[Lewis Thomson (OP)]

Thomas asks:

I once worked in South African gold mines, at over 3km below ground level.  The air pressure in the depths of the mine is deeper than on surface: I knew this from the popping of my ears, and the fact that the empty plastic water bottle would expand in my hand as the air pressure increased as we returned to surface. Back at home, there would be a pssst of releasing air when I took the cap off to fill up the bottle
for the next day.
Deep in the mine, I had more mass above me, and less below me, than I do when I'm on the surface of the earth. What puzzles me is that as gravity is related to mass and with the gravitational effect of the mass above me, I expect that the air pressure would be less underground, not more.  And similarly, I've been to the Tibetan plateau, at altitudes of over 4km above sea level.  In this instance, I have more earth mass (and therefore surely gravity) beneath me than anywhere else I've been, and yet the air pressure there was lower. 
What on earth is going on!?

[chiralSPO]

Very observant!

The earth is nearly 6400 km deep, so going up or down a few km isn't going to change the gravity very much. But altitude obviously has a very strong effect on the pressure, so what's going on?

What you are doing is changing how much air is on top of you. Air is not very massive: at sea level a cubic meter only holds just over one kg of air. But if you think of the whole column of air all the way up to space resting on your shoulders, it turns out it weighs quite a bit. This weight is the atmospheric pressure.

If you have ever gone snorkeling or scuba diving, you will notice the same thing, but on much shorter distances. I can comfortably swim down about 2.5 meters, but any more than that and my ears really hurt, and if I bring a soda bottle of air down, it will get crushed under the weight of the water.

So think of the atmosphere as a deep pool of very light water. At ground level, we are under several dozen km of air. So the pressure falls as we go up (like Tibet, or flying), and the pressure increases if we go down (into a mine).

Now, there is one other complication comparing air to water in this way. Air is very compressible, while water is effectively not. This means that the density of water really doesn't change under different pressures, but the density of air does. This means that air is much denser at the bottom of any vertical column of air than it is at the top, and that therefore you would expect a bigger (absolute) difference in pressure going 1 km down from sea level than you would going 1 km up.

[Colin2B]

Excellent answer. I was typing my response when you posted your’s and it didnt seem necessary to add anything. Other than, start clearing your ears as soon as you leave the surface, don’t wait until you are 2.5m down, it will be much harder then.