## 5 Discussion of the Results and Conclusions  
In this study, we used SBDD to accurately compute the GNSS measurement residuals, and proposed machine learning models, namely tree and Gaussian SVM regression, to effectively model the nominal values of the residuals under fault-free conditions. We evaluated the effectiveness of our suggested residual estimation and modeling method in the light of anomaly monitoring, and compared it to several common residual estimation methods. The assessment validated that the proposed methodology was more accurate than other existing models in estimating the residuals affected by both receiver noise and a multipath environment. Afterwards, the proposed model was used to normalize the training data and compute the IF to over-bound the fault-free measurements. Finally, uncertainty level multipliers were applied to derive the residual thresholds required to satisfy the required system false alarm probability.  
The results of our study demonstrate that the residuals obtained from the test data were successfully bound by the thresholds derived from the trained model. Although the threshold was determined conservatively, the observed false alarm rate was less than but similar in magnitude to the theoretically expected rate, suggesting that the threshold was sufficient for the probability of a false alarm but not too loose for the probability of misdetection. This validates the effectiveness of the proposed GNSS carrier phase residual monitor for enhancing GNSS operation integrity. This monitoring architecture can also be used for code and doppler measurements. The implementation for code measurements would be identical, with the exception of the integer ambiguity. For doppler measurements, neither the integer ambiguities nor the geometric range terms need to be considered.  
While our study primarily focused on modeling residuals under fault-free conditions and conducting a feasibility test of the proposed algorithm, we have future plans to engage in extensive experimental validation across multiple sites over an extended duration. Moreover, our future work will involve investigating methods for detecting faults using the developed model. Specifically, we aim to study the detection of usual fault events such as cycle slips and a multipath at a reference station. By considering the geometric correlation between the two antennas, we hope to estimate the minimum detectable error that could facilitate effective detection and exclusion of multipath effects, which is one of the most unsolved problems.  
All the authors have contributed to the presented work. The first author, [PERSON], suggested and embodied the methodology to correctly model the GNSS measurements and monitor them. Another first author, [PERSON], conceptualized the basic concept and implemented the algorithm. [PERSON] suggested the original concept of the system and supervised its development and the direction of the research. All authors participated in formulating the idea and in discussing the proposed approach and results. All authors have read and agreed to the published version of the manuscript.  
This research was supported with a grant from the National R&D Project \"Development of ground-based Centimeter-level maritime precise PNT technologies\" funded by the Ministry of Oceans and Fisheries (1525013759).  
The data presented in this study are not publicly available due to privacy restrictions.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.