Introduction

Forest fires are classified into surface, canopy, and underground fires. Underground fires occur less frequently than the other types, but the damage can be significant (Watts and Kobziar 2013). Underground forest fire is slow spreading, long-lasting and having low temperature smoldering combustion without flame (Ohlemiller 1985; Rein 2013). These fires cause fatal damage to plant roots (Page et al. 2002; Davies et al. 2013), destruction of soil structure and ground collapse (Hadden et al. 2013), and higher greenhouse gas emissions than other categories of forest fires (Turetsky et al. 2004). Forested peatlands and wetlands with rich organic matter, such as those in boreal and tropical forests, are prone to underground fire (Huang et al. 2016). Soil carbon in these forests accounts for 25% of carbon reserves on land and plays essential roles in ecosystem stability, biodiversity maintenance, and hydrological cycling (Page et al. 2011). Therefore, large underground fires will substantially impact global climate, ecology, and human health (Davies et al. 2013; Hu et al. 2018). Wetlands were previously assumed to be fire-resistant due to their high humidity (Kuhry 1994). However, recent studies reported that smoldering has been widely observed in wetlands (Turetsky et al. 2004; Shetler et al. 2008). Quantitative studies in the field on smouldering events are seldom conducted as they are challenging due to their randomness and unpredictability (Davies et al. 2013). Therefore, combustion studies simulating underground fires are often carried out in the laboratory (Huang et al. 2015).

Large-scale underground fires have been recorded in Indonesia, Russia, Spain, and in other countries in recent years (Page et al. 2002; Cancellieri et al. 2012; Turetsky et al. 2015; Pastor et al. 2018). Underground fires are caused by surface fires or lightning (Davies 2016; Restuccia et al. 2017) with strong concealment and always accompanied by surface fires. There are considerable difficulties in monitoring and extinguishing them (Rein et al. 2008a, b). Their behavior is significantly different from other types of fires (Rein 2013), and could occur with a fuel moisture content of 100% or higher (Reardon et al. 2007; Lin et al. 2019), and persist through heavy rains, weather changes, extinguishing of surface fires, and may last for months or even years (Zaccone et al. 2014). The spread of an underground fire is maintained by heat released during combustion (Pastor et al. 2018), which is key to monitoring (Hartford and Frandsen 1992). Studies of smoldering have mainly focused on combustion characteristics (Pastor et al. 2018), contributing factors (Achtemeier 2006; Wang et al. 2017), combustion spread (Huang and Rein 2017), carbon release (Davies et al. 2013; Hu et al. 2018) and other aspects.

Non-linear mixed effects (NLME) modeling was first proposed by Sheiner and Beal (1980), which included both fixed and random effects parameters, and could identifying average prediction and individual prediction by calculating the estimated value of random effect parameters (Leites and Robinson 2004; Dorado et al. 2006; Yang et al. 2009). A mixed-effects model is a statistical method used in forestry, agriculture, ecology, biomedicine, sociology, economics, and in other fields (Calegario et al. 2005). In forestry, NLME modeling has been used in the studies of tree heights and diameters, growth processes, average growth, and basal area (Budhathoki et al. 2008; Mehtatalo et al. 2014; Ciceu et al. 2020) but has been seldom applied to forest fires. The mechanism of underground fire smoldering is complicated, so its occurrence and development are difficult to predict. The gradually rising temperature of a smoldering fire is like the growth process of a tree, so NLME modeling is applicable to smoldering. NLME modeling can achieve prediction accuracy with fewer independent variables and is suitable for studies on underground fire.

The frequency of underground fires in the boreal forests has increased in recent years (Sinclair et al. 2020), due to global climate change and the impact of human activities (Turetsky et al. 2015). The coniferous forests in the Daxing’an Mountains region, the southernmost margin of the Far Eastern Siberia boreal forests into China, is characterized by the largest number of underground fires in the country. The growth of Larix gmelinii (Rupr.) Rupr. planted in the Tatou wetlands of the Daxing’an Mountains has been poor because of the geographical and climatic conditions, and the decomposition of fallen branches and leaves slow so that the content of organic matter in humus increased, providing abundant fuel for underground fires. Rising temperature is one of the stages during the smoldering (Pastor et al. 2018); one of these is the buildup of temperatures causing significant damages. However, this process is easily monitored. This study focused on the process of rising temperatures in a Larix gmelinii plantation in the Tatou wetlands, analyzed smoldering temperature variations of humus of different particle sizes, and developed a method to predict temperatures of underground fires by non-linear mixed effects (NLME) modeling to provide support for underground fire monitoring and control.

Materials and methods

Study area

Forest fires in the Daxing’an Mountains region occur frequently; the area burned is largest in China. The study area is in the Jiagedaqi Forest Management Technology Promotion Station (123°57′–124°0′ E, 50°20′–50°23′ N), located in the northwest of Heilongjiang province, on the southeast slopes of the Daxing’an Mountains (Fig. 1a). The region has a continental monsoon climate with four distinctive seasons, a changeable climate and large diurnal and seasonal temperature differences. Annual average temperatures are − 1 to 2 °C, the annual effective accumulative temperature 1800–2000 °C, the frost-free period 90–120 days, and annual rainfall 450–500 mm. The technology promotion station was founded in 1973, 15 km to the south of the Jiagedaqi region, the north and west portions of the station is connected with the Dongfeng Forest Station of the Jiagedaqi Forestry Bureau. The total area of the station is 7326 ha2. The main species in the study area are Larix gmelinii, Quercus mongolica Fisch, Betula platyphylla Suk., Populus davidiana Dode, and Betula dahurica Pallas (Tang et al. 2022).

Fig. 1
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a The study area; b, c study location and sample collection; d sample after pretreatment for the combustion experiment

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Sample collection and processing

Three 30 m × 20 m sample fields were established and three 0.5 cm × 0.5 cm quadrats selected (sampling depth was the 46-cm thickness of the humus layer) in the diagonals according to Pastor (2018) (Fig. 1b, c). All the humus in the quadrats was collected, and after removal of surface litter and in the humus layer, taken to the laboratory, stored in kraft paper sacks and dried for 48 h at 105 °C to remove as much moisture as possible. As an uneven distribution of humus particles will affect the temperature measurements accuracy (Huang and Rein 2017), the dried humus was ground and sieved into particle sizes of ≤ 20 mesh (moisture content 0.28%), ≤ 40 mesh (moisture content 0.49%), ≤ 60 mesh (moisture content 0.43%), and ≤ 80 mesh (moisture content 0.49%) for the combustion experiments(Fig. 1d).

Combustion experiments

The combustion furnace was designed for studying a one-dimensional downward combustion process (Fig. 2) (Huang and Rein 2017). The unit was cylindrical (20- cm high with 10- cm walls) of aluminum silicate ceramic fiber with excellent heat insulation. A K-type thermocouple (30 cm long × 2 mm diameter) recorded temperature variations of the humus during combustion. The data were transmitted to a laptop by data acquisition module composed of a 16-channel NI9213 voltage acquisition board card and DAQ-9174 case (4 card slots), with temperature measurement accuracy < 0.25 °C. The data acquisition software (Labview2018) recorded temperature variations collected by the thermocouples. A far-infrared heating plate (30 cm long × 20 cm wide × 5 cm thick) was used as an ignition device. A temperature control meter between the far-infrared heating plate and the power supply kept the temperature of the heating plate constant.

Fig. 2
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Schematic diagram of the smoldering combustion process

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Humus of different particle sizes were placed in the combustion furnace at ambient temperatures and the shape of the particles maintained as much as possible. Holes were arranged every 3 cm on the side of combustion furnace and K-type thermocouples inserted into the humus through the holes. The K-type thermocouples and data acquisition module were connected temperature variation data transmitted to the laptop every 10 s. To ensure combustion continued after removing the heating plate (ignition device), heating time and temperature were set to 1.5 h and 500 °C. A 2-cm gap between the heating plate and combustion furnace allowed for air flow.

Non-linear mixed effects model

The general form of a non-linear mixed effects model is:

$$\begin{gathered} T_{i} = f(\varphi_{i} ,t_{i} ) + \varepsilon_{i} \hfill \\ \varphi_{i} = A_{i} \beta + B_{i} b_{i} \hfill \\ \varepsilon_{i} \sim N(0,\sigma^{2} )\quad b_{i} \sim N(0,D) \hfill \\ \end{gathered}$$
(1)

where \(T_{i}\) is the dependent variable of subject \(i\) and refers to the predicted combustion temperature; \(f\) is the actual value, specific parameter vector \(\varphi_{i}\) and variable value vector \(t_{i}\) of the differentiable function in the subject, and refers to actual combustion temperature of the underground fire; \(\beta\) is the \(p\)-dimensional fixed effect parameter vector; \(b_{i}\) is the \(q\)-dimensional random effect parameter vector; \(D\) is the covariance matrix between random effects; \(A_{i}\) and \(B_{i}\) are correlation matrices with appropriate dimensions ( 0 or 1); \(t_{i}\) is the independent matrix; \(\varepsilon_{i}\) is the random error vector associated with \(T_{i}\); and,\(\sigma^{2}\) is the covariance matrix of the random error.

Base model

Logistic, Richards, and Korf nonlinear models, widely used in forestry to predict the tree growth, height, and DBH (Calama and Montero 2004; Rijal 2012; Sharma et al. 2016; Pan et al. 2020) (Table 1), were selected as the base models to study temperature variations in the combustion process.

Table 1 Base models
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Model selection and validation

The base model was selected according to the values of Akaike’s Information Criterion-(AIC) and Schwarz’s Bayesian Information Criterion-(BIC). Model accuracy was evaluated with different parameters according to the values of Root Mean Square Error (RMSE). Smaller values of AIC and BIC indicate the better fitting degree of the equation; smaller values of RMSE indicates the higher accuracy of the model (Akaike 1974).

$$AIC = - 2\ln L + 2p$$
(2)
$$BIC = - 2\ln L + p\ln n$$
(3)
$$RMSE = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {(T_{ij} { - }\hat{T}_{ij} } )^{2} }}{{n{ - 1}}}}$$
(4)

where \(T_{ij}\) is the actual value, \(\hat{T}_{ij}\) the predicted value, n the sample amount, L the maximum likelihood (ML) value, and p is the number of model parameters.

Statistical analysis

Excel and Origin were used to analyze the combustion temperature variation data with different humus particle sizes and from different depths, and the relationship between the maximum temperature and the combustion depth; NLMIXED and NLINE modules of Statistical Analysis System-(SAS) were used to select, fit, and verify the nonlinear mixed effect models.

Results

Maximum combustion temperature variations of humus of different particle sizes

There was no flame or spark during humus smoldering, and the surface was carbon black and the deep layer brick red after combustion. The maximum combustion temperature of different particle sizes increased with depth. There was a positive linear relationship between depth and temperature (P < 0.05) in both particle ≤ 40 mesh size and ≤ 80 mesh size. The equations were y = 435.68 + 27.87 x and y = 453.82 + 21.82 x respectively. The highest combustion temperature of all particle sizes was at the 15-cm depth with particles ≤ 40 mesh size (897.5 °C), the highest combustion temperature with particle ≤ 80 mesh size was 844.9 °C.

There was a positive linear relationship between depth and temperature (P < 0.01) in both particle ≤ 20 mesh size and ≤ 60 mesh size; the equations were y = 474.76 + 17.42 x and y = 468.84 + 20.21x, respectively. Combustion temperature variations with particle size ≤ 20 mesh at different depths fluctuated the lowest, the highest combustion temperature was 741.6 °C and the lowest was 543.3 °C. The highest combustion temperature with particle size ≤ 60 mesh was 780.4 °C and the lowest was 553.3 °C (Fig. 3).

Fig. 3
figure 3

Maximum combustion temperature of humus for different depths and particle sizes; lower panel are photos of humus after combustion

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Selection and validation of NLME models

The NLME models were selected based on 60% of the humus combustion data with different particle sizes. AIC, BIC evaluated the approximation based on the three models. The AIC (135,343) and BIC (135,373) values of the Richards equation with different particle sizes were the smallest. It may be concluded that Richards equation provided the best approximation and was therefore chosen as the basic equation for the relationship between humus combustion temperature and time (Table 2).

Table 2 Fit statistics
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Richards equations of combustion temperature variation were set with no parameter mixing, and one, two, and three parameters mixing for different particle sizes. All equations with different parameters were convergent, and the RMSE values of models with mixed parameters were smaller than the traditional models. The precision of models with mixed parameters was higher than the conventional models. The RMSE value of the three parameters was the least, therefore, the accuracy of Richards equation with three parameters mixing was the highest (Table 3).

Table 3 Approximation results of mixed-effect models with different parameters
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Mixed-effects model approximation of Richards equation

The relationship between combustion temperature and time at different depths with different particle sizes was fitted by Richards equation with three parameters mixing. According to the temperature variation,\(\beta_{1}\),\(\beta_{2}\) and \(\beta_{3}\) were the estimated values of equations at 15- cm depth for each particle size, and \(b_{ij}\) was the parameter mixing effect at different depths (Table 4). The approximation of Richards equations with different particle sizes is shown in Fig. 4

Table 4 Fitting results of Richards equation
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Fig. 4
figure 4

Approximation of Richards equations of temperature and time at different depths with different humus particle sizes

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Discussion and conclusions

Variation characteristics of humus combustion temperature

Peatland and humus are the main combustibles of underground fires, and peatland fires is of wide concern for the frequent occurrence of large-scale fires (Davies et al. 2013). Commercial peat moss with its homogenous texture and consistent components has been widely used in characterizing smoldering and contributing factors, instead of using peatlands in the field (Huang and Rein 2017). More researchers have started using underground combustibles to approximate reality (Pastor et al. 2018). In this study, actual humus from the Tatou wetlands of the Daxing’an Mountains was used to study temperature variations of vertical smoldering, and the results are reliable and highly applicable.

Smoldering combustion of humus is sustained by the heat released by itself. The maximum temperature was 897.5 °C in this study, close to 900 °C of a similar study using peat (Huang et al 2015). Most studies have reported that the maximum temperature of peat smoldering was around 600 °C (Bar-Ilan et al. 2004; Huang and Rein 2017), lower than our result. Using different experiment devices and materials might account for the difference. It could also reflect the different temperatures of humus and peat smoldering. The mass loss of combustibles was small, and the humus afterwards was brick red due to sufficient burning and high temperatures (Fig. 3) and in addition, there was no ash after combustion, only a thin layer of carbonized humus. The heat release was impeded by the humus of the upper layers when the fire was spreading downwards, so the temperature of humus smoldering was higher and increased with depth. But the mass loss of combustibles in peat smoldering was larger according to Huang and Rein (2014). Without being impeded by upper layers, the heat released quickly so the temperature was lower than the humus.

Combustion depth could go to more than 50 cm (Ballhom et al. 2009). However, the combustion depth was found to be deeper due to the increased aridity of peatlands and wetlands due to global warming (Turetsky et al. 2015). The combustion process is also affected by oxygen levels, so when oxygen decreases as depth increases, combustion might be extinguished. This observation requires further study.

Applicability of NLME in smoldering temperature variation

Unlike surface and canopy fires, the heat released in underground fires is the manifestation of the smoldering process, which is also the important basis for fire monitoring and fighting. Fire intensity and spread rate are both predicted by the heat released (Rein et al. 2008a, b; Kirschke et al. 2013); the heat released, and temperature change of the ground surface are useful for judging the occurrence of underground fires and the direction for digging fire breaks. Therefore, thermos physical models are essential tools for studying the mechanism of underground fire occurrence and development (Huang and Rein 2014), and an important basis for fire monitoring and fighting.

This study was based on NLME modeling to predict temperatures of underground fires and focused on the prediction of temperature during smoldering. Simple statistical models are better for prediction research, and NLME is better than ordinary regression models. Many statistical models have been applied to forest fire prediction (Bem et al. 2018; Nadeem et al. 2020) and have increased prediction accuracy. The variation of temperatures during underground fire combustion showed a slow rise at the beginning, then a rapid rise, and finally, a stable temperature after reaching the maximum temperature. The change curves of the three basic models in this paper are “S” type, which are consistent with the change curves of underground fires, so the models could be applied to the prediction of temperatures during these fires.

In this study, well-replicated underground fire temperatures with varying depths and particle sizes were studied (Hall and Bailey 2001). NLME modeling was shown to be significantly superior in handling longitudinal, multi-level and replicated data (Calegario et al. 2005). Richards equations exhibited the best applicability in fitting humus combustion temperatures after evaluating three models and have been widely used owing to their biological significance, strong adaptability, and high accuracy (Li and Zhang 2010). Hang et al. (1997) and Hall and Bailey (2001) also found high accuracy of Richards equations, consistent with the results of our study.

Conventionally, the least square method was used to estimate model parameters of multilevel data, but this method tended to use data lacking independence, time correlation and space heterogeneity, and would lead to large prediction errors (Mensah et al. 2018). NLME modeling has higher accuracy (Sharma et al. 2016) through set up, fixed effect, and random effect parameters (Kalle 2009). Based on the results of RMSE in this study, the accuracy of models with mixing parameters were much higher than traditional nonlinear models, and the accuracy increased as the number of parameters increased. Ciceu et al. (2020) also reported the high precision of NLME models, although the parameters were different due to the research contents and basic models. Meng et al. (2009) found that a NLME model with two parameters was highly accurate. Pan et al. (2020) reported that one with three parameters had high precision.

Prospect of using NLME modeling in forest fire research

Studies with mixed effects models are emerging research areas in forestry (Calegario et al. 2005), but their applications in forest fire research are rarely reported. Studies of forest fires, based on multi-level aspects, such as combustible characteristics, fire forecasting and fire behavior, should reflect both overall change trend and individual differences, so NLME has more advantages (Guillermo et al. 2006). NLME also has the advantages of high reliability, flexibility and accuracy (Sharma et al. 2016), and have great application potential in the study of forest fire.