Over the past two decades, the use of ambient ground motions for imaging subsurface structures has advanced significantly. This progress is largely driven by the discovery that cross-correlating records between two stations yields waveforms with dispersion characteristics resembling those of surface waves propagating between them . Commonly used ambient seismic sources include microseisms (periods 1–50 s) and the Earth's hum (periods 50–300 s), both generated by interactions between ocean waves and the solid Earth , as well as high-frequency anthropogenic noise (periods <1 s). Dispersion measurements derived from ambient noise correlations have been extensively employed to probe Earth's interior structure .

Beyond ambient noise, late earthquake coda waves contain substantial body wave energy that has traversed deep Earth discontinuities. Consequently, coda correlations are enriched with core-sensitive phases that generally preserve accurate slowness information . The travel times of these extracted phases have been utilized to constrain the fine-scale structure of the Earth's core (; ).

Theoretically, the cross-correlation function (CCF) of noise records converges to the seismic Green's function under ideal conditions – such as a uniform distribution of noise sources enclosing the stations or equipartitioned wavefield energy . However, coda waves often violate these ideal conditions, resulting in anomalously high amplitudes in reconstructed phases compared to earthquake data , as well as persistent features occurring at travel-time differences between conventional phases .

It is also well established that deviations from ideal conditions can introduce travel-time biases in wavefield reconstructions . As a result, it is widely recognized within the coda correlation community that core phases extracted from coda waves do not exactly represent the true inter-station core phases. Their travel times have been shown to be influenced by several factors, including earthquake–station geometry, focal mechanisms, and the specific coda time window selected for correlation. Previous studies have investigated the formation mechanisms of these phases under such variable conditions .

This study aims to clarify under what realistic circumstances the travel times of coda-based core phases correspond to those of true inter-station arrivals. Such an evaluation is essential for assessing the reliability of using extracted travel times as substitutes for direct arrivals in imaging deep Earth structural anomalies. Previous theoretical treatments of noise correlations have often relied on asymptotic techniques – such as the stationary phase method – to derive approximate relationships, which inherently limits the rigor of travel-time accuracy assessments. To overcome this limitation, we introduce a perturbation-based framework to quantify potential travel-time discrepancies.

A key feature of this approach is the decomposition of the problem into a “solvable” reference model and a “perturbation” component. We begin with a bounded homogeneous model representing the solvable part, which is then perturbed to evaluate the accuracy of travel-time reconstruction. The proposed framework is validated through numerical simulations and further demonstrated using real coda correlation data.

We perform a numerical computation to investigate the impact of wave correlation under locally uniform illumination. In the computation, we set the travel time of the reflected wave to 1000 s and wave correlation in the period range of 20–50 s. Then, we obtain the threshold angle θ0=18° as in Eq. (), corresponding to a 2π phase shift relative to the reflected wave. We also compute truncation angles for π and 4π phase shifts, obtaining θ0=13° and θ0=26°, respectively. These truncation angles are used as upper bounds in the integral of Eq. (). For comparison, we compute the accurate arrival of the reflected wave using an upper bound of 180°. The results show: Within a π phase shift, the reconstructed reflected wave and truncation-induced spurious wave interfere, causing phase deviation in the reconstructed reflected wave. Within a 2π phase shift, the two waves align, allowing accurate recovery of the reflected wave's travel time. Within a 4π phase shift, the waves diverge, and the reconstructed reflected wave remains unaffected by integration truncation (Fig. ). This supports the rationality of our non-interference criterion that uses a 2π phase shift to determine the threshold angle.

Given the small threshold angle in our simulation, we further examine correlation under a non-cosine travel-time difference distribution by setting the disturbance term in Eq. () as 24δ(θ,p)=-50θsin⁡2(4θ). This function and its first derivative satisfy the constraints in Eq. (). Integrating over θ from 0 to π and comparing the reconstructed wave with that from a cosine travel-time difference distribution, we find nearly identical phase information (Fig. ). This demonstrates high reconstruction accuracy for travel times even under non-cosine travel-time difference distributions.

We selected 205 large earthquakes (M≥6.8) from 2010 to 2020 with global distribution. For each event, we downloaded broadband waveforms from stations in the USArray Transportable Array. The spatial distribution of the earthquakes and stations is shown in Figs. a and b. For the late coda radiated by each earthquake, instrument responses were removed, and the data were bandpass filtered between 0.02 and 0.07 Hz (15–50 s period) to retain core-sensitive body wave energy. For each earthquake–station pair, we extracted the late coda time window from 10 000 to 40 000 s after the origin time. This window contains dominant multiply scattered waves that have sampled deep Earth structures. For each station pair and each earthquake, we computed the cross-correlation of the coda waveforms following the procedure of . We then calculated the deviation angle ϕ between the earthquake–station plane and the two-station plane. Correlograms were stacked within selected ϕ ranges and inter-station distance bins (bin width 1°). Arrival times of target phases were picked from the stacked correlograms using automated peak detection with manual verification. The number of cross-correlation functions stacked in each bin is statistically summarized in Fig. c.

In the stacked correlograms, prominent deep Earth phases – specifically PcP, ScS, and PKIKP2 – are clearly distinguishable (Fig. ). To quantitatively evaluate whether the illumination condition has been satisfied, we performed a bootstrap analysis by progressively increasing the number of earthquakes in the stack from 10 to 120, in increments of 10. The stabilization of travel times with an increasing event count serves as a diagnostic for meeting the illumination condition. For each subset size, we performed 250 random realizations to compute the mean travel time and standard deviation for the extracted ScS and PKIKP2 phases.

Our results reveal two key findings (Fig. ). First, for both phases, the mean travel time stabilizes and the standard deviation decreases as the number of earthquakes increases, indicating clear convergence toward a stable value. Second, significant travel-time deviations–exceeding ±1 s – are observed when fewer than approximately 20 earthquakes are used, particularly for the ScS phase. This indicates that a minimum event count is required to achieve locally uniform illumination. However, we emphasize that this threshold is not solely dependent on the event count, but also on the azimuthal distribution of sources, their focal mechanisms and the number of correlation traces stacked per distance bin. The cubic scaling relationship derived in Eq. () assumes locally uniform illumination; when this condition is violated – as seen in the smaller earthquake subsets – the scaling relationship breaks down, resulting in the observed high standard deviations.

According to our theoretical framework, when the locally uniform illumination condition is satisfied and the Γ value is large (as it is for both ScS and PKIKP2), correlation signals across different deviation planes should converge toward the true inter-station arrival time. To verify this convergence, we partitioned the correlograms based on specific ranges of the deviation angle ϕ. The resulting stacked correlograms exhibit clear signals in both the ScS and PKIKP2 time windows; however, as predicted, the PKIKP2 signals appear more focused than those of ScS (Fig. ). This trend is further illustrated by comparing the azimuthal deviation (ϕ) ranges of (0, 20°) and (40, 60°) (Fig. a, b): while travel-time deviations for ScS reach up to 3 s, they remain nearly identical for PKIKP2 (Fig. c, d).

This study investigates the reconstruction of inter-station waves under conditions where the incidence of seismic waves is locally uniform along the propagation path. A critical angular threshold θ0, defined in Eq. () as the ratio of the seismic wave period to the inter-station travel time, quantifies the extent of this locally uniform illumination. Equation () further demonstrates that the accuracy of travel time reconstruction scales with the cube of this threshold angle. This cubic scaling relationship reveals that even under ideal uniform illumination, small travel-time deviations persist – a finding consistent with surface wave dispersion studies in inhomogeneous media . The relationship provides a practical criterion for assessing expected travel-time errors in reconstructed body waves based on readily available parameters: wave period and inter-station travel time.

Core phases are characterized by inherently small threshold angles due to their long travel times (e.g., θ0≈11° for PKIKP2 at 10° distance). This property has two important practical implications. First, high-precision travel-time extraction can be achieved under the locally uniform illumination condition according to Eq. (). Second, late codas radiated from a modest number of sparsely distributed earthquakes may satisfy this condition – as demonstrated by our bootstrap analysis showing convergence with approximately 50–100 events (Fig. ). Consequently, our approach relaxes the stringent requirements of traditional seismic interferometry, making it more applicable to practical coda correlation studies where global earthquake distributions are inherently non-uniform.

In Fig. , the reconstructed core phases at near-zero inter-station distances (<4°) exhibit systematically lower signal-to-noise ratios (SNR) than at larger distances. This pattern can be explained within our theoretical framework. The constructive interference required to build a specific phase (e.g., PKIKP2) depends on coda waves with incidence angles nearly identical to that of the target phase. For late coda waves with near-vertical incidence, the propagation path from the earthquake source to the station is substantially longer, resulting in stronger geometric attenuation and thereby reducing the amount of correlated late coda energy available for constructive interference. In addition, as shown in Fig. c, the number of correlation traces stacked in near-zero distance bins is relatively low compared to bins at moderate distances. The reduced stacking fold further limits the enhancement of coherent signals.

The near-vertical incidence of PKIKP2 results in a high Γ value, which in turn facilitates an extremely tight convergence of correlation signals across different deviation planes (Figs.  and ). This geometric advantage ensures that arrival times remain nearly identical regardless of the source deviation angle ϕ. A long-standing debate in coda correlation seismology centers on whether extracted phases represent true inter-station PKIKP2 arrivals (the Green's function) or a modified wavefield (I2*) whose travel times exhibit a systematic dependence on source distribution . Based on our theoretical analysis and empirical results, we propose that the transition from a biased I2* measurement to a true PKIKP2 phase travel time occurs when the following two conditions are jointly satisfied:

Illumination condition for stable reconstruction: The angular range of incident waves meets or exceeds the phase-specific critical angle θ0, ensuring that the stationary phase zone is adequately sampled. This condition depends on the number of earthquakes, focal mechanisms, their azimuthal distribution relative to the inter-station path, and the distribution of inter-station directions stacked per distance bin. The degree to which this condition is satisfied can be evaluated quantitatively through bootstrap convergence analysis (Fig. ), where stabilization of travel times with increasing event count indicates that the illumination condition has been achieved. It can also be assessed by examining the convergence of the correlation signals across different deviation planes.

This study presents a perturbation analysis to evaluate the accuracy of travel-time reconstruction for core phases derived from late earthquake coda correlations under conditions of locally uniform wave incidence along the core-phase propagation path. We introduce a dimensionless parameter, defined as the ratio of seismic wave period to inter-station travel time, to quantify the critical angular threshold for effective reconstruction. Perturbation analysis reveals that the travel time accuracy scales with the cube of this threshold, indicating that localized uniform incidence ensures high-precision reconstruction of core phases, which are inherently characterized by low threshold values. Numerical simulations and empirical coda correlation tests sustain our theoretical findings. Our results demonstrate that accurate travel times of inter-station core phases can be reliably extracted using late coda waves from a sufficiently large number of earthquakes distributed across all deviation angles. This approach provides a practical and robust foundation for coda correlation studies, enhancing confidence in using reconstructed core phases as true empirical arrivals for interferometric imaging of Earth's deep interior.