In tern a tio n a l H yd ro g ra p h ie Review, M o n a c o , LX (1). J a n u a r y 1983
ANALYSIS OF T H E GEOMETRY OF TRANSIT
OF A NAVIGATION SATELLITE
by M a r k S Z Y M O N S K I '* '
IN TR O D U C TIO N
T h e basic p a r a m e te r s ch a ra c te risin g th e c o n d i tio n s in w h ic h m e a s u r e m e n t s to
d e t e r m in e s h ip ’s p o sitio n by m e a n s o f A rtificial Satellites a r e m a d e , a r e d e fin e d a n d
a n a ly s e d in this p a p e r . T h e s e are th e size o f th e se rv ic e a r e a , th e p e r io d o f satellites'
transit t h r o u g h th e serv ice a r e a , the elev a tio n a n g l e in the c u l m in a tio n , a n d the
c o v e r a g e o f th e glo b e w ith th e Satellite N a v ig a tio n S y stem .
T H E G E O M E TR Y O F P A SSIN G
O F A N A VIG A T IO N SA T ELL ITE
T h e u se rs o f Satellite N a v ig a tio n S y s te m s s h o u ld be a w a r e o f th e g e o m e tr y o f
th e m o v e m e n t o f th e Artificial Satellites in th e sh ip 's r a d i o c o m m u n i c a t i o n are a
(service area) a n d the resulting c o n d itio n s o f m e a s u r in g a n d rec eiving th e data
n ec es sary fo r a c c u r a t e d e t e r m in a tio n o f th e sh ip 's p o sitio n in o r d e r to m a x im iz e the
use o f r a d io n a v ig a tio n rec eive rs o f Satellite S y ste m s.
T h e g e o m e tr y is d e p e n d e n t directly u p o n th e o r b it p a r a m e t e r s o f the
navig atio n satellites a n d also th e s h ip ’s p o sitio n rela ted to th e o r b it su rfa ce . B o th the
o r b it p a r a m e te r s o f artificial satellites an d t h e m e t h o d o f m e a s u r e m e n t in flu e n c e the
ex p lo ita tion o f th e ra d io n a v ig a tio n Satellite S y s te m s . T a k in g th e a b o v e into
c o n s id e r a tio n , th e s y s te m s c a n be d iv id e d a s s h o w n in T a b le I.
Table I
C h a ra c te r is tic s o f N a v ig a tio n Satellites
A ltitu de R a n g e s
A ltitu d e R a n g e
A ltitud e (km )
P erio d
L o w ......................................................
M e d i u m ...............................................
S y n c h r o n o u s .....................................
900 - 2,700
13.000 - 20,000
22.000 - 48,000
100 - 150 m in
8 - 12 h
24 h
(*) Institute of Marine Navigation, Gdynia. Poland
T h e A m e r i c a n S y s te m T R A N S I T / N N S S b e lo n g s to th e s y s te m w ith satellites
in lo w altitu d e orbits. T h is s y s t e m is in g e n e r a l use d by m e r c h a n t ships all o v e r th e
w o r l d . T h e te c h n i q u e u se d by T R A N S I T / N N S S m e a s u r e s D o p p le r shift (shift in
f r e q u e n c y d u e to satellite m o ti o n relative to th e n a v i g a to r 's receiver) o f signals
e m a n a t i n g f r o m th e satellite. T h e A rtificial E a r t h Satellites a re p laced in p o la r o r b its
o f a b o u t 1,100 k m a ltitu d e a n d o f ec c e n tric ity e = 0.002 - 0.02. T h e o rb ita l
m o v e m e n t p e r io d o f a n Artificial E a r t h Satellite a m o u n t s to a b o u t 107 min.
T h e S o v ie t S y s te m C Y K A D A , s ta rte d w ith th e satellite KOSMOS-IOOO o n 31
M a r c h 1978, s h o u ld b e m e n t i o n e d h e r e also ( P r a v d a , A p ril 2, 1978 - T A S S
ag e n cy ). It, too , utilizes the d o p p le r te c h n iq u e (“S p u tn ik as N a v ig a to r " by
A . Z e l e z o n o v , L e n i n g r a d P r a v d a , A pril 4, 1978).
T h e s y s te m N A V S T A R / G P S n o w b e in g d e v e lo p e d in th e U S A uses s y n c h r o ­
n ise d c lo c k s in th e satellite a n d sh ip e q u i p m e n t to find th e r a n g e f ro m satellite to
n a v ig a to r . It b e l o n g s to th e sy s te m w ith satellites in m e d i u m aitiiud e orbits. T h e
n a v ig a tio n satellites will be p la ce d in th e ir c irc u la r o r b its a t a n ang le 63° to th e
e q u a t o r , at a n altitu d e o f a b o u t 20,000 k m
T h e S y s te m s M A R I S A T a n d I N M A R S A T a r e s y s t e m s w ith g e o s ta t io n a r y
satellites in high a ltitu d e s y n c h r o n o u s o r b its . A l t h o u g h th e y a r e b eing used e x c l u ­
sively for sea r a d i o c o m m u n i c a t i o n s , t h e possibility o f th e use o f g e o s ta t io n a r y
satellites for n a v ig a tio n p u r p o s e s also c a n n o t be ex c lu d e d .
T h e valu e o f Satellite N a v ig a tio n S y s te m s lies in th e fact th a t th e y p r o v id e
g lo b a l c o v e r a g e w i t h g r e a t a c c u r a c y . T h e g e o m e tr y o f th e re la tio n sh ip b e t w e e n a
sh ip a n d a n a v i g a tio n satellite d e t e r m i n e s th e c o n d itio n s o f receiving r a d io signals
e m a n a t i n g f r o m th e satellite a n d , th e r e f o r e , th e a c c u r a c y o f p o sitio n d e t e r m in a tio n .
T h e c o n d itio n s a s s u ri n g th e r a d i o c o m m u n i c a t i o n o f a satellite w ith a ship a re
c h a r a c t e r i s e d inter alia b y s u c h p a r a m e t e r s as the size o f the service area, the tim e
o f satellite p assing t h r o u g h th e se rv ic e a re a , the e le v a tio n an g le in the c u lm in a tio n ,
a n d c o v e r a g e o f th e g lo b e w ith the Satellite N a v ig a tio n S y ste m .
T o a n a ly s e th e a b o v e p a r a m e t e r s it is c o n v e n ie n t to u se a simpJified m o d e l o f
A rtificial E a r t h Satellite m o v e m e n t s , a n d a s s u m e t h a t satellites m o v e a r o u n d
c i r c u la r o r b its in th e c e n t r e o f w h i c h t h e globe is situated.
F ig u re 1 p r e s e n ts th e se rv ic e a r e a lim ited by a circle w ith ra d iu s 1, a n d
g e o c e n t r ic angle 2 1. T h e p o in t S refers t o th e p o sitio n o f a satellite o n the o r b it at
t h e m o m e n t o f its rise. O u t o f tria n g le OP„S is f o u n d th e f o r m u l a :
(1)
w h e r e lmilx = th e h a l f o f th e g e o c e n tric a n g le SOS, lim iting th e se rv ic e area,
R e = the E a r t h ra d iu s,
Ds = the o rb it altitude.
T i m e t o f satellite p a s sin g in the se rv ic e a r e a is f o u n d f r o m th e e q u a tio n :
t =
w h e r e : Ts = p e r io d o f E a r th passing b y satellite.
/ (Re + D s) 3
Ts = 2 7r V •
(i= c o e ffic ie n t o f th e E a r t h g ra v ita tio n field.
T h e se rv ic e area, h o w e v e r , u n d e r th e ac tu a l c o n d itio n s o f r a d io rec e p tio n , is
a little lim ited. T h is lim itation is d u e to th e in flu e n ce o f t r o p o s p h e r ic refractio n,
in te rru p tin g th e r e c e p tio n o f satellite signals o n e le v a tio n s sm a lle r th a n a b o u t 5°. As
a result o f th e a b o v e th e ang le 1, as it is illustrated by Fig. 1, will be limited a n d will
be f o u n d f r o m th e f o rm u l a :
Re
j^arc cos
(3)
cos h
R* + d s
w h e r e h = e le v a tio n angle lim iting
r e c e ip t o f r a d io signals.
the
n o n - in te r r u p te d
tr o p o s p h e r ic
refra ctio n
T h e g e o m e tr y o f a satellite’s tran sit p r e s e n te d in Fig. 1 refers, h o w e v e r , to a
p a rtic u la r c a se w h e r e th e o b s e r v e r is in th e orbit surface.
F i g u r e 2 illu s tra te s a m o r e g e n e r a l c a s e o f n a v i g a t i o n satellite o b s e r v a t i o n
f r o m a s h i p far f r o m th e o r b i t s u r f a c e at a n a n g l e <P. F o r ¢ = 0 ° th e sa te llite ’s t r a c k
r e f l e c t e d o n th e E a r t h s u r f a c e is t h e a r c A 'L 'B ' o f a g r e a t cir cle. W h e n th e s h i p
m o v e s a w a y f r o m th e o r b i t s u r f a c e th e t r a c k b e c o m e s th e a r c o f a sm all c ircle
( a n a l o g o u s w i t h p arallels). F r o m Fig. 2 :
OL
(4)
cos 1 = ■
w h e r e : O L = a s e g m e n t o n th e o r b i t s u r f a c e f o r m e d f r o m t h e c u t t i n g o f th e h o r i z o n
s u r f a c e w i t h t h e line j o i n i n g t h e E a r t h c e n t e r w i t h th e p o i n t S„ o f
sa tellite c u l m i n a t i o n .
T h e f o l l o w i n g e q u a t i o n s re s u lt f r o m t h i s :
R e cos h
- h
I = arc cos
( R e + D s ) cos 0
F i g . 2. -
T h e s e rv i ce a r e a for t h e s h i p f ar f r o m t h e o r b i t s u r f a c e at a ng l e ¢ .
(5)
for the p as sin g tim e o f th e A rtificial E a r t h Satellite in th e se rv ic e a re a :
Ts J arc cos
r—
R^cosh— n
h
L ( R e + Ds) cos <t>J
t = ---------------------- V ------- --------- 1------------180
(6 )
T h e fo llo w in g p a r a m e t e r c h a r a c te r is in g th e g e o m e t r y o f t h e rela tio n sh ip b e t w e e n a
ship a n d a satellite is th e elev a tio n angle o f the satellite in th e c u l m in a tio n hnuiv.
F r o m Fig. 3. the e le v a tio n hm;u is :
h max = arc ta n
F i g . 3. - T h e g e o m e try o f the relation an d navig atio n satellite (S).
a ship (P0)
w h e r e ; xs, zs = t h e C a r t e s i a n c o o r d in a te s o f th e satellite p o sition.
T h e c u r v e s o f 1, t, hmax f r o m th e a n g le <t> for b o th th e low a n d th e m e d i u m
orb ita l s y s te m s a r e p r e s e n te d in Fig. 4.
T h e p a r a m e t e r s o f th e T R A N S I T s y s te m , as a n e x a m p l e o f solving th e S y s te m
w ith satellites in lo w a ltitu d e orbits, a n d o f N A V S T A R , a s an e x a m p l e o f th e
S y s te m using satellites in m e d iu m altitu d e orbits, w e r e u s e d f o r c o m p u t a t i o n s . T h e
c o r r e s p o n d i n g d a ta are p r e s e n te d in T a b le II.
In th e use o f the S y ste m s m e n tio n e d in T able II, the limitations o f th e service
a r e a o f the n av ig atio n satellites u n d e r the tw o different m e th o d s o f d e te r m in a tio n
o f ship's p o sition o u g h t to be e x a m i n e d separately.
Tn th e first case o f satellites in low altitude orbits (dopp ler te c h n iq u e ) the
service a re a is limited b o th d ue to th e influence o f tro p o s p h e r ic refra ctio n a n d to
th e effect o f cro ss-tra c k e rro r. T h e accuracy o f the d e te r m in a tio n o f a ship's
T ab le II
T h e d a ta for c o m p u t i n g the f o rm u la s
S y ste m
R«
T R A N S I T . 6,370 k m
N A V S T A R . 6,370 k m
(5), (6), (7)
Ds
h
Ts
1,100 km
20,240 km
0°
5°
107 min
12 h
position d e p e n d s p rim a rily on the pass elevation angle. Satellite passes at an
elev a tio n ang le lo w e r t h a n 10° an d hig h e r than 80° at closest a p p r o a c h to the
re c e iv e r d o n o t a f f o rd a sa tisfacto ry a c c u r a c y o f th e positions.
B ut the m e a s u r e m e n t s are c o n tin u o u s ly m a d e d u r in g the w h o le tran sit o f the
satellite o n th e o b s e r v e r 's side. If the e lev a tio n angle at th e p o in t o f clo se st a p p r o a c h
to th e re c e iv e r is h ig h e r t h a n 10° a grea t n u m b e r o f m e a s u r e m e n t s (in relation to
the n u m b e r r e q u ir e d to d e te r m in e th e s h ip ’s p osition) a r e a c c o m p lish e d , m ostly
u n in t e r r u p t e d by tr o p o s p h e r ic r efra ctio n as th e Artificial E a r t h Satellite r e m a in s
lo n g e r o n th e e le v a tio n angle h > 5 ° tha n v e ry low o v e r th e h orizo n .
It s h o u ld be a d d e d th a t the p r o b le m s o f the se rv ic e a r e a lim itation d u e to the
in te r ru p tio n o f th e signal rec ep tio n a n d th o se d u e to th e m inim izing o f the e r r o r o f
the s h ip ’s po sitio n d e t e r m in a tio n are different. T h e c o o r d in a te s o f the ship's p o sitio n
m a y be o b ta in e d if the signal is u n in te r r u p te d . But th e u se rs s h o u ld e v a lu a te the
use fulne ss o f the o b s e r v a tio n s fro m the g e o m e tr y o f tra n sit o f a satellite.
In the s e c o n d case, w h e n s im u lta n e o u s m e a s u r e m e n t s a r e m a d e aga inst
se v eral (3 - 4) Artificial E a r th Satellites, th e r e is a lim ita tio n on th e m i n i m u m
elev a tio n o f the satellites being o b se rv e d . F o r m in im iz in g th e e r r o r o f positio n
d e t e r m in a tio n in the m e a n - s q u a r e d e r r o r sense, the c rite ria G D O P = m in i m u m
s h o u ld be ta k e n into c o n s id e r a tio n , too.
In the ca se w h e n th e ship is in the su r fa c e o f th e o r b it ( ¢ = 0°). in the S y s te m
T R A N S I T th e s e rv ic e a re a is 2 1= 64°, i.e. it c o v e r s a s u r fa c e o f the g lo b e lim ited by
a circle w ith a r a d iu s e q u a l to a b o u t 1,900 n m . It is not la rg e w h e n c o m p a r e d w ith
the service a r e a for th e satellite o f N A V S T A R sy ste m , a m o u n t i n g to 2 1 = 1 4 4 .8 °
w h ic h c o r r e s p o n d s to the surfa ce on a g lo be limited b y a circle w ith a r a d iu s o f
4,300 nm .
It sh o u ld be stressed th a t th e satellite in a lo w a ltitu d e o r b it is in the service
a r e a for a b o u t 19 m in u te s , w hile th e satellite in m e d i u m altitude orb it m a y be
o b s e r v e d for a b o u t 5 hours.
It sh o u ld be n o te d th a t the g e o s ta tio n a r y satellite p laced in the altitu d e o f
25,000 k m c o v e r s an a re a w ith a radius o f a b o u t 4,700 n m , i.e. f ro m th e E q u a t o r
to latitude ± 7 8 ° .
But w hile the satellite in th e low altitude o rb it, a t a given m o m e n t , c o v e r s a
m u c h sm aller service a r e a th a n the s ta tio n a ry satellite, its fast r e v o lu tio n a r o u n d th e
e a r th in 24 h o u r s p e r m its n a vigation o n p ractically the e n tir e global surface.
T h e se rvic e a re a also passes a c c o rd in g to the m o v e m e n t o f the Artificial E a r t h
Satellite. T h e passing is fo u n d fro m :
A1 ( T s) = arc sin
Jjin(
360°
------- •T cos ¢) • sin i
24
s
( 8)
w h e r e qj = g e o g r a p h ic a l latitude,
i = incline o f o r b it plane.
O n the E q u a t o r , d u r in g o n e p erio d o f t h e n a v ig a tio n satellite’s passing a r o u n d
the g lobe the se rvic e a re a in the low altitude orb it will c h a n g e b y 26° (Ts = 107 m in.,
i = 90°), b u t for a satellite in m e d iu m altitu de orb it /11 a m o u n t s to 180° (T, = 12 h,
i = 63°).
T h e c o v e r a g e o f th e globe by R a d io n a v ig a ti o n Satellite S y s te m is t o u n d I r o m
1
2 1 --------A l
A = k ----------------------
(9)
21
w h e r e k = th e n u m b e r o f satellites o n o n e orbit,
N = t h e n u m b e r o f orbits.
T h e c u r v e s o f d e p e n d e n c e A l a n d A f r o m g e o g r a p h ic a l latitude are p r e s e n te d in
fig. 5.
It a p p e a r s th a t o n the E q u a t o r o n e satellite o f th e T R A N S I T S y s te m gives the
c o v e r a g e in 5 8 % , w h ile in the case o f th e N A V S T A R S y s te m th e s a m e c o v e r a g e
will be p ossible for 3 satellites p la ce d o n e by o n e in 3 o rb it surfaces. In b o th c a se s
th e c o v e r a g e is sim ilar, b u t in t h e first case the satellite a p p e a r s period ically o v e r a
c h o s e n p o in t o f the g lob e , a n d in the s e c o n d case it c a n be se en at a g iv e n p o in t o f
th e g lo be a n d th e n at least o n e A rtificial E a r th Satellite is visible c o n t in u o u s ly .
medium
a lt it u.d*
C O N C L U SIO N S
T h e a b o v e c o n s id e r a t io n s p o in t o u t the clear r e la tio n sh ip b e t w e e n th e c h o ic e
o f p a r a m e t e r s o f n a v ig a tio n satellite o r b its a n d th e m e a s u r e m e n t m e th o d u se d in a
g iv e n sy ste m .
One of the basic requirem ents to select a m ethod o f m easurem ent is to kn ow
the orbit param eters and the n u m b e r o f satellites in the system.
T hat is w h y Satellite Systems can be divided in tw o essential groups :
- systems with only periodical observation o f Artificial Earth Satellites.
- systems with continuous observation of one or several satellites sim ulta­
neously.
In using satellites in low altitude orbits quick changes o f m easured param eters
(doppler frequency shift o r range) are achieved. T h e use of the m easurem ent of
doppler frequency shift is very useful here. The transmitting radio station on the
navigation satellite an d receiving equipm ent on a ship are rather simple, and w hat
is im portant is that the requirement concerning ihe stability o f the reference
oscillators in the satellites as well as in the receivers is met by classical crystal
oscillators.
T he main limitations in the use of the low orbit Navigation Satellite Systems
are :
-
relatively great discretion o r care in observations which w ould be
unnecessary if the n u m b e r o f satellites w ere increased;
the dependence on ship's m o v e m e n t in the accuracy of ship's position
determ ined by satellite navigation.
The system of navigation satellites in m edium altitude orbits seems to have the
greatest potential because o f the possibility o f practically continuous observation
and because results are independent o f ship's m ov em ent. T he sim ultaneous
observation o f several satellites in a given region o f the globe is assured with the
establishment of a reasonable quantity of Artificial E a rth Satellites. For exam ple, in
the System N A V S T A R /G P S , 18 satellites moving su b s yn c hron ously six by three
a ro u n d orbit planes assure the observation o f 3 sim ultaneous m easurem ents from
at least 3 satellites w hich are in the service areas (4 = 3.47 for ^ = 0°). and
24 satellites in this system w ould assure the sim ultaneous observation o f at least 5
navigation satellites (4 = 4.63 for ^ = 0 °).
It seems to be m ost useful to use the range m etho d in this case because the
m easurem ents o f doppler frequency shift are limited by a small (in c om parison with
low altitude orbit) relative velocity betw een the satellite and the receiver, and a long
period o f the satellite's passing th ro u g h the service area.
A system with geostationary satellites, in turn, does not assure the coverage
of subpolar regions, and near the equator the accuracy o f position determination
deteriorates. T hus, the system o f geostationary satellites can only be used for sea
navigation purposes in conjunction with satellites in orbits o f a different type.
REFERENCES
[1] B o g e n . A H. (1974) : G eo m e tric p e rf o rm a n c e o f the G lobal Positioning System. 21 Ju n e.
AD-783 210.
[2] H o p f i e l d , Ft.C. (1963): T h e effect o f tro po sp he ric refraction on the d op pler shift o f a
satellite signal. J o u rn a l o f Geophysical R e s e a r c h , Vol. 68, N o . 18, pp. 80-93.
[3]
B a r t h o l o m e w , C.A. (1978): Satellite frequency stan dard s. Navigation, J o u rn a l o f the
in stitu te o f N avigation, Vol. 25, N o. 2, S u m m er, pp. 113-121.
[4]
E a s t o n , R.L. (1975): T he role o f tim e /fre q u e n c y
in N a v y
Proceedings o f the IE E E , Vol. 60. N o . 5, M ay, pp. 1-16.
[5]
M a r t i n , E.H. (1978): G P S user e q u ip m e n t error models. Navigation, J o u rn a l o f the
In stitu te o f Navigation, Vol. 25, No. 2. S u m m e r, pp. 201-211.
[6] S t a n s e l l , T . A . (1978): T h e T r a n s i t N a v i g a t i o n
p. 140.
N av igation
Satel li te S y s t e m ,
Satellites.
R /5 9 3 3 . O c t o b e r ,
[7] Sz y m o n s k i , M ., J u r d z i n s k i , M. (1 980): Satellite system s in sea navigation, G d a n s k ,
ISBN 83-215-1111-2, p. 147.