## 2 Methodology of DLR Measurements
### Telescope Array and Multi-Static Laser Ranging Measurements  
In laser ranging measurements, the laser echoes cover a certain area on the ground, and they can thus be received and detected by an array of several or more telescopes on the ground, which causes increases in the receiving area. These telescopes receive and detect laser echo signals at the same time, significantly increasing the number of laser echoes received by the ground station system per unit time, which is equivalent to improving the receiving capability of a single large-aperture telescope, as shown in Figure 13. The main telescope in the array actively emits laser signals to the target, whereas all the telescopes of the array receive the echo signals. Figure 12: DLR measurements of a target at a distance exceeding 6700 km (catalog ID 28118). Figure 13: The receiving and measurement of laser echo signals by a telescope array. The detection probability and receiving capability of a ranging system consisting of multiple telescopes receiving signals has been studied using dual telescopes that have apertures of 1.56 m and 60 cm and are separated by 60 m to receive laser echo signals [24; 25]. The 60 cm telescope emitted a laser signal to the targets and the two telescopes received an echo signal at the same time. Using the clock of a global navigation satellite system to solve the time synchronization of the two separate telescope systems and considering the actual transmission path of a laser signal, the range gate control of the remote receiving telescope has been realized for laser ranging measurements. Figure 14 illustrates the multi-static measurements of a telescope array. The range gate value is \(\Delta\tau_{0}=\tau_{u0}+\tau_{d0}\) for the main telescope (#0), \(\Delta\tau_{1}=\tau_{u1}+\tau_{d1}\) for the fellow receiving following telescope (#1), \(\Delta\tau_{2}=\tau_{u2}+\tau_{d2}\) for the fellow receiving telescope (#2), respectively. Considering the negligible differences in \(\tau_{u0}\), \(\tau_{u1}\), and \(\tau_{u2}\) for the same target, the range gate values can be calculated based on the laser-signal uplink time from the main telescope and downlink time to the respective following telescopes. Figure 15 shows the real-time combination of laser range residual data recorded by the 1.56 m and 60 cm telescopes, respectively, which increases the total number of laser echoes. Figure 16 shows the DLR measurements for the 1.56 m and 60 cm telescopes, respectively. Compared with the telescope having an aperture of 60 cm alone, the total number of laser echoes received by the two telescopes together has increased by a factor of 4-5. Figure 14: Multi-static laser-ranging measurements of a telescope array. Figure 15: Real-time combination of laser data from the 1.56 m and 60 cm telescopes. Consequently, the receiving ability of the two telescopes is equivalent to that of a telescope system with an aperture of approximately 1.61 m. This result verifies the feasibility and technical advantages of multiple telescopes receiving signals. SHAO has two laser-ranging telescopes with a receiving aperture of 60 cm, separated by 2.5 km, each having a receiving aperture of 60 cm [26]. The main telescope system located at the top of Sheshan Mountain has a laser emitter and receives echo signals at the same time. The fellow telescope located in a science park only receives laser echo signals. DLR measurements were successfully achieved using the dual telescopes. The measurement results of the debris targets given in Table 2 show that the measuring precision of laser ranging of the dual telescopes is comparable. It is concluded for this comparability that the precise control of range gate synchronization is available for the remote receiving telescope system and the detection of debris laser signals can be improved by a long-distance multi-static telescope laser ranging system.