Approximating Altitude From Air Pressure

The ideal gas law implies:

P = ρRT / M

where P is the air pressure, ρ is the air density, T is the air temperature, M is the air molar mass, and, R is the ideal gas constant.  Also:

dP = -ρgdz 

where g is the freefall acceleration, and, z is the vertical coordinate.  Therefore:

dP / P = -gMdz / (RT)

If it is assumed that T = T_o + Lz for some air temperature lapse rate L, where T_o is the ground level air temperature, then:

P = P_o (1 + Lz / T_o)^{-gM / (RL)}

where P_o is the ground level air pressure.  This allows altitude to be approximated from the air pressure!