## 8 Conclusions  
Non-constant systematic TTL error is commonly seen in low-low satellite-to-satellite tracking missions like GRACE and GRACE-FO, and TTL estimation methods with certain calibration maneuvers have previously been investigated in both the KBR and LRI cases. In this paper, an alternative approach for LRI TTL estimation that overcomes the limitation of the time span of the calibration maneuvers has been presented. We introduced the basic working principle of LRI and derived two-way ranging formulations for LRI based on the dual one-way ranging formulations for KBR. The TTL model where the angle biases are taken into account was derived for parameter estimation, requiring data for satellite orbits, attitudes, and non-gravitational accelerations. Therefore, a numerical simulation was performed to generate the simulated GNI1B, SCA1B, ACC1B, LRI1B, and LSM1B data. Cases of S-type and L-type longitudinal offsets were discussed separately and no significant difference was found under low angle noise conditions between the two cases. By comparing the estimation results of the TTL error and the coupling factors, it was found that the impact of the pointing angle noise was more apparent than the impact of the lateral offset and the laser frequency noise. When the pointing angle noise is below \(0.3\,\mathrm{\SIUnitSymbolMicro rad}/\mathrm{Hz}^{1/2}\), the RMS of the TTL correction error will be lower than 4 nm in the cases of both S-type and L-type offsets, which is better than the current TTL level of GRACE-FO.  
For future gravity missions with LRI, it is possible to perform pointing angle measurements at higher precision with advanced attitude sensors and, thus, the approach for TTL estimation introduced in this paper may be implemented. Since the TTL estimation error is not limited by the longitudinal offset, a redundant TMA can be mounted far away from the center of mass and its size can be reduced. Moreover, the daily estimated results of the coupling factors may be used for estimation of the center of mass movement.  
Conceptualization, Z.W. and H.D.; methodology, Z.W.; software, Z.W.; validation, S.Y. and F.L.; formal analysis, Z.W., S.Y., F.J. and K.W.; investigation, Z.W.; resources, H.D.; data curation, Z.W.; writing--original draft preparation, Z.W.; writing--review and editing, H.D. and H.-C.Y.; visualization, Z.W.; supervision, H.D.; project administration, H.-C.Y.; funding acquisition, H.D. and H.-C.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key R&D Program of China (No. 2020 YFC220020 and No. 2022 YFC2204001) and the National Natural Science Foundation of China (NSFC) (No. 12005310).  
The official GRACE-FO Level-1B data products can be found at [[ftp://sdcftp.gfz-potsdam.de/grace-fo/](ftp://sdcftp.gfz-potsdam.de/grace-fo/)]([ftp://sdcftp.gfz-potsdam.de/grace-fo/](ftp://sdcftp.gfz-potsdam.de/grace-fo/)) (accessed on 12 August 2023). The data generated in this study are not publicly available due to [its use in an ongoing study by the authors] but can be made available from the corresponding author upon reasonable request.  
The authors would like to thank the LRI team at the Albert Einstein Institute for providing the GRACE-FO LRI1B (v50-v52, v54-v55) data.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
ACC  
Accelerometer  
AEI  
Albert Einstein Institute  
AF  
Accelerometer frame