GEOMETRY , ACCURACY, AND POSITION OF OCEAN REFLECTING POINTS IN BISTATIC SATELLITE ALTIMETRY

1. 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, 30th August – 3rd September, 2004

2. Authors • 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

3. 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.

4. S (Sender) 2 d 12 a (Receiver) d S 2 2 1 d 1 a 1 g P g r 2 r + h e r 1 r e b b 1 2 Earth (h = ocean height)

5. CHAMP

6. SAC-C

7. Formulae to compute position of the reflecting point on a sphere by approximations

17. 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

20. 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

21. S (Sender) 2 d 12 a (Receiver) d S 2 2 1 d 1 a 1 g P g r 2 r + h e r 1 r e b b 1 2 Earth (h = ocean height)

22. S d 2 S 2 1 d 1 P d' d' 1 2 g g g g P' d' - d d' - d 2 2 1 1

23. S 2 e c S 2 e r e c S 2 a S d' S 1 2 2 1 e r P' d' 1 e r S 1 a 1 g P' g e c P' r e b b 1 2

24. S 2 a d' S 2 2 1 h 2 d' 1 a 1 g h P' 1 g s s P' 2 O s r 1 e b b 1 2

29. Seeking Reflecting Points on Reference Ellipsoid

30. an intersection of 3 quadrics in a special position

31. S 2 S 1 P v q Earth

32. Choice of Cartesian coordinate frame x 2 u O S x 1 1 S 2 x 3

33. Ellipsoid of revolution for reflecting points

35. Rotational cone surfaceof reflected signals S1= vertex

36. Intersection of ellipsoid of revolution with the cone resulting in a plane ellipse P P

37. Cut of plane P with the Earth reference ellipsoid

38. Classification of mutual positions of intersecting ellipses

39. minimum distance between two ellipsoids

40. 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

41. Algorithm of solution matrices of ellipsoids centers of ellipsoids in vector in normal direction tangent vector radius of normal curvature in direction in centre of curvature

42. Iterative solution of minimum distance between two ellipsoids as a progression of distances X0X1 X’0X’1 X’’0X’’1 etc

46. 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.

47. 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

48. 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

49. 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).

50. 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 GAAV ČR number 3003407 • We thank Carl A. Wagner, Cinzia Zuffada, Markus Nitschke, Giulio Ruffini and Martin Wiehl for consultations/literature.