Critical issues for the integrity monitoring of any autonomous navigation applications include adequate uncertainty budget in the observation domain, the redundancy for the determination of the navigational states, as well as the capability of fault detection and exclusion (FDE). Purely statistical evaluation of integrity will not always be adequate due to the presence of remaining systematic uncertainty. This paper aims to contribute to the alternative integrity approach based on interval and set-representations for bounding and propagating system uncertainty. Several key aspects are addressed: how to determine interval bounds to handle GNSS multipath effect in urban environment, how to realize fault detection and exclusion based on constraint satisfaction and set-membership, and how to improve the detector using weighting models. Simulated and real-world experiments are carried out to demonstrate the feasibility of proposed methods. In this way, the alternative integrity approach will contribute to future autonomous navigation applications.