## 4 Analysis of Field Test Results
### Machine Learning Residual Estimation  
The machine learning models for the HPF and LPF have multivariable inputs from both satellites. However, to visualize the models in this section, the parameters for both satellites were set to be identical, and the C/No was fixed at 45 dB-Hz. The results for both the HPF and LPF models are presented in Figure 7. As expected, the modeled residuals are higher at lower elevations, and the residuals in the presence of a multipath increase as well. There are empty spaces in the northern part of the skyplot because no satellites are visible from the ground near 0\({}^{\circ}\) azimuth in the northern hemisphere. Additionally, there are no signals for elevations below 15\({}^{\circ}\) and azimuth between 135\({}^{\circ}\) and 180\({}^{\circ}\) due to obstruction  
Figure 5: Short-baseline double difference residuals after the high-pass filter has split the raw residuals into noise (HPF) and multipath components (LPF). Figure 6: Elevation, azimuth, and C/No corresponding to the satellites used for the residual computation. from a nearby concrete structure, which is shown in the environment overlay provided in Figure 8. We can see that the visibility limitations coincide with the presence of the building, which blocks up to 14.5\({}^{\circ}\) elevation when surveyed using a geodetic GNSS receiver on top of the structure. Between 180\({}^{\circ}\) and 210\({}^{\circ}\) azimuths, the floor of the building rooftop consists of metal, leading to the effects of a multipath at low elevations. From the skylpot and the two-dimensional (2D) projections in Figure 8, we can see that the combined model is successful in estimating the residuals while accounting for the elevation-dependent receiver noise and effects from an environmental multipath. Figure 8: Visualization of the combined modeled data. (**A**) Skylpot of the modeled data with the surrounding environment overlayed. (**B**) Two-dimensional visualization of the modeled data. There are empty spaces where signal is not expected under nominal circumstances due to the location of the receiver and its surroundings. Figure 7: Skyplot visualization of the machine learning models. (**A**) represents the HPF with residuals reaching up to 2.1 mm at low elevations. (**B**) shows the LPF with residuals reaching up to 24 mm due to the presence of a multipath. We assessed the effectiveness of various methodologies in estimating the observed GNSS residuals for anomaly monitoring by comparing the statistical resemblance of each normalized distribution to the standard normal distribution. The models considered were the proposed algorithm that incorporates SBDD and machine learning, ZBDD, elevation-based models (EL), and C/No-based models (CN0). The normalized residuals (\(Z\)) for the proposed algorithm can be calculated using the following equation:  
\[Z=\frac{res_{HPF}+res_{LPF}}{\sqrt{(\sigma_{HPF}^{2}+\sigma_{LPF}^{2})}} \tag{22}\]  
where \(res_{HPF}\) and \(res_{LPF}\) represent the results obtained from a high-pass filter and a low-pass filter, respectively. These filter results are equivalent to receiver noise and a multipath effect. The values \(\sigma_{HPF}\) and \(\sigma_{LPF}\) are sigma values of the noise and multipath for each satellite parameter, which are modeled using the training dataset as described in Section 3.1. The validation was conducted by assessing how well each method modeled the residuals observed for the 1-day training data on 21 July 2020. To evaluate the modeling performance statistically, we normalized the residuals observed from the training data with the residuals modeled by each method. Then, to find the best match, we compared the percentage of normalized samples within the \(1\sigma\) and \(2\sigma\) bounds. This approach is chosen instead of comparing the shapes or other portions of the distribution because most GNSS fault monitoring methodologies define requirements and performances in multiples of \(\sigma\)[51]. Figure 9 displays the normalized residuals of the training data using various modeling approaches.