 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. Characterizing Meltwater 1.Measure the basal meltwater discharge (snow lysimeters) 2.Measure the pathways directly (snow guillotine)

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 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

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. The Snow Guillotine

45. Image Processing Original Image Georeferenced Band Ratio Data Cube

46. 3-Dimensional Data low high Relative dye concentration:

47. Row Results

48. Meltwater Summary (1m 3 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

51. Acknowledgments CU Mountain Research Station / LTER –Andy O’Reilly, Mark Losleben, Kurt Chowanski, Todd Ackerman, Tim Bardsley Green Lakes Valley survey participants Soddie snowpit team and surveyers Snow guillotine experiments Family and friends

52. Significance Testing Compact model: Augmented model: Variance ratio test: F critical Significant! performance statistics

53. Discharge vs. Snow Depth

54. The Snow Guillotine