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Sight reduction

## Summary

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

Steps for measuring and correcting Ho using a sextant.
Using Ho, Z, Hc in intercept method.

Given:

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

First calculate the altitude of the celestial body ${\displaystyle Hc}$ using the equation of circle of equal altitude:

${\displaystyle \sin(Hc)=\sin(Lat)\cdot \sin(Dec)+\cos(Lat)\cdot \cos(Dec)\cdot \cos(LHA).}$

The azimuth ${\displaystyle Z}$ or ${\displaystyle Zn}$ is then calculated by:

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

These values are contrasted with the observed altitude ${\displaystyle Ho}$. ${\displaystyle Ho}$, ${\displaystyle Z}$, and ${\displaystyle Hc}$ 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 ${\displaystyle \sin(Hc)=n-a\cdot (m+n)}$, in which ${\displaystyle n=\cos(Lat-Dec)}$, ${\displaystyle m=\cos(Lat+Dec)}$, ${\displaystyle a=\operatorname {hav} (LHA)}$.

The calculation is:

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


### Ultra compact sight reduction

Haversine Sight Reduction algorithm

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

where ${\displaystyle ZD}$ is the zenith distance,

${\displaystyle Hc=(90^{\circ }-ZD)}$ is the calculated altitude.

The algorithm if absolute values 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[7] was developed for a faster solution without calculation, and with an accuracy of 1°.

Azimuth diagram by Hanno Ix

This diagram could be used also for star identification.[8]

An ambiguity in the value of azimuth may arise since in the diagram ${\displaystyle 0^{\circ }\leqslant Z\leqslant 90^{\circ }}$. ${\displaystyle Z}$ 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[9] should be used:

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

The algorithm if absolute values 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.[10][11]

#### An example

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°

## References

1. ^
2. ^ Pub. 249 Volume 1. Stars; Pub. 249 Volume 2. Latitudes 0° to 39°; Pub. 249 Volume 3. Latitudes 40° to 89°
3. ^ Pub. 229 Volume 1. Latitudes 0° to 15°; Pub. 229 Volume 2. Latitudes 15° to 30°; Pub. 229 Volume 3. Latitudes 30° to 45°; Pub. 229 Volume 4. Latitudes 45° to 60°; Pub. 229 Volume 5. Latitudes 60° to 75°; Pub. 229 Volume 6. Latitudes 75° to 90°.
4. ^ Table de point miniature (Hauteur et azimut), by R. Doniol, Navigation IFN Vol. III Nº 10, Avril 1955 Paper
5. ^ Rudzinski, Greg (July 2015). Ix, Hanno. "Ultra compact sight reduction". Ocean Navigator. Portland, ME, USA: Navigator Publishing LLC (227): 42–43. ISSN 0886-0149. Retrieved 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. ^ "NavList: Re: Longhand Sight Reduction (129172)".
11. ^ Natural-Haversine 4-place Table; PDF; 51kB