| This paper includes in two sections: the first section is about the research and resolution of the parameters of the Earth's liquid core free nutation, and the second section involves in the research and detection of the earth free oscillation. In the first section, we will analyze the free and forced wobble of the earth on the base of the Eular equation of the earth's rotation, and discuss the earth liquid core resonance effect on the factors of diurnal gravity tides. Then, we will explain the observation principle of superconducting gravimeters and illustrate the harmonic analysis of gravity tidal data and the correction of ocean tide loading. At last, we will discuss the tri-frequency spectrum method and results for resolving the parameters of the earth's free core nutation (FCN) by employing the observational factors of diurnal gravity tides. In the second section, we will infer the ordinary differential equation group satisfied by the earth free oscillations (EFO), and then we will give the discussion on the numerical integration of ordinary differential equation groups of torsional and spheroidal oscillations, and apply the Rayleigh principle and perturbation theory to analyze the spectral splitting of EFO. Later we will advise some simple method of looking for the signals of the earth free oscillation and illustrate the pretreatment of the observational data of EFO. Finally we will explain the methods of spectral analysis adopted in this section, and research the spectral analysis results of detecting the earth free oscillation. The main research results gained in this paper are summed up in the following lines:(1) We introduce the tri-frequency spectrum method for the resolution of FCN parameters, this is a new try in this geophysical field. In the tri-frequency spectrum method (TFSM), both the observational amplitude-frequency parameter (OAFP) Awv(ob) and the observational phase-frequency parameter (OPFP) Bwv(ob) have directly geometrical meaning, and both the amplitude-frequency spectrum (AFS) Awv(ml) and the phase-frequency spectrum (PFS) Bwv(ml) have clearly geophysical meaning. And A is shielded in the process of resolving TFCN and Q in TFSM, these help us to analyze the relationships between the resolution results and the observational tidal data distinctly. At the same time, we introduce the extended range of Akp(ob) value (ER(Akp)) as the relatively objective criterion of checking observational tidal data to get rid of those unreasonable data before resolving the FCN parameters, and it is very important for us to improve the quality of resolving FCN parameters objectively and effectively.(2) By comparing the resolution results of FCN parameters obtained by the stacking method from gravity tides with those from VLBI, we can find that TFCN values coming from gravity tides were close to TFCN value from VLBI, but Q values given by gravimetric observation were very different from that by VLBI. So there did not exist an affirmative observation value of FCN parameters before. The resolution values of FCN parameters in the external tri-frequency spectrum method (ETFSM) are TFCN=422.13 ± 23.34(sidereal days) and Q=14781(Q>433.76), which are identical with the observation values of FCN parameters: TFCN=433.1 ±2(sidereal days) and Q=16130±6600 from VLBI. The computational results of inner tri-frequency spectrum method (ITFSM) are TFCN=424.5 ± 5.03(sidereal days) and Q=25993.97 ± 1510.64, which are in agreement with FCN parameters provided by ETFSM and are basically accordant with FCN parameters by VLBI. So the resolution results for FCNparameters in whole tri-frequency spectrum method (TFM) will be relatively significant for the determination of the objective observation value of FCN parameters.(3) In the pretreatment of signals of the earth free oscillations (EFO), we don't follow the digital filters that the former scholars often applied to get rid of gravity tides, but introduce a kind of polynomial fitting technique fragmentally to remove gravity tides...
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