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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.

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    $\begingroup$ You presumably meant to say that you could determine your latitude by measuring the Coriolis force, not the longitude. $\endgroup$ – Red Act Sep 12 '14 at 14:48
  • $\begingroup$ There was some discussion about this during the brief period where they thought they'd detected particles going faster than light (but it turned out they'd miscalculated the position of the detector). Since the detector is, in fact, buried under a mountain to screen out other particles, that's not far different from your situation... except that there they were able to work their way in from known surface locations (which in turn were established by phase measurements from both GPS satellites and surface reference points.) $\endgroup$ – keshlam Sep 13 '14 at 5:09
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    $\begingroup$ If the lab contains a barometer, offer it to one of the guards as a bribe to tell you where you are. $\endgroup$ – David Richerby Sep 13 '14 at 10:16
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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.

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  • $\begingroup$ Wouldn't the difference between magnetic north and true north change when you change the altitude? $\endgroup$ – sampathsris Sep 13 '14 at 1:25
  • $\begingroup$ @Krumia Yes, magnetic declination does change slightly with elevation, so you could be a little more precise in the longitude calculation if you tweaked it based on the altitude estimate. $\endgroup$ – Red Act Sep 13 '14 at 2:46
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    $\begingroup$ "the combination of knowing your latitude and your magnetic declination will give you your longitude" is too simplistically put. Even without known the precise variation of the magnetic declination with longitude for a fixed lattitude, it is bound to be a continuous periodic function, and therefore attain any value except its extremes at least twice. Most likely you will have at least two possible solutions for the longitude. $\endgroup$ – Marc van Leeuwen Sep 13 '14 at 10:58
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    $\begingroup$ @MarcvanLeeuwen That's a valid point, in particular if you don't even have a good guess as to which continent you're on. The magnetic inclination can be used to distinguish between the two possible longitudes in such a case. $\endgroup$ – Red Act Sep 13 '14 at 14:37
  • $\begingroup$ I believe measuring the gravity pull towards the planet's center will also help determine altitude. $\endgroup$ – Thermo's Second Law Sep 14 '14 at 15:01
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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.

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    $\begingroup$ Well, apropos of another recent question, you could build a very large neutrino observatory (AKA super-k) and image the sun in neutrinos. Then, if you knew that your clocks were synchronized to, say, GMT you could get your latitude as well. And seismology on the blasting you need to do to build the tank can get your depth by moderately direct methods. $\endgroup$ – dmckee --- ex-moderator kitten Sep 12 '14 at 15:06
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    $\begingroup$ @dmckee: Ooh, yes, I forgot to measure the depth and seismology would be perfect for that. Are you going to post that as an answer or can I shamelessly pirate the idea to enhance my answer? $\endgroup$ – John Rennie Sep 12 '14 at 15:17
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    $\begingroup$ You can measure the gravitational pull from the Sun superimposed to the one from the Moon, and thus get the local time. $\endgroup$ – Davidmh Sep 12 '14 at 16:54
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    $\begingroup$ @Davidmh - assuming you have an approximate date... $\endgroup$ – Floris Sep 12 '14 at 18:14
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    $\begingroup$ He has a very accurate time piece, he has access to a cesium clock (even if only by radio): in the underground laboratory with lots of equipment. $\endgroup$ – LDC3 Sep 13 '14 at 2:58
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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.

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  • $\begingroup$ Of course the perfect uniformity assumption is wrong, but we can correct for it (barring some degeneracies due to the non-monotonic nature of the curve). $\endgroup$ – user10851 Sep 12 '14 at 23:54
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    $\begingroup$ @ChrisWhite But, if you want to now your depth wrt the local surface, wouldn't that fall within any magnitude of the error that you can obtain with this method? $\endgroup$ – Bernhard Sep 14 '14 at 10:04
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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.

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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.

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  • $\begingroup$ Good answer. Do you mean "Objects . . . fall at 30 feet per second"? $\endgroup$ – HDE 226868 Sep 13 '14 at 18:54
  • $\begingroup$ He didn't say he was a prisoner, but trapped; maybe from a cave-in. Also, if he is under a mountain, then the elevator may not go to the surface, but have an access ramp with several vertical feet up (or thousands) to go to the surface. $\endgroup$ – LDC3 Sep 13 '14 at 19:31
  • $\begingroup$ Given that the OP has presupposed access to "a well equipped lab" and no access to the surface, all this seems ... irrelevant? $\endgroup$ – dmckee --- ex-moderator kitten Sep 13 '14 at 23:28
  • $\begingroup$ "Objects (unaffected by air disturbance like a feather) fall at 300 feet per second." Excuse me? Feathers are massively affected by air resistance. Objects for which air resistance is negligible accelerate at 32 feet per second squared. And, damnit, if you're trapped in an underground laboratory, even in the USA, you're using SI. $\endgroup$ – David Richerby Sep 14 '14 at 11:08
  • $\begingroup$ @LDC3 On the other hand, if you were trapped in a laboratory by a cave-in, the answer is probably "Use your memory of where the laboratory was when you entered it." :-) $\endgroup$ – David Richerby Sep 14 '14 at 11:10
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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.)

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