# Recursive Updates of Wildfire Perimeters Using Barrier Points and Ensemble Kalman Filtering

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## Abstract

This paper shows how the wildfire simulation tool farsite is augmented with data assimilation capabilities that exploit the notion of barrier points and a constraint-point ensemble Kalman filtering to update wildfire perimeter predictions. Based on observations of the actual fire perimeter, stationary points on the fire perimeter are identified as barrier points and combined with a recursive update of the initial fire perimeter. It is shown that the combination of barrier point identification and using the barrier points as constraints in the ensemble Kalman filter gives a significant improvement in the forward prediction of the fire perimeter. The results are illustrated on the use case of the 2016 Sandfire that burned in the Angeles National Forest, east of the Santa Clarita Valley in Los Angeles County, California.

## Keywords

Wildfire Barrier points Ensembles Ensemble Kalman filter farsite## 1 Introduction

The ability to reliably predict fire perimeter propagation during a wildfire event has a large potential in resource allocation and fire fighting planning to help save lives and valuable infrastructure. Studying wildfire dynamics is done by collecting data [3] and combining wildfire simulation tools with experimental data to assimilate or adjust the widfire simulation [11, 12, 13, 14]. Previous work on data assimilation using the farsite [7] wildfire simulation tool combined with ensemble Kalman filtering can be found in [5, 16]. Next to large body of work that use some form of Kalman filtering, [2, 4, 8, 10] there are alternative approaches that use Genetic Algorithms to determine the best set of input parameters to match the measurements [1]. The power of a data driven approach is also confirmed by [17] illustrating improvements to wildfire prediction on data obtained from physical experiments.

Fire perimeters that may be obtained periodically during a wildfire event may be well-suited for periodic or recursive updates of the initial conditions (e.g. initial fire perimeter) and the relevant parameters that govern the wildfire dynamics. For a Kalman filter-based approach it is essential that wildfire perimeter measurements are quantified with a measurement accuracy to find the optimal trade-off in adjusting initial conditions and wildfire parameters. To this extend, the work by [14] uses thermal-infrared imaging to measure the true fire perimeter on a controlled fire experiment done on a (4 m \(\times \) 4 m) patch of land and [17] used ForeFire/Meso-NH simulations produced by [6] as observations. Unfortunately, such methods cannot be employed for time-sensitive and large scale wildfires, where perimeters are obtained with aerial measurements and computations of future wildfire perimeters must be done in near real-time.

To improve the quality of wildfire prediction and to speed up computations, one piece of critical information is often neglected in wildfire data assimilation: stationary points at which a (part of the) fire perimeter remains at the same locations between periodic updates. Clearly, those points can be characterized with a relatively high accuracy and do not require computational updates. In terms of data assimilation, such stationary points can be viewed as constrains in a constrained Ensemble Kalman Filtering [15] formulation. Fortunately, the farsite [7] wildfire simulation tool has the notion of barrier points to account for stationary fire perimeters. Identifying such stationary or barrier points on the fire perimeter and combining this information with ensemble Kalman filtering is the main contribution of this paper.

The results presented are based on the work in [5, 16] to fully use the information of barrier points in the prediction and update steps of the data assimilation tools for farsite. The approach presented in this paper performs recursive data assimilation to estimate the true values of fuel dependent adjustment factors along with wind speed and direction that influence fire spread rate by including them in the state updates. Estimation of these input parameters along with the identification of barrier points further improve the periodic prediction of fire perimeters. The data assimilation tools are tested on actual wildfire perimeter data that was obtained for the 2016 Sandfire that burned in the Angeles National Forest, east of the Santa Clarita Valley in Los Angeles County, California.

## 2 Contour and Stationary Points

### 2.1 Fourier Analysis

*n*-polygon described by a ordered sequence of

*n*piece-wise linear line segments parametrized in Eastern \(e_j\) and Northern \(n_j\) coordinate pairs \((e_j,n_j)\), \(j=0,1,\ldots ,n-1\). To simplify notation, we may represent the

*n*coordinates of the

*n*-polygon as a complex number \(p_j = e_j + i\cdot n_j\) for which we can define a complex Discrete Fourier Transform (DFT)

*n*-polygon should be independent of the starting point \(p_1\) and the (anti)clock wise rotation of the sequence \(p_j\) in the complex plane. A shift in the starting point or rotation can be easily represented in the Fourier series \(\bar{u}_l\) by an additional phase shift of \(u_l\) and given by \(\bar{u}_l = u_l e^{-i \phi }\), where the rotation angle \(\phi \) is determined by an integer shift in the starting point and the binary choice on the anti-clockwise or clockwise rotation of \(p_j\) [9]. The Fourier series representation \(u_l\) of the

*n*-polygon approximation of a fire perimeter allows fire perimeters at subsequent time steps

*k*and \(k+1\) to be compared as parts of the fire perimeter might be stationary. A non-moving or stationary part of the fire perimeter may be due to the presence of a nonburnable surface fuel or explicit fire fighting efforts in which part of the surface fuel has been removed or extinguished. Such information must be taken into account to improve the prediction of fire perimeter progression over time.

*n*-polygon \(p_j(k)\), \(j=0,1,\ldots ,n-1\) at time step

*k*and a

*m*-polygon \(p_j(k+1)\), \(j=0,1,\ldots ,m-1\) at a subsequent time step \(k+1\). As \(n \not = m\) and the starting point \(p_1(k) \not = p_1(k+1)\), the simple check of \(p_j(k) = p_j(k+1)\) will not suffice in determining the stationary points \(p_j(k)\) of the fire perimeter. Instead, we first consider the Least Squares minimization

*m*-polynomial \(p_j(k+1)\) by evaluating the difference between \(s = \min (m,n)\) Fourier coefficients. The end result is a set of Fourier coefficients \(\bar{u}_l(k+1) = u_j(k+1) e^{-i \bar{\phi }}\) for which the inverse DFT will lead to a re-oriented

*m*-polygon \(\bar{p}_j(k+1)\), \(j=0,1,\ldots ,m-1\) of the fire perimeter at time step \(k+1\) that can be compared with the fire perimeter \(p_j(k)\), \(k=0,1,\ldots ,n-1\) at time step

*k*. Stationary points are now defined as the set of points \(p_j(k) \in \mathcal{P}_k\) on the fire perimeter \(p_j(k)\) at time step

*k*for which

*k*and \(\bar{p}_j(k+1)\) at time step \(k+1\) must lie within a distance of \(\varepsilon \) to qualify as stationary points.

### 2.2 Illustration and Boundary Points

With the re-orientation of the fire perimeter points found by the optimization in (2), the location of stationary points identified by the black squares in Fig. 1 become apparent. It should be noted that stationary points along the fire perimeter might persist only for certain amount of time, as the fire could eventually progress. For example, the stationary points shown in Fig. 1 are only valid for the fire perimeter at time step \(k=0\). To identify the stationary points for the fire perimeter at \(k=1\), measurement at time step \(k=2\) are required. Clearly, the information on the set of stationary points is important in predicting the spread of wildfires accurately. The wildfire simulation tool farsite [7] has the notion of a barrier perimeter defined by barrier points to account for the identified stationary points in the fire perimeter by temporarily defining surface fuels as non-burnable.

## 3 Data Assimilation

### 3.1 Ensemble Forward Simulation

*n*real valued eastern- and northern-coordinates of a fire perimeter \(p_j(k)\) at time step

*k*to simulate a fire perimeter \(p_j(k+1)\) at time step \(k+1\). Next to the initial perimeter \(p_j(k)\) specified in a shape file, farsite also uses information on surface fuels, topography, wind speed, wind direction and fuel adjustment factors, collectively combined in the environmental parameter \(\theta _k\) at time step

*k*, to adjust the fire perimeter prediction [16]. In addition, farsite can account for a set of barrier points \(b_j(k) \in \mathcal{P}_k\) as defined in (3) to approximate the stationary points of the fire perimeter. Using only real valued calculations, farsite can be viewed as a non-linear mapping

*X*(

*k*) is a vector of the eastern- and northern coordinates of the wild fire perimeter \(p_j(k)\) and \(B_k\) is a vector of the eastern- and northern coordinates of the barrier points \(b_j(k)\).

It should be noted that the non-linear map \(f(\cdot )\) is not known analytically and both sensitivity or uncertainty of the inputs *X*(*k*), \(\theta _k\) and \(B_k\) can be evaluated numerically via ensemble averaging. For that purpose, random samples (ensembles) are chosen from a probability description of the initial fire perimeter *X*(*k*) and possibly the environmental parameters \(\theta _k\). In this paper only uncertainty on the initial fire perimeter *X*(*k*) is considered in the form of a covariance on the vector *X*(*k*), whereas the environmental parameters \(\theta _k\) and the barrier points \(B_k\) are assumed to be fixed. The later is a reasonable assumption as barrier points are defined as stationary points, whereas variability in \(\theta _k\) can be considered as a possible improvement for the ensemble forward simulation presented in this paper.

*X*(

*k*) for fire perimeter points, the proximity of the eastern- and northern-coordinates to its neighboring points is used. The proximity measure for the covariance \(\sigma _{e_j,n_j}\) for each point \((e_j,n_j)\) on the

*n*-polygon of the fire perimeter is defined by \(\sigma _{e_j,n_j} = h(e_{j-1},y_{j-1},e_{j},n_{j},e_{j+1},n_{j+1})\), where \(h(\cdot )\) is a function that computes the measure of closeness of each point to its neighboring points on the fire perimeter. The value of \(\sigma _{e_j,e_j}\) is inversely proportional to the distance of \((e_j,n_j)\) with its neighboring points. Using the environmental parameters \(\theta _k\), barrier points \(B_k\) and

*N*ensembles \(X^i(k)\) taken from a normal distribution determined by the mean value

*X*(

*k*) and the covariance \(\sigma _{e_j,n_j}\), each ensemble member \(X^i(k)\) at time step

*k*is advanced through the forward model

*recursively*from a measurement of a fire perimeter \(X(k+1)\) at time step \(k+1\) by the optimization in (2) and the definition of the stationary points in (3) using the mean value

*X*(

*k*).

*N*predicted fire perimeters \(X^i(k+1)\) or \(p_j^i(k+1)\) for \(i=1,2,\ldots ,N\) and \(j=0,1,\ldots ,n_i\) have become available. For ensemble averaging, a mean value and a covariance must be estimated, but typically two different fire perimeter boundaries \(X^i(k+1)\) and \(X^j(k+1)\) will not have the same number \(n_i\) of polygon points, starting point of orientation. To address this issue, first all

*N*predicted fire perimeters \(p_j^i(k+1)\), \(i=1,2,\ldots ,N\) are interpolated to the same number of (maximum) points \(n_{max} = \max _i(n_i)\). Subsequently, an optimization similar to (2) is performed to align the starting point and orientation of all the ensembles. Denoting \(u^1_l(k+1)\) as the DFT transform of the first (primary) ensemble \(p_j^1(k+1)\) for \(j,l = 0,1,\ldots ,n_{max}\), the other ensembles are aligned using the minimization

*i*in the exponent \(e^{-i \phi }\) still denotes the complex number \(i^2 = -1\). The \(N-1\) optimizations in (5) leads to a re-oriented set of ensembles represented by the vector \(\bar{X}^i(k+1)\) for \(i=2,\ldots ,N\) and found by the inverse DFT of \(\bar{p}^i_j(k+1) = u^i_l(k+1) e^{-i \phi ^i}\). The end result of this process is set of properly aligned

*N*ensembles \(X^1(k+1)\) and \(\bar{X}^i(k+1)\), \(i=2,\ldots ,n\) for which a mean \(X^\mu (k+1) \in \mathcal {R}^{n_{max}}\) and a covariance \(P^\mu (k+1) \in \mathcal {R}^{n_{max},n_{max}}\) can be computed.

### 3.2 Ensemble Kalman Filter Update

In the ensemble Kalman filter update, a measurement of a fire perimeter obtained at time step \(k+1\) is consolidated with the prediction of the mean \(X^\mu (k+1) \in \mathcal {R}^{n_{max}}\) and the covariance \(P^\mu (k+1) \in \mathcal {R}^{n_{max},n_{max}}\) obtained from the ensemble forward simulation described above. For the optimal consolidation of the measurement and the prediction, we assume that the fire perimeter obtained at time step \(k+1\) is described by its mean \(Y(k+1)\) and a covariance \(P^y(k+1)\). It is worth noting that the covariance \(P^y(k+1)\) may be determined by either the inherent limited accuracy in obtaining the measurement \(Y(k+1)\), the relative distance between the points on the perimeter *Y*(*k*) or estimated by computing a two-dimensional variance from multiple measurements [5].

As the points of the measured fire perimeter \(Y(k+1) \in \mathcal {R}^m\) and typically \(m \not = n_{max}\), the different size of the observation and prediction is handled by linear interpolation of \(Y(k+1)\) and \(P^y(k+1)\) to \(n_{max}\) points. Subsequently, the following steps are done for each ensemble pair contained in \(\tilde{X}\) and \(\tilde{Y}\).

- 1.Find the closest point on \(\tilde{X}^e\) to each point in \(\tilde{Y}^e\) and pair them up, store the pair for which magnitude of distance is minimum and discard others, i.e. capture
*i*that satisfies,along with$$\begin{aligned} \begin{aligned} \min _i (\min _j |y_i^e - x_j^e| ) \quad \forall i = 1,...,m/m_i \\ j = \{1,2,...,n_{max}\}/m_j \end{aligned} \end{aligned}$$*i*store its corresponding*j*in \(m_i\) and \(m_j\) respectively - 2.
Repeat the step above until each point in \(\tilde{Y^e}\) is paired with a unique point in \(\tilde{X^e}\).

- 3.This pairing scheme is used to construct the
*C*matrix that is required to perform the Kalman update step to get the updated perimeter using the Kalman gain*K*via$$\begin{aligned} X_{updated}^e = \tilde{X}^e + K [\tilde{Y}^e - C\tilde{X}^e] \end{aligned}$$(6)

## 4 Results for the 2016 Sand Fire

### 4.1 Illustration for Single Step Data Assimilation

For the illustration of the data assimilation with barrier points, ensemble forward simulations and actual measurements of the 2016 Sandfire in Los Angeles County, California are used at time steps *k* separated by 2 h intervals, as shown in Fig. 2.

In this case, the input fire perimeter given to farsite at step \(k=1\) shown in Fig. 2 is obtained from ensembles of the last fire perimeter depicted earlier in Fig. 1. Note that the variability along the fire perimeter at step \(k=1\) is very small and virtually non-existent in certain parts due to the stationary points identified from the previous time step at \(k=0\).

From Fig. 3 one may recognize the stationary points for the fire perimeter at time step \(k=1\) indicated earlier in Fig. 2. In addition, it can be observed (especially in the bottom right of the graph) that the ensemble forward simulation results may be biased and may have an incorrectly estimated covariance compared to the measurements. It is clear that the forward simulations require a data assimilation step to provide a better fit to the measured fire perimeter \(Y(k+1)\) and adjust the covariance information.

### 4.2 Definition of Fire Coverage Error

### 4.3 Performance Comparison of Data Assimilation with Barrier Points

*with*and

*without*the identification of stationary points.

To summarize the improvement in performance, the mean value of the fire coverage error \(A_e\) as defined in (7) along with the upper and lower bounds due to the variability on the fire perimeters is computer for the different time steps. The results are summarized in Fig. 6 and the improvement in performance measured in fire coverage error is evident from the graph. Both uncertainty and magnitude of the fire coverage error have been reduced when data assimilation is performed, but results are further improved when stationary points are identified and used as barrier points in the ensemble forward simulations. Especially the upper bound on the fire coverage error remains at acceptable levels during several data assimilation steps when using the identified stationary points on the fire perimeter.

## 5 Conclusions

This papers shows the importance of data assimilation combined with the identification of stationary points in correcting the prediction of the spread of wildfires at a large scale. Stationary points are identified by comparison of subsequent fire perimeters and used in farsite to define barrier points to limit fire propagation. Based on the observations of the 2016 Sandfire, it is shown that the combined use of data assimilation and barrier points can significantly improve the fire coverage error. It is worth noting that an Ensemble Kalman Filter (EKF) approach is used to perform data assimilation, where a Gaussian representation of the fire perimeter vector is assumed. However, in certain situations it would be more suitable to implement a particle filter to take into account non-Gaussian distributions of fire perimeter uncertainty for future research directions.

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