## 1 Introduction
TTL correction for LRI is not provided in the GRACE-FO RL04 Level-1B data product [13], though the nadir and cross-track components of the TMA VP positions can be estimated by the so-called center-of-mass calibration (CMC) maneuvers whose initial purpose is to determine the offset between the CM and the center of mass of the ACC proof mass for each satellite [14]. For each CMC maneuver, a periodic rotation around the desired axis is performed on one satellite by activating the corresponding magnetic torque rods. Several attempts have been made to improve the TTL coupling estimation results for KBR and LRI. In [15], additional geometric biases in terms of the pitch and yaw angles of the CM-to-APC vectors were estimated in the process of gravity field retrieval by treating  
Figure 1: Overview of LRI optical components. The master satellite is named SAT1 and the transponder satellite is named SAT2. Each satellite includes at least a laser, a cavity, a fast steering mirror (FSM), a beam splitter (BS), a compensation plate (CP), and a triple mirror assembly (TMA). The vertex point (VP), an intersection point of three mirror planes of the TMA for each satellite, is nominally located at the center of mass (CM) of the satellite in the case of GRACE-FO. them as estimable parameters together with other customary parameters. The time-variable gravity field solutions were significantly improved after adjustment of the geometric biases. Unlike the KBR case, a band-pass filter covering the rotation frequency of CMC maneuvers (83.3 mHz) is directly applied to the LRI range and the pointing angles [10; 14]. In this way, the low-frequency orbital effects can be successfully removed without applying the POD results and polynomial fitting. The reason for not using the POD results is that the LRI TTL errors generated by the CMC maneuvers are much smaller due to the small offset between the CM and VP, but are still higher than the noise level thanks to the low noise ranging measurement of LRI at a Fourier frequency above 40 mHz, which allows parameter estimation at high frequencies. However, the duration of a CMC maneuver is only 180 s, generating relatively little data, part of which is not valid for parameter estimation because of the thruster events on the other satellite [14]. Considering the fact that the TTL error exists in the entire range data since launch, data from calibration maneuvers in a certain restricted time span may be a disadvantage for parameter estimation. In this paper, an alternative approach to TTL coupling estimation for LRI, using the range data without any specific calibration maneuvers for the TTL error, is investigated. This approach does not interrupt the laser link and allows data with longer periods to be included for parameter estimation. Only the TTL coupling parameters are estimated, which is different from the method in [15] in which various types of parameters are estimated in the same process where the parameters may interact with each other. Since the laser ranging interferometer has been adopted as the main satellite-to-satellite tracking instrument for future gravity missions, such as the Next Generation Gravity Mission (NGGM) [16; 17] and the GRACE-ICARUS (GRACE-I) project [18], the KBR case is not discussed in the following. In Section 2, the working principle of LRI is briefly introduced. In Section 3, we discuss the two-way ranging (TWR) formulations for LRI, which represents preparation for the TTL model and the numerical simulation. The basic TTL model for parameter estimation is derived in Section 4. The topic of Section 5 is the numerical simulation process in which five types of data are generated. The TTL estimation results and discussion are given in Section 6. In Section 7, the conclusions are drawn.