A Fast Calculation Method for Electrostatic Fields in Complex Terrain Using NSGA-II and Conformal Mapping

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

See attach. 

Comments for author File: Comments.pdf

Author Response

Comment 1: Abstract: The authors showed some values, but they are general. What is the improvement?? The novelty should be remarked in the abstract
Response: We thank the reviewer for this comment. We agree that the abstract should clearly state the improvements and novelties. Accordingly, we have fully revised the abstract (Page 1) to eliminate general numerical descriptions and explicitly highlight the specific improvements and core novelties of the proposed method. The revised abstract clearly articulates the core innovation of the study: formulating the selection of the critical mapping parameter M as a dual-objective optimization problem (simultaneously minimizing the maximum local error and the mean global error) to achieve automatic and adaptive parameter tuning for conformal mapping, instead of empirical tuning or exhaustive traversal. We also emphasize in the abstract the practical engineering value of the Pareto-optimal front (providing a trade-off space for decision-making under different accuracy requirements). All improvements and novelties are now clear and targeted. The revised abstract can be found on Page 1 of the revised manuscript.

Comment 2: All references should be discussed (line 24, line 28… please review all document, mainly the introduction)

Response: We thank the reviewer for this careful remark. We agree that all references should be properly discussed. Accordingly, we have thoroughly revised the Introduction (Pages 1–3) and now provide a clear, critical discussion for each cited reference, organized by thematic groups. Specifically:

• References on lightning hazards and field calculation necessity [1,2]: We cite Xu (2004) and Albrecht et al. (2016) to establish the practical importance of lightning protection, and we point out that these studies do not address the computational challenges posed by complex terrain.

• References on traditional numerical/analytical methods [3-9]: We discuss the high computational cost of FEM, MoM, FDTD (e.g., Vargas 2022; Lu 2006; Sun & Xu 2015) and the poor adaptability of the charge simulation method and method of images to asymmetric terrain boundaries (e.g., Sun et al. 2021; Inoue 1997). We then note that these limitations motivate the need for a more efficient and adaptive approach.

• References on conventional conformal mapping [10–16]: We analyze the limitation of these methods (e.g., Zhang et al. 2016; Wang 2011; Chen et al. 2009; Harsha & Garner 2021; Wang et al. 2016; O’Connell & Krein 2009; Lévy et al. 2002) – namely, they rely on specific mapping function expressions and cannot handle arbitrarily complex terrain boundaries. We also cite the Schwarz–Christoffel transformation [17] and the improved SSOR iteration method [18] as examples of numerical conformal mapping techniques, while pointing out their sensitivity to initial guesses and high computational cost for dynamic geometries.

• References on optimization algorithms for conformal mapping [19–22]: We discuss Ramakrishnan et al. (2017) [19] (Levenberg‑Marquardt for coordinate matching), Zhao et al. (2023) [20] (genetic algorithms for time‑varying boundaries), Ji et al. (2024) [21] and Li et al. (2025) [22] as representative works that introduced optimization into conformal mapping. We then highlight a common gap: most of these treat parameter tuning as a single‑objective problem (typically minimizing the maximum error) and do not address the trade‑off between local and global accuracy.

• References on the multi-level conformal mapping framework: We cite the foundational work on fractional linear transformation based multi‑level mapping [23]. We note that while these frameworks improve terrain adaptability, they still rely on empirical or exhaustive parameter selection for the key mapping parameter M.

All these discussions have been added to Pages 1-3 of the revised manuscript.

Comment 3: The introduction does not show the need for this research; only 20 references were used. Authors should emphasize more the need for this research and show the goals and main novel at the end of the introduction

Response: We sincerely thank the reviewer for this constructive suggestion. We have substantially revised the Introduction (Pages 1-3) to better articulate the research necessity, enrich the literature review, and explicitly present the research goals and main novelties at the end. The key revisions are as follows:

1. Strengthening the research necessity and practical motivation:
We restructured the opening paragraph to immediately highlight the core technical challenge: “The undulating terrain significantly alters the distribution of electric fields, leading to a substantial increase in the complexity of boundary conditions for field calculations” (Page 1). We then systematically reviewed the limitations of existing method categories:

• Traditional numerical methods (FEM, MoM, FDTD) – high accuracy but computationally prohibitive for engineering practice [3-9].

• Conventional analytical methods (charge simulation, method of images) – efficient but lack adaptability to complex/asymmetric boundaries [8,9].

• Early conformal mapping approaches – restricted to specific mountain profiles that match closed‑form mapping functions [10-12].

• Schwarz–Christoffel transformation and numerical iterative schemes [16–18] – sensitive to initial guesses and expensive for dynamic geometries.

• Single-objective optimization‑aided conformal mapping [19–22] – still ignoring the local‑versus-global accuracy trade-off.

This layered problem analysis clearly demonstrates why a fast, adaptive, and balanced method is urgently needed for complex-terrain lightning electrostatic field calculation.

2. Adding research goals and core novelties at the end of the introduction:
Following the reviewer’s suggestion, we have added a dedicated paragraph at the end of the Introduction (Page 3) that clearly states the research goal: “To address the aforementioned research gaps, this paper proposes a fast calculation framework for lightning electrostatic fields over complex terrain that integrates multi-level fractional linear mapping (FLM) with NSGA-II.” We then explicitly list the three main contributions of this paper in a numbered format:

• (1) A bi-objective optimization model for FLM parameters – with maximum local error and global average error as objectives, solved by NSGA‑II to generate a Pareto‑optimal set.

• (2) A closed-loop adaptive optimization mechanism – automatically matching terrain features to mapping parameters, avoiding empirical tuning and exhaustive traversal.

• (3) Validation on multiple complex terrain types – demonstrating improved accuracy and efficiency while retaining terrain adaptability.

These revisions can be found on Pages 1-3 of the revised manuscript.

Comment 4: Line 59-61“Compared with traditional exhaustive search methods, the proposed method significantly reduces computational time (with an average efficiency improvement of approximately 49.3%) ” This is an introduction. Authors cannot show results. This comparison should be discussed in the section of results and discussion

Response: We fully agree with this comment. Accordingly, the above result-related content has been completely deleted from the Introduction section (Page 1-3). All quantitative comparisons of computational efficiency (49.3% improvement) and calculation accuracy (5% mean error reduction) have been moved to the Simulation Results and Comparative Analysis section (Pages 9-11) and Discussion section (Pages 12-14), where detailed analysis is combined with experimental data, Pareto front comparison (Fig.8) and time efficiency comparison (Fig.9). The Introduction now only focuses on research motivation, literature review and research objectives, without any experimental result description, fully complying with academic writing norms.

Comment 5: Section1 is after the introduction section. Please revise the renumbering of the manuscript

Response: We apologize for the confusion caused by the unclear numbering. Upon re-examination, the actual section structure of our manuscript is: 1. Introduction → 2. Section 1 (Theoretical Framework) → 3. Section 2 (Simulation Results) → 4. Discussion. This numbering is consistent with the journal's format (Introduction numbered as Section 1). We have carefully reviewed and verified the numbering of all sections, subsections, figures, tables, and cross-references throughout the manuscript to ensure completeness and correctness. No further numbering errors were found. This verification has been noted on Pages 1-14 of the revised manuscript.

Comment 6: Really it is destructured. Please, authors, should be careful

Response: We thank the reviewer for this critical comment. We have carried out a comprehensive restructure and optimization of the entire manuscript to significantly improve its structural clarity and logical coherence (Pages 1-14). The specific revisions are as follows:

1. Standardize the overall structure: The manuscript now follows a clear and logical flow: Abstract → Keywords → 1. Introduction → 2. Theoretical Framework → 3. Simulation Results → 4. Discussion → Acknowledgements → References. Each section now has a clear core theme, and transitional statements have been added to strengthen the logical connections between sections. For example, at the end of the Introduction, we explicitly state the research objectives and three main contributions, which naturally leads to the theoretical framework in Section 2.

2. Optimize subsection division: Overly long and content-dense sections have been split into thematic subsections with clear research focuses:

• Section 2 (Theoretical Framework) is now divided into: 2.1 Conformal Mapping Algorithm and 2.2 NSGA-II Optimization Framework;

• Section 3 (Simulation Results) is now divided into: 3.1 Pareto Front Analysis, 3.2 Efficiency Comparison, 3.3 Verification of Electric Field Calculation Based on Optimized Parameters, and 3.4 Robustness Verification for Different Terrain Configurations;

• Section 4 (Discussion) is divided into: 4.1 Practical Engineering Implications of the Synergistic Improvements, 4.2 Limitations and Critical Analysis of Engineering Applicability, and 4.3 Future Extension Steps for Realistic Engineering Applications.

3. Unify expression style and terminology: We have standardized the description of all key concepts (e.g., Multi-Level Conformal Mapping, NSGA-II, fractional linear mapping parameter M), deleted redundant and irrelevant content, and ensured coherence and compactness throughout the manuscript.

4. Optimize figure/table layout and captions: All figures (Fig. 1-13) now have clear, standardized captions that accurately reflect the content. The figure order has been adjusted to align with the narrative logic of the methods and results. All cross-references to figures and tables have been checked and corrected.

Comment 7: The authors define the theory, but there is no real discussion without references

Response: We thank the reviewer for this important observation. We agree that theoretical definitions must be supported by appropriate references and in-depth discussion. To address this issue, we have substantially revised the theoretical framework section (Section 2) by adding three key references that provide a complete derivation chain for the conformal mapping formulas used in our work. The specific revisions are as follows:

We have added the following two classic monographs to support the theoretical derivation of mapping complex terrains to the unit circle using the Schwarz-Christoffel (S-C) transformation in conjunction with Cauchy's integral theorem:

• Carroll, T. (2024). Geometric Function Theory [1]. This textbook provides a rigorous and self-contained account of conformal mappings, the Riemann Mapping Theorem, and the uniformisation of planar domains. We specifically cite this work to support the theoretical basis of mapping simply connected domains (our complex terrain profiles) to the unit disk, a critical step in our multi-level conformal mapping framework.

• Gakhov, F.D. (1977). Boundary Value Problems [2]. This classic monograph presents the fundamental theory of boundary value problems and integral equations. We use this reference to support the initial derivation steps of the conformal mapping formulas, particularly the treatment of boundary conditions in complex domains.

Additionally, we have added a contemporary reference that bridges theory to application: Li et al. (2025) [3], which systematically refines the derivation process of the conformal mapping formulas in polar coordinates.

These references and the associated discussions have been integrated into Section 2 (Pages 3-4) of the revised manuscript.

Comment 8: There is no conclusion

Response: We thank the reviewer for this comment. We have added a dedicated conclusion section integrated at the end of the Discussion section (Page 8) (complying with the journal’s concise writing norm), which fully summarizes the entire study and includes four core parts:

• A concise summary of the main research work (developing an NSGA-II-Multi-Level Conformal Mapping integrated framework for lightning electric field calculation in complex terrain);

• A clear statement of the key experimental results and their practical engineering value (5% mean error reduction, 49.3% efficiency improvement, providing an efficient solution for lightning protection engineering);

• An objective analysis of the study’s limitations (only electrostatic component considered, 2D terrain approximation, no stochastic lightning attachment);

• Specific prospects for future research directions (extending to 3D terrain, incorporating transient electromagnetic effects, validating against field measurements).

This conclusion section fully summarizes the research content, results and implications of the study, and the conclusions are highly consistent with the front content, forming a complete logical closed loop. The added conclusion can be found on Page 12-14 of the revised manuscript.

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents an interesting attempt to combine multi-level conformal mapping with NSGA-II for efficient electric-field calculation over complex terrain. The topic is relevant, and the proposed framework appears to improve optimization efficiency compared with exhaustive search. However, several aspects of the methodology, validation, and presentation should be strengthened before the manuscript can be considered for publication. 

1.The novelty of the proposed method should be clarified more explicitly. The manuscript explains that the main contribution is the integration of NSGA-II with multi-level conformal mapping for adaptive parameter selection, but the distinction from prior conformal-mapping-based and optimization-assisted electric-field studies remains somewhat insufficient. The authors should clarify more concretely what methodological advance is achieved beyond replacing exhaustive search with NSGA-II. 

2. The validation is too limited to demonstrate the general applicability of the method. At present, the verification appears to rely mainly on a single mountainous terrain profile and comparison with a COMSOL benchmark. While this is a useful first step, additional cases with different terrain shapes, geometric complexity, or source conditions would be necessary to establish the robustness and broader applicability of the proposed framework. 

3. The optimization procedure should be reported in greater detail for reproducibility. Since the contribution strongly depends on the NSGA-II-based optimization process, the manuscript should provide more implementation details, including population size, number of generations, crossover and mutation settings, termination criteria, and sensitivity to random initialization. Without such information, it is difficult for readers to assess the reproducibility and stability of the reported results. 

4. The practical significance of the reported accuracy improvement should be discussed more carefully. The manuscript reports an approximately 5% reduction in mean error and about 49.3% improvement in computational efficiency compared with exhaustive search. While the computational gain is clearly meaningful, the engineering significance of the 5% error reduction should be interpreted more critically, especially since the reduction in maximum error is acknowledged to be limited. A more in-depth discussion on when this level of improvement becomes practically important would strengthen the paper. 

5. The limitations and future applicability should be expanded in the discussion section. The manuscript appropriately acknowledges that the present framework is limited to electrostatic field analysis, 2D terrain approximation, and does not include stochastic lightning attachment behavior. However, these limitations are only briefly mentioned. The discussion would benefit from a more critical explanation of how these assumptions may affect applicability in real engineering problems and what concrete steps are required to extend the method toward more realistic 3D and transient analyses.

Author Response

Comment 1: The novelty of the proposed method should be clarified more explicitly. The manuscript explains that the main contribution is the integration of NSGA-II with multi-level conformal mapping for adaptive parameter selection, but the distinction from prior conformal-mapping-based and optimization-assisted electric-field studies remains somewhat insufficient. The authors should clarify more concretely what methodological advance is achieved beyond replacing exhaustive search with NSGA-II.

  Response 1: Thank you for pointing this out; we agree with this comment. Therefore, we have supplemented more explicit descriptions of novelty in the abstract (Page 1) and at the end of the introduction (Page 3) of the revised manuscript, and added specific comparative analyses of differences from existing studies, with the following detailed revisions:

• Clearly emphasize in the abstract: The core innovation of this study is not merely replacing exhaustive search with NSGA-II, but for the first time formulating the parameter selection of multi-level fractional linear mapping (FLM) as a dual-objective optimization problem (simultaneously minimizing the maximum local error and the mean global error), breaking through the limitation of existing single-objective optimization that only focuses on a single error indicator;
• Supplement in the "main contributions" section at the end of the introduction (Page 3): Compared with existing optimization-assisted conformal mapping studies (e.g., single-objective optimization by Ramakrishnan et al. 2017, time-varying boundary mapping by Zhao et al. 2023), this study establishes a closed-loop adaptive matching mechanism between "terrain features and mapping parameters", eliminating the need for empirical parameter tuning and providing a decision space for multi-scenario adaptation through the Pareto-optimal solution set— a methodological breakthrough not achieved by existing methods;
• Add a quantitative comparison with existing studies in the discussion section (Page 12): Clearly state that under the premise of maintaining comparable maximum error, the proposed method achieves a 5% reduction in mean global error and a 49.3% improvement in efficiency, with robustness verification covering 3 types of complex terrain, far exceeding the single-terrain verification scope of existing studies. All revisions focus on methodological innovations "beyond merely replacing the search algorithm", making the novelty more concrete and quantifiable

Comment 2: The validation is too limited to demonstrate the general applicability of the method. At present, the verification appears to rely mainly on a single mountainous terrain profile and comparison with a COMSOL benchmark. While this is a useful first step, additional cases with different terrain shapes, geometric complexity, or source conditions would be necessary to establish the robustness and broader applicability of the proposed framework

Response 2: We agree. Accordingly, we have revised the manuscript by adding validation experiments for 3 types of typical complex terrains (Section 3.4, Page 11 of the revised version) to emphasize the robustness and generalizability of the method. The specific revisions are as follows:  

1. Add a new subsection "3.4 Robustness Verification for Different Terrain Configurations" (Page 11), supplementing robustness verification for 3 types of complex terrains with distinct morphological characteristics: scarp slope terrain, double-peak terrain, and multi-peak terrain, covering a terrain gradient from simple to complex;
2. Provide corresponding error contour maps for each terrain type (Figs. 11-13, Page 11) and analyze the relationship between terrain complexity and error distribution— although the maximum relative error increases moderately with the increase in terrain irregularity, the method still maintains stable and reliable calculation accuracy, verifying its adaptability to different terrains;
3. Retain the original high-fidelity benchmark verification with COMSOL (Page 9, Fig. 7) and supplement the parameter optimization stability analysis under different terrains, proving that the method can quickly converge to the optimal parameters under various terrain boundary conditions. The revised validation system covers "a single benchmark terrain + 3 types of typical complex terrains", fully supporting the broad applicability of the method.

Comment 3: The optimization procedure should be reported in greater detail for reproducibility. Since the contribution strongly depends on the NSGA-II-based optimization process, the manuscript should provide more implementation details, including population size, number of generations, crossover and mutation settings, termination criteria, and sensitivity to random initialization. Without such information, it is difficult for readers to assess the reproducibility and stability of the reported results.

Response 3: Thank you for this critical comment; we fully agree. Accordingly, we have supplemented complete implementation details of the NSGA-II optimization process in Section 2.2.2 (Pages 6-8 of the revised version) to ensure result reproducibility. The specific revisions are as follows:

1. Clarify core parameter settings (Page 7):
   • Population size: 100 (balancing diversity and computational efficiency);
   • Number of objective functions: 2;
   • Number of decision variables: 1 (only mapping parameter M);
   • Selection strategy: Binary tournament selection (tournament size = 2);
   • Crossover operation: Simulated Binary Crossover (SBX) with crossover probability pc​=0.7 and distribution index η1​=2;
   • Mutation operation: Polynomial mutation with mutation probability pm​=0.05 and distribution index η2​=5;
   • Elite preservation mechanism: Select the top 50 optimal individuals from both the parent population (100 individuals) and the offspring population (100 individuals) to form the next generation population (100 individuals in total);
    
2. Supplement termination criteria (Page 7): The iteration stops when the relative change in the average Euclidean distance of the Pareto front is less than 10−6 for 10 consecutive generations, with the maximum number of iterations set to 100 (to avoid redundant computations);
3. Add sensitivity analysis of random initialization (Page 7, Table 1): Verification using 4 sets of independent random seeds (rng(1), rng(10), rng(100), rng(1000)) shows that the maximum relative deviation of the optimal parameter M is only 3.6%, the deviation of the maximum local error is < 2.6%, and the deviation of the mean global error is < 2.2%, proving the method's low sensitivity to random initialization and result stability. All details are supplemented in accordance with reproducibility standards, ensuring readers can fully replicate the optimization process.

Comment 4: The practical significance of the reported accuracy improvement should be discussed more carefully. The manuscript reports an approximately 5% reduction in mean error and about 49.3% improvement in computational efficiency compared with exhaustive search. While the computational gain is clearly meaningful, the engineering significance of the 5% error reduction should be interpreted more critically, especially since the reduction in maximum error is acknowledged to be limited. A more in-depth discussion on when this level of improvement becomes practically important would strengthen the paper.

Response 4: Thank you for this constructive comment; we fully agree. Accordingly, we have added a special discussion on the engineering significance of the 5% error reduction in Section 4.1 (Page 12 of the revised version). The specific revisions are as follows:  

1. Clarify the applicable scenarios for the 5% reduction in mean error (Page 11): In lightning protection of precision electronic equipment (e.g., mountainous aerospace test bases, precision instrument workshops), this error reduction can effectively reduce cumulative deviations in electric field distribution prediction, avoid "under-protection" risks caused by misalignment of lightning rods/grounding grids, directly improve the reliability of lightning protection schemes, and reduce equipment damage and system downtime;
2. Analyze in combination with actual engineering requirements: In electromagnetic compatibility (EMC) assessments, the 5% error reduction can avoid misjudgment of EMC compliance thresholds for communication base stations, radar antennas, and other equipment, reducing unnecessary retrofitting costs;
3. Emphasize the synergistic value of "accuracy + efficiency": Although the 5% accuracy improvement is moderate, when combined with the 49.3% efficiency improvement, it can shorten the preliminary design cycle by 30%-40% in large-scale projects (e.g., lightning protection design of mountainous high-voltage transmission corridors involving hundreds of terrain cross-sections) while ensuring design accuracy meets engineering standards— an engineering value that cannot be achieved by a single accuracy or efficiency improvement;
4. Objectively explain limitations: Clearly state that this error reduction has limited improvement for "extreme risk control scenarios" (e.g., lightning protection in ultra-high-risk areas) but is applicable to most general engineering scenarios, complementing the maximum error control scheme. The revised discussion clearly illustrates the practical engineering significance of the 5% error reduction, avoiding mere numerical descriptions.

Comment 5: The limitations and future applicability should be expanded in the discussion section. The manuscript appropriately acknowledges that the present framework is limited to electrostatic field analysis, 2D terrain approximation, and does not include stochastic lightning attachment behavior. However, these limitations are only briefly mentioned. The discussion would benefit from a more critical explanation of how these assumptions may affect applicability in real engineering problems and what concrete steps are required to extend the method toward more realistic 3D and transient analyses

Response 5: We agree. Accordingly, we have revised the manuscript by greatly expanding relevant content in Section 4.2 (Limitations) and 4.3 (Future Work) (Pages 13-14 of the revised version). The specific revisions are as follows:  

1. Deepen the analysis of the impact of limitations on engineering applications (Section 4.2, Page 13):
   • Limitation of electrostatic field: Clarify that the existing model is only applicable to lightning leader propagation analysis and cannot capture transient electromagnetic effects of lightning discharges (e.g., electromagnetic pulse radiation, terrain reflection), resulting in its use only as a "preliminary assessment tool" that cannot support the detailed design of transient protection systems (e.g., surge protection for electronic equipment);
   • Limitation of 2D terrain approximation: Point out that 2D cross-sectional approximation cannot reflect 3D terrain complexity (e.g., undulating ridges, multi-valley terrain), which may lead to inaccurate boundary mapping in the third dimension and non-negligible errors in electric field prediction for large-scale 3D terrain;
   • Limitation of stochastic lightning attachment: Explain that the randomness of lightning attachment (variations in lightning strike points, discharge intensity) is not incorporated, making the deterministic model unable to reflect the statistical characteristics of lightning strike risks and limiting its application in quantitative lightning risk assessment;
    
2. Supplement specific and feasible future extension steps (Section 4.3, Page 13-14):
   • Extension to 3D terrain analysis: ① Establish a 3D multi-level mapping framework for unstructured 3D terrain boundaries by combining FLM with 3D conformal mapping methods (e.g., spherical or cylindrical conformal transformation); ② Extend the single parameter M to a multi-dimensional parameter vector and reconstruct the dual-objective optimization model to adapt to 3D geometric adaptability and error control; ③ Integrate Digital Elevation Model (DEM) data to realize automatic extraction and mapping of 3D terrain boundaries;
   • Integration of transient electromagnetic effects: ① Couple the optimized conformal mapping framework with the Finite-Difference Time-Domain (FDTD) or Partial Element Equivalent Circuit (PEEC) method to realize the transformation of transient field calculation from complex 3D terrain to regular domains, reducing the computational complexity of transient simulation; ② Introduce lightning channel transient models (e.g., transmission line model, modified transmission line model) to simulate the time-varying characteristics of lightning current;
   • Integration of stochastic lightning attachment behavior: ① Combine the proposed deterministic electric field model with statistical lightning attachment models (e.g., electrogeometric model, leader progression model) to establish a stochastic field-risk coupling model; ② Introduce random variables (e.g., lightning strike point coordinates, discharge current amplitude) and conduct Monte Carlo simulations to realize quantitative assessment of lightning strike risks in complex terrain;
   • Experimental validation: Future work will focus on calibrating model parameters and improving engineering prediction accuracy using field measurement data of lightning-induced electric fields in mountainous areas. The revised limitation analysis is more critical, and the future extension steps are specific and feasible, fully supporting the method's potential for extension to practical engineering scenarios.

Reviewer 3 Report

Comments and Suggestions for Authors

Title: A Fast Calculation Method for Electrostatic Fields in Complex Terrain Using NSGA-II and Conformal Mapping

NSGA-II (Non-dominated Sorting Genetic Algorithm) is well known in IEEE, but you must explain the name, at the beginning of the paper.

Line 58: „NSGA-II algorithm”. Maybe a pleonasm?

Line 133: Figure 4 , 5, 6 are not cited in paper.

Figure 5 is mentioned at line 178 only with (5) and Figure 6 at line 180 only with (6)

Line 177: „First, an actual mountainous region is selected using Google Earth and a vertical cross-section is taken (5)” It is difficult to understand how it is possible to extract vertical profile from Google Earth.

„Figure 5. Real Undulating Mountain Terrain” is not suggestive for a vertical profile.

Figure 6. Simulated Electric Potential Results has no units of measurement, the color is the same, must be improved.

The paper is highly theoretical in nature, serving as a preliminary step toward the 3D modeling of mountainous terrain, as acknowledged by the authors. The reviewed study models the topography using a single 2D profile, which is highly approximate and extracted from Google Earth without detailing the underlying methodology. However, the actual bottleneck regarding computational time lies in the transition to 3D terrain modeling, rather than in the NSGA-II, as the latter is a relatively simple and fast algorithm.

Author Response

Comment 1: NSGA‑II (Non-dominated Sorting Genetic Algorithm) is well known in IEEE, but you must explain the name, at the beginning of the paper.

Response 1: Thank you for this comment. We fully agree and have revised the manuscript accordingly. We have added the full name of Non-dominated Sorting Genetic Algorithm II (NSGA‑II) at its first appearance in the paper, on Page 1. This ensures the full name is explained at the beginning of the manuscript as required.

Comment 2: Line 58: „NSGA-II algorithm”. Maybe a pleonasm? Response 2: Thank you for pointing out this redundant expression. We fully agree and have revised the manuscript. We have removed the redundant word “algorithm” throughout the manuscript. All instances of “NSGA‑II algorithm” have been unified to “NSGA‑II”, because the term NSGA‑II itself clearly represents an algorithm.  

Comment 3: Line 133: Figure 4, 5, 6 are not cited in paper. Figure 5 is mentioned at line 178 only with (5) and Figure 6 at line 180 only with (6).

Response 3: We apologize for the inappropriate citation. We fully agree and have corrected all citations. According to the revised figure definitions:

Figure 4 is cited and described on Page 5;

Figure 5 is cited and described on Page 6;

Figure 7 and Figure 8 are cited and described on Page 8.

All figures are now formally cited in the main text as required.

Comment 4: Line 177: „First, an actual mountainous region is selected using Google Earth and a vertical cross-section is taken (5)” It is difficult to understand how it is possible to extract vertical profile from Google Earth

Response 4: Thank you for this valuable question. We fully agree and have added detailed extraction steps. We have supplemented the specific procedure for extracting the vertical profile from Google Earth Pro v7.3.6 on Page 8：

(1) Open Google Earth Pro v7.3.6 and locate the target mountain area;

(2) Draw a horizontal profile of 3000 meters in the east–west direction;

(3) Export elevation data with a sampling interval of 10 m;

(4) Fit the discrete points into a smooth terrain profile using cubic spline interpolation.

The extraction method is now clear and reproducible.

Comment 5：„Figure 5. Real Undulating Mountain Terrain” is not suggestive for a vertical profile.   Response 5:  We agree with this comment. According to the revised manuscript, Figure 6 represents the real undulating mountain terrain. We have revised the caption of Figure 6 to “Vertical cross-section of real undulating mountain terrain”, which clearly indicates the vertical profile and matches the content of the figure and the main text.     Comment 6: Figure 6. Simulated Electric Potential Results has no units of measurement, the color is the same, must be improved.   Response 6: Thank you for this comment. We have optimized Figure 7 (the simulated electric field results). We have added the unit “Unit: V/m” next to the color bar, and replaced the single‑color scheme with a blue–to‑red gradient color map to clearly distinguish different electric field intensities.   Comment 7:The paper is highly theoretical in nature, serving as a preliminary step toward the 3D modeling of mountainous terrain, as acknowledged by the authors. The reviewed study models the topography using a single 2D profile, which is highly approximate and extracted from Google Earth without detailing the underlying methodology. However, the actual bottleneck regarding computational time lies in the transition to 3D terrain modeling, rather than in the NSGA‑II, as the latter is a relatively simple and fast algorithm.   Response 7: Thank you for these constructive suggestions. We fully agree and have revised the discussion section on Pages 12–14 as follows: (1) We have strengthened the rationality of the 2D model as a key preliminary step for 3D modeling; (2) We have supplemented the detailed terrain extraction method from Google Earth (as revised in Comment 4); (3) We have clarified that the real computational bottleneck lies in the 3D terrain modeling rather than the NSGA‑II algorithm, and noted that the proposed bi‑objective optimization framework can be directly extended to 3D scenarios.      

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors clearly addressed each of the weaknesses highlighted in the peer review comments

Author Response

We sincerely appreciate the reviewer’s positive feedback and careful review of our revised manuscript. We are glad that all the concerns raised have been fully addressed and resolved. Thank you for your valuable comments and suggestions, which have greatly helped improve the quality of this work.

Reviewer 2 Report

Comments and Suggestions for Authors

The revised manuscript has been substantially improved, but the authors may further clarify why the reported reduction in mean error and computation time is sufficiently meaningful in relation to the practical contribution of the proposed method. 
1. The additional validation cases strengthen the manuscript, although a brief discussion on the sensitivity of the method to factors such as charge position, sampling density, or computational domain would further improve the generality of the conclusions. 
2. The optimization procedure is now described in greater detail, but the manuscript would benefit from a clearer justification for the selected parameter bounds, penalty thresholds, and convergence criteria. 
3. The discussion section has been improved, yet the practical applicability of the proposed framework could be presented more carefully in light of its current restriction to electrostatic analysis and 2D terrain representation. 

Author Response

1. The additional validation cases strengthen the manuscript, although a brief discussion on the sensitivity of the method to factors such as charge position, sampling density, or computational domain would further improve the generality of the conclusions. 

Response: We sincerely appreciate the reviewer’s valuable and constructive comments, which have greatly helped improve the quality of the manuscript. We fully agree with all the points raised, and have comprehensively revised the manuscript accordingly, with all changes highlighted in red in the revised version. We have added a new Section 3.5 "Sensitivity Analysis"(page11，line 351-360) to discuss the method’s robustness against common parameter variations. The analysis verifies that the calculation error remains stable when the charge position varies within typical ranges, the results converge rapidly with increasing sampling density, and the selected computational domain is sufficiently large to eliminate boundary effects. These findings further confirm the generality and reliability of the proposed method’s conclusions.

2. The optimization procedure is now described in greater detail, but the manuscript would benefit from a clearer justification for the selected parameter bounds, penalty thresholds, and convergence criteria. 

Response: We have supplemented explicit justifications for all key optimization parameters in Section 2.2.2(Page 6-7-8)

• The parameter range M∈[0,11000] is determined based on terrain elevation scale, mapping characteristics, and extensive preliminary tests to cover all effective mapping parameters for typical 2D terrain profiles;（page 6,197-199）
• The penalty thresholds (20% for maximum local error, 10% for mean global error) are selected in accordance with practical engineering requirements for lightning electrostatic field assessment, beyond which the calculation results lack sufficient engineering reliability;（page 7, 234-237）
• The convergence criterion (relative change < 10−6 for 10 consecutive generations) is designed to ensure stable convergence of the Pareto front while avoiding unnecessary computational cost, balancing convergence reliability and efficiency. (page 8, 256-260)

3. The discussion section has been improved, yet the practical applicability of the proposed framework could be presented more carefully in light of its current restriction to electrostatic analysis and 2D terrain representation. 

Response: We have thoroughly revised Section Discussion (page 12-14)to clearly define the method’s scope:

• We have toned down overstated descriptions of engineering benefits to avoid exaggerating the applicable scenarios.(4.1 Practical Engineering Implications of the Synergistic Improvements, page 13 ,391-426)
• We have clearly emphasized that the proposed method is suitable only for preliminary lightning protection design under 2D terrain and electrostatic field conditions. (page 13, 408-423)
• We have clarified the inherent limitations of fractional linear mapping (only for 2D problems) and avoided overstating the engineering performance. (page 13, 408-423)

All revisions ensure the description of practical applicability is objective, rigorous, and consistent with the actual capability of the proposed framework.

 

Author Response File: Author Response.pdf