fractal dimensions of wildﬁre spreadingmulti-agency coordination (geomac) database (geomac, 2012)...

Nonlin. Processes Geophys., 21, 815–823, 2014www.nonlin-processes-geophys.net/21/815/2014/doi:10.5194/npg-21-815-2014© Author(s) 2014. CC Attribution 3.0 License.

Fractal dimensions of wildfire spreadingS.-L. Wang, H.-I. Lee, and S.-P. Li

Institute of Physics, Academia Sinica 128 Sec. 2, Academia Rd., Nankang, Taipei, 11529, Taiwan (ROC)

Correspondence to:S.-L. Wang ([email protected])

Received: 2 December 2013 – Revised: 22 April 2014 – Accepted: 24 June 2014 – Published: 7 August 2014

Abstract. The time series data of 31 wildfires in 2012 in theUS were analyzed. The fractal dimensions (FD) of the wild-fires during spreading were studied and their geological fea-tures were identified. A growth model based on the cellularautomata method is proposed here. Numerical study was per-formed and is shown to give good agreement with the fractaldimensions and scaling behaviors of the corresponding em-pirical data.

1 Introduction

Wildfire is one of the main disturbances in an ecosystem andhas been of much concern in many countries over the years.Historically, it has been the cause of numerous and possiblyirreversible damages, with deep ecological as well as soci-ological and economic impacts. In some extreme cases, thewhole ecosystem might be wiped out and result in large num-bers of human casualties. It is therefore understood that theneed for designing and developing effective ways of deal-ing with wildfires is constantly increasing, as such phenom-ena appear ever more often. In fact, the area burned annuallyby wildfire has always been a factor that influences policydecisions and future land-use planning in countries such asthe United States and Canada. There are indeed many fac-tors that can affect the spreading of wildfire and the area thatwill be burned once the fire has started. Among them, cli-mate (Kitzberger et al., 2007), forest type (McKenzie et al.,1996), wind (Weise and Biging, 1996), landscape (Romme,1982) and also human activities are some of the major fac-tors that can influence the occurrence and spreading of wild-fires. Among these factors, climate seems to be a major fac-tor that governs the rate of spread and the burned area ofwildfires. For example, there are significantly more cases ofwildfires reported during summer time, when the tempera-ture in a region is considerably higher. Early wildfire history

evidence from diverse climate regions and vegetation alsosuggests that ignitions and spreads of wildfires have strongrelations to climate changes (Matthew et al., 2010; Littell etal., 2009; Westerling and Bryant, 2008). In recent years, therehave also appeared studies that investigate the relationshipbetween global and regional warming in wildfire activities(Westerling et al., 2006; Lee et al., 2013). Over the years,there have been many studies on the modeling of wildfirespreading by various approaches (Sullivan, 2009a) to assistpolicy makers in designing and developing effective waysof dealing with wildfires. In recent years, it has also caughtthe attention of physicists, who want to understand this sub-ject from the physical point of view (Bak et al., 1990; Chenet al., 1990). In the study of complex dynamical systemssuch as wildfires, physicists are interested in investigatingwhether they are critical systems. In these studies, the au-thors modeled the wildfires as self-organized critical systemsand studied their properties. A more interesting feature ofwildfires is that satellite images show that the spreading ofwildfires looks fractal (Caldarelli et al., 2001). A natural ap-proach would therefore be to use some physical models suchas percolation models (Caldarelli et al., 2001) to mimic thespreading of wildfires. One would then be able to calculatethe statistical properties of wildfires by using these physicalmodels and comparing them with empirical data. Previousworks (Caldarelli et al., 2001) only analyzed the final imageof wildfires because of data availability. With the advance-ment of modern technology, more and better wildfire imagedata are now available. Some of the large wildfires can nowbe recorded daily once they are discovered, and one can in-deed observe the spreading of a particular wildfire as a func-tion of time. The images have a spatial resolution of 5 m,and thus provide much more accurate data for analysis. Onewould then be able to analyze the dynamical and statisticalproperties of wildfires with better precision.

Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.

816 S.-L. Wang et al.: Fractal dimensions of wildfire spreading

Table 1.The list of 31 wildfires analyzed in the paper.

Name Date Position State Size Duration(mm/dd) (W, N) (USA) (km2) (days)

County Line 04/05 82.5, 30.5 FL 141.5 17.7Sunflower 05/13 111.5, 34.0 AZ 65.2 8.7Gladiator 05/14 112.3, 34.2 AZ 61.8 7.5Little Sand 05/24 107.2, 37.4 CO 100.9 43.3High Park 06/09 105.4, 40.6 CO 354.3 20.0Little Bear 06/09 105.8, 33.4 NM 179.8 17.0Poco 06/15 110.9, 34.1 AZ 48.4 7.2Russells Camp 06/19 105.8, 42.5 WY 22.1 9.9Waldo Canyon 06/23 104.9, 38.9 CO 73.9 12.0Weber 06/23 108.3, 37.3 CO 41.0 9.0Wood Hollow 06/25 111.5, 39.7 UT 192.0 4.2NeighborMountain 06/26 78.4, 38.7 VA 8.6 9.3Point 06/26 78.3, 38.9 VA 4.7 7.7Ash Creek 06/27 106.5, 45.7 MT 1005.2 7.5Seeley 06/27 111.2, 39.5 UT 192.7 14.2Arapaho 06/28 105.5, 42.2 WY 396.7 13.4Clay Springs 06/28 112.4, 39.3 UT 436.5 11.4Wolf Den 06/30 109.2, 39.8 UT 80.4 7.0Shingle 07/03 112.6, 37.5 UT 32.8 7.1Mill 07/08 122.7, 39.3 CA 119.5 12.8MillerHomestead 07/10 119.2, 42.8 OR 647.1 7.2Chrandal Creek 07/11 114.2, 45.5 MT/ID 10.1 20.0Flat 07/12 123.4, 40.8 CA 7.0 6.5Robbers 07/12 120.9, 39.0 CA 10.7 8.5Chips 07/30 121.3, 40.0 CA 305.3 37.7Trinity Ridge 08/04 115.4, 43.7 ID 593.5 64.0Barry Point 08/07 120.8, 42.1 OR 376.0 27.4Merino 08/12 114.9, 44.7 ID 31.9 37.0Cascade Creek 09/01 121.5, 46.1 WA 81.2 36.0Sheep 09/07 116.3, 45.5 ID 196.7 33.7Pole Creek 09/11 121.6, 44.2 OR 108.4 25.9

In this paper, we provide the first analysis of wildfirespreading time series. We show that wildfire spreading ex-hibits a universal scaling behavior that has never been re-ported before. We also show that the satellite images revealvery interesting statistical and dynamical properties as thewildfire spreads.

2 Analysis of empirical data

We construct here the digital perimeters of the wildfire eventsthat happened in America that are available from the Geospa-tial Multi-Agency Coordination (Geomac) database (Geo-mac, 2012) as a form of geographic information system(GIS). In the Geomac database, these digital data are recon-structed from Global Positioning System (GPS) and infraredimagery. We analyze a total of 31 wildfires that either spreadover a large area or persisted for a long enough time beforethey were extinguished. For example, the wildfire that oc-curred in Ash Creek, Montana had the largest burned area of1005 km2, and the wildfire in Trinity Ridge, Idaho persistedfor a total of 64 days. Table 1 above is the list of wildfiresanalyzed in this paper.

2 S. -L. Wang et al.: Fractal Dimensions of Wildfire Spreading

Table 1. The list of 31 wildfires analyzed in the paper

Name Date Position State Size Duration(mm/dd) (W,N) (USA) (km2) (Days)

County Line 04/05 82.5, 30.5 FL 141.5 17.7Sunflower 05/13 111.5, 34.0 AZ 65.2 8.7Gladiator 05/14 112.3, 34.2 AZ 61.8 7.5

Little Sand 05/24 107.2, 37.4 CO 100.9 43.3High Park 06/09 105.4, 40.6 CO 354.3 20.0Little Bear 06/09 105.8, 33.4 NM 179.8 17.0

Poco 06/15 110.9, 34.1 AZ 48.4 7.2Russells Camp 06/19 105.8, 42.5 WY 22.1 9.9Waldo Canyon 06/23 104.9, 38.9 CO 73.9 12.0

Weber 06/23 108.3, 37.3 CO 41.0 9.0Wood Hollow 06/25 111.5, 39.7 UT 192.0 4.2

Neighbor Mountain 06/26 78.4, 38.7 VA 8.6 9.3Point 06/26 78.3, 38.9 VA 4.7 7.7

Ash Creek 06/27 106.5, 45.7 MT 1005.2 7.5Seeley 06/27 111.2, 39.5 UT 192.7 14.2

Arapaho 06/28 105.5, 42.2 WY 396.7 13.4Clay Springs 06/28 112.4, 39.3 UT 436.5 11.4

Wolf Den 06/30 109.2, 39.8 UT 80.4 7.0Shingle 07/03 112.6, 37.5 UT 32.8 7.1

Mill 07/08 122.7, 39.3 CA 119.5 12.8Miller Homestead 07/10 119.2, 42.8 OR 647.1 7.2Chrandal Creek 07/11 114.2, 45.5MT/ID 10.1 20.0

Flat 07/12 123.4, 40.8 CA 7.0 6.5Robbers 07/12 120.9, 39.0 CA 10.7 8.5Chips 07/30 121.3, 40.0 CA 305.3 37.7

Trinity Ridge 08/04 115.4, 43.7 ID 593.5 64.0Barry Point 08/07 120.8, 42.1 OR 376.0 27.4

Merino 08/12 114.9, 44.7 ID 31.9 37.0Cascade Creek 09/01 121.5, 46.1 WA 81.2 36.0

Sheep 09/07 116.3, 45.5 ID 196.7 33.7Pole Creek 09/11 121.6, 44.2 OR 108.4 25.9

very interesting statistical and dynamical properties as thewildfire spreads.

2 Analysis of Empirical Data75

We here construct the digital perimeters of the wildfire eventshappened in America that are available from the GeospatialMulti-Agency Coordination (Geomac) database (Geomac,2012) as a form of geographic information system (GIS). InGeomac database, these digital data are reconstructed from80

Global Positioning System (GPS) and infrared imagery. Weanalyze a total of 31 wildfires which either spread a large areaor persist for a long enough time before they extinguished.For example, the wildfire that occurred in Ash Creek, Mon-tana had the largest burnt area of 1005 km2 and the wildfire85

in Trinity Ridge, Idaho persisted for a total of 64 days. Table1 below is the list of wildfires analyzed in this paper.

Fig. 1. The time sequence of Little Bear wildfire in New Mexico,U.S.A. in 2012, (a)Jun 18, 21:04 (b)Jun 19, 13:33 (c)Jun 19, 21:42(d)Jun 20, 08:42 (e)Jun 21, 20:57 (f)Jun 22, 20:46. Each of the fig-ures (a) - (f) has width of 2750 pixels and height 4450 pixels, cor-responding to an actual size of 13.75 km × 22.25 km.

In the empirical wildfire datasets, both the burnt and burn-ing sites are given by their longitudinal and latitudinal co-ordinates. To construct the digital perimeters of the wildfire90

events, we first project the empirical wildfire data recordedin geographic coordinates onto a two-dimensional square lat-tice, transforming the wildfire data into digital graphs. Eachlattice site corresponds to one pixel of the digital image ofthe wildfire from the database with a precision of 25 m2 per95

pixel. Figure 1 is an example of a wildfire occurred in LittleBear, New Mexico, U.S.A. in 2012 after we project its ge-ographic coordinate onto a two dimensional lattice. The im-ages correspond to a time sequence (a)Jun 18, 21:04 (b)Jun19, 13:33 (c)Jun 19, 21:42 (d)Jun 20, 08:42 (e)Jun 21, 20:57100

(f)Jun 22, 20:46. The white portion in each figure corre-sponds to the burnt area while the black portion is the un-burnt surroundings. Each of the figures (a) - (f) in Fig.1 haswidth of 2750 pixels and height 4450 pixels, correspondingto an actual size of 13.75 km × 22.25 km.105

From the digital graph, we can compute the burnt area (A)of the wildfire under study. To proceed, we label lattice siteswith 1 for the burnt sites and 0 for the unburnt sites. Figure 2is an illustration of a digital graph of wildfire with 1 for theburnt sites and 0 for the unburnt sites. By counting the num-110

ber of lattice sites with value 1, we will obtain an approxi-mate value of the burnt area of the wildfire. We will then de-

Figure 1.The time sequence of the Little Bear wildfire in New Mex-ico, USA in 2012:(a) 18 June, 21:04 LT,(b) 19 June, 13:33 LT,(c) 19 June, 21:42 LT,(d) 20 June, 08:42 LT,(e) 21 June, 20:57 LT,(f) 22 June, 20:46 LT. Each of the panels(a)–(f) has a width of2750 pixels and a height of 4450 pixels, corresponding to an actualsize of 13.75km× 22.25km.

In the empirical wildfire data sets, both the burned andburning sites are given by their longitudinal and latitudi-nal coordinates. To construct the digital perimeters of thewildfire events, we first project the empirical wildfire datarecorded in geographic coordinates onto a two-dimensionalsquare lattice, transforming the wildfire data into digitalgraphs. Each lattice site corresponds to one pixel of the dig-ital image of the wildfire from the database, with a precisionof 25 m2 per pixel. Figure 1 is an example of a wildfire thatoccurred in Little Bear, New Mexico, USA in 2012 after weproject its geographic coordinate onto a two-dimensional lat-tice. The images correspond to a time sequence: (a) 18 June,21:04 LT, (b) 19 June, 13:33 LT, (c) 19 June, 21:42 LT,(d) 20 June, 08:42 LT, (e) 21 June, 20:57 LT, and (f) 22 June,20:46 LT. The white portion in each figure corresponds tothe burned area, while the black portion is the unburned sur-roundings. Each of the panels (a)–(f) in Fig. 1 has a width of2750 pixels and a height of 4450 pixels, corresponding to anactual size of 13.75km× 22.25km.

From the digital graph, we can compute the burned area(A) of the wildfire under study. To proceed, we label lat-tice sites with 1 for the burned sites and 0 for the unburnedsites. Figure 2 is an illustration of a digital graph of wild-fire, with 1 for the burned sites and 0 for the unburned

Nonlin. Processes Geophys., 21, 815–823, 2014 www.nonlin-processes-geophys.net/21/815/2014/

S.-L. Wang et al.: Fractal dimensions of wildfire spreading 817S. -L. Wang et al.: Fractal Dimensions of Wildfire Spreading 3

Fig. 2. Example of a binary graph. For this graph, A = 19, and itsaccessible perimeter L = 18.

duce the radius of gyration (Rg) from its burnt area. The ra-dius of gyration of the burnt area can be obtained as follows:Let the location of the i-th site of the burnt area be (xi,yi).115

The loocation of the center of the burnt area is then equal to(xrc,yrc) = (

∑ixi/A,

∑i yi/A) , and the radius of gyration

would then be equal to√

(∑i(x

2i + y2

i )/A− (x2rc + y2

rc) .In a straightforward way, one can also calculate the ac-

cessible perimeter (L) of the wildfire as it spreads. We here120

compute the accessible perimeter in the following way. Inthe original digital graph, when a burnt site (1) has all 8 near-est and next nearest neighbors being burnt sites (1), we willchange its label to 0. After we do this, we can count the num-ber of 1’s in the new graph and obtain the accessible perime-125

ter (L) of the wildfire. Figure 3 is an illustration of this pro-cedure. In Figure 3 (a), the sites in yellow and red are burntsites but only the red site is surrounded by burnt sites as itsnearest and next nearest neighbors. We then re-label this siteas 0 as shown in Figure 3 (b). After we carry out this relabel-130

ing, we will then count the sites labeled 1 to obtain L. Onecan of course do the perimeter measurement in other ways.As an example, let us again refer to Fig. 3 (a). One may ne-glect the corner sites (yellow) in the neighborhood of the redsite. let us take the upper left corner site as an illustration. In135

this case, we neglect the site but instead take the length of theperimeter around this yellow corner site as the length of thedistance of the centers on the left and top of this corner site.The perimeter around this corner would then be 2

√2 instead

of 3. This would give an error of about 5%. The difference140

by using these two different ways of counting would differby much less than 5%. We should mention here that the dif-ference by using different counting methods would mainly

Fig. 3. An illustration of the evaluation of L. In (a), the sites inyellow and red are burnt sites (1) but only the red site is surroundedby burnt sites as its nearest and next nearest neighbors. We re-labelthis site as 0 as shown in (b) for the evaluation of L.

affect the proportional constant but has very little effect onthe exponent.145

There are a total of 717 temporal datasets of wildfire im-ages in the database for the 31 wildfires listed in Table 1.Each of the temporal datasets corresponds to one data pointwhen one plots e.g., the burnt area as a function of Rg . Asmentioned above, the resolution of these wildfire images is150

good to a scale of 5 meters. With such a high resolution andlong periods of observation, one would be able to study thechange of fractal dimensions as the wildfire spreads over alarge area. We will here study the fractal dimensions of theburnt area (Df ) and perimeter (Dp) of the recorded wildfires155

by using the box-counting method (Caldarelli et al., 2001).The fractal dimension (Df ) for a regular two-dimensionalobject such as a square is equal to 2. We will therefore plot2−Df vs.Rg and see how the burnt areas of wildfires deviatefrom regular objects. Fig. 4 is a plot of the (2−Df ) vs. Rg for160

all the wildfires listed in Table 1. The green dots are the datapoints of the longest wildfire event (Little Sand) with a totalof 50 points. One can see that as Rg grows, Df approaches2. The slope of a linear fit is equal to −0.72± 0.02. Thissuggests that as the burnt area grows larger, the perimeter of165

this burnt area becomes smoother and closer to a regular two-dimensional object. This behavior was already observed in apercolation model studied a long time ago (Niessen and Blu-men, 1986). Fig. 5 is a plot of Dp vs. Rg . Dp remains to bewithin the range of 1 and 1.15 as Rg grows. The average of170

Dp for all the data points is about 1.06.Fig. 6 shows the result of the burnt area (A) as a function

of Rg. It is interesting to note that all the 717 data pointsobey an empirical function, A∼ 4.51R2

g . The fact that A isproportional to the square of Rg is in agreement with the175

empirical behavior ofDf (going to 2 as its asymptotic limit).What is more striking is the fact that they have approximatelythe same proportional coefficient. The error estimate of theproportional coefficient and the exponent of Rg are respec-

Figure 2. Example of a binary graph. For this graph,A = 19, andits accessible perimeterL = 18.

sites. By counting the number of lattice sites with value 1,we will obtain an approximate value of the burned area ofthe wildfire. We will then deduce the radius of gyration(Rg) from its burned area. The radius of gyration of theburned area can be obtained as follows: let the location ofthe ith site of the burned area be (xi , yi). The location ofthe center of the burned area is then equal to(xrc,yrc) =(∑

i xi/A,∑

i yi/A), and the radius of gyration would then

be equal to√(∑

i

(x2i + y2

i

)/A −

(x2

rc + y2rc

).

In a straightforward way, one can also calculate the ac-cessible perimeter (L) of the wildfire as it spreads. We com-pute here the accessible perimeter in the following way. Inthe original digital graph, when a burned site (1) has alleight nearest and next-nearest neighbors as burned sites (1),we will change its label to 0. After we do this, we can countthe number of 1s in the new graph and obtain the accessibleperimeter (L) of the wildfire. Figure 3 is an illustration of thisprocedure. In Fig. 3a, the sites in yellow and red are burnedsites, but only the red site is surrounded by burned sites as itsnearest and next-nearest neighbors. We then re-label this siteas 0, as shown in Fig. 3b. After we carry out this relabeling,we will then count the sites labeled 1 to obtainL. One canof course make the perimeter measurement in other ways.As an example, let us again refer to Fig. 3a. One may ne-glect the corner sites (yellow) in the neighborhood of the redsite. Let us take the upper left corner site as an illustration.In this case, we neglect the site, but instead take the lengthof the perimeter around this yellow corner site as the lengthof the distance of the centers on the left and at the top ofthis corner site. The perimeter around this corner would thenbe 2

√2 instead of 3. This would give an error of about 5 %.

The difference by using these two different ways of count-ing would differ by much less than 5 %. We should mentionhere that the difference by using different counting methods

S. -L. Wang et al.: Fractal Dimensions of Wildfire Spreading 3

Fig. 2. Example of a binary graph. For this graph, A = 19, and itsaccessible perimeter L = 18.

duce the radius of gyration (Rg) from its burnt area. The ra-dius of gyration of the burnt area can be obtained as follows:Let the location of the i-th site of the burnt area be (xi,yi).115

The loocation of the center of the burnt area is then equal to(xrc,yrc) = (

∑ixi/A,

∑i yi/A) , and the radius of gyration

would then be equal to√

(∑i(x

2i + y2

i )/A− (x2rc + y2

rc) .In a straightforward way, one can also calculate the ac-

cessible perimeter (L) of the wildfire as it spreads. We here120

compute the accessible perimeter in the following way. Inthe original digital graph, when a burnt site (1) has all 8 near-est and next nearest neighbors being burnt sites (1), we willchange its label to 0. After we do this, we can count the num-ber of 1’s in the new graph and obtain the accessible perime-125

ter (L) of the wildfire. Figure 3 is an illustration of this pro-cedure. In Figure 3 (a), the sites in yellow and red are burntsites but only the red site is surrounded by burnt sites as itsnearest and next nearest neighbors. We then re-label this siteas 0 as shown in Figure 3 (b). After we carry out this relabel-130

ing, we will then count the sites labeled 1 to obtain L. Onecan of course do the perimeter measurement in other ways.As an example, let us again refer to Fig. 3 (a). One may ne-glect the corner sites (yellow) in the neighborhood of the redsite. let us take the upper left corner site as an illustration. In135

this case, we neglect the site but instead take the length of theperimeter around this yellow corner site as the length of thedistance of the centers on the left and top of this corner site.The perimeter around this corner would then be 2

√2 instead

of 3. This would give an error of about 5%. The difference140

by using these two different ways of counting would differby much less than 5%. We should mention here that the dif-ference by using different counting methods would mainly

Fig. 3. An illustration of the evaluation of L. In (a), the sites inyellow and red are burnt sites (1) but only the red site is surroundedby burnt sites as its nearest and next nearest neighbors. We re-labelthis site as 0 as shown in (b) for the evaluation of L.

affect the proportional constant but has very little effect onthe exponent.145

There are a total of 717 temporal datasets of wildfire im-ages in the database for the 31 wildfires listed in Table 1.Each of the temporal datasets corresponds to one data pointwhen one plots e.g., the burnt area as a function of Rg . Asmentioned above, the resolution of these wildfire images is150

good to a scale of 5 meters. With such a high resolution andlong periods of observation, one would be able to study thechange of fractal dimensions as the wildfire spreads over alarge area. We will here study the fractal dimensions of theburnt area (Df ) and perimeter (Dp) of the recorded wildfires155

by using the box-counting method (Caldarelli et al., 2001).The fractal dimension (Df ) for a regular two-dimensionalobject such as a square is equal to 2. We will therefore plot2−Df vs.Rg and see how the burnt areas of wildfires deviatefrom regular objects. Fig. 4 is a plot of the (2−Df ) vs. Rg for160

all the wildfires listed in Table 1. The green dots are the datapoints of the longest wildfire event (Little Sand) with a totalof 50 points. One can see that as Rg grows, Df approaches2. The slope of a linear fit is equal to −0.72± 0.02. Thissuggests that as the burnt area grows larger, the perimeter of165

this burnt area becomes smoother and closer to a regular two-dimensional object. This behavior was already observed in apercolation model studied a long time ago (Niessen and Blu-men, 1986). Fig. 5 is a plot of Dp vs. Rg . Dp remains to bewithin the range of 1 and 1.15 as Rg grows. The average of170

Dp for all the data points is about 1.06.Fig. 6 shows the result of the burnt area (A) as a function

of Rg. It is interesting to note that all the 717 data pointsobey an empirical function, A∼ 4.51R2

g . The fact that A isproportional to the square of Rg is in agreement with the175

empirical behavior ofDf (going to 2 as its asymptotic limit).What is more striking is the fact that they have approximatelythe same proportional coefficient. The error estimate of theproportional coefficient and the exponent of Rg are respec-

Figure 3. An illustration of the evaluation ofL. In (a), the sitesin yellow and red are burned sites (1), but only the red site is sur-rounded by burned sites as its nearest and next-nearest neighbors.We re-label this site as 0, as shown in(b) for the evaluation ofL.

would mainly affect the proportional constant, but has verylittle effect on the exponent.

There is a total of 717 temporal data sets of wildfire im-ages in the database for the 31 wildfires listed in Table 1.Each of the temporal data sets corresponds to one data point,e.g., in the plot of the burned area as a function ofRg. Asmentioned above, the resolution of these wildfire images isgood to a scale of 5 m. With such a high resolution and longperiods of observation, one would be able to study the changein fractal dimensions as the wildfire spreads over a large area.We will study here the fractal dimensions of the burned area(Df) and perimeter (Dp) of the recorded wildfires by usingthe box-counting method (Caldarelli et al., 2001). The frac-tal dimension (Df) for a regular two-dimensional object suchas a square is equal to 2. We will therefore plot 2− Df vs.Rg and see how the burned areas of wildfires deviate fromregular objects. Figure 4 is a plot of 2− Df vs. Rg for allthe wildfires listed in Table 1. The green dots are the datapoints of the longest wildfire event (Little Sand), with a totalof 50 points. One can see that asRg grows,Df approaches 2.The slope of a linear fit is equal to−0.72± 0.02. This sug-gests that as the burned area grows larger, the perimeter ofthis burned area becomes smoother and closer to a regulartwo-dimensional object. This behavior was already observedin a percolation model studied a long time ago (Niessen andBlumen, 1986). Figure 5 is a plot ofDp vs.Rg. Dp remainswithin the range of 1 to 1.15 asRg grows. The average ofDpfor all the data points is about 1.06.

Figure 6 shows the result of the burned area (A) as a func-tion of Rg. It is interesting to note that all the 717 data pointsobey an empirical function,A ∼ 4.51R2

g. The fact thatA isproportional to the square ofRg is in agreement with the em-pirical behavior ofDf (going to 2 as its asymptotic limit).What is more striking is the fact that they have approxi-mately the same proportional coefficient. The error estimatesof the proportional coefficient and the exponent ofRg are,respectively, 4.51± 0.10 and 2.00± 0.02. For comparisonwith some of the known regular two-dimensional figures, weknow that the relationship betweenA andRg for the circleis A = 2πR2

g, while those of the square and an equilateral

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818 S.-L. Wang et al.: Fractal dimensions of wildfire spreading4 S. -L. Wang et al.: Fractal Dimensions of Wildfire Spreading

10−1

100

101

10−2

10−1

100

Rg (km)

2−

Df

data

α=−0.72 ±0.02

Little Sand

Fig. 4. Plot of (2−Df ) vs. Rg for the 31 wildfires listed in Table1. The green dots are the data points of the longest wildfire event(Little Sand) with a total of 50 points. The broken red line has aslope of −0.72± 0.02.

10−1

100

101

−0.05

0

0.05

0.1

0.15

0.2

Rg (km)

Dp

−1

data

Fig. 5. Plot of Dp vs. Rg for the 31 wildfires listed in Table 1.

tively 4.51± 0.10 and 2.00± 0.02. To compare with some180

of the known regular two-dimensional figures, we know thatthe relationship between A and Rg for circle is, A= 2πR2

g ,while that of the square and an equilateral triangle are 6 R2

g

and 5.19R2g respectively. For a star polygon with coefficient

of 7/3 heptagram(Coxeter, 1973), the coefficient is 4.49, as185

shown in Figure 7. The result from the empirical data thusindicates that the burnt areas of all the wildfires that one hasrecorded have shapes close to that of a two-dimensional con-cave polygon. Fig. 8 is a plot of the boundaries (L) of thewildfires as a function of Rg . Again, the data points follow190

10−1

100

101

10−2

10−1

100

101

102

103

104

Rg (km)

Are

a (

km

2)

data

Fig. 6. Burnt area (A) as a function of Rg , A∼ 4.51 R2g . The broken

red line has a slope of 2 for comparison.

an empirical function, L∼ 19.57R1.17g with an error estimate

of 19.57± 1.10 and 1.17± 0.04 for the proportional coeffi-cient and exponent of Rg respectively.

To summarize, we have observed that both the burnt areaand the accessible perimeter of wildfire reveal power law be-195

havior as a function of Rg. Moreover, the burnt areas of allthe wildfires under study could be described by the same em-pirical function A∼ 4.51R2

g .

3 Model and Numerical Simulation

In this section, we introduce our model for the spreading200

of wildfires. To try to model wildfire dynamics by takinginto account all possible factors such as temperature, pre-cipitation, wind speed, vegetation, landscape, etc. is a verycomplex task to achieve. In fact, a model for predicting awildfire spread would inevitably have to take into account205

these external environmental factors in a certain way. We alsonotice that there are already many attempts to model wild-fire spreading, ranging from physical models to empiricalmodels(Sullivan, 2009; Bak et al., 1990; Chen et al., 1990;Rothermel, 1972; Anderson et al., 1982). Each of these pro-210

posed models has some success but a complete picture ofhow to effectively describe the dynamics of wildfires is stillnot available for the moment. For our purpose, we will ana-lyze the empirical data by employing a statistical approach.We here set an easier task by introducing a simple model215

to investigate the wildfire dynamics to reveal the geographicfeatures of the wildfires under study. This is based on a cellu-lar automata approach by introducing a 2-dimensional latticeto mimic the forests that the wildfires would spread. Thereexist works using cellular automata or lattice model approach220

Figure 4. Plot of 2−Df vs.Rg for the 31 wildfires listed in Table 1.The green dots are the data points of the longest wildfire event (Lit-tle Sand), with a total of 50 points. The broken red line has a slopeof −0.72± 0.02.


10−1

100

101

10−2

10−1

100

Rg (km)

2−

Df

data

α=−0.72 ±0.02

Little Sand


10−1

100

101

−0.05

0

0.05

0.1

0.15

0.2

Rg (km)

Dp

−1

data





g




10−1

100

101

10−2

10−1

100

101

102

103

104

Rg (km)

Are

a (

km

2)

data







g .







Figure 5. Plot ofDp vs.Rg for the 31 wildfires listed in Table 1.

triangle are 6R2g and 5.19R2

g, respectively. For a star polygonwith a coefficient of 7/3 heptagram (Coxeter, 1973), the co-efficient is∼ 4.49, as shown in Fig. 7. The result from theempirical data thus indicates that the burned areas of all thewildfires that one has recorded have shapes close to that of atwo-dimensional concave polygon. Figure 8 is a plot of theboundaries (L) of the wildfires as a function ofRg. Again,the data points follow an empirical function,L ∼ 19.57R1.17

g ,with an error estimate of 19.57±1.10 and 1.17±0.04 for theproportional coefficient and the exponent ofRg, respectively.

To summarize, we have observed that both the burned areaand the accessible perimeter of wildfire reveal power law be-havior as a function ofRg. Moreover, the burned areas of


10−1

100

101

10−2

10−1

100

Rg (km)

2−

Df

data

α=−0.72 ±0.02

Little Sand


10−1

100

101

−0.05

0

0.05

0.1

0.15

0.2

Rg (km)

Dp

−1

data





g




10−1

100

101

10−2

10−1

100

101

102

103

104

Rg (km)

Are

a (

km

2)

data







g .







Figure 6. Burned area (A) as a function ofRg, A ∼ 4.51R2g. The

broken red line has a slope of 2 for comparison.

all the wildfires under study could be described by the sameempirical functionA ∼ 4.51R2

g.

3 Model and numerical simulation

In this section, we introduce our model for the spreadingof wildfires. To try to model wildfire dynamics by takinginto account all possible factors such as temperature, pre-cipitation, wind speed, vegetation, landscape, etc. is a verycomplex task to achieve. In fact, a model for predicting awildfire spreading would inevitably have to take into accountthese external environmental factors in a certain way. Wealso notice that there have already been many attempts tomodel wildfire spreading, ranging from physical models toempirical models (Sullivan, 2009a; Bak et al., 1990; Chenet al., 1990; Rothermel, 1972; Anderson et al., 1982). Eachof these proposed models has had some success, but a com-plete picture of how to describe the dynamics of wildfireseffectively is still not available for the moment. For our pur-poses, we will analyze the empirical data by employing astatistical approach. We set here an easier task by introduc-ing a simple model to investigate the wildfire dynamics toreveal the geographic features of the wildfires under study.This is based on a cellular automata approach by introduc-ing a two-dimensional lattice to mimic the forests that thewildfires would spread in. There exist works using a cellu-lar automata or lattice model approach to study wildfire; see,e.g.,Bak et al.(1990), Caldarelli et al.(2001), andPorterieet al.(2007). For example, people have used hexagonal cel-lular automata to model wildfire spreading (Trunfio, 2004;Hernandez Encinas et al., 2007; Avolio et al., 2012). Thereare also works that incorporate meteorological factors suchas wind speed and direction, type and density of vegetation,etc. into the dynamical rules of cellular automata, and which


S.-L. Wang et al.: Fractal dimensions of wildfire spreading 819S. -L. Wang et al.: Fractal Dimensions of Wildfire Spreading 5

10−2

10−1

100

101

10−2

10−1

100

101

102

103

Rg (km)

Are

a (

km

2)

data

(7/3)

a)

b)

Fig. 7. An illustration of a two-dimensional concave polygon, (a) a7/3 heptagon. (b) scaling of 7/3 heptagon showed A ∼ 4.49 R2

g .

to study wildfire, see e.g.,(Bak et al., 1990; Caldarelli etal., 2001; Porterie et al., 2007). For example, people haveused hexagonal cellular automata to model wildfire spread-ing(Trunfio, 2004; Hernandez Encinas et al., 2007; Avolio etal., 2012). There are also works which incorporate meteo-225

rological factors such as wind speed and direction, type anddensity of vegetation, etc. into the dynamical rules of cellularautomata and compared the numerical results to the wildfireon Spetses Island in 1990(Alexandridis et al., 2008). As afirst step of our numerical study, we will not take into ac-230

count these factors into the dynamical rules of our model.In this way, we are trying to capture the qualitative behaviorof wildfire dynamics by using this simple model. To makeit more specific, we here only consider a two-dimensionalsquare lattice. Other types of lattices such as triangular lat-235

tices can be treated in a similar manner. On each lattice site,there is only one unit of plant, or we simply say one tree/site.In order to simulate the spreading of wildfires, we incorpo-rate dynamical rules that can mimic the wildfires under study.

10−1

100

101

100

101

102

103

Rg (km)

Le

ng

th (

km

)

data

α=1.17±0.04

Fig. 8. L vs. Rg for the 31 wildfires listed in Table 1. The brokenred line is a fit of the empirical data with a slope of 1.17± 0.04.

The rules for the model are as follows. There are indeed 3240

states that each site can take, namely, unburnt (U), ignited (I)and burnt (B). If the site is in state U, it will remain in stateU if it has no neighboring site in state I. In our simulation,we only consider nearest neighbors. Therefore, each latticesite will only be affected by 4 nearest neighbors during our245

simulation. On the other hand, if the site has a neighboringsite in state I, it will then have a certain probability q thatwill be ignited and if it is ignited, it will be in state I in thenext time step. Furthermore, for any cell in a state I, it willbecome burnt in the next time step, i.e. in a state B. Once the250

cell reaches state B, it will remain to be in state B throughoutthe rest of our simulation and a burnt site cannot ignite anyof its neighboring sites once it reaches such a state. The rulesof the spreading of wildfire at each time step t can now besummarized as follows, for a lattice site i, its state Si(t+ 1)255

is given by

Si(t) = U → Si(t+ 1) = U (1)(no neighboring site in state I at timestep t)

Si(t) = U → Si(t+ 1) = I (2)260

(a probability q from each of its neig-hboring site in state I at time t)

→ Si(t+ 1) = U (3)(a probability 1-q from each of its ne-ighboring site in state I at time t)265

Si(t) = I → Si(t+ 1) =B (4)(the site is in state I at time t)

Figure 7. An illustration of a two-dimensional concave polygon,(a) a 7/3 heptagon.(b) Scaling of the 7/3 heptagon showedA ∼

4.49R2g.

compared the numerical results with the wildfire on SpetsesIsland in 1990 (Alexandridis et al., 2008). As a first step ofour numerical study, we will not take into account these fac-tors in the dynamical rules of our model. In this way, we aretrying to capture the qualitative behavior of wildfire dynam-ics by using this simple model. To make it more specific, weonly consider a two-dimensional square lattice here. Othertypes of lattices such as triangular lattices can be treated in asimilar manner. On each lattice site, there is only one unit ofplant, or we simply say one tree/site. In order to simulate thespreading of wildfires, we incorporate dynamical rules thatcan mimic the wildfires under study. The rules for the modelare as follows. There are indeed three states that each site cantake, namely, unburned (U), ignited (I), and burned (B). If thesite is in state U, it will remain in state U if it has no neighbor-ing site in state I. In our simulation, we only consider nearestneighbors. Therefore, each lattice site will only be affectedby four nearest neighbors during our simulation. On the otherhand, if the site has a neighboring site in state I, it will then


10−2

10−1

100

101

10−2

10−1

100

101

102

103

Rg (km)

Are

a (

km

2)

data

(7/3)

a)

b)

Fig. 7. An illustration of a two-dimensional concave polygon, (a) a7/3 heptagon. (b) scaling of 7/3 heptagon showed A ∼ 4.49 R2

g .

to study wildfire, see e.g.,(Bak et al., 1990; Caldarelli etal., 2001; Porterie et al., 2007). For example, people haveused hexagonal cellular automata to model wildfire spread-ing(Trunfio, 2004; Hernandez Encinas et al., 2007; Avolio etal., 2012). There are also works which incorporate meteo-225

rological factors such as wind speed and direction, type anddensity of vegetation, etc. into the dynamical rules of cellularautomata and compared the numerical results to the wildfireon Spetses Island in 1990(Alexandridis et al., 2008). As afirst step of our numerical study, we will not take into ac-230

count these factors into the dynamical rules of our model.In this way, we are trying to capture the qualitative behaviorof wildfire dynamics by using this simple model. To makeit more specific, we here only consider a two-dimensionalsquare lattice. Other types of lattices such as triangular lat-235

tices can be treated in a similar manner. On each lattice site,there is only one unit of plant, or we simply say one tree/site.In order to simulate the spreading of wildfires, we incorpo-rate dynamical rules that can mimic the wildfires under study.

10−1

100

101

100

101

102

103

Rg (km)

Le

ng

th (

km

)

data

α=1.17±0.04

Fig. 8. L vs. Rg for the 31 wildfires listed in Table 1. The brokenred line is a fit of the empirical data with a slope of 1.17± 0.04.

The rules for the model are as follows. There are indeed 3240

states that each site can take, namely, unburnt (U), ignited (I)and burnt (B). If the site is in state U, it will remain in stateU if it has no neighboring site in state I. In our simulation,we only consider nearest neighbors. Therefore, each latticesite will only be affected by 4 nearest neighbors during our245

simulation. On the other hand, if the site has a neighboringsite in state I, it will then have a certain probability q thatwill be ignited and if it is ignited, it will be in state I in thenext time step. Furthermore, for any cell in a state I, it willbecome burnt in the next time step, i.e. in a state B. Once the250

cell reaches state B, it will remain to be in state B throughoutthe rest of our simulation and a burnt site cannot ignite anyof its neighboring sites once it reaches such a state. The rulesof the spreading of wildfire at each time step t can now besummarized as follows, for a lattice site i, its state Si(t+ 1)255

is given by

Si(t) = U → Si(t+ 1) = U (1)(no neighboring site in state I at timestep t)

Si(t) = U → Si(t+ 1) = I (2)260

(a probability q from each of its neig-hboring site in state I at time t)

→ Si(t+ 1) = U (3)(a probability 1-q from each of its ne-ighboring site in state I at time t)265

Si(t) = I → Si(t+ 1) =B (4)(the site is in state I at time t)

Figure 8.L vs.Rg for the 31 wildfires listed in Table 1. The brokenred line is a fit of the empirical data with a slope of 1.17± 0.04.

have a certain probabilityq that will be ignited, and if it isignited, it will be in state I in the next time step. Furthermore,for any cell in state I, it will become burned in the next timestep, i.e., in state B. Once the cell reaches state B, it will re-main in state B throughout the rest of our simulation, and aburned site cannot ignite any of its neighboring sites once itreaches such a state. The rules of the spreading of wildfireat each time step t can now be summarized as follows: for alattice sitei, its stateSi(t + 1) is given by

Si(t) = U → Si(t + 1) = U (1)

(no neighboring site in state I at time stept)

Si(t) = U → Si(t + 1) = I (2)

(a probabilityq from each of its neighboring

site in state I at timet)

→ Si(t + 1) = U (3)

(a probability 1− q from each of its

neighboring site in state I at timet)

Si(t) = I → Si(t + 1) = B (4)

(the site is in state I at timet)

From the above rules, one can easily see that for a site thatis in state U, and having n neighboring sites in state I at timet , the probability that it will be ignited at timet+1 is equal to1−(1−q)n. This is the basic model that one can start with. Inreality, the parameterp depends on many factors, such as theclimate, vegetation, landscape, etc. of the region under study.Indeed, this basic set of rules would leave too many unburnedlattice sites within the burned area, which is different fromwhat we observed in the case of real wildfires, e.g., the LittleBear wildfire shown in Fig. 1. This is illustrated in Fig. 11a,which is a sample simulation of wildfire spreading on a two-dimensional square lattice with size 1024× 1024 using the


820 S.-L. Wang et al.: Fractal dimensions of wildfire spreading

Figure 9. An illustration of the two-dimensional lattice with oneburning site in the middle, with its neighbors denoted by N, M, NNand MM by their distance from the site.

above set of basic rules. We will therefore modify the dy-namical rules of the model to mimic this empirical behavior.In the case of real wildfires, the threshold for the unburnedplants to be on fire within the burned region could likely belower than the area outside the burned region, thus leavingvery few unburned areas within the burned region. To takethis observation into account, we therefore need to introduceadditional dynamical rules that could mimic such an effect.In general, it can be argued that a burning tree can ignite notonly its nearest neighboring trees, but also its next neighbor-ing trees, and sometimes even next to next-neighboring trees,etc. We therefore incorporate this scenario into our dynami-cal rules by taking the second, third and fourth nearest neigh-bors into account. This is illustrated in Fig. 9 below. The first,second, third and fourth nearest neighbors are denoted by N,M, NN and MM, respectively. We will here choose the prob-ability of ignition to be inversely proportional to the squareof the distance between sites in the numerical simulations ofthe spreading of wildfires. The probability that the site wouldnot be ignited by a neighboring burning site labeledN is thenequal to (1−q), while the probability that it would not be ig-nited by a burning site labeledM is equal to(1− q)1/2. Theoverall ignition probabilityf is

f = 1− (1− q)(n+m/2+nn/4+mm/5),

wheren, m, nn and mm denote the number of burning sitesfor N, M, NN and MM, respectively.

Another possible extension of the set of dynamical rulesis to introduce into the model some kind of nonlinear effectthat simulates the various factors mentioned above. We sug-gest here two sets of nonlinear rules, namely the fire den-sity effect (FDE) and the burned site effect (BSE). The jus-tification for introducing these sets of rules is the following.As the wildfire spreads, the temperature and the humidity in


From the above rules, one can easily see that for a site thatis in state U, and having n neighboring sites in state I at timet, the probability that it will be ignited at time t+1 is equal to270

1− (1− q)n. This is the basic model that one can start with.In reality, the parameter p depends on many factors such asthe climate, vegetation, landscape, etc. of the region understudy. Indeed, this basic set of rules would leave too manyunburnt lattice sites within the burnt area, which is differ-275

ent from what we observed in the case of real wildfires, e.g.,the Little Bear Wildfire shown in Figure 1. This is illustratedin Figure 11 (a), which is a sample simulation of wildfirespread on a two-dimensional square lattice with size 1024 x1024 using the above set of basic rules. We will therefore280

modify the dynamical rules of the model to mimic this em-pirical behavior. In the case of real wildfires, the thresholdfor the unburnt plants to be on fire within the burnt regioncould likely be lower than the area outside the burnt region,thus leaving very few unburnt areas within the burnt region.285

To take into account this observation, we therefore need tointroduce additional dynamical rules that could mimic suchan effect. In general, it can be argued that a burning tree canignite not only its nearest neighboring trees, but also its nextneighboring, and sometimes even next to next neighboring290

trees, etc. We therefore incorporate this scenario into our dy-namical rules by taking the second, third and fourth nearestneighbors into account. This is illustrated in Figure 9 below.The first, second, third and fourth nearest neighbors are de-noted N, M, NN and MM respectively. We will here choose295

the probability of ignition to be inversely proportional to thesquare of the distance between sites in the numerical simu-lations of the spreading of wildfires. The probability that thesite would not be ignited by a neighboring burning site label-ing N is then equal to (1− q), while the probability that it300

would not be ignited by a burning site labeling M is equal to(1− q)1/2. The overall ignition probability f is

f = 1− (1− q)(n+m/2+nn/4+mm/5)

(where n, m, nn and mm denotes the number of burning sitesfor N, M, NN and MM respectively.)305

Another possible extension of the set of dynamical rulesis to introduce some kind of nonlinear effects which simu-late the various factors mentioned above into the model. Wehere suggests two sets of nonlinear rules, namely, the firedensity effect (FDE) and the burnt site effect (BSE). The jus-310

tification to introduce these sets of rules is the following. Asthe wildfire spreads, the temperature and the humidity in theneighborhood of the burning (FDE) and burnt trees (BSE)would change. This would then change the probability forthe unburnt trees to be ignited, which usually increases. To315

imitate these effects, we set the probability for an unburntsite to be ignited by FDE to be f = 1− (1− q)(nk). k = 1is the basic percolation model. In a similar fashion, we setthe probability of an unburnt site to be ignited by BSE to be

Fig. 9. An illustration of the two-dimensional lattice with one burn-ing site in the middle with Its neighbors denoted by N, M, NN andMM by their distance from the site.

Fig. 10. Set of dynamical rules used in the simulationthroughout this paper. Basic Percolation (BP) f = 1− (1− q)n.Fire Density Effect(FDE)f = 1− (1− q)(n

k), k 6= 1. Burnt SitesEffect(BSE)f = 1−(1−q)(n(1+zn(b))), z 6= 1. Correlated(Cor)f =1− (1− q)(n+m/2+nn/4+...).

f = 1− (1− q)(n(1+zn(b))), where n(b) denotes the number320

of burnt sites in nearest neighbor. z = 0 corresponds to thebasic percolation. One can see that the exponent of (1− q)increases as z increases. This in turn means that the proba-bility that the site would not be ignited by neighboring siteswill drop faster as z increases. In our simulations, we always325

set z to be larger than or equal to 0. Figure 10 summarizesthe set of dynamical rules used in our simulation.

Figure 11 is an illustration of our simulation on a 1024x 1024 square lattice using different sets of dynamical rulesin the model. As an example, we use the following set of330

parameters in the simulation (a)BP, p=0.5, q=0.38; (b)Cor,

Figure 10.Set of dynamical rules used in the simulation throughoutthis paper. Basic percolation (BP)f = 1− (1− q)n. Fire density

effect (FDE)f = 1− (1− q)(nk), k 6= 1. Burned sites effect (BSE)

f = 1− (1− q)(n(1+zn(b))), z 6= 1. Correlated (Cor)f = 1− (1−

q)(n+m/2+nn/4+...).

the neighborhood of the burning (FDE) and burned (BSE)trees will change. This would then change the probability ofthe unburned trees being ignited, which usually increases.To imitate these effects, we set the probability of an un-burned site being ignited by FDE atf = 1−(1−q)(n

k). k = 1is the basic percolation model. In a similar fashion, we setthe probability of an unburned site being ignited by BSE atf = 1− (1− q)(n(1+zn(b))), wheren(b) denotes the numberof burned sites in nearest neighbor.z = 0 corresponds to thebasic percolation. One can see that the exponent of(1− q)

increases asz increases. This in turn means that the probabil-ity that the site will not be ignited by neighboring sites willdrop faster asz increases. In our simulations, we always setz to be larger than or equal to 0. Figure 10 summarizes theset of dynamical rules used in our simulation.

Figure 11 is an illustration of our simulation on a 1024×

1024 square lattice using different sets of dynamical rulesin the model. As an example, we use the following setof parameters in the simulation: (a) BP,p = 0.5, q = 0.38;(b) Cor,p = 0.5, q = 0.11 (correlation length is five latticesites); (c) FDE,p = 0.5, q = 0.34,k = 3; (d) BSE,p = 0.5,q = 0.29, z = 3. In general, the set of rules based on BSEgives better results as compared with the empirical data suchas in Fig. 1.

In the remaining part of this section, we will thereforeanalyze statistical features of the model by performing nu-merical simulations using BSE rules. All simulations wereperformed on a 2048× 2048 square lattice, with each z av-eraged over 100 simulations. In each of the simulations, weput four burning sites at the center and let the wildfire spreadaccording to the BSE rules with different values ofz. In per-forming the simulations, one can use any algorithm for asyn-chronous updates for the lattice. In our simulation, we usethe Metroplis algorithm of the asynchronous update, whichis commonly known as the Monte Carlo method in statisti-cal physics (Metropolis et al., 1953). Figure 12 is an illus-tration of the spreading of burned areas with BSE for differ-ent values ofz. One can see that asz increases, so does thespreading of burned areas. Note that forz ≤ 1, the burned


S.-L. Wang et al.: Fractal dimensions of wildfire spreading 821


Fig. 11. (a) BP, p = 0.5, q = 0.38 (b) Cor, p = 0.5, q = 0.11,(correlation length = 5 lattice sites (c) FDE, p = 0.5, q = 0.34, k =3 and, (d) BSE, p = 0.5, q = 0.29, z = 3

p=0.5, q=0.11 (correlation length = 5 lattice sites) ; (c)FDE,p=0.5, q=0.34 and k=3, (d)BSE, p=0.5, q=0.29 and z=3. Ingeneral, the sets of rules based on BSE gives better results ascompared to the empirical data such as in Figure 1.335

In the remaining part of this section, we will therefore ana-lyze statistical features of the model by performing numericalsimulations using BSE rules. All simulations were performedon a 2048 x 2048 square lattice with each z averaged over100 simulations. In each of the simulations, we put 4 burning340

sites at the center and let the wildfire spread according to theBSE rules with different values of z. In performing the simu-lations, one can use any algorithms for asynchronous updatesfor the lattice. In our simulation, we use the Metroplis algo-rithm of the asynchronous update which is commonly known345

as the Monte Carlo method in statistical physics (Metropolis,1953). Fig.12 is an illustration of the spreading of burnt areawith BSE for different values of z. One can see that as z in-creases, so does the spreading of burnt area. Note that for z≤ 1, the burnt area stops growing after a few hundred Monte350

Carlo steps. The result also shows that the burnt area for thesame number of Monte Carlo steps increases rapidly as z in-creases. In performing the numerical simulation, one MonteCarlo step usually refers to one sweep of update for the lat-tice, i.e., if the lattice is an N×N lattice, there will be an av-355

erageN2 update for each Monte Carlo step. In our simulationhere, one only needs to perform updates for the burning sitesand their neighboring sites in each Monte Carlo step. Further-

100

101

102

103

100

101

102

103

104

105

106

107

Rg

Are

a

0, α=1.9317

0.5, α=2.0773

1, α=2.0899

1.5, α=2.0493

2, α=2.0544

2.5, α=2.0631

3, α=2.0496

3.5, α=2.0365

4, α=2.0302

empirical data

100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6x 10

5

Time step

Are

a

0

0.5

1

1.5

2

2.5

3

3.5

4

a)

b)

Fig. 12. (a) The spreading of burnt area with BSE for different val-ues of z. The burnt area for the same number of Monte Carlo stepsincreases rapidly as z increases (Change the labeling of steps). (b)The corresponding result of the burnt area as a function of Rg withBSE and fitting A∝Rα

g . Empirical data points are also includedfor comparison.

more, since one only performs about 1000 Monte Carlo stepsfor each wildfire spread in Fig. 12, one can employ any type360

of boundary conditions on the lattice since the burnt sites willnever get to the boundaries of the lattice. Note that we alsoinclude the empirical data points in Fig.12 for comparison. Ingeneral, one would consider a forest to have about 10 trees ormore for each hectare (10,000 m2), we therefore assume that365

the distance between two adjancent lattice sites is equivalentto a separation of 25 meters. Fig. 13 presents the numericalresults of the L versus Rg with BSE for different values of z.One can see that there is a gradual decrease of β(L∝Rβg ) asz increases.370

Figure 11. (a) BP, p = 0.5, q = 0.38; (b) Cor, p = 0.5, q = 0.11(correlation length is five lattice sites);(c) FDE,p = 0.5, q = 0.34,k = 3; and(d) BSE,p = 0.5, q = 0.29,z = 3.

area stops growing after a few hundred Monte Carlo steps.The result also shows that the burned area for the same num-ber of Monte Carlo steps increases rapidly as z increases. Inperforming the numerical simulation, one Monte Carlo stepusually refers to one sweep of updates for the lattice; i.e., ifthe lattice is anN × N lattice, there will be an averageN2

update for each Monte Carlo step. In our simulation here,one only needs to perform updates for the burning sites andtheir neighboring sites in each Monte Carlo step. Further-more, since one only performs about 1000 Monte Carlo stepsfor each wildfire spread in Fig. 12, one can employ any typeof boundary condition on the lattice, since the burned siteswill never get to the boundaries of the lattice. Note that wealso include the empirical data points in Fig. 12 for compar-ison. In general, one would consider a forest to have about10 trees or more for each hectare (10 000 m2). We thereforeassume that the distance between two adjacent lattice sites isequivalent to a separation of 25 m. Figure 13 presents the nu-merical results ofL vs. Rg with BSE for different values of

z. One can see that there is a gradual decrease inβ(L ∝ Rβg )

asz increases.Figure 14 showsDp vs.Rg, with BSE for different values

of z. When the simulated burned area is small, the perime-ter is more rugged, and the increase in the burned area doesnot change the perimeter as fast as a regular two-dimensionalobject. As the burned area increases, so doesDp, and it willeventually approach an asymptotic value close to that of aregular two-dimensional object.


Fig. 11. (a) BP, p = 0.5, q = 0.38 (b) Cor, p = 0.5, q = 0.11,(correlation length = 5 lattice sites (c) FDE, p = 0.5, q = 0.34, k =3 and, (d) BSE, p = 0.5, q = 0.29, z = 3

p=0.5, q=0.11 (correlation length = 5 lattice sites) ; (c)FDE,p=0.5, q=0.34 and k=3, (d)BSE, p=0.5, q=0.29 and z=3. Ingeneral, the sets of rules based on BSE gives better results ascompared to the empirical data such as in Figure 1.335

In the remaining part of this section, we will therefore ana-lyze statistical features of the model by performing numericalsimulations using BSE rules. All simulations were performedon a 2048 x 2048 square lattice with each z averaged over100 simulations. In each of the simulations, we put 4 burning340

sites at the center and let the wildfire spread according to theBSE rules with different values of z. In performing the simu-lations, one can use any algorithms for asynchronous updatesfor the lattice. In our simulation, we use the Metroplis algo-rithm of the asynchronous update which is commonly known345

as the Monte Carlo method in statistical physics (Metropolis,1953). Fig.12 is an illustration of the spreading of burnt areawith BSE for different values of z. One can see that as z in-creases, so does the spreading of burnt area. Note that for z≤ 1, the burnt area stops growing after a few hundred Monte350

Carlo steps. The result also shows that the burnt area for thesame number of Monte Carlo steps increases rapidly as z in-creases. In performing the numerical simulation, one MonteCarlo step usually refers to one sweep of update for the lat-tice, i.e., if the lattice is an N×N lattice, there will be an av-355

erageN2 update for each Monte Carlo step. In our simulationhere, one only needs to perform updates for the burning sitesand their neighboring sites in each Monte Carlo step. Further-

100

101

102

103

100

101

102

103

104

105

106

107

Rg

Are

a

0, α=1.9317

0.5, α=2.0773

1, α=2.0899

1.5, α=2.0493

2, α=2.0544

2.5, α=2.0631

3, α=2.0496

3.5, α=2.0365

4, α=2.0302

empirical data

100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6x 10

5

Time step

Are

a

0

0.5

1

1.5

2

2.5

3

3.5

4

a)

b)

Fig. 12. (a) The spreading of burnt area with BSE for different val-ues of z. The burnt area for the same number of Monte Carlo stepsincreases rapidly as z increases (Change the labeling of steps). (b)The corresponding result of the burnt area as a function of Rg withBSE and fitting A∝Rα

g . Empirical data points are also includedfor comparison.

more, since one only performs about 1000 Monte Carlo stepsfor each wildfire spread in Fig. 12, one can employ any type360

of boundary conditions on the lattice since the burnt sites willnever get to the boundaries of the lattice. Note that we alsoinclude the empirical data points in Fig.12 for comparison. Ingeneral, one would consider a forest to have about 10 trees ormore for each hectare (10,000 m2), we therefore assume that365

the distance between two adjancent lattice sites is equivalentto a separation of 25 meters. Fig. 13 presents the numericalresults of the L versus Rg with BSE for different values of z.One can see that there is a gradual decrease of β(L∝Rβg ) asz increases.370

Figure 12. (a)The spreading of burned areas with BSE for differentvalues ofz. The burned area for the same number of Monte Carlosteps increases rapidly asz increases (changing the labeling of thesteps).(b) The corresponding result of the burned area as a functionof Rg, with BSE and fittingA ∝ Rα

g . Empirical data points are alsoincluded for comparison.

Note that in Figs. 12–14,Rg is measured in terms ofthe lattice length unit (one unit is the distance between twoneighboring lattice sites), while the areaA is measured interms of the number of unit lattice cells.

4 Conclusions

In this paper, we have analyzed time series data of 31 wild-fires in the US that occurred in 2012, with a total of 717 datasets of wildfire images. With a spatial resolution of 5 mand daily data for these wildfire incidents, one is able tocarry out a better analysis of the dynamical and statisticalproperties of wildfires. The behavior of fractal dimensions(FD) during the spreading was studied and their geologicalfeatures were identified. The result shows that the fractal


822 S.-L. Wang et al.: Fractal dimensions of wildfire spreading8 S. -L. Wang et al.: Fractal Dimensions of Wildfire Spreading

100

101

102

103

100

101

102

103

104

105

Rg

L

0, α=1.68

0.5, α=1.68

1, α=1.72

1.5, α=1.69

2, α=1.55

2.5, α=1.39

3, α=1.27

3.5, α=1.19

4, α=1.14

Fig. 13. L versus Rg with BSE for different values of z

100

101

102

103

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rg

Dp

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 14. Dp vs. Rg with BSE for different values of z

Figure 14 shows Dp vs. Rg with BSE for different valuesof z. When the simulated burnt area is small, the perimeteris more rugged and the increase in the burnt area does notchange the perimeter as fast as a regular two-dimensionalobject. As the burnt area increases, so does Dp and it will375

eventually approach an asymptotic value close to that of aregular two-dimensional object.

Note that in Figures 12-14, Rg is measured in terms of thelattice length unit (1 unit is the distance between two neigh-boring lattice sites) while the area A is measured in terms of380

the number of unit lattice cells.

4 Conclusions

In this paper, we have analyzed time series data of 31 wild-fires in the U.S. that occurred in 2012, with a total of 717datasets of wildfire images. With a spatial resolution of 5385

meters and daily data for these wildfire incidents, one is ableto carry out a better analysis on the dynamical and statis-tical properties of wildfires. The behavior of fractal dimen-sions (FD) during the spreading was studied and their ge-ological features were identified. The result shows that the390

fractal dimension Df of the burnt areas approaches 2 whileDp of the corresponding accessible perimeter ranges be-tween 1 and 1.15 as the wildfire spreads. More interestingly,all data points follow empirical relations for the burnt area(A∼ 4.51R2

g) and accessible perimeter (L∼ 19.57R1.17g ). In395

a spatially-uniform landscape, the fractal dimension of wild-fire would be 2. The observation that the fractal dimensionof wildfire spreading is less than 2 could mean that its dy-namics is driven by spatial heterogeneity in the environment,e.g., different types of plants, different distances between400

patches, etc. may reduce the spatial spread of wildfire. Thismay well imply that the fractal dimension of wildfires ob-served in the data is a proxy for spatial heterogeneity, whichmay come about because of differences in environmental, cli-matic and other factors between locations. The fact that wild-405

fire spreading exhibits fractal behavior suggests that the un-derlying dynamics can be characterized by the fractal struc-tures. Thus, one can study models whose dynamics incor-porate such fractal structures without knowing the details ofeffects from ecology, climate, etc as a first step to understand410

the dynamics of wildfire spreading. Taking this as a startingpoint, a growth model based on cellular automata methodis proposed in this paper to mimic both the dynamical andstatistical properties of the spreading of wildfires. Numericalresults are compared with the fractal dimensions and scal-415

ing behaviors of the corresponding empirical data and showgood agreement between the simulated data from the modeland the empirical data from the database(Geomac, 2012).The results suggested that the simple model proposed hereis able to capture the qualitative behavior of wildfire dynam-420

ics. It would be interesting to see if one can further extendthe model to incorporate the meteorological factors as men-tioned above.

References

Alexandridis, A., Vakalis D., Siettos, C. I., and Bafas, G. V.: A cel-425

lular automata model for forest fire spread prediction: The caseof the wildfire that swept through Spetses Island in 1990, Appl.Math. Comput., 204, 191–201, 2008

Anderson, D. G., Catchpole, E. A., DeMestre, N. J., and Parkes, E.:Modeling the spread of grass fires, J. Austral. Math. Soc. (Series430

B), 23 451–466, 1982Avolio M. V., Di Gregorio S., Spataro W., and Trunfio G. A.: A

theorem about the algorithm of minimization of differences for

Figure 13.L vs.Rg, with BSE for different values ofz.


100

101

102

103

100

101

102

103

104

105

Rg

L

0, α=1.68

0.5, α=1.68

1, α=1.72

1.5, α=1.69

2, α=1.55

2.5, α=1.39

3, α=1.27

3.5, α=1.19

4, α=1.14

Fig. 13. L versus Rg with BSE for different values of z

100

101

102

103

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rg

Dp

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 14. Dp vs. Rg with BSE for different values of z

Figure 14 shows Dp vs. Rg with BSE for different valuesof z. When the simulated burnt area is small, the perimeteris more rugged and the increase in the burnt area does notchange the perimeter as fast as a regular two-dimensionalobject. As the burnt area increases, so does Dp and it will375

eventually approach an asymptotic value close to that of aregular two-dimensional object.

Note that in Figures 12-14, Rg is measured in terms of thelattice length unit (1 unit is the distance between two neigh-boring lattice sites) while the area A is measured in terms of380

the number of unit lattice cells.

4 Conclusions

In this paper, we have analyzed time series data of 31 wild-fires in the U.S. that occurred in 2012, with a total of 717datasets of wildfire images. With a spatial resolution of 5385

meters and daily data for these wildfire incidents, one is ableto carry out a better analysis on the dynamical and statis-tical properties of wildfires. The behavior of fractal dimen-sions (FD) during the spreading was studied and their ge-ological features were identified. The result shows that the390

fractal dimension Df of the burnt areas approaches 2 whileDp of the corresponding accessible perimeter ranges be-tween 1 and 1.15 as the wildfire spreads. More interestingly,all data points follow empirical relations for the burnt area(A∼ 4.51R2

g) and accessible perimeter (L∼ 19.57R1.17g ). In395

a spatially-uniform landscape, the fractal dimension of wild-fire would be 2. The observation that the fractal dimensionof wildfire spreading is less than 2 could mean that its dy-namics is driven by spatial heterogeneity in the environment,e.g., different types of plants, different distances between400

patches, etc. may reduce the spatial spread of wildfire. Thismay well imply that the fractal dimension of wildfires ob-served in the data is a proxy for spatial heterogeneity, whichmay come about because of differences in environmental, cli-matic and other factors between locations. The fact that wild-405

fire spreading exhibits fractal behavior suggests that the un-derlying dynamics can be characterized by the fractal struc-tures. Thus, one can study models whose dynamics incor-porate such fractal structures without knowing the details ofeffects from ecology, climate, etc as a first step to understand410

the dynamics of wildfire spreading. Taking this as a startingpoint, a growth model based on cellular automata methodis proposed in this paper to mimic both the dynamical andstatistical properties of the spreading of wildfires. Numericalresults are compared with the fractal dimensions and scal-415

ing behaviors of the corresponding empirical data and showgood agreement between the simulated data from the modeland the empirical data from the database(Geomac, 2012).The results suggested that the simple model proposed hereis able to capture the qualitative behavior of wildfire dynam-420

ics. It would be interesting to see if one can further extendthe model to incorporate the meteorological factors as men-tioned above.

References

Alexandridis, A., Vakalis D., Siettos, C. I., and Bafas, G. V.: A cel-425

lular automata model for forest fire spread prediction: The caseof the wildfire that swept through Spetses Island in 1990, Appl.Math. Comput., 204, 191–201, 2008

Anderson, D. G., Catchpole, E. A., DeMestre, N. J., and Parkes, E.:Modeling the spread of grass fires, J. Austral. Math. Soc. (Series430

B), 23 451–466, 1982Avolio M. V., Di Gregorio S., Spataro W., and Trunfio G. A.: A

theorem about the algorithm of minimization of differences for

Figure 14.Dp vs.Rg, with BSE for different values ofz.

dimensionDf of the burned areas approaches 2, whileDpof the corresponding accessible perimeter ranges between1 and 1.15 as the wildfire spreads. More interestingly, alldata points follow empirical relations for the burned area(A ∼ 4.51R2

g) and accessible perimeter (L ∼ 19.57R1.17g ).

In a spatially uniform landscape, the fractal dimensionof wildfire would be 2. The observation that the fractaldimension of wildfire spreading is less than 2 could meanthat its dynamics is driven by spatial heterogeneity in theenvironment, e.g., by different types of plants, differentdistances between patches, etc., which may reduce thespatial spreading of wildfire. This may well imply that thefractal dimension of wildfires observed in the data is a proxyfor spatial heterogeneity, which may come about becauseof differences in environmental, climatic and other factorsbetween locations. The fact that wildfire spreading exhibitsfractal behavior suggests that the underlying dynamics can

be characterized by the fractal structures. One can thus studymodels whose dynamics incorporate such fractal structureswithout knowing the details of effects from ecology, climate,etc. as a first step to understanding the dynamics of wildfirespreading. Taking this as a starting point, a growth modelbased on the cellular automata method is proposed in thispaper to mimic both the dynamical and statistical propertiesof the spreading of wildfires. Numerical results are comparedwith the fractal dimensions and scaling behaviors of thecorresponding empirical data, and show good agreementbetween the simulated data from the model and the empiricaldata from the database (Geomac, 2012). The results suggestthat the simple model proposed here is able to capturethe qualitative behavior of wildfire dynamics. It would beinteresting to see if one can further extend the model toincorporate the meteorological factors as mentioned above.

Edited by: A. GangulyReviewed by: two anonymous referees

References

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Anderson, D. G., Catchpole, E. A., DeMestre, N. J., and Parkes,E.: Modeling the spread of grass fires, J. Austral. Math. Soc.,Series B, 23, 451–466, 1982.

Avolio, M. V., Di Gregorio, S., Spataro, W., and Trunfio, G. A.: Atheorem about the algorithm of minimization of differences formulticomponent cellular automata, Lect. Not. Comp. Sc., 7495,289–298, 2012.

Bak, P., Chen, K., and Tang, C.: A forest-fire model and somethoughts on turbulence, Phys. Lett. A, 147, 297–300, 1990.

Caldarelli, G., Frondoni, R., Gabrielli A., Montuori, M., Retzlaff,R., and Ricotta, C.: Percolation in real wildfires, Europhys. Lett.,56, 510, doi:10.1209/epl/i2001-00549-4, 2001.

Chen, K., Bak, P., and Jensen, M. H.: A deterministic critical forestfire model, Phys. Lett. A, 149, 207–210, 1990.

Coxeter, H. S. M.: Regular polytopes, Dover Publications, NY,1973.

Hernandez Encinas, L., Hoya White, S., Martin del Rey, A., and Ro-driguez Sanchez, G.: Modelling forest fire spread using hexago-nal cellular automata, Appl. Math. Model., 31, 1213–1227, 2007.

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Kitzberger, T., Brown, P. M., Heyerdahl, E. K., Swetnam, T. W.,and Veblen, T. T.: Contingent Pacific-Atlantic Ocean influenceon multicentury wildfire synchrony over western North America,P. Natl. Acad. Sci. USA, 104, 543–548, 2007.

Lee, H.-I., Wang, S.-L., and Li, S.-P.: Climate Effect on WildfireBurned Area in Alberta (1961–2010), Int. J. Mod. Phys. C, 24,1350053, doi:10.1142/S0129183113500538, 2013.

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http://dx.doi.org/10.1063/1.1699114

http://dx.doi.org/10.1088/0305-4470/19/5/013

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