## 7 Discussion
The blue and magenta traces in Figure 15a are flat over a wide range of lateral offsets due to the limitation of the coupling factor error in the last two terms of Equation (54), while the flat part of the green trace is narrower, and the red trace shows no flat parts since it is dominated by the first two terms of Equation (54), which is proportional to the pointing angle noise and the lateral offset. The flat parts of the blue and magenta traces in Figure 15b are mainly limited by the coupling factor error, which is not induced by the laser frequency noise but other error sources, such as the accelerometer noise, while the red and green ones are limited by the pointing angle noise. As the lateral offsets grow, the slopes of the traces in Figure 15a (e.g., the red, blue, and magenta traces) will no longer be an approximate zero value since the first two terms, which are proportional to the lateral offsets, start to dominate the estimation error, while the coupling factor error in the last two terms does not vary a lot. In contrast, the rising parts of the traces in Figure 15b (e.g., the blue and magenta traces) are due to the increasing coupling factor error in the last two terms caused by the increasing laser ranging noise. Figure 14: RMS values of estimation error under different pointing angle noise. Figures 14 and 15 do not show much difference between the results of S-type and L-type offsets except for the red and green traces in Figure 15a. The pointing angles used in TTL estimation contain biases of several hundreds of \(\upmu\)rad, inducing additional linear coupling terms in the case of the L-type offset. Since the input angle biases were different from each other, varying from zero to several hundreds of microns, even if the input lateral offsets were set to zeros, the coefficients of the additional linear coupling terms would behave like the lateral offsets and contribute to the coupling factors \(p_{y,i}\) and \(p_{z,i}\) in the first two terms of Equation (54), which is responsible for the flat part of the green trace. The additional linear terms in the case of the S-type offset can be omitted so the red trace behaves normally. In addition, the slope of the rising part of the green trace is slightly larger than 1, mainly due to the non-negligible terms \(\theta_{y,i}p_{x,i}\) and \(\theta_{z,i}p_{x,i}\) in Equation (54), where the coupling factor \(p_{x,i}\) is up to \(1.5\,\mathrm{m}/\,\mathrm{rad}^{2}\). The influence of these two terms will be much smaller in the case of the S-type offset; thus, the slope of the rising part of the red trace is almost 1. Assuming that the CM-VP lateral offset will not exceed 0.5 mm and the performance of the laser frequency stability will be the same as GRACE-FO, the main limitation for the investigated approach to TTL estimation is the pointing angle noise. One drawback of this research is that the input attitude data were obtained from the official SCA1B data of GRACE-FO, which may deviate the actual cases in future gravity missions. The actual pointing angles at the passband may be lower, mandating stronger restrictions on the accuracy of angle measurement. All in all, the pointing angle noise level is absolutely the key to successful TTL estimation in this study. However, the official LSM1B data product of GRACE-FO only provides angle measurement with a noise level of a few \(\upmu\mathrm{rad}/\mathrm{Hz}^{1/2}\), which is not precise enough for the estimation method in this paper. Therefore, attitude sensors with higher precision, such as an advanced position sensor system for the FSM based on the one in GRACE-FO or an interferometer readout system for FSM actuation angles, may be implemented in future gravity missions. The accelerometer noise model in this study plays a similar role as the laser frequency noise since the ACC noise enters when subtracting the differential non-gravitational range, and we found the contributions of the ACC noise and the laser frequency noise to the ranging data in the passband are at the similar level.