Intercept method: Difference between revisions

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TheIn "Intercept[[astronomical Method"navigation]], orthe "Marcq St Hilaire'''intercept method"''', asalso itknown isas also'''Marcq ratherSt. inaccuratelyHilaire knownmethod''', is an astronomical [[navigation]]a method of calculating an observer's position on earthEarth ([[geopositioning]]). It was originally called the ''azimuth intercept'' method because the process involves drawing a line which intercepts the [[azimuth]] line. This name was shortened to ''intercept'' method and the ''intercept distance'' was shortened to "'intercept"'.

The method yields a [[line of position]] (LOP) on which the observer is situated. The intersection of two or more such lines will define the observer's position, called a "fix". Sights may be taken at short intervals, usually during hours of twilight, or they may be taken at an interval of an hour or more (as in observing the Sun during the day). In either case, the lines of position, if taken at different times, must be advanced or retired to correct for the movement of the ship during the interval between observations. If observations are taken at short intervals, a few minutes at most, the corrected lines of position by convention yield a "fix". If the lines of position must be advanced or retired by an hour or more, convention dictates that the result is referred to as a "running fix".

== Summary ==

The intercept method is based on the following principle.

The actual distance from the observer to the geographical position ('''GP''') of a celestial body (that is, the point where it is directly overhead) is "measured" using a [[sextant]]. The observer has already estimated his position by [[dead reckoning]] and calculated the distance from the estimated position to the body's GP; the difference between the "measured" and calculated distances is called the intercept.

[[Image:Diagram showing GP distance = ZD.jpg|thumb|right|500px]]

The diagram on the right shows why the zenith distance of a celestial body is equal to the angular distance of its GP from the observer's position.

The rays of light from a celestial body are assumed to be parallel (unless the observer is looking at the moon, which is too close for such a simplification). The angle at the centre of the earthEarth that the ray of light passing through the body's GP makes with the line running from the observer's [[zenith]] is the same as the zenith distance. This is because they are [[corresponding angles]]. In practice it is not necessary to use zenith distances, which are 90° minus altitude, as the calculations can be done using observed altitude and calculated altitude.

Taking a sight using the intercept method consists of the following process:

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==Methodology==

[[Image:Intercept Sight Reduction-00.png|thumb|304px|right|Diagram illustrating the intercept sight reduction process]]

Suitable bodies for celestial sights are selected, often using a Rude Star Finder. Using a [[sextant]], an altitude is obtained of the sunSun, the moonMoon, a star or a planet. The name of the body and the precise time of the sight in [[UTC]] is recorded. Then the sextant is read and the altitude (''Hs'') of the body is recorded. Once all sights are taken and recorded, the navigator is ready to start the process of "[[sight reduction"]] and plotting.

The first step in sight reduction is to correct the sextant altitude for various errors and corrections. The instrument may have an error, IC or index correction (Seesee article on [[sextant#AdjustmentSextant adjustment|adjusting a sextant]]). Refraction by the atmosphere is corrected for with the aid of a table or calculation and the observer's height of eye above sea level results in a "dip" correction, (as the observer's eye is raised the horizon dips below the horizontal). If the Sun or Moon was observed, a semidiameter correction is also applied to find the centre of the object. The resulting value is "observed altitude" (''Ho'').

Next, using an accurate clock, the observed celestial object's geographic position (''GP'') is looked up in an almanac. That's the point on the Earth's surface directly below it (where the object is in the [[zenith]]). The latitude of the geographic position is called declination, and the longitude is usually called the [[Hourhour angle]].

The first step in sight reduction is to correct the sextant altitude for various errors and corrections. The instrument may have an error, IC or index correction (See article on [[sextant#Adjustment|adjusting a sextant]]). Refraction by the atmosphere is corrected for with the aid of a table or calculation and the observer's height of eye above sea level results in a "dip" correction, (as the observer's eye is raised the horizon dips below the horizontal). If the Sun or Moon was observed, a semidiameter correction is also applied to find the centre of the object. The resulting value is "observed altitude" (Ho).

Next, the altitude and azimuth of the celestial body are computed for a selected position (assumed position or AP). This involves resolving a spherical triangle. Given the three magnitudes: local hour angle (''LHA''), observed body's declination (''dec.''), and assumed latitude (''lat''), the altitude ''Hc'' and azimuth ''Zn'' must be computed. The local hour angle, ''LHA'', is the difference between the AP [[longitude]] and the hour angle of the observed object. It is always measured in a westerly direction from the assumed position.

Next, using an accurate clock, the observed celestial object's geographic position (GP) is looked up in an almanac. That's the point on the Earth's surface directly below it (where the object is in the [[zenith]]). The latitude of the geographic position is called declination, and the longitude is usually called the [[Hour angle]].

The relevant formulas (derived using the [[Spherical trigonometry#Identities|Sphericalspherical trigonometric identities]]) are:

Next, the altitude and azimuth of the celestial body are computed for a selected position (assumed position or AP). This involves resolving a spherical triangle. Given the three magnitudes: local hour angle (LHA), observed body's declination (dec.), and assumed latitude (lat), the altitude Hc and azimuth Zn must be computed. The local hour angle, LHA, is the difference between the AP [[longitude]] and the hour angle of the observed object. It is always measured in a westerly direction from the assumed position.

: {{<math|size=large|>\sin(Hc) {{=}} \sin(lat) &middot;\cdot \sin(dec) + \cos(lat) &middot;\cdot \cos(dec) &middot;\cdot \cos(LHA)}} </math>

The relevant formulas (derived using the [[Spherical trigonometry#Identities|Spherical trigonometric identities]]) are:

: <math>\begin{align}

: {{math|size=large|sin(Hc) {{=}} sin(lat) &middot; sin(dec) + cos(lat) &middot; cos(dec) &middot; cos(LHA)}}

:\tan(Zn) = <math>\mathrm{tan(Zn \pm 180) &= \frac{\sin(LHA)}{\sin(lat) \cdot \cos(LHA) - \cos(lat) \cdot \tan(dec)}}</math> \triangleq tanZ \\

Z &= arctan(tanZ) \text{ } \in [-90,+90] \\

Zn &= \begin{cases}

Z & \text{if }LHA \in [0,90] \\

Z+180 & \text{if }LHA \in [90,270]\\

Z+360 & \text{if }LHA \in [270,360] && \equiv \text{ mod 360}\\

\end{cases} \\

\end{align}</math>

:The adjustment from Z to Zn (which is in <math>[0,360]</math>, and measured from North) has two reasons:

: <math>\mathrm{tan(Zn) = \frac{sin(LHA)}{sin(lat) \cdot cos(LHA) - cos(lat) \cdot tan(dec)}}</math>

::(1)The angles in [0,360] with the same <math>\tan</math> is not unique (since <math>\tan(X) = tan (X \pm 180)</math>), but <math>\arctan</math> is defined only in <math>[-90,90]</math>.

::(2)The negative angle must be adjusted to positive angle.

or, alternatively,

: <math>\mathrmbegin{ align}
\cos(\pm Zn) &= \frac{\sin(dec) - \sin(lat) \cdot \sin(Hc)}{\cos(lat) \cdot \cos(Hc)}}</math> \triangleq cosZ \\

Z & = \arccos(cosZ) \text { } \in [0,180] \\

Zn &= \begin{cases}

+Z & \text{if }LHA \in [180,360] \\

-Z+360 & \text{if }LHA \in [0,180] \\

\end{cases} \\

\end{align}</math>

:The adjustment for disambiguating <math>\cos</math> values has similar reasons.

Where

:''Hc'' = Computed altitude

:''Zn'' = Computed azimuth (Zn=0 at North)

:''Z'' = preliminary result for Zn (in some nautical almanacs)<ref name="Training_Movie_CelNav">{{cite web

| title = Celestial Navigation

| website = youtube.com

| date = 11 January 2015

| url = https://www.youtube.com/watch?v=fn9xMkNUMmY&t=1592s

| at = about 26m32s (1h33m31s)

| language=en

| access-date=July 1, 2022

}}</ref>

:''lat'' = Latitude

:''dec'' = Declination

:''LHA'' = Local Hour Angle

Since recently theseThese computations can be done easily done using electronic calculators or computers but traditionally there were methods which used loglogarithm or haversine tables. Some of these methods were H.O. 211 (Ageton), Davies, [[haversine]], etc. The relevant [[haversine]] formula]] for ''Hc'' is

: <math>\mathrmoperatorname{ haversinhav}(\overline{Hc}) = haversin\operatorname{hav}(LHA) \cdot cos(lat) \cdot cos(dec) + haversin\operatorname{hav}(lat \pm dec) }</math>

Where ''{{overline|Hc}}'' is the zenith distance, or complement of ''Hc''.

''{{overline|Hc}}'' = 90° - ''Hc''.

The relevant formula for Zn is

: <math> \mathrmoperatorname{ hav}(Zn) = \frac{ \cos(lat - Hc) - \sin(dec)}{2 \cdot \cos(lat) \cdot \cos(Hc)} }</math>

When using such tables or a computer or scientific calculator, the navigation triangle is solved directly, so any assumed position can be used. Often the dead reckoning DR position is used. This simplifies plotting and also reduces any slight error caused by plotting a segment of a circle as a straight line.

With the use of astroastral navigation for air navigation, faster methods needed to be developed and tables of precomputed triangles were developed. When using precomputed sight reduction tables, selection of the assumed position is one of the trickier steps for the fledgling navigator to master. Sight reduction tables provide solutions for navigation triangles of integral degree values. When using precomputed sight reduction tables, such as H.O. 229, the assumed position must be selected to yield integer degree values for ''LHA'' (local hour angle) and latitude. West longitudes are subtracted and east longitudes are added to ''GHA'' to derive ''LHA'', so AP's must be selected accordingly. When using precomputed sight reduction tables each observation and each body will require a different assumed position.

Professional navigators are somewhat splitdivided in usage between sight reduction tables on the one hand, and handheld computers or scientific calculators on the other. EitherThe methodmethods isare equally accurate. It is simply a matter of personal preference which method is used. An experienced navigator can reduce a sight from start to finish in about 5five minutes using nautical tables or a scientific calculator.

The precise ___location of the assumed position has no great impact on the result, as long as it is reasonably close to the observer's actual position. An assumed position within 1 degree of arc of the observer's actual position is usually considered acceptable.

The calculated altitude (''Hc'') is compared to the observed altitude (''Ho'', sextant altitude [(''Hs]'') corrected for various errors). The difference between ''Hc'' and ''Ho'' is called "intercept" and is the observer's distance from the assumed position. The resulting line of position (''LOP'') is a small segment of the [[circle of equal altitude]], and is represented by a straight line perpendicular to the azimuth of the celestial body. When plotting the small segment of this circle on a chart it is drawn as a straight line, the resulting tiny errors are too small to be significant.

Navigators use the memory aid "computed greater away" to determine whether the observer is farther from the body's geographic position (measure intercept from ''Hc'' away from the azimuth). If the ''Hc'' is less than ''Ho'', then the observer is closer to the body's geographic position, and intercept is measured from the AP toward the azimuth direction.

The last step in the process is to plot the lines of position ''LOP'' and determine the vessel's ___location. Each assumed position is plotted first. Best practise is to then advance or retire the assumed positions to correct for vessel motion during the interval between sights. Each LOP is then constructed from its associated AP by striking off the azimuth to the body, measuring intercept toward or away from the azimuth, and constructing the perpendicular line of position.

To obtain a fix (a position) this ''LOP'' must be crossed with another ''LOP'' either from another sight or from elsewhere e.g. a bearing of a point of land or crossing a depth contour such as the 200 metre depth line on a chart.

==Sights==

Until the age of satellite navigation ships usually took sights at dawn, during the forenoon, at noon (meridian transit of the Sun) and dusk. The morning and evening sights were taken during twilight while the [[horizon]] was visible and the stars, planets and/or moonMoon were visible, at least through the telescope of a [[sextant]]. Two observations are always required to give a position accurate to within a mile under favourable conditions. Three are always sufficient.

==Running fix==

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== See also ==

* [[celestialCelestial navigation]]

* [[Circle of equal altitude]]

* [[navigationSight reduction]]

* [[Intersection (air navigation)]]

* [[latitude]]

* [[Intersection (land navigation)]]

* [[longitude]]

* [[Navigation]]

* [[latitudeLatitude]]

* [[Longitude]]

* [[Haversine formula]]

* [[Longitude by chronometer]]

==References==

{{Reflist}}

*''Nicholls's Concise Guide, Volume 1'', by Charles H. Brown F.R.S.G.S. Extra Master

*''Norie's Nautical Tables'', edited by Capt. A.G. Blance

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==External links==

* ''Navigational Algorithms'' http://sites.google.com/site/navigationalalgorithms/

*[https://web.archive.org/web/20080430060810/http://www.smartcomsoftware.com/winastro.html WinAstro sight reduction software]

* Correction to the sextant altitude http://es.wikipedia.org/wiki/Archivo:CorrecionHs.jpg

* Marcq St Hilaire intercept for the line of position http://es.wikipedia.org/wiki/Archivo:MarcqSaintHilaire.jpg

*[http://www.smartcomsoftware.com/winastro.html WinAstro sight reduction software]

*[http://www.navigation-spreadsheets.com/navigation_triangles.html Navigation Spreadsheets: Navigation Triangles]