Detection of Earth's free oscillations utilizing TianQin

TL;DR

This work addresses the challenge of observing Earth's free oscillations from space by leveraging TianQin's high-Earth orbit to suppress gravity-field noise. It develops a closed-form analytic waveform for Earth’s spheroidal normal modes as observed in the TianQin TDI-X channel and validates it against numerical simulations, enabling efficient Bayesian parameter estimation of mode amplitudes from simulated events. The study demonstrates that a magnitude $7.9$ Wenchuan-like earthquake could yield a SNR of about $73$, with roughly nine modes resolvable, illustrating TianQin's potential to probe Earth's interior and earthquake mechanisms independently of ground gravimetry. By combining orbital modulation with a robust statistical framework, the work provides a pathway for space-based detection of geophysical signals and cross-validation with terrestrial observations, while outlining directions to generalize to additional TDI channels and more sophisticated models.

Abstract

The measurement of Earth's free oscillations plays an important role in studying the Earth's large-scale structure. Space technology development presents a potential method to observe these normal modes by measuring inter-satellite distances. However, the disturbance from the Earth's low-degree gravity field makes it challenging for low Earth orbit gravity measurement satellites such as Gravity Recovery and Climate Experiment (GRACE) and TianQin-2 to extract signals from Earth's free oscillations directly. Here, we propose that by taking advantage of the high Earth orbit, the TianQin satellites can effectively avoid this disturbance, enabling direct measurement of Earth's free oscillations. We derive an analytical waveform to describe the response of Earth's free oscillations in TianQin. Based on this waveform, we use Bayesian analysis to extract the normal modes from numerical simulation data and perform parameter estimation. Our findings reveal that for a magnitude 7.9, Wenchuan-like earthquake, the resulting free oscillations will generate a signal that signal-to-noise ratio (SNR) is 73 in TianQin, and approximately 9 different modes can be distinguished. This result shows TianQin can open a new window to examine the Earth's free oscillations and study the Earth's interior and earthquakes independently from ground-based gravity measurement.

Figures (4)

• Figure 1: The figure illustrates the Earth's free oscillation response within the TianQin TDI-X channel, derived from both numerical simulation and analytical modeling approaches. The earthquake that induces the Earth's free oscillations occurs on day 30. The numerical simulation result is depicted by the blue solid line, while the analytical model result is represented by the orange dashed line. The green solid line illustrates the residual resulting from the subtraction of the aforementioned two results. With the exception of a minor region near the occurrence of the earthquake, the numerical result closely aligns with the predictions of the analytical formula.
• Figure 2: The figure illustrates the Earth's free oscillations response within the TianQin TDI-X channel in the frequency domain. Both numerical simulation data (blue solid line) and analytical modeling data (orange dashed line) are plotted in this figure. In the zoomed-in plot, the spectrum of the ${}_0 S_2$ mode is depicted. ${}_0 f_2$, $f_e$ and $f_o$ denote frequencies of ${}_0 S_2$ mode, Earth's rotation, and satellites' rotation respectively. It can be seen that this mode is split into 10 distinct spectral lines due to the influence of Earth's and satellites' rotation. Spectra splitting also occurs in other modes, and they can all be described by the analytical model (Eq. \ref{['eq: omega-relationship']}\ref{['eq: TDI-X-channel-final-result']} ). Different modes could add up deconstructive since they have opposite phases, making the overall peak appear to be slightly shifted. This is exactly what happens in certain frequencies like ${}_0 f_2 - f_e + 2f_o$, ${}_0 f_2 -f_e + 2f_o$, ${}_0 f_2 + f_e -2f_o$ and ${}_0 f_2 + f_e + 2f_o$. The green line represents the noise amplitude of TianQin TDI-X channel Li_2023_GWSpace.
• Figure 3: This figure shows the parameter estimation results and accuracy. The blue lines mark the true values we use to generate simulation data.
• Figure 4: The corner plots show the two-dimensional posterior distribution between different parameters. The blue lines show the injected values of parameters in simulated data.