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
• 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
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.
S1
S2
P
re
d1
d2
(Sender)
(Receiver)
re+ h
Earth (h = ocean height)
d12
r1
r2
CHAMPCHAMP
SAC-C SAC-C
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.
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
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
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
S1
S2
P
re
d1
d2
(Sender)
(Receiver)
re+ h
Earth (h = ocean height)
d12
r1
r2
P
P'
d1d2
d'2d'1
S1S2
d'2 - d2d'1 - d1
sinrdd ii
sin21212 dddd
rPP
S1
S2
P'
re
d'1
d'2cS1
rS1
c
rS2
S2
P'r
P'c
S1
S2
P'
O
sP' s2
s1 re
h2
h1
d'1
d'2
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
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
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
an intersection of 3 quadrics in a special position
S 1
S 2
v
Earth
P
O
u
x2
x1
x3
S 1 S2
Choice of Cartesian coordinate frame
)(||
11
11 OS
OSe
)(||
11
13 ue
uee
132 eee
-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
-2
-1
0
1
2
-1
0
1
2
-2
-1
0
1
2
-2
-1
0
1
2
-1
0
1
2
-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
-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
Cut of plane P with the Earth reference ellipsoid
0cos)()( 12211 aa
exuxuex
Classification of mutual positions of intersecting ellipses 0)()( 22 xgxf
minimum distance between two ellipsoids
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
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
Iterative solution of minimum distancebetween two ellipsoids
as a progression of distances X0X1
X’0X’1 X’’0X’’1 etc
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.
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
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
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).
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.
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