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OK, I give it a try.

Frank questioned whether his hypothetical model of a series of balloon
soundings along the line of sight would allow an improved calculation
of the refraction near the horizon and Bill B. questions whether this
would help with thermal inversion/abnormal lapse rates causing
refraction anomalies at my visible horizon?

First I should mention that the abnormal refraction phenomena depend
on the viewing angle; they appear near the astronomical horizon, i.e.
when looking horizontally. If the position of the eyes is considerably
above or below the the large temperature gradient they may - depending
on the height difference - actually not even notice that there exists
somewhere an abnormal layer, this because the angle of incidence is
too large. It is thus important from where I observe e.g. the sun near
the horizon.

If the observation is done from a boat with the eyes a few meters
above the sea, the apparent horizon is fairly close to the
astronomical horizon; the dip can therefore be subjected to anomalous
refraction phenomena but the sun still well above the horizon not. The
measured angular distance between sun and horizon is erroneous because
of the wrong position of the horizon (dip) not because of the sun's
position. If, in the other case, the sunset is observed from a high
point above the sea the viewing angle of the horizon is already too
large (too distant from the astronomical horizon) to be disturbed by
anomalous atmospheric layers. If in this case the sun is near the
astronomical horizon and the position of the eyes close to e.g. an
inversion the image of the sun is "disturbed". The wrong result
obtained from measuring the distance between sun and horizon is in
this case due to the wrong position of the sun and not the position of
the dip. If both, sun and apparent horizon, are close to the
astronomical horizon both can also be affected by anomalous
atmospheric layers.

A word on weather balloons which measure pressure, temperature and
relative humidity of the atmosphere. These measurements are only
available at certain selected heights, which, depending on the
situation, may have been selected too coarse to catch certain
anomalous features. Imagine a "S"-shaped vertical temperature profile
which may eventually only have been measured with two arbitrary
selected data points; depending where those two data points are one
may obtain "any" temperature gradient. I also intend to use weather
balloon data, but from those I try to obtain from statistical analyses
"typical" values for a certain location at a given local time for the
various height layers of the atmosphere.

If you are more interested in refraction: have a look at the over 100
Web-pages which Andrew T. Young dedicated to this subject. There are
two entries into this labyrinth, the alphabetic one
http://mintaka.sdsu.edu/GF/bibliog/alphindex.html
and the tables of content
http://mintaka.sdsu.edu/GF/bibliog/toc.html

Marcel

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