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The subject of this thread touches really what I'm working at since a
few years. It might be helpful/useful to add a few comments without
going into great details.

George wrote:
>
>  First,  I should apologise for confusing, in Navlist 4635, between the works
>  of two specialists on atmospheric optics, Brad Schaefer and Andy Young, in
>  writing- "Brad Schaefer is a recognised authority on optics in the
>  atmosphere, and more recently has produced an authoritative series of papers
>  titled "Sunset Science"".
>  I thank Marcel Tschudin for pointing that out. Thanks to Marcel for noting
>  corrections to Schaefer's paper "Refraction near the horizon",  Publ.
>  astron. society of the Pacific, vol 102, pages 796 - 805 (July 1990). I now
>  have a .pdf copy of that paper, which I, too, could pass on to anyone
>  showing an interest.

You are welcome George. We are all prone to make mistakes. I'm
thankful to those who show the ones I produce.

>
>  Second, I suggest that Greg Rudzinski missed the important difference
>  between refraction at and near the horizon, and that at much higher angles,
>  when he wrote, in Navlist 4649- .
>
>  " I base my 6 minutes of arc figure on the refration variables as
>  seen in the A4 table of the Nautical Almanac. The temperature and
>  pressure extremes are -7.3' to +6.9' from the mean."
>

I think that's not really Greg's fault. It's the fault that over
several decades refraction tables have been published providing
refraction values for large zenith distances which are (only) based on
the standard atmosphere. I should mention the one from D. H. Sadler,
published in 1952, providing refraction values at the horizon for
observers at heights up to 55'000ft which - at least up to a few years
ago - were still published in almanacs. This situation shows that
apparently no one was really interested in finding out how the
refraction near the horizon could be better predicted.

George wrote:
>  Yes, Greg can correct that 34 arc-minute figure for local air density if he
>  wishes, but that is no more than a tiny fraction of the overall fluctuation
>  in refraction at the horizon.

This really depends on the height and latitude of the observer. Most
of my observations are done at about 3 to 4m height and at about 40N.
The height is probably not much different to navigational
observations. The above mentioned 34 arc minutes are only marginally
larger than the sun's diameter. The duration of the sunset, i.e. the
time between the lower and the upper limbs touch the horizon is here a
bit less than 3 minutes. Delayed sunsets by half a minute, a full
minute or so are quite common; larger delays are rather rare. These
observations don't confirm your above comment related to the 34 arc
min "...but that is no more than a tiny fraction". Things start to be
different when observing from higher up.

George wrote:
>
>  I ask on what evidence Frank bases his assertion that-
>
>  | "under
>  | certain very common circumstances, e.g. at sea in temperate climates, the
>  | day-to-day variability in the refraction at the horizon is relatively
>  small
>  | (a few minutes of arc), "
>
>  And even if that assertion is valid, how on Earth does a navigator know,
>  when trying to use a sunset time to ascertain his position, whether Frank's
>  restrictions apply or not, within the "variation within the variability"
>  that he conjures up?

My suggestion would be that one measures the temperature of the water
and of the air. If this temperature difference is "considerable",
extreme refractions may be likely. Unfortunately I'm not in a position
to quantify the "considerable". From my own experience at my present
location these effects are more likely during spring and autumn. In
spring the water is still cold from winter and the air may already be
warm; this leads at the surface to inversions with ducting. In autumn
the sea is still warm from summer and the air already cold; this leads
to "Omega" sunsets or inferior mirages.

George wrote:
>   If we are talking about use of timing a sunset for real navigation,
>  requirements for position knowledge can be very relaxed in mid ocean. It is
>  only when a land mass is being approached that navigation becomes critical.
>  So observations made at coastal sites, with a view over the ocean, will be
>  particularly relevant in assessing the navigator's problem. And those are
>  exactly the sites from which Schaefer and Liller have timed sunsets, at sea,
>  in temperate climes.

Cerro Tololo is not what I would call a coastal site; it's about 70km
from the coast.

Side note: Looking at temperature profiles derived statistically from
a lot of sounding profiles in this area suggest that the sunset
observations over the Pacific from Vina del Mar and Cerro Tololo tend
to have inversions (water of the Pacific colder than the air) and thus
may be prone to delayed sunsets by ducting.

George wrote:
>  ... I understand (personal communication from Schaefer) that "our
>  atmosphere always have rapidly and wildly varying thermal structure (i.e.,
>  temperature inversions come and go on all time scales and are generally
>  present at some level)." In that, he does not distinguish between over-land
>  and over-sea, though no doubt such effects are significantly greater over
>  land.

It depends probably on the scale one looks at it. At a very fine scale
temperature variations may likely correspond to white noise. However,
on a larger scale the temperature gradients do follow to a certain
extend a diurnal pattern at coastal and continental locations, this as
a result of the diurnal temperature cycle within the planetary
boundary layer of the atmosphere.

Imagine a pendulum. The length of the pendulum corresponds to the
height of the planetary boundary layer and its deflection to the
diurnal cycle of the temperature range at the surface. The temperature
at the pivot remains roughly constant. Before or at sunrise the
pendulum is at minimal temperature, indicating an inversion, and in
the early afternoon the pendulum reaches maximal temperature,
indicating a (super) adiabatic temperature gradient. Its these
differences which are responsible for the refraction differences
between sunrise and sunset.

George wrote:
>  Half of Schaefer's events we are considering were sunrises, so presumably
>  were in the early morning, and all were close to shore, in that

Liller's observations at Vina del Mar and Schaefer's observation at
Cerro Tololo were made at sunset (around 22h/23h UT). As far I
remember from looking up these places in Google Earth they wouldn't
have been able to observe sunrises over the sea from their locations.
In addition to sunsets Schaefer mentions a few set events of Moon and
Venus.

Frank wrote:
>  | Of course, navigators at
>  | sea do not generally have access to current atmospheric temperature
>  | profiles. Some naval navigators might have been able to observe the real
>  | temperature profile (by launching balloons or even aircraft), but the
>  | computing power to generate refraction tables "on the fly" probably
>  arrived
>  | too late for naval celestial navigation.

and George wrote:
>  What would be particularly interesting would be a measurement of horizontal
>  refraction and temperature profile at the same place and time, but I doubt
>  if that's ever happened.

The measurements of a single balloon sounding doesn't seem to be
useful for calculating refraction. However I still believe that useful
profiles can be derived from statistical analysis of a large number of
such sounding data.

In navlist 4617, Peter Fogg suggested-
>
>  "Here's a thought: if the time of apparent sunrise/sunset was observed
>  regularly;
>  the extent of the difference or inaccuracy shown by observation
>  compared to calculated data could be evaluated on a regular basis and
>  contrasted with other information about position."

and George replied:
>  And indeed, that procedure might well add useful information to Schaefer's
>  body of data on horizontal refraction, for locations taken far from land. On
>  those occasions when the sky was clear right down to the horizon, with no
>  sign of distant cloudbanks, the time of the last (or first) glimpse of the
>  Sun could be logged to the GMT second, from a well-known height of eye, with
>  a precise GPS position at that moment, with a note of the datum in use. With
>  the closeness of horizon that can be seen from a small craft, it could only
>  work reliably in calm conditions without significant swell. How about that
>  for a project on the next cruise, one that calls for no special
>  instrumentation? It wouldn't be hard to collect that data together from a
>  number of navlist members, and boil it down. Perhaps that's been done
>  already, but if it has I haven't heard of it.

At present I'm doing something like this (in addition to a lot of
other things on this subject). My observations are sunsets over the
sea at 3 to 4m height. I found that a good way to document the
observations is to take exactly timed photos during the sunset event.
From those photos the height of the sun above the apparent horizon can
be measured in fractions of the sun's horizontal diameter. Before each
sighting the watch is adjusted to a time server. The timing is
expected to be within 1 to 2 seconds. Coordinates from each sighting
location are taken from Google Earth; it could be within about three
meters but I use a set of coordinates for an area within about 50m.
The meteorological data reported in half hourly intervals are from the
airport at the same coast but about 50km away and daily balloon
soundings are done at a place about 10km from here. So far there are
about 120 days of sunset observations. Each day with an average of may
be 4 to 5 photos ... My problem is that I haven't yet found time to
document all this information. It requires a lot of work since each
photo has to be measured several times to obtain some statistical
information to go along with it. The location here has one
disadvantage: from spring till late summer the sun sets at a location
where a distant part of the metropolis protrudes from behind the
apparent horizon to about a quarter of the sun's diameter above it. It
was noticed that all these observations provide very, very small
refraction values. I don't understand (yet) the reason for this; it
could e.g. be caused by the sea-land-transition and/or the efficient
heat exchange at large house surfaces of the metropolitan area.

All these (systematically) documented sunsets are from
mid-latitude-locations. May be there is someone here who has the
possibility to do such observations at a high latitude location, e.g.
60 deg or higher. Who is living in a castle on top of a hill at the
edge of a long headland? He might eventually observe sunrises and
sunsets.

Marcel

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