Sight reduction

<h2 id="algorithm">Algorithm</h2>

Given:

<ul><li>
 
 
 
 L
 a
 t
 
 
 {\displaystyle Lat}
 
, the <a href="/facts/Latitude/IBYqkjhU">latitude</a> (North - positive, South - negative), 
 
 
 
 L
 o
 n
 
 
 {\displaystyle Lon}
 
 the <a href="/facts/Longitude/HsC61ML9">longitude</a> (East - positive, West - negative), both approximate (assumed);</li>
<li>
 
 
 
 D
 e
 c
 
 
 {\displaystyle Dec}
 
, the <a href="/facts/Declination/5X4S4k9u">declination</a> of the body observed;</li>
<li>
 
 
 
 G
 H
 A
 
 
 {\displaystyle GHA}
 
, the <a href="/facts/Greenwich_hour_angle/rj9KG0Sz">Greenwich hour angle</a> of the body observed;</li>
<li>
 
 
 
 L
 H
 A
 =
 G
 H
 A
 +
 L
 o
 n
 
 
 {\displaystyle LHA=GHA+Lon}
 
, the <a href="/facts/Local_hour_angle/rj9KG0Sz">local hour angle</a> of the body observed.</li></ul>
First calculate the altitude of the celestial body 
 
 
 
 H
 c
 
 
 {\displaystyle Hc}
 
 using the equation of <a href="/facts/Circle_of_equal_altitude/Luhv147W">circle of equal altitude</a>:

 
 
 
 sin
 ⁡
 (
 H
 c
 )
 =
 sin
 ⁡
 (
 L
 a
 t
 )
 ⋅
 sin
 ⁡
 (
 D
 e
 c
 )
 +
 cos
 ⁡
 (
 L
 a
 t
 )
 ⋅
 cos
 ⁡
 (
 D
 e
 c
 )
 ⋅
 cos
 ⁡
 (
 L
 H
 A
 )
 .
 
 
 {\displaystyle \sin(Hc)=\sin(Lat)\cdot \sin(Dec)+\cos(Lat)\cdot \cos(Dec)\cdot \cos(LHA).}

The azimuth 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 or 
 
 
 
 Z
 n
 
 
 {\displaystyle Zn}
 
 (Zn=0 at North, measured eastward) is then calculated by:

 
 
 
 cos
 ⁡
 (
 Z
 )
 =
 
 
 
 sin
 ⁡
 (
 D
 e
 c
 )
 −
 sin
 ⁡
 (
 H
 c
 )
 ⋅
 sin
 ⁡
 (
 L
 a
 t
 )
 
 
 cos
 ⁡
 (
 H
 c
 )
 ⋅
 cos
 ⁡
 (
 L
 a
 t
 )
 
 
 
 =
 
 
 
 sin
 ⁡
 (
 D
 e
 c
 )
 
 
 cos
 ⁡
 (
 H
 c
 )
 ⋅
 cos
 ⁡
 (
 L
 a
 t
 )
 
 
 
 −
 tan
 ⁡
 (
 H
 c
 )
 ⋅
 tan
 ⁡
 (
 L
 a
 t
 )
 .
 
 
 {\displaystyle \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).}

These values are contrasted with the observed altitude 
 
 
 
 H
 o
 
 
 {\displaystyle Ho}
 
. 
 
 
 
 H
 o
 
 
 {\displaystyle Ho}
 
, 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
, and 
 
 
 
 H
 c
 
 
 {\displaystyle Hc}
 
 are the three inputs to the <a href="/facts/Intercept_method/F8bI0XWF">intercept method</a> (Marcq St Hilaire method), which uses the difference in observed and calculated altitudes to ascertain one's relative location to the assumed point.

<h2 id="tabular-sight-reduction">Tabular sight reduction</h2>
The methods included are:

<ul><li>The Nautical Almanac Sight Reduction (NASR, originally known as Concise Tables for Sight Reduction or Davies, 1984, 22pg)</li>
<li>Pub. 249 (formerly H.O. 249, Sight Reduction Tables for Air Navigation, A.P. 3270 in the UK, 1947–53, 1+2 volumes)<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>Pub. 229 (formerly H.O. 229, Sight Reduction Tables for Marine Navigation, H.D. 605/NP 401 in the UK, 1970, 6 volumes.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>The variant of HO-229: Sight Reduction Tables for Small Boat Navigation, known as Schlereth, 1983, 1 volume)</li>
<li>H.O. 214 (Tables of Computed Altitude and Azimuth, H.D. 486 in the UK, 1936–46, 9 vol.)</li>
<li>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.)</li>
<li>H.O. 208 (Navigation Tables for Mariners and Aviators, known as Dreisonstok, 1928, 113pg.)</li></ul>
<h2 id="longhand-haversine-sight-reduction">Longhand haversine sight reduction</h2>
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.

<h3>Doniol</h3>
The first approach of a compact and concise method was published by R. Doniol in 1955<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> and involved <a href="/facts/Haversine/MllE4ISd">haversines</a>. The altitude is derived from 
 
 
 
 sin
 ⁡
 (
 H
 c
 )
 =
 n
 −
 a
 ⋅
 (
 m
 +
 n
 )
 
 
 {\displaystyle \sin(Hc)=n-a\cdot (m+n)}
 
, in which 
 
 
 
 n
 =
 cos
 ⁡
 (
 L
 a
 t
 −
 D
 e
 c
 )
 
 
 {\displaystyle n=\cos(Lat-Dec)}
 
, 
 
 
 
 m
 =
 cos
 ⁡
 (
 L
 a
 t
 +
 D
 e
 c
 )
 
 
 {\displaystyle m=\cos(Lat+Dec)}
 
, 
 
 
 
 a
 =
 hav
 ⁡
 (
 L
 H
 A
 )
 
 
 {\displaystyle a=\operatorname {hav} (LHA)}
 
. 
The calculation is:

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

<h3>Ultra compact sight reduction</h3>

A practical and friendly method using only <a href="/facts/Haversine/MllE4ISd">haversines</a> was developed between 2014 and 2015,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> and published in <a href="http://fer3.com/arc/">NavList</a>.
A compact expression for the altitude was derived<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> using haversines, 
 
 
 
 hav
 ⁡
 (
 )
 
 
 {\displaystyle \operatorname {hav} ()}
 
, for all the terms of the equation:

hav
        ⁡
        (
        Z
        D
        )
        =
        hav
        ⁡
        (
        L
        a
        t
        −
        D
        e
        c
        )
        +
        
          (
          
            1
            −
            hav
            ⁡
            (
            L
            a
            t
            −
            D
            e
            c
            )
            −
            hav
            ⁡
            (
            L
            a
            t
            +
            D
            e
            c
            )
          
          )
        
        ⋅
        hav
        ⁡
        (
        L
        H
        A
        )
      
    
    {\displaystyle \operatorname {hav} (ZD)=\operatorname {hav} (Lat-Dec)+\left(1-\operatorname {hav} (Lat-Dec)-\operatorname {hav} (Lat+Dec)\right)\cdot \operatorname {hav} (LHA)}

where 
 
 
 
 Z
 D
 
 
 {\displaystyle ZD}
 
 is the <a href="/facts/Zenith_distance/tcm21e6t">zenith distance</a>,

 
 
 
 H
 c
 =
 (
 
 90
 
 ∘
 
 
 −
 Z
 D
 )
 
 
 {\displaystyle Hc=(90^{\circ }-ZD)}
 
 is the calculated altitude.
The algorithm if <a href="/facts/Absolute_value/dwJBpEs5">absolute values</a> are used is:

if same name for latitude and declination (both are North or South)
 n = hav(|Lat| − |Dec|)
 m = hav(|Lat| + |Dec|)
if contrary name (one is North the other is South)
 n = hav(|Lat| + |Dec|)
 m = hav(|Lat| − |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<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> was developed for a faster solution without calculation, and with an accuracy of 1°.

This diagram could be used also for star identification.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
An ambiguity in the value of azimuth may arise since in the diagram 
 
 
 
 
 0
 
 ∘
 
 
 ⩽
 Z
 ⩽
 
 90
 
 ∘
 
 
 
 
 {\displaystyle 0^{\circ }\leqslant Z\leqslant 90^{\circ }}
 
. 
 
 
 
 Z
 
 
 {\displaystyle 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<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> should be used:

 
 
 
 hav
 ⁡
 (
 Z
 )
 =
 
 
 
 hav
 ⁡
 (
 
 90
 
 ∘
 
 
 ±
 |
 D
 e
 c
 |
 )
 −
 hav
 ⁡
 (
 |
 L
 a
 t
 |
 −
 H
 c
 )
 
 
 1
 −
 hav
 ⁡
 (
 |
 L
 a
 t
 |
 −
 H
 c
 )
 −
 hav
 ⁡
 (
 |
 L
 a
 t
 |
 +
 H
 c
 )
 
 
 
 
 
 {\displaystyle \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)}}}

The algorithm if <a href="/facts/Absolute_value/dwJBpEs5">absolute values</a> are used is:

if same name for latitude and declination (both are North or South)
 a = hav(90° − |Dec|)
if contrary name (one is North the other is South)
 a = hav(90° + |Dec|)
m = hav(|Lat| + Hc)
n = hav(|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.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h4>An example</h4>
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°
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Navigation/sGGAJ0AQ">Navigation</a></li>
<li><a href="/facts/Celestial_navigation/pCpbhzJt">Celestial navigation</a></li>
<li><a href="/facts/Circle_of_equal_altitude/Luhv147W">Circle of equal altitude</a></li>
<li><a href="/facts/Intercept_method/F8bI0XWF">Intercept method</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Navigational Algorithms: <a href="https://sites.google.com/site/navigationalalgorithms/software/Windows">AstroNavigation - Free App for Windows</a></li>
<li>Navigational Algorithms: <a href="https://sites.google.com/site/navigationalalgorithms/downloads">resources for Longhand Haversine Sight Reduction</a></li>
<li><a href="http://fer3.com/arc/">NavList</a> A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Position-Finding</li>
<li><a href="http://celestialtools.webs.com/">Celestial Tools for the USPS/CPS JN/N Student</a></li>
<li><a href="https://tube.geogebra.org/m/1531651">Graphical all-haversine Hc reduction</a> <a href="https://web.archive.org/web/20160516013424/https://tube.geogebra.org/m/1531651">Archived</a> 2016-05-16 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li>
<li><a href="https://play.google.com/store/apps/details?id=com.Sightreduction">Sight Reduction - free App for android</a></li>
<li><a href="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</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">The American Practical Navigator (2002) <a href="/wiki/The_American_Practical_Navigator" target="_blank">/wiki/The_American_Practical_Navigator</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Pub. 249 Volume 1. Stars Archived 2020-11-12 at the Wayback Machine; Pub. 249 Volume 2. Latitudes 0° to 39° Archived 2022-01-22 at the Wayback Machine; Pub. 249 Volume 3. Latitudes 40° to 89° Archived 2019-07-13 at the Wayback Machine <a href="https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%201-2020-Dec.pdf" target="_blank">https://www.thenauticalalmanac.com/Pub.%20249%20Vol.%201-2020-Dec.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Pub. 229 Volume 1. Latitudes 0° to 15° Archived 2017-01-26 at the Wayback Machine; Pub. 229 Volume 2. Latitudes 15° to 30°; Pub. 229 Volume 3. Latitudes 30° to 45°; Pub. 229 Volume 4. Latitudes 45° to 60° Archived 2017-01-30 at the Wayback Machine; Pub. 229 Volume 5. Latitudes 60° to 75° Archived 2017-01-26 at the Wayback Machine; Pub. 229 Volume 6. Latitudes 75° to 90° Archived 2017-02-11 at the Wayback Machine. <a href="https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_1/Pub229Vol1.pdf" target="_blank">https://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_1/Pub229Vol1.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Table de point miniature (Hauteur et azimut), by R. Doniol, Navigation IFN Vol. III Nº 10, Avril 1955 Paper <a href="http://fer3.com/arc/m2.aspx/Table-De-Point-Miniature-R-Doniol-FrankReed-jul-2015-g32063" target="_blank">http://fer3.com/arc/m2.aspx/Table-De-Point-Miniature-R-Doniol-FrankReed-jul-2015-g32063</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Rudzinski, Greg (July 2015). "Ultra compact sight reduction". Ocean Navigator (227). Ix, Hanno. Portland, ME, USA: Navigator Publishing LLC: 42–43. ISSN 0886-0149. Retrieved 2015-11-07. <a href="http://issuu.com/navigatorpublishing/docs/on227_download_edition" target="_blank">http://issuu.com/navigatorpublishing/docs/on227_download_edition</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Altitude haversine formula by Hanno Ix http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121 <a href="http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121" target="_blank">http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Azimuth diagram by Hanno Ix. http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-HannoIx-jun-2015-g31689 <a href="http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-HannoIx-jun-2015-g31689" target="_blank">http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-HannoIx-jun-2015-g31689</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Hc by Azimuth Diagram http://fer3.com/arc/m2.aspx/Hc-Azimuth-Diagram-finally-HannoIx-aug-2013-g24772 <a href="http://fer3.com/arc/m2.aspx/Hc-Azimuth-Diagram-finally-HannoIx-aug-2013-g24772" target="_blank">http://fer3.com/arc/m2.aspx/Hc-Azimuth-Diagram-finally-HannoIx-aug-2013-g24772</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Azimuth haversine formula by Lars Bergman http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441 <a href="http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441" target="_blank">http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">"NavList: Re: Longhand Sight Reduction (129172)". <a href="http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29172" target="_blank">http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29172</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Natural-Haversine 4-place Table; PDF; 51kB <a href="https://yadi.sk/i/4MmOYyXhUshbxA" target="_blank">https://yadi.sk/i/4MmOYyXhUshbxA</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Sight reduction open-in-new

Sight reduction