As humanity's exploration of the Earth's internal structure deepens, Earth's free oscillations, serving as crucial "fingerprints" for revealing the large-scale structure and dynamic processes within the Earth, have always been a core subject in geophysics. Ground-based station observations are currently the mainstream method for measuring Earth's free oscillations. With the advancement of space technology, high-precision inter-satellite distance measurement holds the potential to become a novel method for detecting these oscillations.
In a recent paper published in Space: Science & Technology, a research team from the School of Physics and Astronomy at Sun Yat-sen University, in collaboration with the TianQin Research Center for Gravitational Physics, proposed a novel detection and analysis method for Earth's free oscillations utilizing the "TianQin" space-borne gravitational wave detector. The study constructed a theoretical response model for Earth's free oscillations within the TianQin detector and derived their analytical waveform for high-orbit satellite laser interferometric measurements. Through numerical simulation and Bayesian parameter estimation, the research team demonstrated that for a major seismic event like the 2008 Wenchuan earthquake, TianQin could achieve a clear detection with a signal-to-noise ratio as high as 73 and independently distinguish at least nine different free oscillation modes.
This research not only methodologically demonstrates for the first time the feasibility of directly detecting Earth's free oscillations using high-orbit gravitational wave detectors, but also pioneers a new interdisciplinary pathway integrating space-based gravity measurement with geophysical research. By leveraging the frequency splitting effect introduced by satellite orbital motion, this method can circumvent calibration errors inherent in traditional multi-station observations, enabling more independent and precise probing of Earth's internal structure. This work significantly expands the interdisciplinary application scope for China's autonomous space science mission—the TianQin project—and provides novel theoretical tools and technical reserves for future space-based exploration of Earth's internal structure and seismic mechanisms.
First, building upon the unique orbital configuration and measurement principles of the TianQin space gravitational wave detector, the authors systematically established a theoretical response model for Earth's free oscillations in inter-satellite laser interferometric measurements. Traditional detection of Earth's free oscillations primarily relies on ground-based gravimeter networks, which are constrained by local noise, instrument calibration errors, and limited spatial coverage. Meanwhile, low-Earth orbit gravity satellite missions have struggled to directly extract free oscillation signals due to strong interference from Earth's high-degree gravity field. Operating at an altitude of approximately 100,000 kilometers, TianQin, as a space-borne gravitational wave detector, employs a three-satellite equilateral triangle constellation. It measures changes in inter-satellite distances using laser interferometry with picometer-level precision. This makes it highly sensitive to low-frequency gravitational field variations, and its high-orbit characteristic effectively avoids coupling interference from Earth's high-degree gravity field. In this study, based on Kaula's linear perturbation theory, the authors derived an analytical expression for the strain signal induced by Earth's free oscillations in the Time-Delay Interferometry (TDI)-X channel. This model describes Earth's free oscillations as a superposition of a series of damped oscillatory modes, where each mode corresponds to a set of coefficients in the spherical harmonic expansion, incorporating frequency splitting effects caused by Earth's rotation and satellite orbital motion. Using the numerical simulation program TQPOP to generate satellite orbital data, the authors compared the analytical waveform with the numerical results. Figure 1 shows the Earth's free oscillation response curves in the TDI-X channel obtained from the analytical model (orange dashed line) and numerical simulation (blue solid line). The two show high consistency throughout the entire period except for the initial phase immediately after the earthquake occurrence, with the residual (green solid line) maintained at an extremely low level, validating the effectiveness of the analytical model. Furthermore, Figure 2 displays the characteristic strain spectrum of the signal in the frequency domain, clearly revealing frequency splitting phenomena in multiple oscillation modes due to Earth's rotation and satellite orbital motion. This aligns perfectly with the model's predictions. This section of the work provides a solid theoretical foundation and a reliable signal model for the direct and independent detection of Earth's free oscillations from space.
Secondly, leveraging Bayesian statistical inference and the Markov Chain Monte Carlo (MCMC) method, the authors conducted a systematic study on signal extraction and parameter estimation from simulated observation data containing signals and noise, aiming to quantify TianQin's actual detection capability for Earth's free oscillations. The research assumed the presence of Gaussian stationary noise in the detector and constructed a likelihood function based on the inner product between the waveform template and the data. For parameter estimation, the authors fixed the frequency and quality factor for each oscillation mode as given by the Preliminary Reference Earth Model, treating the spherical harmonic coefficients describing the oscillation amplitudes as the parameters to be estimated. For a simulated seismic event with a moment magnitude of 7.9, resembling the Wenchuan earthquake, the authors injected numerically simulated Earth's free oscillation signals into the TDI-X channel data and added TianQin's theoretical noise. Subsequently, they performed MCMC sampling using the emcee software to obtain the posterior distributions of the parameters. Table 2 lists the true values, MCMC-estimated values, and their uncertainties for the spherical harmonic coefficients of the nine primary modes. These results were compared with error estimates derived from the Fisher Information Matrix, showing good consistency, which validates the reliability of the MCMC estimation. Figure 3 intuitively presents the posterior distributions of these parameters in the form of violin plots, with blue vertical lines indicating the true parameter values. The analysis results indicate that for an earthquake of this magnitude, the Earth's free oscillation signal can achieve a signal-to-noise ratio as high as 73 in TianQin's TDI-X channel, allowing for the clear distinction of approximately nine different oscillation modes. Figure 4 displays the two-dimensional posterior distributions and correlations among some key parameters through a corner plot, revealing the constraints and interdependencies of different mode parameters during joint estimation. This part of the work, for the first time from the perspective of data analysis and parameter estimation, quantitatively demonstrates that TianQin possesses reliable detection capability for Earth's free oscillations excited by major earthquakes. It takes a solid step towards the future practical application of high-orbit gravity satellites for geophysical research.
Finally, the authors extend their research perspective to different TDI combination channels and explore the potential relationship and discrimination methods between Earth's free oscillation detection and TianQin's primary scientific objective—gravitational wave detection. The authors point out that the derived analytical waveform can be conveniently extended from the TDI-X channel to the TDI-Y and TDI-Z channels by adjusting the satellite orbital phase. Furthermore, the study analyzes the behavior of Earth's free oscillation signals in the widely used AET channels for GW detection, which are orthogonal linear combinations of the XYZ channels. Unlike GW signals, which are significantly suppressed in the T channel, numerical simulations show that Earth's free oscillations can also generate a high signal-to-noise ratio response in the TDI-T channel. This difference stems from the fact that GWs can typically be approximated as plane waves, whereas Earth's gravitational perturbations (including free oscillations) originate from a near-field source and do not satisfy the plane wave assumption, leading to different response characteristics in the T channel. The authors acknowledge that the current theoretical waveform derivation cannot perfectly describe the response in the T channel, and a more complex model needs to be developed in the future. This finding holds significant application value: in future TianQin observations, when both GW events and major seismic signals are captured simultaneously, a global analysis method utilizing their distinct response characteristics in the A, E, and T channels can be developed to achieve signal separation. In summary, this study not only pioneers a novel interdisciplinary direction for detecting Earth's free oscillations using TianQin and validates its detection capability, but also establishes a methodological foundation for the integrated processing of multi-physical field signals (Earth's gravitational perturbations and cosmic gravitational waves). It comprehensively expands the scientific implications and application prospects of China's autonomous space science mission.
