Ocean Wave Spectral Form

Again, another justification of my NASA Job : Scatterometry

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.

Geometry of Scatterometry

Global Wind field from QuikSCAT

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。

《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

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.

ro

2 sin

Bragg Scattering from the Ocean Surface

                                                        ,

ro

2 sin

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 •

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.

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.

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.

Phillips’s Equilibrium Spectrum

• Based on dimensional analysis, Phillips proposed the asymptotic form for spectra as

2

o5

o

g( n ) , for n n ; (n) = 0, otherwise.

ng : gravitation acceleration

n : frequency; n : the peak frequency

: Phillips Equilibrium range constant

Laboratory Data

Kaitaigorodskii 1962

42

5o

Based on Phillips's for, Kitaigorodskii proposed

g n (n) = exp '

n n

: an additional constant

This gives spectrum a smooth and continuous functional form.

Curve-Fitting Modifications

JONSWAP Spectrum :Hasselmann et al (1973)

2

omaxnPM 2 2

max o

a o b o

n

42

5o

n n( n ) = ; exp ;

( n ) 2 n

= , when n n ; = , when n > n .

g 5 n(n) = exp 4 nn

Detailed Spectral Shape

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.

Observations

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

• Therefore, we have

21a cos + ak cos 2

2

222 2

oo

o o

21 loglog a log a kak2

m = 1log n log 2n log2

Observations

1 / 22

o

2

1/ 22

oo

For a narrowband spectrum, we also have

S= , the Significant Slope

log 2 S m = .

log 2

a 2 ; k ;

2

Verification

Wallops Spectrum

m2o o

5o

Based on the above observations, we propose

n ng (n) = exp

n n n

with three parameters : , and to be determined.

Observation

o

( n )Since 0 at n = n ,

n

we must have m .

mTherefore, if we choose = 4, then .

4

Observation

2

0

1( m 1 )2

4

1( m 5 )

4

By definition, = ( n ) dn ;

2 S m 1therefore, = .

1( m 1 )4

4

Wallops Spectrum

42

m 5 mo o

o

Finally ,we have

g m n (n) = exp '

n n 4 n

This gives spectrum a smooth and continuous

functional form with only two parameters: n and

significance slope ss.

Validation

Validation

Validation

Validation

The relevance of the Significant Slope

We found the key;

the truth is actually very simple.

Validation

Validation

Validation

Validation

Validation

Validation

Validation

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.

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!?).

*4

gu( n )

n

Epilog : Toba’s asymptotic form

Detailed Spectral Shape

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.