Field-based generic empirical flame length–fireline intensity relationships for wildland surface fires

Data sources

The BONFIRE worldwide database (Fernandes et al. 2020) was the source of data for this study. BONFIRE comprises outdoors fire behaviour characteristics and the corresponding fire environment descriptors gathered from an exhaustive search of the peer-reviewed and grey literature. Included are data pertaining to the forward spread of individual experimental fires in any vegetation type, as well as from prescribed fire operations and wildfires, but excluding fires initiated by exceedingly narrow ignition lines (<2-m length) or featuring potential interaction between fire fronts.

Each BONFIRE record includes data reliability ratings as per Cheney et al. (2012) and was attributed a generic vegetation type (forest, shrubland, grassland), a broad fuel type (e.g. long needle conifer) and a fuel complex defined by the fuel layer(s) carrying the fire (e.g. litter + shrub). The compiled records were considered eligible for our analysis if they pertained to an experimental study represented by a minimum of three fires and if rated with high reliability. After applying these restrictions, we identified just 797 fires for which all variables required by the analysis (w, R and L_f) were available. Nonetheless, this BONFIRE subset spans worldwide and covers a diverse array of fuel types and fuel complexes, as can be seen in Table 1, where the original data sources are indicated.

Fireline intensity and flame length

I_f was computed as per Eqn 1. H can be obtained as a fraction of fuel heat of combustion determined in a bomb calorimeter, accounting for the heat losses occurring during outdoors combustion, like moisture vaporisation, incomplete combustion and radiative heat losses by the convection column (Byram 1959). We retrieved average values for the heat of combustion of forest and shrubland fuels (22 111 kJ kg⁻¹) and grass fuels (19 850  kJ kg⁻¹) from Susott (1982). We considered that the fraction of the heat of combustion assumed to be H was 0.75, which was roughly the value calculated by Byram (1959) and Nelson and Adkins (1986, 1988). As a result, we obtained H = 16 583  kJ kg⁻¹ for forest and shrubland fuels and H = 14 888  kJ kg⁻¹ for grass fuels.

Quantifying precisely how much fuel is consumed in the active flame zone of a field fire is nearly impossible. We therefore adopted the common and reasonable assumption that w equals surface fine (<6 mm thickness) fuel load (live and dead) and that most of the consumption of coarser fuels (when existent) occurs in the form of residual or glowing combustion and, therefore, has a minor contribution to the flame generated at the fire front (Alexander 1982).

Although L_f is typically measured above the fuel bed by default (Alexander 1982), Eqn 2 uses L_f measured from the base of the fuel bed, being assessed from the tip of the flame along its axis (Fig. 1). Because in many fires flame angle was not provided, we estimated L_f by adding fuel height h, instead of h/sin(flame angle), to the reported flame length. With increasing wind or slope, flames tilt towards the unburned fuel, flame angle diminishes and L_f becomes progressively underestimated. However, because L_f is expected to increase, the error also becomes less significant. The number of fires was unevenly distributed across the I_f range, with many more low-I_f fires. For this reason, for each vegetation type, we ordered the experiments by increasing I_f, grouped the fires into 100 kW m⁻¹ bins ([0–100], [100–200], etc.) and averaged L_f and I_f within those groups to obtain a better distribution of data points across the I_f range, and thus more reliable fits.

Data analysis

Coefficients a and b were determined by fitting the log-transformed form of Eqn 2 by least-squares (e.g. Cheney et al. 2012). We fitted Eqn 2 using the full dataset and examined the effect of vegetation type (forest, shrubland, grassland) as a categorical variable (significant at P < 0.05) to probe for the possibility of using the same coefficients for more than one fuel complex, and we proceeded with model development based on the results from this analysis. The bias from model back-transformation was corrected according to Snowdon (1991).

Predictions were evaluated based on the coefficient of determination (R²), root mean square error (RMSE), mean absolute error (MAE) and mean bias error (MBE). Residuals were checked for normality with the Shapiro–Wilk test (P > 0.05) or, when significance was below the threshold value, for approximate normality by visually inspecting their histograms. Independence from predicted values was evaluated by correlation analysis. We determined the reciprocals of the developed models (L_f as a function of I_f). We also plotted Byram’s (1959) L_f − I_f relationship with the present models for a graphical comparison.

Parameter ranges and data points

The wide span in fuel bed structure and flame dimensions inherent to the dataset can be inferred from the h and L_f ranges (Table 1): 0.1–1.0 m and 0.1–8.9 m for forest (n = 406), 0.16–4.8 m and 0.5–16.7 m for shrubland (n = 207) and 0.04–0.9 m and 0.2–6 m for grassland (n = 184). Wind speed and slope angle were not reported in all studies. For those who did, ranges were 0.5–22 km h⁻¹ (measured at 1.5–2-m height) and 0–22° in forest, 0.3–33.9 km h⁻¹ and 0–30° in shrubland and 2.5–53.1 km h⁻¹ and 0–7° in grassland. As a result of averaging L_f and I_f within 100 kW m⁻¹ intervals, the distribution of data points across the I_f range became much more balanced. We obtained 45 L_f − I_f data points for forest, 73 for shrubland and 85 for grassland.

Model development and evaluation

In the joint analysis of the full dataset, vegetation type was significant, yet differences between forest and shrubland were small, with the confidence interval estimates for the latter totally included within the confidence interval for the former, so we separately fitted Eqn 2 to forest–shrubland data and to grassland data. The back-transformed forest–shrubland equation (Fig. 2) accounted for 76.6% (Table 2) of the existing variability, with an MAE of 1812 kW m⁻¹. The explanation of variability (82.1%) was higher for grasslands (Fig. 3) and MAE was slightly lower (1734 kW m⁻¹). MBE was approximately zero in both cases. Residuals were approximately normally distributed for forest–shrubland, normally distributed for grassland, and weakly correlated with predicted values. Reciprocals of Eqn 2 are L_f = 0.04001 I_f^0.5846 for forest–shrubland and L_f = 0.03888 I_f^0.5111 for grassland. Byram’s (1959) equation is L_f = 0.0775 I_f^0.46; its reciprocal yields similar results to the equation derived for grassland (Fig. 4) but severely, and increasingly, overestimates I_f for flames longer than ~2 m in forest and shrubland.

Fig. 2.

Eqn 2 fit and evaluation for forest–shrubland after back transformation: (a) Fireline intensity (I_f) as a function of flame length (L_f), dashed lines are the 95% CI; and (b) observed vs predicted I_f values. Fitted coefficients and evaluation metrics are given in Table 2.

Fig. 3.

Eqn 2 fit and evaluation for grassland after back-transformation: (a) Fireline intensity (I_f) as a function of flame length (L_f), dashed lines are the 95% CI; and (b) observed vs predicted I_f values. Fitted coefficients and evaluation metrics are given in Table 2.

Fig. 4.

Graphical representation of Byram’s (1959) fireline intensity (I_f)–flame length (L_f) relationship (I_f = 259.8 L_f^2.174) and the corresponding models developed in the present study (Table 2).

Model performance and influence of fuel properties

The L_f − I_f models performed well, explaining about 80% of the observed variability and providing unbiased estimates. In forest and shrubland fires, I_f can be estimated from L_f using the same empirical relationship. Fires in grassland need different fitted coefficients, thus revealing differences in flame characteristics, which have already been noted by Cheney (1990). However, L_f scales approximately with the square root of I_f in both cases, as expected from theory (Nelson 1980).

Fuel consumption will not exactly equal fine fuel load. Consequently, I_f calculation with the latter in lieu of the former is expected to introduce uncertainty when fitting a L_f − I_f relationship, but judging from model evaluation statistics, this simplification was not an influent factor. The wide data range in w and fuel bed density (ρ_b), shown by their interquartile ranges for forest (0.66–1.3 kg m⁻², 2.9–11.0 kg m⁻³), shrubland (0.76–1.7 kg m⁻², 1.3–3.7 kg m⁻³) and grassland (0.25–0.67 kg m⁻², 1.0–2.0 kg m⁻³), and the fact that these metrics are not required as predictors to produce accurate L_f − I_f models, leaves no doubt about their reduced influence on flame structure. Also, it is reasonable to assume that variation in fuel bed structure descriptors (like h, w and ρ_b) would similarly affect flame properties in forest–shrubland or grassland. On the other hand, a fuel metric that fundamentally distinguishes forest–shrubland and grassland, and significantly influences R, is the surface area-to-mass ratio of the fuel particles (S_m) (Rossa and Fernandes 2018a). R increases with S_m, which is much higher for grassland. In fact, the S_m of mediterranean forest–shrubland foliage follows a normal distribution, with an average value of 8.2 m² kg⁻¹, contrasting with the much higher values of 20–40 m² kg⁻¹ observed for herbaceous fuels (Rossa and Fernandes 2018b).

L_f is a visible manifestation of the hot combustion gases, and the chemical composition of wildland fuels is very similar, thus producing little variation between H values. Thus, the same ‘amount of flame’ should be produced for the same I_f, regardless of the fuel bed. Contrasting S_m fuels will produce structurally different flames, which seem to become more compact as S_m increases. Additional evidence supports this idea: based on laboratory experiments in conifer slash fuel, Anderson et al. (1966) obtained an L_f − I_f relationship that predicts L_f = 10 m to yield I_f = 2000 kW m⁻¹, whereas our models result in L_f = 4 m for forest–shrubland (Fig. 2a) and L_f = 2 m for grassland (Fig. 3a). Even the thinnest woody fuels within a slash fuel bed have very low S_m; for example, a 4 mm round eucalypt twig will have S_m = 1.5 m² kg⁻¹, contrasting with typical values of 8.2 m² kg⁻¹ for foliar fuels and 30 m² kg⁻¹ for grass fuels (Rossa and Fernandes 2018b).

Model applicability and application

Although our models performed better within the fire behaviour range most commonly associated to surface fire spread, say up to an I_f of 4000 kW m⁻¹, a relevant feature of the fitted equations is the inclusion of high-intensity fire data in their development. For more flammable fuel complexes, namely tall shrublands and forests with a well-developed woody understorey, this is expected to improve L_f or I_f estimates over those afforded by the previously available relationships. Flame size quantification in forest crown fires is scarce, but a cursory inspection of the predictive ability of our L_f − I_f relationship in such cases is warranted. Flame height data is available for five active crown fires in jack pine stands in Canada (Stocks et al. 2004; Butler et al. 2004a), for which mean I_f varied between 39 896 and 78 533  kW m⁻¹, calculated as previously described. Assuming that L_f equals flame height, i.e. inputting underestimates of L_f, our L_f − I_f equation estimates I_f with a mean absolute percentage error of 13.2% (range 1.8–35.2%), which is encouraging.

Beyond the already mentioned putative S_m effect, the supposed influence of fuel structure or other fuel bed-specific effects on fire behaviour properties does not preclude generic L_f − I_f relationships from producing useful estimates, especially for operational fire management purposes. For low-I_f fires, MAE represents a greater percentage of the observed I_f values. Thus, the present models are more appealing for use in high-I_f fires (>~2000 kW m⁻¹), which typically occur under low fuel moisture content and strong winds (high R) in tall fuel complexes (high w). To obtain accurate I_f estimates for low-I_f fires, the use of fuel bed-specific L_f − I_f relationships may be a better option. Finally, different L_f − I_f relationships have been reported for backing and heading fires in the same fuel type (Clark 1983; Fernandes et al. 2009). The L_f of fires spreading under calm conditions theoretically scales with the 2/3 power of I_f (Fons et al. 1963; Thomas 1963; Albini 1981), which also applies to fires backing into the wind (Nelson 1980). Consequently, our L_f − I_f relationships should not be extrapolated to backing fires.

The L_f − I_f relationships can be applied for various purposes and in different ways. The reciprocals of Eqn 2 can be integrated in fire behaviour prediction schemes to produce estimates of L_f from I_f whenever the former is a variable of interest, e.g. in the frame of fire suppression and firebreak construction (Alexander 2000), or for fire hazard or fire risk assessment and mapping (e.g. Thompson et al. 2011). The usefulness of I_f estimates is thoroughly covered by Alexander and Cruz (2020). Estimation of I_f from L_f is expedient when the latter can be measured or calculated based on visual observation, either during or after the fire. If fire-spread rate is concurrently assessed, fine fuel consumption can be estimated as well (as per Eqn 1), without the need for destructive pre-burn and post-burn sampling, which is relevant for prescribed burning operations and fire effects studies. Estimates by personnel in the fireline can thus assist decision-making during fire control or fire use operations and can be used to anticipate fire effects such as crown scorch height and tree mortality. Likewise, post-fire surveys are able to translate fire effects in terms of the associated fireline intensity, which is useful for prescribed burning monitoring and wildfire study cases.

Comparison with Byram’s relationship

The pioneer L_f − I_f relationship of Byram (1959) is the best known and most widely used. Oddly, although Byram’s model is most commonly viewed as applicable to surface heading fires (Nelson 1980; Alexander and Cruz 2012), it was presumably based mostly on backing fires (34 out of a total of 41). Interestingly, it was derived from outdoor fires in a forest fuel type with a grass component, which might explain why it compares better with our grassland equation instead of that for forest–shrubland. Our results highlight the inadequacy of using the Byram (1959) model for forest and shrubland fires with L_f above ~2 m, consistent with the findings of recent laboratory experiments (Finney and Grumstrup 2023).

Rothermel (1991) stated that Byram’s equation severely underestimates the L_f of crown fires, and as such, recommended the more realistic equation of Thomas (1963), where L_f is proportional to I_f^0.67, as in the equation of Butler et al. (2004b) for crown-fire L_f. The L_f − I_f relationship for the forest–shrubland variant displays b = 0.58, midway of the coefficients of Byram (b = 0.46) and Thomas. This may be an outcome of the substantial presence in the dataset of high-intensity observations in fuel complexes dominated by an elevated and aerated component, either in shrubland or in forest.

Study relevance

The discrepancies between previous results suggesting that generic L_f − I_f functions are unviable (Alexander and Cruz 2012) are most likely explained by the following reasons: (1) individual studies are usually limited in the number of fires and observed I_f range, which may lead to substantial differences in predicted values when models are extrapolated much beyond the development data range; (2) flame pulsation makes L_f evaluations subjective and originates discrepancies between measurements taken in real time or using video images (allowing for a visual average) and measurements based on photographs (capturing a single snapshot); and (3) most studies do not specify how the flame is measured and lack a standard method of measurement. We advocate that L_f should be assessed from the fuel base level to enable comparability between fires in different fuel heights and adequacy to heat transfer modelling (Nelson and Adkins 1986; Anderson et al. 2006), whereas Alexander (1982) proposes measuring L_f from the flame-depth midpoint at the fuel surface level to the tip of the flame.

I_f is a versatile fire metric that can be used for a wide variety of purposes; it can be estimated from L_f, and vice versa. Although L_f assessment is subjective and greatly depends on its definition and mode of observation and calculation, it is a readily apparent descriptor (Rothermel 1991) and therefore practical to use. Particular situations may benefit from the use of fuel bed-specific L_f − I_f relationships to assure increased accuracy of I_f estimates, e.g. for prescribed burning planning purposes for which the most accurate relationships between fire behaviour and fire effects are required (Fernandes et al. 2012; Hiers et al. 2020). However, deriving specific models for all existing fuel complexes is not feasible. As a consequence, the generic relationships developed in the present study will be of interest in many situations. Moreover, our results are based on a great amount of data from very diverse literature sources reporting field fires conducted worldwide – in a wide range of fuel species, vegetation structure, wind speed, slope angle and flame dimensions, and thus are robust.