## 1 Introduction  
Tightly-coupled inertial navigation system/global navigation satellite system (INS/GNSS) integration systems are an attractive positioning option in many navigation service applications [1, 2]. Although considerable studies have been conducted to improve the performance or reduce the computational burden, the algorithms of the optimal adaptive filtering and satellite selection are still not theoretically and practically perfect and warrant further investigations. A tightly-coupled system uses the GNSS pseudo-range and pseudo-range rate measurements as reference to evaluate and correct the INS error [3]. In practice, in order to cope with the unstable measurement noise covariance \(\mathbf{R}\), caused by the instabilities of the environment and the receiver [4, 5], the adaptive Kalman filter (AKF) which can estimate \(\mathbf{R}\) online should be utilized to guarantee the navigation accuracy [6, 7, 8]. The innovation and residual are the most commonly used information to estimate \(\mathbf{R}\) adaptively, and the corresponding algorithms are known as innovation-based adaptive estimation (IAE) [9] and residual-based adaptive estimation (RAE) [10]. Nevertheless, the innovation and residue are coupled with the state estimation error, which can affect the accuracy of **R** and filtering or even cause divergence [11], especially in a biased state estimation situation. Besides, abnormal measurements arise in areas like urban and canyons [12]. In these areas, GNSS suffers from large errors due to the multipath, poor geometry and high noise. [PERSON] [13] proposed a novel adaptive filter by assuming that the measurement errors follow a heavy-tailed distribution. The user acceleration derived from GNSS Doppler measurements and the direct output of inertial measurement unit (IMU) are compared to generate a scalar to adjust **R**. [PERSON] [14; 15] introduced the robust estimation technique to INS/GNSS tightly-coupled systems to identify and reject aberrant measurements. Unfortunately, these algorithms are not theoretically quantitative and will be also affected by inaccurate state estimates such as IAE/RAE. Satellite selection is an important element to guarantee positioning accuracy in INS/GNSS tightly-coupled systems. Geometric dilution of precision (GDOP) [16; 17; 18; 19; 20; 21], signal to noise ratio (SNR) [22; 23] and carrier to noise ratio (CNR) [24; 25] are standard indexes utilized to evaluate the positioning accuracy from the views of geometry constraints and signal quality. However, the GNSS measurement precision level is a crucial factor to be considered in satellite selection and has not been well studied due to a lack of effective approaches. In current GDOP- and SNR/CNR-based satellite selection algorithms, the satellites which have accurate measurement can be excluded due to low SNR/CNR or high GDOP and vice versa, the satellite suffering from large measurement error can possibly be taken into consideration to derive the navigation solution for a high SNR/CNR or low GDOP. Hence, the satellite selection algorithm faces the drawback that it may involve improper satellites or reject a suitable satellite and as such lead to negative effects on the solution. The satellite elevation angle is another factor that affects the satellite selection. The satellite elevation-dependent weight is adopted for the a priori variance for GNSS observations [26], which is based on the assumption that a lower elevation angle introduces higher measurement noise due to its increased possibility of multipath delay. However, the deficiency of lacking a real evaluation of measurement noise still exists in this approach. To overcome the aforementioned limitations of these existing approaches, the present work proposes a novel redundant measurement noise covariance estimation (RMNCE) approach, which employs redundant information obtained from two independent measurement systems [27] to estimate their corresponding measurement noise properties.