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Bruce, you informed us on:
>
> My very  limited theodolite measurements of dip from land seems to  agree
> with the value of 0.89 (more or less considering scatter especially below
> eye height of 25 ft).   Peters recommends against refraction measurements at
> eye heights below 18 ft.   I'm still gathering and checking my dip
> measurements, but will eventually give the data to the NavList community.

I encourage you to continue with your direct measurements of the dip,
hopefully arriving at a data set covering all seasons, with dips
resulting from positive and negative temperature differences between
air and sea.

You then asked us:

> I'm herewith asking the NavList community to look at one or two of their
> best observations from known locations and heights (above 16-20 ft if
> possible) and determine if their "fix" would be closer to the true location
> if smaller dip were used.  I'm particularly hoping that Jeremy could do this
> from his ship. I've tried using my own data and I've come to no definite
> conclusion because of my own scatter/technique. By the way,  at eye height
> below 14 ft, even with much scatter in the data, dip = about 0.8 sqrt H
> feet....... I think.

This would unfortunately not help much because one would use for this
verification the (astronomical) refraction which itself is not exact.
The "wrong" dip in combination with the "wrong" (astronomical)
refraction could actually be more accurate than the "true" dip in
combination with the "wrong" refraction.

May be this is an occasion to show here some intermediate results from
analysing my Hs-measurements (i.e. always refraction plus dip) of the
setting sun over the Marmara Sea. Below I show how the sum of squared
height differences (in moa) between measurement and calculation of
3076 observations differ for different calculation models. (Note that
a smaller value of this sum indicates a better agreement between
calculation and measurement.)

Model 1: Refraction with Bennett formula and Dip with k-factor=0.163.
This k-factor corresponds together with the mean HoE and the mean
Earth radius to a dip-factor=1.76' (for HoE in m). This Model 1
corresponds to Nautical Almanac data. The sum of squared differences
is 8530. When determining the k-factor by best fit reduces it only
very marginally to k-factor=0.161: This small difference changes the
dip-factor and the sum of squared differences only very marginally at
some non-important decimals. This indicates that the formula in the
Nautical Almanac for the dip appears indeed to be the best for using
in combination with Bennett's refraction formula.

Model 2: Refraction with Sinclair formula for low altitudes and
k-factor by best fit. This results in a slightly better agreement than
Model 1, i.e. it has the slightly reduced sum of squared differences
of 8283, and the fitted k-factor=0.469 corresponds to a
dip-factor=1.40' (for HoE in m).

Model 3: Refraction from a ray tracing calculation using the
US-Standard Atmosphere and k-factor by best fit. This results in a
further reduced sum of squared differences of 8151 and a fitted
k-factor=0.312 corresponding to a dip-factor=1.60' (for HoE in m).

Note that the above models do not yet consider the influence of e.g.
- the temperature difference between air and sea,
- the wind velocity and
- the temperature dependence from altitude which starts to show up
below about 5 degrees
on dip and refraction. By considering these additional dependencies
the sum of squared differences reduces to less than 5000.

In conclusion: The "Nautical Almanac"-model (1) is not the best one.
However, the differences between model 1 and 3 are so marginal that
they are generally not noticeable (in the noise), except in large data
sets. A noticeable improved estimation can be attained by considering
additional environmental parameters.

A large data set with directly measured dips would allow finding the
atmospheric model producing the best matching refraction values.

Marcel

   

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