The hypsometric equation, as discussed in Metr 201, expands out to
where Ęz is the 1000-500 mb thickness. However, the equation is more general than that (as you will see when you derive it) and the thickness between any two pressure levels can be related to the mean temperature of that layer by changing the upper and lower bounds.
Solving for the mean temperature of the 1000-500 mb layer yields
where R is the gas constant of dry air. ln2=.693147; g=9.8 ms-2; R=287.04 J kg-1 K-1
Take the local derivative of both sides of the equation
which states that the changes in thickness in a column of air that extends from 1000 to 500 mb at a fixed location will be proportional to the local changes in the mean temperature of that layer (in this case, the 1000-500 mb layer). If these changes are smoothly distributed through the whole layer, than this can be used to forecast the surface temperature change.
A thickness change of 3 dm will result, according to this equation, in a temperature change of 1.5o K (or 1.5oC or about 3oF)