Mathematicians to the Rescue!

Transcription

1 Mathematicians to the Rescue! Modelling Fire Spread in the Boreal Forest Department of Mathematics and Statistics University of Victoria, CANADA Visiting UCant 07/08-12/08 August, 2008

2 Outline 1 Forest Fire in Canada The Problem with Forest Fire Prediction. The Project 2

3 The Problem with Forest Fire Prediction. The Project Figure: Chisholm fire, fires accounting for 2.5 million ha burned per year Suppression costs of million per year.

4 The Problem with Forest Fire Prediction. The Project Figure: Fire History, Western Canada, by Ignition Source

5 The Problem with Forest Fire Prediction. The Project Figure: Fire History, Canada

6 The Problem with Forest Fire Prediction. The Project Canada s wildfire spread modelling software Development ; Operational 2002; Current version Participated in the 2006 Industrial Problem Solving Workshop (Pacific Institute for the Mathematical Sciences PIMS ) Ongoing collaboration with the academic community and a major project with Mathematics of Information Technology and Complex Systems (MITACS) program.

7 The Problem with Forest Fire Prediction. The Project Other software systems based on the same mathematics include FARSIGHT (US Forest Service) and SIROFIRE (Australian Bushfire simulator from CSIRO). Cellular automata models exist (FIRESTATION (Portugal no commercial release)) Heat transfer and combustion models (mostly under development).

8 Phenomenological: Elliptical fire spread shapes. WIND c b a Figure: A simple forest fire a + c = head rate of spread= HROS b = flank rate of spread = FROS a c = back rate of spread = BROS. Tables prepared by the Canadian Fire Behavior Prediction system.

9 Huygens Principle Figure: Front Propagation via Huygens Gwynfor Richards (1990) wrote down a set of equations based on this model.

10 Richards Equations x t (s, t) = b2 cos θ(x s sin θ+y s cos θ) a 2 sin θ(x s cos θ y s sin θ) a 2 (x s cos θ y s sin θ) 2 +b 2 (x s sin θ+y s cos θ) 2 y t (s, t) = b2 sin θ(x s sin θ+y s cos θ) a 2 cos θ(x s cos θ y s sin θ) a 2 (x s cos θ y s sin θ) 2 +b 2 (x s sin θ+y s cos θ) 2 subject to the initial conditions x(s, 0) = x 0 (s) y(s, 0) = y 0 (s). + c sin θ + c cos θ

11 Richards Equations for Mathematicians ( xt y t ) ( ) cos θ sin θ R θ = sin θ cos θ = AT A n A n + c A = ( b 0 0 a ) R θ n = (ys, x s ) T, c = R T θ (0, c) T.

12 Validation (rather late in the game) Steadily expanding ellipse IS a solution of Richards equations in homogeneous conditions.

13 Current Implementation Marker method (or bead method) Local fuel/moisture provided by gridded data from the Canadian Fire Hazard Database.... combined with local wind and SLOPE data, and fed into the Fire Behaviour Prediction System. Output is the spatial data a(x, y), b(x, y), c(x, y) and θ(x, y) Richards Equations discretized and the updated front computed from a, b, c, θ and t, s

14 Many Complications... Tangles and singularities quickly develop in the moving front. (The IPSW problem) (Solution: An unwinding routine based on the 2-colour theorem) Markers concentrate and dissipate along the front due to topological effects. (Solution: redistribute markers using a de Boor s type algorithm.) As the front grows, the number of markers grows and hence the dimension of the problem. There exist a number of numerical nonlinear bottlenecks in which begin to show up for large time simulations. (Solution: ongoing, endless... ).

15 Tangles in Merging Fires R2 R1 R2 Pi R1 R2 Pi R1 Pi

16 nightmare P i P i P i

17 Figure: One simulation

18 )*+*,!-*./ &$! &(! &#! &'! &"! &&! &!!! "! #! $! %! &!! &"! &#! Figure: Another simulation

19 &$! )*+*,!-*./ &(! &#! &'! &"! &&! &!!! "! #! $! %! &!! &"! &#! Figure: Marker method left, Level set method right

20 Level Set Method Look for a surface φ(x, y, t) = 0 in 3D whose contours at level t form the evolving front Γ(t) in the xy plane at time t.

21 Assuming the vector field v = v (x, y, t) is given for the evolution (x t, y t ) = v (x, y, t), substituting into φ and differentiating d dt yields φ t + v x,y φ = 0 for the unknown function φ subject to initial condition Γ(0) = {(x, y) : φ(x, y, 0) = 0}. Richards evolution: v = A T A n A n + c, n = x,y φ and hence φ t + A x,y φ + c x,y φ = 0

22 Numerical solution of the level set equation proceeds layer by layer (beginning with a discrete approximation to the signed distance function to Γ(0) on the zero-th layer). There are a number of subtle points to be attended to eg: upwinding to avoid instabilities/shocks and CFL conditions A good reference which is also fun to read J. Sethian, Level Set Methods and Fast Marching Methods, Cambridge U. Press, 2003 or on Sethian s website.

23 Our Group CB Anne Bourlioux (U de Montréal) John Braun, (U of Western Ontario) Thomas Hillen (The boss, U of Alberta) Dave Martell, (U of Toronto) Cordy Tymstra ()

24 Sponsors MITACS and the GEOIDE project (Cordy Tymstra, Robert Bryce) Alberta Sustainable Resource Development Branch Natural Sciences and Engineering Research Council of Canada

25 Forest Fire in Canada Thanks! Figure: Fire (prescribed burn), July 28, TODAY, Canmore Alt.