## 3 Results
### Gravity Field Solutions  
#### 3.3.1 Solutions estimated from 71 days of observations  
We estimated four satellite gravity field models up to d/o 130 using the point-wise acceleration method from the accelerations derived in the previous section. The EIGEN-6C4 model up to d/o 130 was used as the reference model to validate the estimated gravity field models. This model was characterized by a higher accuracy because it was constructed by a combination of multi-source gravity data such as GOCE, GRACE, and LAGEOS satellite observations, satellite altimetry, and ground gravity data [42]. Note that the empirical variance-covariance matrices used here were constructed directly by using residual accelerations.  
From the perspective of gravity spectroscopy, Figure 3 depicts the residual spherical harmonic spectra with a log10 scale of the estimated solutions when compared with the EIGEN-6C4 model. According to Figure 3, the distribution of all of the residual harmonic coefficient spectra were similar. The noise in the spherical harmonic coefficients increased with the spherical harmonic degree and the sectorial and near sectorial harmonic coefficients had better accuracy, which is an inherent feature of the SST-hl observations used for gravity field modeling [29]. The near sectorial harmonic coefficients based on the Graz-orbit differentiated accelerations performed best, and the accuracy of the zonal and near zonal coefficients was worse. Furthermore, the high degree harmonic coefficients estimated by the carrier phase method performed the best.  
To assess the performance of the recovered gravity field models in the spectral domain, the Degree-Error Root Mean Square (DE-RMS) values and cumulative geoid height differences (\(\sigma_{N_{n}}^{(C)}\)) were computed according to the following formula [18]:  
\[DE-RMS_{n}=\sqrt{\frac{1}{2n-8}\sum_{m=5}^{n}\left[\left(\overline{C}_{nm}^{ est}-\overline{C}_{nm}^{ref}\right)^{2}+\left(\overline{S}_{nm}^{est}-\overline{S}_{ nm}^{ref}\right)^{2}\right]} \tag{16}\]  
Figure 3: Spherical harmonics triangle of the estimated geopotential coefficients when compared with EIGEN-6C4.  
\[\sigma_{N_{n}}^{(C)}=R\sqrt{\sum_{n}\frac{1}{2n-8}\sum_{m=5}^{n}\left[\left( \overline{C}_{nm}^{est}-\overline{C}_{nm}^{ref}\right)^{2}+\left(\overline{S}_{ nm}^{est}-\overline{S}_{nm}^{ref}\right)^{2}\right]} \tag{17}\]  
where \(\left(\overline{C}_{nm}^{est},\overline{S}_{nm}^{est}\right)\) are the estimated spherical harmonic coefficients, and \(\left(\overline{C}_{nm}^{ref},\overline{S}_{nm}^{ref}\right)\) denote the coefficients of the reference gravity field model; \(R\) denotes the Earth's mean radius. Additionally, to account for the polar gap problem, neglecting m \(<\) 5 is applicable for the computation of the DE-RMS and cumulative geoid height differences.  
Figure 4 shows the DE-RMS of the four recovered models. Table 5 presents the cumulative geoid height differences for the degrees from 10 to 130. From Figure 4 and Table 5, for the whole frequency band of the recovered model, the carrier phase differentiation method had a comparable performance with the orbit differentiation method. The cumulative geoid height difference of the carrier phase solution up to 130 was the smallest, which was 37.02 cm. The geoid height differences of the recovered gravity field solutions are shown in Table 6, and their spatial distributions are shown in Figure 5. We applied a 500-