1. Snow and Ice Modeling
Ted Kim
April 24, 2002
COMP 259

2. Overview Large Scale Snow Modeling Medium Scale Snow Modeling
Mountain Ranges
Medium Scale Snow Modeling
Not just a height field
Ice Modeling
Phase field representation

3. Large Scale Modeling Geospecific Rendering of Alpine Terrain
Premože, Thompson, Shirley – Eurographics Rendering Workshop 1999

4. Large Scale Modeling (2)
Realism of terrain based on real-world elevation data can be enhanced using aerial photos

5. Problems
United States Geological Survey (USGS) aerial photos in grayscale
Stationary shadows
Fixed time of day
Fixed season
Cannot show less snow for warmer seasons
Flat features
Trees and brush visibly 3D even from large distance

6. System Overview Image processing Scene creation Shadow removal
Image segmentation
Scene creation
Adding seasonal effects
Adding 3D trees and vegetation

7. Image Processing Maximum likelihood Bayesian classifiers
Won’t go into detail (COMP 254)

8. Image Processing (2) Segments image into snow, trees, rock, etc.
Locate trees so 3D tree models can be inserted later
Vegetation location important for snow melt calculations
Need to know what is underneath snow in case simulation melts snow off

9. Scene Creation Terrain is divided into grid
Grid resolution is same as input height field
Accumulation
Snow is dumped into grid cells according to elevation
Melt Simulation
Snow melts off based on cell height and ‘available radiation’

10. Accumulation User specifies at base elevation:
Rate of precipitation
Precipitation density
Ambient air temperature
Change in 3 values with change in elevation (linear factor appears to work)
Snow accumulates if temperature is below Tsnow threshold

11. Simulation Melting equation: M = Cm (Tair – Tmelt)
M daily melt (mm / day)
Cm melt rate (mm / C0 day)
Tair ambient temperature
Tmelt melting temperature

12. Melt Rate Cm = km kv RI (1 – A) km ‘proportionality constant’
(not explained)
kv vegetation transmission coefficient
RI solar radiation index
A snow albedo

13. Melt Rate Terms Albedo, proportion of reflected energy:
A = 0.4[1 + e-kt]
t days
k ‘time constant’ (0.2 / day)
‘Available Radiation’ (RI) – Effect of sun based on cell orientation and shadowing: established equations from hydrology

14. Rain New snow fall raises albedo to 0.8, rain lowers albedo to 0.4
With rain, Cm is:
Cm(rain) = Cm Prain
Where Prain is amount of rainfall in mm

15. Rendering Snow is zero thickness
Regions with any accumulation rendered as snow
Adding snow to texture is straightforward
Subtracting snow – assume rock beneath (most photos are from summer)
Insert trees and brush according to segmentation information

16. Early spring
Late spring

17. Snow Modeling
Computer Modeling of Fallen Snow
Fearing - SIGGRAPH 2000

18. Problem
Model snow on more local scale with realistic physical behavior
Photo Simulation

19. Approach Accumulation stage Stability stage
Determine how much snow accumulates per surface
Stability stage
Determine how snow avalanches downward

20. Accumulation Stage Core idea: Implementation:
Like ray tracing, trace snowflakes backwards from ‘launch sites’ to sky
Unlike ray tracing, flakes flutter, so path is not straight
Implementation:
10-15 flakes per sites gives good estimate
Refine number of samples per region to get better estimate in interesting regions
Elevate underlying mesh by amount of snow

21. Snowflake Tracing No existing theory on how flakes flutter
Flake path randomly perturbed in XY direction
Radius and direction chosen from normal probability distribution

23. Sky Buckets
If many flakes hit same region of sky, region does not have infinite snow
Sky divided into buckets of size skyarea, and snow in each bucket is divided evenly over all incident flakes

25. Flake Dusting Regions with small amount of snow
Procedural noise textures

26. Refinement
If adjacent launch sites have different amounts of ‘visible sky’, refine launch site
Add launch site slightly offset from midpoint of two launch sites
If adjacent sites are similar, can eliminate a launch site

27. Stability Stage
Use angle of repose (AOR) to calculate where snow avalanches
AOR measures static friction of a granular material
AOR is angle between snow on top of adjacent launch sites, not angle of launch site itself

28. Angle of Repose AOR changes with type of snow
900 for fresh snow
150 for slush
Can model probability of stability around AOR

30. Stability Algorithm For each launch site Calculate AOR with neighbors
Perform obstacle test with neighbors
(test shown on next slide)
Shift snow to unblocked neighbors
Repeat until all neighbors stable or site no longer has snow

31. Obstacle test Avalanche can be blocked by Scene object
Snow on top of scene object
Partially by snow on scene object

32. Snow Over Edges Snow can avalanche over object edges
Several particles (< 5) thrown over the edge
Where particles come to rest, snow is placed

33. Stability Termination
Simulation runs out of time
List of unstable launch sites is empty
Very little snow moved on last pass
Usually first few passes resolve most interesting features

34. Looks good … could use some ice
(Video)
Looks good … could use some ice

35. Ice Modeling
Modeling and numerical simulations of dendritic crystal growth
Kobayashi, Physica D 62 (1993)

36. Chemistry Review ‘Supercooled’ solution Slowly cool water
Can go below freezing and still be liquid
Add a ‘seed crystal’
Ice forms outwards in interesting patterns

37. Problem Crystal growth is a complex boundary tracking problem
Conventional ‘level set’ methods have had some success [Osher and Sethian], but does not produce realistic dendrite growth.
Newer versions have had better success, but have not done reading yet
‘Phase field model’ has had success

38. Phase Field Model Most methods treat phase boundary as infinitely thin
Kobayashi treats boundary as finitely thin, a ‘field’
Something can be 63% water, 37% ice (???)

39. Validity Treating boundary as finite seems arbitrary
As finite thickness approaches zero, ‘Stefan problem’, basis for most theories, is recovered
Approach has been asymptotically validated against lower level approach, ‘microscopic solvability theory’

40. Phase PDE
Need a PDE for phase p that describes how an element transitions from water (p = 0) to
ice (p = 1) over time t
Once we have the PDE, we can solve using FEMs

41. Magic According to Kobayashi, the necessary phase equation is:
Derived from energy equations used in superconductivity (note that  is a constant)
term term term 3

42. Intuition Intuitively: Term 1: Still working on it
Term 2: Introduces anisotropy, so ice does not grow symmetrically
Term 3: Allows temperature to regulate phase change speed
term term term 3

43. Term 2 Term 2 is nonlinear diffusion, where are constants
thickness of finite interface
‘mode of anisotropy’
‘strength of anisotropy’
fixed direction

44. Term 2(2) Think in terms of FEMs:
Only variable is , which is the normal direction of the phase field at a finite element
Thickness of the the interface will change as normal direction changes: anisotropy

45. Term 3 Third term introduces temperature dependence
Where m is the temperature dependant term
,  constants
Te Freezing temperature of water (constant)
T temperature of current finite element

46. Term 3(2) How does term 3 affect ?
If T < Te, cell is below freezing and, m > 0
If T = Te, cell is at freezing temp and, m = 0
In the 3rd order term 3:
there are three roots: one each at 0 and 1, and one that can be moved by m

47. Contribution of term 3 to Red: Below freezing Blue: At freezing
positive
contribution
negative
contribution
water
ice
phase

48. Contribution
water
ice
phase
Think in terms of FEMs again
The solution must be cold for water to transition to ice
Near Te, only if element is already mostly ice, p > 0.5, will it transition
Colder, elements that are mostly water will transition too

49. Final Equation Must also model heat diffusion
First term is normal heat diffusion Laplacian
Second term adds heat produced by phase changes, K = latent heat
T feeds back into term 3, so regions that are too hot can no longer change phase

50. Implementation Can ignore the complexity of the math
2 uniform grids, one for phase, one for temperature
Temperature starts as supercooled (0)
Phase starts as water (p = 0) except for seed crystal
Iterate

51. Further Reading
"Adaptive Mesh Refinement Computation of Solidification Microstructures using Dynamic Data Structures”
Provatas et al, J. Comp. Phys. vol 148, 265 (1999).
“Recent Developments in Phase-Field Models of Solidifcation”
Wheeler et al, Advances in Space Research vol 16, 163 (1995)