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In [[astronavigation]], '''sight reduction''' is the process of deriving from a [[Sight (device)|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.<ref>''[[The American Practical Navigator]]'' (2002)</ref>

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 ==
[[File:Corrections for Sextant Altitude.en.jpg|thumb|Steps for measuring and correcting {{mvar|Ho}} using a [[sextant]].]]
[[File:MarcqSaintHilaire.en.jpg|thumb|Using {{mvar|Ho}}, {{mvar|Z}}, {{mvar|Hc}} in intercept method.]]
Given:
* <math>Lat</math>, the [[latitude]] (North - positive, South - negative), <math>Lon</math> the [[longitude]] (East - positive, West - negative), both approximate (assumed);
* <math>Dec</math>, the [[declination]] of the body observed;
* <math>GHA</math>, the [[Greenwich hour angle]] of the body observed;
* <math>LHA = GHA + Lon</math>, the [[local hour angle]] of the body observed.

First calculate the altitude of the celestial body <math>Hc</math> using the equation of [[circle of equal altitude]]:

<math display=block>\sin(Hc) = \sin(Lat) \cdot \sin(Dec) + \cos(Lat) \cdot \cos(Dec) \cdot \cos(LHA).</math>

The azimuth <math>Z</math> or <math>Zn</math> (Zn=0 at North, measured eastward) is then calculated by:

<math display=block>\cos(Z) = \frac{\sin(Dec) - \sin(Hc) \cdot \sin(Lat)}{\cos(Hc) \cdot \cos(Lat)} = \frac{\sin(Dec)}{\cos(Hc) \cdot \cos(Lat)} - \tan(Hc) \cdot \tan(Lat).</math>

These values are contrasted with the observed altitude <math>Ho</math>. <math>Ho</math>, <math>Z</math>, and <math>Hc</math> 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&ndash;53, 1+2 volumes)<ref>[https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%201-2020-Dec.pdf Pub. 249 Volume 1. Stars] {{Webarchive|url=https://web.archive.org/web/20201112002241/https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%201-2020-Dec.pdf |date=2020-11-12 }}; [https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%202.pdf Pub. 249 Volume 2. Latitudes 0° to 39°] {{Webarchive|url=https://web.archive.org/web/20220122221200/https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%202.pdf |date=2022-01-22 }}; [https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%203.pdf Pub. 249 Volume 3. Latitudes 40° to 89°] {{Webarchive|url=https://web.archive.org/web/20190713223352/http://thenauticalalmanac.com/Pub.%20249%20Vol.%203.pdf |date=2019-07-13 }}</ref>
* Pub. 229 (formerly H.O. 229, Sight Reduction Tables for Marine Navigation, H.D. 605/NP 401 in the UK, 1970, 6 volumes.<ref>[https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_1/Pub229Vol1.pdf Pub. 229 Volume 1.  Latitudes 0° to 15°] {{Webarchive|url=https://web.archive.org/web/20170126050410/http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_1/Pub229Vol1.pdf |date=2017-01-26 }}; [https://web.archive.org/web/20140308052740/http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_2/Pub229Vol2.pdf Pub. 229 Volume 2. Latitudes 15° to 30°]; [https://web.archive.org/web/20140327104102/http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_3/Pub229Vol3.pdf Pub. 229 Volume 3. Latitudes 30° to 45°]; [https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_4/Pub229Vol4.pdf Pub. 229 Volume 4. Latitudes 45° to 60°] {{Webarchive|url=https://web.archive.org/web/20170130204843/http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_4/Pub229Vol4.pdf |date=2017-01-30 }}; [https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_5/Pub229Vol5.pdf Pub. 229 Volume 5. Latitudes 60° to 75°] {{Webarchive|url=https://web.archive.org/web/20170126050601/http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_5/Pub229Vol5.pdf |date=2017-01-26 }}; [https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_6/Pub229Vol6.pdf Pub. 229 Volume 6. Latitudes 75° to 90°] {{Webarchive|url=https://web.archive.org/web/20170211062914/http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_6/Pub229Vol6.pdf |date=2017-02-11 }}.</ref>
*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&ndash;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&ndash;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<ref> Table de point miniature (Hauteur et azimut), by R. Doniol, Navigation IFN Vol. III Nº 10, Avril 1955 [http://fer3.com/arc/m2.aspx/Table-De-Point-Miniature-R-Doniol-FrankReed-jul-2015-g32063 Paper]</ref> and involved [[haversine]]s. The altitude is derived from <math>\sin (Hc) = n - a \cdot (m + n)</math>, in which <math>n = \cos (Lat - Dec)</math>, <math>m = \cos (Lat + Dec)</math>, <math>a = \operatorname{hav}(LHA)</math>. 

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

=== Ultra compact sight reduction ===
[[File:Haversine Sight Reduction.jpg|thumb|Haversine Sight Reduction algorithm]]

A practical and friendly method using only [[haversine]]s was developed between 2014 and 2015,<ref name="Rudzinski_2015">{{cite journal |title=Ultra compact sight reduction |author-first1=Greg |author-last1=Rudzinski |others=Ix, Hanno |journal=Ocean Navigator |publisher=Navigator Publishing LLC | location=Portland, ME, USA |date=July<!-- /August --> 2015 |issue=227 | issn=0886-0149 |pages=42–43 |url=http://issuu.com/navigatorpublishing/docs/on227_download_edition |access-date=2015-11-07}}</ref> and published in [http://fer3.com/arc/ NavList].

A compact expression for the altitude was derived<ref>Altitude haversine formula by Hanno Ix http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121</ref> using haversines, <math>\operatorname{hav}()</math>, for all the terms of the equation:
<math>\operatorname{hav}(ZD) = \operatorname{hav}(Lat - Dec) + \left( 1 - \operatorname{hav}(Lat - Dec) - \operatorname{hav}(Lat + Dec) \right) \cdot \operatorname{hav}(LHA)</math>

where <math>ZD</math> is the [[zenith distance]],

<math>Hc = (90^\circ - ZD)</math> is the calculated altitude.

The algorithm if [[absolute value]]s are used is:
{{pre|1=
if same name for latitude and declination (both are North or South)
  ''n'' = hav({{abs|Lat}} − {{abs|Dec}})
  ''m'' = hav({{abs|Lat}} + {{abs|Dec}})
if contrary name (one is North the other is South)
  ''n'' = hav({{abs|Lat}} + {{abs|Dec}})
  ''m'' = hav({{abs|Lat}} − {{abs|Dec}})
''q'' = ''n'' + ''m''
''a'' = hav(''LHA'')
hav(''ZD'') = ''n'' + ''a'' · (1 − ''q'')
''ZD'' = archav() -> inverse look-up at the haversine tables
''Hc'' = 90° − ''ZD''
}}
For the azimuth a diagram<ref>Azimuth diagram by Hanno Ix. http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-HannoIx-jun-2015-g31689</ref> was developed for a faster solution without calculation, and with an accuracy of 1°.
[[File:Azimuth diagram by Hanno Ix.jpg|thumb|Azimuth diagram by Hanno Ix]]
This diagram could be used also for star identification.<ref>Hc by Azimuth Diagram http://fer3.com/arc/m2.aspx/Hc-Azimuth-Diagram-finally-HannoIx-aug-2013-g24772</ref>

An ambiguity in the value of azimuth may arise since in the diagram <math>0^\circ \leqslant Z \leqslant 90^\circ</math>. <math>Z</math> 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<ref>Azimuth haversine formula by Lars Bergman http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441</ref> should be used:

<math>\operatorname{hav}(Z) = \frac{\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)}</math>

The algorithm if [[absolute value]]s are used is:
{{pre|1=
if same name for latitude and declination (both are North or South)
  ''a'' = hav(90° − {{abs|Dec}})
if contrary name (one is North the other is South)
  ''a'' = hav(90° + {{abs|Dec}})
''m'' = hav({{abs|Lat}} + ''Hc'')
''n'' = hav({{abs|Lat}} − ''Hc'')
''q'' = ''n'' + ''m''
hav(''Z'') = (''a'' − ''n'') / (1 − ''q'')
''Z'' = archav() -> inverse look-up at the haversine tables
if Latitude ''N'':
  if ''LHA'' > 180°, ''Zn'' = ''Z''
  if ''LHA'' < 180°, ''Zn'' = 360° − ''Z''
if Latitude ''S'':
  if ''LHA'' > 180°, ''Zn'' = 180° − ''Z''
  if ''LHA'' < 180°, ''Zn'' = 180° + ''Z''
}}
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.<ref>{{Cite web|url=http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29172|title = NavList: Re: Longhand Sight Reduction (129172)}}</ref><ref>[https://yadi.sk/i/4MmOYyXhUshbxA Natural-Haversine 4-place Table]; PDF; 51kB</ref>

==== An example ====
{{pre|1=
Data:
  ''Lat'' = 34° 10.0′ N (+)
  ''Dec'' = 21° 11.0′ S (−)
  ''LHA'' = 57° 17.0′
Altitude ''Hc'':
  ''a'' = 0.2298
  ''m'' = 0.0128
  ''n'' = 0.2157
  hav(''ZD'') = 0.3930
  ''ZD'' = archav(0.3930) = 77° 39′
  ''Hc'' = 90° - 77° 39′ = 12° 21′
Azimuth ''Zn'':
  ''a'' = 0.6807
  ''m'' = 0.1560
  ''n'' = 0.0358
  hav(''Z'') = 0.7979
  ''Z'' = archav(0.7979) = 126.6°
  Because ''LHA'' < 180° and Latitude is ''North'': ''Zn'' = 360° - ''Z'' = 233.4°
}}

== See also ==
* [[Navigation]]
* [[Celestial navigation]]
* [[Circle of equal altitude]]
* [[Intercept method]]

== References ==
{{Reflist}}

==External links==
* Navigational Algorithms: [https://sites.google.com/site/navigationalalgorithms/software/Windows AstroNavigation - Free App for Windows]
* Navigational Algorithms: [https://sites.google.com/site/navigationalalgorithms/downloads resources for Longhand Haversine Sight Reduction]
* [http://fer3.com/arc/ NavList] A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding
* [http://celestialtools.webs.com/ Celestial Tools for the USPS/CPS JN/N Student]
* [https://tube.geogebra.org/m/1531651 Graphical all-haversine Hc reduction] {{Webarchive|url=https://web.archive.org/web/20160516013424/https://tube.geogebra.org/m/1531651 |date=2016-05-16 }}
* [https://play.google.com/store/apps/details?id=com.Sightreduction Sight Reduction - free App for android]
* [https://play.google.com/store/apps/details?id=com.Vector_solution_2_CoP Vector Solution for the intersection of two Circles of Equal Altitude - free App for android]

[[Category:Navigation]]
[[Category:Celestial navigation]]