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Refraction determination

Using a derivative of Hohenkerk&Sinclair's implementation (1985) of Auer&Sandish (1979) refraction model, one can see what the effect is of a changing temperature gradient near ground level (Reijs, 2007). Several temperature gradient values are typical for certain atmospheric conditions (Sun's insolation, wind speed, cloud cover, cloud height and time of day) and these are defined as stability class A to G. The temperature gradients belonging to these stability classes range from <-19 (stability class A) to >40 C/km (stability class G).

The variability due to ranging temperature gradients (as defined by for instance stability classes) is given in this graph (Reijs, 2007):

The apparent altitude in above picture is related to two different horizon configurations:

• Positive apparent altitudes are for astronomical refraction over a real/imaginative terrestrial marker (terrestrial refraction).
• Negative apparent altitudes are for astronomical refraction over a vast plane surface (aka terrestrial refraction; sometimes called: levelling refraction)

Stability class G can happen normally during midnight and Stability class A can happen during noon. Depending on the atmospheric conditions the temperature gradient around sunrise or sunset can be close to these temperature gradients (aka stability classes). Sunrise conditions can be more varying than the sunset temperature gradients (Lapworth 2003, 2006).

Dynamics of the temperature gradient and its influence on the ray's curvature radius

An example of refraction due to a temperature inversion

The Sun can be disfigured due to the temperature profile and one of such happens when there is an inversion in the height-temperature profile causing a mock-mirage. See in below picture for the profile:

Sun at -66' in benchamerk3 (eye at 350m)	Sun at -67' in benchamerk3 (eye at 350m)
Sun at -97' in benchamerk3 (eye at 350m)	Sun at -98' in benchamerk3 (eye at 350m)

Variation in profiles

If we average the refraction seen due to varying temperature gradients (given by the 950 event observations [three years] at the Pilgrim power plant during consecutive days), we see the following graph (the averages: single lines (axis in the middle); the standard deviation: double lines (axis on the right side)).

One could distinguish three apparent altitude regions that are resulting due to variability of the temperature gradient:

1. Positive Apparent Altitude Refraction: PAAR
   above AppAlt=0.25, the standard deviation (1 sigma), due to variation of temperature gradient, is smaller than 6'
2. Negative Apparent Altitude Refraction: NAAR
   below AppAlt=-0.5, the standard deviation is larger than 30' and slightly decreasing
3. Around Zero Apparent Altitude Refraction: AZAAR
   between the above two boundaries, the standard deviation changes between 6' and 30'.

Refraction sensitivity due to parameter changes

Comparing VR's refraction implementation (in PAAR region)

Beside the webmaster's Auer&Standish (1979) implementation (VR); Marcel Tschudin (MET) and Andrew Young (ATY) also have an implementation of Auer&Standish, while Siebren van der Werf (SvdW) (2003, 2008) has his own methodology and implementation that allows for non-spherical symmetry in the HT (Height-Temperature) profiles (which can be very handy when have more complex landscapes, as described here). The three implementations have been benchmarked and they provide within arcsecs the same results.
The HT profiles were provided by Steve McCluskey (SMcC), who derived them from the CASES99 study. Four HT-profiles (being outliners during CASES99 sun rise events) were used for comparing the three different implementations. The results from the four HT profiles within these three implementation can be seen in below graph:

The four implementation provide similar outputs for these four events, this shows that the results of the implementations and methodologies are reproducible. Such similarity happens both for ground elevations of 0m (NAAR, AZAAR and PAAR) and 433m (PAAR), so the four implementation are also handling the elevation of the ground properly.

For around zero and negative altitudes (AZAAR and NAAR) the HT profile can cause problems (ducting) in the Auer&Standish implementations; van der Werf's method is able to overcome this problem by changing the HT profile depending on the distance from the observer (aka allowing non-spherical symmetry). But several benchmark tests have been performed with a HT profiles that produce ducting and the four implementations provide within arcsec agreement.

Comparing refraction variation

Refraction measurements

• RS based on photometric measurements of Sunsets on 14 days (Sampson, 2008)
  Location: Barbados, West Indies
  Climate zone: Af
  Start of HTPR profile: 0m, station TPR (unknown), observer at 2.5m: CABL
  Weather conditions: clear
  Time: before sunset (Sun's upper rim for 2 to 0 deg)
  Standard deviation derivation: 1 sigma in the refraction at the SS/SR event.
  Refraction determination: observations
• Other measurements in PAAR, AZAAR and NAAR regions can be seen in this link. And here is that graph repeated and compared to calculations using: Sinclair (using USA76 model) and Stability classes A and G (using MUSA76 model). All the measurements are based on their own station heights, and station TPR and all involved a vast plane surface (levelling refraction).
  

References

Acknowledgements

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