Several factors beyond the heat needed to dry-out and ignite the fuel ahead of a flaming front have been attributed to the M -damping effect on R ( Catchpole & Catchpole, 1991 ), such as the entrainment of moisture into the combustion zone and the attenuation of infra-red radiation by water vapour released from unburnt fuel. Still, not discarding those effects, in the present work Q _i will be assumed as the main responsible for slowing down fire spread. Q _i is given by ( Rossa & Fernandes, 2018b ):

where c _f , c _w , T _i , T _f , T _v , and Q _w , are, respectively, fuel specific heat, water specific heat, fuel igniting temperature, fuel initial temperature, water boiling temperature, and water latent heat of evaporation. In physically-based formulations ( Thomas, 1971 ; Rothermel, 1972 ), Q _i is commonly used to account for the M -damping, as opposed to field-derived models. The relative M -effect on R , i.e. , f _M , results from dry-to-wet fuel ignition needs ratio:

A total of 282 laboratory fires were retrieved from several sources (Table 1). R ₀ tests ( n = 181) compiled in Rossa & Fernandes (2018a) include experiments from Rossa (2009) and Oliveira (2010) , and pertain to fire spread in litter, slash, and shrub-like fuel beds, i.e. , vertically placed tree branches with or without a surface litter layer. Fuel beds were built using quasi-live, i.e. , collected live with M decreasing as a function of storage time, and dead vegetation of several species ( Pinus pinaster Ait., Eucalyptus globulus Labill., Eucalyptus obliqua L'Her., Acacia mangium Willd., Quercus robur L., Pinus resinosa Sol. ex Ait.).

R _S burns ( n = 50) with slope angle ( S ) set to 20° were retrieved from Rossa et al. (2016) . Fuel beds were made of vertically positioned quasi-live shrub and tree branches of four species: Acacia dealbata Link., Cytisus striatus (Hill) Rothm., P. pinaster , and E. globulus . In the A. dealbata tests, air-dried leaves had contracted folioles, because they fold inward when branches are cut from the plant and surface-to-volume ratio is greatly diminished, attaining a fire behaviour similar to the remaining fuel species.

The R _U experiments ( n = 51) from Rossa & Fernandes (2017a) were carried out under constant wind speed ( U ) of 8 km h ^-1 wind in shrub-like fuel beds, composed of vertically placed quasi-live tree branches over a dead litter layer. P. resinosa and P. pinaster needles were over-layered by P. pinaster branches, and E. globulus leaves were over-layered by E. globulus branches. In all laboratory trials ( R ₀ , R _S , R _U ), only the foliar fuel component was considered for computing oven-dry fuel bed load ( w ) and density (ρ _b ) in vegetation containing woody elements.

The applicability of f _M to real-world fire spread was tested based on 123 outdoors fires (experimental and wildfires). A comprehensive data set ( n = 100), representative of global shrubland fire behaviour, was retrieved from Anderson et al. (2015) , which compiled data from Catchpole (1987) , Vega et al. (1998) , Fernandes (2001) , Vega et al. (2006) , (2009) , and Cruz et al. (2010) .

Wildfires in fully cured grasslands ( n = 23), com­piled by Cheney et al. (1998) , were used to test f _M for fire spread in very low M conditions, seldom attained in experimental fires. Data provenance was Cheney et al. (1998) own observations, McArthur (1966) , Finocchiaro et al. (1970) , Douglas (1970) , McArthur et al. (1982) , Rawson et al. (1983) , Keeves & Douglas (1983) , Maynes & Garvey (1985) , and Noble (1991) . Fuel beds were undisturbed, cut or grazed, and eaten-out pastures. Because Cheney et al. (1998) did not report M , the Noble et al. (1980) equation describing the McArthur (1977) model: M _d = (97.7 + 4.06 RH ) / ( T _a + 6.0) - 0.00854 RH , where RH is relative humidity, was used to obtain M estimates.

f _M was used to develop R ₀ , R _S , and R _U models from the laboratory fire spread data. In the Rossa & Fernan­des (2018a) R ₀ formulation based on fuel bed height ( h ) and M , the h -exponent is close to unity. So, for the sake of simplicity, a linear h -effect was assumed. The present R ₀ model was obtained by linear fitting R ₀ to h f _M . In the case of R _S and R _U data, structural fuel bed metrics of most trials were close to the experimental mean, despite some variation between observed minimum and maximum h and w values. Also, both S and U were kept constant. As a result, M was the parameter with most influence on R _S and R _U , and both models were obtained by establishing a linear relationship between R and f _M .

Both studies where field fires were compiled (Che­ney et al. , 1998; Anderson et al. , 2015 ) pro­vide R models accounting for the M -effect through an exponential decay f _M , which, like Eq. [3] f _M , varies in the 0-1 range. Thus, the concept of using a generic f _M -function was tested by using the original R models, substituting their original (specific) f _M by the proposed generic f _M . In mixed live and dead fuel complexes, this exchange can only be done if the specific f _M -function accounts for both M _d and M _l , as in Anderson et al. (2015) . Specific f _M were plotted against generic f _M -values and predictions using both f _M -functions were evaluated for comparison.

Goodness of fit of linear regressions was assessed based on the coefficient of determination ( R ² ). All predictions (laboratory and field fires) were evaluated using deviation measures: root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and mean bias error (MBE) ( Willmott, 1982 ).

Laboratory-based R models built with the generic f _M showed good agreement with data. They yielded R ² slightly below those obtained using the original power law f _M -based models (0.667-0.947), but significantly above the 0.566 and 0.665 values obtained for the R _S and R _U models using exponentials ( Rossa et al. , 2016 ; Rossa & Fernandes, 2017a , 2018a ). Despite a small decrease in accuracy, when compared with the use of power laws, the generic f _M provides important benefits, such as not becoming extremely sensible at very low M -values and allowing extrapolation to moisture-free conditions. The generic f _M allowed improved prediction ability in relation to the specific f _M -functions used in existing field-based models for shrubland experimental fires and grassland wildfires.

Laboratory data included a great number of tests in several fire spread conditions over a wide M -range, and fuel beds were very diverse in terms of species and structure. R ₀ laboratory tests are representative of field R ₀ and a reasonable surrogate for backing fires R ( Rossa, 2017 ; Rossa & Fernandes, 2018a ). That is not the case of slope and wind-driven laboratory trials, in which R is limited by the fire front width ( Fernandes et al. , 2009 ). Shrubland and grassland outdoors fires enabled the positive testing of f _M in R _U conditions free of scaling issues. There is no apparent reason for f _M not to hold for slope-driven field fires as well. Not excluding the need of further assessing f _M with additional field data, its overall performance in all tested fire spread situations lends strong support to its ability of successfully incorporating empirically-based R models in generic fire spread conditions.

If Eq. [3] were developed without assuming that moisture-free conditions will be attained at T _f = 100 °C, i.e. , with the numerator becoming 1.72 (320 - T _f ) instead of 378.4, using the generic f _M in the field-derived R models would yield MBE of 3.92 and 41.7 m min ^-1 , respectively for shrubland and grassland fires. The arising of this substantial over-prediction bias lends support to the supposition that f _M -functions based only on M implicitly account for T _f . In other words, this means that in a hypothetical situation of fire spread through a dry fuel bed at, for example, T _f = 20 °C, predicted R using typical empirical field-based models would be higher than observed because the M -functions were fitted in conditions where the decrease in M is concurrent with increasing T _f . As a result, estimated f _M attains its maximum, i.e. , fire spread attenuation is minimum, although fuel conditions will delay fuel ignition more than expected in an extrapolation to M = 0%, where T _f was supposed to grow concomitantly with diminishing M . It is important to notice that this rationale was derived from results using a limited field data set, hence further testing with additional data would benefit its confirmation.

M _d of field fuels is easy to sample. Overall M determination requires measuring both M _d and M _l ( Rossa et al. , 2015b ), as well as assessing dead and live fuel mass fractions, which may be problematic in very heterogeneous fuel complexes. This is a limitation of using the generic f _M , when compared to f _M -functions accounting for only the M _d -effect. Most empirical fuel-dependent models rely on the sole use of M _d ( Cruz et al. , 2015 ) to provide a satisfactory R explanation, which restricted the data available to test the specific f _M -function proposed in the present work. Field-based models based only on M _d work well because, usually, M _l is either constant or correlated with M _d for a given fuel complex ( Rossa & Fernandes, 2017b ).

Nevertheless, especially for experimental programs composed of a limited number of tests, possible diffi­culties in assessing overall M might pay-off in terms of the advantages of using a generic f _M . The use of experimental outdoors fires as a source of development data is appealing because of the strong resemblance to real-world fire-spread. However, this option is often challenged by heterogeneity in fuel bed properties and correlated fuel descriptors, which elude the correct quantification of specific effects ( Rossa & Fernandes, 2017b ). Establishing a priori the M -effect through the use of f _M significantly simplifies the proper assessment of the remaining influent variables.