A Digital Twin Approach to Forest Fire Re-Ignition: Mechanisms, Prediction, and Suppression Visualization

Abstract

Statistics indicate that over 90% of large forest fires experience re-ignition after initial extinction. However, research on the mechanisms triggering forest fire rekindling remains largely empirical, lacking an intuitive 3D mathematical model to elucidate the process. To fill this gap, this study proposes a digital twin-based forest fire re-ignition trigger model to investigate the transition from smoldering to flaming combustion. Leveraging digital twin technology, a virtual forest environment was constructed to assess the influence of ambient wind conditions and terrain slope on the smoldering-to-flaming (StF) transition based on historical rekindling data. Subsequently, logistic regression was employed in a reverse iterative process to update the model parameters, thereby establishing a matching mechanism between the model predictions and the observed rekindling states. This approach enables the adaptive adjustment of the weights assigned to key variables (e.g., wind speed and slope) and facilitates the prediction of forest fire rekindling probability within the virtual environment. Additionally, digital twin simulations are employed to assess the 3D firefighting effectiveness of unmanned aerial vehicles (UAVs) deploying hydrogel and solidified foam extinguishing agents. This visualization of the firefighting process provides valuable insights, aiding in the development of more effective strategies for preventing and controlling fire re-ignition.

1. Introduction

Statistics indicate that over 90% of large forest fires undergo re-ignition after initial extinction [1]. Forest fires, as a prevalent form of natural disaster [2], can be classified into two primary categories: smoldering and open flame. Smoldering is characterized by an exothermic combustion process occurring in subsurface fires, which is primarily governed by solid-phase reactions at relatively low temperatures. Conversely, open flame represents a rapid combustion phenomenon that is dominated by gas-phase reactions at elevated temperatures. Under certain conditions, an underground fire may reach its ignition threshold, thereby initiating a new fire, a process referred to as smoldering-to-flaming (StF) transition (Figure S1) or the re-ignition of forest fires [3,4]. This transition results in intensified combustion, which has profound implications for the atmosphere, ecosystems, and human health [5].

The phenomenon of re-ignition of forest fires (hereinafter referred to as ‘StF’) is characterized by gas-phase ignition that is initiated by the self-acceleration of carbon oxidation within a smoldering environment, ultimately reaching the ignition threshold. Peng et al. [6] developed a comprehensive model to describe the transition from smoldering to flaming combustion under conditions of natural diffusion. Their investigation focused on various factors impacting the triggering mechanism of StF transition, including the concentration of combustible gases, oxygen diffusion, fuel characteristics, residual layer thickness, and temperature. They subsequently derived a mathematical expression outlining the conditions required for the initiation of flaming combustion under natural diffusion conditions. However, the integral model was primarily based on fixed parameter assumptions, limiting its ability to account for the dynamic interactions of complex environmental variables. Given that smoldering combustion exhibits different behaviors in terms of heat transfer, oxygen transport, and gas flow dynamics in both windy and calm environments, as well as on sloped versus flat terrains, this study further explores the impacts of ambient wind and terrain slope angle on StF transition based on the foundational work conducted in Peng et al. [6].

The complexity of forest environments, coupled with the inherent risks associated with StF transition experiments, complicates the acquisition of pertinent fire scene data through actual re-ignition experiments and the execution of firefighting operations. Consequently, the simulation and prediction of forest fires, as well as the reconstruction of fire scenes, emerged as significant focal points of contemporary forestry research [1]. In contrast, simulation studies of fire occurrence and suppression in 1D and 2D forest scenarios [7,8,9] employ visualization processes that are relatively coarse and constrained by low-dimensional spatial assumptions. These limitations impede the realistic representation of dynamic fire interactions in 3D terrains, the intuitive presentation of multi-dimensional triggering mechanisms from muddy combustion to open-flame combustion, and the immersive validation of fire suppression strategies. In this regard, 3D forest scenarios are essential for effectively simulating both the ignition and extinction phases of a fire during StF transitions. Therefore, the application of advanced digital twin technology is proposed to develop a 3D visualization simulation of the forest floor. This approach leverages original historical experimental data to facilitate the investigation of StF transition through multi-dimensional data within a virtual environment.

The study of 3D visualization concerning the state transition of forest fires involves the modeling and state-driven analysis of complex substances. The fundamental objective of state prediction is to forecast future conditions based on the current state. Currently, numerous prediction models and analyses of the factors impacting forest fire occurrences have been developed both domestically and internationally. The predominant modeling techniques employed include artificial neural networks [10], maximum entropy algorithms [11,12], classification trees [13], Poisson regression, negative binomial regression, zero-inflated Poisson models, zero-inflated negative binomial models, and logistic regression models [14,15,16,17]. Although machine learning techniques have been applied to fire prediction, their “black box” nature limits their physical interpretability. In response, this study proposes a hybrid approach that uses logistic regression to bridge the gap and inversely embed historical re-ignition data into the parameter space of a physical model [6]. By integrating logistic regression-driven parameter inversion with digital twin scenarios, efficient closed-loop optimization can be achieved, enabling iterative alignment of model predictions with real-world data, from historical data to physical models.

This study examines the gas-phase reaction associated with the re-ignition of forest fires by first addressing the primary conditions that impact the StF transition process, specifically oxygen concentration and heat conduction. The research investigates the efficacy of drones deploying hydrogel extinguishing agents and solidified foam extinguishing agents at the re-ignition point, utilizing a constructed digital twin scenario of the forest environment. This approach facilitates the visualization of firefighting techniques and their respective results, thereby providing valuable support for the training of firefighters and administrators. Additionally, the results may serve as a decision-making aid in firefighting operations [18,19].

The contributions of this paper are outlined as follows:

(1)

Utilizing digital twin technology to integrate multi-source data from topography, meteorology, and vegetation, a dynamic 3D forest scene was constructed to visualize the entire process of the StF transformation. The system supports the adjustment of dynamic parameters, such as ambient wind, and simulates the delivery of extinguishing agents, thereby providing a high-fidelity platform for fire deduction in complex environments and compensating for the limitations of traditional 1D and 2D models in representing spatial interactions.

(2)

A logistic regression algorithm was incorporated into the mathematical model of forest fire recombination [6]. Key parameters, including wind speed and slope, were calibrated via reverse iteration of historical data, resulting in the construction of an interpretable 3D trigger model for recombination state-driven and probabilistic prediction of the virtual scene.

(3)

3D simulation experiments revealed the fire extinguishing effects of hydrogel and cured foam extinguishing agents, offering intuitive data support for the design of fire extinguishing strategies.

The structure of the paper is organized as follows:

Section 1 provides an introduction to the background and significance of the research. Section 2 offers a review of pertinent studies related to 3D modeling of forest fires. Section 3 summarizes the research content and method proposed in this study. Section 4 describes data integration, scenario construction, heat transfer principles, state driving, and fire suppression principles. Section 5 details the design and implementation of the algorithm. Section 6 presents and discusses the research results. Section 7 evaluates the strengths and weaknesses of the research based on the relevant results and offers a comprehensive outlook for future investigations.

2. Related Research

Previous research revealed that StF in forest ecosystems is impacted by static factors, including terrain and vegetation type, as well as dynamic variables, such as ambient wind, temperature, and organic matter content. This study employs digital twin technology to create a virtual representation of a forest, drawing upon results from prior investigations. Additionally, it integrates a logistic regression algorithm to further examine the impacts of ambient wind and terrain slope angle on StF transition. The paper proposes a model for triggering StF within a 3D environment and further evaluates the 3D fire suppression efficacy of hydrogel and solidified foam fire extinguishing agents.

2.1. Impact Factors of Smoldering

At present, the critical conditions for smoldering extinction, smoldering ignition, and direct open flame ignition are extensively investigated [20,21,22]. However, investigations into the critical conditions for StF transition remain in the preliminary stages. Yin et al. [23] suggested that the smoldering process can be self-sustaining at a moisture content of 20% when the upper humus layer exhibits low moisture levels. Xin et al. [24] revealed that the critical moisture content of peat necessary for a positive StF transition is also 20%, with a critical wind speed of 4 m/s. Yang et al. [25] found that the high temperature required for StF transition is approximately 450 °C based on their analysis of experimental data. Aldushin et al. [26] and Wang et al. [27] demonstrated that StF transition can only occur when the peat layer exceeds a certain critical length or thickness. The critical value for StF transition varies significantly depending on different fuel types, environmental conditions, and smoldering conditions [28]. Therefore, a singular critical value for StF transition is not universally applicable [29], making it challenging to accurately represent the objective laws governing StF transition. Further research is necessary to explore the various factors impacting smoldering and the critical boundary relationships of StF transition that arise from their interactions (

Table 1. Impact factors of StF transition in forest fires (Grouped by category).

Category	Impact Factor
Meteorology	Ambient wind	Temperature and humidity	Oxygen concentration	Light intensity
Terrain	Slope	Elevation	Aspect	Longitude and latitude
Fireground	Thickness of residual layer	Intensity of heat	Degree of carbonization	Area of fire
Vegetation	Water content	Organic content	Inorganic content	Vegetation cover

).

The combustible water content, inorganic content, and organic content are regarded as critical factors impacting the occurrence and sustainability of smoldering combustion [4,20]. Yang et al. [30,31] conducted systematic experiments on peat smoldering and discovered that inorganic content is directly correlated with the formation of the ash layer during the smoldering process. They suggested that fine ash particles occupy the pores of the peat, thereby altering its porosity and permeability, which subsequently impacts the supply of oxygen. Additionally, ambient wind conditions facilitate the transformation of StF by enhancing oxygen transport within the smoldering and gas-phase ignition zones. Valdivieso et al. [32] revealed that the likelihood of StF transition in pine needles increases with higher wind speeds.

The slope gradient angle is a critical geomorphological parameter that significantly impacts the StF transition of forest fires, particularly in relation to topographic conditions. However, there remains a notable gap in the existing research regarding the impact of slope gradient angle on StF transition of forest fires [33]. The buoyancy effect causes the spread of smoldering on slopes to differ from that on flat surfaces, particularly in terms of heat transfer, oxygen transport, and gas flow mechanisms. Walther et al. [34] suggested that the buoyancy effect plays a more pronounced role during the critical phase of smoldering ignition under natural convection conditions (Figure S2). The gravitational forces acting on slopes can induce a complex buoyancy flow that directly impacts the oxygen supply in the char zone, thereby impacting the StF transition.

The aforementioned studies highlight the critical smoldering factors that require further investigation for the transformation of StF: fuel conditions (e.g., combustible moisture content, inorganic content, and organic content), ambient wind patterns, and terrain slope angle.

2.2. Digital Twins

The rapid development of technologies such as internet of things (IoTs), big data analytics, and simulation has facilitated the emergence of digital twins. As an increasing amount of data digitally mirror the real world, it empowers scientists and information technology specialists to optimize deployments for enhanced efficiency and to explore various hypothetical scenarios [35]. Nită et al. [36] highlighted that digital twins offer a viable solution for digitization within the forestry sector. However, the predominant focus of research on digital twin technology tends to center on areas such as smart manufacturing and factory operations [37]. Zhang et al. [38] integrated digital twin and virtual simulation technologies to propose a human-computer-physical system (HCPS) interaction mechanism aimed at the intelligent control of underground equipment in coal mines. This development is crucial for advancing intelligent systems in mining operations. Conversely, the application of digital twin technology in environmental simulation and monitoring research remains limited. Saihanba Forest Fire Command Center promoted the implementation of “digital twin” technology [39], while Tang et al. [40] proposed its application in intelligent energy systems, representing a notable instance in this domain.

Buonocore et al. [41] introduced a framework for the development of a digital twin of forests, which creates a virtual representation of the forest to facilitate the reporting of forest-related data by integrating various state variables at both the tree and forest levels. Similarly, Li et al. [42] proposed a digital twin architecture for a virtual poplar plantation forest system based on an experimental plantation. The utilization of the digital twin allows for the comprehensive importation of forest regeneration data, enabling the generation of digital maps and establishing a precise linkage between the physical and digital realms [43]. This integration facilitates the intelligent simulation of scenarios related to StF transition. Consequently, the feasibility and rationality of the design scheme can be validated in a digital and virtualized context, providing an effective approach for implementing the trigger mechanism for the re-ignition of forest fires. This method also aids in addressing the operational challenges encountered by the Institute of Re-ignition of Forest Fires.

2.3. State-Driven Model

The StF transition driving model for forest fires serves as a fundamental framework for the investigation of simulation techniques related to the re-ignition of forest fires. Currently, there exists a limited number of semi-empirical models addressing StF transition, with Peng’s mathematical model being one of the most widely utilized in the field [6]. In a comprehensive review of gas-phase ignition, Peng et al. developed an integral model that describes the propagation of smoldering and its transition to flaming combustion under 1D natural diffusion conditions. This model incorporates various factors impacting the triggering mechanism of StF transition. The integral method was employed to solve the conservation equations for mass and energy within the dynamic regions, leading to the derivation of mathematical expressions that define the smoldering conditions present in the flames as follows [6]:

$$\left(1-\alpha \left(\infty \right)\right)\left({\rho }_0-{\rho }_1\right){\displaystyle \frac{2k_0}{{\rho }_0{c}_{p,0}}}{\displaystyle \frac{1}{{\alpha }_0\left(\infty \right)}}>{\displaystyle \frac{D{\epsilon }_c{Y}_{gas,f}}{{\delta }_c}}$$

(1)

where $\mathsf{\alpha }(\mathrm{\infty })$ is the proportion of fuel involved in oxidative pyrolysis; ${\rho }_0$ is the initial density of fuel; ${\rho }_0-{\rho }_{\mathrm{c}}$ is the density difference of fuel; ${\mathrm{k}}_0$ is the heat transfer coefficient of fuel; $c_{p,0}$ is the specific heat capacity of fuel; $D$ is the diffusion coefficient of oxygen; ${\epsilon }_c$ is the porosity of charcoal; $Y_{gas,f}$ is the critical combustible gas concentration; and $\mathsf{\delta }\mathrm{c}$ is the thickness of the residual layer.

The factors impacting the transformation of StF are not confined to those previously mentioned. Therefore, an algorithm is required to explore additional impacting factors based on the existing model. Currently, logistic regression is extensively utilized for probabilistic predictions. Zhou et al. [44] developed a static-dynamic coupled landslide displacement prediction model, employing logistic regression to address the limitations of traditional static prediction models in capturing landslide evolution. To examine the spatial distribution characteristics of gully landforms on the Loess Plateau, Fan Tiancheng et al. utilized logistic regression models [45] to analyze the spatial distribution of gully landforms and their environmental control factors within the Yanhe River basin. Additionally, they proposed a probabilistic prediction model for the risk of glacial lake outbursts in the Tibetan region. Ma et al. [46] introduced a hazard probability prediction model for glacial lake outbursts in the Tibet region, conducting logistic regression analysis on glacial lake samples from the region.

In this study, the factors impacting the rekindling process were digitized, and a logistic regression model was integrated with Peng’s mathematical model. This integration facilitated the analysis and prediction of the StF transition state, taking into account variables such as ambient wind conditions and terrain slope angle.

2.4. Firefighting Materials and Equipment

Presently, the materials utilized for extinguishing forest fires are primarily categorized into three types: chemical fire extinguishing agents, hydrogel fire extinguishing agents, and foam fire extinguishing agents. Within the category of foam fire extinguishing agents commonly employed in forest fire management, there are three principal types: water-based foam fire extinguishing agents, gel foam fire extinguishing agents, and solidified foam fire extinguishing agents [47] (

Table 2. Comparison of relevant properties for five extinguishing agents.

Forest Fire Suppression Materials
TYPE	Raw Material	Fire Suppression Principle	Advantage	Disadvantage
FOREST CHEMICAL FIRE SUPPRESSANT	Diammonium phosphate, ammonium polyphosphate [48]	Interrupts combustion chain by free radical pyrolysis	Wide range of applications Flexible and easy to use	High cost Hard to biodegrade Potential human toxicity [49]
HYDROGEL FIRE SUPPRESSANT	Various polymers, natural ingredients, additives	Moisture evaporation removes heat Forms an insulating gel film [50]	Fast cooling speed Low amount of extinguishing agent [51]	High cost Insufficient mobility and permeability Limited scope of application
WATER-BASED FOAM FIRE EXTINGUISHER	Foaming agent, foam stabilizer, deionized water, additive	Forms a water film to block oxygen Evaporation absorbs heat	Easily accessible and cost-effective Good fluidity and permeability	Residual components can have an impact on the environment Stability and adhesion require enhancement Limited scope of application
GEL FOAM FIRE SUPPRESSANT	Water-based foam, gelling agent	Cover combustible surfaces and isolate from oxygen	Strong adhesion and diffusion Prolonged fire suppression	High cost Limited application scenarios
SOLIDIFIED FOAM FIRE EXTINGUISHER	Water-based foam, silicate-based inorganic powder	Forms a hardened protective layer, preventing oxygen penetration Moisture release provides cooling	Low environmental impact High thermal stability and durability Environmentally friendly, wide range of applications	High cost

).

Forest firefighting equipment serves multiple functions, including extinguishing fires, preventing their spread, and facilitating detection and monitoring. In recent years, aerial firefighting technology has emerged as an essential method for combating forest fires [52], offering benefits such as early detection and rapid response [53]. Due to their high efficiency, flexibility, intelligence, and multifunctionality, unmanned aerial vehicles (UAVs) have become a significant asset in contemporary firefighting efforts and are expected to assume an even more critical role in future firefighting operations.

This study does not address the large-scale re-ignition of forest fires. Instead, it focuses on the characteristics of various extinguishing materials, utilizing UAVs to deploy hydrogel and solidified foam extinguishing agents for simulated firefighting operations. This approach facilitates a comprehensive examination of the 3D suppression impacts of the two agents and offers reliable data to inform decision-making for firefighters in subsequent rescue operations.

3. Overview

This study aims to investigate the mechanisms underlying the re-ignition of forest fires by employing digital twin technology to create a virtual forest environment based on Peng’s mathematical model. The research further examines the impact of ambient wind conditions and terrain slope angles on StF transition. Additionally, a logistic regression algorithm is utilized to propose a model that describes the re-ignition of forest fires within a 3D context. Furthermore, the effectiveness of UAV-delivered hydrogel fire extinguishing agents and curing foam fire extinguishing agents is evaluated. By visualizing the suppression process, the extinguishing efficacy of various dosages and methods on re-ignited flames is assessed, thereby enhancing the realism and immersion of the firefighting experience. This research provides valuable data to support decision-making in firefighting and rescue operations. The overall structure of the paper is illustrated in Figure 1.

4. Materials and Methods

4.1. Data Integration and Scenario Building

Analysis of forest fire data focused on the re-ignition phenomenon influenced by variations in multi-dimensional factors. The study relies on a long-term historical collection of forest fire databases from Xichang and Panzhihua, which include historical fire imagery, meteorological conditions, topographical features, and geomorphological characteristics. Selected data were integrated for model development. Additionally, a Neo4j (https://neo4j.com/) graph database was employed to organize fire-related data (e.g., temperature, humidity, wind speed, and wind direction) recorded during historical StF transitions, thereby facilitating model construction and enhancing data retrieval to provide comprehensive baseline support for the study (

Table 3. Partial dataset of environmental and fire conditions in impacted regions.

Variable	Mean	Deviation	Minimum	Maximum	25% Quartile	75% Quartile
Oxygen concentration (%)	4.2	2.1	1	9	3	6
Slope (°)	32.4	3	5	83	25	65
Wind speed (m/s)	4.3	2.5	1	10.1	3.6	8
Residual layer thickness (cm)	4.1	2.1	2	8	3.2	6.4
Temperature (°C)	410	224	45	850	236	615
Humidity (%)	26	40	2	64	15	52

).

Various datasets are utilized in this study, including terrain, fire, weather, vegetation, and other impact-related information, to create a data-driven virtual representation of a forest scene within a digital twin framework. To accurately simulate the StF transition scene, IoT technology is employed to facilitate changes in the model scene and to access data through a message queuing telemetry transport (MQTT)-type interface in J3D (https://www.fulima.com/). The associated processes and initial data configurations are depicted in Figure 2 and

Table 4. Initial parameter settings for fire simulation.

Ambient Light	Light Rotation	Light Intensity	Temperature	Humidity	Oxygen Content	Residual Layer Thickness
Cloudy plain	15°	1.05lx	120 °C	25%	16%	7 cm
Flame particle volume	Emission direction	Emission range	Maximum lifetime	Minimum lifetime	Interface response form	Timing duration
1	Spread outwards	Round	5 ms	2 ms	Json	10 ms

.

Subsequently, the model is constructed. Historical meteorological, forest topographic and geomorphological, and vegetation distribution data were integrated using Lithium Code Cloud J3D Digital Twin Designer and RBI Business Intelligence Designer to develop a digital model that accurately reflects the development pattern of forest fire recombustion. By incorporating multi-dimensional environmental data (e.g., slope, elevation, and other relevant information pertaining to the macro-geomorphology and vegetation distribution of the forested area), a comprehensive large-scale forest scene is generated. This scene encompasses the geometric and structural modeling of trees, terrain, and various other elements. Additionally, multiple localized models were integrated to create a cohesive forest scene. The integration accounts for the effects of light, weather variations, and other factors to ensure precise optimization and to provide a standardized database and virtual environment for subsequent physical modeling and simulation. The dynamic transformation process simulated in this study, referred to as the StF transition, comprises four states: unburned (e.g., vegetation intact, no pyrolysis, sufficient combustibles, and no ignition points), smoldering (e.g., slow internal oxidation, absence of visible flame on the surface, and smoke release), open flame (e.g., intense combustion, visible flashpoints, and high-temperature gas release), and extinguished (e.g., vegetation in a visibly charred state, with the open fire extinguished). The method for modeling the forest elements is illustrated in Figure 3.

4.2. Heat Transfer Principles and State Transitions

The 3D geometric model presented herein integrates heat transfer, ambient wind, and topography to simulate the combustion process. A joint physical-data driven model of fire evolution is then constructed by combining heat transfer equations, historical data, and coupled wind-slope correction terms within a digital twin environment that effectively links data inputs with the dynamic process.

4.2.1. Principle of Heat Conduction

The StF transformation is a gas-phase reaction that is strongly influenced by oxygen concentration and heat transfer between adjacent vegetation. In the shaded combustion state, the vegetation wood undergoes decomposition into carbon and combustible gases, with oxidative reactions occurring when carbon comes into contact with oxygen [54]:

$$Wood (solid)+Heat\to Fuel (solid)+Pyrolyzate (gas)$$

(2)

$$Fuel (solid)+O_2 (gas)\to Heat+other gases+Ash (solid)$$

(3)

The heat generated simultaneously is transferred between neighboring vegetation. When the conditions for rekindling are met, the shaded combustion is exposed to the heat and ignites, as illustrated by the following equation for calculating heat transfer:

$${\displaystyle \frac{dQ}{dt}}=\lambda A{\displaystyle \frac{T_a-T_b}{H_c}}$$

(4)

The heat absorbed by the vegetation module is calculated using the above equation based on Fourier’s law. In this equation, λ denotes the thermal conductivity, A represents the contact area between vegetation, the temperature difference between adjacent vegetation is expressed as $T_a-T_b$, and the carbonaceous thermal insulation parameter, $H_c\in [0.01,0.1]$.

4.2.2. Vegetation State Transition

A mathematical model-driven framework facilitates data-driven and dynamic iteration of vegetation states within 3D twinned scenes. In this study, StF transition data are derived from a historical database lookup. Ambient wind and terrain slope angle significantly impact oxygen concentration and heat transfer, which in turn impacts the StF transition process. Consequently, logistic regression is integrated with the mathematical model to further explore the influence of ambient wind and terrain slope angle on the dynamic transformation process. Equation (1) is revised as follows:

$$P\ast (1-\alpha (\infty )) ({\rho }_0-{\rho }_{\mathrm{c}}){\displaystyle \frac{2{\mathrm{k}}_0}{{\rho }_0{\mathrm{c}}_{\mathrm{p},0}}}{\displaystyle \frac{1}{{\alpha }_0(\infty )}}>{\displaystyle \frac{D{\epsilon }_c{Y}_{gas,f}}{{\delta }_c}}$$

(5)

$$\mathrm{P}={\displaystyle \frac{1}{1+{\mathrm{e}}^{-\mathsf{\beta }\mathrm{X}}}}$$

(6)

$$\beta =a_0+a_1{x}_1+a_2{x}_2$$

(7)

where $\beta$ is the linear function of wind speed (m/s) and terrain slope angle (°) corresponding to the variables, while $x_1$, $x_2$, $a_0$, $a_1$, and $a_2$ are the regression coefficients of each impact factor.

StF transition can be conceptualized as a collection of behaviors exhibited by individual vegetation modules. Each vegetation module possesses a different set of discrete states that vary over time, impacted by a multitude of factors [55]. When considering the forest scene as a coordinate system, the horizontal axis is represented by X units and the vertical axis by Y units, resulting in a total of X × Y units. Each unit (Unit size: 1 m × 1 m [55]) corresponds to varying environmental conditions and static attributes, including diverse wind directions, wind speeds, and terrain characteristics. Therefore, this study establishes a digital twin model that incorporates the impact of multi-dimensional data. To facilitate simple interactions among vegetation modules located in different areas, the vegetation is categorized into four states. Furthermore, to account for the impact of multi-dimensional data (e.g., ambient wind, slope, and thermal conduction from neighboring cells) on StF transition of the target cell, a rule governing the state transition of the cell element (i, j) is formulated for the subsequent time step, which is elaborated upon in the following sections (Figure 4):

• If the state of the target cells (i, j) at time t is “0” and there are no surrounding cells in the burning state, the state of “t + 1” is still “0”.
• If the state of the target cells (i, j) at time t is “0” and there are surrounding cells in the burning state, the state at “t + 1” will be “1”.
• The state of the target cells (i, j) at time t is “1” and satisfies the modified mathematical expression, then the state at “t + 1” will be “2”.
• The state of the target cells (i, j) at time t is “2”, and as the fuel runs out, the state at “t + 1” will change to “3”.
• The state of the target cells (i, j) at time t is “1” or “2”, and the operation of the UAV to drop the fire extinguishing agent is carried out, and depending on the different doses of the drop, the state of “t + 1” will change to “0” or “1”.
• The state of the target cells (i, j) at time t is “2”, and if the UAV drops the extinguishing agent to form a barrier, the state at “t + 1” will change to “3”.

After the importation of the database, the logistic regression model, guided by StF state transition rules, can delineate a binary relationship between the impact factor and the probability of re-ignition. In this context, the impact factor serves as the independent variable, while the change in vegetation state is treated as the dependent variable. The model is fitted using “Maximum Likelihood Estimation”, which facilitates the identification of optimal bias parameter values and weights based on the differential data derived from virtual reality. This process ultimately culminates in the development of the forest fire StF transition model.

4.3. Suppression Principles

StF transition is a gas-phase reaction that is significantly impacted by oxygen concentration and heat transfer. For fire suppression methods, this study focuses on the primary gas-phase reaction’s direct impact factors. Drones are used to deliver water gel extinguishing agents and curing foam extinguishing agents from the air, effectively obstructing the conditions necessary for oxygen and temperature that could lead to re-ignition. This approach aims to achieve effective fire suppression.

4.3.1. Physical Asphyxiation

When applied to a burning surface, hydrogel and solidified foam extinguishing agents rapidly expand, forming a dense gel film or foam layer that enhances fire suppression by creating an effective thermal and oxygen barrier. This physical barrier adheres closely to the surface of the combustible material, effectively isolating it from oxygen and thereby interrupting the supply of oxygen essential for the gas-phase reaction. This process results in a phenomenon known as physical asphyxiation. The formula for calculating the oxygen transmission rate (OTR) in the upper layer is expressed as follows:

$$OTR={\displaystyle \frac{K\ast \Delta P\ast A}{d}}$$

(8)

where K is the oxygen transmission coefficient, which describes the material’s ability to transmit oxygen (cc/(m²·24 h·0.1 MPa)); $\Delta P$ is the difference in oxygen partial pressure on both sides of the gel or foam film ($\mathrm{M}\mathrm{P}\mathrm{a}$); $A$ is the experimental test area (m²); and $d$ is the thickness of the film or foam ($\mathrm{m}\mathrm{m}$).

The thickness of the film or foam varies depending on the dosage of the extinguishing agent applied, directly impacting the rate of oxygen transmission and, consequently, the effectiveness of fire suppression. This variation may ultimately disrupt the balance of the combustion reaction, resulting in the transformation of vegetation from an open-flame state to a cloudy or unburned state.

4.3.2. Cooling Down

During the firefighting process, the evaporation of water from the hydrogel extinguishing agent, along with the expansion and curing of the foam, absorbs a significant amount of heat. This absorption prevents the plants undergoing pyrolysis from receiving adequate heat to sustain their temperature, thereby decelerating the combustion rate and extinguishing the flames. Additionally, this cooling effect contributes to the extinguishment of the fire by diminishing the conditions necessary for re-ignition and inhibiting the reactivation of the vegetation cells. The evaporation of moisture and its associated cooling effect can be quantified using the following equation:

$$m_0\ast L=m_1\ast \Delta T\ast C_p$$

(9)

where $m_0$ is the mass of the extinguishing agent ($\mathrm{k}\mathrm{g}$); $L$ is the heat of vaporization of water ($2.26\times {10}^6 \mathrm{J}/\mathrm{k}\mathrm{g})$; $m_1$ is the mass of the vegetation (kg); $\Delta \mathrm{T}$ is the value of the temperature reduction (°C); and $C_{\mathrm{p}}$ is the specific heat capacity of vegetation (Table S1 [56]).

4.3.3. Isolation and Blocking

The application of an extinguishing agent in unburned areas can create a foam isolation zone, which serves to obstruct the interaction between burning materials and adjacent combustible materials. This intervention significantly impacts the heat conduction processes, preventing the transfer of thermal energy to neighboring vegetation. Consequently, the vegetation within the isolation zone will transition into an extinguished state as its fuel is depleted, while the vegetation outside this zone will remain unable to ignite due to insufficient heat exposure. This mechanism indirectly contributes to fire extinguishment, as well as prevention and control efforts.

The variation in mass of the combustible material during the combustion process is given in Equation (8), where ${\mathrm{H}}_{\mathrm{c}}$ and $\mathrm{A}$ are both partially dependent on the geometry of the combustibles and changes throughout the combustion process, $\mathsf{\alpha }$ is a dimensionless coefficient that adjusts the reaction result, and ${\mathrm{f}}_{\mathrm{T}\mathrm{M}}$ is the rate of the pyrolysis reaction [57].

$${\displaystyle \frac{dm}{dt}}+\alpha f_{TM}{H}_{c}A=0$$

(10)

$$f_{TM}=\eta (u)\ast \left\{\begin{array}{l}0, T_M\le T_0\\ S ({\displaystyle \frac{T_M-T_0}{T_1-T_0}}\\ 1, T_M\ge T_1\end{array}\right.) , T_0<T_M<T_1$$

(11)

$$\eta (u)=({\eta }_{\max }-1)S({\displaystyle \frac{u}{u_{ref}}})+1$$

(12)

$$S(x)=3x^2-2x^3$$

(13)

where S(x) denotes an S-shaped curve that transitions smoothly from 0 to 1 for temperatures between $T_0$ = 350 °C and $T_1$ = 650 °C. The temperature $T_M$ is determined by $T_M=T-\Delta T$, where $\Delta T={\displaystyle \frac{m_0\ast L}{m_1\ast C_p}}$ and $T$ are dynamically retrieved from the historical database. The function η(u) modulates the reaction rate based on wind speed; in the absence of wind, η = 1, whereas under wind conditions, η(u) is adjusted according to the magnitude and direction of the wind speed. u(x,t) represents a time-varying vector that characterizes the wind [57], dynamically sourced from a historical database, with its values corresponding directly to the actual recorded wind speed (v(t)) and wind direction. In the present model, the η(u) function (Equation (12)) modulates the rate of pyrolysis (Equation (11)) based on the magnitude of wind speed, while wind direction influences the process indirectly through the spatial distribution of the dynamic wind field u(x,t) in the process of oxygen supply and heat transfer to neighboring vegetation units.

The oxygen concentration within the fuel, ambient temperature, and fuel loss in the presence of an extinguishing agent can be determined using Equations (8)–(13). Subsequently, the variations in these factors can be integrated to initiate the combustion equilibrium. This process allows for the visualization of the suppression impact of physical asphyxiation caused by the extinguishing agent.

4.4. Visualization

To effectively convey the characteristics of environmental data related to the re-ignition of forest fire disaster areas and the analysis of StF transition probabilities, this study employs RBI Business Intelligence Designer to create a visualization and interaction interface. This interface is capable of automatically conducting data drilling, integrating heterogeneous data sources, and facilitating 2D/3D integration. The visualization technology enables the representation of the digital twin model through 3D images, videos, and other formats. The visualization interface presents a simulation of the actual scene via the user interface, enabling the display of negative combustion monitoring, re-ignition probability prediction, fire warning, and fire-extinguishing processes, and providing intuitive tools for decision makers.

5. Algorithm Implementation

This study employs Fulima Cloud J3D Digital Twin Designer and RBI Business Intelligence Designer as tools for the development of digital twin applications. The implementation of related functionalities, logical design, and computations is conducted using Python (3.10) programming language (Algorithm 1).

Algorithm 1. The reignition of forest fires is driven by dynamic historical data.
Input: Vegetation Data {T1, T2, …, Tn} (position, organic content, moisture content, initial state), Meteorological and Topographic Data (wind speed, slope angle, temperature, humidity), Historical Dataset ($D={\{(v^{(j)},{\alpha }^{(j)},y^{(j)})\}}_{j=1}^m, y\in \left\{\left.0,1\right\}\right.$), Suppression Parameters.
Output: Final states of all vegetation units, optimal mathematical model.
1:	Initialize: Load digital twin scene (Figure 3), Model parameters (${\mathrm{a}}_0,{\mathrm{a}}_1,{\mathrm{a}}_2$, k0, D…), Bind historical data to virtual scene (Figure 5)
2:	While not converged and iterations < max_iterations then
3:		For each sample in D: Calculate predicted probability by Equation (5), Compute the loss by Equation (16) and backward propagation, and update by Equations (17) and (18);
4:		If $\left|\Delta J\right|<1\times {10}^{-5}$ then
5:			Output optimized coefficients ($a_0^{*}, a_1^{*} a_2^{*}$);
6:			end if
7:			for each i = 1, 2, …, n do
8:				Update the surface temperature and oxygen concentration and organic matter content of the Ti according to the relative humidity of the environment by Equations (4) and (8)–(10);
9:				If Ti in smoldering (State 1) then
10:					If Equation (5) holds and adjacent flames exist then
11:						Transition to flaming (State 2);
12:					else if conduct fire-fighting operations and update the data of fire scene by Equations (4) and (8)–(10) then
13:						Transition to unburned (State 1);
14:					end if
15:				If Ti in flaming (State 2) then
16:					if conduct fire-fighting operations and update the data of fire scene by Equations (4) and (8)–(10) then
17:						Transition to smoldering (State 1);
18:					else if fuel depletion in Ti or extinguish the fire by creating firebreaks Equation (10) then
19:						Transition to extinguished (State 3);
20:					end if
21:				end if
22:				If Ti in fextinguished (State 3) and mass drops to zero by Equation (10) then
23:					Remove Ti from {T};
24:				end if
25:			end if
26:			Update the vegetation state according to Equation (14);
27:		end for
28:	end for
29:	Get the termination status of all vegetation and optimal mathematical model.

Based on the above principles and methods, the constructed virtual forest scene incorporates a predefined set of initial weights within the mathematical model to drive the vegetation state changes. The model-driven state is then compared with the original vegetation state from the database, and the gradient descent algorithm in logistic regression is utilized to update the weight parameters in the mathematical model. This process enables the refinement of the StF triggering mechanism, allowing for accurate predication of forest fire rekindling probabilities. Ultimately, this approach facilitates the visualization of both rekindle and fire suppression processes within the virtual forest environment (Figure 6).

The initial weights of the mathematical model are predetermined to facilitate the simulation of changes in vegetation states. The model-driven states are then compared with the original vegetation states from the database. The gradient descent algorithm associated with logistic regression is utilized to update the weight parameters within the mathematical model. This process ultimately leads to the establishment of the StF triggering mechanism, which is used to predict the probability of forest fire re-ignition.

6. Experimental Results and Analysis

6.1. Simulation Validation of Impact Factors

This study aims to investigate the impacts of ambient wind and slope angle on StF transition, utilizing the mathematical model [6] as a foundational framework. To ensure the accuracy of the simulation results, the initial phase of the research involves manipulating the temperature and increasing the carbon layer thickness at the smoldering point within a controlled, flat, and windless experimental area. Subsequently, the study verifies the efficacy of the mathematical model. 3D visualization of the digital twin StF transition effects, along with the results of the simulation experiments, is presented in Figure 7.

The results indicate that an increase in temperature leads to an acceleration of StF transition, resulting in the uniform appearance of an open flame around the afterburning point. Conversely, an increase in the thickness of the carbon layer results in a deceleration of StF transition, accompanied by the phenomenon of smoldering disappearance. These observations are consistent with the state changes documented in the original database and further validate the efficacy of Peng’s mathematical and theoretical modeling in facilitating StF state transition.

The ambient wind significantly impacts the oxygen concentration at the smoldering point, thereby impacting the StF transition process. This study simulates the relevant results under varying wind speeds and directions, taking this critical factor into account. The measured wind speed under natural convection conditions, as recorded by an anemometer, is 3.4 m/s. According to the wind power level classification [58], this corresponds to a class 2 wind (Table S2), categorized as a light wind, during which no transition from smoldering to open flame occurs throughout the entire process. Consequently, this research investigates StF transition effects at wind speeds classified as classes 3 and 4. In the context of uniform fuel and without accounting for slope effects, the results of the 3D visualization and simulation experiments of the digital twin StF transformation effect are shown in Figure 8.

The results indicate that an increase in wind speed correlates with a more rapid StF transition and an increase in the number of open flames observed between each time step. Furthermore, for varying wind directions, the orientation of the occurrence of open flames progressively aligns with the prevailing wind direction.

The mechanisms of heat and oxygen transport differ between sloped and flat-slope smoldering. Therefore, variations in slope angles can lead to changes in wind direction, which directly impact the oxygen supply within the charcoal layer zone and, subsequently, StF transition. This study investigates the impact of StF transition by increasing the terrain slope angle in the test area under flat and windless conditions, utilizing uniform fuel. The results indicate that StF transition occurs more rapidly in the experimental area as the terrain slope angle increases (Figure 9).

Considering the effects of ambient wind and terrain slope on the StF transition, a logistic regression algorithm based on the mathematical model [6] was employed for the preliminary fitting of the StF state transition. Initial weights were set as [0, 0, 0], yielding a modified fire-driven mathematical model. Discrepancies between the experimental results and 3D simulation outcomes were used as feedback for further corrections, ultimately determining optimal values for a₀, a₁, and a₂, namely (5.333, 0.0158, −0.094). The data of a₀, a₁, and a₂ during gradient descent varies as shown in Figure 10.

The final corrected mathematical expression is expressed as follows:

$${\displaystyle \frac{1}{1+e^{5.333+0.0158x_1-0.094x_2}}}\ast (1-\alpha (\infty )) ({\rho }_0-{\rho }_{\mathrm{c}}){\displaystyle \frac{2{\mathrm{k}}_0}{{\rho }_0{\mathrm{c}}_{\mathrm{p},0}}}{\displaystyle \frac{1}{{\alpha }_0(\infty )}}>{\displaystyle \frac{D{\epsilon }_c{Y}_{gas,f}}{{\delta }_c}}$$

(14)

6.2. Model Accuracy Validation

The loss function serves as a quantitative measure of the discrepancy between the predictions generated by the logistic regression model and the actual observed values. Given that logistic regression addresses a classification problem, the re-ignition of forest fires is 1, while the non-re-ignition is 0. In contrast to traditional regression models, which yield deterministic results, the logistic regression model produces probabilistic predictions. These probabilities are utilized to ascertain whether the estimated result is 1 or 0. This framework encompasses two scenarios as follows:

$$\mathrm{J}=\left\{\begin{array}{c}\begin{array}{c}-\log \left(\mathrm{p}\right) \mathrm{i}\mathrm{f} \mathrm{y}=1\\ -\log \left(1-\mathrm{p}\right) \mathrm{i}\mathrm{f} \mathrm{y}=0\end{array}\end{array}\right.$$

(15)

The integration of the aforementioned two cases with the function results in the following cross-entropy loss for logistic regression:

$$\mathrm{J}(\mathsf{\theta })={\displaystyle \frac{1}{\mathrm{m}}}{\sum }_{\mathrm{i}=1}^{\mathrm{m}}\left[-{\mathrm{y}}^{(\mathrm{i})}\log ({\mathrm{h}}_0({\mathrm{x}}^{\mathrm{i}}))-(1-{\mathrm{y}}^{(\mathrm{i})})\log (1-{\mathrm{h}}_0({\mathrm{x}}^{\mathrm{i}}))\right]+{\displaystyle \frac{\mathsf{\lambda }}{2\mathrm{m}}}{\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathsf{\theta }}_{\mathrm{j}}^2$$

(16)

In this study, the model parameters are updated using the gradient descent method throughout the experimental process to identify the optimal weights and biases of StF transition model. This approach aims to ensure that 3D driving results of the model closely approximate the original state changes. The gradient descent function is presented as follows:

$${\displaystyle \frac{\partial \mathrm{J}(\mathsf{\theta })}{\partial }}={\displaystyle \frac{1}{\mathrm{m}}}{\sum }_{\mathrm{i}=1}^{\mathrm{m}}({\mathrm{h}}_0({\mathrm{x}}^{(\mathrm{i})})-{\mathrm{y}}^{(\mathrm{i})}){\mathrm{x}}_{\mathrm{j}}^{(\mathrm{i})} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}=0$$

(17)

$${\displaystyle \frac{\partial \mathrm{J}(\mathsf{\theta })}{\partial }}=({\displaystyle \frac{1}{\mathrm{m}}}{\sum }_{\mathrm{i}=1}^{\mathrm{m}}({\mathrm{h}}_0({\mathrm{x}}^{(\mathrm{i})})-{\mathrm{y}}^{(\mathrm{i})}){\mathrm{x}}_{\mathrm{j}}^{(\mathrm{i})})+{\displaystyle \frac{\mathsf{\lambda }}{\mathrm{m}}}{\mathsf{\theta }}_{\mathrm{j}} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{j}\ge 0$$

(18)

During the gradient descent process, this study employed GridSearchCV for hyperparameter tuning to better capture variations in ambient wind and terrain slope angle. The search determined a learning rate of 0.004, and the model was trained for 20,000 iterations. As a result, the test set accuracy exceeded 85.1% (Table S3), representing an approximately 4.3% improvement over conventional direct data fitting methods. These findings highlight the strong predictive performance of the StF transition trigger model (Figure 11).

6.3. Fire Suppression Visualization

Different dosages of extinguishing agents yield varying suppression impacts. Previous studies indicate that when the mixing ratio is maintained between 0.2% and 0.5%, commonly utilized extinguishing agents demonstrate enhanced fire-extinguishing efficacy, with optimal performance observed at mixing ratios of 0.3% and 0.5% [59]. This study specifically examines the impacts of hydrogel extinguishing agents and solidified foam extinguishing agents at a mixing ratio of 0.3% on fire suppression efficacy under identical reignited flame conditions. A total of 100 mL of extinguishing agent was administered at 2.5-min intervals, and 3D fire suppression visualizations and results are illustrated in Figure 12. As the quantity of extinguishing agent increases, the fire suppression effectiveness of both agents exhibits an approximately exponential growth before stabilizing. Notably, the dosage required to achieve stable fire suppression with the solidified foam extinguishing agent is lower than that needed for the hydrogel extinguishing agent.

The strategic placement of extinguishing agents significantly impacts their effectiveness in fire suppression. This study examines the impact of extinguishing operations conducted by UAVs utilizing hydrogel and solidified foam extinguishing agents. Specifically, it investigates the impacts of flame base targeting, rekindling obstruction, and isolation zone establishment on extinguishing efficacy across three delivery modes (Figure 13).

For a fixed quantity of extinguishing agent, the solidified foam extinguishing agent demonstrates superior fire suppression efficacy compared to the hydrogel extinguishing agent across all three delivery methods, with the most effective results observed when extinguishing at the flame’s base.

7. Conclusions and Outlook

This study investigated the triggering mechanisms of forest fire re-ignition by implementing forest scene simulations based on digital twin technology. A logistic regression algorithm is applied to analyze the impacts of ambient wind and terrain slope angle on the re-ignition process, building upon previous research. The study ultimately proposes a 3D model for StF transition in forest fires. The key conclusions are summarized as follows:

(1)

A virtual scene was created using digital twin technology. The simulation accounts for various factors, including combustible materials, static elements, and dynamic factors. These factors are integrated to visualize the re-ignition behavior within a 3D forest environment.

(2)

Based on Peng’s mathematical model, a logistic regression algorithm was utilized to examine the impacts of ambient wind and terrain slope on StF transition. The resulting model provides a framework for predicting the triggering of forest fire re-ignition in a virtual environment, using state-driven and probabilistic predictions.

(3)

The study conducted an initial exploration of the suppression efficacy of UAVs equipped with hydrogel and solidified foam extinguishing agents. The 3D visualizations generated offer valuable insights for developing more effective fire suppression strategies on the ground, providing a scientific foundation for firefighter decision-making.

Despite the valuable insights provided by this research, several limitations must be acknowledged:

(1)

The study focused on constructing a virtual scene based on historical data. The real-time interactions between the virtual scene and real-world entities were not considered, and the spatial discretization for forest fire recombustion was not further examined.

(2)

To broaden the applicability of the StF transformation mechanism, additional forest scenarios are required. This paper simulated only the forest scenarios of Xichang and Panzhihua, which imposes significant limitations on its scope. Given the complex physical and chemical processes of forest fire re-ignition, employing digital twins to develop multiple forest scenarios could yield more comprehensive validation.

(3)

The transformation from cloudy ignition to open fire is highly complex. Consequently, beyond the influences of combustibles, wind speed, slope, and other factors addressed herein, further investigation into additional factors affecting forest fire re-ignition is warranted. For instance, the potential for wind to alter the distribution of combustibles (i.e., bringing them into contact with external oxygen and triggering secondary re-ignition) has not been considered.

Based on the above discussion, this study will further investigate the synergistic application of multiple hyperparametric methods and the potential for establishing numerical threads as the volume of data continues to increase. Additionally, the model will be enhanced by explicitly incorporating wind direction parameters and integrating vector wind field models, such as computational fluid dynamics (CFD) simulations. We will also introduce additional influencing factors to improve the model’s accuracy in predicting outcomes in complex terrain and multi-wind scenarios, thereby facilitating more efficient data utilization and more precise simulations. Furthermore, the model will be refined through parametric sensitivity analyses. The impact of discretization on the results will be quantitatively assessed via parametric sensitivity analysis. Adaptive grid techniques will be explored to optimize computational efficiency, and the effect of varying spatial cell sizes on prediction outcomes will be systematically examined.