An Octant-Based Multi-Objective Optimization Approach for Lightning Warning in High-Risk Industrial Areas

Abstract

Lightning strikes are a major hazard in tropical regions, especially in northern Brazil, where open-area industries such as mining are highly exposed. This study proposes an octant-based multi-objective optimization approach for spatial lightning alert systems, focusing on minimizing both false alarm rate (FAR) and failure-to-warn (FTW). The method uses NSGA-III to optimize a configuration vector consisting of directional radii and alert thresholds, based solely on historical lightning location data. Experiments were conducted using four years of cloud-to-ground lightning data from a mining area in Pará, Brazil. Fifteen independent runs were executed, each with 96 individuals and up to 150 generations. The results showed a clear trade-off between FAR and FTW, with optimal solutions achieving up to 16% reduction in FAR and 50% reduction in FTW when compared to a quadrant-based baseline. The use of the hypervolume metric confirmed consistent convergence across runs. Sensitivity analysis revealed spatial patterns in optimal configurations, supporting the use of directional tuning. The proposed approach provides a flexible and interpretable model for risk-based alert strategies, compliant with safety regulations such as NBR 5419/2015 and NR-22. It offers a viable solution for automated alert generation in high-risk environments, especially where detailed meteorological data is unavailable.

1. Introduction

Lightning strikes represent one of the most lethal natural hazards in tropical regions. According to Pinto Jr and Pinto [1], it is estimated that 50 lightning strikes occur every second on Earth (i.e., about 1.5 billion per year). The majority, around 1 billion, occur in the tropical region over the continents. Of this total, around 30% are lightning strikes, that is, around 500 million lightning strikes occur each year on the planet, with around 150 million in the tropical region. Around 25,000 people die each year from lightning strikes, and in Brazil alone, an average of 130 people die each year. Most lightning accident victims are young people during outdoor activities. Such accidents have serious consequences to people, such as physical damage, unconsciousness, or death [2]. For industries, lightning may cause global losses of billions of dollars annually, especially for those that operate in open fields, such as mining, which needs to both protect workers and maximize production [3].

According to data from the Brazilian National Institute for Space Research (INPE), Brazil leads the world in absolute number of cloud-to-ground (CG) lightning events, with more than 77 million discharges annually. The Pará state, located in the northern Amazon region, is consistently among the most affected. This region combines high atmospheric instability with a vast extension of forested, rural, and industrial zones, which makes it especially vulnerable to lightning-related fatalities and infrastructure losses.

In the Amazonian state of Pará, lightning strikes are among the most prevalent hazards, combining high natural vulnerability with occupational exposure. Open-air activities such as mining, logging, fishing, and (subsistence) farming expose thousands of workers daily to lightning risk. Particularly, mining operations, with their expansive, exposed areas and metallic structures, face heightened danger of human injury and operational disruption [4]. These conditions highlight the urgent need for more effective, real-time lightning alert systems capable of supporting emergency protocols and protecting outdoors workers.

Due to this, several studies have proposed solutions to mitigate lightning-related risks across various domains, including failure in power transmission lines [5] and safety in mining operations [3]. A first attempt was proposed in [3], where a warning system for a mining area in Pará, Brazil, using a quadrant-based segmentation of the monitoring region was detailed. The approach involves a two-step grid search for four objectives: false alarm rate (FAR), failure-to-warn (FTW), operational downtime, and lead time. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [6] multicriteria method was used to rank the solutions.

However, this approach can be improved in some respects, such as the following: (i) refining the spatial granularity by using octants instead of quadrants; (ii) excluding operational downtime as an explicit objective, since it is directly correlated with FAR (greater FAR implies in greater operational downtime), and (iii) replacing the grid search with an evolutionary algorithm, which can more efficiently explore the solution space.

Thus, our proposed approach builds on this foundation by applying Non-dominated Sorting Genetic Algorithm-III (NSGA-III) [7] to optimize both FAR and FTW in a localized lightning alert system for mining zones in Pará, which is in line with the technical standards NBR 5419/2015 [8] (aligned with IEC 62305, defines guidelines for the design, installation, and maintenance of lightning protection systems in Brazil) and NR-22 [9] (specifically regulates lightning protection in mining operations, including requirements for detection systems and evacuation protocols), and offers a feasible solution in terms of worker safety and reduced downtime in operations. The use of octant-based spatial division increases the granularity of warning zones, potentially allowing for more tailored and timely alerts. Although limited in geographic scope, the method provides a flexible model that can be adapted to similar high-risk, data-rich environments.

Given these challenges and regulatory demands, this study proposes a methodology for lightning alert optimization, which is an octant-based spatial model combined with a bi-objective NSGA-III optimization algorithm, aiming to improve the balance between alert accuracy (minimizing FAR) and safety (minimizing FTW) using only historical lightning location data. The objective is to lay the foundation for an automated, self-adaptive alert system capable of being deployed in mining zones and other high-risk areas without reliance on external weather or sensor data.

The remainder of this paper is organized as follows: Section 2 reviews literature in lightning prediction and multi-objective spatial optimization. Section 3 details the proposed methodology, including data preparation, NSGA-III configuration, objective definitions, and implementation details. Section 4 presents and discusses the results, including comparisons with a baseline, sensitivity analyses, and convergence assessments. Finally, Section 5 concludes the study and outlines directions for future research.

2. Related Works

Effective lightning alert systems require a combination of atmospheric science understanding and computational decision-making tools. In recent years, different approaches have been proposed to improve the accuracy and operational relevance of lightning warnings, particularly in high-risk regions such as the Amazon. These approaches vary from signal-based classification methods to multi-objective spatial optimization techniques. This section reviews some studies that support and motivate the methodology proposed in this work.

In Brazil, several studies have explored lightning behavior and its interaction with environmental and anthropogenic factors. Santos et al. [4] analyzed the influence of land use and topography on lightning incidence in the Carajás mineral province, highlighting how mining activities and elevated terrain alter local atmospheric conditions, increasing lightning vulnerability. They optimized spatial analysis using regression and least-squares adjustment, confirming the need for location-aware warning systems in complex landscapes.

Other regional studies focused on lightning detection and classification. For instance, Almeida et al. [10] focused on eastern Pará, applying empirical optimization to locate and characterize CG lightning using VAISALA network data. Their study helped define key physical parameters for regional lightning behavior but did not result in an alerting mechanism. Leal et al. [11] proposed a deep learning classifier to analyze lightning waveform data collected from Tucumã in Pará, optimizing architecture to improve classification accuracy. This work improved signal interpretation but did not explore operation integration or spatial segmentation for alerts.

One of the most directly related works is by Alves et al. [3], who developed a quadrant-based warning system for a mining site in Pará. Their model combined a grid search approach with the TOPSIS to optimize four indicators. Despite its practical relevance, the quadrant division limits spatial granularity, and the grid search has limited efficiency in exploring better solution spaces.

To address these limitations, recent research has increasingly adopted evolutionary multi-objective optimization algorithms. Notably, NSGA-III [7] has proven effective in various environmental and spatial planning contexts. For example, Zare et al. [12] applied NSGA-II and III to evaluate groundwater vulnerability under uncertain climate scenarios. They modeled tradeoffs between pollution risk, cost, and resilience and validated their solutions using the hypervolume (HV) metric for convergence-diversity. The COPRAS multicriteria method was used to identify the best compromise solution. Qu et al. [13] adapted NSGA-III for urban waste management planning, confirming its applicability in real-world, spatially distributed systems. Trivedi and Singh [14] proposed a multi-objective model to optimize the location of temporary shelters in disaster-affected regions, considering uncertainties in damage extent and access routes. Their approach balanced proximity, coverage, and risk mitigation, illustrating how evolutionary optimization can support spatial decisions in high-risk scenarios involving human safety. More recently, Liu et al. [15] applied NSGA-III to optimize net-zero energy buildings, balancing energy efficiency and thermal comfort while maintaining strong Pareto front performance.

A summary of these related works is presented in

Table 1. Comparative table of key studies related to this work.

Ref.	Focus	Method Used	Domain	Relevance to This Work
[3]	Lightning alert system	Quadrant-based optimization + TOPSIS	Pará mining site	Direct baseline for this study.
[4]	Land use and topography vs. lightning	Regression, GIS	Carajás (Pará), mining	Confirms spatial variability of risk.
[10]	CG lightning location optimization	Empirical + VAISALA data	Eastern Pará	Defines spatial patterns, no alerts.
[11]	Signal classification	Deep Learning	Tucumã (Pará)	Improves signal accuracy.
[12]	Groundwater vulnerability	NSGA-II/III	Environmental risk mapping.	Shows NSGA-III for uncertainty modeling.
[14]	Shelter location optimization	NSGA-III	Disaster zones	Balances spatial constraints and safety.

. It is possible to realize that these works demonstrate the wide applicability of multi-objective optimization to warning systems and emergency logistics, especially when spatial variability and operational constraints are involved. While none of them address lightning alerts in the Amazon region, they offer methodological insights that support the use of evolutionary optimization in this study.

Thus, the presented literature confirms the relevance of both spatial modeling and evolutionary optimization in environmental alerts systems. Our work extends these concepts by applying them to a real-world lightning alert scenario in a high-risk industrial region, leveraging spatial asymmetry and historical strike data to minimize alert errors.

3. Material and Methods

This section presents the data used in the study, the formulation of the multi-objective optimization problem using NSGA-III, the evaluation strategy based on HV, and the computational setup used for the experiments. Figure 1 illustrates the methodology used in this paper.

3.1. Data Acquisition

The dataset used in this work is called BrasilDAT [16], with data from a geostationary lightning mapper (GLM) on board GOES-16 satellite. BrasilDAT, implemented in 2011 by INPE’s Atmospheric Electricity Group (ELAT), uses Earth Networks technology to detect both cloud-to-ground and intracloud lightning. It operates via the time-of-arrival (TOA) method in the 1 Hz to 12 MHz range, using at least three sensors to measure the time differences in electromagnetic signal arrival. These differences define hyperbolic curves for each sensor pair, and the intersection of these curves indicates the lightning strike location [4]. To be fair, this dataset was the same as the one used in [3] for the quadrant-based optimization technique. Each row represents a lightning record, with a timestamp (hour/minute/second) and position (latitude/longitude). BrasilDAT compiles multiple data sources to determine the exact time, type, and location of lightning, having already been validated and used in previous research [3,4].

The experiments were conducted using lightning strike data collected between September 2020 and April 2024 in the vicinity of a mining operation in the state of Pará, Brazil. After preprocessing, only valid CG discharges within a 50 km radius from the mining site were retained. Each strike was associated with one of eight octants around the central point of interest. The octant structure allowed for directional analysis and adaptive warning strategies based on spatial patterns of lightning incidence.

The dataset was segmented temporally (5 min interval) to simulate real-time operation, and the ground-truth event classification was used to compute alert performance metrics.

3.2. Science About Lightning

CG lightning is a result of intense electric field buildup between the cloud base and the Earth’s surface, commonly observed in convective storm systems. It is a complex atmospheric phenomenon resulting from charge separation within cumulonimbus clouds, intensified by vertical motion and microphysical processes. In tropical regions such as the Amazon, abundant moisture, intense solar heating, and convective instability contribute to a high frequency of lightning events. Brazil, in particular, ranks among the countries with the highest CG lightning activity globally [1,17].

Lightning incidence is not homogeneous across space. It tends to cluster over certain terrain types, such as elevated areas (e.g., on average, the statue of the redeeming Christ, in Rio de Janeiro, Brazil, is struck by lightning six times per year) or metallic infrastructures, due to enhanced electric field concentration. This spatial heterogeneity, confirmed in prior studies [3,10], supports our model’s use of direction-sensitive alert radii (octants) and the assumption that past CG lightning strikes are statistically indicative of future risk zones. Furthermore, empirical studies have shown that terrain features and surface conditions modulate the formation and propagation of discharges, which aligns with the design of an asymmetric warning model that reflects real-world exposure patterns.

In particular, Santos et al. [4] observed that convective systems and lightning in South America exhibit strong spatial and seasonal variability, with high activity zones frequently aligned with orographic features and surface heating gradients. These findings support the notion that even static historical data can carry embedded signatures of environmental influences, which can be leveraged in spatial alert systems.

By leveraging four years of CG lightning location data, this work builds on atmospheric science principles to design a model that, while computational, remains rooted in the observed physical behavior of lightning events. This ensures that the alert system is not only operationally efficient but also scientifically grounded in the mechanisms of lightning occurrence.

3.3. NSGA-III Formulation

To explore tradeoffs between issuing unnecessary alerts and missing critical lightning events, we modeled the alert configuration problem as a multi-objective optimization task and solved it using the NSGA-IIII algorithm [7].

Each individual in the population encodes a vector of nine parameters: eight radii (one for each octant, ranging from 10 to 30 km) and a threshold value (1 to 3 strikes) that determines the minimum number of discharges required to issue a warning. The protected region is defined as a 10 km-radius circle centered on the operation site.

3.3.1. Decision Variables

Each solution $x\in Z^9$ is a vector that represents a complete lightning alert configuration for the region (1):

$$x=\left[r_1,r_2,\dots ,r_8,t\right]$$

(1)

where $r_1\in \{10,11,\dots ,30\}$ represents the radius (in km) for alert activation in octant $i$, and $t\in \{1,2,3\}$ is the minimum number of lightning strikes within any active octant needed to trigger a warning.

This representation enables asymmetric sensitivity across directions, reflecting the natural variability in lightning exposure.

To define the bounds and behavior of these decision variables, we considered several topographic and environmental characteristics of the study area. Specifically, the mining site is surrounded by varied terrain, including elevated zones to the northwest and dense vegetation to the southeast. Historical data indicate that certain octants, particularly those aligned with elevated or deforested regions, consistently recorded higher lightning densities. Consequently, we allowed directional radii to adapt within the optimization to capture this variability. Furthermore, local microclimate patterns, likely influenced by topography and land cover (e.g., heat islands from metallic infrastructure or moisture retention in forested zones), may alter convection behavior, increasing lightning incidence in specific sectors. By permitting individual radii per octant, the model configuration accounts for these spatial asymmetries without explicit external inputs. These factors, therefore, indirectly shaped the parameter space explored by the optimization algorithm.

3.3.2. Objective Functions

Two conflicting objectives are minimized, following [3] descriptions:

• FAR: proportion of alerts that were issued without any real lighting threat within the defined protected region; see (2). This type of alert directly disrupts production, as an alert issued forces operations to be halted until the risk condition ceases.

  $$f_1\left(x\right)=FAR\left(x\right)$$

  (2)
• FTW: proportion of true lightning threats for which the system failed to issue any alert; see (3). This type of alert poses a direct danger to people in the open field, leaving them unprotected.

  $$f_2\left(x\right)=FTW\left(x\right)$$

  (3)

Both metrics are computed by simulating the alert system on historical data using each candidate configuration $x$. The goal is to find configurations that balance proactive warning without overwhelming the system with false positives.

3.3.3. Constraints

The optimization problem can be defined as Equations (4) and (5) for FAR and FTW, respectively. The constraints are described in Equation (6).

$$\mathrm{Minimize}{f}_1\left(x\right)=FAR\left(x\right)$$

(4)

$$\mathrm{Minimize}{f}_2\left(x\right)=FTW\left(x\right)$$

(5)

$$\begin{gathered} \mathrm{subject}\mathrm{to}:10\le r_i\le 30,{\forall }_i=1,\dots ,8 \\ 1\le t\le 3 \end{gathered}$$

(6)

No additional constraints were imposed, but configurations were filtered to ensure basic feasibility during evaluation (e.g., at least one octant must be within a realistic activation range).

3.3.4. Parameter Settings

Finally, the parameters used for NSGA-III were based on the original recommendations by [7]. The algorithm was run for a maximum of 150 generations, with a population size of 96 individuals. The crossover probability was set to 0.8, and the mutation probability to 0.2.

The stop criterion was defined by two conditions: (i) a maximum of 150 generations, and (ii) no improvement in HV for 25 consecutive generations.

Additionally, the elitism mechanism for selecting individuals to the next generation followed the standard NSGA-III strategy based on reference point associations and perpendicular distances in objective space. To promote exploration and avoid premature convergence, 5% of the individuals in each new generation were randomly generated to maintain diversity.

3.3.5. Programming Environment and Computational Resources

The system was implemented in Python 3.9.6 with the following components: Numpy [18] and Pandas [19] for data processing and metric calculations; Geopy for geographic distance computation between strikes and the target; Matplotlib v.3.10.3 and Seaborn v. 0.13.2 for visualizations.

In this project, the team had access to a cluster (set of computers), so we could use 96 dedicated cores (corresponding to the number of individuals per generation). In total, we ran 15 rounds of experiments and stored the last generation and the HV across generations.

4. Results

This section presents the results obtained with the NSGA-III evolutionary algorithm applied to optimize lightning alert zones for the protection of a mining region in Pará, Brazil. This area is known for its high incidence of cloud-to-ground lightning and hosts open-pit mining operations with many exposed workers. In contrast to the quadrant-based grid search approach [3], this work introduces a novel spatial segmentation using octants, enabling asymmetric and adaptive coverage across eight directional slices of the monitoring area. Each solution is evaluated according to its ability to reduce FAR and FTW simultaneously.

The NSGA-III algorithm was executed 15 times independently, with 96 individuals per run. In all cases, the stopping criterion was reached due to lack of HV improvement over 25 generations. The smallest execution reached generation 25 and the longest 125. The non-dominated solutions obtained from the last generation of each run are shown in Figure 2. As illustrated, the 1440 solutions (96 × 15) exhibit a clear trade-off between FAR and FTW. The FAR ranges from 0.424 to 0.728, while FTW varies from 0.077 to 0.305. These results reveal the inherent conflict in optimizing for both objectives simultaneously: minimizing misses often requires broader coverage, which increases false positives—a well-known challenge in risk-based alert systems. Notably, solutions that aggressively reduce FTW typically incur higher FAR, reflecting the cost of safety-oriented configurations.

These patterns are consistent with trade-offs observed in related multi-objective hazard management studies. For example, Refs. [12,14] showed similar inverse relationships when optimizing between protection coverage and false interventions in water and disaster shelter contexts. Moreover, the variability across octants introduces a spatial layer of complexity absent in prior grid-based methods, such as those of presented in [3], reinforcing the importance of directional tuning in regions with high lightning activity, such as Pará.

To represent a single Pareto front approximation, Figure 3 shows the final set of solutions from the 1st execution. It is possible to observe that the FAR ranged from 0.424189 to 0.682434, while the FTW ranged from 0.086169 to 0.298451. The individual that achieved the lowest FAR (FAR = 0.424189, FTW 0.298452) was $x^{*}=\left[18191614131011133\right]$. Meanwhile, the configuration that obtained the lowest FTW (FAR = 0.682434, FTW = 0.086169) was $x^{*}=\left[11262628202621221\right]$.

These two extremes demonstrate the diversity and flexibility of the NSGA-III approach. Decision-makers can explore and select configurations that best align with operational priorities—whether reducing operational interruptions (lower FAR) or maximizing worker protection (lower FTW).

To evaluate the convergence behavior and quality of the Pareto fronts, we computed the HV metric at each generation for all 15 runs. As shown in Figure 4, the average HV shows a consistent upward trend, particularly in the first 40–50 generations. This indicates effective learning and improvement in the population’s ability to span the objective space. The shaded band represents the minimum and maximum HV observed across runs and highlights the variability inherent to stochastic algorithms. After generation 60, the HV curve begins to plateau, indicating stabilization of the Pareto front and convergence toward optimal trade-offs. In several runs, HV remained unchanged for 25 consecutive generations, triggering the termination criterion. While some variation remains due to the stochastic nature of initialization and selection, the overall trend confirms that NSGA-III was able to discover stable and diverse sets of non-dominated solutions within reasonable computational effort.

The increasing HV and stable Pareto structure align with findings from Refs. [13,15], who also applied NSGA-III in spatially distributed, real-world domains. Their studies similarly observed fast early gains followed by convergence, which is a desirable pattern in multi-objective evolutionary optimization.

4.1. Comparison with the Baseline

As previously discussed, our baseline is the quadrant-based approach proposed in [3]. In their method, the monitoring area is divided into four quadrants, and the optimal fixed radius for alert activation is selected via grid search. To ensure a fair comparison, we implemented their approach over the same period of historical lightning data used in our experiments (about 4 years). The results obtained using the quadrant-based model were FAR = 0.511933 and FTW = 0.196013.

When comparing these metrics with those obtained from the first NSGA-III execution (see Figure 3), we identified four octant-based solutions that outperform the baseline in both objectives simultaneously. These configurations are detailed in

Table 2. Octant-based solutions that outperform the quadrant-based baseline in both FAR and FTW.

Approach	Configuration	FAR	FTW
Octant-based (Proposed)	[17 19 20 19 19 11 16 17 2]	0.494448	0.195028
	[18 17 23 19 19 11 16 15 2]	0.498291	0.192963
	[17 19 23 19 19 11 16 15 2]	0.499318	0.191969
	[17 19 23 19 19 11 16 17 2]	0.503544	0.189532
	[15 23 23 19 19 11 16 15 2]	0.510326	0.186947
	[17 22 23 19 19 11 16 15 2]	0.510483	0.186844
	[18 17 20 19 19 18 17 16 2]	0.510927	0.186226
Quadrant-based (Baseline)	[16 16 16 16 12 12 12 12 1]	0.511933	0.196013

.

These solutions are also highlighted in Figure 5, which shows all solutions from the first execution. The red dot indicates the quadrant-based configuration, while the blue points represent the octant-based solutions that outperform the baseline in both FAR and FTW simultaneously, confirming the benefits of increased spatial resolution.

It is generally expected that increasing the spatial resolution of monitored areas, such as transitioning from quadrants to octants, improves alert precision by allowing for better localization of lightning strike patterns. In our results, the octant-based segmentation showed reductions in both FAR and FTW compared to the quadrant-based. However, this trend is subject to practical limitations. Excessive refinement may eventually lead to overfitting or sparsity issues, especially in regions with low lightning frequency. While our study did not explore subdivisions beyond octants, future work could investigate the limiting scale of segmentation, identifying the point at which additional granularity ceases to produce meaningful gains in alert accuracy.

In addition, one of the stopping conditions used in our NSGA-III implementation was the absence of improvement in the hypervolume (HV) metric over a predefined number of generations. This criterion was consistently met across all optimization runs, which indicates that the solution space had stabilized and no further meaningful improvements could be achieved, even with refined spatial granularity. This behavior reinforces the idea that there exists a practical limit to the benefits gained from increasing segmentation resolution, aligning with the hypothesis that warning accuracy tends to plateau beyond a certain spatial scale.

4.2. Discussion

The results obtained through the octant-based NSGA-III optimization provide evidence of meaningful improvements in alert configuration over traditional quadrant-based strategies, especially in spatially complex and risk-prone environments such as Pará. The ability to fine-tune directional coverage by assigning distinct radii to each of the eight octants allowed the algorithm to explore alert strategies that are more context-sensitive, adapting to potential anisotropies in lightning distribution.

From an operational perspective, this adaptability is particularly relevant for the mining industry, where workers are often spread across different zones of a facility and exposed to open areas. In such scenarios, the ability to tailor alert sensitivity to specific directions offers tangible benefits, potentially reducing both unnecessary evacuations and undetected threats.

Moreover, the results demonstrate that an evolutionary approach is not only effective in managing FAR vs. FTW tradeoff but also capable of generating a diverse portfolio of solutions. These solutions can be selected post hoc by decision-makers depending on the risk tolerance and operational constraints. This flexibility is a key strength of NSGA-III and aligns with the findings of Refs. [12,15], which reported similar advantages in environmental and building optimization contexts.

However, some important limitations must be noted. First, although the octant-based model provided improvements, the absolute performance margin compared to the quadrant-based baseline was not drastic. The best octant-based solutions outperformed the baseline in both objectives, but many others remained comparable or inferior. This suggests that the gains come from selecting configurations rather than systemic superiority and emphasizes the need for careful solution selection. Second, while the hypervolume analysis confirmed convergence, the stagnation threshold used to halt optimization may have prevented the exploration of potentially superior solutions in later generations. Incorporating more adaptive stopping conditions or hybrid local search methods might enhance performance in future iterations. Additionally, this study focused on a specific regional context—a mining facility in Pará. While the approach is generalizable, the transferability of the optimized configurations to other domains (e.g., airports, urban centers, or agriculture) must be verified with additional datasets. In contrast to [11], who used machine learning for lightning classification and prediction, our method focuses on spatial decision optimization, and both approaches could be integrated in future work for hybrid predictive alert systems.

The results point to a practical recommendation: in high-risk areas with complex geography or human exposure, spatial asymmetry in alerts should be prioritized, as it can lead to safer and more efficient emergency responses. Tools such as NSGA-III can assist planners and safety engineers in exploring the trade-off space rather than relying on fixed thresholds.

Finally, it is important to highlight that the directional flexibility enabled by octant-based segmentation also captures environmental asymmetries present in the study region. The northwest octants, for instance, consistently required larger alert radii in optimal configurations, which may be attributed, for instance, to higher terrain elevation and modified surface characteristics from mining activities. These geographic conditions are known to affect convective processes and local lightning behavior. Therefore, the model’s ability to internalize such spatial heterogeneity without explicit geophysical inputs reinforces the suitability of multi-objective evolutionary optimization for geographically complex risk scenarios.

4.3. Sensibility Analysis

Each solution $x^{*}\in Z^9$ in the optimization process encodes eight directional radii $r_i\in \left[\mathrm{10,30}\right]$ (one for each octant) and one threshold $t\in \{\mathrm{1,2},3\}$, representing the minimum number of strikes within any octant to trigger a warning.

An analysis of the final Pareto-optimal solutions revealed non-uniform patterns in the values of the radii across octants. In particular (i) octants facing west and northwest tended to receive larger radii more frequently in dominant solutions, and (ii) eastern octants often showed smaller optimal radii, suggesting different lightning densities or strike behaviors depending on direction, possibly influenced by local topography or prevailing wind patterns in the Pará region. This asymmetry reinforces the benefit of the octant-based segmentation, which allows the alert system to allocate risk coverage non-uniformly, unlike the quadrant-based model where all areas share the same activation radius.

These findings support the argument made in previous works (e.g., [4]) that local topography and environmental factors should influence the configuration of lightning warning systems, unlike purely data-driven classification models (e.g., [11], this work demonstrates the value of interpretable, spatially structured optimization).

Regarding the alert threshold t, most of the best-performing solutions converged to a value of 2, suggesting that requiring at least two lightning strikes in each octant strikes a better balance between minimizing FAR and ensuring some evidence of risk before issuing a warning alert. Configurations with t = 1 generally led to lower FTW, but achieved higher FAR, as alerts were triggered too frequently. In contrast, t = 3 often missed critical events, increasing FTW to out-of-scope levels. This highlights the importance of calibrating this parameter carefully based on risk tolerance.

While the octant-based model increases flexibility, the sensitivity results indicate that certain octants consistently receive similar radius values across multiple solutions. This may point to redundant variables or low-sensitivity directions, which can be simplified in future studies through dimensionality reduction techniques, such as clustering octants with similar behavior or applying principal component analysis (PCA) in the decision space.

5. Conclusions

This study presents a new approach for optimizing localized lightning alert configurations in mining areas in Pará, Brazil. The proposed method is based on an octant-based spatial model and the NSGA-III evolutionary algorithm. The results show that the octant-based approach, when optimized with NSGA-III, was able to generate diverse Pareto-optimal solutions that outperformed the baseline method in both objectives. Specific configurations were identified, which provided lower FAR and FTW simultaneously, demonstrating the benefits of finer spatial resolution and adaptive alert parameters. The use of HV as an evaluation metric further confirmed the convergence and quality of the algorithm across multiple independent runs.

In addition, the sensitivity analysis revealed important patterns in the distribution of optimal radii per octant and the behavior of the threshold parameter, highlighting opportunities for simplifying the model without compromising performance. These findings are particularly relevant for operational applications in mining, energy, and other sectors, where rapid and accurate lightning alerts can prevent injuries, equipment loss, and productivity interruptions.

Despite these promising results, the study has some limitations. The optimization was based on historical data from a single location and time period. Although the model does not include explicit environmental or topographic variables explicitly, the spatial asymmetries observed in optimized configurations suggest that such influences were captured in the dataset throughout the years and reflected in lightning patterns. Future work should validate the approach in other geographic contexts and consider the integration of real-time prediction models (e.g., neural networks or radar-based nowcasting) or other sources to complement the optimization framework. Moreover, exploring hybrid algorithms, dynamic thresholds, and multi-location coordination strategies could further enhance the robustness and applicability of the method.

In conclusion, this study demonstrates that combining spatially adaptive segmentation with evolutionary multi-objective optimization offers a promising path forward for lightning alert systems, especially in complex, exposed environments where human safety is a priority.