Like the Future Air Navigation System (FANS), the Performance-Based Navigation (PBN) system is a concept of how we ensure our aircraft are where we say they are, when we say they are. The designation is mostly consistent, but not completely. How you qualify and how you prove your qualification is pretty much up to the state. But that approval isn't listed as "PBN" but as the individual levels of compliance, i.e., RNP-4.
Everything here is from the references shown below, with a few comments in an alternate color.
Figure: Lateral Navigation Errors, from ICAO Doc 9613, figure ii-a-2-1.
[ICAO Doc 9613, p. I-(iii)]
This isn't to say PBN systems cannot use a particular type of sensor, only that the sensor isn't the defining characteristic of the specification. An acceptable solution could be GPS-based or even DME/DME-based. But approval for that solution for the specification depends on more than just what sensor is used.
[ICAO Doc 9613, ¶1.2.] The navigation specification is used by a State as a basis for the development of their material for airworthiness and operational approval. A navigation specification details the performance required of the RNAV system in terms of accuracy, integrity, availability and continuity; which navigation functionalities the RNAV system must have; which navigation sensors must be integrated into the RNAV system; and which requirements are placed on the flight crew.
[ICAO Doc 9613, ¶126.96.36.199] On-board performance monitoring and alerting is the main element that determines if the navigation system complies with the necessary safety level associated to an RNP application; it relates to both lateral and longitudinal navigation performance; and it allows the aircrew to detect that the navigation system is not achieving, or cannot guarantee with 10–5 integrity, the navigation performance required for the operation.
The 10-5 integrity is the 0.9999 probability concept. The monitoring and alerting is the key component of a PBN specification.
[ICAO Doc 9613, ¶188.8.131.52] Both RNAV and RNP specifications include requirements for certain navigation functionalities. At the basic level, these functional requirements may include:
[ICAO Doc 9613, ¶1.2.5]
This gets messy, here are two examples:
[ICAO Doc 9613, ¶3.4.2]
You do not get approved for PBN, rather you get approved for each specification. Details here:
ICAO Doc 9613 - Performance Based Navigation (PBN) Manual, International Civil Aviation Organization, Fourth Edition, 2013
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