# Relation between altimeter setting (QNH) formula and the barometric formula

I'm searching for a reliable source for a formulization of the QNH adjustment to the barometric altitude. I found this document from weather.gov that gives a formula for the Altimeter Setting, which is QNH if I'm correct. If I replace the formula's values with the appropriate constant names, it becomes this:

$$Altimeter Setting (QNH?) = \left ( P - 0.3 \right )\left ( 1 + \left ( \left ( \frac{P_{0}^{\frac{L.R}{g}} L}{T_{0}} \right ) \left ( \frac{H}{\left ( P - 0.3 \right )^{\frac{L.R}{g}}}\right ) \right ) \right )^{\frac{g}{LR}}$$

I also know the following formula from the US Standard atmosphere (which is identical to the ISA standard atmosphere up to 50 km):

Which can be expresses like this (for troposphere):

$$P = P_{0} \left [ \frac{T_{0}}{T} \right ]^{\frac{g.M}{R.L}}$$

Symbols and their meanings for extra information:

• $$P$$: Pressure at the airport
• $$H$$: Altitude at the airport
• $$P_{0}$$: Standart pressure (1013.25 $$hPa$$)
• $$T_{0}$$: Standart temperature (288.15 $$K$$)
• $$L$$: Temperature lapse rate
• $$R^{*}$$: Universal/ideal gas constant
• $$R$$: Characteristic gas constant
• $$g$$: Gravitational acceleration of Earth.
• $$M$$: Molar mass of dry air

So, my questions are twofold:

1. Are these formulas actually different reperesentations of the same formula? I tried but I haven't been able to acquire the second formula from the first one.

2. What is the $$-0.3$$ for in $$(P-0.3)$$ in the first formula? Why is it there?

• If you plug in the figures and get the same answer, or very close, the practical value is confirmed. Unless your aim is to understand the math. I'd try some known rules of thumb like 1 hPa ~ 27 ft ie 27 ft with Std Atmosphere for the rest of the variables should lead to a stn pressure of 1014.25. – skipper44 Nov 16 '20 at 14:46
• I've tried it, their results were about 5 hPas apart, which equates to about 40-50 meters of altitude difference. But I am trying to understand the math as well. If I'm gonna use this formula from weather.gov, I need to be able show where it originated from. A reliable source/formula. – Melih Durmaz Nov 19 '20 at 8:53
• Which one seems to be most correct when compared to the ISA -- International Standard Atmosphere - Table ? – skipper44 Nov 19 '20 at 14:31