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Transition altitude refers to the altitude above which all altimeters are set to a standard pressure setting (typically 29.92 inHg) instead of to the local surface pressure, and altitudes are given in flight levels (the standard-pressure altitude, divided by 100; for instance, with one’s altimeter set to standard pressure and reading 52,900 feet, one would say “flight level 529” rather than “52,900 feet”) rather than in feet above mean sea level (MSL). The height of the transition altitude depends on where you’re flying; for instance, transition altitude is at 18 kft in the United States, 10 kft in Australia, and as low as 6 kft or even lower in parts of Europe.

Low transition altitudes raise the possibility of an airport’s height above sea level being close to, or even above, the transition altitude; this wouldn’t be an issue in the United States (where only the very highest one or two mountain peaks reach above the 18-kft transition altitude, and this altitude is well into the Death Zone anyways unless one is exceptionally well-acclimated to very high altitudes) or Australia (where even the highest peaks are less than 8 kft above MSL, well below the 10-kft transition altitude), but could easily become a problem in regions with both high terrain and low transition altitudes (such as Europe, which has many high mountains - and, more to the point, many airports high up in mountainous areas - combined with transition altitudes that go at least as far down as 6 kft), as maintaining terrain clearance requires knowing what one’s altitude actually is, rather than knowing what altitude the pressure at the static ports would correspond to if the local altimeter setting was 29.92 inHg.

How do airports located at elevations near to, or above, transition altitude handle this? Does the airspace around these airports use a local transition altitude higher than elsewhere? Do approaches and landings at, and takeoffs from, these airports rely solely on radar, rather than barometric, altimeters? Something else?

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I don't have an authoritative source for what the various regulatory agencies in the various regions use for publishing their transition altitudes, but your assumption is pretty close: airspace is planned so that no airport is "that close" to the transition level.

Some TAs are defined across large swaths, such as the US, Canadian, Australian TAs. Some TAs are more broken up, even within political boundaries. I believe Brazil (or at least one of the South American countries) has different TAs for different "terminal areas".

There has been on-again/off-again discussions in the European region of establishing a common TA, but (as of this date) that has not happened. You will see fairly low TAs in low, flat areas like The Netherlands, but higher TAs in the mountainous regions.

Again, all of these TAs are "planned out" so that there is no danger of being too low while assigned to a flight level, and so that you are back on QNH well before you are on approach to an airport.

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