A Finsler geodesic spray paradigm for wildfire spread modelling

doi:10.1016/j.nonrwa.2015.09.011

Nonlinear Analysis: Real World Applications

Volume 28, April 2016, Pages 208-228

https://doi.org/10.1016/j.nonrwa.2015.09.011 Get rights and content

Abstract

One of the finest and most powerful assets of Finsler geometry is its ability to model, describe, and analyse in precise geometric terms an abundance of physical phenomena that are genuinely asymmetric, see e.g. Antonelli et al. (1993, 2003), Yajima and Nagahama (2009), Bao et al. (2004), Cvetič and Gibbons (2012), Gibbons et al. (2007), Astola and Florack (2011), Caponio et al. (2011), Yajima and Nagahama (2015). In this paper we show how wildfires can be naturally included into this family. Specifically we show how the celebrated and much applied Richards’ equations for the large scale elliptic wildfire spreads have a rather simple Finsler-geometric formulation. The general Finsler framework can be explicitly ‘integrated’ to provide detailed–and curvature sensitive–geodesic solutions to the wildfire spread problem. The methods presented here stem directly from first principles of $2$ -dimensional Finsler geometry, and they can be readily extracted from the seminal monographs Shen (2001) and Bao et al. (2000), but we will take special care to introduce and exemplify the necessary framework for the implementation of the geometric machinery into this new application — not least in order to facilitate and support the dialog between geometers and the wildfire modelling community. The ‘integration’ part alluded to above is obtained via the geodesics of the ensuing Finsler metric which represents the local fire templates. The ‘paradigm’ part of the present proposal is thus concerned with the corresponding shift of attention from the actual fire-lines to consider instead the geodesic spray–the ‘fire-particles’–which together, side by side, mould the fire-lines at each instant of time and thence eventually constitute the local and global structure of the wildfire spread.

Introduction

Every day the World is confronted with wildfires in various regions of our globe. Any wildfire is a highly nonlinear phenomenon, which is in pertinent demand for multidisciplinary and multi-scale analysis and better understanding. Detailed understanding is needed — both for emergency planning, which depends severely on quick and reliable predictions of the wildfire spread in time, as well as for the proper understanding of global issues concerning the CO₂ releases and biological and physical changes to the land surface [1]. Such phenomena obviously present scientific opportunities with no shortage of social significance. This fact is repeatedly stressed and documented in every paper that is concerned with the understanding, predicting, and modelling of wildfires, see e.g. [2]. Correspondingly there are several explicit and recent calls from the fire fighter community for new appropriate and effective first principles, i.e. new mathematical models, to handle and understand better the spreading mechanism of the wildfires in forests, grasslands, and wheat fields — with wind, slope, varying fuel properties across the domain and in geographically complicated terrain, see for example the description of the wildfire simulator Prometheus in [3], the comparison of various simulators in [4], [5], and the general surveys as in e.g. [6], [7], [8].

As already alluded to in the abstract, Finsler geometry is a very strong tool for modelling physical phenomena that are genuinely asymmetric and/or non-isotropic, see e.g. [9], [10], [11], [12], [13], [14], [15], [16], [17]. In this paper we show how the geometric analysis of wildfires can be naturally added to this long list of applications of Finsler geometry.

We briefly describe the standard modelling of wildfires including Huyghens’ principle in Section 2. In Section 3 we emphasize and illustrate how to set up a general fire template field in a parameter domain. The principles of Finsler metrics, the ensuing first variation of arc-length, and the important notion of $F$ -geodesics are surveyed in Sections 4 Finsler metrics, 5 First variation of. The resulting $F$ -geodesic spray, its enveloping properties, and the induced exponential wildfires are constructed in Sections 6 Enveloping, 7 Finsler induced wildfires. In Sections 8 The Randers–Zermelo elliptic wildfires, 9 Richards’ equations the Richards’ equations are discussed in terms of their Randers–Zermelo equivalents, and we show that for elliptic wildfires the Richards’ equations are solved by the corresponding Finsler-geodesic sprays. Specific examples of $F$ -geodesic spray driven wildfires are constructed and illustrated in Sections 10 A simple example, 11 A hemispherical elliptic wildfire, 12 A non-elliptic example. The final two Sections 13 Conclusion, 14 Discussion present the main conclusions from the present paper together with a brief suggestion for further work.

Section snippets

Huyghens’ principle

Following the pioneering works of G. D. Richards [18], [19], [20], [21], [22], van Wagner [23], Anderson et al. [24], and Glasa–Halada [25], [26], [27], [28], [29], we will apply a number of assumptions to be satisfied by the wildfires. We will only consider 2-dimensional, regular, smooth and deterministic wildfires ignited at time $t = 0$ on a smooth and regular ignition fireline (or at an ignition point). The fire spread is then represented by a smooth and regular vector function $γ (s, t)$ in a $($

Parametric domains

A real world fuel domain in a geographic region is usually not directly given as a flat domain $U$ in $R^{2}$ . The precise representation of the fuel domain in such a flat parameter domain therefore needs some consideration.

The specific choice of fire template (indicatrix, or pointed oval) at each point $(u, v)$ in the parameter domain depends on the fuel condition, the wind, and the topography (the slope) of the actual real world fuel domain at the corresponding point $r (u, v)$ . For example, the slope in

Finsler metrics

By classical definition, see [41], [42], a Finsler metric on a domain $U$ is a smooth family of Minkowski norms on the tangent planes, i.e. a smooth family of indicatrix templates which in each tangent plane $T_{p} U$ at the respective points $p = (u, v)$ in the parameter domain $U$ is determined by a nonnegative function $F$ as follows:

1.
$F$ is smooth on the punctured tangent plane $T_{p} U - {(0, 0)}$ .
2.
$F$ is positively homogeneous of degree one: $F (k V) = k F (V)$ for every $V \in T_{p} U$ and every $k > 0$ .
3.
The following bilinear symmetric

First variation of $F$ -arclength

Following [41, Chapter 5] we survey the derivation of the important first variation formula for the $F$ -length functional in a domain $U$ with a given Finsler metric $F$ . It is stated here in its most general ( $n$ -dimensional) form — for notational convenience only — but will be restricted and applied to the two-dimensional cases of wildfires below.

The first variation formula will give us the ODE differential equation conditions for a curve to be an $F$ -geodesic in $U$ , i.e. the analytic condition for a

Enveloping

The converse to Proposition 5.1 also holds — at least locally — in the following sense:

Proposition 6.1

Suppose $c$ is an $F$ -geodesic from a point $p$ to a not too far away point $q$ in $U$ — i.e. $c$ satisfies the geodesic equation (30) all the way — then $c$ is the $F$ -shortest curve from $p$ to $q$ .

The proof of Proposition 6.1 does not follow directly from the first variation formula, but involves an application of the so-called exponential map (that we will define and apply also below) together with the Finsler version

Finsler induced wildfires

With these ingredients we are now ready to define formally how a given template field induces a unique wildfire spread from a given ignition line or ignition point which is in accordance with Huyghens’ enveloping principle.

Definition 7.1

Let $I_{p}, p \in U$ , denote a given fire template field in $U$ with induced Finsler metric $F$ and let $η_{0}$ denote a simple closed regular curve in $U$ . Then the $I$ -induced wildfire spread from the ignition curve $η_{0}$ is the net $γ (s, t)$ defined by the $F$ -geodesic exponential map: $γ (s, t) = {exp}_{η_{0} (s)} (t \cdot$

The Randers–Zermelo elliptic wildfires

We consider a surface $S$ in $R^{3}$ with parametrization $r (u, v)$ so that $r : U \mapsto S$ , as for example in Fig. 1, Fig. 2, Fig. 6.

As part of our general assumptions, all fuel information on the surface is intrinsically encoded into a pointed oval, an indicatrix, in each tangent plane $T_{(u, v)} U$ at the point $(u, v)$ of the parameter domain. In this section we assume that all indicatrices are pointed ellipses. In practice they are found and determined in the way already described in the introduction. Finsler metrics

Richards’ equations

In this section we observe how Richards’ equations for the spread of elliptic wildfires fit naturally into the Finsler geodesic spray paradigm, and in particular that they are in fact equivalent to the Zermelo version of the Hamilton orthogonality conditions in Corollary 8.1.

Theorem 9.1

Richards [18], Glasa and Halada [26]

A wildfire $γ (s, t)$ on a given ellipse template field with Zermelo equivalent data $a (u, v), b (u, v), C (u, v) = (c_{1} (u, v), c_{2} (u, v))$ , and $θ (u, v)$ is determined by the following equations for the partial derivatives $γ_{s}^{'} (s, t) = (u_{s}^{'}, v_{s}^{'})$ and

A simple example

With the following simple choice of ellipse field, i.e. Zermelo data, we obtain the corresponding wildfire spread (from ignition at $(u, v) = (0, 0)$ ) as indicated in Fig. 4: $a (u, v) = 1$ $b (u, v) = 3$ $C (u, v) = (0, 2)$ $θ (u, v) = u - (2 / 5) .$

The figure has been constructed from a numerical solution of the geodesic spray equations for the $F$ -exponential map in (42).

Remark 10.1

We observe in Fig. 5 (see also Fig. 8), that the $F$ -geodesic fire discs centred at the points on the next-outermost fire line will envelope the outermost fire line,

A hemispherical elliptic wildfire

The following example is based on a very recent work on concrete Randers spaces with constant curvature [61], where the corresponding wildfire problem has a simple analytic solution. It is of particular interest for us because the resulting wildfires in this metric are analytically solvable and thus they represent unique possibilities for comparing the analysis with the results of numerical methods and simulations that are applied to solve the wildfire spread problems. This example is thence

A non-elliptic example

There is, of course, an abundance of other strongly convex ovals in the plane, than just the ellipses, that can be used for setting up an indicatrix field — and thence a Finsler metric — in a given parameter domain; see e.g. the nice constructive approaches to the analysis of various relevant ovals in [62], [63], [64] and the previous works towards the generalization of the elliptic fire-template fields in [65], [66], [28].

The Matsumoto metric is a non-elliptic Finsler metric whose

Conclusion

We have shown that each specific choice of a smoothly varying strongly convex pointed oval field (modelling small time linearized firelets) in the parameter fuel domain produces a Finsler metric $F$ with this given indicatrix field, and under the Huyghen’s enveloping Ansatz the corresponding wildfires are governed by the $F$ geodesic spray equations.

Specifically, for the elliptic fire template fields we have embedded the well known Richards’ equations into the Finsler geodesic spray paradigm:

Theorem 13.1

The

Discussion

Our discussion is so far only semi-global in the sense that we do not in this paper consider the formation of cut loci, the so-called ‘bear hugs’, which typically appear during the long time spreading of wildfires in non-constant fuel domains. A beginning discussion of these aspects of global Finsler geometry and their possible applications for the wildfire spread modelling can be found in [67], [68].

Even in the semi-global regular setting the accumulation of errors, which is induced by using

Acknowledgements

The author would like to thank the referees for encouraging comments and for suggestions which have improved the presentation of this work.

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View full text

A Finsler geodesic spray paradigm for wildfire spread modelling

Abstract

Introduction

Section snippets

Huyghens’ principle

Parametric domains

Finsler metrics

First variation of F-arclength

Enveloping

Finsler induced wildfires

The Randers–Zermelo elliptic wildfires

Richards’ equations

Richards [18], Glasa and Halada [26]

A simple example

A hemispherical elliptic wildfire

A non-elliptic example

Conclusion

Discussion

Acknowledgements

Nonlinear Anal. RWA

Ann. Physics

Nonlinear Anal. RWA

Forest Ecol. Manag.

Math. Comput. Simul.

Int. J. Heat Mass Transfer

Int. J. Therm. Sci.

Nonlinear Anal. RWA

J. Math. Anal. Appl.

Rep. Math. Phys.

Ann. Physics

Ecol. Model.

Ecol. Model.

Estimates of CO2 from fires in the United States: Implications for carbon management

Carbon Balance Manag.

Development and Structure of Prometheus: the Canadian Wildland Fire Growth Simulation Model, Information Report NOR-X-417

A study of simulation errors caused by algorithms of forest fire growth models, Forest Research Report 167

A comparative review on wildfire simulators

IEEE Syst. J.

Wildland surface fire spread modelling, 1990–2007. 1: Physical and quasi-physical models

Int. J. Wildland Fire

Wildland surface fire spread modelling, 1990–2007. 2: Empirical and quasi-empirical models

Int. J. Wildland Fire

Wildland surface fire spread modelling, 1990–2007. 3: Simulation and mathematical analogue models

Int. J. Wildland Fire

Finsler geometry of seismic ray path in anisotropic media

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.

Zermelo navigation on Riemannian manifolds

J. Differential Geom.

General very special relativity is Finsler geometry

Phys. Rev. D

Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging

Int. J. Comput. Vis.

On the energy functional on Finsler manifolds and applications to stationary spacetimes

Math. Ann.

Elliptical growth model of forest fire fronts and its numerical solution

Internat. J. Numer. Methods Engrg.

Properties of elliptical wildfire growth for time dependent fuel and meteorological conditions

Combust. Sci. Technol.

A general mathematical framework for modeling 2-dimensional wildland fire spread

Int. J. Wildland Fire

A computer algorithm for simulating the spread of wildland fire perimeters for heterogeneous fuel and meteorological conditions

Int. J. Wildland Fire

The mathematical modelling and computer simulation of wildland fire perimeter growth over a 3-dimensional surface

Int. J. Wildland Fire

A simple fire-growth model

For. Chron.

Modelling the spread of grass fires

J. Aust. Math. Soc. Ser. B

First variation of $F$ -arclength

Estimates of $C O_{2}$ from fires in the United States: Implications for carbon management