How to determine your position underground?

Question

Crossed my mind after random rant on wikipedia that lead me to articles about chronometers and measuring position.

Let's assume I were trapped in the underground laboratory with lots of equipment but without any access to the surface. Would I be able to properly determine my position (latitude, longitude and altitude), and if so, what instruments are needed? (and mny what's the coolest way to do it :)

I thought about measuring Coriolis effect, which could lead to latitude measurement, and earth's gravity map could give more hints, but it's still far too imprecise.

Answer 1

A precise measurement of the Coriolis force will not only give you your latitude, but will also tell you which direction is true north. A compass will tell you which direction is magnetic north, and the combination of knowing your latitude and your magnetic declination will give you your longitude. Measuring the long-term average air pressure, assuming there's a direct air path between you and the surface that doesn't involve fans, will give you a rough idea of your altitude.

Answer 2

A Foucault pendulum will tell you your latitude. The rate of rotation per day relative to the Earth is:

$$ \omega = 2\pi\sin\theta $$

where $\theta$ is the latitude (positive for north and negative for south).

To a first approximation determining the longitude is impossible because to a first approximation the Earth is axially symmetric. If you have an accurate chronometer you could use strain gauges to detect the tides and work out the longitude from the time and the measured position of the Moon. If you're not allowed a chronometer then the only thing I can think of is to measure the acceleration due to gravity and use the geoid, though whether this would give you a unique position I'm not sure.

Answer 3

John's answer gives some ideas for latitude and longitude. You could measure your altitude (read depth) by measuring the weight of a known mass. In a perfectly uniform, spherical Earth, the weight is proportional to your distance from the center.

Answer 4

I hear that surveyors who work underground in mines and tunnels use a gyro-theodolite. My understanding is that a gyrocompass is based on the same physical principles and will also work underground.

When either device is turned on, it spins up an internal gyroscope. Then the rotation of Earth leads to torque-induced gyroscopic precession of that gyroscope, which leads to a determination of the direction of true north.

The inclination of that true north vector, relative to "straight down" measured by a plumb bob, can give a good estimate of latitude.

Answer 5

You did not say that you did not have an Internet connection and a computer with a browser. If you knew the name of the underground facility and they where in USA, the guards and the staff probably have a licensed surface transmitter licensed by the American FCC. The transmitter is for surface perimeter security and staff trips to the local store for supplies. You can go to the FCC database and look up the license by name and it will give you the lat/long coordinates and physical address. However, if you have the physical address of the facility there are Internet programs that convert physical address to lat/long coordinates.

You also did not say whether you were allowed to stick your head up out of the surface doorway to look at the sun. Knowing the correct time and the local sun angle you could compute rough position. You could use your hand as a rudimentary protractor. If you knew what hemisphere you where in then you could compute the sun's angle toward the equator. (Note: if there is an overcast day and you can not see the sun use a polarized light filter to locate the sun behind the clouds - similar to the Viking SunStone or cordierite crystal)

If you had an airplane altimeter you might be able to determine your sea level. You can make a homemade altimeter at: http://www.ehow.com/way_6467568_homemade-altimeter.html

However, if there is a elevator (or vertical ventilation shaft), go the the shaft and whistle loudly. The time it takes for the return echo is the distance to the top. Sound travels at 1,100 feet per second. So divide the time it takes from you whistling until you hear the return by half. And then do the math to compute your depth below ground. The echo is bouncing off of the elevator's floor or the ventilation shaft's hood. Send the elevator to top floor first.

Or you could drop a medium-sized rock or metal ball bearing into the shaft from the top floor. Listen for the sound of the crash. Objects (unaffected by air disturbance like a feather) fall at 300 feet per second. The return sound traveled at 1,110 feet per second. If you have a stopwatch you could compute the distance to the bottom. I don't feel like doing the algorithm for that. You can do it I'm sure.

Answer 6

Altimeters based on barometric pressure should work equally well below the surface as above the surface. (You would either have to calibrate the altimeter based on the known local atmospheric pressure, or average the reading over a long period.)