Mathematics of Information Technology and Complex Systems


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Forest Fires and Spread in Heterogeneous Landscapes


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.
All projects mentioned here are closely linked to the GEOIDE.

Project 1: Incorporating Randomness (J. Braun and D. Martell):

The variability inherent in fire growth and spread is not captured by deterministic models. 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 a, b, c and q 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 q , (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):

Fire managers 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.