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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.
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