fixed points in survex

[Lev Bishop]

On Thu, 6 Feb 2003, John Halleck wrote:

> On Thu, 6 Feb 2003, Lev Bishop wrote:
> 
>   It is nice to see a literate post, from someone that appears
>   to actually be familiar with the issues, as irrelevant as I
>   may consider most of them to be in this context.

Thanks for your nice words. I do agree that most of them are irrlevant,
but worth considering and then ruling out so at least you know the limits
of applicability of your software and don't try to use survex in a regime
where the assumptions are invalid.

> > 1) Tidal effects on the earth. The two sources of earth tides are the sun,
> > causing effects of 17cm amplitude, and the moon, with 36cm (the next most
> > influential body is venus, whose effects are less than 0.05% of the
> > above). However, as long as our measurements cover an area of limited
> > dimensions then the effects will be small. For example, over a 100km
> > distance, the tides should come to less than a cm.
> 
>    Where does this figure come from?  It is higher than what I've been told.
>    (Unless you mean "MUCH less than a cm".)

Rough calculation as follows: take combined tidal effect to be sinusoidal
and of 50cm amplitude (eg a spring tide when sun & moon line up
36cm+17cm=53cm). assume we're on the equator and feel the full +-53cm
displacement over the course of a day.  100km on the equator implies an
angle of roughly 100/6400 radians subtended at the centre of the earth. we
must double this to get the phase angle for the tide (because there are 2
tides per day, one on the side closest to the sun/moon one on the opposite
side). assuming we're midway between tides at the current moment in time
then the difference in vertical position is 50cm * sin(2*100/6400) or
using small angle approx 50cm*2*100/6400=1.5cm, actually a little more
than a cm (for this worst-case scenario). However, this calculation is for
vertical positions referenced to the ellipsoid - presumably the Geoid will
distort almost the same amount as the earth tides (perhaps the earth tides
will slightly lag the Geoid since its the effective gravity (gravity plus
centrifugal forces) that drives the earth tides in the first place). So
for Geoid-referenced heights the differences should indeed be MUCH less
than a cm. I guess what you've been told was for land surveying - 
measuring using levels and so on it is the geoid height that matters. I 
was thinking in terms of high-accuracy DGPS in which case its the 
ellipsoid height that matters, although thinking about it a little more 
carefully I suppose that the software which does the postprocessing for 
such types of phase-ranging dual-frequency long-baseline DGPS probably has 
the option to make tidal corrections for you - its something that you 
wouldn't be able to ignore in such cases (unless you made observations for 
several days on the same point and let time-averaging take care of it). 

> > Assumption: the directions "north" and "down" are constant.
> > 
> > Violated because: the earth is ellipsoidal. At the 10^-5 level the earth
> > is roughly an ellipsoid. The mean radius is about 6400km. The "down"  
> > directions for 2 points on the surface of the earth and separated by 100km
> > differ by almost a degree.
> 
>   Don't forget refraction.  Because the air closer to the ground 
>   is denser than the air farther up, in isothermal conditions the the
>   line of sight drops.  (Thereby making up somewhat for the curviture
>   of the earth.
>
>   If I remember aright, the earth curvature drops 1foot/mile, and
>   the refraction drops line of sight 1/2 ft/mile, for a net APPARENT
>   curviture of about 0.6 feet/mile.

Actually I had forgotten refraction - thanks for bringing that up. The
earth curvature does drop about 0.7feet in 1mile, but it depends on the
*square* of the length of the sighting (ie the correct unit is
0.7foot/mile-squared). So for short legs, as used in cave surveying, it
really should be negligible. Eg, for a 50m leg (a very long leg - I
usually only carry a 10m or 20m tape) the curvature would be 0.2mm, 
although it could be relevant to surface surveying if people are using 
theodolites, etc, to make long legs between mountaintops. I don't know 
much about refraction but I'd be surprised if it were so predictable in a 
drafty, humid, etc, cave.

What I'm really thinking about is a little different, though. When you're
calculating the directions of legs in cartesian space you can't really
assume (as survex does) that up is the +z direction, north is the +y
direction, east is the +x direction. To take an extreme example, lets say
you started on the equator with that definition of your coordinate system.  
In that coordinate system, a point 1/4 of the way east around the equator,
you would still have north in the +y direction, but up would now be +x and
east would be -z. This is a direct result of how we define directions on
the globe.Refraction shouldn't play much part so long as the legs are
short, since the refraction curvature will depend on the square of the leg
length.

> > NOTE: because the direction "north" is actually a direction which is 
> > perpendicular to the direction "down" rather than just the direction 
> 
>   I'm not sure I agree with the argument of this paragraph.
>   But private email is probably more suited for a discussion of this.
>
> > parallel to the earth's axis (conventional or instantaneous) any errors in 
> > measuring either the gravity direction or the axis direction will be 
> > multiplied by tan(lattitude) when calculating "north". Ie a 1' deflection 

As an example, lets imagine we're at a point 1m south of the north pole.  
We only have to walk 3.14m east (or west) for the direction "north" to
change by almost 180 degrees - remember walking east or west just takes us
around a circle of radius ~1m centred on the pole. Or to show the role of
the deflection of the vertical: Lets assume again we're very close to the
north pole and we're attempting to determine the north direction by
astronomical observations and we've already worked out which point in the
sky is the north celestial pole and lets pretend there's a star there. If
you want to walk north you want to "follow the star" but since the star is
almost vertically above you its very hard to know which direction is north
- anything which interferes with your idea of "vertical" will strongly
affect your idea of north. If that still doesn't make sense then feel free
to send me an email.

> > Final comment: My guess is that almost everything I've said so far is
> > actually completely irrelevant because my feeling is that real cave
> > surveys are actually dominated neither by random nor by systematic errors
> > but rather by blunders/transcription errors.
> 
>   Or by the fact that angles are read to an accuracy much worse than most
>   of the effects you mention.

Actually, that doesn't matter. As long as the legs are not all in the same 
direction, the quantisation error can be dominated by systematic error of 
much smaller magnitude. For example, lets say we read to the nearest 
degree and we have no other errors - ie we *always* read to the nearest 
whole degree, never make a mistake, even when the true direction is very 
close to a half-integer number of degrees. In this case the only random 
error we have is quantisation error, in a uniform distribution  with zero 
mean and extending out to +-0.5 degrees. This implies a standard deviation 
of 1/sqrt(3)*0.5=0.29 degrees. If we take n measurements from this 
distribution then the central limit theorem says that as n gets large the 
mean of the errors approaches a normal distribution with standard 
deviation 0.29/sqrt(n). Ie, if our survey has 400 legs, the quantisation 
errors will be dominated by systematic errors if the latter are not kept 
below 0.015deg or about 1'.

Lev