1. Ocean Wave Spectral Form Again, another justification of my NASA Job : Scatterometry

2. Scatterometry The Scatterometer is the same radar as the altimeter but side looking. The measured parameter is the radar backscattering cross section (intensity), which is related to the local wind stress. Wind stress over the global ocean is critically important in meteorological and oceanographic studies.

3. Geometry of Scatterometry

4. Global Wind field from QuikSCAT

7. 謁金門 馮延巳（ 903 年－ 960 年） 风乍起，吹绉一池春水。闲引鸳鸯香径里， 手捋红杏蕊。 斗鸭阑干独倚，碧玉搔头斜 坠。终日望君君不至，举头闻鹊喜。 903 年 960 年 《南唐書‧馮延巳傳》： 「元宗嘗戲延巳曰：『吹皺一池春水，干卿何事 ? 』延 巳對曰：『未如陛下小樓吹徹玉生寒。』元宗悅。」 * 元宗 : 李璟, 南唐中主

8. Principle of Scatterometry Scatterometry is a side-looking radar measuring the backscattering power that is related to the local wind stress. For a side-looking radar, Backscattering is primarily caused by Bragg scattering: Therefore, we have to know the intensity of ocean surface wave at a specific wave length.

9. Bragg Scattering from the Ocean Surface,

10. Bragg’s Law n is an integer determined by the order given, λ is the wavelength of x-rays, d is the spacing between the planes in the atomic lattice, θ is the angle between the incident ray and the scattering planes

11. Sir William Lawrence Bragg Sir William Lawrence Bragg and his father, Sir William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS and Diamond. He played a major part in the 1953 discovery of the structure of DNA, in that he provided support to Francis Crick and Watson who worked under his aegis at the Cavendish. 1962 Crick, Watson and Maurice Wilkins Nobel Prize.

12. Other needs for Ocean Wave Spectrum Nonlinear wave study Air-sea interaction study Probability study of the wave field Ship design Oceanic structures: oil platforms.

13. History Lacking actual data, computation resources and method*, the early spectral forms were empirical and ad hoc: Even the dimension was wrong. Phillips (1958) published an asymptotic form based on dimensional analysis similar to what Kolmogorov did for turbulence studies. This asymptotic form became the base for all the subsequent developments. Phillips, however, changed his mind in 1985. * This was before Cooley and Tukey invented the FFT in 1965.

14. Phillips’s Equilibrium Spectrum Based on dimensional analysis, Phillips proposed the asymptotic form for spectra as

15. Laboratory Data

16. Kaitaigorodskii 1962

17. Curve-Fitting Modifications

18. JONSWAP Spectrum : Hasselmann et al (1973)

19. Detailed Spectral Shape

20. Observations The line connecting the spectral peak and the peak of the second harmonics seems to form a good overall asymptotic line. If that is the case, the slope of the asymptotic line could be different from the constant -5, but its value could be determined theoretically.

21. Observations Note that the water wave forms are nonlinear. To the second order approximation, we have Therefore, we have

22. Observations

23. Verification

24. Wallops Spectrum

25. Observation

27. Wallops Spectrum

28. Validation

32. The relevance of the Significant Slope We found the key; the truth is actually very simple.

33. Validation

40. Conclusion Apparently, the spectral form is not that complicate. The two-parameter model works well for the energy containing wave range of the spectrum. Future research should concentrate on relating the internal parameters with exterior environmental conditions. Unfortunately, the relationships might not be unique.

41. Epilog The proposed spectral form is not adequate for the application to Scatterometry. Toba had proposed an asymptotic form in 1973 that was wind dependent: This form had gain increasing acceptance and even converted Phillips in 1985. But the form works only for the high frequency spectral tail range (The tail waggles the dog!?).

42. Epilog : Toba’s asymptotic form

43. Detailed Spectral Shape

44. Epilogue Whatever the spectral form it may be, it is still not useable for Scatterometry, for the Bragg scattering based Scatterometry needs wave number spectrum. But most spectrum is in frequency space. It is not possible to convert frequency to wave number with the dispersion relationship at this spectral range. Meanwhile, my problem with the traditional spectral analysis became increasingly irresolvable.