is estimated as in section 3.2.1 and the area is computed from the number of
grid cells for which the fire arrival time is less than or equal to the
discrete time step.

  2. 2.

If, additionally, infrared perimeter data is available, a subset of the points
in the perimeter are used in the same way as active fire detections to obtain
an estimated fire arrival time.

  3. 3.

When using artificial fire arrival times as the “ground truth" for
experiments, the area at each time step is computed from the number of grid
cells for which the fire arrival time is less than or equal to the discrete
time step.

  4. 4.

If a large number of infrared perimeter observations are available for a real-
world fire, time may discretized by the time of the observations and the area
at each time step computed as the area within the polygon defined by the
perimeter observation.

The growth rate of the actual fire and the model fire growth can be compared
visually to give a sense of how well the simulation matched the actual fire.
Figure 3.32 shows an example of such a comparison where the slope of the
curves depicting the size of the fire over time are roughly the same of the
first four days of the fire, but are very different after that time. This
divergence indicates that the model is failing to capture the behaviior of the
real fire after the fourth day. For testing purposes, when evaluating many
simulations or estimates of fire arrival times, for example, it is convenient
to compute a single number, the relative growth error

$RGE=\frac{||A_{e}-A_{g}||_{2}}{||A_{g}||_{2}}$ (3.3.1)

where $A_{e}$ is the estimated fire area at each discrete time step, $A_{g}$
is the “ground truth" fire area at the corresponding time step, and the norm
is the Euclidean. The $RGE$ gives a sense of how well the estimate or
simulation matches the real-world fire growth but does not give information
about whether the estimated growth rate was too high or too low.

Figure 3.32: Plot of estimated fire area over time. The blue line indicates
the size of the actual fire, estimated from satellite detection data. The red
line is the size of the modeled fire. It can be seen that the model does not
capture the explosive growth that occurred.

####  3.3.2 Measure of Effectiveness

The measure of effectiveness (MOE) as presented in Warner _et al._ (2004) is a
two-dimensional statistic that evaluates a model output with a comparison of
the known state of the system, describing the decreasing amount of either
false positives or false negatives in the forecast or estimate. In terms of
fire simulations, a false positive represents the model predicting the fire to
arrive at a location where no fire was actually observed. Conversely, a false
negative indicates a location where fire was actually observed but the model
indicated that location remained unburnt. It is computed as

$MOE=(x,y)=\left(\frac{A_{OV}}{A_{OB}},\frac{A_{OV}}{A_{PR}}\right)=\left(1-\frac{A_{FN}}{A_{OB}},1-\frac{A_{FP}}{A_{PR}}\right),$
(3.3.2)

where $A_{FN}$ and $A_{FP}$ given the area where false negatives and false
positives occurred, respectively. $A_{OB}$ and $A_{PR}$ are the areas of the
real-world observation and the model prediction, respectively, and $A_{OV}$ is
the overlap area, where the observation and prediction coincide. Figure 3.33
shows the relationship of these regions for a simulation that overestimated
the size of the fire and had a score $MOE=(x,y)=(0.899359,0.873599)$.
Importantly, the region of the observed fire is the union of all blue and
orange pixels and the region of the forecast fire is the union of all blue and
black pixels. If the forecast and observations coincided exactly, the
graphical representation would be a solid blue color. If the fire areas of the
forecast and observations were disjoint, there would be no blue in the figure
since all locations would indicate either a false positive or a false
negative.

The MOE gives a good sense of how well a simulation or estimation of a fire
arrival time matches the actual fire in terms of the locations where the fire
occurred at a single time, but it cannot give a sense of how well the
simulation evolved over the period of the event. When using the MOE to asses
how well a method works it is possible to consider separately the tendency to
produce false positive or false negative indications at locations in the fire
domain. For operational use, it might be preferable to have a model that makes
errors on the side of caution by producing more false positives than false
negatives.

Figure 3.33: Illustrating the measure of effectiveness (MOE). The MOE is
computed from the areas of false positives, false negatives, and overlap. The
overlap is the blue region where the model and real-world observations have
both indicated burning. The observed area of the actual fire is the union of
all blue and orange pixels. The area of the simulated fire is the union of the
blue and black pixels. In this example, the area of the modeled fire is close
to that of the observed fire.

####  3.3.3 Sørenson Index

The Sørenson index is another method of assessment made by comparison of
regions contained within a fire perimeter. In the case of a real fire example,
infrared fire perimeters as observed from overflying aircraft will be used to
compare against the region of an estimated fire arrival time. For artificial
data, perimeters are established by finding the location of all fire arrival
times before a specified time associated with a perimeter. This method of
evaluating simulations is algebraically equivalent to the method employed by
in Cardil _et al._ (2019a) and was proposed by Sørenson as measure of the
similarity between biological samples (Sørensen, 1948). A score $S_{a}$ is
assigned to each comparison of the forecast with the perimeter according to
the formula

$S_{a}=\frac{2\times A(forecast\cap perimeter)}{A(forecast)+A(perimeter),}$
(3.3.3)

where $A$ is the area of a region. Thus $A(observed)$ is the area of the fire
contained within the observed perimeter, $A(forecast)$ is the area enclosed
within the model forecast perimeter at the time the perimeter observation was
made, and $A(forecast\ \cap\ observed)$ gives the area of the intersection of
the forecast and observed perimeter areas. When the forecast and observed
perimeters encompass disjoint areas, we have $A(forecast\ \cap\ observed)=0$
so that $S_{a}=0.$ This is the lowest score which can be assigned. Conversely,
when the forecast and observed perimeters exactly coincide, then $2\times
A(forecast\ \cap\ observed)=A(forecast)+A(observed)$ so that $S_{a}=1$, the
maximum score possible.

In the case that there are $n$ perimeter observations during a simulation
period, the score $S_{a}$ can be computed as the average

$S_{a}=\frac{1}{n}\sum_{i=1}^{n}s_{i}.$ (3.3.4)

The Sørenson index is similar to the MOE in that is gives a sense of how well
a simulation or estimate of the fire arrival time compares with the actual
fire in terms of area, but it does not give a sense of the goodness of fit in
a temporal sense. The chief advantage of the Sørenson index is the simplicity
of the measure. The single number is easy to interpret since it measures how
well the forecast fire area matches that of the real fire but it does not
provide information on whether the predicted fire area was overestimated or
underestimated.

####  3.3.4 Relative Error

The relative error (RE) is the norm of the difference between the estimated
and actual fire arrival times divided by the norm of the actual fire arrival
time. This score gives a good sense of how well the simulation or estimate of
the fire arrival time was able to match the data. This method of assessment
was used for tests using artificial fire scenarios to validate the methods
being developed. Since the fire arrival times of real fires are generally not
known, this method of assessment is not used when working with real fire
scenarios. The relative error is calculated as

$RE=\frac{||T-T_{e}||}{||T||}$ (3.3.5)

where $T$ is the known fire arrival time, recorded as days since simulation
start, and $T_{e}$ is an estimate derived from observational data.

####  3.3.5 Rate of Spread Error

This assessment score seeks to measure how well the average ROS in a forecast
or estimate of fire arrival time compares with a known rate of spread. The
ultimate intention is to be able to use this score to make a judgment about
the accuracy of the burn model in SFIRE and improve forecasts by adjusting a
key model parameter, the fuel moisture content (FMC). Because the total FMC is
dependent on the underlying fuels in the fire domain, comparison between the
forecast ROS and the known ROS are compared at each grid point in the fire
domain where the forecast and the data both agree that fire had occurred.
Thus, the score is computed only in the “overlap region" that is used to
compute the MOE and corresponds to the blue region seen in Figure 3.33.

The ROS is computed using the Matlab function “gradientm" (MathWorks, 2021
(accessed March 10, 2021)) that uses the latitude and longitude of a location
and a scalar field, in this case the fire arrival time, to compute the
gradient with units in meters and seconds. Since $T=T(x,y)$, the gradient of
the fire arrival time gives the reciprocal of the ROS, and is, in fact, the
Eikonal equation

$\left\|\nabla T\right\|=\frac{1}{R}.$ (3.3.6)

The ROS at a location can be found by the reciprocal of the gradient’s norm.

Although the ROS is computed from the fire arrival time $T=T(x,y)$ the rate of
spread is more complicated than a function of only position. Other inputs such
a fuel properties, weather, and terrain slope play a role in determining the
ROS. Indeed, since the weather varies with time, the ROS is partially a
function of the fire arrival time itself. Thus, making adjustments to the
model based on an estimate of the ROS computed only from position is
problematic.

####  3.3.6 Spread Direction Error

When the ROS is computed from the gradient of the fire arrival time, the
partial derivatives $(T_{x},T_{y})=\nabla T$ allow for computation of the
direction of the fire spread. The arctangent function is used to compute an
angle

$\theta=\arctan\frac{T_{y}}{T_{x}}$ (3.3.7)

that describes the direction of the gradient in radians counterclockwise from
the direction east.

###  3.4 Examples and Testing the Method

This section investigates how the method for estimating the fire arrival time
works and shows some of the tests that were performed to find the best
parameters that control how different parts of the method are accomplished.
Because the fire arrival time for real fires is usually not known, we use the
strategy of constructing some artificial fire arrival time and artificial data
so that a “ground truth," known solution to a problem exists to be compared
against the estimate produced by the proposed method.

Artificial fire arrival times were constructed by a process that first created
several cones with a common vertex and height, but varied in the width of the
cone at the top. The cones were then combined by taking the maximum value
among all the cones at all points in the domain. For example, if the cones
could be determined by the functions $f_{1}(x,y),f_{2}(x,y),f_{3}(x,y)$, the
artificial fire arrival time $T(i,j)$ would be determined at location $(i,j)$
in the fire domain by

$T(i,j)=\max\\{f_{1}(x,y),f_{2}(x,y),f_{3}(x,y)\\}$ (3.4.1)

Artificial fire detections were randomly scattered on the surface of the
artificial fire arrival time comes at locations where the fire arrival time
was less than the time corresponding to the “flat top" of the cone. A random
number was generated for each potential location and if that number was below
a chosen threshold, an artificial fire detection was created at that point. In
the experiments that follow, roughly 5% of the potential detection locations
were assigned an artificial detection time. The top row of panels in Figure
H.14 shows a sampling of some these artificial data. The random process
produced a number of different shapes of the artificial fires as can be seen.
Some shapes were similar to simple cones, but others can be seen as disjoint
regions implying the presence of more than a single fire in the domain. Figure
H.6 shows one such example.

####  3.4.1 Testing the Method

##### Choosing the Number of Clusters

When clustering data, the number of clusters $k$ must be chosen by the user.
There exist algorithms to find an optimal number $k$ by seeking to minimize
the squared distances from cluster members to the centroid of the cluster.
Those algorithms have not been used in this research. For the purpose at hand,
the optimal number of clusters to use will be that that allows the method for
estimating the fire arrival time from satellite detections to be most
accurate.

To find an optimal number of clusters to use, the method for estimating the
fire arrival time was repeatedly employed, using a varying number of clusters
and the estimate produced was compared with the “ground truth". Directed
graphs and sets of shortest paths using between one and forty clusters were
used. For each cluster size, the norm of the difference between the “ground
truth" and the estimated fire arrival time was computed. Figure 3.34 shows the
results of varying the number of clusters in a single example case. In
general, when the cluster size reaches approximately 15, the error reaches a
plateau.

This test was repeated for several different “ground truth" scenarios,
involving at least 100 points of data used as artificial detection locations
and the results were similar. For that reason 20 clusters has been used as a
default value for the work. There are a few exceptional cases where using a
different number of clusters would make sense. If the number of fire
detections is small, the number of clusters should be reduced. Indeed, if the
number of fire detections is less than the number of clusters, the k-means
algorithm cannot complete. If the number of detections equals the number of
clusters, then each detection will belong to its own cluster and the purpose
of clustering the data will not be fulfilled. For large fires, involving many
thousands of fire detections, a larger number could be used although that has
not been tested in the current work.

Figure 3.34: The effect of cluster size on the estimated fire arrival time of
a single example case produced from graphs derived using various numbers of
clusters. The horizontal axis shows the numbers of clusters used to partition
the detection data and the vertical axis shows the norm of the difference
between the “ground truth" fire arrival time $T$ and the estimated fire
arrival time $T_{a}$ derived from the detection data. The norm of the
difference decreases generally when more clusters are used, but then reaches a
plateau when approximately 15 clusters are used. Other example cases showed
similar characteristics.

##### Testing with Artificial Fire Arrival Time and Data

Because the fire arrival time of real-world fire is not known, to test the
method we first begin with construction of artificial fire arrival times and
artificial fire detection data. The method for estimation fire arrival time is
then employed and the resulting estimated fire arrival time is compared with
the artificial fire arrival time used as “ground truth." In the first example,
hundreds of artificial fire arrival time cones were generated, along with
artificial detection data, and the interpolation method was performed using
different parameters. The assessment methods outlined above were then used to
find those parameters that gave better results.

The purpose of the experiment was to determine the optimal way to estimate the
fire arrival time from only satellite data. In particular, choosing whether to
use a multigrid approach or to work on a single grid was a goal. A secondary
consideration was whether to interpolate extra points along the shortest paths
in the directed graph and how many extra points to use. The conclusion of the
test was that using the multigrid approach with extra data points interpolated
along the paths at a distance of 2000 meters was the the best method.

In the experiment, 280 artificial fire arrival times were created. Artificial
active fire detections were were randomly scattered across the fire arrival
surface in such a way that approximately 5% of the grid cells within the outer
perimeter would have an active fire detection. When the locations of the fire
detections were determined, the time associated with each was adjusted upwards
to a discrete set of times to mimic the way that real satellite data is
received in granules.

The assessment of the estimated fire arrival times was done in two categories

  1. 1.

Comparison of the fire arrival time in terms of its growth, final size and
shape, as well as a relative error.

  2. 2.

Comparison of the rate of spread and the direction of the of the fire spread.

Acronyms for the assessment scores in first category described above.

  * •

AGE - Average growth error.

  * •

MRE - Mean relative error.

  * •

SRE - Standard deviation of relative errors

  * •

MOE X - $x$-component of the measure of effectiveness. A higher score indicate
less false negatives.

  * •

MOE Y - $y$-component of the measure of effectiveness. A higher score
indicates less false positives.

  * •

||MOE|| - Norm of the the measure of effectiveness

  * •

Sørenson index

Acronyms for the assessment scores described above

  * •

MRD - Mean ROS difference

  * •

SRD - Standard deviation of the differences in ROS

  * •

MDD - Mean direction of spread difference

  * •

SDD - Standard deviation of the differences in the the direction of spread

Several different scenarios were tested. The goal was to find whether a
multigrid approach was useful and to determine how interpolation of extra
points along the paths in the graph affects the iterative interpolation
process. This secondary goal was accomplished by making estimates of the fire
arrival time without using interpolation along the paths as well as using
extra points at various different spacings. Spacing is taken to be the
distance along the path between two original detection data. Extra points were
interpolated along the paths so that the distance between any two points on
the path was no greater than the spacing. For example, if the 1000 meter
spacing was used but all detections on a path were 500 meters from their
neighbors, no extra points would be interpolated. When the spacing indicated
interpolation of extra points was to be made, three extra points were added to
the path between the adjacent, original detections.

The following list describes them various strategies for estimating the fire
arrival time used to test the method.

  1. 1.

Multi - NP: Multigrid with no extra points interpolated

  2. 2.

Multi - 500: Multigrid with extra points spaced at 500 meters

  3. 3.

Multi - 1000: Multigrid with extra points spaced at 1000 meters

  4. 4.

Multi - 2000: Multi grid with extra points spaced at 2000 meters

  5. 5.

Multi - 3000: Multigrid with extra points spaced at 3000 meters

  6. 6.

Multi - 4000: Multigrid with extra points spaced at 4000 meters

  7. 7.

Single - NP: Single grid used with no extra points interpolated

  8. 8.

Single - 1000: Single grid used with extra points spaced at 1000 meters

Table H.1 shows the results of this test relating the estimate to the “ground
truth" fire arrival time in terms of its growth, final size and shape, as well
as a relative error. The rows of the table indicate various strategies for
estimating the fire arrival time used to test the method. The columns of the
table give the average test score for the estimated fire arrival time for all
280 artificial fire data scenarios created. The columns “Best" and “Worst"
correspond to the specific artifical data scenario for which the strategy had
the highest or lowest average of all test scores, respectively. At a glance,
it can be seen that the mean relative error (MRE) is found to be lower when
using the multigrid approach. Table 3.2 ranks the averages of the scores
according to which strategy produced the best results, with a lower rank
indicating a better score. The column “Rank Sum" adds up the test rankings for
each strategy and the column “Rank" rank shows these sums ordered by
increasing values. The lowest value of this final rank corresponds to the
strategy for estimating the fire arrival time by used of the multigrid
approach with additional points interpolated along the paths with a spacing of
2000 meters.

Scenrario | AGE | MRE | SRE | MOE X | MOE Y | ||MOE|| | Sorenson | Rank sum | Rank
---|---|---|---|---|---|---|---|---|---
Multi - NP | 2 | 6 | 6 | 1 | 6 | 5 | 5 | 31 | 4
Multi - 500 | 7 | 1 | 4 | 8 | 1 | 8 | 8 | 36 | 6
Multi - 1000 | 3 | 3 | 1 | 5 | 5 | 2 | 2 | 21 | 2
Multi - 2000 | 1 | 2 | 5 | 4 | 2 | 1 | 1 | 16 | 1
Multi - 3000 | 4 | 4 | 2 | 6 | 3 | 3 | 3 | 25 | 3
Multi - 4000 | 6 | 5 | 3 | 7 | 4 | 4 | 4 | 32 | 5
Single - NP | 8 | 7 | 7 | 2 | 8 | 6 | 7 | 44 | 7
Single -1000 | 5 | 8 | 8 | 3 | 7 | 7 | 6 | 44 | 8
Table 3.2: Ranking of the test results for estimating the fire arrival time
from artificial data in terms of growth, final size and shape, as well as
relative error. All of the 280 results of the experiment for estimating the
fire arrival time from data are compiled into a single table. The best method
for estimating fire arrival time from data was to use the multigrid method
with interpolation of extra points along the paths at a distance of 2000
meters. This table summarizes and ranks the results in Table H.1.

Table H.2 gives the results of this test in terms of the rate of spread and
the direction of the of the fire spread. The rows of the table indicate
various strategies for estimating the fire arrival time used to test the
method. The columns of the table give the average test score for the estimated
fire arrival time for all 280 artificial fire data scenarios created. The
second and third columns give the mean and standard deviation of the
differences in the average ROS for each fire arrival time. The fourth and
fifth columns give the mean and standard deviation of the differences in the
angle of the gradient for each fire arrival time. It can be seen that the
magnitude of the mean ROS difference (MRD) is lowest for the strategy using
the multigrid approach and spacing of interpolated data points at 2000 meters.
Table 3.3 ranks the average scores in the same was was done for the first part
of the test. Again, it was found that using the multigrid approach with 2000
meter spacing of extra points produced the best results.

Scenario | MRD | SRD | MDD | SDD | Rank sum | Rank
---|---|---|---|---|---|---
Multi - NP | 4 | 6 | 2 | 5 | 17 | 3
Multi - 500 | 2 | 4 | 3 | 4 | 13 | 2
Multi - 1000 | 3 | 5 | 6 | 3 | 17 | 4
Multi - 2000 | 1 | 3 | 1 | 6 | 11 | 1
Multi - 3000 | 5 | 8 | 5 | 2 | 20 | 5
Multi - 4000 | 6 | 7 | 7 | 1 | 21 | 7
Single - NP | 8 | 1 | 4 | 7 | 20 | 6
Single - 1000 | 7 | 2 | 8 | 8 | 25 | 8
Table 3.3: Ranking the estimation approaches by how well they were able to
estimate the average ROS in the fire and how well the direction of the
gradients matched that of the “ground truth" fire arrival time. The best
strategy was to use the multigrid approach with extra points interpolated
along the paths at a 2000 meter spacing. This table summarizes and ranks the
results in Table H.2.

Finally, the conclusions from the first and second parts of this test,
referred to as “Area Tests" and “ROS tests," respectively, are combined in
Table 3.4. The right column ranks the strategies for estimating the fire
arrival time. The multigrid strategy using interpolation with a 2000 m spacing
was found to produced the best results for this experiment using artificial
data. Interestingly, not using the multigrid approach produced the worst
results, even when extra points were interpolated along the paths.

Scenario | Area Tests | ROS Tests | Total | Rank
---|---|---|---|---
Multi - NP | 31 | 17 | 48 | 5
Multi - 500 | 36 | 13 | 49 | 4
Multi - 1000 | 21 | 17 | 38 | 2
Multi - 2000 | 16 | 11 | 27 | 1
Multi - 3000 | 25 | 20 | 45 | 3
Multi - 4000 | 32 | 21 | 53 | 6
Single - NP | 44 | 20 | 64 | 7
Single - 1000 | 44 | 25 | 69 | 8
Table 3.4: Table for all ranks for the experiment using artificial fire
arrival times and fire detections. This table summarizes all results from
Table H.1 and Table H.2. Using a multigrid approach with interpolation of
extra points with a spacing of 2000 meters provided the best estimates of the
fire arrival time.

##### WRF-SFIRE Output Files Used as Ground Truth

In this test, 115 fire arrival times output from WRF-SFIRE were used to make
“ground truth" fire arrival times and artificial detections were scattered on
the surface in the same way they were in the previous test, around 5% of the
possible locations had an artificial fire detection. Figure H.11 shows the
shape of 6 of these fire arrival times and the locations of the artificial
detections. In comparison with the artificial fire arrival times used in the
previous experiment, the fire arrival times from WRF-SFIRE output have rougher
outer perimeters, but none indicate that more than a single fire was burning
in the domain.

These simulations play the role of cross validation in the experiment, with
the goal to determine that the methods developed do not result in an
overfitting of data when applied to a different set of inputs. The testing
method used was exactly the same as that producing the best results in the
experiment using 280 artificial fire arrival times. Only the multigrid method
with interpolation of extra points along the paths with spacing of 2000 meters
was used in this test. The results of the tests, in Table 3.5, relating to
estimated fire arrival time in terms of its growth, final size and shape, as
well as the relative error the were similar to the tests using artificial fire
arrival times, with the exception that the average growth error was worse.
Table 3.6 gives results for the tests relating to the ROS and the direction of
the fire spread. In this case, we find that the difference in the average ROS
is higher than it was for the tests using 280 artificial fire arrival times.

The results in these tables represent averages of the scores for all 115
testing scenarios. To get a better sense of the associated variances of these
statistics, Figure H.8 shows a scatter plot of the MOE, with MOE X on the
horizontal axis and MOE Y on the vertical. Figure H.9 and Figure H.10 show
histograms of the relative errors and the differences in average ROS for all
115 tests.

Scenrario | AGE | MRE | SRE | MOE X | MOE Y | ||MOE|| | Sorenson | Best | Worst
---|---|---|---|---|---|---|---|---|---
Multi - 2000 | 0.2690 | 0.0056 | 0.0015 | 0.9475 | 0.6453 | 1.15 | 0.7621 | 111 | 99
Table 3.5: Results from estimating the fire arrival time using the multigrid approach with additional points interpolated along the paths at a distance of 2000 meters. The scores relate to the growth, final size and shape, as well as relative error of the estimated fire arrival time. For this test, the “ground truth" fire arrival time for each of the 115 estimates made was taken to be the output fire arrival time from a fire simulation made with the WRF-SFIRE model. Compare with the results in Table H.1 where comparison was made with artificial fire arrival times. The results are similar, with the exception of the average growth error score. The results for this score are worse.  Scenario | MRD | SRD | MDD | SDD | Best ROS | Worst ROS | Best $\theta$ | Worst $\theta$
---|---|---|---|---|---|---|---|---
Multi - 2000 | 0.1546 | 0.1027 | -0.0451 | 0.1463 | 111 | 6 | 61 | 90
Table 3.6: Results from estimating the fire arrival time using the multigrid
approach with additional points interpolated along the paths at a distance of
2000 meters. The scores relate to the ROS and direction of fire spread in
estimated fire arrival time. For this test, the “ground truth" fire arrival
time for each of the 115 estimates made was taken to be the output fire
arrival time from a fire simulation made with the WRF-SFIRE model. Compare the
results to Table H.2. In this test, the difference in the estimated average
ROS was much worse that when using artificial fire arrival times as the
“ground truth."

####  3.4.2 Examples and Testing with Real Fires

##### Finding the Parameter $p$

To determine the optimal value $p$ in the interpolation of new data points
along the paths (Equation 3.2.6), a test was devised. As a first step,
estimates of the fire arrival time were made for set of the artificial fire
arrival times used in the experiments of Section 3.4.1 using various values of
$p$. For each set of detection data, 11 estimates of the fire arrival time
were made using values of $p$ ranging from 0.0 to 1.0 in increments of 0.1.
All estimates were made on a common computational grid without using a
multigrid approach to save time. Each fire estimated arrival time was then
compared against the artificial “ground truth" fire arrival time and the
relative error, MOE, and the Sørenson index were computed. The average of each
score for all tested values of $p$ was computed and the results plotted in
Figure 3.35. The MOE and Sørenson index scores generally increase with $p$,
except for MOE Y which exhibited a decease for $p=1.0$. The scores for the
relative error generally decreased, indicating a better performance, when $p$
was increased, but and increase occurred for $p=1.0$. Thus, it was determined
that the optimal value is $p=0.9$.

Figure 3.35: Assessment of estimated fire arrival times of 280 artificial fire
arrival times using varied values of the interpolation parameter $p$. Each
panel represents the average score of all 280 estimates for varying values of
$p$. On the top row, from left to right are the plots of MOE X and MOE Y. On
the bottom are the plots of relative error, and the Sørenson index. These
plots suggests that using $p=0.9$ is optimal.

Further investigation of the optimal value $p$ to use was accomplished by
making estimates of the fire arrival times of real-world fires. Estimates of
the fire arrival time for the the Patch Springs Fires and the Cougar Creek
Fire were made using varying values of $p$ between 0.0 and 1.0. For each value
used, the resulting fire arrival time was compared to an infrared perimeter
observation and the MOE and Sørenson index were computed. The results are
summarized in the tables and figures that follow. Tables H.3 and H.4 show the
results which are summarized in figures H.12 and H.13. The effect of $p$ in
these individual cases was similar to that of the average effect determined in
280 cases using artificial data. The scores for the estimates generally
increased with $p$ but showed a decrease for $p=1.0$. In both specific cases,
using $p=0.9$ for the parameter was found to be the best, although results
were similar for $p=1.0$. These results closely match those found when using
the suite of artificial fire arrival times in the testing previously detailed.

##### Real Fire Example: Patch Springs Fire with Perimeter Data

The Patch Springs Fire occurred in August 2013. In this section, we use the
method of Section 3.2.1 to create an estimate of the fire arrival time over
the first six days of the event. The estimated fire arrival time used two
infrared perimeters and 39 granules of satellite data. For each perimeter
observation, 200 fire data points were used. Although perimeter observations
typically contain 1000 points or more, the larger grid spacing using for the
interpolation technique makes use of more than a few hundred points
unproductive. Figure 3.36 shows the clustering of the data, the graph
structure with shortest paths, and the resulting estimate of the fire arrival
time. Comparison of the estimated fire arrival time was made with a perimeter
observation from August 16 at 09:47 UTC. When compared with the estimated fire
arrival times made without perimeter observations, in Table H.3, the
simulation here is marginally better. The results are summarized in Table 3.7.
Without a known fire arrival time, only the MOE and Sørenson index have been
used to evaluate the reults. For this test, the results are consistent with
the experiments in Section 8.2.1, indicating that the method of using infrared
perimeters in addition to satellite fire detections is a useful strategy for
estimating the fire arrival time of a real-world fire.

MOE X | MOE Y | ||MOE|| | Sorenson
---|---|---|---
0.9232 | 0.8550 | 1.2858 | 0.9089
Table 3.7: Assessment of the estimated fire arrival time of the Patch Springs
fire made using infrared perimeter data and satellite data. The addition of
the extra data source has made the estimate better than any of the estimates
in Table H.3.

Figure 3.36: Estimation of fire arrival time for the Patch Sprigs fire using
infrared perimeter observations. In the upper left, active fire detection data
and points from infrared perimeter observations used for estimation of the
fire arrival time of the Patch Springs fire. In upper right, the shortest
paths in the directed graph for the Patch Fire simulation, initialized from
satellite data and infrared perimeter data. At the bottom, the estimated fire
arrival time. 200 data points were used in each perimeter. The perimeters are
visible as the rings with large amounts of data at approximately day 2.5 and
day 3 of the estimate. Figure 3.37: Assessment of the estimated fire arrival
time for the Patch Springs fire using infrared perimeter observations as a
data source. Two perimeter observations were used as a data source and the the
estimate was compared to an infrared perimeter from August 16 at 09:47 UTC.
The addition of the perimeter data made for a small improvement of the
estimate in comparison with the estimates from Section 3.4.2 that only used
satellite data.

##### Real Fire Example: Cougar Creek Fire

In some cases, real fires are intensely observed and many infrared perimeter
observations are made available. In this section, many estimates of the fire
arrival time for the Cougar Creek Fire were made and evaluated by comparison
with an infrared perimeter observation. Both a single-pass and the multigrid
method with interpolation of points with 2000 meter spacing were used. In each
estimate, all satellite fire detection data up to the time of a perimeter
observation was used to estimate the fire arrival time. In total, 27
perimeters were used to form comparisons. For each estimate, the MOE and the
Sørenson index were computed. Additionally, the area within the perimeters of
the observation and estimate were made at each of the 27 times. All result
computations were made on grids of 250 meter resolution. This grid spacing is
well below the resolution of the typical perimeter observation but above the
resolution of the fire detection data.

Given a sequence of infrared observations, its is possible to get an estimate
of the growth rate of the real fire by computing the area within each
perimeter observation. The area within these perimeters is plotted along with
the area found by estimation using satellite data for both the single-pas and
multigrid approaches Figure 3.38. The blue lines indicate area of the fire
estimated from data and the red lines indicate the area within the infrared
perimeters. Both estimation strategies show similar growth to that indicated
by the infrared data, but the single-pass method tends underestimate the area
and the multigrid approach tends to overestimate it. Figure 3.39 shows a
comparison of the assessment of the tow methods for an estimate of the fire
arrival time at August 14, 07:00:00 UTC. At this time step, both strategies
both strategies had similar results, with the MOE and Sørenson index scores
differing less than 0.05. In both cases, the strategies slightly
underestimated the size of the fire. Figure H.18 shows the final estimates of
the sequence. Again, both strategies delivered similar results, but the
multigrid method gives less false negatives and more false positives.

Figure 3.38: Comparison of fire sizes from fire arrival time estimates of the
Cougar Creek Fire using a one-pass method (left) and multigrid method (right).
The one-pass method underestimated the size of the fire while the multigrid
method overestimated it.

Figure 3.39: Assessing the estimated fire arrival time of the Cougar Creek
Fire using a one-pass method (left) and multigrid method (right) by comparison
with an infrared perimeter observation. The estimates are derived for the
available satellite data up to the perimeter time of August 14, 07:00:00 UTC.
Both methods ted to underestimate the size of the fire. In this case, some
satellite fire detections lie outside the infrared perimeter represented by
the union of the orange and blue areas.

Table H.5 and Table H.6 show the results of assessment of the estimated fire
arrival times using the single-pass and multigrid strategies, respectively. In
general, both strategies had similar results with the typical score for both
components of the MOE and the Sørenson index remaining above 0.80. Some lesser
scores were recorded in the early stages of the fire and can most likely be
attributed to the a relatively small amount of satellite data data being
available. For example, the first estimate had the score $S=0.6082$ using the
single-pass strategy. The corresponding score for the multigrid strategy was
also low, $S=0.6704.$ Scores for later estimates remained largely constant,
implying that once sufficient satellite data were available, both strategies
produced acceptable results. Table 3.8 compares the average scores made by
both strategies across all 27 estimated fire arrival times. The two strategies
produced similar results. The largest differences exit in the scores for MOE X
and MOE Y. The largest score for MOE X was achieved by the multigrid strategy,
indicating it produced less false negatives. The single-pass strategy produced
less false positives.

Method | MOE_X | MOE_Y | |MOE| | S
---|---|---|---|---
One Pass | 0.8267 | 0.9051 | 1.2283 | 0.8607
Multigrid | 0.8898 | 0.8457 | 1.2299 | 0.8640
Table 3.8: Comparison of the average assessment scores for the one pass method
and multigrid method used to estimate the fire arrival time of the Cougar
Creek Fire. The scores are very similar, with the multigrid showing a slight
advantage in having a final burn area closer to that indicated by the infrared
perimeter observations.

##### Real Fire Example: Camp Fire

The Camp Fire was a large and destructive fire that took place in Califormia
during November, 2018. Extreme weather cause the fire to expand rapidly
(Brewer and Clements, 2020) and it is thought that the spread mechanism was
largely due to burning embers being carried by winds in advance of the main
fire front (Cal Fire, 2018 (Accessed March 3, 2020)). It is estimated that the
fire grew to 28,000 ha in the first 24 hours. Figure 3.41 shows a scatter plot
of the clustering of the satellite data and the structure of shortest paths
from an assumed ignition point. The “flat bottom" of this graph is further
evidence of the explosive growth of this fire in its early stages. Five
infrared observations were made during the first two days of this fire and
estimates of the fire arrival time were made to compare with each of them.
Figure 3.40 shows how the area of the estimates compared with the are with in
the infrared perimeters. In general, the estimated area was smaller, but the
slope of the two curves are similar, indicating rates of growth that are
largely the same. Figure H.22 shows comparison of the estimated fire
perimeters with the infrared perimeters at four times. The estimated
perimeters differ from the infrared observations mostly in the southwest
region of the fire that active fire front during the first two days. Figure
3.42 shows the assessment of the estimate made by comparison to the final
infrared perimeter observation made on November 10, at 09:04 UTC. In the
figure, the area of in the southwest showing false negatives may be due to the
spread mechanism of the fire. Winds from the northeast carried burning embers
ahead of the main fire front, causing many small spotting fires to occur.
These spotting fires may have been observed by infrared camera onboard
aircraft but not by satellites, given the region of false negatives at the
edge of the fire front. The scores for this estimate were high, with both
components of the MOE and the Sørenson index all exceeding 0.90. These scores
were as good as those for estimates of both the Patch Springs and Cougar Creek
fire previously discussed.

Figure 3.40: Growth of the Camp fire during the first several days. The growth
rate of the area was 28,000 ha/day and the estimated growth rate was 30,000
ha/day. The estimated are of the fire was smaller than that observed, but the
rate of growth was very similar. Interestingly, the computed area within the
infrared perimeters showed a small decrease after 1.5 days.

Figure 3.41: Clustered data for the Camp Fire (left) and the shortest paths in
the directed graph (right). Note how the low angles of the paths near the
bottom of the graph indicate the high ROS observed in the explosive early
hours of this fire. Figure 3.42: Assessment of the estimated fire arrival time
of the Camp Fire using the MOE and the Sørenson index. The fire started in the
northeast region of the domain and progressed rapidly towards the southwest.
The union of the blue and orange area indicate the region within the infrared
perimeter.

####  3.4.3 Limitations of the Method

The method currently cannot give good estimates of the fire arrival time under
all circumstances. Some considerations are outlined in this section.

##### More than one fire in the domain

On some occasions it can happen that more than one fire is present in the
simulation domain but only one fire is being modeled. With very large
simulation domains during fire season this is more likely. This presents
problems to the method since the fire detections of both fires will be treated
as belonging to a single fire with a unique ignition point. During the process
of constructing the set of shortest paths from the ignition point to all the
other fire detections in the domain, paths will be constructed to connect the
two distinct fires. When more points are interpolated along these paths, the
method will then be assigning fire arrival times to areas that have not
burned. There are several options to avoid this kind of problems.

  1. 1.

Attempt to detect distinct fires and ignore detections not belonging to the
fire not being simulated. This has been explored in the process of the current
research. Presently, after construction of the distance matrix, the active
fire detections are clustered into two groups with k-means clustering. If the
number of detections in one group is significantly smaller than the other, the
distance between each detection in the smaller cluster and all other
detections in the entire fire domain is set to zero. This has the effect of
“disconnecting" these detections from all others when the directed graph is
constructed. Figure 3.43 shows a fire domain with two distinct fires inside it
but the detections in the smaller cluster will not be considered as part of
the fire being studied.

  2. 2.

Eliminate connections between fire detections too far apart. This method
attempts to exclude detections outside of the fire being simulated by imposing
a “speed limit" related to the path segements that could connect to fire
detections in a directed graph. After the distance matrix is constructed, the
ROS between any two possible connected detections can be computed by dividing
the distance between them by the time the observations were made. Some
threshold value can be chosen so that if the ROS between two detections is
above the threshold value, the distance between them will then be set to zero,
eliminating the possibility of a connection between them in the directed
graph.

These methods can work with a relatively small fire domain with just one or
two fires inside it, but will not work on larger domains with more distinct
fires inside. A first step to handling the problem would be devise a technique
to identify individual fire complexes using the satellite active fire
detections. If this could be done, then the methods developed in this research
could be applied to the identified individual fires all at the same time. With
parallel computing resources being used, it would be possible to work with all
of the fires in a large region simultaneously.

Figure 3.43: Two distinct fires in a single simulation domain. In total, there
are 361 active fire detections in the fire domain. The active fire detections
were first clustered into two sets using k-means clustering and then the
members of the smaller cluster were automatically removed from the directed
graph used to make the paths seen in the larger cluster.

##### Size of the Fire

The size of the fire is an important consideration. Large fires that have
burned for a long time present a challenge because the amount of data can be
quite large. For example, construction of the distance matrices used for
making the path structure is expensive. This limitation could be overcome by
only estimating the last several days of the fire arrival time for large,
lengthy fires. Conversely, small fires may be easier to work with
computationally but the sparsity of data is problematic. If only a small
number of active fire detections are present, the method of spatially
clustering the detection locations may fail.

##### Ignition Point Estimation

In some cases the estimation of an ignition point is difficult. This is likely
to occur for fires that have an ignition followed by explosive growth, as was
the case with the Camp Fire. In such cases, averaging the locations of the
fire set of detections is likely to lead to an inaccurate ignition location
estimate since the winds responsible for the explosive growth would indicate
the location was near the edge of the region containing the detections used in
the estimate. The method for estimating the ignition point by grid search to
find the point giving the maximum data likelihood may not work with such cases
because the WRF-SFIRE model is used to run an ensemble of short-term
forecasts, but the spread mechanism of such a fire might be poorly modeled by
WRF-SFIRE.

###  3.5 Uncertianty in the Estimation of The Rate of Spread

The path structure constructed as part of the method for estimating the fire
arrival time serves as a convenient tool for estimating ROS at many locations
within the fire domain. Under the assumption that fire has spread in a
straight line between detection locations along a path in the structure,
computation of the ROS is a simple matter of dividing the distance between the
points by the difference of the times at which the locations were observed to
be burning. However, as previous discussed in Section 2.2, the actual location
of fire on the ground is uncertain. Likewise, the time that fire arrived at
the locations cannot be known precisely. Thus, the estimated ROS derived from
satellite observations should be treated as a random variable. Some of the
assumptions and uncertainties when working the estimated ROS are listed below.

  * •

Geolocation errors. The geolocation error depends on the scan angle by which
the satellite observed the pixel. Higher angles correspond to larger possible
geolocation errors (Nishihama, 1997). It will be assumed that the geolocation
error for any observation follows a normal distribution with a standard
deviation of 125 meters. This corresponds to an observation made with a scan
angle of approximately 35 degrees.

  * •

The geolocation error is more complicated than being assumed. The variance
depends on the scan angle, but the components of this error are different in
the track direction (parallel to the flight path) and the across track
direction (roughly perpendicular to the flight path.) At nadir view (zero scan
angle), the region containing the 3$\sigma$ error is roughly circular. As scan
angle increases towards a maximum of $55^{\circ}$ the region containing the
3$\sigma$ becomes more elongated in the across track direction.

  * •

It will be assumed that the uncertainty in the fire arrival time at any active
fire detection will be uniformly distributed over the previous 12 hours. This
assumption follows from the probability of satellite detection seen in Figure
2.7. The probability of detection is almost 100% for many hours after the fire
has arrived at a location but there is no reason to assume any time during
this period is more likely than another.

  * •

In principle, the length of period for which the detection probability is high
depends on factors not being accounted for. For example, fire in a dry, grassy
region would be detectable for a relatively short time if windy conditions
were present because the fuels would be consumed quickly. Fires moving slowly
in regions with wetter fuels my be detectable for longer periods.

####  3.5.1 Estimation of Uncertainty for Points Along the Paths in the Graph

With shortest paths connecting an assumed ignition point to active fire
detections in the fire domain, an estimate of the rate of spread (ROS) between
two can be calculated by dividing the distance between them by the difference
in the times the observations were recorded. The calculated ROS is subject to
error because the location of the fire detections always have an associated
geolocation error. Further, even if there was no geolocation error involved,
the time the fire arrived at any particular location cannot be known
precisely. If a fire is reported at a position during the satellite overpass,
there is no way to determine if the fire has just arrived there or if it has
been there for many hours. Figure 3.44 shows how the distance between two
satellite observations will be handled. In the figure, the orange stars
represent locations of active fire detections. The dashed lines around the
detections represent the boundaries of 3$\sigma$ geolocation. Panel (a) shows
two bivariate normal distributions with axes aligned according to the flight
path of the observation platform. Panel (b) shows an approximation obtained by
assuming equal variance in the across-track and along-track directions of each
detection. Panel (c) shows the use of symmetry to reduce the problem to one
dimension, resulting in reduced complexity by working with two normally
distributed random variables.

Figure 3.44:  Visualizing and simplifying the the uncertainty in the distance
between two active fire detections.

Since the ROS between two active fire detections is always computed from
observations made at different times (and possibly by different platforms) it
will be assumed that the two observations are independent. The distance
between the points on the ground can then be considered a random variable.
Similarly, the difference in the time between observations will be considered
a random variable derived from the difference of two iid uniform random
variables. The ROS we will denote $R$ and the distance and time difference
between observations will be denoted $D$ and $T$, respectively.

$R=D/T$ (3.5.1)

To simplify the mathematics and make an analytic variance possible to compute,
the assumptions shown in Figure 3.44 are used to orient two fire detections in
a way that the distance between them can be expressed as the difference of
Gaussian random variables. The first detection is placed at the origin and has
distribution $N(0,\sigma_{1}^{2})$. The second is along the $x$-axis at
location $(d,0)$ with distribution $N(d,\sigma_{2}^{2})$. Thus the pdfs are

$f_{1}(x)=\frac{1}{\sqrt{2\pi}\sigma_{1}}\exp\left(\frac{-x^{2}}{2\sigma_{1}^{2}}\right)\hskip
14.22636ptf_{2}(x)=\frac{1}{\sqrt{2\pi}\sigma_{2}}\exp\left(\frac{-(x-d)^{2}}{2\sigma_{2}^{2}}\right).$
(3.5.2)

The pdf of the distance between these points is computed by

$f(z)=\int_{-\infty}^{\infty}f_{1}(x-z)f_{2}(x)dx=\frac{1}{\sqrt{2\pi(\sigma_{1}^{2}+\sigma_{2}^{2})}}\exp\left(\frac{-(d-z)^{2}}{2(\sigma_{1}^{2}+\sigma_{2}^{2})}\right)$
(3.5.3)

The random variable describing the time between fire arrival at the detection
locations is the sum of two uniform random variables $T=Y+(-X)$ with $X\sim
U(t_{1}-l,t_{1})$ and $Y\sim U(t_{2}-l,t_{2})$, where $t_{1}$ and $t_{2}$ are
the times of the first and second detections and $l$ is the length of time
before the reported time when the fire could have arrived at the location. The
density is found using the convolution method for obtaining the sum of random
variables.

$f(t)=\left\\{\begin{array}[]{ll}f_{l}(t)=\frac{1}{l^{2}}\left(t-[(t_{2}-t_{1})-l]\right)&\quad(t_{2}-t_{1})-l\leq
t\leq(t_{2}-t_{1})\\\
f_{r}(t)=-\frac{1}{l^{2}}\left(t-[(t_{2}-t_{1})+l]\right)&\quad(t_{2}-t_{1})\leq
t\leq(t_{2}-t_{1})+l\\\ 0&\quad\text{otherwise}\end{array}\right.$ (3.5.4)

To get the joint distribution describing the random variable that is the ROS
between two active fire detections along a path, we need to divide a distance
by the time. We now compute the reciprocal of the density in Equation 3.5.4.
Letting $T$ be the random variable describing the time between two detections,
define $S=\frac{1}{T}$. The PDF for $S$ is computed by first finding its CDF
and then differentiating.

$F_{S}(s)=P(S\leq s)=P\left(\frac{1}{T}\leq s\right)=P\left(\frac{1}{s}\leq
T\right)$ (3.5.5)

The resulting CDF is found by integration of the piecewise linear function in
Equation 3.5.4.

$F_{S}(s)=\left\\{\begin{array}[]{ll}0&s<\frac{1}{t_{2}-t_{1}+l}\\\
\frac{\left(1/s-(t_{2}-t_{1}+l)\right)^{2}}{2l^{2}}&\frac{1}{t_{2}-t_{1}+l}\leq
s\leq\frac{1}{t_{2}-t_{1}}\\\
1-\frac{\left(1/s-(t_{2}-t_{1}-l)\right)^{2}}{2l^{2}}&\frac{1}{t_{2}-t_{1}}\leq
s\leq\frac{1}{t_{2}-t_{1}-l}\\\ 1&\frac{1}{t_{2}-t_{1}-l}<s\\\
\end{array}\right.$ (3.5.6)

Differenting the CDF gives the PDF

$f_{S}(s)=\left\\{\begin{array}[]{ll}\frac{-1/s+(t_{2}-t_{1}+l)}{(ls)^{2}}&\frac{1}{t_{2}-t_{1}+l}\leq
s\leq\frac{1}{t_{2}-t_{1}}\\\
\frac{1/s-(t_{2}-t_{1}-l)}{(ls)^{2}}&\frac{1}{t_{2}-t_{1}}\leq
s\leq\frac{1}{t_{2}-t_{1}-l}\\\ 0&\text{otherwise}\\\ \end{array}\right.$
(3.5.7)

Letting $c=t_{2}-t_{1}$, the time between detections, the expected value of
$S$ is found with

$\begin{split}E[s]=\int_{\frac{1}{c+l}}^{\frac{1}{c-l}}f_{S}(s)sds&=\frac{c+l}{l^{2}}\ln\left(\frac{c+l}{c}\right)-\frac{c-l}{l^{2}}\ln\left(\frac{c}{c-l}\right)\\\
&=\frac{1}{l^{2}}\left[(c+l)\ln\left(\frac{c+l}{c}\right)+(l-c)\ln\left(\frac{c}{c-l}\right)\right]\\\
&=\frac{1}{l^{2}}\left[l\ln\left(\frac{c+l}{c-l}\right)+c\ln\left(\frac{c^{2}-l^{2}}{c^{2}}\right)\right]\\\
&=\frac{1}{l^{2}}\ln\left[\left(\frac{c+l}{c-l}\right)^{l}\left(\frac{c^{2}-l^{2}}{c^{2}}\right)^{c}\right]\end{split}$
(3.5.8)

$E[S^{2}]=\int_{\frac{1}{c+l}}^{\frac{1}{c-l}}f_{S}(s)s^{2}ds=\frac{1}{l^{2}}\ln\left(\frac{c^{2}}{c^{2}-l^{2}}\right)$
(3.5.9)

The variance is then

$\mathrm{Var}[S]=E[S^{2}]-E[S]^{2}=\frac{1}{l^{2}}\ln\left(\frac{c^{2}}{c^{2}-l^{2}}\right)-\left(\frac{1}{l^{2}}\ln\left[\left(\frac{c+l}{c-l}\right)^{l}\left(\frac{c^{2}-l^{2}}{c^{2}}\right)^{c}\right]\right)^{2}.$
(3.5.10)

To get the variance of the ROS, expressed as $R=DS,$ we compute

$\begin{split}\mathrm{Var}[R]&=E[R^{2}]-E[R]^{2}\\\ &=E[(DS)^{2}]-E[DS]^{2}\\\
&=E[D^{2}S^{2}]-E[D]^{2}E[S]^{2}\\\ &=E[D^{2}]E[S^{2}]-E[D]^{2}E[S]^{2}\\\
&=(\mu_{1}^{2}+(\sigma_{1}^{2}+\sigma_{2}^{2}))\frac{1}{l^{2}}\ln\left(\frac{c^{2}}{c^{2}-l^{2}}\right)-\mu_{1}^{2}\left(\frac{1}{l^{2}}\ln\left[\left(\frac{c+l}{c-l}\right)^{l}\left(\frac{c^{2}-l^{2}}{c^{2}}\right)^{c}\right]\right)^{2}.\end{split}$
(3.5.11)

Figure 3.45 explores the variance in the ROS graphically. The left panel shows
a surface giving the ROS as a function of the differences in time and distance
and the distance between detections. The panel on the right shows the standard
deviation of the ROS, according to Equation 3.5.11. It is notable that the
standard deviation is large when the ROS is large. This should serve as a
warning that a high estimated of the ROS should be treated with caution. High
rates of spread between points on a path will be encountered when active fire
detections a distant spatially but were recorded at similar times.

Figure 3.45: The ROS (left) and standard deviation of ROS (right), assuming
the actual fire arrival time at the detection location is uniformly
distributed over the 6 hours before the satellite imaging and assuming a
geolocation error with standard deviation of 335 meters. In both panels, the
horizontal axes give the spatial and temporal distances between two fire
detections. The standard deviation is high when the distance between
detections is large but the time separating them is less than ten hours.

Figure 3.46: The probability distribution and 95% confidence region of the ROS
with two fire detections separated by 21 hours and 1500 meters. On the left is
the probability distribution of the ROS when considered a random variable. On
the right is a color plot of the ROS showing the 95% confidence region for the
ROS.

##  Chapter 4 ADJUSTING THE FIRE ARRIVAL TIME IN THE MODEL

This chapter explains how satellite fire data is used to make corrections to a
running wildfire simulation by a data assimilation technique that makes use of
the data likelihood function and method for estimating the fire arrival time
from data that were previously discussed. The first part of the chapter gives
an explanation of the method and its rationale. Hypothetical examples are used
to illustrate the method and a discussion of its limitations is given. The
second part of the chapter makes use of the data assimilation method in
modeling three real-world wildfires. The final part of the chapter details a
workflow for modeling wildfires that uses infrared perimeter observations and
satellite fire data to make an estimate of the fire arrival time to be used to
initialize a wildfire simulation. After short-term forecast is made, data
assimilation techniques are then used to update the model state and a longer
forecast is then produced.

###  4.1 DataAssimilation

Data assimilation seeks to make adjustments to the fire arrival time in a way
that balances the uncertainties of the forecast with the uncertainties in the
observational data. In a typical forecasting usage, a model is run forward for
a length of time during which inaccuracies in predictions are likely to
increase. During this initial model run, real-world observation of the
wildfire may become available that can be used to correct the course of the
model. The model is stopped and data is used to make corrections to the fire
arrival time and the model is then restarted from the updated estimate. A
cycling approach where a model is successively started and then corrected with
new data can be used to help steer an in-progress simulation. An early example
of the approach was used in Coen and Schroeder (2013). The Coupled Atmosphere-
Wildland Fire Environment (CAWFE) (Clark _et al._ , 1996) was used to simulate
a fire from the ignition and fire perimeters estimated from VIIRS active fire
detections were then used to re-initialize the fire simulation with each
successive overpass of the satellite. In this experiment, the satellite data
was used directly, without effort made to account for the inherent
uncertainties in. Progress towards a more robust method using the cycling
concept was made in Mandel _et al._ (2014b). Instead of using satellite
directly, a Bayesian approach was used to combine satellite data with the
model forecast to obtain an updated state of the fire. This updated state of
the fire was then used to form the restart conditions in a cycling routine
similar to that used by Coen et al. The data assimilation method proposed in
this thesis follows the cycling strategy employed by both Coen et al. and
Mandel et al. but employs a different method for obtaining an update of the
fire arrival time used to restart the simulation. The method follows closely
from that described in Section 3.2 that is used to estimate the fire arrival
time from satellite data alone. Instead of using a rough estimate of the fire
arrival time as a starting point for the iterative interpolation process, the
model forecast fire arrival time is used and adjustments made to it according
to the data the data likelihood function from 2.3. Both VIIRS and MODIS data
are used as a source of observations.

####  4.1.1 Method

The data assimilation method is similar to that used for estimation of fire
arrival time from satellite data except that the initial estimate used is the
model forecast and the data likelihood function regulates the adjustments made
to the forecast. Both active fire detections and non-fire pixels are used to
make adjustments to the forecast fire arrival time in order to make the
analysis fire arrival time that blends forecast with satellite data. Figure
4.1 shows the difference that using the data likelihood function makes when
using satellite data and an initial estimate of the fire arrival time to make
an updated estimated. In the bottom panel of the figure are contours of an
initial estimate of the fire arrival time, here taken to be the model
forecast. In the upper left and the right are contours of two versions of the
analysis fire arrival time. The analysis in upper left has been made without
use of the data likelihood function and the iterative interpolation method has
fit the satellite data closely as can be seen by the jagged fire perimeter. On
the right, the analysis has been made with use of the data likelihood function
and the iterative interpolation has blended the forecast and data differently,
achieving a smoother result.

Figure 4.1: Illustrating the effect of using the data likelihood function for
weighting the forecast fire arrival time and time of the active fire
detections. The top left and top right panels represent two versions on the
analysis fire arrival time made without and with use of the data likelihood
function, respectively. The bottom panel shows the forecast.

##### Active Fire Detections

Active fire detections are used to adjust the forecast fire arrival time. The
data likelihood function from Section 2.3 controls how the forecast fire
arrival time is adjusted at locations of satellite active fire detections in a
manner similar to that used in Section 3.2. Figure 4.2 shows how the data
likelihood function determines how far the iterative interpolation process
moves the forecast fire arrival time up or down towards the time of active
fire detections. The orange stars indicate the time and location of active
fire detections. The black line represents the forecast fire arrival time.
Detection A has a time before the forecast fire arrival time, detection B has
a time exactly coinciding with the forecast, and detection detection C has a
time after the forecast. The analysis fire arrival time at detection A has the
largest difference from the forecast at this location because the forecast is
inconsistent with data. At location C, the difference between the forecast and
analysis is smaller because the data and forecast fire arrival time are both
plausible. The fire could have arrived at location C at the time of the
forecast, but the satellite was not present to witness that arrival.

Figure 4.2: The data likelihood function determines how far the iterative
interpolation process moves the forecast fire arrival time up or down towards
the time of active fire detections.

At each location of an active fire detection $d_{i}$, the analysis fire
arrival time $T_{a}$ is a weighted average of the forecast fire arrival time
$T_{f}$ and the time of the satellite fire detection $T(d_{i})$ with a weight
given by the data likelihood. The analysis fire arrival time is computed as

$T_{a}=\alpha(T_{f}-T(d_{i}))+T(d_{i})$ (4.1.1)

where $\alpha$ is the computed from the exponential of the likelihood function
of Equation 2.3.14. Thus the weight $\alpha$ is a number between zero and one,
given that the false detection rate used in the likelihood function is set to
zero. If $\alpha=1$, then $T_{a}=T_{f}$ and $\alpha=0$ gives $T_{a}=T(d_{1})$.
In practice, the false detection rate is set to several percent and $\alpha$
is never zero, meaning that the satellite data is never fully trusted and the
forecast fire arrival time always influences the the analysis fire arrival
time $T_{a}$ at every location in the fire domain.

With the analysis fire arrival time computed at each location of an active
fire detection, the iterative interpolation technique from Section 3.2 is then
used to make adjustments to the fire arrival time at nearby locations with a
local averaging method. Figure H.3 shows a sequence of panels depicting the
method on a one-dimensional fire line. In this example, no additional data
points have been interpolated along the paths. Figure H.4 shows an identical
scenario except that additional points have been interpolated along the paths.
The interpolation of more points helps this analysis better resolve the
gradient information implied by the data that is seen on the left side of the
figures at approximately time $t=5.5.$

##### Non-fire Pixels

In the same way that the data likelihood function determines the weighting of
the forecast fire arrival time and the satellite fire detection time at the
fire detection locations, the data likelihood determines how the analysis fire
arrival time will be a weighted average of the forecast fire arrival time and
the end-time of the estimation period. Figure 3.30 shows how the forecast fire
arrival time at the locations of non-fire pixels is moved upwards towards the
“flat top" of the fire arrival time cone that marks the end-time of the
simulation. The non-fire pixels are only used to make adjustments to the
forecast fire arrival outside of a polygon that is drawn around the active
fire detections in the domain. Figure 4.4 shows the effect of using non-fire
pixels to form the analysis fire arrival time. The bottom panel shows the
forecast fire arrival time and the upper left and upper right panels show the
analysis formed without using and with using ground non-fire pixels,
respectively.

Figure 4.3: Using non-fire pixels to help constrain the size of a simulation
or estimate of fire arrival time. The original perimeters of the forecast fire
arrival time are on the left. Note that the contours lines in the northeast of
the fire area extend into areas without any active fire detections. On the
right is the effect of moving the fire arrival time upward for non-fire pixels
outside of the polygon containing active fire detections. The iterative
process shrinks the perimeter and smooths the fire arrival time.

Figure 4.4: Comparison of the analysis made with and without use of non-fire
pixel data. One the bottom are the original forecast fire arrival time and
detection locations. The upper left panel shows analysis formed without use of
non-fire pixels. Note the perimeters extend into regions not containing any
fire detection pixels. The upper panel on the right shows the analysis formed
using non-fire pixel data. The outer perimeter more closely matches the
locations where active fire detections are located.

####  4.1.2 Limitations of the Method

The data assimilation method developed has some limitations in its utility due
to the way its was designed and implemented. We outline some of those
limitations below and offer potential remedies.

##### Use of Non-fire Pixels

The use of non-fire pixels to form the analysis fire arrival time is
restricted to a region outside of a polygon drawn around the locations of the
active fire detections. This represents a loss of of information about the
behavior of the fire. It is expected that some regions within the outer
perimeter of the fire may remain unburned. Indeed, there may be regions
contained within the fire perimeters, such as lakes or sand dunes, that
contain no fuels to be burned. As implemented, the method developed here can
assign a fire arrival time to a location that can never burn. One possible
remedy for this shortcoming would be to use information about the fuels in the
fire domain in addition to the non-fire pixels. The fire arrival time at
locations where no fuels are found and non-fire pixels were recorded could be
set to the end-time of the estimation period.

##### Data Likelihood Usage

For the examples in this thesis, the data likelihood used the same parameters
for all locations in the fire domain, regardless of the effects of weather,
terrain, and fuel properties. In particular, the fuel types in the fire domain
would have an effect on the probability of the satellite detection that is a
component of the data likelihood. For example, one would expect that fire in a
dry, grassy region would be detectable by satellite for a shorter time than
fire in a forest with large trees because the mass of fuel in the grassy
region would be less and the fuel would be completely consumed by fire more
rapidly. Figure 4.5 shows the difference in the shapes of curves giving the
probability of detection since fire arrival time for a region with a fast
burning fuel and for a region with a slow burning fuel. The curves are
hypothetical, based on Equation 2.3.9 and setting the parameter $a$, which can
be determined by Equation 2.3.11, so that the probability of detection being
20% after 10 hours after the fire arrival time for the fast burning fuel and
20% after 20 hours for the slow burning fuel. The data likelihood function
described in this work does have the capability to account for such
differences. Additional research is required to adapt it for the varying fuel
types that are used by the fire spread model.

Figure 4.5: Illustrating the difference in the probability of satellite fire
detection for regions with different fuel types. On the left is the
probability of detection for a region with a fast burning fuel. On the right
is the probability of detection for a region with a slow burning fuel.

###  4.2 Operational Usages

In an operational setting, a fire simulation is initialized from an ignition
point or from an estimate of the fire arrival time. After some time of running
the model forward, new data becomes available and is used to make adjustments
to the model output in a way that balances the uncertainties that exist in
both the forecast and observational data. This analysis fire arrival time then
becomes the starting point from which the next forecast is initialized. A
typical simulation includes several cycles of making a model forecast followed
by performing data assimilation.

In this section, two forecasting strategies using data assimilation will be
demonstrated. In the first, only satellite observations will be used as a a
data source. With rare exception, satellite data can be obtained for any fire
to be simulated and this first strategy can be always be used. The second
strategy will make use of infrared perimeter observations in addition to
satellite data. In general, these types of observations will not be available
for every fire. However, for large fires that pose a significant risk to life
and property, they are often made available by the responding authorities.
When available, these observations can be used to make an estimated fire
arrival time that can be used to initialize a fire simulation. Both strategies
are employed to simulate real-world fires and the results of those simulations
are assessed by comparison with infrared perimeter observations, using the
tools described in Section 3.3.

####  4.2.1 Forecast Cycling with Data Assimilation

When only satellite observations are available, the forecast method that uses
data assimilation starts with a simulation initialized from an ignition point
or estimated fire arrival time and then periodically updates the model state
with assimilation of satellite fire observations. A sequence of short-term
forecasts are made in a cyclic manner where the forecasts are followed by
updates. The basic method is outlined below.

  1. 1.

Begin a simulation from an ignition point or a fire arrival time that has been
estimated using satellite fire data.

  2. 2.

Run the model forward for a specified amount of time. In an operational
setting, this might be a simulation spanning one day or up to a week.
Typically, simulations for periods longer than two days begin to lose
accuracy. See the results from Section 4.2.1 to see some results.

  3. 3.

Collect all the satellite data made available since the initial data
collection in the first step.

  4. 4.

Construct the analysis fire arrival time from the directed graph with shortest
paths as detailed in Section 4.1. Insert the analysis into the WRF-SFIRE
restart file for the simulation.

  5. 5.

Restart the simulation from the restart file, using the analysis in place of
an ignition point or estimated fire arrival time as prescribed in step 1.

  6. 6.

Go to step 2 and repeat steps 3 and 4 until the simulation end time.

This procedure was used to simulate three real-world fires. In each case,
forecast cycling with data assimilation produced forecasts that better matched
infrared perimeter observations than simulations that made no attempt to
adjust the course of an in-progress fire with data assimilation techniques.

##### Patch Springs Fire

The Patch Springs fire took place in August 2013 approximately 60 kilometers
southwest of Salt Lake City, Utah. The fire is estimated to have started early
in the evening of August 10 and was finally 100% contained by October 23 (USDA
Forest Service and Management, 2015 (accessed April 20, 2020) . Initially
starting slowly, the fire advanced rapidly on August 14(Gabbert, 2019). The
following discussion pertains to modeling this fire with WRF-SFIRE during its
first eight days.

Using WRF-SFIRE, the Patch Springs fire was simulated over the period of
August 11 to August 18. the ignition location was chosen as
$40.37^{\circ}N,-112.64^{\circ}W$, with the ignition time of 06:00 UTC. The
simulation was run on one computational domain with a relatively coarse grid
spacing of 225 meters. Five data assimilation cycles were incorporated into
the simulation. Figure 4.6 illustrates the timing of the simulation periods.
An explanation of the cycles of the simulation follows.

  * •

Cycle 0 \- The simulation was started from an ignition point and run for eight
days. Satellite data from the first two days was collected and combined with
the first two days of the simulation forecast, forming the analysis fire
arrival time that covered the first two days of the fire simulation.

  * •

Cycle 1 \- The analysis fire arrival time from cycle 0 was used to spin-up the
model, replaying the first two days of the fire and bringing the state of the
atmosphere into synchronization with the state of the fire. After the two days
of spin-up time, the model took over and produced an additional six days of
forecast. Satellite data from the first day of this forecast was collected and
combined with the first day of the simulation forecast, forming the analysis
fire arrival time that covered the first three days of the fire simulation.

  * •

Cycle 2 \- The analysis fire arrival time from cycle 1 was used to spin-up the
model, replaying the third day of the fire. After the one day of spin-up time,
the model took over and produced an additional five days of forecast.
Satellite data from the first day of this forecast was collected and combined
with the first day of the simulation forecast, forming the analysis fire
arrival time that covered the first four days of the fire simulation.

  * •

Cycle 3 \- The analysis fire arrival time from cycle 2 was used to spin-up the
model, replaying the fourth day of the fire. After the one day of spin-up
time, the model took over and produced an additional four days of forecast.
Satellite data from the first day of this forecast was collected and combined
with the first day of the simulation forecast, forming the analysis fire
arrival time that covered the first five days of the fire simulation.

  * •

Cycle 4 \- The analysis fire arrival time from cycle 3 was used to spin-up the
model, replaying the fifth day of the fire. After the one day of spin-up time,
the model took over and produced an additional three days of forecast.
Satellite data from the first day of this forecast was collected and combined
with the first day of the simulation forecast, forming the analysis fire
arrival time that covered the first six days of the fire simulation.

  * •

Cycle 5 \- The analysis fire arrival time from cycle 4 was used to spin-up the
model, replaying the fifth day of the fire. After the one day of spin-up time,
the model took over and produced an additional two days of forecast.

Figure 4.6: Schematic for several data assimilation cycles run consecutively
for simulation of the Patch Springs fire. For the sake of comparison, all
simulations were run from the end of the spin-up period the end time of the
entire simulation period. Operationally, the cycle run would be terminated
when new data was assimilated and a restart of the simulation initiated.

Figure 4.7: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the first cycle of the Patch Fire simulation. The data
assimilation was performed on a computational grid with 250 meter spacing.
Note that the ignition point has been moved towards the south by the process.
The outer perimeter of the analysis fir arrival time was shifted eastward over
much of the fire domain to better match the satellite fire detections.

Figure 4.8: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the second cycle of the Patch Fire simulation. The data
assimilation was performed on a computational grid with 250 meter spacing.
Minimal changes were made to the forecast during this cycle.

Figure 4.9: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the third cycle of the Patch Fire simulation. The data
assimilation was performed on a computational grid with 250 meter spacing.
Minimal changes were made to the forecast during this cycle.

Figure 4.10: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the fourth cycle of the Patch Fire simulation. The data
assimilation was performed on a computational grid with 250 meter spacing. The
outer perimeter of the forecast fire arrival time was expanded in the
northwest region of the fire domain due to the presence of many active fire
detections in the region.

Figure 4.11: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the fifth cycle of the Patch Fire simulation. The outer
perimeter of the forecast fire arrival time was expanded in the northwest
region of the fire domain due to the presence of many active fire detections
in the region.

Figures 4.7-4.11 show how the data assimilation method was able to adjust the
forecast fire arrival time to make use of the satellite data collected. As can
be seen in Figure 4.7, the initial simulation required significant adjustment
to to account for the large number of fire detections that were outside of its
outer perimeter on the east side of the active fire region. During the
remainder of the simulation, smaller adjustments were made to the fire arrival
time by the data assimilation method. Each of the cyclic data assimilation
adjustments incorporated 24 hours of additional satellite data. The minimal
adjustments to the forecast indicate the model was providing good short-term
forecasts.

To assess the method, an infrared perimeter observation was used to evaluate
how well the forecasts matched the actual fire. Although the data assimilation
routine made periodic adjustments and restarts of the model, every new
forecast start was allowed to run to at least the time of the infrared
perimeter observation on August 16, at 09:47 UTC. Table 4.1 shows the results
obtained as assessed using the MOE and Sørenson index. Graphically, the
results are summarized in Figure H.23. The forecasts tended to overestimate
the size of the fire in the northeast and southeast areas of the fire domain
and underestimate it in the northwest.

Also included in the table of results and in Figure H.23 are results for a
simulation labeled “Estimated Start." This model run was initialized from an
estimated fire arrival time that was made using satellite fire detection data.
This short-term forecasts did not exhibit the overestimation of the size of
the fire in the northeast and southeast areas of the fire domain like the
other forecasts did, but it also failed to predict the fire growth in the
northeast are of the fire domain.

Cycle | Hours of Forecast | MOE X | MOE Y | ||MOE|| | Sørenson
---|---|---|---|---|---
0 | 130 | 0.4920 | 0.7629 | 0.9078 | 0.5982
1 | 82 | 0.7543 | 0.6413 | 0.9901 | 0.6932
2 | 58 | 0.7009 | 0.6476 | 0.9543 | 0.6732
3 | 34 | 0.7733 | 0.6394 | 1.0034 | 0.7000
4 | 10 | 0.8993 | 0.6297 | 1.0978 | 0.7407
Estimated Start | 12.5 | 0.6143 | 0.9149 | 1.1020 | 0.7350
Table 4.1: Scores for the cycling of the Patch Springs fire when compared with
an infrared perimeter observation made August 16, at 09:47 UTC. The results in
this table are summarized visually in Figure H.23

##### Cougar Creek Fire

The Cougar Creek fire took place in August, 2015. The fire is thought to have
started naturally due to lightning strike at approximately 6:00 PM local time
on August 10. Overall growth of the fire was slow due to high relative
humidity and cloud cover. More than 400 fire fighting personnel were deployed
and by September 14 the fire was 97% contained. The fire was fueled by timber
(some beetle-killed), light logging slash, and tall grass. The total size of
the eventually grew to more than 53,000 acres (USDA Forest Service and
Management, 2013 (accessed April 20, 2020).

We simulated the first 4.5 days of the Cougar Creek fire. The simulation was
performed on a three-domain computational grid, with the fire model running at
a resolution of roughly 30 meters between grid nodes. For comparison, all
cycles were run to produce a fire arrival time of the entire 4.5 days of
simulation. Periodic restarts of the simulation were made after 24 hours of
simulation time. The schematic in Figure H.24 shows how the cycling was
accomplished. Further explanation of the timing follows.

  * •

Cycle 0 \- The simulation was started from an ignition point and run for 4.5
days. Satellite data from the first day was collected and combined with the
first day of the simulation forecast, forming the analysis fire arrival time
that covered the first day of the fire simulation.

  * •

Cycle 1 \- The analysis fire arrival time from cycle 0 was used to spin-up the
model, replaying the first day of the fire and bringing the state of the
atmosphere into synchronization with the state of the fire. After the one day
of spin-up time, the model took over and produced an additional 3.5 days of
forecast. Satellite data from the first day of this forecast was collected and
combined with the first day of the simulation forecast, forming the analysis
fire arrival time that covered the first two days of the fire simulation.

  * •

Cycle 2 \- The analysis fire arrival time from cycle 1 was used to spin-up the
model, replaying the second day of the fire. After the one day of spin-up
time, the model took over and produced an additional 2.5 days of forecast.
Satellite data from the first day of this forecast was collected and combined
with the first day of the simulation forecast, forming the analysis fire
arrival time that covered the first three days of the fire simulation.

  * •

Cycle 3 \- The analysis fire arrival time from cycle 2 was used to spin-up the
model, replaying the third day of the fire. After the one day of spin-up time,
the model took over and produced an additional 1.5 days of forecast. Satellite
data from the first day of this forecast was collected and combined with the
first day of the simulation forecast, forming the analysis fire arrival time
that covered the first four days of the fire simulation.

  * •

Cycle 4 \- The analysis fire arrival time from cycle 3 was used to spin-up the
model, replaying the fourth day of the fire. After the one day of spin-up
time, the model took over and produced an half day of forecast.

Figures 4.12-4.15 show how the data assimilation made adjustments to the
forecast fire arrival time. The first two cycles show fairly large adjustments
being made to the forecast and the final to cycles show more moderate
adjustments. Most likely this fire exhibited some uneven growth due to weather
or fuel conditions the model was unable to account for. As was done in the
case of the Patch Springs fire, each forecast from the cycles of the
simulation was allowed to run for the entire duration of the simulation
period. These forecasts were compared with an infrared perimeter observation
made on August 15 at 10:33 UTC. The numeric results are summarized in Table
4.2 and graphically represented in Figure H.26. Each forecast from the cycles
produced a better result, but most of the forecasts tended to overestimate the
size of the fire.

For comparison, the fire was also simulated by initializing the model with
satellite detection data. Only 9 hours of forecast were produced by this
simulation and it can be compared to the short forecast made as part of the
cycling routine. The results of these two forecasts are quite similar. Most
likely, the reason for the overestimation of the fire area by these forecasts
was due to the fuel moisture content used by the model being too dry. Section
5.1.3 shows results from a simulation that used an adjusted FMC to achieve
better results.

Another possible contributing factor to the overestimation of fire size when
using data assimilation is related to the replay of the analysis fire arrival
time during the restart of the fire simulation for the next forecast cycle.
Although the analysis does provide an improved estimate of the state of the
fire in terms of the fire arrival time, it does not necessarily give an
accurate picture of where the fire is hottest and growing most rapidly. For
example, in the simulation of the Cougar Creek fire, during the second
forecast cycle, the forecast did not predict the growth in the southwest
region of the fire, as is seen in Figure 4.13. Apparently, the growth of the
fire was mostly in a southwesterly direction during this time, with little
growth appearing elsewhere. However, the data assimilation strategy treats all
parts of the fire perimeter as if equal fire growth was occurring everywhere.
This equal growth can be seen in the forecast produced in the third cycle of
the Cougar Creek fire, shown in Figure 4.14. In this forecast, the fire had
continued to expand towards the northwest and southeast, despite some
knowledge that most active are of the fire was in the southwest. Possibly this
shortcoming of the data assimilation method could be overcome by adjusting the
properties of the underlying fuels in the areas where the data assimilation
has contracted the fire perimeter, indication the fire was less active in the
region than the model had predicted.

Figure 4.12: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the first cycle of the Cougar Creek Fire simulation. The
data assimilation was performed on a computational grid with 250 meter
spacing. The forecast had underestimated the size of the fire and the data
assimilation expanded the outer perimeter to contain more of the active fire
detections.

Figure 4.13: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the second cycle of the Cougar Creek Fire simulation. The
data assimilation was performed on a computational grid with 250 meter
spacing. Again, the outer perimeter was expanded to contain more of the active
fire detections.

Figure 4.14: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the third cycle of the Cougar Creek Fire simulation. The
data assimilation was performed on a computational grid with 250 meter
spacing. The outer perimeter was contracted slightly by the data assimilation
method in this cycle. The forecast had shown more growth in the northeast and
southest sections of the fire region than was indicated by the satellite data.

Figure 4.15: Contours of the forecast the fire arrival time (left) and the analysis (right) for the third cycle of the Cougar Creek Fire simulation. The data assimilation was performed on a computational grid with 250 meter spacing. Minimal adjustment were made to the fire arrival time during the data assimilation at the end of this cycle. Cycle | Hours of Forecast | MOE X | MOE Y | ||MOE|| | Sorenson
---|---|---|---|---|---
0 | 105 | 0.6638 | 0.6647 | 0.9394 | 0.6642
1 | 81 | 0.9787 | 0.4966 | 1.097 | 0.6589
2 | 57 | 0.98733 | 0.4773 | 1.097 | 0.6435
3 | 33 | 0.9580 | 0.5446 | 1.1019 | 0.6944
4 | 9 | 0.9533 | 0.6284 | 1.1418 | 0.7575
Estimated Start | 9 | 0.9560 | 0.6177 | 1.1382 | 0.7505
Table 4.2: Scores for the cycling of the Cougar Creek fire when compared with
a perimeter observation made August 15, at 10:33 UTC.

##### Camp Fire

The Camp fire was a large and destructive fire that took place in California
in 2018. Drought conditions, dry fuels, and high winds combined to produce a
fire that moved 25 kilometers in the first 12 hours (Cal Fire, 2018 (Accessed
March 3, 2020; Brewer and Clements, 2020) and consumed over 100,000 acres
during its first 2 days (USDA Forest Service and Management, 2018 (accessed
April 20, 2020)). The left panel of Figure 4.16 shows the path structure
derived from satellite detections during the first two days. The shallow
angles of the paths indicate a high rate of spread was present. It is thought
that a major component of the spread mechanism of this fire was from the wind
transport of burning embers from vegetation and man-made structures (Brewer
and Clements, 2020; Syifa _et al._ , 2020). The spot fires initiated by these
wind-borne firebrands were observed to occur a mile in advance of the main
bulk of the fire (Cal Fire, 2018 (Accessed March 3, 2020)). This spread
mechanism is not a part of the Rothermel model that WRF-SFIRE uses for
computing the fire spread, making data assimilation an essential component of
simulating such a fast-moving fire. Indeed, the right panel of Figure 4.16
shows a histogram of the difference between the ROS implied by the satellite
detections and that of the model forecast at those detection locations. One
average, the model was underestimating the ROS by about 0.4 m/s.

We used WRF-SFIRE to simulate the critical first two days of the Camp Fire.
The simulation was run in WRF-SFIRE using three nested domains with
resolutions of 5000m, 1667m, 500m. The simulation used four data assimilation
cycles to incorporate observations from the MODIS and VIIRS active fire
products. Figure H.25 shows the timing of the cycles. For assessment, the
cycle forecasts were compared with an infrared perimeter observation made on
November 10 at 07:00 UTC. More details of the individual forecast cycles
follow.

  * •

Cycle 0 \- The simulation was started from an ignition point and run for 43
hours. Satellite data from the first 16 hours of the simulation was collected
and combined with the first 16 hours of the simulation forecast, forming the
analysis fire arrival time that covered the first 16 hours of the fire
simulation.

  * •

Cycle 1 \- The analysis fire arrival time from cycle 0 was used to spin-up the
model, replaying the first 16 hours of the fire and bringing the state of the
atmosphere into synchronization with the state of the fire. After the spin-up
period, the model took over and produced an additional 27 hours of forecast.
Satellite data from the first 8 hours of this forecast was collected and
combined with the first 8 hours of the simulation forecast, forming the
analysis fire arrival time that covered the first 24 hours of the fire
simulation.

  * •

Cycle 2 \- The analysis fire arrival time from cycle 1 was used to spin-up the
model, replaying hours 16 to 24 of the fire. After the spin-up period, the
model took over and produced an additional 19 hours of forecast. Satellite
data from the first 8 hours of this forecast was collected and combined with
the first 8 hours of the simulation forecast, forming the analysis fire
arrival time that covered the first 32 hours of the fire simulation.

  * •

Cycle 3 \- The analysis fire arrival time from cycle 2 was used to spin-up the
model, replaying hours 24 to 32 of the fire. After the spin-up period, the
model took over and produced an additional 11 hours of forecast. Satellite
data from the first 8 hours of this forecast was collected and combined with
the first 8 hours of the simulation forecast, forming the analysis fire
arrival time that covered the first 40 hours of the fire simulation.

  * •

Cycle 4 \- The analysis fire arrival time from cycle 2 was used to spin-up the
model, replaying hours 32 to 40 of the fire. After the spin-up period, the
model took over and produced an additional 3 hours of forecast.

Figures 4.17-4.20 show the adjustments made to the forecast fire arrival time
by data assimilation. The initial cycle, Figure 4.17, shows a massive
expansion of the fire perimeter to reflect the large number of active fire
detections that extended beyond forecast perimeter. Indeed, this type of
adjustment is necessary for a fire that spread largely by winds carrying
embers ahead of the main fire front. The WRF-SFIRE model does not have the
capability to advance fire by this mechanism and it is not surprising that it
could not accurately predict the movement of the fire front during this period
of rapid growth. In Figure 4.18, the forecast showed fire growth in the
southern part of the fire domain that was not observed by satellite, leading
to a contraction of the perimeter here during the data assimilation process.
Interestingly, there’s a large gap in the satellite fire detections between
two “forks" of the fire in the south and southwest region of the fire domain.
Most likely fire was present in this gap but was not observed. Possibly the
region was obscured from satellite view by smoke. Table 4.3 summarizes the
results of comparison with the cycling forecasts with an infrared perimeter
observation. Figure H.27 give a graphical representation of these results. The
forecasts tended to overestimate the size of the fire.

For comparison, a forecast of 19 hours duration, initialized from an estimated
fire arrival time, was also produced. This short-term forecast ended at the
same time of the simulations produced during the cycling routine and can be
compared with the forecast produced in cycle 2 of the main simulation. As can
be seen in the rightmost panels in Figure H.27, the results are similar. The
simulations both overestimated the size of the fire.

Figure 4.16: Two figures showing the high rate of growth of the Camp Fire
during its early stages. On the left, the shallow angles of paths in the graph
indicate a high ROS. On the right, a histogram of the difference between the
ROS forecast by the model and the ROS estimated from satellite data indicate
the modeled fire was spreading much slower than real-word fire.

Figure 4.17: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the first cycle of the Camp Fire simulation. The data
assimilation was performed on a computational grid with 250 meter spacing. The
initial simulation underestimated the size of the fire, causing the analysis
fire arrival time to have an expanded outer perimeter when compared to that of
the forecast.

Figure 4.18: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the second cycle of the Camp Fire simulation. The data
assimilation was performed on a computational grid with 250 meter spacing.

Figure 4.19: Contours of the forecast the fire arrival time (left) and the
analysis (right) for the third cycle of the Camp Fire simulation. The data
assimilation was performed on a computational grid with 250 meter spacing.
According to satellite data, very little growth in the fire had occurred
during this cycle, but the model continued to show expansive growth to the
south of the main region of the fire domain. The data assimilation routine
contracted the perimeter back to a similar position it was in at the end of
the previous cycle.

Figure 4.20: Contours of the forecast the fire arrival time (left) and the analysis (right) for the fourth cycle of the Camp Fire simulation. The data assimilation was performed on a computational grid with 250 meter spacing. In this final cycle, the forecast fire perimeter was again contracted to account for growth no observed by satellite. Cycle | Hours of Forecast | MOE X | MOE Y | ||MOE|| | Sorenson
---|---|---|---|---|---
0 | 43 | 0.6111 | 0.8124 | 1.0166 | 0.6976
1 | 27 | 0.9709 | 0.4664 | 1.0771 | 0.6301
2 | 19 | 1.0000 | 0.4149 | 1.0827 | 0.5865
3 | 11 | 0.9975 | 0.5393 | 1.1340 | 0.7001
4 | 3 | 0.9012 | 0.7658 | 1.1826 | 0.8283
Estimate | 19 | 1.0000 | 0.4278 | 1.0877 | 0.5993
Table 4.3: Assessments of the simulation of the Camp Fire. Each cycle was run
until November 10, 07:00 UTC so that a comparison with an infrared perimeter
observation could be made. Even very short forecasts had a difficult time with
predicting the behavior of this fire. In general, data assimilation helped
improve the model output, not as much as could be expected. The final row in
the table represents the a simulation initialized from an estimated fire
arrival time produced from satellite fire data. The output from this
simulation is very similar to that of the cycle 2 simulation. Figure H.27
gives a set of graphics that help visualize these results.

####  4.2.2 48-Hour Forecast Workflow

This forecast strategy works by assimilating the most recent satellite data to
update a short-term forecast initialized from an estimated fire arrival time
made using infrared perimeter observations. The goal is to start a simulation
from the best possible estimate of the state of the fire, derived from
infrared perimeter observations, and then use the fire model and additional
satellite fire observations to make minimal corrections to a short-term
forecast. The model is then restarted from this updated estimate of the state
of the fire and allowed to run for 48 hours.

Typically, infrared perimeter observations are made by flying an aircraft over
the fire around midnight, local time. Because of the accuracy of these
observations, they provide the best available snapshot of the state of the
fire and can be used to initialize a simulation. These perimeter observations
may only become available to researchers sometime in the afternoon of the
following day. During the period between when the infrared observations are
made and when they become available to fire forecasters, paossibly many hours
will have passed and additional satellite observations will have been made.
The forecasts strategy in this section details a method to take advantage of
these latest satellite observations by making a short-term forecast,
initialized from the perimeter observations, and then assimilating the most
recent satellite data using the same method previously outlined. The workflow
for this forecast strategy is as follows.

  1. 1.

At 5:00 PM local time, collect satellite and infrared perimeter data.
Typically the infrared perimeter observations made around midnight the
previous day become available at this time.

  2. 2.

Estimate the fire arrival time from the data.

  3. 3.

Run the model forward until the model time matches real time. This short-term
forecast can be run in about two hours real time.

  4. 4.

Collect all the satellite data covering the time of the short-term forecast.

  5. 5.

Make an analysis fire arrival time by using the data assimilation techniques
from this chapter.

  6. 6.

Restart the simulation from the analysis fire arrival time.

##### Patch Fire Example

A simulation of the Patch Springs fire was run using the 48-hour forecast
workflow. The simulation was started from an estimated fire arrival time
derived from satellite fire data and an infrared perimeter observation. In
total, 7 satellite granules and one infrared perimeter were used. The these
observations spanned approximately the first two days of the fire. Figure 4.22
shows these observations and the estimated fore arrival time made for the
simulation initialization. The simulation was run forward in time to produce
several days of forecast, but in an operational setting, this simulation would
produce only about 20 hours of forecast.

To emulate the forecasting strategy, data assimilation was performed to make
minor adjustments to the model output using the latest observations collected
in the 21 hours since the time of the previous infrared perimeter. During this
window, four granules of satellite data became available. Figure 4.23 shows
hows the effect of data assimilation. The Analysis has made minor changes to
the fire arrival time, contracting the final perimeter over much of the fire
domain, but expanding it slightly in northwest region of the fire. The
forecast produced after assimilating 21 hours of data showed improvement over
a the forecast made from the original initialization of the fire from the
perimeter and satellite data when compared to an infrared perimeter. The
results of this comparison are presented in Table 4.1 and graphically in
Figure H.28. The 48-hour strategy produced better results than the data
assimilation cycling strategy from Section 4.2.1. The results of this
comparison are presented in Table 6.2.

Figure 4.21: Timelines for the Patch Fire, 48-hour simulation experiment. The
orange arrow indicates timings of simulations and the blue arrow indicates the
progression of real-time tasks taken to produce and process the simulations in
the red timeline.

Figure 4.22: Satellite fire data organized into clusters and paths from the
data used to initialize the initial simulation of the 48-hour workflow
experiment for the Patch Springs Fire. The upper panels show the clustering
and shortest paths. The bottom panel shows the estimated fire arrival time
used to initialize the model.

Figure 4.23: Data assimilation of 21 hours of satellite and perimeter data for the Patch Fire simulation. In the top panels, contours for the forecast and analysis fire arrival times can be seen. In the bottom, the final perimeters of both the forecast and analysis fire arrival times can be seen together.  Cycle | Hours of Forecast | MOE X | MOE Y | ||MOE|| | Sorenson
---|---|---|---|---|---
0 | 79 | 0.7027 | 0.6609 | 0.9647 | 0.6812
1 | 58 | 0.6450 | 0.7559 | 0.9937 | 0.6961
Table 4.4: Results from the simulation of Patch Springs Fire using the 48-hour
forecast strategy. Assesments were made by comparison was an infrared
perimeter observation 10 hours after the the normal 48 hour forecast period.
The assessment scores show that assimilating even a small number of additional
satellite observations helped produce a better forecast.

##### Cougar Creek Fire Example

A simulation of the Cougar Creek Fire was run using the 48-hour forecast
workflow. The simulation was started from an estimated fire arrival time
derived from satellite fire data and an infrared perimeter observation. In
total, 24 satellite granules and four infrared perimeters were used. The these
observations spanned approximately the first two days of the fire. Figure 4.25
shows these observations. The simulation was run forward in time to produce
several days of forecast, but in an operational setting, this simulation would
produce only about 20 hours of forecast.

To emulate the forecasting strategy, data assimilation was performed to make
minor adjustments to the model output using the latest observations collected
in the 16 hours since the time of the previous infrared perimeter. During this
window, three granules of satellite data containing 7 active fire detections
became available. Figure 4.26 shows the effect of data assimilation. The
analysis has made minor changes to the fire arrival time, contracting the
final perimeter over much of the fire domain. Table 4.5 gives the results of
this test. Figure H.29 shows graphical representations of these results.
Assessments were made by comparison was an infrared perimeter observation.
Comparison between the cycle 0 and cycle 1 scores show that assimilating even
a small number of additional satellite observations helped produce a better
forecast.

Figure 4.24: Timeline for Cougar Creek Fire, 48-hour simulation experiment.
The orange arrow indicates timings of simulations and the blue arrow indicates
the progression of real-time tasks taken to produce and process the
simulations in the red timeline.

Figure 4.25: Satellite fire data organized into clusters and paths from the
data used to initialize the initial simulation of the 48-hour workflow
experiment for the Cougar Creek Fire. The upper panels show the clustering and
shortest paths. The bottom panel shows the estimated fire arrival time used to
initialize the model.

Figure 4.26: Data assimilation of 21 hours of satellite and perimeter data for the Cougar Creek Fire. In the top panels, contours for the forecast and analysis fire arrival times can be seen. In the bottom, the final perimeters of both the forecast and analysis fire arrival times can be seen together.  Cycle | Hours of Forecast | MOE X | MOE Y | ||MOE|| | Sorenson
---|---|---|---|---|---
0 | 61 | 0.7316 | 0.8349 | 1.1101 | 0.7799
1 | 45 | 0.7305 | 0.8722 | 1.1377 | 0.7972
Table 4.5: Results from the simulation of Cougar Creek Fire using the 48-hour
forecast strategy. Assessments were made by comparison was an infrared
perimeter observation. The assessment scores show that assimilating even a
small number of additional satellite observations helped produce a better
forecast. Figure H.29 shows graphical representations of these results.

####  4.2.3 Comparison of the Strategies

Each forecast strategy outlined in this chapter was evaluated by comparing a
final forecast perimeter with an infrared perimeter observation using the MOE
and the Sørenson index. For simplicity, a score for each strategy was computed
from the sum of these measures as

$\text{SCORE}=\text{MOE X}+\text{MOE Y}+S.$ (4.2.1)

where MOE X and MOE Y are the two components of the measure of effectiveness
describing decreasing false negatives and false positives, respectively, and
$S$ is the Sørenson index.

Additionally, for the purpose of comparison, evaluation of the initial
forecast produced in the data assimilation cycling strategy was also
evaluated. These forecasts used no data assimilation and serve as a baseline
for comparison. The chief aim for data assimilation is to produce better
forecasts than would otherwise be possible. That aim was met, as both the data
assimilation cycling and the 48-hour forecast strategies produced better
results. The results are summarized in Table 4.6, according to Equation 4.2.1.

Forecast Strategy | Patch Springs | Cougar Creek
---|---|---
Single Forecast | 1.8940 | 1.9927
Data Assimilation | 2.0217 | 2.1081
48-Hour Strategy | 2.0970 | 2.3999
Table 4.6: Overall results of simulation strategies for the Patch Springs and
Cougar Creek Fire simulations. The scores were computed according to Equation
4.2.1. For both fires, the 48-hour forecast strategy produced the best
results. Using a single simulation produced the worst results.

##  Chapter 5 ADJUSTING FUEL MOISTURE CONTENT IN THE MODEL

###  5.1 Modifying Fuel Moisture Content

Fire behavior is affected by weather, terrain, and the properties of the
available fuels. If the fire model has accurate terrain maps and the weather
component of the coupled model supplies accurate forecasts, improving the
fidelity of the fuel properties used by the model is a plausible strategy to
produce better fire forecasts. In this section we will discuss a method to
improve the estimate of the fuel moisture content of the fuels in the domain
of a fire simulation. The method works by comparing the ROS that is computed
from an estimated fire arrival time with the the ROS computed from the
forecast fire arrival time. An adjustment to the fuel moisture content (FMC)
is made so that the ROS of the forecast fire arrival time better matches the
ROS of the estimated fire arrival time that has been made with the use of
satellite fire detections.

The rate of spread in the model can be adjusted by various means. The WRF-
SFIRE model has several parameters that can be changed to affect the way the
factors like wind and terrain influence the ROS. Adjusting the ROS in this
manner would be similar to using the multipliers in the equations determined
by fire personnel at the fire front as was done by Rothermel and Rinehart
(1983) and Finney (1998). Another method for making adjustments to the ROS in
the model in Cardil _et al._ (2019b) for the Wildfire Analyst software.
Changes are made based on comparison of the forecast fire arrival time in
specific locations where the fire has been observed. The method developed here
is similar in the sense that comparison of model output and observational data
is used to make affect the change of the ROS. However, the method presented
here to change the ROS differs. The adjustment factors available in WRF-SFIRE
are not changed; instead, the the fuel moisture content in the input files
used by the WRF-SFIRE is adjusted. By adjusting the FMC, it is possible to
make the ROS faster or slower, as required. For example, drier fuels will make
fires grow faster than comparatively wetter fuels.

Of particular concern is the FMC of dead fuels found in the fire domain. The
FMC of these fuels is influenced greatly by the weather (Mandel _et al._ ,
2014a). These dead fuels are organized into classes defined by the time it
takes for 63% of the difference of the FMC and an equilibrium FMC (that
depends on the weather) to vanish. Thus “1 hour" fuels are small-sized fuels
like dry grass or pine needles that react to the changes in the weather
relatively quickly. The “100 hour" fuels are the size of smaller tree
branches, up to 10 cm in diameter, and react to changes in the weather more
slowly. The fuel moisture estimates used in the WRF-SFIRE system are the
product a data assimilation method developed in Vejmelka _et al._ (2016).

####  5.1.1 Model sensitivity to changes in the FMC

Even small changes in the fuel moisture content can lead to large changes in
the behavior of the modeled fire. Therefore the proposed method takes a
conservative approach to adjusting the FMC. To demosntarte the effect of
changes to the FMC in the model, two simulations of the Patch Springs Fire
were made using WRF-SFIRE, each having identical starting conditions except
for the FMC. One simulation had a fuel moisture content 1% drier than the
other. The results for the fire simulations are shown Figure 5.1. The figure
shows the area of the fire forecasts after 48 and 72 hours of simulation time.
These are simulation times that could be used in an operational setting. It
can be seen that a change in the initial FMC has a large effect on the
eventual size of the fire. Drying the fuel out by one percent resulted in a
simulated fire that was nearly twice the size of the simulation using wetter
fuels.

Fuel class | Base FMC | Revised FMC
---|---|---
1 hr | 0.040 | 0.030
10 hr | 0.060 | 0.050
100 hr | 0.045 | 0.035
1000 hr | 0.050 | 0.040
Live Fuel | 0.270 | 0.26
Table 5.1: Changes to the FMC for an experiment to show model sensitivity to
changes in FMC. The left column shows the fuel class and the center and right
columns show the initial and revised FMC contents.

Figure 5.1: Effect of changing the FMC of the model output. Two simulations
were run with identical initial conditions with the exception of the fuel
moisture content. One the left are contours of the fire arrival time for the
initial simulation. On the right are the contours of the fire arrival time for
a simulation that had decreased the fuel moisture content by 1% in all fuel
categories. The drier fuels resulted in a fire whose area was nearly twice the
size of the fire in the initial simulation.

####  5.1.2 Method

The proposed method for finding an adjustment to the FMC is simple and should
be considered more of an initial proof of concept rather than established an
algorithm. The ROS can be influenced by many factors and this method works by
comparing a forecasted ROS and a ROS estimated from satellite data. It is an
open question how to best change the FMC in a way that accounts for the
difference in these two rates of spread. Therefore, the method makes small,
conservative changes only. Changes are made in the course of a simulation
routine that uses data assimilation cycling. If a change to the fuel moisture
is made, its effect will occur during the forecast period of the cycle
following the data assimilation. The adjustment of the FMC is accomplished by
the following steps.

  1. 1.

Compute the final fire areas within the estimated and forecast fire arrival
time perimeters.

  2. 2.

Compute the ROS in the fire domain from the gradient of the estimated fire
arrival time and the forecast fire arrival time.

  3. 3.

Consider only ROS less than 2 m/s. When computing the ROS from the gradient of
the fire arrival time, it is typical that in many places the ROS will be
greater than 2 m/s. These are very fast and not consistent with fires the
spread by a mechanism modeled by WRF-SFIRE, so we only consider a ROS below
the 2 m/s threshold. Additionally, when this threshold is not enforced, the
presence of high rates of spread, as computed from the gradient of the fire
arrival time, act as statistical outliers that can have a large effect on the
average ROS. The upper left panel of Figure 5.4 shows a surface plot of the
ROS containing locations in the fire domain where the computed ROS was more
that $2\times 10^{1}1$ m/s.

  4. 4.

Compute the mean difference in the ROS for forecast and estimate fire arrival
time in areas where both fires showed burning took place.

  5. 5.

There should be agreement between observations about the area of the fire and
the differences in ROS to suggest making a change to the FMC. For example, if
the the forecast area is smaller than the estimate area but the the estimated
ROS in the forecast is greater than the estimated ROS, no changes should be
made.

  6. 6.

Collect fuel information about the fire domain and find the inverse function
for the burn curve in the primary fuel type found in along the paths in the
graph. In this research, the inverse function was found by making a cubic
spline of the function seen in Figure 5.2.

  7. 7.

Compute adjustment of FMC so that the mean ROS for the forecast matches that
of the estimate, according to the burn curve.

  8. 8.

Adjust the fuel moisture content used by the model. In WRF-SFIRE, this means
adjusting the variable FMC_GC in the input file used for the next cycle.

Figure 5.2: Tools for adjusting the FMC in the simulation. On the left, a
histogram of the differences in the model ROS and the ROS implied by the data.
The forecast was underestimating the ROS. On the right, curves giving the
relationship between fuel moisture content and ROS can be used to adjust the
fuel moisture content for the next simulation cycle.

Figure 5.3: Fuels underlying the shortest paths in the detection graph for the
Patch fire simulation. On the left, we see that the majority of the fire took
place in the type 2 fuels. On the right, the paths though the detections
placed on a map of the underlying fuels are seen. The color bar indicates the
fuel the type found at each location.

##### ROS of the Estimate and Forecast

Figure 5.4: The ROS seen spatially in the fire domain. In the upper left,
several large spikes of the ROS in forecast fire arrival time can be seen. For
the estimation of changes in the FMC, only ROS less than 2 m/s are considered.
The upper right and bottom panel show the location of these lower rates of
spread in the forecast and estimate from satellite data, respectively.

####  5.1.3 Examples of Forecasts Using Adjustment of Fuel Moisture Content

Simulations of the Patch Springs and Cougar Creek fires were made using a
workflow and starting conditions that were identical to those detailed in
Chapter 4, with the exception that adjustments of the FMC were made during the
data assimilation phase of each cycle. The results were mixed, in the case of
the Patch Springs Fire, the assessment of the simulation with 58 hours of
forecast time shows a result that was a little worse than that of a simulation
where the FMC was not changed at any stage. The simulation of the Cougar Creek
Fire using adjustment of the FMC was a little better than the simulation where
no changes were made to the FMC. The following sections detail the experiments
and show how these conclusions were derived.

##### Patch Fire Simulation with FMC Adjustments

A simulation of the Patch Springs fire was run using the cycling strategy as
in Section 4.2.1, except that a change to the FMC was made during the data
assimilation phase, according to the procedure outlined above. For simplicity,
if a change to the FMC was suggested by the algorithm, it was applied to all
fuels in the entire fire domain. Figure 5.5 and Figures H.31-H.33 show how the
data assimilation made adjustment to the fire arrival time during the
simulation cycles as well as a histogram showing how the ROS in the forecast
differed from the estimated ROS. Table 5.2 shows a comparison of the forecast
and estimated fire areas as well as the average ROS in the forecast and
estimated fire arrival time for each cycle. When any adjustment to the FMC was
made by the algorithm, that adjustment is also indicated.

Assessment of the simulation was made by comparison with and infrared
perimeter and the results are summarized in Table 5.3. Figure H.34 gives a
graphical representation of these results. When looking at the results from
producing 58 hours of forecast, after the FMC adjustment, the results are a
little bit worse compared to using data assimilation without changing the FMC.
The increase in the FMC caused the estimated fire area to be smaller than it
should have been. However, as the simulation progressed, the scores for later
cycles eventually became higher than those for the simulation without changes
made to the FMC. It is likely that the adjustments made to the 100-hour fuels
in the fire domain played a part of this improvement.

Figure 5.5: Cycle 0 of the Patch Fire simulation with adjustment of FMC. In the upper left and right are contours of of the fire arrival time for the forecast and the analysis, respectively. On the bottom is the histogram showing the difference between the ROS at mesh points in the fire domain. The area of the forecast is smaller, but the ROS was higher in the forecast so that no changes to the FMC were made for the next simulation cycle. Cycle | Estimate Area | Forecast Area | Estimate ROS | Forecast ROS | FMC Adjustment %
---|---|---|---|---|---
0 | 767 | 1126 | 0.1818 | 0.1949 | 0
1 | 1603 | 1684 | 0.1755 | 0.2051 | 0.1967
2 | 1998 | 1950 | 0.0994 | 0.1802 | 0
3 | 2734 | 2343 | 0.0867 | 0.1048 | 0
4 | 3602 | 3212 | 0.0912 | 0.1087 | 0
Table 5.2: Criticial data for making adjustments to the FMC in a simulation of the Patch Springs fire. Computations made on a 250m grid. No FMC adjustments were made during the initial cycles because the size of the forecast fire was too large but the ROS in the forecast was lower than that in the estimate from data. Cycle | Hours of Forecast | MOE X | MOE Y | ||MOE|| | Sorenson
---|---|---|---|---|---
0 | 130 | 0.4920 | 0.7629 | 0.9078 | 0.5982
1 | 82 | 0.7543 | 0.6413 | 0.9901 | 0.6932
2 | 58 | 0.6247 | 0.7157 | 0.9500 | 0.6671
3 | 34 | 0.6873 | 0.6869 | 0.9717 | 0.6871
4 | 10 | 0.8925 | 0.6684 | 1.1150 | 0.7643
Table 5.3: Assesment of the Patch Fire Simulation With Adjustment of FMC.
Compare with the results in Table 4.1. The first two cycles had identical
results since no changes were made to the FMC. The simulation with 58 hours of
forecast was slightly worse than if no changes had been made to the FMC. The
final forecasts were better, using the FMC adjustment.

##### Cougar Creek Fire with FMC Adjustment

A simulation of the Cougar Creek fire was run using the cycling strategy as in
Section 4.2.1 except that a change to the FMC was made during the data
assimilation phase. If a change to the FMC was suggested by the algorithm, it
was applied to all fuels in the entire fire domain. Figures 5.6,5.7, H.35, and
H.36 show how the data assimilation made adjustment to the fire arrival time
during the simulation cycles as well as a histogram showing how the ROS in the
forecast differed from the estimated ROS. Table 5.4 shows a comparison of the
forecast and estimated fire area as well as the average ROS in the forecast an
estimate for each cycle. When any adjustment to the FMC was made by the
algorithm, that adjustment was indicated too.

Assessment of the simulation was made by comparison with and infrared
perimeter and the results are summarized in Table 5.5. Figure H.37 gives a
graphical representation of these results. Compared to using data assimilation
without changing the FMC, these results are a little better. In particular,
the change of the FMC in the second cycle made for wetter fuels that helped
constrain the growth of the fire, making for a better ultimate forecast.

Figure 5.6: Cycle 0 of the Cougar Creek Fire simulation with adjustment of
FMC. In the upper left and right are contours of of the fire arrival time for
the forecast and the analysis, respectively. On the bottom is the histogram
showing the difference between the ROS at mesh points in the fire domain. The
forecast underestimated the size of the fire and the ROS so that the FMC was
adjusted downward by -0.0574%

Figure 5.7: Cycle 1 of the Cougar Creek Fire simulation with adjustment of FMC. In the upper left and right are contours of of the fire arrival time for the forecast and the analysis, respectively. On the bottom is the histogram showing the difference between the ROS at mesh points in the fire domain. The forecast overestimated the size of the fire and the ROS so that the FMC was adjusted upward by 0.1947% Cycle | Data Area | Forecast Area | Data ROS | Forecast ROS | FMC Adjustment %
---|---|---|---|---|---
0 | 18498 | 12924 | 0.1379 | 0.1006 | -0.0574
1 | 78038 | 122154 | 0.1397 | 0.186 | 0.1947
2 | 151647 | 148920 | 0.1111 | 0.1517 | 0
3 | 161939 | 135567 | 0.0948 | 0.1066 | 0
Table 5.4: Adjustments made to the FMC in the simulation of the Cougar Creek Fire. Adjustments were only made after cycle 0 and cycle 1. In cycles 2 and 3, the larger fire area implied by the data contradicted the larger average ROS found in the forecast. The algorithm makes no adjustments under these circumstances. Cycle | Hours of Forecast | MOE X | MOE Y | ||MOE|| | Sorenson
---|---|---|---|---|---
0 | 105 | 0.6638 | 0.6647 | 0.9394 | 0.6642
1 | 81 | 0.982 | 0.3736 | 1.0507 | 0.5413
2 | 57 | 0.7025 | 0.7353 | 1.0169 | 0.7185
3 | 33 | 0.6918 | 0.8971 | 1.1329 | 0.7812
4 | 9 | 0.8919 | 0.8045 | 1.2011 | 0.8459
Table 5.5: Assessment of the Cougar Creek Fire simulation using adjustment of
the FMC. Compare with the results in the test without adjustment of FMC in
Table 4.2.

####  5.1.4 Limitations of the method

The proposed method for adjusting the FMC used by the model has some
limitations and drawbacks that are explored in the following list. Some
possible remedies are suggested as well. The goal of adjusting the FMC is to
make the ROS in the simulation more closely match what is observed in the real
world. Because fire behavior is such a complicated phenomena, other approaches
for producing a better ROS in the model could be explored as well as what was
presented in this research.

  1. 1.

The method does not consider all inputs that effect the ROS of a fire such as
fuel types, terrain, etc. Controlling for these other inputs would allow for a
more precise, and likely more conservative, adjustment of the FMC. For
example, if strong winds were present in the simulation, it could be the case
that the effect of FMC on the ROS of the fire played a comparatively minor
role when compared to the wind.

  2. 2.

Comparison between the forecast ROS and the estimated ROS is made at specific
locations within the fire domain without regard to the changing properties of
the fuels. For example, if the forecast fire arrival time at a specified
location differs from the estimated fire arrival time at that location by 12
hours, it is likely that the underlying fuels in that area should be expected
to have different FMC because of the diurnal cycle of FMC increasing during
the cooler period of night and decreasing during hotter days. These diurnal
changes have a larger affect on the the 1-hour and 10-hour fuels, but in
principle, all fuels should be expected to have different properties at
different times. The adjustment of FMC might be more accurate if comparison of
the ROS was done only for location where the difference in the forecast and
estimated fire arrival time was was not too large. A simpler method would be
to only make adjustments to the FMC for the 100-hour and 1000-hour fuels that
are less sensitive to changes in the weather and are observed more sparsely.

  3. 3.

The cut-off of ROS above 2 m/s is arbitrary. No testing was done to arrive at
this number. With certain combinations of weather, topography,and fuels
properties some fires could exhibit such rapid growth. A better cut-off for
the ROS could be possibly be found by using the estimates of the variance for
the ROS found in Section 3.5.

##  Chapter 6 CONCLUSION AND FURTHER WORK

###  6.1 Conclusion

####  6.1.1 Main results

The goal of this research was to develop techniques for using satellite data
to provide improvements to wildfire modeling. The first of three three goals
was to develop methods to obtain better initial conditions for simulating a
wildfire by estimating either the ignition point or fire arrival time of a
wildfire. The estimation of the fire arrival time should be considered the
most useful of these two methods since it paves the way for starting better
wildfire simulations and the estimate obtained may be used to infer other
properties about a fire such as its ROS. The second goal was to develop a
method for using satellite data to update a running fire simulation by data
assimilation in a way that is justifiable by the known properties and
uncertainties of the those data. The third goal was use satellite data to
adjust the fuel moisture content used by the model to make better fire
forecasts possible. The first two goals have been met, but the third goal
needs more development before it can be called an unqualified success. Some of
the main results are summarized in this section.

####  6.1.2 Estimation of Igbition Point and Fire Arrival Time

The first part of this thesis detailed methods used to estimate the ignition
point location and the fire arrival time from satellite fire data. Because the
the ignition point location and the fire arrival time of real wildfires are