| 20 Oct 2025
Perturbation Analysis of Travel-Time Accuracy for Core Phases Reconstructed from Seismic Interferometry
Abstract. Correlating late coda waves from large earthquakes produces stable waveforms that approximate inter-station core phases. However, the properties of these coda waves often violate the strict assumptions underlying classical Green's function retrieval, raising doubts about the physical correspondence of the reconstructed arrivals to true inter-station phases and limiting their utility in seismic imaging. In this study, we present a perturbation analysis of core-phase interferometry and show that accurate travel-time information can be recovered under locally uniform wave incidence along the inter-station path. We introduce a dimensionless parameter – defined as the ratio of the seismic wave period to the inter-station travel time – which establishes a critical angular threshold. Our perturbation analysis reveals that the travel-time reconstruction accuracy scales with the cube of this threshold, allowing high-precision recovery of core phases, particularly those associated with small threshold values. Numerical simulations validate the theoretical predictions. By applying the proposed framework to real coda correlation data, we demonstrate that core phases can be reliably reconstructed using a sufficiently large number of global earthquakes – even without the traditionally assumed uniform source distribution. These results establish a rigorous theoretical foundation for extracting high-precision core-phase travel times from coda correlations, enhancing the reliability of seismological imaging of Earth's deep interior.
This manuscript tackles a pivotal issue in seismology: enhancing the accuracy of core-phase travel-time reconstruction through coda wave interferometry, a topic of paramount importance for advancing the imaging of Earth's deep interior. The authors employ a perturbation analysis to quantify travel-time deviations and establish a critical angular threshold, constituting a valuable theoretical contribution that addresses persistent uncertainties regarding the physical correspondence between reconstructed coda-derived phases and true inter-station core phases. However, several substantive issues pertaining to literature integration, theoretical rigor, completeness in data analysis, and contextualization with prior research must be resolved to bolster the manuscript's robustness and impact. Based on the following assessment, a major revision is recommended.
Major revisions:
1, Insufficient Quantitative Analysis of Data Volume, Distribution, and Deviation Angle Dependence on Reconstruction Stability:
A central conclusion of this study is that "a sufficiently large number of global earthquakes" enables reliable core-phase reconstruction. However, this claim lacks the necessary quantitative foundation and fails to systematically assess how the stability of travel-time retrieval depends on key data characteristics. Specific shortcomings include:
a) Undocumented Processing Parameter: The manuscript does not specify the coda time window (e.g., start and end times relative to the origin time) used for the correlation analysis. This critical parameter directly influences the extracted waveforms and must be reported to ensure reproducibility.
b) Unquantified Impact of Event Count: While 205 earthquakes were used, the study provides no analysis of how reconstruction stability degrades with smaller datasets. To objectively define data sufficiency, a sensitivity analysis is required. For instance: How does the standard deviation of the reconstructed travel times for key phases (e.g., ScS, PKIKP²) evolve as the number of events used in the stack is progressively reduced from 205 to, for example, 150, 100, or 50 (e.g., via bootstrap resampling)? Does a minimum event count threshold exist, below which the stability deteriorates significantly or the proposed cubic scaling relationship between accuracy and the angular threshold breaks down?
c) Incomplete Investigation of Deviation Angle Dependence: While Figure 10 presents a valuable analysis of convergence with increasing maximum deviation angle (φ), the current approach of using cumulative ranges (e.g., 0-10°, 0-20°, etc.) limits its power to validate the theoretical framework. A more rigorous, binned analysis is needed. The data should be stacked and analyzed within discrete, non-overlapping ranges of the deviation angle φ (e.g., 0-10°, 10-20°, 20-30°, etc.).
This binned comparison is critical for directly testing the theoretical prediction of whether travel times remain accurate and unbiased across all directions of wave incidence under the local uniformity assumption.
Such an analysis would reveal: 1) If travel-time biases exist in specific ranges of the deviation angle, and 2) The magnitude of any such biases as a function of φ.
A finding of consistent travel times across all discrete angular bins would strongly corroborate the theoretical model. Conversely, identifying systematic biases in certain angular ranges would provide invaluable insights into the limits of the local uniformity condition and guide future data selection.
2, Missed Opportunity for Theoretical Validation through Comparative Analysis of ScS and PKIKP² Phases
The manuscript estimates a critical angle of 18° for the ScS phase and applies this framework in the real-data analysis. However, it does not fully leverage the contrasting behaviors of ScS and PKIKP² phases observed in Figure 10 to rigorously test and validate the underlying theory. Specifically, the ScS travel time shows a clear dependence on the deviation angle φ, while the PKIKP² travel time remains stable. This striking discrepancy represents a critical opportunity to strengthen the study's conclusions, yet it remains largely unexplained.
To transform this observation into a powerful validation of the theoretical framework, the authors should:
a) Perform Phase-Specific Theoretical Calculations: The critical angle and the expected travel-time deviation are functions of the wave period (T) and the phase-specific travel time (t(p)). The manuscript must present the theoretically predicted critical angle and the scaling of travel-time accuracy specifically for the PKIKP² phase, rather than implicitly assuming the ScS-derived value applies.
b) Provide a Physically Consistent Explanation for the Contrast: The fundamental difference in the sensitivity of ScS and PKIKP² to the deviation angle φ must be explained within the proposed theoretical framework. This discussion should explicitly link the distinct ray paths of the two phases (e.g., ScS reflecting off the core-mantle boundary versus PKIKP² traversing the inner core) to the potential magnitude of the deviation function δ(θ, p) and its higher-order derivatives in Eq. 19. For instance:
Does the more complex path of PKIKP² through the inner core lead to a different "smoothness" of δ(θ, p) near θ=0, resulting in smaller higher-order terms and thus greater robustness to a limited deviation angle range?
Conversely, does the ScS path make it more susceptible to structural heterogeneity near the core-mantle boundary, amplifying the higher-order derivatives and making its reconstruction more sensitive to the angular distribution of sources?
By quantitatively calculating the phase-specific theoretical parameters and then using them to explain the empirically observed difference in stability between ScS and PKIKP², the authors can demonstrate that their model not only predicts general behavior but also accurately captures the specific physics governing different core phases. This would significantly elevate the impact of the study by providing a unified and predictive theoretical explanation for the key observational result in Figure 10.
3, Unaddressed Discrepancies in Figures 8 and 9: Both figures indicate that most core phases exhibit obvious waveform differences as the inter-station distance approaches 0°. The manuscript does not investigate the origin of this systematic pattern, which could stem from physical phenomena (e.g., near-field effects, 3D structural complexities) or methodological artifacts (e.g., inadequate azimuthal coverage at very short distances). Explaining this observation is vital for affirming the method's reliability across the entire distance range.
4, Need for a Unifying Theoretical Discussion on I2* versus True Phase Reconstruction for PKIKP²:
The manuscript reports stable PKIKP² travel times across a wide range of deviation angles (Fig. 10). This finding appears to contradict a body of prior work (e.g., Wang & Tkalčić, 2019, 2020; Costa de Lima et al., 2022) which argues that coda correlations typically retrieve an I2* wavefield—a modified Green's function whose travel times exhibit a dependence on the distribution of seismic sources (e.g., varying with deviation angle). The authors have a critical opportunity to use their theoretical framework to explain and reconcile these differing observations, thereby making a seminal contribution to the debate on what is physically extracted from coda correlations.
To achieve this, the authors must:
a) Explicitly Discuss the Discrepancy within Their Theoretical Context: The discussion should directly engage with the findings of the aforementioned I2* studies. The core argument should posit that the critical difference lies in whether the condition of "local uniform wave incidence" (quantified by the critical angle θ₀) is met. The prior conclusions—that travel times vary with deviation angle—likely stem from datasets or phases where this local uniformity condition was not satisfied. In contrast, the current study, potentially by leveraging a massive global dataset for PKIKP², may have met this condition, thus successfully retrieving the true inter-station travel time.
b) Provide a Physical Mechanism for PKIKP² Stability Based on the Perturbation Analysis: The authors must use their theoretical framework to explain why the PKIKP² phase in their study is robust. The key lies in Eq. (19): the travel-time error scales with θ₀³ and the higher-order derivatives of the deviation function δ(θ,p).
The authors should argue that the specific ray path of PKIKP² (traversing the inner core) results in a "smoother" δ(θ,p) near θ=0 (i.e., very small higher-order derivatives). Combined with its specific period-to-travel-time ratio yielding a small θ₀, this makes the phase inherently robust to variations in the deviation angle φ once a basic illumination threshold is crossed.
This provides a physical mechanism for why their method, under the right conditions, avoids the deviation-angle-dependent biases characteristic of I2* retrieval.
c) Propose a Generalized Criteria for True Travel-Time Extraction: The manuscript should synthesize these points into a clear proposition: The transition from retrieving a biased I2 to the true PKIKP² travel time occurs when the angular range of incident waves meets or exceeds the phase-specific critical angle θ₀ and the structural setting leads to a sufficiently smooth deviation function. This would powerfully contextualize their results, suggesting that the previous I2* observations and their own stable result are not fundamental contradictions but are explained by the degree to which the conditions of their unified theory are met.
Minor comments:
1, The Introduction's discussion of the research gap concerning travel-time deviations for coda-based core phases requires sharper focus and better integration with key literature. Notable studies on core-phase extraction (e.g., Wang & Tkalčić, 2019, JGR-Solid Earth; Poli et al., 2017, Earth and Planetary Science Letters; Phạm & Tkalčić, 2022, Nature Communications) should be cited to delineate the knowledge gap and underscore the novelty of this work.
2, To strengthen the practical motivation, the Introduction should explicitly mention applications of coda-based core phases in imaging Earth's interior. Citing relevant studies (e.g., Wang, Song, & Xia, 2014, Nature Geoscience; Tkalčić & Phạm, 2018, Science; Wang & Tkalčić, 2022, Nature Astronomy) would highlight the significance of accurate travel-time reconstruction.
3, Several conclusions would benefit from citations to post-2020 research to enhance timeliness. For instance, referencing recent comparative studies on core-phase travel-time accuracy (e.g., Costa de Lima et al., 2022, JGR-Solid Earth) would help contextualize the findings within the latest advancements.
4, The terminology for the angle of wave arrival at a station is inconsistent, alternating between "incident angle" and "incidence angle." The standard seismological term "incidence angle" should be used consistently throughout the manuscript.
5, The manuscript uses "azimuth" to describe the orientation of the earthquake-station plane relative to the inter-station plane. This usage is not explicitly distinguished from the standard seismological definition of azimuth (the angle from true north to the source-station great-circle path). To prevent confusion, the authors should clearly define their custom azimuth (or use “deviation angle”) and clarify its relationship to the conventional term.
6, “microseisms (1–50 period)” should be “microseisms (periods 1–50 s)”
7,“205 large earthquakes (≥M 6.8)” should be “205 large earthquakes (M ≥ 6.8)”