## 1 Introduction  
As the high-speed railway network becomes denser and the number of trains running on the network increases, the safety of high-speed trains has become an issue of great concern [1, 2]. There is a requirement for real-time, continuous, accurate, and reliable navigation information for the operation control system to control, schedule, and monitor train movements [1, 3, 4]. In tunnels, the GNSS signal is lost and there is a lack of other available information, which presents a challenge to these navigation systems [4]. The issue has drawn particular attention from academics, with both China and the European community funding related research projects [5]. Traditional train positioning uses speed and distance measurement methods such as track circuits, transponders, axe-counting equipment, and Doppler radar to unify the speed and location of the moving train under one-dimensional coordinates [4].Currently, the positioning accuracy of distances required by the European Train Control System (ETCS) and the Chinese Train Control System (CTCS), based on balise groups placed on the track and odometer readings, is \(\pm\) 5 m \(+\) 5% measured distance [5]. However, there has been a gradual trend towards the reduction of expensive and complex trackside equipment in train positioning technology development [5]. Traditional methods are associated with poor positioning accuracy, less train information, complicated system maintenance, and inability to meet the requirements of high-speed train positioning in long tunnels [6]. Today, the most commonly used train navigation sensors are GNSS, INS, camera, Doppler speedometer, barometric altimeter, odometer, lidar, electronic maps, and specially designed equipment [6]. However, none of these sensors can independently meet the requirements of navigation accuracy in all environments, and multisensor combined navigation that can integrate their respective advantages has become the development trend of train navigation [7; 8]. Due to complementary nature of GNSS and INS, they form the basis of integrated train navigation systems. However, in long tunnels where there is no GNSS signal, it is difficult to ensure navigation accuracy by relying only on the INS composed of MEMS devices [5; 9; 10]. In recent years, many scholars have conducted research on train navigation in tunnels. Some scholars have proposed methods for establishing reference information in tunnels. [PERSON] proposed a received signal strength indication (RSSI) positioning method based on ZigBee technology [11], but the error increased significantly as the speed increased. [PERSON] conducted an INS and radio frequency identification (RFID) integrated navigation simulation experiment and achieved an astonishing 1.5 m accuracy over a 500 s GNSS outage [12]. However, it is difficult to provide precise locations for RFID tags in long tunnels, and the high speed of trains is also a challenge to RFID [13]. [PERSON] proposed a method of combining INS and map matching (MM) [4], but did not report the positioning accuracy during GNSS outage. It is also uncertain whether MM can suppress the error of INS. Establishing a high-precision digital map is difficult in a long tunnel. Some studies using multisensor information fusion methods have achieved meter-level accuracy in tunnels [4; 5; 10; 14]. However, these systems are very complicated and costly, and there are cumbersome installation and calibration processes for sensors. Some scholars have proposed methods based on track and train motion constraint information. [PERSON] proposed a method based on a train motion model and track constraints [15], which had poor accuracy. [PERSON] used a combination of a train motion model and a gyroscope to locate trains in tunnels [16], which produced a horizontal position error of 31 mover 10 s and a large position error. [PERSON] used train motion constraints to suppress the INS error [17], which significantly improved the accuracy of train navigation.