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SCALING A SATELLITE T R I A N G U L A T I O N NET
WITIt LASER RANGE MEASUREMENTS*)
KURT LAMBECK
Smithsonian Astrophysical Observatory, Cambridge**)
INTRO DUCq'ION
An i m p o r t a n t characteristic o f the satellite t r i a n g u l a t i o n m e t h o d is that a n u n s c a l e d g e o m e t r i c
reference s y s t e m c a n be e s t a b l i s h e d that is entirely free f r o m terrestrial geodetic m e a s u r e m e n t s .
Because this characteristic s h o u l d be m a i n t a i n e d , because the p r o p a g a t i o n o f v a r i a n c e s t h r o u g h
large terrestrial t r i a n g u l a t i o n s o f t e n b e c o m e s u n c e r t a i n , a n d because limited practical d a t a are
c u r r e n t l y available o n the p r o p a g a t i o n o f v a r i a n c e s t h r o u g h the satellite t r i a n g u l a t i o n nets,
it is desirable to keep terrestrial geodetic a n d satellite geodetic m e a s u r e m e n t s as s e p a r a t e entities
until the final m e r g i n g o f t h e two s y s t e m s into a single unified reference s y s t e m is a t t e m p t e d .
Terrestrial m e a s u r e m e n t s s h o u l d therefore n o t be u s e d to scale the spatial t r i a n g u l a t i o n . A n
i n d e p e n d e n t m e t h o d s h o u l d be used t h a t can, at a later stage, be c o m p a r e d with the s u r f a c e
m e a s u r e m e n t s so as to give the greatest possible insight into the variance p r o p a g a t i o n c h a r a c t e r istics o f b o t h s y s t e m s .
A simple a n d precise m e t h o d for scaling
o b s e r v a t i o n s m a d e s i m u l t a n e o u s l y f r o m two
a n d lasers for t h e ranges. C u r r e n t l y the m o s t
c a m e r a s , a n d o n e or b o t h h a v i n g laser r a n g e
t h e spatial net is to i n t r o d u c e r a n g e a n d direction
or m o r e s t a t i o n s u s i n g c a m e r a s for the directions
likely a p p r o a c h is to use two s t a t i o n s , b o t h h a v i n g
equipment.
To e n s u r e s i m u l t a n e o u s events, the o b s e r v a t i o n s f r o m the two s t a t i o n s s h o u l d be m a d e over
the s a m e s h o r t arc of a visible satellite pass a n d a n interpolation m a d e for a synthetic o b s e r v a t i o n
at a fictitious instant. If m o r e directions t h a n range m e a s u r e m e n t s are m a d e d u r i n g t h e arc,
a greater c o m p a t i b i l i t y o f the accuracies o f the two types of o b s e r v a t i o n c a n also be achieved:
laser range o b s e r v a t i o n s are currently a c c u r a t e to a few m e t e r s [3] while single direction m e a s u r e m e n t s are reliable to a b o u t 1.5 to 2 arcsec, a n d s y n t h e t i c directions are accurate to a b o u t 0.7
to 1-0 arcsec.
1. A C C U R A C Y O F D E T E R M I N I N G
THE DISTANCE
BETWEEN TWO STATIONS
Synthetic simultaneous observations are assumed to have been computed, and range
measurements from one station only are first considered. With the two stations
denoted by P. and Pb, the satellite position by Ps, the observed range from the first
station to the object by r~,, and the angles subtended at any one of the three positions
by the other two points by O, the !ength r.b of the station-station vector can be expressed by r.b = G~[sin (0~ + Ob)/sin Obj.
*) P r e s e n t e d at the C o n f e r e n c e on Scientific R e s e a r c h e s U s i n g O b s e r v a t i o n s o f Artificial
Earth Satellites, Prague, C z e c h o s l o v a k i a , April 1968.
**) Address: C a m b r i d g e , M a s s a c h u s e t t s 02138, U . S . A .
StLIdia geoph, et geod. 12 (1968)
339
K. Lambeck
Applying the law of propagation of variances to this expression gives
() t;
z=
7~. .b
(1)
zz + COt" ( 0 . +
r .s
0b)~0,,+
2
sin z 0.
.
sin 2 0 b sin" (0 a + Oh) ~ " "
Instead of the angles 0, the directions of the station-satellite vector are observed,
k----/
and the accuracy of the direction of the station-station vector can be expressed by
a.b = a~n
, then a 0.
z = sin -2 0.(a 2 + aj0 ) and gob
- = sin-2 00(~r2 _, aob)
.2"
If the direction of the station-station vector has been previously determined from
simultaneous direction observations that are independent of the directions observed
to determine the scale of this vector, the accuracy of the station-station vector direction can be approximated by [1] a~b = ~r~/0"lVn, where n is the n u m b e r of pairs
of observations used to determine the direction of the station-station vector.
Equation (1) can now be written as
r/.b
\r/.s
[_
sin 20.
sin*0bsin z ( 90 ~
0hi
1 +
0"
1
,
Or, assuming that n is large so that 1/(0-17n) ~ I, and that the range a n d direction
observations are of equal accuracy, that is (a,/r).s = % the simplified expression
for the accuracy of r.b becomes
(2a)
(/[
~ "
\ r/.b
1 + cot-' (0. + 0b)
sin 20.
+
sin-" 0.
sin* 0b sin-" (0. + 0b)
-
,
or per definition
(2b)
(a,/r)~b = S' 2 .
The factor S' is a measure of the precision of tile transfer of the scale o f the stationsatellite vector P~ to the scale of the station-station vector.
A similar expression to equation (2) can be derived if ranges have b e e n observed
[ 2h = S " a:,
"~ where S t ' = ( A C - B'-)/(A + C
f r o m b o t h s t a t i o n s . Now,(a~,r).
2B) and
A=I+
cot 2 (0. + 0b)
sin 20.
+
sin 20~
sin* 0h sin 2 (0. + Oh)'
B=-co:(0o+ 0~)+ sin* 0.
+ sin* 0~
sin ~ 0. sin 30 b
C=I+
cot-' (0. + 0b)
s i l l 2 0t,
340
+
sin-' 0b
sin 4 0 a sin z
(0 a + Oh)
S t u c H a g e o p h , e t g e o d . 1:~ (1968)
Scaling a SatetHte TriangulationNet...
The S factors, or "scale-transfer precision factors", can be evaluated for variable 0,
or alternativel? for variable satellite height, H, distance of the vector PAPA. L, and
zenith distance z, of the observation. Fig. 1 indicates results for the ratio L/H = 1
and for both the S' and S" factors.
ji.1
/
t2
"~" ~.
20 /
/
//i
//,
ii
\
\\
~,l
Fig, I. Scale-transfer factor as a function of tke subsatellite positions. T o p figure is for range
measurements from P ~ only. Lower figure is for range measurements from both stations.
From figures such as Fig. l, the optimum geometry could be selected for any
particular combination of stations and satellites; in most cases, however, of greater
interest that the discrete values of S are the average values of these factors for observations distributed throu~tlout the coverage area c o m m o n to the two stations.
Both the average scale-transfer precision factors, S, and the size of the c o m m o n
coverage area, 8A, are functions of the parameters L, H, and the maximum zenith
distances, : . . . . at which observations can be made. The frequency with which
Studia geoph, et geod. 12 (1968)
341
K. Lambeck
a particular object can be observed from a pair of stations is to a first approximation
proportional to the size of the common coverage area, so that an o p t i m u m combination for these parameters can be reached by minimizing the ratio S/6A.
While direction observations to satellites can be made for zenith distances up to
about 75 ~ laser measurements are generally restricted to zenith distances below 60 :,
either by atmospheric absorption of the laser signal or by the orientation of the
satellite's corner reflectors relative to the line of sight.
If, for range observations from one station (P~) only, the laser measurements are
restricted to zenith distances below (z,)m~, for each value of L/H there exists an
optimum value of the maximum
i
zenith distance (%),,,~ below which
observations from the second station
(Pb) should be avoided. This is the
value for which the ratio "3'/~A' is a
minimum for given values of (z~),,,x
and L/H. Fig. 2 illustrates the results.
If range observations are observed
60 ~
from the two stations, the maximum
zenith distances of observations will
generally be the same f o r both stations.
On the assumption t h a t for each
60
70
(Za).,ax the optimum value of (zb),,. ~
(ZB)m o x
is selected, the ratio g/6A can be
evaluated for different values of L/H,
Fig. 2. O p t i m u m c o m b i n a t i o n s o f zmax for range
from one station.
and the optimum c o m b i n a t i o n of
values for L and H determined.
Results for S'/'6A' - ranges observed from one station only - are presented in
Fig. 3; and results for S"/6A" ranges observed from both. stations - a r e given in
Fig. 4. The optimum values of L. H, and z,,,x are those for which the a b o v e ratios
are mimima], and both cases indicate an almost linear relationship between the
optimum values of L/H and (z,) . . . . which for all practical purposes is independent of H. This relationship can for both instances be written as (z in degrees)
L/H(I + 0"25) = 0"03(z,)m~x - 0-3, 500 < H < 2000 km, 50~ < ( z , ) , ~ < 70 ~
For'laser observations confined to (z~)m,X = 60 ~ the optimum value of L/H is
about 1.5 4- 0.4. Unfortunately, however, the station locations and available satellite
orbits are predetermined so that this optimization of parameters is not always possible.
The relationships between S and L/H are expressed in Fig. 5 for ranges observed
from one station (P.) only, and in Fig. 6 for ranges observed from b o t h stations.
For practical purposes these relationships can be expressed by two or three linear
equations. For example, if the range is observed from one station ordy and the
342
Studia geoph, et geod. IZ (1968)
Scalinq a Satellite Triangu[atiot Net...
m e a s u r e m e n t s f r o m this station are confined to elevations above 30 ~ R' is given by
(3)
$' =
- 9 ' 8 ( I - 0"75L/H)
5-5
s.o(1 - o-75c/;H)
2.0 < L/H < 3"0
1-2 < L/H < 2-0
0-s < L/H < a.2
and the accuracy with which ,',~ can be determined is given by (a,/r)~ o = S / n x ~r2.
For ranges observed from both stations and confined to elevations above 30 ~ .~"
is given by
s'=
4.i(1 - 0-25c/H)
[19-3(1
0"70L/H)
1'2 < L/H < 3.0
0-7 < L/H < 1.2.
~0.-/
1-23~
o
5
30 iJ
zma , : 65 ~
kin
!
kC
= 500
o
5
~m
-
'
50F
kzH
20I
z ma~
= ~5 ~
x
"~
= 500
km
000
2000
5
9
J-~
S'/SM
Fig. 3. S'/SA' as a function of L and H for three values of z,n=~.
Studia
geoph,
et geod.
12 (1963)
3d.3
K. L a n l b e c k
Expressions like the above are important for assessing the accuracy with which the
scale of geometrical triangulations can be determined, for deciding the optimum
configurations, and for estimating the total number of observations required to scale
the net with a preassigned accuracy.
3OF-
f
L/H
L_
~
iza)mox 50~
=
o-
-
'~
5CO
~m
/
~ =500 ~m
2000
o
:
. . . . . . .
~
.....................
~o
i
1
29
~z~)nax 75~
=
L/N
~I 1
o ~~ o
,
,,-__~
;-
~
o 5oo
~"/3A"
k~
_ _ _
_ _ J
Fig. 4. S"/~5.-4 expressed in
terms of L , H, and zm~x.
2. FREQUENCY OF OBSERVING SATELLITES SIMULTANEOUSLY
FROM THE TWO STATIONS
The fl'equency, f, with which a satellite can be observed from two stations can be
expressed as f = 1 4 4 0 p ~ p , ( A t ) - t passes/day, where p~ is the probability of the
satellite being in the c o m m o n coverage area of the two stations, Pv is the average
344
Studia
geoph,
et
geod.
12 (1968)
Scaling a Satellite Triagnulation Net...
probability of the satellite being visible, and At is the average duration of a satellite
pass through the common coverage area.
Fig. 7 presents the average frequencies of satellite passes simultaneously visible
from two stations for bottt passive and active observations confined to zenith distances
below 60 ~ These results are valid for a mean station latitude of 30 ~ and art object
in a near polar orbit. Data for computing similar frequencies for different station
latitudes and orbita! inclinations are given in [2].
3o
~
H
)
_98
Q,L ~'=55
3
L
......................
-D - - - - - -
~___
J
30
20
7 ILIH)~ 55
.,oL
Z mQ1
~'~
L__
.
.
.
.
.
255
(1_0.7
.
=
50~
L/HI
y-
-
,
Fig. 5. Function ~" expressed with (za)~ x as independent parameter.
i
+
t.!~
S ~4
i . 0 25 I,./H)
--~......
-S"-~ 93
zma
i ~
_O7
_,H
50
z
O
0-'-
I
_
20
Fig. 6. S" expressed as a function of L, H, and z'm~.
S t u d i a g e o p h , et g e o d .
1Z
(1968)
345
K. Lambeck
If the scale is to be d e t e r m i n e d w i t h an a p r i o r i
accuracy estimate of
r e q u i r e d n u m b e r o f o b s e r v a t i o n s u n i f o r m l y d i s t r i b u t e d t h r o u g h the c o m m o n
err~r, t h e
cove)age
a r e a is g i v e n by
(4)
'~
r 7
S- t cri/(erHr)-,
n =
where er, is the a c c u r a c y o f a single o b s e r v a t i o n . Since the d i r e c t i o n c o m p o n e n t s
o f the o b s e r v e d s t a t i o n - s a t e l l i t e v e c t o r will g e n e r a l l y be less precise t h a n the o b s e r v e d
m a g n i t u d e o f this vector, the a c c u r a c y o f the f o r m e r q u a n t i t y s h o u l d b e used.
T h e total o b s e r v i n g time, N,
required
to
obtain
these
4~ F
n
k
o b s e r v a t i o n s is g i v e n by N =
=
n/[f(l
- ,:~)],
the ' p r o b a b i l i t y
where
of
3 O
~\
is
.~
f a i l i n g to
PASSIVE SATELLITE
zm~ , = 6 0 "
i \\
LIH
Z
%
O
~
= 30 ~
i
= 90*
o b t a i n successful o b s e r v a t i o n s
a n d w o u l d i n c l u d e losses incurred
by
weather,
mal-
functioning of camera,
and
timing
lack
of
laser
equipment,
simultaneity
of
0
and
O5
I;5
I,O
20
AVERAGE NUMBER O~ VISIBLE PASSES/DAY
the
7
'k \
observations.
,aCTIVE SATELLITE
i
= 90 ~
L/H
t
Fig. 7. Frequency of a satellite
pass simultaneously visible fro.m
two stations.
3.
A
0
(3~
0.5
I0
1.5
2D
AVERAGE NUMBER OF VISIBLE PASSES/DAY
THEORETICAL
EXAMPLE
Consider the two SAO Baker-Nunn station sites at New Mexico and Florida, the former
having also a laser. The chord distance between them is about 2600 kin, and their mean latitude
is about 30 ~ north. The distance between these stations is to be determined to an accuracy of about
1 in 500,000, or 5.0 m, by means of combined laser and direction observations. The direction
between these stations has previously been determined to a high accuracy by use o f simultaneous
observations. As already mentioned, each laser range measurement is precise t o a few meters
or less, whereas the accuracy of synthetic Baker-Nunn observations is about 1.0 arcsec. Observations are confined to zenith distances below 60 ~.
The satellite ro be used in Geos 1, which is in an orbit of 60 ~ inclination with a n average height
of about 1700 kin. Thus, L/H is about 1'5, or near the optimum value. Then t h e average scaletransfer precision factor as given by Eq. (3) is S' = 5-5. The number of observations required
is therefore (Eq. (4)) n = (5.5 • 25)/4 = 35.
346
Studia
geoDl'x, e t
g e o d . I Z (1968)
Scaling a Satellite Trianoulation ,Vet...
Table 1. S u m m a r y o f d a t a used in solution.
Station-station vector directton solution
Covariance matrix
(arcsec)2
Direction cosines
Station-pair
9001--9010
Event
a n d station
l
m
n
6
.9654357
--.169444
--,2001585
~[(cos 6) R . A.]
0.0523
--0,0402
--0.0402
0.0739
Simultaneous direction and range observations
.....................
G
Direction
cosines
Range
Along
Across
c,
l
~n
(Mm}
track
track
(m)
l
9001
9010
.04295280
--.74037368
---76021340
.35207940
2-0866960
3-4
0-8
0.9
0-9
20
2
9001
9010
.54823639
--,47916428
---53516116
--.33229905
2.3777680
0.7
l-1
0.6
0.8
20
3
9001
9010
-10317028
---71112051
--.71131741
--.33336747
2.1219470
1.2
1.0
0.6
0,5
2-0
4
9001
9010
"26667862
--.60702012
---5470~_8
--,26326729
2-2995600
l-0
1.0
0'4
0-8
2-0
5
9001
9010
"31376141
--'56729617
---49048331
---23550709
2.3798430
1.2
0.8
0-3
1-5
2,0
6
9001
9010
'48832989
---52707160
---54177821
---30673143
2"2959200
0.9
1"4
1.2
1.0
2-0
A p p r o x i m a t e station coordinates ( M m )
9001
9010
X
Y
Z
--1.5357590
0.9762820
--5.1669950
--5"6013890
3.4010410
2.8802420
Studia geoph, et geod, 12 (196a)
347
K. Lambeck
Table 2. S u m m a r y of results of the computation.
Corrections
9010
to
station coordinates
AX
AY
AZ
(m)
(m)
(m)
--27.7
--56
t5.0
Computed distan~c
900t--9010
2 , 6 0 2 005 ~. 12 m
Covariance matrix o f adjusted station-station vector
a~ir
~r/r
8(c~/r)
-d(tT./r)
- - 2 2 " 8 ~'z 10 - 1 2
(err/r) d
--2'8
(~r r)
x 10 - 1 2
--2-5 x
10 - 1 2
--3"4
X 10 - 1 2
--1.8
•
10 - t 2
~-3"2 x
10 - t 2
T h e average frequency of an intervisible Geos 1 pass is 0.37 per day if observations are made
to a passive object, and 0-46 per day if observations are made to an active objec t (that is, the
flashes are observed for the direction determination, and blind firing methods are used to obtain
the ranges).
T h e number of observing nights required is therefore: N ,= 3 5 / [ 0 . 3 7 ( 1 - - ~t)] = 95 nights
for passive observations, and N =: 3 5 / [ 0 . 4 6 0 - - ~)1 = 76 nights for active observations.
4. S O M E
RESULTS
Before the cameras and laser at the two stations mentioned in the previous section
were moved to new sites, they made a few simultaneous range and direction observations to the satellites Geos 1 and DI-C. Six successful simultaneous events were
recorded over an observing period or about 1 month. Although this number is small,
it serves to illiustrate how powerful this method of scaling is, the data yielding an
accura.cy that is of the same order as the scale determination from conventional
triangulation over continental distances, i.e., about five parts in 10 6.
Numerous simultaneous direction observations have been made between the two
stations over a period of several years, and these data have been used t o determine
the direction of the station-station vector. This solution, together with its covariance
matrix, and the simultaneous range and direction observations have been used to
determine new estimates for the components of the station-station vector. Table 1
gives the relevant observational data, and Tab. 2 presents a summary o f the results
of the computations. The distance obtained from the laser measurements is
348
Studla geoph, et geod. 12 (1968)
Scaling a Satellite Trianqulation Vet...
2,602,005 4- 12 m, scaled to the velocity of light. The distance of the same line can
also be c o m p u t e d from the S m i t h s o a i a n Institution 1966 Standard Earth I-4] as
2,601,976 scaled to a gravitational c o n s t a n t of G M - 398 603.2 x l0 ~ m 3 sec -2.
The terrestrial survey between the two stations yields and arc distance of 2,601,970 m.
Because of the limited available data. the only conclusion that can be d r a w n is that
the results of the laser range m e a s u r e m e n t s are in good agreement with the S t a n d a r d
Earth scale. O n l y when more m e a s u r e m e n t s become available, particularly from
stations in different parts of the world, will a precise scale d e t e r m i n a t i o n be possible.
The SAO now has lasers operating at three stations - Hawaii, Arizona, and Athens,
a n d it is anticipated that several more distance determinations will be made in the
fut-tre to enable the satellite t r i a n g u l a t i o n net to be precisely scaled.*)
Reciewer: 3,[. Bur.~a
Received 29.4. 1968
ReJ?rences
[I] K. Lambeck: Optimum Station-Satellite Configurations for Simuhaneous Direction
Observations. SAO Spec. Rep. No 231, (I966).
[2] K. Lambeck: Precise Geodetic Position Determination with the Aid of Artificial Earth
Satellites: the Geometric Solution. Dissertation, Dept of Surv. and Geod., Oxford 1967.
[3] C. G. kel-.r.L.A. Maestre, P. H. A n d e r s o n : Satellite Ranging with a Laser and the Correction for Atmospheric Refraction. Proc. of the Int. Syrup., Figure of the Earth and Refraction. Vienna, March 1967, Sonderheft Osterr. ZfV. 25 (1967), 163.
[4] C. A. L u n d q u i s t , G. Veis editors: Geodetic Parameters for a 1966 Smithsonian Institution
Standard Earth. SAO Spec. Rep. 200, vol. 1, (1966), 11.
Pe3roste
O I - I P E ~ E 2 I E H H E M A C U I T A B A CH-YTHI4KOBOIY[ TPI, I A H F Y S I I t ~ H I 4
H O i I A 3 E P H B I M H A B ~ t O ~ E H H I I M PACCTOIIHI417f
KuRr LAMBECK
C.utuncot,oaclca.,~ AcmpooSu~uuecna.e 06cepeamopu.'L [<e.udptld34r Macca~ycemc
O6cy',Ka.~rorca pe3y,-mrarbt CuKxpo)iKbLXaa6nro.a~'a~tR Hanpam'IeuHit )t paccToRmtgt IzlC3, BI,Inom~emu,tx C~wrcouoBcKoR Acrpod~3n~ecKo.R O6cep~trop~e~ (CAO)co cny'rm~o~bix CTaHtt~fff ~- 9001 (Opra~ Hac) rt J~_ 9010 (lOrmTep), a /Iaerca non~trKa gcnora,3osas~st ;maepmax
ua6.tmamt~i'~,manKonrpo;cqCeTltCIIyTI-II[KOBOt~Tp~a.~ryJLm.g~tCAO. PaCCMaTp!4BalOTCItaeKOTOpble
BoIIpocb[ TOt~HOCTI~ c~Iy~HI[KOBbIX Tpaaltry;Lm.ia~, [IOCTpOeHHbIX rio orFrgqecKm'vt Ha6moaemam,~
TOIIO~ZKTpttt~CKtIX uanpaB;teKrtRs no na3eph'bLM Ha6JItO/J, ellHItM TOriotleNrp~NecmlxpaccTom{ai:L
Hocryanno 29.4.1968
*) This work was supported in part by grant no. NsG 87-60 from the National Aeronautics
and Space Administration.
S t u d l a g e o p h , et g e o d . 12 (1968)
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