Sight reduction

File:Corrections for Sextant Altitude.en.jpg.]]

File:MarcqSaintHilaire.en.jpg

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:

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

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.

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)[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 }}
• Pub. 229 (formerly H.O. 229, Sight Reduction Tables for Marine Navigation, H.D. 605/NP 401 in the UK, 1970, 6 volumes.[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 }}.
• 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.)

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.

The first approach of a compact and concise method was published by R. Doniol in 1955 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] and involved haversines. The altitude is derived from $\backslash sin\; (Hc)\; =\; n\; -\; a\; \backslash cdot\; (m\; +\; n)$, in which $n\; =\; \backslash cos\; (Lat\; -\; Dec)$, $m\; =\; \backslash cos\; (Lat\; +\; Dec)$, $a\; =\; \backslash operatorname\{hav\}(LHA)$.

The calculation is:

n = cos(Lat − Dec)

m = cos(Lat + Dec)

a = hav(LHA)

Hc = arcsin(n − a ⋅ (m + n))

File:Haversine Sight Reduction.jpg

A practical and friendly method using only haversines was developed between 2014 and 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 2015 |issue=227 | issn=0886-0149 |pages=42–43 |url=http://issuu.com/navigatorpublishing/docs/on227_download_edition |access-date=2015-11-07}} and published in [http://fer3.com/arc/ NavList].

A compact expression for the altitude was derivedAltitude haversine formula by Hanno Ix http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121 using haversines, $\backslash operatorname\{hav\}()$, for all the terms of the equation:

$\backslash operatorname\{hav\}(ZD)\; =\; \backslash operatorname\{hav\}(Lat\; -\; Dec)\; +\; \backslash left(\; 1\; -\; \backslash operatorname\{hav\}(Lat\; -\; Dec)\; -\; \backslash operatorname\{hav\}(Lat\; +\; Dec)\; \backslash right)\; \backslash cdot\; \backslash operatorname\{hav\}(LHA)$

where $ZD$ is the zenith distance,

$Hc\; =\; (90^\backslash circ\; -\; ZD)$ is the calculated altitude.

The algorithm if absolute values 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 diagramAzimuth diagram by Hanno Ix. http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-HannoIx-jun-2015-g31689 was developed for a faster solution without calculation, and with an accuracy of 1°.

File:Azimuth diagram by Hanno Ix.jpg

This diagram could be used also for star identification.Hc by Azimuth Diagram http://fer3.com/arc/m2.aspx/Hc-Azimuth-Diagram-finally-HannoIx-aug-2013-g24772

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

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

The algorithm if absolute values 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.{{Cite web|url=http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29172|title = NavList: Re: Longhand Sight Reduction (129172)}}[https://yadi.sk/i/4MmOYyXhUshbxA Natural-Haversine 4-place Table]; PDF; 51kB

{{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°

}}

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

{{Reflist}}

• 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