Content uploaded by Kathryn Leonard

Author content

All content in this area was uploaded by Kathryn Leonard on May 19, 2016

Content may be subject to copyright.

Error Correction for Fire Growth Modeling

Kathryn Leonard and Derek DeSantis

CSU Channel Islands, Camarillo, CA, USA

kleonard.ci@gmail.com

Abstract. We construct predictions of ﬁre boundary growth using level set meth-

ods. We generate a correction for predictions at the subsequent time step based

on current error. The current error is captured by a thin-plate spline deformation

from the initial predicted boundary to the observed boundary, which is then ap-

plied to the initial prediction at the subsequent time step. We apply these methods

to data from the 1996 Bee Fire and 2002 Troy Fire. We also compare our results

to earlier predictions for the Bee Fire using the FARSITE method. Error is mea-

sured using the Hausdorff distance. We determine conditions under which error

correction improves prediction performance.

1 Introduction

Developing accurate models for the growth of forest ﬁres is a vexing problem with

life-and-death implications. The physical interactions between variables in a ﬁre are

too complex to be captured by any solvable mathematical formulation. For example,

humidity plays an important role in rate of ﬁre spread (ROS), but the ﬁre itself alters

the humidity of the surrounding air. Additionally, collecting reliable measurements of

these variables is often impossible during a ﬁre event.

The model currently used by US ﬁre departments, FARSITE [3], propagates ﬁres

locally along ellipses normal to the ﬁre boundary via functions based on topography,

atmosphere, vegetation, and elevation above ground level of the ﬁre [6]. Implemen-

tation often relies on coarse approximations to real-time input parameters, or none at

all. Recently, level set methods have been applied to model ROS [5]. Level set meth-

ods develop a global model of ﬁre growth that depends on the geometry of the ﬁre

front as well as an external vector ﬁeld that can encompass the aforementioned exter-

nal variables. Again, implementations of the the level set model rarely account for the

complications of the physical reality of the ﬁre.

Not surprisingly, none of these models produces very accurate predictions. As a

result, attention has turned to error correction [4], [7], whereby correcting error in a

current prediction relies on errors at earlier times. The hope is that we can account for

the barriers to sophistication in our implementations by learning from their failures.

In one such approach, Fujioka deﬁnes a notion of bias in [4] based on a polar rep-

resentation of the ﬁre boundary, and generates a correction based on removing that bias

from the prediction estimates. His work concludes that the uncorrected estimates are

more accurate. We compare our results with his in Section 3.2. In another approach,

Rochoux, et. al, develop a probabilistic framework using simulations and controlled

burns to generate a best linear unbiased estimator (BLUE) of the correction using tech-

niques based on the Kalman Filter [7]. To date, the methodology has not been applied

to real-world ﬁre prediction.

Our approach uses ﬁre intensity data from the California Troy and Bee ﬁres to ex-

plore the idea that past ﬁre behavior can realistically inform future ﬁre prediction on the

time scales for which data is available during an actual ﬁre event. We also explore the

relative accuracy of level set and FARSITE methods for modeling ﬁre spread for the

Bee ﬁre, data that captures the ﬁrst 105 minutes of ﬁre growth.

We apply the level set method as implemented by Sumengen [9] to an initial ﬁre

boundary. We determine a mismatch between the predicted ﬁre boundary and the ob-

served ﬁre boundary using thin plate spline (TPS) point matching as implemented by

Chui and Rangarajan [2]. The level set prediction at the following time step is then

adjusted according to the TPS deformation. Accuracy is measured using the Hausdorff

distance between the observed and predicted boundaries. We present accuracy results

for the Troy and Bee ﬁres, and compare our results for the Bee ﬁre with those found in

Fujioka [4].

2 Methods

Troy ﬁre data consists of 13 heat intensity aerial images at approximately 10 minute

intervals beginning in the afternoon of June 19, 2002. The 1996 Bee ﬁre data consists

of three images at 15, 45, and 105 minutes after ignition. All data was obtained from the

USDA Forest Service. In addition, our data includes predicted boundaries generated by

Fujioka’s method described in [4] for the Bee Fire. Fujioka’s method uses FARSITE, a

Rothermel-based method, with unusually detailed wind data to generate predictions.

2.1 Preprocessing

Bee ﬁre data consists of UTM coordinates of the ﬁre boundary points, which we scaled

down. To extract boundaries from the heat intensity images comprising the Troy ﬁre

data, we segment the ﬁre area using k-means clustering with k= 2, then extract the

coordinates of the boundary contour. Given the boundary coordinates, we compute the

signed distance function of the boundary for input to the level set method.

2.2 Level Set Method

The level set method models contours evolving in time as the zero level sets of a func-

tion φt(x, y). The level set function satisﬁes the level set equation:

dφ

dt +V· ∇φ= 0

where V(x, y)is a continuous vector ﬁeld encoding both external forces and intrinsic

geometry of the curve [8]. For the Troy ﬁre, Vcontains coarse wind information and

constant rate of spread normal to the curve. For the Bee ﬁre, Vis just the constant

normal rate of spread (wind data was not available). We compute a ﬁrst-order Godunov

numerical solution as implemented in [9] with square mesh width dx =dy = 0.5, 1.5

iterations per minute of prediction, and α= 0.5in determining dt.

2.3 Thin-plate Spline Matching

Thin-plate splines (TPS) approximate smooth planar deformations mapping one con-

tour onto another by deﬁning a function f(v) = Pn

i=1 aiφ(v−xi)based on pairings

of control points {(xi, yi)}n

i=1 that minimizes the energy functional [1]:

n

X

i=1

||f(xi)−yi|| +λZZ "∂2f

∂x22

+ 2 ∂2f

∂x∂ y +∂2f

∂y22#.

Given two sets of boundary points {xi}and {yj}, the energy minimization is dif-

ﬁcult to compute. As implemented in Chui and Rangarajan [2], an approximate min-

imization is found using deterministic annealing. At high temperatures, point sets are

matched based on geometric features. As the temperature decreases, points are matched

based on proximity. The output of the implementation is a point matching between the

two sets of boundary points, and the weights and coefﬁcients for computing the result-

ing transformation for any new input points. In our implementation, initial temperatures

range from 400 to 7500 and ﬁnal temperatures range from 12 to 1000, based on magni-

tude of the coordinates.

2.4 Error Correction

At time t=t0, we input points on the initial ﬁre boundary B0to the level set method,

producing an estimate P1for the observed boundary B1at t=t1. In this ﬁrst stage,

there is no history available to construct a corrected prediction, so the corrected pre-

diction Q1=P1. We then compute the TPS matching between the level set prediction

P1and the observed boundary B1, producing a function f1:R2→R2. We begin the

iteration at t=ti,i > 0, with the observed boundary Biand the TPS mapping ficap-

turing the error between the level set prediction Piand Bi. We then generate the level

set prediction Pi+1 of the observed boundary Bi+1 at t=ti+1 and correct it according

to fi, generating a revised prediction Qi+1 of Bi+1, where Qi+1 =fi(Pi).

2.5 Error Measurement

We use the Hausdorff distance to measure the error between predicted and observed ﬁre

boundaries. The Hausdorff distance captures the maximum Euclidean distance between

two boundaries. Given two boundaries B1, B2,

dH(B1, B2) = max

i,j=1,2max

p∈Bi

min

q∈Bj,i6=jd(p, q)

where d(p, q)is the standard Euclidean distance between points p, q in the plane.

3 Results

We ﬁnd that neither the level set predictions {Pi}nor the corrected predictions {Qi}

satisfactorily capture the ﬁre behavior. For the Troy ﬁre data, both methods provide

adequate predictions, with better accuracy sometimes with correction and sometimes

without. For the Bee ﬁre, neither corrected nor uncorrected method is satisfactory. We

judge Fujioka’s predicted boundaries to be superior. In other words, correction after the

fact does not adequately compensate for the inability of the original implementation to

adequately model the physical complexities of the ﬁre.

3.1 Troy Fire Results

The Hausdorff distances between the observed boundary and the predicted boundaries

are given in Table 1. We also show a sampling of the level set predictions without cor-

rection (cyan), with correction (green), and the observed boundaries (yellow) in Figures

1-9. Note that the corrected and uncorrected predictions for the boundary at time t= 2

are the same because there is not yet a history of error. Note also in Figures 7and 9how

signiﬁcant new growth areas emerge but are not captured at all by either corrected or

uncorrected predictions.

Time-step Level Set Corrected

2 51.99 51.99

3 13.92 51.30

4 19.10 11.64

5 14.14 14.79

6 22.36 16.21

7 15.13 19.50

8 28.07 22.63

9 24.18 15.10

10 16.12 15.83

11 26.00 19.45

12 13.03 15.72

Table 1. Hausdorff distance between level set predictions of ﬁre boundaries and the observed

boundaries, with and without TPS correction based on error at previous time step.

In certain cases, the correction contributes to a substantially more accurate predic-

tion (Figures 3,8), but often the two predictions are roughly the same distance from

the boundary. As the images in Figures 1-10 show, the level set method errors tend to

underestimate growth while the TPS-corrected predictions tend to overestimate growth.

3.2 Bee Fire Results

For the Bee ﬁre, only three time steps of data are useable (Figure 11) giving a very small

sample. We include results nonetheless because we are able to directly compare our

predictions with FARSITE predictions. The Hausdorff distances between the observed

boundary and the predicted boundaries are given in Table 2for level set predictions,

0 50 100 150 200 250 300 350 400

120

140

160

180

200

220

240

260

280

300

320

Fig. 1. Troy ﬁre t= 2: level set prediction (cyan), TPS-corrected level set prediction (green), and

observed boundary (yellow).

0 50 100 150 200 250 300 350 400

140

160

180

200

220

240

260

280

300

Fig. 2. Troy ﬁre t= 3: level set prediction (cyan), TPS-corrected level set prediction (green), and

observed boundary (yellow).

0 50 100 150 200 250 300 350 400

120

140

160

180

200

220

240

260

280

300

Fig. 3. Troy ﬁre t= 4: level set prediction (cyan), TPS-corrected level set prediction (green), and

observed boundary (yellow).

0 50 100 150 200 250 300 350 400

100

120

140

160

180

200

220

240

260

280

300

Fig. 4. Troy ﬁre t= 5: level set prediction (cyan), TPS-corrected level set prediction (green), and

observed boundary (yellow).

0 50 100 150 200 250 300 350 400

100

120

140

160

180

200

220

240

260

280

300

Fig. 5. Troy ﬁre t= 6: level set prediction (cyan), TPS-corrected level set prediction (green), and

observed boundary (yellow).

0 50 100 150 200 250 300 350 400 450

100

120

140

160

180

200

220

240

260

280

300

Fig. 6. Troy ﬁre t= 6: level set prediction (cyan), TPS-corrected level set prediction (green), and

observed boundary (yellow).

0 50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

Fig. 7. Troy ﬁre t= 7: level set prediction (cyan), TPS-corrected level set prediction (green), and

observed boundary (yellow).

0 50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

Fig. 8. Troy ﬁre t= 10: level set prediction (cyan), TPS-corrected level set prediction (green),

and observed boundary (yellow).

0 50 100 150 200 250 300 350 400 450

50

100

150

200

250

300

Fig. 9. Troy ﬁre t= 11: level set prediction (cyan), TPS-corrected level set prediction (green),

and observed boundary (yellow).

0 50 100 150 200 250 300 350 400 450

0

50

100

150

200

250

300

Fig. 10. Troy ﬁre t= 12: level set prediction (cyan), TPS-corrected level set prediction (green),

and observed boundary (yellow).

corrected level set predictions, and FARSITE predictions. We applied TPS correction

to the FARSITE predictions but found no improvement. We do not include those results

here. Again, recall that at t= 60 minutes, no history of error exists and so the corrected

prediction is equal to the original prediction. Note also that the scale of these errors is

different than the results for the Troy ﬁre, as the coordinate systems are different.

−500 0 500 1000 1500 2000

0

500

1000

1500

2000

2500

Fig. 11. Bee ﬁre after 30, 60 and 105 minutes.

Time Level Set Corrected FARSITE

60 591.7 591.7 790.5

105 1907.7 1876.5 954.2

Table 2. Hausdorff distance between FARSITE or level set predictions of ﬁre boundaries and the

observed boundaries, with and without TPS correction based on error at previous time step.

At the early stage, the two methods are comparable. At the later stage, however,

FARSITE error is half the error for either of the level set predictions.

We show the FARSITE predictions (white), level set predictions without correction

(green), and the observed boundaries (yellow) in Figures 12 and 13.

4 Discussion

We have shown that the level set method with and without TPS error correction mod-

els the Troy ﬁre reasonably well. For time steps where the ﬁre growth remains stable,

error correction does improve estimates meaningfully. An adaptive scheme where the

−500 0 500 1000 1500 2000

−400

−200

0

200

400

600

800

1000

1200

1400

1600

Fig. 12. Bee ﬁre t= 60 minutes: FARSITE prediction (white), level set prediction (cyan), and

observed boundary (yellow).

−1000 −500 0 500 1000 1500 2000 2500

−500

0

500

1000

1500

2000

2500

3000

Fig. 13. Bee ﬁre t= 105 minutes: FARSITE prediction (white), level set prediction (cyan), TPS-

corrected level set prediction (green), and observed boundary (yellow).

decision to correct or not is based on degree of change in atmospheric, terrain, or tem-

perature factors may be desirable.

We also show that the more accurate approximation for the Bee ﬁre is the Rothermel-

based model. We believe this is largely due to the superiority of the FARSITE method

with detailed wind data to our implementation of the level set method with no wind

data.

Some of the most unpredictable behavior arises from newly formed bulges in the

ﬁre, so-called “ﬁre ﬁngers,” which are among the most dangerous of ﬁre behaviors.

Current work is underway to better model these localized events. Predicting when and

where ﬁre ﬁngers are likely to arise will be helpful, even if precise modeling of their

boundary evolution eludes us.

Our work demonstrates the challenges of applying historical error to correct current

prediction of ﬁre boundaries. Particularly during the early stages of a ﬁre, or for a ﬁre

with unusual physical constraints, the orientation and magnitude of boundary evolution

is rapidly changing. Sophistication equal to that desired in the original model of bound-

ary evolution is likely necessary to produce useful correction. Past error alone, at least

as globally measured, is not reliably informative. Future work will consider local error

measures.

Acknowledgements

The authors gratefully acknowledge Francis Fujioka for introducing us to the ﬁre mod-

eling problem, sharing data and predictions from the Bee ﬁre, and predictions for the

Troy ﬁre, Robert Tissell for sharing data from the Troy ﬁre, and the National Science

Foundation IIS-0954256 for funding our work.

References

1. Bookstein, F. Principal Warps: Thin-plate Splines and the Decomposition of Deformations.

IEEE Transactions in Pattern Analysis and Machine Intelligence, 14(2):239-256, 1992. 3

2. Chui, H., Rangarajan, A. A New Algorithm for Non-rigid Point Matching. IEEE Conference

on Computer Vision and Pattern Recognition, 2000. 2,3

3. Finney, M. FARSITE: Fire Area Simulator-Model, Development and Evaluation. Report

RMRS-RP-4, US Department of Agriculture Forest Service, Rocky Mountain Research Sta-

tion Paper, 1998. 1

4. Fujioka, F. A New Method for the Analysis of Fire Spread Modeling Errors. International

Journal of Wildland Fire, 11:193-203, 2002. 1,2

5. Mallet, V., Keyes, D., Fendell, F. Modeling Wildland Fire Propagation with Level Set Meth-

ods. Journal of Computers & Mathematics with Applications, 57(7): 1089-1101, 2009. 1

6. Rothermel, R. A Mathematical Model for Predicting Fire Spread in 17 Wildland Fuels. Re-

search Paper INT-115, US Department of Agriculture Forest Service, 1972. 1

7. Rochoux, M., Delmottea, B., Cuenot, B., Riccia, S., Trouv, A. Regional-scale simulations

of wildland ﬁre spread informed by real-time ﬂame front observations Proceedings of the

Combustion Institute, 34(2):2641-2647, 2013. 1,2

8. Sethian, J. Level Set Methods and Fast Marching Methods : Evolving Interfaces in Compu-

tational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge

University Press, 1999. 2