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

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

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

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

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

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

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

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

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’

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

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

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

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

Rain New snow fall raises albedo to 0.8, rain lowers albedo to 0.4 With rain, Cm is: Cm(rain) = Cm + 0.00126 Prain Where Prain is amount of rainfall in mm

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

Early spring Late spring

Snow Modeling Computer Modeling of Fallen Snow Fearing - SIGGRAPH 2000

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

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

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

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

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

Flake Dusting Regions with small amount of snow Procedural noise textures

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

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

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

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

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

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

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

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

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

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

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

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 (???)

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’

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

Magic According to Kobayashi, the necessary phase equation is: Derived from energy equations used in superconductivity (note that  is a constant) term 1 term 2 term 3

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 1 term 2 term 3

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

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

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

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

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

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

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

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

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)