How can I compute indicated altitude?

Question

I would like to compute what an altimeter would display given a true altitude and atmospheric conditions. Specifically:

Given:

• True altitude, in feet
• Outside air temperature, in degrees C
• Local altimeter setting, in inHg (e.g. from nearest metar)
• Kollsman setting, in inHg (which may or may not be set correctly to the nearest metar)

I would like to compute indicated altitude. If it simplifies things, assume I only care about altitudes below FL300.

I've tried applying the formula from here in reverse:

$$A_I = \dfrac{1000 \cdot A_T}{4 \cdot \Delta_{ISA}+1000}$$

but I'm not sure if this accurately represents how altimeters work or if it is meant to be an approximation. Also, it does not take into account the difference between Kollsman setting and local altimeter setting.

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I suppose this question can be split into two separate parts:

1. A pressure altimeter takes as input the ambient pressure and Kollsman setting and outputs an altitude. While this isn't done electronically (at least in the "steam gauge" type), what formula encapsulates what it is doing?
2. Given true altitude, local altimeter setting, and ambient temperature, what will be the ambient pressure?

Answer 1

To answer your question #1, you're right, altimeters take 2 things in input:

• QNH, a.k.a. altimeter setting
• Static pressure P coming from the pitot system

Altimeters assume that both the QNH and the air pressure are values within the International Standard Atmosphere (ISA). I'm not sure how this is carried out mechanically in the altimeter but this is the equation they solve:

$$h=\frac{T_0}{L}\left[\left(\frac{\mathrm{QNH}}{P_0}\right)^\frac{R_sL}{g}-\left(\frac{P}{P_0}\right)^\frac{R_sL}{g}\right]\tag{1}$$

where:

• $h$: Altitude reading in meters
• $P$: Air pressure in Pascal
• $\mathrm{QNH}$: Altimeter setting in Pascal
• $L$: Temperature lapse $=0.0065 \mathrm{~K/m}$
• $T_0$: Standard temperature $=288.15 \mathrm{~K}$
• $P_0$: Standard pressure $=101325 \mathrm{~Pa}$
• $g$: Gravitational acceleration $\approx 9.81 \mathrm{~m/s}^2$
• $R_s$: specific gas constant for dry air $\approx 287.058 \mathrm{~J \cdot kg^{−1}K^{−1}}$

To answer question #2 all we need to do is to invert equation 1:

$$P=P_0\left[\left(\frac{\mathrm{QNH}}{P_0}\right)^\frac{R_sL}{g}-\frac{Lh}{T_0}\right]\tag{2}$$

just be aware that in this case $h$ is not the true altitude, but the altitude indicated in the altimeter. If you want to use true altitude you first need to compute it, inverting the relation you also mention in your question:

$$h_{true} = h + \frac{4\cdot h}{1000}\cdot\left(T_\mathrm{OAT}-T_\mathrm{ISA}\right)\tag{3}$$