Sight reduction explained

In astronavigation, sight reduction is the process of deriving from a sight (in celestial navigation usually obtained using a sextant) the information needed for establishing a line of position, generally by intercept method.

Sight is defined as the observation of the altitude, and sometimes also the azimuth, of a celestial body for a line of position; or the data obtained by such observation.^[1]

The mathematical basis of sight reduction is the circle of equal altitude. The calculation can be done by computer, or by hand via tabular methods and longhand methods.

Algorithm

Given:

Lat , the latitude (North - positive, South - negative), Lon the longitude (East - positive, West - negative), both approximate (assumed);Dec , the declination of the body observed;GHA , the Greenwich hour angle of the body observed;LHA = GHA + Lon , the local hour angle of the body observed.

First calculate the altitude of the celestial body

Hc using the equation of circle of equal altitude:

$$\backslash sin(Hc)\; =\; \backslash sin(Lat)\; \backslash cdot\; \backslash sin(Dec)\; +\; \backslash cos(Lat)\; \backslash cdot\; \backslash cos(Dec)\; \backslash cdot\; \backslash cos(LHA).$$

The azimuth

Z or Zn (Zn=0 at North, measured eastward) is then calculated by:

$$\backslash cos(Z)\; =\; \backslash frac\; =\; \backslash frac\; -\; \backslash tan(Hc)\; \backslash cdot\; \backslash tan(Lat).$$

These values are contrasted with the observed altitude

Ho . Ho , Z , and Hc are the three inputs to the intercept method (Marcq St Hilaire method), which uses the difference in observed and calculated altitudes to ascertain one's relative location to the assumed point.

Tabular sight reduction

The methods included are:

• The Nautical Almanac Sight Reduction (NASR, originally known as Concise Tables for Sight Reduction or Davies, 1984, 22pg)
• Pub. 249 (formerly H.O. 249, Sight Reduction Tables for Air Navigation, A.P. 3270 in the UK, 1947 - 53, 1+2 volumes)^[2]
• Pub. 229 (formerly H.O. 229, Sight Reduction Tables for Marine Navigation, H.D. 605/NP 401 in the UK, 1970, 6 volumes.^[3]
• The variant of HO-229: Sight Reduction Tables for Small Boat Navigation, known as Schlereth, 1983, 1 volume)
• H.O. 214 (Tables of Computed Altitude and Azimuth, H.D. 486 in the UK, 1936 - 46, 9 vol.)
• H.O. 211 (Dead Reckoning Altitude and Azimuth Table, known as Ageton, 1931, 36pg. And 2 variants of H.O. 211: Compact Sight Reduction Table, also known as Ageton - Bayless, 1980, 9+ pg. S-Table, also known as Pepperday, 1992, 9+ pg.)
• H.O. 208 (Navigation Tables for Mariners and Aviators, known as Dreisonstok, 1928, 113pg.)

Longhand haversine sight reduction

This method is a practical procedure to reduce celestial sights with the needed accuracy, without using electronic tools such as calculator or a computer. And it could serve as a backup in case of malfunction of the positioning system aboard.

Doniol

The first approach of a compact and concise method was published by R. Doniol in 1955^[4] and involved haversines. The altitude is derived from

\sin (Hc) = n - a ⋅ (m + n) , in which n = \cos (Lat - Dec) , m = \cos (Lat + Dec) , a = \operatorname{hav}(LHA) .

The calculation is: n = cos(Lat − Dec) m = cos(Lat + Dec) a = hav(LHA) Hc = arcsin(n − a ⋅ (m + n))

Ultra compact sight reduction

A practical and friendly method using only haversines was developed between 2014 and 2015,^[5] and published in NavList.

A compact expression for the altitude was derived^[6] using haversines,

\operatorname{hav} , for all the terms of the equation:\operatorname{hav}(ZD) = \operatorname{hav}(Lat - Dec) + \left( 1 - \operatorname{hav}(Lat - Dec) - \operatorname{hav}(Lat + Dec) \right) ⋅ \operatorname{hav}(LHA)

where

ZD is the zenith distance,Hc = ( 90 ^{\circ} - ZD) is the calculated altitude.

The algorithm if absolute values are used is:For the azimuth a diagram^[7] was developed for a faster solution without calculation, and with an accuracy of 1°.This diagram could be used also for star identification.^[8]

An ambiguity in the value of azimuth may arise since in the diagram

0 ^{\circ} \leqslant Z \leqslant 90 ^{\circ} . Z is E↔W as the name of the meridian angle, but the N↕S name is not determined. In most situations azimuth ambiguities are resolved simply by observation.

When there are reasons for doubt or for the purpose of checking the following formula^[9] should be used:

\operatorname{hav}(Z) = (\operatorname{hav)/(( 90) ^{\circ} \pm \vert Dec\vert) - \operatorname{hav}(\vert Lat\vert - Hc)}{ 1 - \operatorname{hav}(\vert Lat\vert - Hc) - \operatorname{hav}(\vert Lat \vert + Hc)}

The algorithm if absolute values are used is:This computation of the altitude and the azimuth needs a haversine table. For a precision of 1 minute of arc, a four figure table is enough.^{[10] [11]}

An example

See also

• Navigation
• Celestial navigation
• Circle of equal altitude
• Intercept method

External links

• Navigational Algorithms: AstroNavigation - Free App for Windows
• Navigational Algorithms: resources for Longhand Haversine Sight Reduction
• NavList A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding
• Celestial Tools for the USPS/CPS JN/N Student
• Graphical all-haversine Hc reduction
• Sight Reduction - free App for android
• Vector Solution for the intersection of two Circles of Equal Altitude - free App for android

Notes and References

1. The American Practical Navigator (2002)
2. https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%201-2020-Dec.pdf Pub. 249 Volume 1. Stars
3. https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_1/Pub229Vol1.pdf Pub. 229 Volume 1. Latitudes 0° to 15°
4. Table de point miniature (Hauteur et azimut), by R. Doniol, Navigation IFN Vol. III Nº 10, Avril 1955 Paper
5. Ultra compact sight reduction . Greg . Rudzinski . Ix, Hanno . Ocean Navigator . Navigator Publishing LLC . Portland, ME, USA . July 2015 . 227 . 0886-0149 . 42–43 . 2015-11-07.
6. Altitude haversine formula by Hanno Ix http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121
7. Azimuth diagram by Hanno Ix. http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-HannoIx-jun-2015-g31689
8. Hc by Azimuth Diagram http://fer3.com/arc/m2.aspx/Hc-Azimuth-Diagram-finally-HannoIx-aug-2013-g24772
9. Azimuth haversine formula by Lars Bergman http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441
10. Web site: NavList: Re: Longhand Sight Reduction (129172).
11. https://yadi.sk/i/4MmOYyXhUshbxA Natural-Haversine 4-place Table