Altitude difference by barometric formula

I think no one will object to the statement that the air is thinner at an altitude of two kilometers, and the atmospheric pressure is less than at sea level. If we put these words in a scientific form, it turns out that the pressure (density) of the gas depends on its altitude in a gravitational field. On this phenomenon method of barometric leveling is built.

Barometric leveling - the method of determining the height difference between two points by atmospheric pressure measured at these points. Since the atmospheric pressure and the altitude above the sea level are also dependent on the weather, for example, on the water vapor content of air, if possible, the method is applied to make measurements at points with the smallest interval between the measurements. The points themselves should not be located too far from each other.
The difference in altitude is calculated as follows.
There is a rather complicated formula of Laplace:

It is, in addition to temperature and pressure also takes into account the absolute humidity and latitude of the measuring point, that is, in practice it seems to be not in use.
And use a simple Babinet formula
,
where - Gas expansion factor equal to

Indeed, in an era without computers and calculators, even this formula was ... well, not difficult but hard for calculations. To determine the height difference, people used auxiliary barometric level tables.

Barometric stage - the height at which we must ascend so the pressure drop by 1 mm Hg
That is, we took and simplified the Babinet formula to expression

and calculate h for different values of temperature and pressure.
We acquired tables similar to barometric pressure tables

Thus, by measuring, for example, the pressure difference at the average temperature t and the average pressure p, Meteorologists could find the value of the barometric stage from the table and multiply it by the amount of pressure difference.
It is clear that the formula gives the result with a margin error, but at the same time, it is approved that the error does not exceed 0.1 - 0.5% of the measured altitude.

The barometric leveling method allows determining the height of a point above sea level without resorting to geodetic leveling.
In practice, the height of the point above sea level is determined using the closest ranging mark, height above sea level known.
For example, the ranging mark is at 156 meters. The barometer shows that the ranging mark at 748 mmHg, being transferred to the defined point, the barometer shows 751 mmHg. The average temperature is 15 degrees Celsius. Using the Babinet formula, obtain -33.78 m, i.e., the point below the ranging mark at 33.78 meters, and its height is approximately 122.22 m. Taking the average pressure of 748 mmHg and using the barometric table, we get -33.85, i.e., the height is approximately 122.15 m.
The calculator below illustrates everything said above.

Barometric leveling

Pressure at the first point (mmHg)

Pressure at the second point (mmHg)

Average air temperature (degrees Celsius)

Calculation precision

Digits after the decimal point: 2

Calculate

Altitude difference

 

Altitude difference (using the barometric stage formula )

 

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