1. A Prelude to HHT Analysis My work led to but before the HHT

2. Probabilistic Structure of the Ocean Surface Justification for my NASA job

3. Historic events The Sputnik satellite (4 October 1957) shocked the United States. President Eisenhower (1 October 1958) established the national Aeronautics and Space Administration (NASA). President Kennedy (25 May 1961) proposed the challenge for a race to the moon within 1960s. Apollo program succeeded in the first moon landing on 20 July 1969.

4. Historic events What else to do with the space program? NASA had been wandering till now. One of the suggestions was to study the Earth with radar. As more than 70% of the Earth surface is ocean, we have to understand what the backscattering radar signals from the ocean meant. Or, how the ocean surface influence the radar backscattering.

5. Historic events Instruments proposed for Earth system studies: –Altimeter –Scatterometer –Synthetic Aperture Radar –Radiometer (visual, IR and microwave) All instruments need to know the ocean surface probability structure of the ocean surface.

6. My Personal Locus 1960 : graduated (BS) from NTU majoring structure theory in Civil Engineering. 1967 : graduated (Ph D) from the Johns Hopkins University majoring fluid mechanics and mathematics with a thesis on random ocean waves. 1969 : started my teaching job in Oceanography and conducting wave research for NASA. 1975 : joining NASA to participate in the Seasat-1 project.

7. Seasat-I 1978

9. Applications of Satellite Altimetry Navigation Prediction of seafloor depth Plate tectonic Geoid Sea level changing, global warming Ocean circulation Wave and sea state monitoring

10. Geoid Equi-potential surface. Local fluctuation could reach 0.1 M/ KM.

11. ECGM96 Geoid (30’x30’): the equi-potential surface

12. Local Geoid from ERS-1

13. Geoid 1. Ocean 2. Ellipsoid 3. Local plumb 4. Continent 5. Geoid (important military applications)

14. Sea Level Changes A global warning related topic

17. Sea Level Change measured by Altimetry

18. Ocean Circulation Geo-strophic balance

19. Mean absolute sea surface height estimated from 5 years of T/P data relative to the EGM-96 geoid.

20. TOPEX : 4 year mean

21. Principle of Altimetry Pulse-limited radar

24. Specular Reflection

26. Traditional Approach Surface waves are important in determining the sea surface height, if the precision is to be within a couple of centimeters. As the number of ocean waves is so large, and the waves are nearly independent, therefore, by Central Limit Theorem, the probability density of the surface elevation should be normal. NO!!

27. The correct approach Waves are nonlinear, which produce the up-down asymmetry. Altimeter get its return from the specular reflection, which depends on joint- distribution of elevation-slope distribution. Asymmetry in elevation-slope distribution could induce sea-sate bias.

28. Ocean Surface

30. The basic law of the seaway is the apparent lack of any law -- Lord Rayleigh

31. Surface Probability Model

35. Fundamental Theorem of Probability governing transform of random variables

36. Surface Probability Model

37. Non-Gaussian Elevation Distribution

38. Verifications Laboratory studies in NASA Wind-Wave Experimental Facility

40. Non-Gaussian Elevation Distribution

42. Joint Distributions Elevation-Slope

43. Verification

47. Conditional Distributions Joint Elevation-Zero-Slope

51. Sea Surface Bias Verification

52. Theoretic and Observed

53. Conclusion Based on simple nonlinear wave model, we have established a non-Gaussian joint elevation-slope distribution. But this model is only an approximation, for we did not consider the small scale riding waves, which may be pretty uniformly covering the surface of the large waves. This uniform coverage makes the large wave approximation works well.

54. Conclusion Indeed, comparison with laboratory observations showed good agreements, which support the hypothesis stated above. I believe the agreement would also be good for field data (Because ocean surface uniformly covered with small waves). The results could be used to determine sea-state bias in altimetry and other applications.