On the Use of Semi-empirical Flame Models for Spreading Chaparral Crown Fire

Flame geometry plays a key role in shaping fire behavior as it can influence flame spread, radiative heat transfer and fire intensity. For wildland fire, a thorough understanding of relationships between flame geometry including flame length, flame height and flame tilt can help advance the derivation of comprehensive models of wildfire behavior. Within the fire community, a classical flame modeling approach has been the development of semi-empirical models. Many of these models have been derived for surface fuels or for pool fire configurations. However, few have sought to model flame behavior in chaparral crown fires. Thus, the objective of this study was to assess the applicability of existing semi-empirical models on observed chaparral crown fire geometry. Semi-empirical models of flame tilt, flame height and flame length were considered. Comparison with experimental observation of crown fuel layer flame height showed good agreement between two-fifths power law that relates flame height to heat release rate. Predictions of flame tilt were obtained from application of semi-empirical power-law correlations relating flame tilt angle to Froude number. Observed flame tilt values exhibited low correlation with predicted values. Thus, two new power-law correlations were proposed. Coefficients for new models were obtained from regression analysis.


INTRODUCTION
The occurrence of large fres has increased signifcantly in many regions around the world. One region particularly impacted by wildfres is southern California, where the terrain, highly fammable fuels, dry ambient conditions and fast foehn type winds (known locally as Santa Ana winds) generate conditions highly favorable to wildfre (Rothermel and Philpot, 1973). Thus, fuel and weather conditions exist such that in the event of an ignition event, the potential for wildfre is high. In the southern California case, growing wildfre potential, and fast population growth have occurred in parallel. This coupled growth has prompted changes to the so-called wildland urban interface, that is, the region separating the wildland from urban settlements. The growth of the wildland urban interface coupled with increased fre risk, places people and their property closer to fre. Because of the growing threat, the ability to accurately predict fre behavior has become paramount. This is contingent on thorough understanding of physical mechanisms driving fre spread and intensity. Because wildfre behavior is shaped by its environment, it is important to defne the key conditions shaping fre behavior in a regional landscape and climate. In mediterranean climates, chaparral fres typically burn as crown fres (Barro and Conard, 1991), a category of fre consisting of two fuel layers, an above ground surface fuel layer and an elevated fuel layer known as a crown layer. In chaparral crown fres, fres typically start in the easily ignitable surface fuels and spread in the crown fuel layer (Tachajapong et al., 2014). Before a fre can spread in the crown, the fre must move vertically from the surface fuels to ignite crown fuels, a process defned as transition (Weise et al., 2018a).
Little is known of the exact mechanisms which produce eective transition and spread in chaparral crown fres. Transition and spread in crown fre, involves a dynamic energy exchange between the surface and crown fuel layers. Spread in the crown fuel layer may require energy to be supplied from the surface fuel layer, as in passive or dependent crown fres; or may rely on the crown fuel layer alone to maintain successful spread, as in independent fres (Van Wagner, 1977). In the case of crown fre spread where energy is partially or solely supplied by the crown fuel layer, identifying the mechanism through which energy is exchanged from the crown fame to unburned fuel is necessary to better understand mechanisms for successful crown fre spread. Hence, assessing fame properties in the crown fuel layer, particularly fame geometry, may be a key step in generating a rigorous characterization of chaparral crown fre behavior.
Flame geometry has been shown to infuence fame spread through radiative heat transfer (Albini, 1985). Thus, numerous groups have focused on assessing fame geometry properties as they relate to fre spread. It is pertinent then to present a brief review of studies examining fame geometry characteristics for wildland fuels; this now follows. Byram (1959) conducted foundational work to characterize combustion and fre behavior in forest fuels. The Byram intensity which defnes heat release rate per unit time per unit length, is perhaps the most widely accepted expression for fre intensity. Byram proposed early correlations relating fame length to fre intensity. Since the frst formulation, numerous groups have derived semi-empirical expression of fame length as a function of fre intensity for various fuels. Thomas proposed correlations relating fame height to fuel supply rate and burner dimensions in conditions without wind (Thomas et al., 1961) and with wind (Thomas, 1963). In work by Nelson (1980) theoretical formulations for fame length, height, tip velocity and tilt angle as a function of Byram intensity were examined for light southern pine fuels. In addition to theoretical modeling, the work presents results from semi-empirical power-law modeling of fame length and tip velocity as a function of Byram intensity. Steward (1970) derived mathematical expressions relating mass fow rate to fame height. Zukoski et al. (1980) examined entrainment characteristics in methane diusion fames and proposed powerlaw correlations of fame height as a function of heat release rate and burner diameter. Similarly, Heskestad (1983Heskestad ( , 1984 related fame height to heat release rate and burner diameter. Other recent studies of fame conditions and fame spread include those by Gang et al. (2017) and Zhou et al. (2018). Fernandes et al. (2009) derived empirical correlations of fame length and fame height for head and back wildfres. They expressed fame length and height of head fres as a function of Byram intensity and fuel loading. For back fres, both fame length and height were expressed as a function of Byram intensity. Alexander and Cruz (2012) surveyed expressions of fame length presented as function of fre intensity. Alexander and Cruz (2012) identify the signifcance of fame length to crown fuel layer ignition behavior and highlight power-law expressions relating Byram intensity to fame length for various fuels. Fernandes et al. (2000) derived a powerlaw expression relating Byram intensity for fame length in shrublands. Other recent studies of fame spread include those by Gang et al. (2017) and Zhou et al. (2018).
Works focusing on fame geometry for shrub and chaparral fuels include computational evaluation of fame properties such as the one by Padhi et al. (2016) in which fame geometry in a stationary shrub fre was considered. Moreover, a numerical analysis of fame tilt angle and height, in a spreading shrub fre was presented by Morvan (2007). Recent work by Weise et al. (2018b) compared predictions from fame models to results from experimental circular and line fre confgurations of chaparral fre. Model predictions of fame height and fame tilt angle, were compared against experimental values in work by Nelson et al. (2012). Laboratory scale work by Weise and Biging (1996) evaluated the eect of wind and slope on fame properties. Importantly, the previous experimental studies did not include a dual layer, crown fre confguration.
Results from the works reviewed above include semi-empirical correlations which show promise in predicting fre spread behavior. However, few of these semi-empirical models have been produced through the study of chaparral fre modeled as crown fre, as done when modeling chaparral fre with distinct fuel layers for surface and crown fuels. Thus, the aim here is to examine crown fame geometry and to survey the applicability of semiempirical models of fame geometry to chaparral fres modeled as dual-layer crown fres. To the knowledge of the authors, no prior work has attempted to use established models of semi-empirical fre spread for chaparral fuels modeled with distinct layers for the surface and crown fuel beds. We consider that modeling chaparral fres with a dual layer confguration will more precisely replicate spread behavior as it can capture the dynamic energy exchange between the surface and crown fres. To this purpose, this paper compares models of fame geometry to observations of fame data obtained from wind tunnel experiments in which the surface and crown fuel beds were modeled as separate fuel beds. Data from experiments with wind-blown spread are examined. The next section describes the experimental procedure and modeling approach.

METHODOLOGY Experimental
Experiments were conducted in a specialized wind tunnel located at the USDA Forest Service Pacifc Southwest Research Station fre laboratory in Riverside, California. The wind tunnel study area was composed of two distinct fuel beds representing the dead fuel surface layer and the live fuel crown layer. The surface fuel layer was constructed on the wind tunnel foor and a platform mounted on the top of the tunnel frame contained the crown fuel bed (see Figure 1). Aspen (Populus tremuloides Michx.) excelsior (shredded wood) served as the surface fuel; crown fuels consisted of chamise (Adenostoma fasciculatum Hook & Arn.) branches and foliage harvested locally. Custom instrumentation was developed to measure mass loss from the crown fuel layer; full details of this system can be found in Cobian-Iñiguez et al. (2017). Surface fuel mass loss was measured using an electronic scale placed under a portion of the excelsior fuel bed. Fires were started by igniting the surface fuel bed (excelsior) using a butane torch and ethyl alcohol as lighter fuid. Wind was activated simultaneously with surface fuel bed ignition. Once ignited, the excelsior fuel bed developed a fame and the fre spread. The surface fre spread under the crown fuel thus preheating it to the point of ignition, at this point the fre transitioned to the crown fuel layer. Thereafter, a fame developed in the crown fuel layer and the fre in the crown fuel layer was allowed to spread until extinction. Before explaining the properties that were measured using the image processing techniques here, it is necessary to note some basics of fame geometry. In microgravity, experiments have shown and demonstrated that a laminar diusive fame has a spherical shape. When gravity is applied, a good approximation is to assume that gravity forces will stretch the spherical shape of the fame like a candle and the resulting shape will be an ellipsoidal. Following the same logic, when analyzing the shape of a turbulent diusive fame, an ellipsoidal (in the case of 2D projection, an ellipse) can be used as a reference for fame characteristics such as fame length and fame tilt.
Flame geometry was obtained from video recordings obtained using a Sony Handicam 1 at 30 frames per s. We used two dierent algorithms for video data processing: one for fame height, H, see Figure 1, and another one for fame tilt angle, f , and fame length, L f , see Figure 2F. The fame height algorithm was generated in MATLAB. The script was designed to convert raw red-green-blue (RGB) images to black and white images through thresholding in order to isolate the fame and generate a fame perimeter image. Flame height was obtained from the fame perimeter image. Video data was resampled from 30 to 1 Hz. Once the datum were re-sampled, fame height was obtained at 1 s intervals. The resulting data were used to obtain one absolute maximum fame height value for each experiment. The complete fame height dataset included both the surface and crown fame. Therefore, for the purposes of the analysis shown here, the surface fame height was cropped out. Computationally, this was done by identifying the vertical location of the crown fuel bed in a sample image of the experimental setup. The pixel value at this location was extracted and selected as a threshold. A script was developed to flter out values falling under the threshold thus isolating the crown fuel bed.
An algorithm based on computer vision was developed to obtain fame tilt angle and fame length from an experiment video. The use of this methodology is motivated by advancements in computer vision over the past decade through which image processing for fre imaging has improved. Edge detection has been used to identify fame edge contours (e.g. Gupta and Gaidhane, 2014). The fundamental parameters that can be obtained from visual images are fame height, fame tilt, and fame length. The algorithm and process used here to obtain such parameters follow. At frst, images were preprocessed to obtain edges of the fame. To do so, frst, a homography and prospective transformation was applied to the raw image. The transformation corrected the perspective of the images. Later, the RGB channeled image was converted to hue-saturation-value images (HSV) format and the value channel (V) was extracted from the image. Subsequently, a threshold value was selected to convert the images to a binary image. Once the image was converted to binary, the fame edge, or perimeter was obtained using an edge detection algorithm, Sobel edge detection.
After obtaining the fame edge, the binary or edge images (Figures 2D,E) were labeled and segmented into discrete fames ( Figure 2F) (in case of both surface and crown fuel fames) distinct from each other and the background. This step essentially established what were known in image processing as regions. The regions, the crown fame and surface fame, region 1, and region 2, were now the computational objects of interest. Once the regions were established, the image features, fame length and orientation, were computed. This was done by calculating the moments of the region as described by Burger and Burge (2008). Calculations of the second moment returned orientation and major axis of the region. The coordinates and dimensions of the major axis and orientation were used to produce an ellipsoid using the OpenCV (Bradski, 2000) library in Python. This produced an ellipsoid which enveloped the fame and had a major axis equal to the fame length and an orientation equal to the fame tilt angle. The last processing step leading up to the generation of the ellipsoid was visualized in Figure 2G. For the purposes of the study here, only region 1, the crown fame was analyzed.

Modeling Techniques
Data obtained experimentally was compared to predictions from existing semi-empirical models to be described in this section. The goal was to assess whether currently available models accurately describe the chaparral crown fre system modeled. Flame geometry properties were defned according to naming and measuring conventions described by Figures 1, 2. Predicted fame height was calculated from heat release rate (Q). Theoretical heat release rate was obtained from mass loss rate according to Equation (1)   (1) where h represented the low heat of combustion (for chamise h = 14.71 KJ/g). Following Zukoski et al. (1980), we modeled fame height using a semi-empirical power-law correlation of the forms in Equations 2 and 3, where H max and Q max represented maximum fame height and maximum heat release rate, respectively. The second approach was to use the power-law correlation proposed by Sun et al. (2006) We obtained maximum heat release rate using the two methods proposed by Sun et al. (2006) which, for consistency, we name following their convention such that in Method 1, maximum heat  release rate was defned as the heat loss rate occurring at the time of maximum mass loss rate (Q max, Method 1 ˘ ṁ (t ṁ max )). In Method 2, maximum heat release rate was defned as the heat loss rate occurring at the time of maximum fame height Q max, Method 2 ˘ ṁ (t fame height, max ). In addition to fame height, we estimated fame tilt from power law and log-log correlations. Next, we obtained predicted fame tilt values. Predicted fame tilt as a function of Froude number, a dimensionless measure of the relative importance of buoyant and inertial forces Williams (Williams, 1985), was compared to experimental data. The general form for Froude number is given by (4) where U is the gas velocity, g is the gravitational constant and D is the characteristic length (Drysdale, 2011). To correlate Froude � tan f = Fr .
number to fame tilt angle, some have used fame height, H, as a characteristic length (Albini, 1981) while others have used fame length, L f , (Putnam, 1965). In the approach here the latter was used, hence the resulting Froude number expression used was of the fnal form given by Equation (5) (5) The empirical correlation between fame tilt angle, f , and Froude number, Fr, was of the form given by Equation (6) (Albini, 1981;Nelson and Adkins, 1986;Weise and Biging, 1996) In Equation 5, f is the fame tilt angle as measured from the vertical as presented in Figure 2B. The coeÿcient and power dependence can be estimated following the regression analysis in Weise and Biging (1996) (Figure 2F). Coeÿcients to ft Equation (6) to the data from chaparral crown fre experiments were obtained through regression analysis.

Error Analysis
Agreement between the observed and predicted values of fame height and fame tilt was quantifed using the measures identifed in Cruz and Alexander (2013) and Weise et al. (2018b). These error analysis schemes have been previously used in analyzing results from wildland fre behavior studies (Cruz and Alexander, 2013). Perhaps the most elemental form of dierence is simply the dierence between observed and predicted values or where we have adopted notation from Willmott (1982) to represent observed values by O and predicted values by P. If N is the number of samples, then the mean bias of the error (MBE), the mean absolute error (MAE) and the root mean square error (RMSE) can be given as  In this way we aimed to provide RMSE as a percentage error. The mean absolute percent error (MAPE), Equation 12, was also measured and it provides an additional form of percentage error (12) According to Cruz and Alexander (2013), percentage error as measured by MAPE is optimized as it nears zero and an acceptable range for good values is 10%.

Experiment Classifcation
Four experiment classes were used to quantify the eect of wind and the separation distance between the surface and crown fuel layer, which following Van Wagner (1977) is called crown base height (CBH) in this work. Table 1 summarizes the conditions for each experimental class. The eect of wind and crown base height on fame height was examined for all experimental classes. A total of 18 experiments were considered for fame height analysis. Flame tilt was primarily observed in wind driven fame spread, experiment classes 1 and 2. For this stage of the study, we focused only on the eect of wind on fame tilt, therefore we focused only on one experimental class for the fame tilt analysis, class 4. Two experiments conducted on the same day were examined. This enabled greater uniformity in fuel conditions as fuels burned for both experiments were collected on the same day under the same ambient conditions. Experiment A (experiment burn time = 171 s, RH = 52%, FMC = 54%) was the frst experiment analyzed, Experiment B (experiment burn time = 385 s, RH = 28%, FMC = 54%) was the second.

Flame Height
Data from experiments with and without wind with two crown base height values (CBH 1 = 60 cm and CBH 2 = 70 cm) were considered for the analysis. Power law relationships as described by Equations (2) and (3) were used to estimate maximum fame heights from maximum heat release rates for each experiment. For fame height analysis, experiments from all classes were considered.
Comparison of the data with Equations (2) and (3) did not show signifcant dierences between models which is not surprising (Figure 3). When comparing observed values to theoretical values using Method 1 to estimate maximum heat release rate (Figure 3A), it was observed that just under 70% of experiments were over-estimated by the model. Theoretical values estimated using method 2, over 75% of experiments considered were under-estimated ( Figure 3B). Comparison of observed and predicted values showed that for Method 1, only 10% of experiments fell outside 70% of accuracy ( Figure 4A).
In the case of power law predictions using method 2, 17% of experiments fell outside the bounds of 70% of accuracy ( Figure 4B). Model statistics resulting from the power-law predictions of maximum fame height using method 1 and method 2 are shown in Table 2

Predicted Flame Tilt
Experimental fame tilt angle was obtained from videos by using the computer vision algorithm described in the Methods section. The analysis here represents fame tilts in windblown fames. Only confgurations with CBH 2 are included. We explored derivation of new semi-empirical correlations applicable for fame tilt angles in chaparral crown fre. Two experiments of wind-blown fames with CBH 2 were analyzed. Power law regression coeÿcients were obtained from linear regression performed on a log-log plot computed using the Python scipy stats linear regression library. In the frst experiment analyzed, hereby called experiment A, the power-law relationship obtained was, Observations compared against the power-law given by Equation (13) are presented in Figure 5A. A log-log plot of the data with the corresponding correlation is presented in Figure 5C. A linear regression was performed on the log-log plot in order to obtain the required coeÿcients. The curve shown in Figure 5A shows reasonable agreement between the power-law ft given by Equation (13) and observed data (R 2 = 0.85). Moreover, observed-vs.-predicted analysis showed that only 10% of fame tilt samples considered fell outside of the 70% accuracy bounds when using this modeling method, see Figure 5B. A sound degree of agreement was consequentially also observed for the log-log analysis (Figures 5C,D). In the second experiment analyzed, hereby called experiment B, the power-law relationship obtained was, Observations compared against the power-law given by (14) are presented in Figure 6A. A log-log plot of the data with the corresponding correlation is presented in Figure 6C. Similarly, to Experiment A, a linear regression on the log-log plot was used to obtain the required coeÿcients for modeling. The power-law correlation represented in Figure 6, shows a reasonable correlation between observed values for Experiment B and Equation (14) (R 2 = 0.84). Evaluation of observedvs-predicted values showed that only 30% of fame tilt samples considered fell outside the model 70% accuracy bounds. Reasonable agreement was also observed in the loglog analysis.
Model statistics resulting from the power-law ft on Experiment A and Experiment B are shown in Table 3

DISCUSSION
Flame height results are discussed in terms of model choice and method of heat release rate calculation. In terms of model choice, we evaluated the use of the two-ffths power law given in Equation 2, and the power law derived for dead Fall fuels proposed by Sun et al. (2006) given in Equation 3. Both power laws express fame height in terms of heat release rate. Our results indicate good agreement between fame height values observed experimentally and predicted fame height. Little variation between the two empirical models (Equations 2 and 3) was observed as exemplifed by the almost coinciding curves in Figure 3. This suggests the validity of the Fall fuels model, Equation 3, proposed by Sun et al. (2006) for experiments conducted in fre season for chaparral fres modeled as chaparral crown fres.
In terms of heat release calculation method, results showed some variation with respect to fame height obtained from powerlaw correlations of heat release rate using the time at maximum mass loss rate, Method 1 (Q max, Method 1 ˘ ṁ (t ṁ max )), and   From the analysis presented here, it can be argued that like in other fre spread applications, power-law semi-empirical models may be used to represent fre spreading in the crown fuel layer of chaparral crown fres. The relatively low variation in error between the two models derived here indicate that with further optimization and by considering an expanded dataset, a unifed power-law correlation of fame tilt as a function of Froude number could be derived for fames in chaparral crown fres. In assessing results on fame tilt, it was also observed that when estimating fame tilt angle as a function of Froude number in the form given by Equation (6), wind speed did not change and hence, the only varying parameter was fame length or fame height.
Moreover, our results exhibited what could be considered small Froude numbers Fr << 1. The small values points to the dominance of buoyancy forces in governing fame structure. Establishing this feature of fre behavior for the fre system modeled here is signifcant as it provides information on the modes of heat transfer governing fre spread behavior. This is important as in recent years a great deal of attention has been invested to studying the role of convective and radiative heat transfer in wildland fre behavior. Recent studies examining this aspect of wildfre behavior include those by Finney et al. (2015), Morvan and Frangieh (2018), and Maynard et al. (2016). Results from our work may thus follow others in indicating the role of buoyancy forces driving fame structure and consequently fre behavior. Particular to the work here is a diagnosis on chaparral crown fre fuel beds which illustrates the infuence of buoyancy forces on the specifc case of crown fre spread and fame behavior in the chaparral. To further understand the role of buoyancy forces in this chaparral crown fre system, future work would beneft from fow visualization such as Schlieren which has been recently used for visualization of convective fow in wildland fre systems (Aminfar et al., 2019).

SUMMARY AND CONCLUSIONS
The work here aimed to serve as proof of concept on the applicability of certain established models of fame properties to spreading chaparral crown fres. Predictions of fame geometry, particularly fame height and fame tilt angle, were compared to observed values obtained from wind tunnel experiments. Maximum fame height was predicted as a function of maximum heat release rate using power law correlations. Additionally, following Sun et al. (2006), we used two methods to calculate maximum heat release rate. Method 1 where maximum heat release is defned at the time of maximum mass loss rate [Q max, Method 1 ˘ ṁ (t ṁ max )] and Method 2 where maximum heat release rate is defned at the time of maximum fame heat Q max, Method 2 ˘ ṁ (t fame height, max ). A good degree of agreement was found between the two-ffths power law correlation of maximum fame height as a function of maximum heat release rate. Similar agreement was found when considering the powerlaw derived for Fall fuels proposed by Sun et al. (2006).
Error and statistical analysis refected the positive agreement between predicted and observed values and highlighted some nuances in the predictive potential of the models. Particularly, it was found that Method 1 and Method 2 for maximum heat release estimation showed similar results, but that Method 2 resulted in some degree of underprediction of observed values. However, most other measures of error showed reasonable agreement with observed data. For this reason, it may be concluded that for the conditions tested here, it was shown that the two-ffths power law may in fact be used to predict fame height from maximum heat release rate in chaparral crown fre spread. Fundamental work in studies including that of Thomas (1963) have successfully applied two-ffths power law correlation to spreading natural fres (without wind). More recent studies have successfully applied of these correlations in chamise chaparral burns in pool fre confgurations. Despite the fact that the study here was not conducted for pool fre confgurations but instead investigated spread fre, the wind conditions tested (1 m/s) are potentially ft for ensuring that for the conditions tested and the fuels considered, the two ffths power law correlation relating fame height to heat release rate does not fail. In future work it would be worth examining whether increasing wind speed would eect any changes in the applicability of this type of correlation. Additionally, here we considered wind-driven and non-wind driven fames as well as two experimental CBH confgurations in our assessment of fame height prediction, future work should examine dierences in model agreement between the dierent experimental conditions.
We derived two sample power-law correlations from selected experiments. Error analysis of fame tilt angle predictions obtained from these new power-law correlations showed good agreement between observed and predicted values. The fndings in our study lead to the conclusion that in fact, new semi-empirical power-law correlations may be used to express fame height in spreading crown fres as a function of heat release rate. Finally, the results presented here were obtained from selected experiments and as such are representative of the particular conditions tested. We recognize that the models tested and derived here may be limited to the operational conditions assessed and described in the methodologies section of this work. Nonetheless, they represent important steps toward the derivation of new fame property models for chaparral crown fre applications.

DATA AVAILABILITY
The datasets generated for this study are available on request to the corresponding author.

AUTHOR'S NOTE
This manuscript was prepared, in part, by a U.S. Government employee on oÿcial time, is not subject to copyright and subject to copyright is in the public domain.

AUTHOR CONTRIBUTIONS
JC-I designed and conducted the experimental study and analyzed data. AA developed codes and computer vision algorithms. DW and MP provided guidance in experimental design and data analysis.

FUNDING
This work was supported in part by agreement 13JV11272167-062 between USDA Forest Service PSW Research Station and the University of California-Riverside. This research was partially supported by funding from the DOD/DOE/EPA Strategic Environmental Research and Development Program project RC-2640 administered through agreement 16JV11272167026