## 1 Introduction  
Global Navigation Satellite Systems (GNSS) provide position, navigation, and time (PNT) information to users all around the world. The concept of GNSS is based on processing known GNSS satellite positions and times with the distances between the satellites and the user to compute estimated user position, velocity, and time (PVT). There are two major types of GNSS positioning: Single Point Positioning (SPP) and Differential Positioning (DP). The standalone method uses only the GNSS measurement observations available to the receiver, while the Differential GNSS (DGNSS) method uses additional information from nearby monitoring stations. Types of information provided by the monitoring stations include GNSS measurements, and the corresponding integrity information. In case of GPS, using the uncorrected measurements results in an expected horizontal accuracy of an over 5 m root mean square error (RMSE), while the DGNSS method provides an expected accuracy of a 1 m RMSE [1; 2; 3; 4]. Although the DGNSS has the potential to provide improved accuracy and integrity PVT, the system accuracy and integrity are dependent on how well the monitoring stations can assess the quality of the measurements and detect anomaliessuch as cycle slips, a non-pre-surveyed multipath, or a signal blockage prior to reporting them to the user [5]. Therefore, all the GNSS measurements must be monitored for faults in real time to ensure safe and reliable navigation for various applications including autonomous ground vehicles and unmanned aerial vehicles (UAV) [6; 7]. To monitor the validity of the carrier phase measurement, modeling the residual under a normal condition should be preceded so that thresholds for the nominal expected residual can be defined to monitor outliers. One form of residual modeling is using the theoretically expected carrier phase noise from the third-order phase-lock loop (PLL) used in the carrier phase tracking [8]. Although it is a theoretical and straightforward way for the modeling, it has the limitation of not accounting for other normal errors, such as a tolerable multipath under typical signal reception environments, i.e., open-sky, sub-urban, and urban [9]. Another usual way to estimate the expected normal residuals is using the elevation angle of the satellites. The elevation angle is generally inversely proportional to the carrier phase noise due to an increased path length between the satellites and the receiver [10; 11]. Even a multipath error under a typical condition can also be roughly modeled with the elevation angles [12; 13]. However, the error of each satellite cannot be properly modeled according to the azimuth because it is based on the assumption that a noise and multipath error have the same magnitude at the same elevation angle. It cannot describe a non-uniform multipath error for the same elevation angle due to the site visibility-related effect. There also exist several methodologies for modeling and defining the thresholds based on linear or differential combination of GNSS measurements. First, there are code carrier divergence tests that monitor the variations in the code measurement noise and ionosphere [14]. Although this method is effective for monitoring code measurement quality and the health status, carrier measurement noise should be assumed to be negligible without any outlier residuals, so carrier monitoring is difficult based on the code carrier divergence test. Carrier residual computation methodologies including the zero-baseline double difference (ZBDD), carrier detrending, and short-baseline double difference (SBDD) can be utilized for monitoring [15]. All of these differential methods are used to detrend the geometry terms between a satellite and a user in the carrier phase measurements. ZBDD detrends the geometry term but also removes common noise terms and even a multipath present between the satellite and the antenna splitter and accordingly underestimates the noise present in the signals.