A positioning system or synthetic phase processor or rover station for providing high integrity positions with graduated accuracies. The positioning system includes one or more real time kinematic (RTK) reference stations for receiving GPS signals at established reference positions and measuring reference phases. The positioning system or the rover station selects a synthetic offset vector and uses the synthetic offset vector for inferring synthetic reference phases to a synthetic position. The rover station uses the synthetic reference phases with actual or virtual reference positions for determining a rover position having an added positional error controlled by the synthetic offset vector. For another embodiment a secure rover station dithers a secure position with a synthetic offset vector for providing an unsecure rover position having the added positional error.