# How is true altitude derived? [duplicate]

I read everywhere that the true altitude is computed as:

True Altitude = Indicated Altitude + (ISA Deviation × 4/1000 × Indicated Altitude)

But why is that the case? How is that derived? Where does that 4/1000 come from?

The exact formula to compute true altitude is:

$$h_{true} = h + \frac{h}{T_0}\cdot\left(T_\mathrm{OAT}-T_\mathrm{ISA}\right)\tag{1}$$

where you can see that the value usually represented by $$\frac{4}{1000}$$ is in fact $$\frac{1}{T_0}=\frac{1}{288.15}\approx0.00347$$. You can check how it's derived here

• It is not an exact formula. Taking 288.15 K is a cruder approximation, but taking one value is still an approximation. Jun 9 at 15:02
• For more see, e.g., The Hydrostatic Equation in the Evaluation Algorithm for Radiosonde Data Your link also correctly notes that it is not truly a "true" altitude. Jun 9 at 15:08
• I say it's exact because in order to derive it, you don't need to take any shortcut (approximations, Taylor expansions or similar). It can be derived directly from the barometric equation and the definition of the ISA atmosphere. But the ISA itself is an approximation so in that sense, you're right, it's not exact: despite the name, it doesn't really represent the true altitude.
– fab
Jun 9 at 16:24

It isn't. Indicated altitude is a function of just the outside pressure and the altimeter setting. The pressure at a given absolute altitude is given by:

$$P=P_0\exp\left(\int^{\rm altitude}_0-\frac{g}{R(h)T(h)}{\rm d}h\right)$$

where $$P_0$$ is the ground-level pressure, $$R$$ is the real gas constant for air (which depends on humidity) and $$T$$ is the at a given height AGL.

You don't have to evaluate this integral: the point here is that it depends on the temperature and humidity at every altitude between you and the ground. In other words, the relationship between pressure and absolute/true altitude depends on not only the local outside air temperature, but also the temperature lapse rate and the humidity at every level of the atmosphere below you.

The formula you've quoted is an approximation that works when the atmosphere is "kinda standard," i.e. when the temperature is non-standard but the temperature lapse rate and humidity are standard. It is done by evaluating an integral like this, given specific assumptions about the temperature profile, and then approximating the resultant equation as a straight line.

In short, it is impossible to compute your true or absolute altitude using only barometric information, outside air temperature, and altimeter setting. Having an altimeter setting and temperature as measured at a nearby airport helps a lot, but it isn't sufficient to determine the true altitude unless your plane is sitting on the ground at said airport.

• Yes this is the "truly true altitude", but it's impossible to compute using a barometer.
– fab
Jun 9 at 17:54
• "it is impossible to compute your true or absolute altitude using only information available in the plane.". Well, that's not entirely true. The GPS can give you that information.
– fab
Jun 9 at 18:07
• @fab Yes, that's my point. I've added a paragraph to make that more clear. Any formula picked using only local information can be dramatically wrong under unusual meteological conditions. Jun 9 at 18:07
• @fab Fair point. (Though there are a lot of weird subtleties to GPS-derived altitude). Is this wording better? Jun 9 at 18:09
• Yes, that's 100% correct now.
– fab
Jun 9 at 18:10