Incrementally maintaining a qualitative understanding of physical system behavior based on observations is crucial to real-time process monitoring, diagnosis, and control. This paper describes the DATMI theory for dynamically maintaining a pinterp-space, a concise representation of the local and global interpretations consistent with observations over time. Each interpretation signifies an alternative path of states in a qualitative envisionment. DATMI can use domain-specific knowledge about state and transition probabilities to maintain the best working interpretation. By maintaining the space of alternative interpretations as well, DATMI avoids the need for extensive backtracking to handle incomplete or faulty data.