[NatureNS] Determining Elevation the hard way

Date: Sun, 11 Feb 2007 17:05:24 -0400
From: Stephen Shaw <srshaw@dal.ca>
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Hi Dave and other McGyverites who have got into this...
  I forget exactly who, but think it was a German who said that the average
respondent "would rather use another man's toothbrush than his ideas", and I'm
therefore not unduly shocked that Dave without evidence rejects my proposed
sighting method for determining height as having a large expected instrumental
error, while favouring his own even more improbable alternative.  In a winter
contest to do just-possible thought experiments that we guess Jamie will never
follow through on anyway, it's really hitting below the belt to get 
serious and
start talking about errors, but I feel I've got to rise to the bait here.
   Dave, I couldn't follow your pressure formulation as given, but essentially,
this seems to be that as you go up the hill and air pressure drops, the volume
in the closed end of the manometer will rise proportionally, if temperature
stays the same (I think it's Boyle's Law, PV=RT?).  This is sure to work as a
nice thought experiment but since you are getting finecky, the question is
whether any such change would be detectable in practice.  Air pressure is
halved at about 18,000 feet which should be detectable, but it seems doubtful
that Jamie lives that high up; more likely the house is (say) 200 meters above
sea level (~600 feet).  I looked up Atmospheric Pressure on Wikipedia and if
you use the second equation given there for calculating air pressure against
elevation (for the lowest zone, starting at sea level), an increase in
elevation of 200 m would generate a change in atmospheric pressure of only 2.4
percent.  I doubt that a change in volume (= air column height) of 2.4% could
even be detected reliably never mind accurately measured with your 
Tygon tubing
method, even if you could avoid water droplets on the wall as you jog up the
hill carrying it (Tygon is quite wettable by water, which I think is 
partly why
such instruments often use mercury in glass).
   If you go back to my suggested sighting method, the two angles are eminently
measurable (unlike in your method), and the distance between the two rocks
should be easily measurable to better than 1 percent with a calibrated 
rope, so
the main expected error would be in the inaccuracy of reading the two angles
from a home-made scale.  Because tangents are involved, the actual errors
depend strongly on the actual dimensions and angles involved, and I doubt if
you can make, fit and read a home-made scale to better than 1 degree.  If the
house is 200 m up, and the first rock is 1000m out and the second 500 m
further, a 1 degree error in setting or reading the scale on the telescope for
the two angles would tranlate into a 21-34 percent error in height estimation
(worst cases, where the errors run in opposite directions rather than partly
cancel out).  You could improve this a bit by repeatedly sighting and 
averaging
the angles but I doubt if you could get it down below +/- 10 percent, because
of difficulty making and accurately centering the scale. This would not 
be that
bad for a first try.  With your method, you too perhaps could improve
detectability a bit by running up and down the hill repeatedly, carrying the
maonometer, but I think my method of averaging is more restful.
Commenting on methods for measuring distance, Lois' method using 3-4-5
Pythagorean triangles that are 'similar' (not 'congruent'): this OK for short
distances, as in the example given (factor of 10), but invites the 
comment that
if you have a measuring stick 5 units long already, it would be actually
quicker to simply tumble it 6 times to measure out the 30 units required than
to set up all the sighting sticks and to make sure that they are vertical. How
accurate would it be if you wanted to measure something at, say, 1000 units
distant, abandoning the 3-4-5 system but keeping the 4 unit baseline? --
sighting errors over the sticks would be much more significant.  The website
mentioned by Ray may be interesting historically but doesn't help because
ultimately one's paces, thumb or whatever have to be calibrated with a 
ruler to
pull out dimensions that we currently use, like feet or meters. Can't estimate
if you can't calibrate.  One method not mentioned so far is that years ago,
Polaroid cameras introduced a range-sensing method that worked over near to
middle distances, that I think employed the reflection of an ultrasonic sound
pulse from the desired object, presumably by measuring the time delay.  Maybe
this is how the hand-held devices used by real estate agents to measure the
sizes of rooms work - I'm not sure.  Not sure either how my camera
autofocusses, either.
Final advice to Jamie would be to listen to Peter and rent a surveying
instrument, or if he really wants a project, make his own survey system with a
laser diode (a few dollars these days), angle scales, tripod, level and and a
calibrated measuring pole.  Methods using triangulation on maps are 
practically
sensible, but essentially amount to cheating in this McGyver context, because
the map has already been calibrated by experts.
Steve

Quoting David & Alison Webster <dwebster@glinx.com>:
> Hi Steve & All,                Feb 8, 2007

>    Your formulation looks good but I would expect instrument error to 
> be large & have consequently looked at a second approach; an 
> improvized differential water manometer. At least that seems like a 
> reasonable name. Having never used or made anything similar, it would 
> be best to check this method out before using it to establish runway 
> elevation for instrument landing purposes.
>
>    Theory: This procedure makes use of the change in air pressure 
> with elevation. At the initial elevation (house or mean sea level as 
> convenient) water height in a transparent U-tube (both arms the same) 
> is recorded just after one end is closed to the air such that air 
> volume of the closed end is not changed. As rapidly as possible (to 
> avoid temperature changes) the unit is moved to the other location 
> (mean sea level or house) and the height of water in the two arms is 
> recorded. If the initial point is not about half way between the two 
> final points then there has been a change in water volume or enclosed 
> air volume due to temperature changes and it may be desirable to 
> start again. As a rule of thumb, a 5 cm difference from sea level 
> would represent an elevation difference of about 3.9 metres.
>
>    [DIGRESSION: If I have thought this through correctly, the 
> difference in water level between the two arms will slightly or 
> seriously underestimate the difference in air pressure between the 
> two locations, because change in volume of the air on the closed side 
> will change air pressure on the closed side. A smaller air pressure 
> on the closed will be increased and a larger pressure on the closed 
> side will be decreased by an amount proportional the fractional 
> change in volume. It might be possible to correct for this it one had 
> to. ]
>
>    From observed difference (h; cm) in water level between the two 
> arms (ig