Since the inception of the optimal sequence estimation (OSE) method, various research teams have substantiated its efficacy as the optimal stacking technique for handling array data, leading to its successful application in numerous geoscience studies. Nevertheless, concerns persist regarding the potential impact of aliasing resulting from the choice of distinct station distributions on the outcomes derived from OSE. In this investigation, I employ theoretical deduction and experimental analysis to elucidate the reasons behind the immunity of the Y_l'm'-related common signal obtained through OSE to variations in station distribution selection. The primary objective of OSE is also underscored, i.e., to restore/strip a Y_l'm'-related common periodic signal from various stations. Furthermore, I provide additional clarification that the ‘Y_l'm'-related common signal’ and the ‘Y_l'm'-related equivalent excitation sequence’ are distinct concepts. These analyses will facilitate the utilization of the OSE technique by other researchers in investigating intriguing geophysical phenomena and attaining sound explanations.

Different from other normal modes of the Earth's free oscillation that depend on all the six components (M, M, M, M, M, and M) of the centroid moment tensor, the amplitudes of the radial modes depend on the M component (e.g., scalar moment (M₀), dip (δ), and slip (λ)) and hypocenter depth of the focal mechanism, and hence can be easily used to constrain these parameters of the focal mechanism. In this study, we use the superconducting gravimeter (SG) records after the 2011 Tohoku earthquake to analyze the radial modes ₀S₀ and ₁S₀. Based on the solutions of the focal mechanism provided by the GCMT and USGS, we can obtain the theoretical amplitudes of these two radial modes. Comparing the theoretical amplitudes with the observation amplitudes, it is found that there are obvious differences between the former and the latter, which means that the GCMT and USGS focal mechanisms cannot well represent the real focal mechanism of the 2011 event. Taking the GCMT solution as a reference and changing the depth and the three parameters of the M moment, the scalar moment (M₀) and the dip (δ) have significant influences, but the effects of the slip (λ) and the depth are minor. After comparisons, we provide a new constraint (M₀ = 5.8 ± 0.09 × 10²² N·m, δ = 10.1 ± 0.08°, λ = 88°, and depth = 20 km) for the focal mechanism of the 2011 event. In addition, we further determine the center frequency (1.631567 ± 2.6e^-6 mHz) and quality factor (2046.4± 50.1) of the ₁S₀ mode.

The period and quality factor of the Chandler wobble (CW) are useful for constraining the Earth's internal structure properties, such as the mantle elasticity. It has been shown that the CW is mainly excited by a combination of atmospheric and oceanic processes; hence based on a deconvolution method, we can remove them from the excitation sequence of the CW to estimate its period P and quality factor Q. We finally re-estimate P = 432.3 ± 0.8 days and Q = 85 ± 15 for the CW. Based on those two estimates, we investigate the relationship between the geomagnetic jerks and the excitation sequences of the CW. The geomagnetic jerks or jerk bounds are well consistent with the sudden changes of the excitation sequences of the CW. This demonstrates that the geomagnetic jerks could be a possible excitation source of the CW. It is crucial for understanding the excitation of the CW and for deeper geophysical insights into the geomagnetic jerks.

We review three derivative-free methods developed for uncertainty estimation of non-linear error propagation, namely, MC (Monte Carlo), SUT (scaled unscented transformation), and SI (sterling interpolation). In order to avoid preset parameters like as these three methods need, we introduce a new method to uncertainty estimation for the first time, namely, SCR (spherical cubature rule), which is no need for setting parameters. By theoretical derivation, we prove that the precision of uncertainty obtained by SCR can reach second-order. We conduct four synthetic experiments, for the first two experiments, the results obtained by SCR are consistent with the other three methods with optimal setting parameters, but SCR is easier to operate than other three methods, which verifies the superiority of SCR in calculating the uncertainty. For the third experiment, real-time calculation is required, so the MC is hardly feasible. For the forth experiment, the SCR is applied to the inversion of seismic fault parameter which is a common problem in geophysics, and we study the sensitivity of surface displacements to fault parameters with errors. Our results show that the uncertainty of the surface displacements is the magnitude of ±10 mm when the fault length contains a variance of 0.01 km².