Not true, 95% of commercial aircraft have a TAWS system [0] that can show relative terrain on their navigation displays and provides FLTA alerts (forward looking terrain avoidance) by using an onboard terrain database. This is an FAA requirement for any aircraft with 6 or more passengers.
The terrain following radar feature that has been in fighter jets since the late 60s uses radar -- as the name implies, and serves a similar purpose. It allows the pilot to set an altitude and a "comfort" level regarding how aggressive the autopilot can be with regards to avoiding danger (basically, how quickly the plane can pull Gs to avoid terrain, and how many Gs it can pull).
That works well when you're straight and level, attentive, and the radar can point at the ground. This system can't rely on radar exclusively though because the aircraft may not have its radar pointing at the ground (as in the video, the aircraft is inverted in a pretty steep dive).
So, they have to factor in precision INS/GPS and known topology to assess terrain altitude in order to perform collision avoidance.
The F-16 system doesn't rely on radar, although it can use it, because fighters often fly with radar off. It tells the enemy you're coming. It's based on INS/GPS and a terrain database obtained from radar scans of the Earth made from the Space Shuttle in the 1990s.
Dumb question, maybe, but how long before that data becomes inaccurate? Or rather, are there any areas where the change in elevation for the purpose of this system could be big enough in a 30ish year timescale that it would cause problems?
I assume no geological process alters the land drastically enough, quickly enough, that you'd notice, but what about water-level changes (dammed rivers?), melting glaciers, etc? Is "hard" ground consistent enough that no human processes are going to cause the data to diverge from the database drastically without the chance to update the database with new topographical surveys?
Right, the Auto-GCAS feature doesn't rely on radar -- but the normal TFR system does, so the OP was half-correct in that there is a system that can use the radar to do ground collision avoidance... just not this particular system.
