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Parametrisation the height-temperature profiles

Several height-temperature (HT) profiles, that should be close to each other, have been evaluated for the refraction cases (using ray tracing implemented by Reijs (Hohenkerk&Sinclair, 1985)), this to see if a parametrisation of the HT profile can provide realistic predictions of the refraction:
  1. Low height range of CASES99Bench (blue dots)
    A logarithmic of quadratic curve (depending which has the maximum goodness of fit [R2/coefficient of determination closest to 1]).
  2. High height range of CASES99Bench (orange dots)
    A linear curve (showing a lapse rate).
    The CASES99 HT profiles look to have above the surface layer an average lapse rate (up to 58.1m) of 0.036K/m for Sunrises and 0.009K/m for Sunsets. This behavior is indeed expected due to the well developed NBL of the night.
  3. A surface layer point
    The surface layer point between the two ranges (in below graph at 10m) was optimised to fit an overall smooth result (gray dots).
    There is no significant difference in the height of the surface layer: 18m for Sunrises and 17m for Sunsets. One would expect a higher surface layer for Sunrise due to the well developed NBL of the night.
Determination of
        CASES99 approximation

Looking at Astronomical refraction

The eye is at 1.65m height (Near). All profiles are extended up to 85km with MUSA76 profile (average temperature gradient=0.0065K/m between 58.1m and Hr at -56.5C [around 11km]).

Results

Below one can see the differences (in arcmin) with the reference data (CASES99Bench):
 Difference astronomical refraction between HT profiles
The following can be deducted:

Looking at levelling refraction

The eye is at 58.1m (Near) and the vast level surface is at 10m (Distant). If using a vast level surface at 1.65m, the CASES99 profiles give refractions that make the ground go high up around zero (AZAAR) and negative (NAAR) apparent altitudes. because the near ground lapse rate (between 0.23 and 0.63m) is lower than -0.11K/m (see below picture). So a Fata Morgana (superior mirage) is what happens.

Distribution of lpase rate for CASES99 profiles
      (sunset/rise)

To get enough data to compare, the vast level surface has been put on 10m and some profile change was made to MUSA76:

Results

Below one can see the differences (in arcsec) with the reference data (CASES99Bench):

Difference levelling refraction between HT profiles
The following can be deducted:

Looking at terrestrial refraction

The eye is at 1.65m (Near) and the terrestrial object is at 58.1m (Distant).

Results

Below one can see the differences (in arcsec) with the reference data (CASES99Bench):
 Difference between HT profiles
The following can be deducted:

Observations

If we check the differences from the CASES99Bench results, we get the following overview (between bracket the average value):


Refraction difference [% from CASES99Bench refraction]
Astronomical
Levelling
Terrestrial
CASES99Approx 2
10
0-15 (7.5)
CASES99OneStep
Not valid
2
0-70 (35)
MUSA76 10
10
0-80 (40)
Formula
0-15 (8)
0-15 (8)
0-50 (25)
Maximum of above
10
10
40

Approximate absolute values
AppAlt of CASES99Bench [']
9'
-11'
60'
Refraction of CASES99Bench [']
30'-45' (40')
70"-330" (4')
15"-100" (1')

<Green means the best mapping profile>
<Red means the on average worse mapping profile>

CASES99Approx is the best solution; except for levelling refraction where it is CASES99OneStep. MUSA76 looks not to be the best method.
The Formula perform in an average way.
All of these methods (including CASES99Bench) have the drawback that for former times (like in archaeoatronomy) the parameters (like [approximated] HT profile, Near&Distant temperature or average lapse rate) are unknown and very difficult to predict.
So perhaps we need to try our best to predict the parameters (by knowing the events [rise/set], the location, a general idea of the HT profile and the temperatures), and this might give the accuracy quoted in the above table in the row Maximum of above for the different refractions. This is not that far from the astronomical refraction variation seen in this web page. (around 5' variation of the astronomical refraction at AppAlt=9')

References

Hohenkerk, Catherine Y., and A.T. Sinclair. 1985. "The computation of angular atmospheric refraction at large zenith angles." ed. by HM nautical almanac office. Cambridge.
Werf, Siebren Y. van der. 2003. 'Ray tracing and refraction in the modified US1976 atmosphere', Applied optics, Vol 42: pp. 354-66.

Acknowledgements

I would like to thank people, such as Stephen McCluskey, Marcel Tschudin and others for their help and constructive feedback. Any remaining errors in methodology or results are my responsibility of course!!! If you want to provide constructive feedback, let me know.

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Major content related changes: December 21, 2017