the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Aug 2026
Development and application of a physics-constrained adaptive decomposition model for displacement sequences of step-like landslide
Abstract. Addressing the nonlinear evolution of step-like landslides in China's Three Gorges Reservoir area, which is driven by the strong coupling of reservoir water level fluctuations and rainfall pulses, this paper proposes a highly adaptive and physics-constrained displacement decomposition framework. First, a Chebyshev-Lévy flight-enhanced sparrow search algorithm (CLF-SSA) is developed to adaptively optimize the core parameters of variational mode decomposition (VMD). Concurrently, an integrated objective function—incorporating mathematical sparsity (envelope entropy) and a physical correlation penalty for the periodic component—is constructed. This framework ensures that the decomposition process is mechanistically-driven from its foundational level, maintaining a strictly physics-constrained architecture. Secondly, a multi-dimensional quantitative reconstruction strategy for intrinsic mode functions (IMFs) is established, integrating central frequencies, correlations with multi-source environmental factors (reservoir water level, rainfall, and elevation), and seasonal energy distribution. This strategy enables the robust decoupling of cumulative displacement sequences into distinct trend, periodic, and random components, effectively mitigating the phenomenon of mode mixing. Finally, the surface displacement of a representative step-like landslide in the TGR area was decomposed using the proposed framework. The results demonstrate that the model achieves sub-millimeter reconstruction precision, with an average root mean square error (RMSE) of approximately 0.5 mm. Notably, the decoupled periodic components exhibit exceptional consistency with geomechanical responses, achieving a maximum correlation gain of 4283.60 %. By incorporating the PyLith continuous-slip numerical model for physical validation, this study effectively elucidates the displacement step-triggering mechanism induced by seepage pressure during reservoir drawdown. This transition from statistical identification to mechanistic interpretation provides robust physical support for overcoming the pervasive time-lag bottlenecks in landslide early warning engineering. Future research will focus on integrating this physics-constrained decomposition framework with deep learning models to enhance the real-time predictive capability for diverse landslide types.
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Status: open (until 01 Oct 2026)
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RC1: 'Comment on egusphere-2026-2913', Anonymous Referee #1, 25 Aug 2026
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AC1: 'Reply on RC1', Yimin liu, 11 Sep 2026
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Detailed revision:
Comment 1: Some methodological descriptions should be clarified. The authors are encouraged to provide a brief explanation of the selection of the main parameters involved in the proposed method, particularly the physical constraint weight λ in the joint fitness function and the criteria used to classify the decomposed modes into trend, periodic, and stochastic components. A short description of the parameter-setting procedure or the corresponding decision criteria would improve the clarity and reproducibility of the study.
Response: We thank the reviewer for this helpful suggestion. We agree that the parameter-setting procedure and the quantitative criteria used for IMF-like mode reconstruction were insufficiently described in the original manuscript. We have therefore revised the Abstract, Sections 2.4 and 3.1 to improve the transparency and reproducibility of the proposed framework.
Specifically, the physical constraint weight in the joint fitness function is now explicitly reported as λ = 10, which was kept fixed for all monitoring stations rather than tuned separately for individual datasets. We have also clarified that the VMD mode number was evaluated over K∈{3, 4 ,5, 6}, while the penalty factor α was searched within [300,2000], with a population size of 30 and 30 CLF-SSA iterations.
In addition, explicit hierarchical criteria have been added for IMF-like mode classification. Modes with center frequencies below 1/365d-1 are identified as long-term trend modes. For the remaining modes, a mode is assigned to the periodic component when either its absolute correlation with reservoir water level exceeds 0.15 or its seasonal spectral-energy ratio exceeds 0.15. Modes showing rainfall correlation above 0.15, together with the remaining non-seasonal high-frequency residual modes, are assigned to the stochastic component. Elevation correlation is additionally used as an auxiliary criterion for identifying long-term deformation modes. The same thresholds are applied to all monitoring stations.
These additions provide explicit parameter ranges, fixed thresholds, and a reproducible decision sequence without changing the underlying decomposition framework.
Furthermore, in response to the reviewer’s suggestions regarding methodological clarity and reproducibility, the corresponding codes have been revised and upgraded, including improvements in parameter setting documentation and IMF-like component classification procedures. The updated source code has been publicly uploaded to GitHub according to the journal’s data and code-sharing requirements: https://github.com/liyix39/Physics-Constrained-VMD-Landslide-Displacement-Decomposition/tree/revision/reviewer-1
Comment 2: The discussion of the results could be slightly improved. Some of the reported results, such as the very large percentage increase in correlation, would benefit from a brief explanation of how these values are calculated and interpreted. In addition, when discussing the relationships between displacement components and environmental factors, the authors should use slightly more cautious wording to distinguish statistical correlation from physical causation.
Response: We sincerely thank the reviewer for this valuable suggestion. We agree that the interpretation of the quantitative results and the relationships between displacement components and environmental factors should be presented more cautiously.
To address this comment, we have revised Section 5.3 to clarify the calculation and interpretation of the reported PCC increase values. Specifically, the relative PCC increase is now explicitly defined as the percentage change in the PCC between the reconstructed periodic displacement and reservoir water level (RWL) relative to that between the original cumulative displacement and RWL. The corresponding equation and variable definitions have been added. Because the PCC values of the original cumulative displacement with RWL are very small, the resulting relative percentage increases can be numerically large. These percentages therefore represent relative changes from a small baseline and should not be interpreted as absolute PCC values exceeding the theoretical range of ([-1, 1]).
In addition, we have carefully revised the discussion in Sections 5.2 and 5.3 by replacing overly deterministic expressions with more cautious descriptions. Specifically, statements implying direct physical causation have been modified to emphasize statistical associations between decomposed displacement components and environmental variables. For example, the relationships between periodic displacement variations and reservoir water-level fluctuations, as well as between short-term stochastic components and rainfall variations, are now described as statistical correlations or potential associations rather than direct causal mechanisms.
Furthermore, a clarification has been added to the discussion section indicating that the identified relationships represent statistical associations and should be interpreted together with geological and hydrological knowledge rather than as definitive evidence of direct causation.
These revisions improve the scientific rigor of the discussion and provide a more balanced interpretation of the proposed decomposition results.
Comment 3: The literature review and references should be carefully checked and updated where necessary. The authors should ensure that the literature review adequately reflects recent research related to landslide displacement decomposition, VMD-based methods, and physics-informed approaches. At the same time, all in-text citations and reference entries should be checked for consistency with the journal's required format. Overall, some of the literature are relatively outdated, please update with some latest relevant literature.
Response: We sincerely thank the reviewer for this valuable suggestion. We agree that the literature review should be further updated to better reflect recent developments in landslide displacement decomposition, VMD-based methods, and physically constrained modelling strategies.
Following the reviewer’s suggestion, we have carefully revised the Introduction section by incorporating recent studies on advanced displacement decomposition and VMD-based hybrid frameworks. Specifically, several recent studies related to VMD-based landslide displacement analysis and prediction have been added to better illustrate the recent progress from traditional signal decomposition methods toward adaptive and intelligent decomposition frameworks. These newly introduced references include recent works focusing on the integration of VMD with optimization algorithms and machine learning models, as well as studies emphasizing the decomposition of cumulative displacement into physically meaningful components such as trend, periodic, and stochastic terms.
In addition, the discussion of the limitations of existing decomposition approaches has been refined to better motivate the proposed physics-constrained framework. Recent developments in physically guided and physics-informed strategies have been considered to highlight the increasing demand for incorporating physical knowledge into data-driven models and to clarify the contribution of the proposed method.
Furthermore, Table 1 (“Summary of Landslide Deformation Decomposition Methods”) in Page 3 has been revised accordingly. Recent representative studies have been added to the VMD-related categories, including newly developed VMD-based hybrid decomposition frameworks. The characteristics and limitations of different decomposition methods have also been updated to provide a more comprehensive comparison between traditional approaches and recent intelligent decomposition strategies.
Finally, all in-text citations and reference entries have been thoroughly checked and revised according to the journal’s formatting requirements. The consistency of author names, publication years, journal information, citation styles, and DOI formats has been verified throughout the manuscript.
These revisions improve the timeliness, completeness, and consistency of the literature review and provide a clearer research context for the proposed physics-constrained displacement decomposition framework.
Comment 4: The manuscript requires careful language and formatting revision. Several grammatical, typographical, and formatting issues remain throughout the manuscript. The authors should carefully check the English expression, abbreviations, mathematical symbols, figure and table captions, equation formatting, and terminology consistency. Some sentences are relatively lengthy and could be simplified to improve readability. A thorough proofreading of the entire manuscript before resubmission is recommended.
Response: We sincerely appreciate the reviewer’s careful reading and valuable suggestions regarding language and formatting improvements. Following the reviewer’s recommendation, the entire manuscript has been thoroughly proofread and revised to improve readability, clarity, and consistency.
Specifically, grammatical and typographical errors have been corrected throughout the manuscript, and several lengthy sentences have been reorganized or simplified to improve academic expression. The use of abbreviations has been standardized, with all abbreviations defined at their first occurrence and consistently applied thereafter. In addition, mathematical symbols, equation expressions, figure and table captions, and formatting styles have been carefully checked and revised where necessary.
Furthermore, terminology related to displacement decomposition, environmental variables, and modal classification has been unified throughout the manuscript to avoid inconsistent expressions. The formatting of references, citations, and other manuscript elements has also been checked according to the journal requirements.
These revisions improve the overall language quality, presentation clarity, and technical consistency of the manuscript.
Citation: https://doi.org/10.5194/egusphere-2026-2913-AC1
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AC1: 'Reply on RC1', Yimin liu, 11 Sep 2026
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The manuscript presents a physics-constrained displacement decomposition framework based on CLF-SSA and VMD for step-like landslides. The topic is relevant and the manuscript provides a relatively complete methodology and case study. The results are generally convincing. I recommend minor revision. The following issues should be addressed:
The authors are encouraged to provide a brief explanation of the selection of the main parameters involved in the proposed method, particularly the physical constraint weight λ in the joint fitness function and the criteria used to classify the decomposed modes into trend, periodic, and stochastic components. A short description of the parameter-setting procedure or the corresponding decision criteria would improve the clarity and reproducibility of the study.
Some of the reported results, such as the very large percentage increase in correlation, would benefit from a brief explanation of how these values are calculated and interpreted. In addition, when discussing the relationships between displacement components and environmental factors, the authors should use slightly more cautious wording to distinguish statistical correlation from physical causation.
The authors should ensure that the literature review adequately reflects recent research related to landslide displacement decomposition, VMD-based methods, and physics-informed approaches. At the same time, all in-text citations and reference entries should be checked for consistency with the journal's required format. Overall, some of the literature are relatively outdated, please update with some latest relevant literature.
Several grammatical, typographical, and formatting issues remain throughout the manuscript. The authors should carefully check the English expression, abbreviations, mathematical symbols, figure and table captions, equation formatting, and terminology consistency. Some sentences are relatively lengthy and could be simplified to improve readability. A thorough proofreading of the entire manuscript before resubmission is recommended.
Overall, the manuscript is generally well structured and the proposed approach is interesting. The above issues are mainly related to clarification, presentation, and formatting, and can be addressed through revision without substantial changes to the current methodology. Therefore, I recommend publication after minor revision.