geometry , accuracy, and position of ocean reflecting points in bistatic satellite altimetry
DESCRIPTION
GEOMETRY , ACCURACY, AND POSITION OF OCEAN REFLECTING POINTS IN BISTATIC SATELLITE ALTIMETRY. J. Klokočník, J. Kostelecký , M. Kočandrlová. IAG International Symposium: Gravity, Geoid and Space Missions – GGSM2004, Porto, Portugal, 30 th August – 3 rd September, 2004. Authors. - PowerPoint PPT PresentationTRANSCRIPT
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GEOMETRYGEOMETRY, ACCURACY, AND POSITION , ACCURACY, AND POSITION OF OCEAN REFLECTING POINTS IN OF OCEAN REFLECTING POINTS IN BISTATIC SATELLITE ALTIMETRYBISTATIC SATELLITE ALTIMETRY
J. Klokočník, J. Kostelecký, M. Kočandrlová
IAG International Symposium:Gravity, Geoid and Space Missions – GGSM2004,Porto, Portugal, 30th August – 3rd September, 2004
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• Jaroslav Klokočník, CEDR - Astronom. Inst. Czech Acad. Sci., Ondřejov Obs., Czech Republic, [email protected]
• Jan Kostelecký, CEDR- Res. Inst. Geod. Zdiby & CTU Prague, Fac. Civil Eng., Czech Republic, [email protected]
• Milada Kočandrlová, CTU Prague, Fac. Civil Eng., Dept. Mathem., Czech Republic, [email protected]
Authors
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Abstract• We analyse time and space distribution of specular points P in
bistatic altimetry (BA) between LEO (e.g. CHAMP or SAC-C) and HEO (GPS, GALILEO).
• We clearly demonstrate significantly higher number and density of reflecting points P in the case of BA in a comparison with traditional monostatic radar nadir altimetry.
• We present accuracy assessments for position of reflecting points, accounting for measurement (delay) error and orbit errors of senders (GPS) and receiver (CHAMP)
• First attempts at determination of position of P on a reference surface different from a sphere.
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S1
S2
P
re
d1
d2
(Sender)
(Receiver)
re+ h
Earth (h = ocean height)
d12
r1
r2
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CHAMPCHAMP
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SAC-C SAC-C
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Formulae to compute position of the reflecting point on a sphere by
approximationsIterative solution for position of reflecting point PP, on the sphere –
see Wagner and Klokočník, 2003:
11
1221
211
21121
,sincos/sincossinsin
,.cos
,sin/sinsinsinsinsin
P
P
P
rr
the angles i are computed from measurements and orbit informationiteratively, the radius-vectors ri follow from POD of Si.
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Accuracy assessment for height of reflecting points on a sphere accounting for measurement (delay) error
and orbit errors of senders (GPS) and receiver (CHAMP)
approach I
given:
error of τ = t1+t2-t12
orbit errors of senders and receiver
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irrhh ,,,, 21
hh
rr
hr
r
hh 2
21
1
222
2
2
1
2
2
2
2
2
1
2 ,2,1
21 m
h
r
mh
r
mhm
r
hm
r
hm xx rr
rrh
sin2
cos2
cossin
212
22
1 rrrr
rr
r
hii
i
i
with δ = 1, -1 for i = 1, 2
sin2
cos2
sincos
212
22
1
122
rrrr
rr
h
sin2
1h
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Accuracy assessment for height of reflecting points on a sphere accounting for measurement (delay) error
and orbit errors of senders (GPS) and receiver (CHAMP)
approach II
given:
error of (d1+d2),
orbit errors of senders and receiver
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S1
S2
P
re
d1
d2
(Sender)
(Receiver)
re+ h
Earth (h = ocean height)
d12
r1
r2
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P
P'
d1d2
d'2d'1
S1S2
d'2 - d2d'1 - d1
sinrdd ii
sin21212 dddd
rPP
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S1
S2
P'
re
d'1
d'2cS1
rS1
c
rS2
S2
P'r
P'c
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S1
S2
P'
O
sP' s2
s1 re
h2
h1
d'1
d'2
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sinrdd ii
sin21212 dddd
r
222
2
2121sin4
1ddddr mmm
2
1
22
2
2
1
222222
2 41
sin4
121
ii
cS
ii
cSi
rSddr iii
mk
mmmm
w h e r e 1
21,cosh
hhk
hr
r
ie
ei
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Vertical position error of reflecting points between GPS and CHAMP, std of measurement = 20 mm
0
20
40
60
80
100
120
140
160
180
200
20 30 40 50 60 70 80 90
gamma [deg]
err
or
[mm
]
sd: CH=100/200, GPS=50/100
sd: CH=50/200, GPS=50/100
sd: CH=50/100, GPS=50/100
sd: CH=20/50, GPS=50/100
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Seeking Reflecting Points on Reference Ellipsoid
-5
0
5
-2
0
2
4
6
-6
-4
-2
0
2
-5
0
5
-2
0
2
4
6
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an intersection of 3 quadrics in a special position
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S 1
S 2
v
Earth
P
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O
u
x2
x1
x3
S 1 S2
Choice of Cartesian coordinate frame
)(||
11
11 OS
OSe
)(||
11
13 ue
uee
132 eee
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-20
2
0
5
10
15
20
-2
0
2
-20
2
0
5
10
15
20
1:2
23
2
22
2
21
1 b
x
b
x
a
xQ
Ellipsoid of revolution for reflecting points
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-2
-1
0
1
2
-1
0
1
2
-2
-1
0
1
2
-2
-1
0
1
2
-1
0
1
2
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-10
-5
0
5
10-5
0
5
10-5
0
-10
-5
0
5
10
23
22
21
222211 )(cos))( xxexuxuex
||cos|)(:| 112 SXuSXQ
Rotational cone surface of reflected signals
S1= vertex
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-10
-5
0
5
10
0
10
20
-5
0
-10
-5
0
5
10
21
222211 )(cos))(( a
a
exuxuex
Intersection of ellipsoid of revolution with the cone resulting in a plane ellipse P
P
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Cut of plane P with the Earth reference ellipsoid
0cos)()( 12211 aa
exuxuex
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Classification of mutual positions of intersecting ellipses 0)()( 22 xgxf
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minimum distance between two ellipsoids
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Principle of solution
Correct [theoretical] result:
touch of two ellipsoids Q0 and Q1
• Practical result (due to observing errors): imaginary or real intersection of the two ellipsoids
• Possible solution: seeking of minimum distance between the two ellipsoids
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Algorithm of solution
10 , FF matrices of ellipsoids 10 ,QQ
10 ,OO centers of ellipsoids 10 ,QQ
1,0,10 iXQOO ii
1||, ii nn vector in normal direction iQ in iX
ii nvtnnv 10 tangent vector iQ
Tiii
Tiii
i tFt
nFxr radius of normal curvature
iQ in iX in direction it
iiii nrXO centre of curvature iQ
RPXQXQXXXXX 011001010 ,,inf
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Iterative solution of minimum distancebetween two ellipsoids
as a progression of distances X0X1
X’0X’1 X’’0X’’1 etc
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Conclusion• BA between LEO and HEO may yield many more
reflecting points than traditional altimetry of LEO• If the technology can be proven, the space BA promises
a distinct gain in coverage of the oceans at fine scales in time and space in comparison with traditional altimetry
• Accuracy of reflecting points decreases only slowly with off-nadir angles γ
• In total error budget at a centimeter level, the orbit errors of HEO and LEO must be accounted for together with a measurement error
• cont.
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cont., Conclusion II
• Mathematical model for determination of position of reflecting point on reference rotational ellipsoid utilizes mutual position between two ellipses. Ellipse 1 is intersection of cone of rotation (with vertex in S1) and ellipsoid of rotation around S1S2. Ellipse 2 is in the same plane as Ellipse 1 and is intersection of this plane and reference ellipsoid of the Earth. Position of P on this ellipsoid is found iteratively.
• Another iterative solution (without any cone): distance between two ellipsoids
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BA has potentially many geo-applications: mesoscale eddies, ocean surface roughness, winds, mean sea surface, sea-ice, namely in polar areas
Space data of sufficient accuracy is urgently needed
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Literature• Komjathy A., Garrison J.L., Zavorotny V. (1999): GPS: A new tool for Ocean science, GPS World, April, 50-56.
• Lowe et al (2002): 5-cm precision aircraft ocean altimetry using GPS reflections, Geophys. Res. Letts. 29:10.
• Martin-Neira, M. (1993): A passive reflectometry system: application to ocean altimetry, ESA Journal 17: 331-356.
• Ruffini, G., Soulat, F. (2000): PARIS Interferometric Processor analysis and experimental results, theoretical feasibility analysis, IEEC-CSIC Res. Unit., Barcelona, PIAER-IEEC-TN-1100/2200, ESTEC Contr. No. 14071/99/NL/MM, ftp://ftp.estec.esa.nl/pub/eopp/pub/
• Truehaft, R., Lowe, S., C. Zuffada, Chao, Y. (2001): 2-cm GPS-altimetry over Crater Lake, Geophys. Res. Letters 28:23, 4343-4346.
• Wagner, C., Klokočník, J. (2001): Reflection Altimetry for oceanography and geodesy, presented at 2001: An Ocean Odyssey, IAPSO-IABO Symp.: Gravity, Geoid, and Ocean Circulation as Inferred from Altimetry, Mar del Plata, Argentina.
• Wagner, C., Klokočník, J. (2003): The value of ocean reflections of GPS signals to enhance satellite altimetry: data distribution and error analysis, J. Geod. (in print).
• Zuffada, C., Elfouhaily, T., Lowe, S. (2002a): Sensitivity Analysis of Wind Vector Measurements for Ocean Reflected
GPS Signals, it Remote Sensing Env. (in print).
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Acknowledgments
• This research has been supported by the grant LN00A005 (CEDR) provided by Ministry of Education of the Czech Republic and by the grant of GA AV ČR number 3003407
• We thank Carl A. Wagner, Cinzia Zuffada, Markus Nitschke,
Giulio Ruffini and Martin Wiehl for consultations/literature.
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Reflection Point Problemspherical and ellipsoid case
in Bistatic Satellite Altimetry
anonymous FTP: sunkl.asu.cas.cz
cd pub/jklokocn/ PPT_BA_PORTO.ppt
The End