Forest Fires and Spread in Heterogeneous Landscapes
Each year, thousands of forest fires burn in Canada, consuming millions of hectares.
Although important for the ecology of a forest ecosystem, fires threaten native species and
human life and property. It is therefore important that the fire management community
have access to decision support tools to help fight and manage forest fires.
Since 1925, Canadian researchers have investigated the influence of weather, fuel type,
fuel moisture, and landscape topography on the spread of forest fires. Such research led
to the publication of fire hazard tables. Since the 1970s, computer guided prediction
tools have been developed. In May 2002, C. Tymstra (Alberta Sustainable Resource
Development, ABSRD) and R. Bryce released the first version of Prometheus , which has
now become the state of the art computational fire growth prediction tool.
Our objective is to develop a complete multiscale approach to the mathematical modelling
of forest fire spread and control. The dynamics of a forest fire can be roughly
divided into three scales. On the microscopic scale we consider local information such
as fuel type, moisture and wind and weather conditions, which are expressed through
indices such as FWI (Fire Weather Index) and the FBP (Canadian Forest Fire Behavior
Prediction System). We plan to study stochastic effects and to include local wind
conditions (Projects 1 and 3). On the mesoscopic scale (Lagrangian approach) we use
the local information to compute a rate of spread (ROS) of the fire front. The models
of Richards from 1990, 1995, and 1999 and the software package Prometheus are mesoscopic
models. We plan to further develop these models (Projects 2, 4, and 5). On the
macroscopic scale (Eulerian approach) are mathematical models which still use local information,
however, the unknown function is a macroscopic object. Here we follow two
approaches, the level set method to describe the evolution of the fire front (Project 3) and
reaction-advection-diffusion equations for the energy release rate and for the temperature
distribution (Project 4).
On all scales we obtain interesting and open mathematical questions. The MITACS
team will work on these questions, and focus on the linkage between the scales. The
results will be discussed with the partners at ABSRD and will be implemented into the
existing software package of Prometheus.
Project 1: Incorporating Randomness (J. Braun and D. Martell):
inherent in fire growth and spread is not captured by the deterministic Prometheus
model. Maps of probability contours will be more useful to fire managers than the type
of output that is currently provided by the model.
One approach is to apply a residual-based block-bootstrap procedure, whereby the
values are initially smoothed spatially and temporally. Residuals from the
smooths can then be block re-sampled and added back to the smooths, giving rise to
randomly perturbed local parameter values which, in turn, give rise to random perturbed
solutions of the differential equations (1).
We will work with a post-doctoral fellow to
(i) validitate this type of two-dimensional bootstrap procedure, while modelling spatial autocorrelation,
(ii) find appropriate smoothing techniques for circular data
(iii) include fuel type discontinuities, and
(iv) validate with a large number of fire data sets.
Preliminary work on (iii) using additive spline models with indicator variables shows promise.
Another approach is the stochastic cellular automata model of Boychuk et al (2007).
The accuracy of the Prometheus simulator depends on the accuracy of the FBP system
relations. Some of these relations were developed originally using ad hoc strategies. Improvements
in accuracy should be achievable using methods designed to handle random
and mixed effects models. Measures of uncertainty on the micro and mesoscale will also
be incorporated into the FBP system.
Project 2: Delooping and the Marker Method (C. Bose):
The existing Prometheus
algorithm can be made more efficient and more stable. Improvements can be realized by
(i) smoothing, (ii) re-gridding, and (iii) de-looping. These problems arise through the
incorporation of microscopic information into a mesoscopic model. Here we study how
the microscopic information can be used more accurately. All of the three approaches
(i)-(iii) have seen preliminary investigation at the 2006 PIMS Industrial Problem Solving
Workshop (IPSW). Prometheus programmers implemented first ideas and reported a
nearly five-fold improvement in processing time.
(i) Smoothing is necessary, when the local conditions destroy the smooth structure of
the fire front. Smoothing will also be studied within the inclusion of random effects in
Project 1 and the diffusion approach in Project 5.
(ii) Re-gridding becomes necessary, when, after a number of iterations the nodes have
clustered or diluted along the fire front. A very natural idea to choose new nodes is
to implement a de-Boorís-type algorithm after each time step, where a correct penalty
function has to be found. This requires close collaboration between mathematicians and
the Prometheus development team.
(iii) De-looping is a critical issue for this project. The basic problem is to prevent
the fire front from evolving into already burnt regions such that the fire front consist of a
finite number of disjoint, simple closed curves with no crossings (tangles). A de-looping
algorithm based on a winding number calculation is not robust and fails in some simple
instances. A new algorithm is proposed, based on the Two-Colour Theorem for planar
curves. It works much better on many test cases, however, it is still not robust and fails
when the topological complexity of the tangled front becomes too high. The delooping
problem is related to a well-known polygonal clipping problem in computer graphics (Vatti
1992). For example, two disjoint fires can merge, and the problem is to determine, at the
first time of overlap, the frontier of the union of the two burnt areas. Our two-colour
untangling algorithm appears to be very robust and efficient for this polygonal clipping
problem. Further studies into the two-colour approach and polygonal clipping is needed.
We also need to study alternative methods to solve the de-looping problem satisfactorily.
Further ideas for de-looping might arise in collaboration with Project 3, since level sets
intrinsically have no tangles.
Project 3: Level Set Methods (A. Bourlioux):
This project concerns the level-set
approach to interface computations, and the effect of a multi-scale advection velocity field
on the effective propagation of an interface.
Level-set type methods have been developed over the last 15-20 years precisely to
handle robustly the tangles which arise from use of the marker method. A preliminary
effort at implementing a level-set algorithm within the framework of the Prometheus
approach was carried out during IPSW 2006. The results were very promising.
The first task is to investigate the level-set approach as an alternative or complement
to the current Prometheus algorithm. Specific questions in the forest fire context include:
(i) robustness: Develop a robust interface tracking algorithm under noisy atmospheric
and environmental data and random wind bursts. (ii) efficiency: Make a level-set type
approach as numerically efficient as the existing marker method. (iii) adaptivity: the
time and spatial scales characterizing fire propagation can vary tremendously within one
simulation and so does the need for accuracy. We plan to design a fully automatic adaptive
strategy to solve a level-set type equation that accounts both for the features of the
solution as well as the needs of the end-users.
The second task is to model the effect of bursts of wind (spatial heterogeneity and
intermittency) and other microscopic and mesoscopic heterogeneities. The modelling task
proposed here will be carried out within the thin-interface (level-set) framework, which
relates micro- and macroscopic scales. Both, deterministic and stochastic generic wind
flow perturbations will be investigated. In the presence of topography, the Wind Wizard
solver will be used to compute the fundamental unstable modes for the wind flow. Those
modes can be stimulated through stochastic forcing to investigate the impact on the fire
propagation. Finally, based on the numerical and theoretical studies above, we plan to
formulate an effective deterministic or stochastic approach to predict not only an average
front location but also worst and best case scenarios reflecting the uncertainty of the wind.
This part is closely related to Project 1.
Project 4: Diffusion Models (T. Hillen):
Richardsí model (1) has been derived
using a certain approximation. A graduate student, Jon Martin, just finished a MITACS
Internship where he derived the higher order correction terms to Richardsí model. He
showed that the correction terms are curvature terms, which appear as diffusion terms
along the fire front. They become relevant in areas of high curvature. Implementation of
this second order term forms another possible method to smooth out the fire front and to
prevent tangles (see Project 2).
The microscopic mechanisms of fire spread involve (I) convection, (II) radiation, (III)
conduction, and (IV) spotting. Spotting describes fire brands which are launched high
into the air by local convection winds. They can travel up to two miles and start a
new fire. In this project we use reaction-advection-diffusion equations to include these
microscopic effects (I)-(IV) into a macroscopic model. In collaboration with the postdoc
Petro Babak we follow two approaches: (i) a formulation using energy balance, and (ii) a
formulation based on temperature distribution. Dr. Babak started already to work with
these models and he has some promising preliminary results. We plan to develop a first
testable model in collaboration with A. Bourlioux very soon.
The graduate student, J. Martin, will now focus on spotting. He will study stochastic
and deterministic models (see also Project 1) and investigate if spotting does speed up
an advancing fire front.
Finally, the numerical solution of the diffusion models will provide a third numerical
solver for fire fronts. The numerics will then be compared with the marker method and
with the level set method from Project 3.
Project 5: Management and Optimization (D. Martell, J. Braun):
must develop and evaluate, in the presence of uncertainty, alternative containment
strategies. They must consider projected fire growth given the local fuel, forecast weather,
topography, and the potential impact of alternative strategies on suppression costs and
the social, economic and ecological impact of the fire.
It is difficult to model fire suppression such that fire managers can quickly and easily
evaluate the cost effectiveness of their actions. There are exceptions (e.g. Fried and Fried
1996), but these are based on relatively simple fire growth models. More recently,
Donovan and Rideout 2003 formulated the suppression of a large fire as an integer programming
problem. Incorporating suppression effects in stochastic models and into Prometheus is a
major objective of this project.
While Projects 1-5 focus on various aspects of forest fire growth, the connecting
scheme is a full multiscale approach. Uncertainty arises on various scales (see Project 1
and 3), and macroscopic models are based on microscopic information (see Projects 2,3,4).
A detailed knowledge of the underlying mechanisms of convection, radiation, conduction
and spotting is necessary to derive useful macroscopic forest fire models.