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, jklokocn@asu.cas.cz

• Jan Kostelecký, CEDR- Res. Inst. Geod. Zdiby & CTU Prague, Fac. Civil Eng., Czech Republic, kost@fsv.cvut.cz

• Milada Kočandrlová, CTU Prague, Fac. Civil Eng., Dept. Mathem., Czech Republic, kocandrlova@mat.fsv.cvut.cz

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