389
1 INTRODUCTION
From the time of French Admiral Marcq St. Hilaire
sailors inherited and successfully used a method of
determining the position of the ship, a method that
bears her name and is known in the Anglo-Saxon
language as the "Intercept Method".
The classical method, as it exists today, consists in
calculating the elements of two astronomical lines of
position (LOP) that can be determined on the basis of
simultaneous observations, as a result the ship’s
position being a “fix” or on the basis of time-delayed
observations, the ship’s position being a "running
fixed".
As it is known, the calculations are based on a DRP
(Dead Reckoning Position). In the case of
simultaneous observations, the calculations
performed to determine the two LOPs are affected by
the error induced by DRP. This means that we
introduce this error for twice.
The method proposed by this paper reduces the
errors by half given that the second LOP is calculated
starting from a point located on the first LOP which
means that it can be assimilated to a fixed position.
This point is the intersection point of the intercept
with the azimuth.
The introduction of Intercept Point coordinates in
the calculation of the second LOP leads to an increase
in the accuracy of the final calculation of the ship's
position coordinates.
Intercept Point coordinates are calculated using the
Plane Navigational Triangle from the mathematics
used by navigators and known as "Sailings".
The elements to be determined are difference of
latitude and difference on longitude which will be
applied to DRP coordinates.
Combined Method Of Sight Reduction
A. Buslă
Constanta Maritime University, Constanta, Romania
ABSTRACT: As ships and maritime transport have evolved, knowledge of navigation methods has also
evolved, reaching today modern means that require less of the skills and time of navigators to determine the
position of the ship on sees and oceans.
However, the IMO resolutions maintain the obligation for seafarers to know the procedure for deter-mining the
position of the ship based on the use of astronomical position lines, a process known simply as the "Intercept
Method".
As is well known, the classical "Intercept Method" involves a graphical stage aimed to determine the
geographical coordinate of Fix position.
This paper presents a combined method which eliminates the graphical construction which may involve
plotting errors. The method introduces mathematical computation of fix geographical coordinates.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 15
Number 2
June 2021
DOI: 10.12716/1001.15.02.16
390
With the obtained results the second Plane
Navigational Triangle to find the coordinates of the
ship's fix.
As can be seen, the graphic constructions are
missing.
The concept of the problem can be the object of a
computer program. [1–4]
2 PRINCIPLE OF THE METHOD
The classical “Intercept Method” involves the
following steps:
− taking the sight of two celestial bodies and then
sight reduction; as a result, the intercept and
azimuth are obtained
− the two lines of position (LOP) are plotted using
intercept and azimuth
− the fix position of the vessel is given by the
intersection point of the two LOPs.
− the geographical coordinates of the fix are
extracted using graphical procedure
The method proposed by this paper takes the first
steps presented above and eliminates the graphical
constructions. The geographical coordinates of the fix
are determined by computation. In this way the
graphical plotting errors are eliminated and accuracy
of fix determination increases enough.
In many cases the sight reduction starts from
estimated position (EP), from dead reckoning position
(DRP) or from assumed position (AP). All of them
include errors. The both LOPs are affected by the
errors of EP, DRP or AP.
This method uses geographical coordinates of DRP
only for the first LOP. To reduce the second sight the
coordinates of the first intercept point (IP1) will be
used. As a result, the origin of the second azimuth
(Zn2) will be placed just in the first intercept point.
The IP is plotted on the azimuth toward the
celestial body (CB) if the intercept is positive or away
the CB if intercept is negative.
To compute the Fix coordinates simple
trigonometric formulas are used. They are the same as
those used by “Plane Sailing”. A few words about the
“Plane sailing”.
The Earth surface is considered being a plane
surface. In this way the navigational triangle (see
picture bellow) is a plane right triangle. Its elements
are:
Figure 1. The navigational triangle
A – departure position
B – arrival position
C – course angle
Dist – distance to be travelled
Dep – departure = distance measured on the arrival point
parallel between the meridians of departure and arrival
positions
DLat – difference of latitude
Formulas:
( )
( )
( )
cos
sin
/
DLat Dist C
Dep Dist C
DLon Dep sec Lat med
=
=
=
where:
DLon = difference of longitude
Latm = half the aritmetical sum of the departure
position latitude (LatA) and the arrival position
latitude (LatB).
The arrival point coordinates are computed as
follows:
B
A
B
A
Lat Lat DLat
Lon Lon DLon
=+
=+
3 CLASSICAL METHOD
On November 20, 2020, at the chronometer time
CT=19h20m16s in DR position: Lat =41°47.8’ N ; Lon =
029°34.6’ W the following sights were taken and
recorded: Altair star - observed altitude Ho= 51°30.2'
and Vega star – observed altitude Ho=58°39.9'. Height
of eye is 14 m and index correction IC = +1.5'. The
chronometer correction is CC= + 00m00s. The
geographical fix coordinates are required.
Table 1. Shorted computation
_______________________________________________
Date Nov. 20, 2020 Nov. 20, 2020
_______________________________________________
Body ALTAIR VEGA
Ho 51
o
30’2 58
o
39’9
Hc 51
o
28’3 58
o
37’7
Intercept(a) +1,8 Nm +2,2 Nm
Zn 217
o
,9 278
o
,5
_______________________________________________
391
1
1
1
2
2
2
41 47',8
029 34',6
1,8
217 9
2,2
278 5
Lat N
DPR
Lon W
a Nm
LOP ALTAIR
Zn
a Nm
LOP VEGA
Zn
=
=
=+
=
=+
=
Figure 2. Graphic representation
0',5
3',1
DLat
Diff
DLon
=−
=−
Solution:
41 47',8
-0',5
41 47',3
-029 34',6
-3',1
029 37',7
41 47',3
029 37',7
DRLat
DLat
Lat N
DRLon
DLon
Lon W
Lat N
Fix
Lon W
= +
+=
=
=
+=
=
=
=
4 COMBINED METHOD
As we have stated above, the second azimuth will be
plotted from the first intercept point (IP1). As a result,
the graphical constructions will be changed.
Figure 3. Graphic representation
We can see the DRP, the first azimuth (Zn1), the
first intercept (a1), the first intercept point (IP1) and
the first LOP (LOP1).
The second azimuth (Zn2) is plotted from the first
intercept point (IP1) not from DRP. Being a point on
the first LOP (LOP1) the IP1 is more accurate than
DRP.
Algorithm:
1. reduce the first sight using the sin Hc
2. compute the geographical coordinates of the first
intercept point (IP1)
3. reduce the second sight by means of the sin Hc
formula using the IP1 coordinates instead of DRP
coordinates
4. solve the right triangle formed by IP1, IP2 and Fix
points to compute the size of Dist2
5. compute the geographical coordinate of Fix
position.
1. Reduce the first sight
The computation was done separately and recorded in
the Shorted Computation table: intersect (a1) = 1,8 Nm
and azimuth (Zn1) = 217°9.
2. Compute the geographical coordinates of the first
intercept point (IP1)
We will consider DRP as departure point (DP) and IP1
as arrival point. Also, the first intercept (a1 = 1,8 Nm)
will be considered as distance to be travelled (Dist1)
and the direction of first azimuth (Zn1=217°,9) as
course angle (C1).
− DLat computation:
1
1
cos DLat Dist C=
( ) ( )
1 ,8 cos 217 ,9 1 ,8 0,789084 1',4DLat= = − =−
392
− Departure computation:
1
1
sin DLat Dist C=
( ) ( )
1 ,8 sin 217 ,9 1 ,8 0,6142852 1',1Dep= = − =−
− Latitude of IP1 computation:
( )
1
1
41 47',8 1',4 41 46',4
Lat DRLat DLat
IPLat
=+
= + − =
− DLon computation:
( )
sec
m
DLon Dep Lat=
1
41 47',8 41 46',4 83 34',2
= 41 47',1
2 2 2
m
DRLat IPLat
Lat
+
+
= = =
( )
-1',1 sec 41 47',1 -1',1 1,3411109 -1',5DLon = = =
1
34',6-1',5 - 36',1-029 029LonIP = =
1
41
029
46',4
36',1
N
I
o W
Lat
Ln
P
=
=
3. Reduce the second sight
(separate computation using sin Hc formula)
2
2
1,3
278 ,6
a Nm
LOP
Zn
=+
=
4. Solving the right triangle formed by IP1, IP2 and
Fix points
We can observe that this triangle is a right triangle.
We know:
− the side IP1/IP2 is equal to
2
1,3a Nm=+
− the angle formed by LOP1/LOP2/Fix is equal to
difference in azimuths
21
278 ,6 217 ,9 60 ,7Zn Zn Zn = − = − =
− the hypotenuse LOP1/Fix (labeled Dist2) can be
computed as follows:
( )
22
1',3 60 ,7 1,3 1,1466978 1,5Dist a cosec Z cosec Nm= = + = = +
− orientation of LOP1 can be considered the direction
of course from DRP to Fix (C2)
21
90 217 ,9 90 307 ,9C Zn= + = + =
5. Compute the geographical coordinate of Fix
position
To compute the Fix geographical coordinates, we
will consider the first intersect point (IP1) as a
departure point (DP) and the Fix position as an arrival
point (AP).
We have the following elements:
− departure point coordinates:
41 46',4 ; 029 36',1Lat N Lon W= =
− distance to be traveled (Dist2=1,5Nm )
− course angle (C2=307°,9)
1. DLat computation:
( )
( )
22
cos
1,5 cos 307 ,9 1,5 0,61422852 0',9
DLat Dist C
DLat
=
= = =
2. Dep2 computation:
( )
( ) ( )
22
2
sin
1,8 sin 307 ,9 1,5 0,789084 1',2
DLat Dist C
Dep
=
= = − = −
3. Lat of Fix computation:
provides the fixed position of the ship based on
measured inputs.
1
41 46',4 0',9 41 47',3
Fix
Fix
Lat IPLat DLat
Lat N
=+
= + =
d. DLon computation:
( )
( )
2
1
41 46',4 41 47',3 83 33',7
= 41 46',9
2 2 2
-1',2 sec 41 46',9 -1',2 1,3410412 -1',6
-029 36',1-1',6 029 37',7
41 47',3
029 37',7
m
Fix
m
Fix
DLon Dep sec Lat
IPLat Lat
Lat
DLon
Lon W
Lat N
Fix
Lon W
=
+
+
= = =
= = =
= =
=
=
5 CONCLUSION
The results obtained by using the two methods are
equal which demonstrates the accuracy of the
combined method.
The proposed calculation method brings two major
advantages:
− downsize of calculation errors of the two
simultaneous LOPs
− elimination of errors created by graphic plotting.
The content of the method is in fact a step forward
in the process of improving the use of astronomical
navigation in our days by increasing the degree of
accuracy in determining the astronomical position of
the ship.
To facilitate the use of method, it is recommended
to prepare by the time observation and computation
sheets whose fields should be completed step by step
in real time.
Mastering the method but especially its application
does not mean a return to the past but rather an
additional safety measure in keeping the navigation
accurate.
393
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