First of all, I see some discussion of the shape of the geoid.
Deviations from sphericity are interesting, but are probably so small
that they will not show up in such simple experiments. If you want info
on the geoid, the NOAA website has lots of good stuff:
http://www.ngs.noaa.gov/GEOID/geoid_def.html
For directions and ranges on the ellipsoid, see
http://www.ngs.noaa.gov/cgi-bin/Inv_Fwd/inverse2.prl
Regarding the issue of how far you can see, I have some info at the end
of my page
http://aty.sdsu.edu/explain/atmos_refr/horizon.html
that points out that surveyors have actually used baselines as long as
192 miles -- between elevated sites, of course.
Regarding "compensating for" refraction: There's no realistic way to do
this. You'd need temperature data at least an order of magnitude more
accurate than air temperatures can be measured, especially under field
conditions. Aspirated thermometers claim an accuracy of about 0.1
degree; but remember it is the vertical temperature *gradient* that you
need. So you have to construct a very accurate temperature profile.
For this reason, the optical people have repeatedly pointed out that the
sensible way to go is the other direction: use the observed ray bending
to infer the temperature profile. The meteorologists have shown no
interest in this idea, mostly because conditions clear enough to allow
it to be used are relatively uncommon.
Please note that humidity is a small contributor to the refractivity of
air at *optical* wavelengths; so unless the laser was in the IR, you can
(to a good approximation) ignore humidity effects. See my page on the
refractivity of air:
http://aty.sdsu.edu/explain/atmos_refr/air_refr.html
The remark
"However, if the surface of the lake is flat, and the laser
is level, then why would there be any refraction? Vertical
refraction of a straight and level laser over water will
only occur because the water surface, and hence the
temperature gradient above it, is curved. Hence this
demonstration suggests the surface of the lake is curved."
makes no sense to me. Refraction depends on the density gradient of the
air, regardless of the shape of the lake surface. The density gradient
depends on both the temperature gradient and the vertical pressure
gradient due to hydrostatic equilibrium; see my page
http://aty.sdsu.edu/explain/atmos_refr/bending.html
for how to calculate ray bending.
Most people are quite unaware of how much the ray bending varies with
the local meteorological situation. There are well-known diurnal
effects, first noticed in 1674:
http://aty.sdsu.edu/bibliog/bibliog.html#Perrault1967a
-- Perrault's observations showed effects of about a quarter of a degree
on a 2 km path. So of course they are quite visible to the naked eye,
if you have a fixed eye position and a fixed reference point 50 or 100
meters away, and watch the diurnal motion (and the day-to-day motion,
which is considerably larger) of a landmark a few km away.
So, although the "standard" refraction can be computed from a standard
atmospheric model, the real atmosphere -- especially over a body of
water! -- is always different. Sometimes the ray is concave toward the
surface of the Earth; sometimes it is convex. These one-shot experiments
are useless for determining the shape of the Earth.
The nice night-time picture of the laser beam and the lights on the far
shore very clearly miraged (by an inferior mirage) is correctly
described by the lines
"It shows the laser bending upwards, and it looks like it does
actually hit something on the other side. What is fascinating is
that the return light is then split in two, and we get a
"reflection" underneath it, just just like with the lights on
the right."
However, the next line:
" Here it is with some reference lines. The red lines show better
where it actually would be without refraction."
is not true, unless the Earth is flat. The curvature of the Earth makes
the inferior mirage shrink vertically as the height of the eye above
the surface increases. This was first pointed out by Bravais in 1853
(see the reference to his paper in Ann. Met., in the bibliography I
cited above). Unfortunately, because of the complicated path of the
rays in the inferior mirage [see my paper in Appl. Opt. 54, no. 4,
pp. B170-B176 (Feb. 1,2015)], the "reflection" near the surface in the
inferior mirage is not at all like a simple reflection at a solid surface;
so it is not possible to infer the shape of the Earth from this effect.
Anyway, I was glad to see that someone had put "reflection" in quotes,
as it certainly is not a simple reflection at all. Remember that the
temperature profile in the inferior mirage is a logarithmic one (again,
see my Applied Optics paper for the details.)
I don't have time to wade through the rest of this thread, but maybe I
have pointed out where you can find information about these things.
By the way, there are much better time-lapse movies of mirages and
varying refraction during the day on YouTube than the one in this
discussion. Here's one:
https://www.youtube.com/watch?v=AJc3kOxiV6c
on the experiment day, especially at daytime when our laser will not be well seen on long distance.
