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A Preludeto
HHT Analysis
My work led to but before the HHT
Probabilistic Structure of the Ocean Surface
Justification for my NASA job
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.
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.
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.
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.
Applications of Satellite Altimetry
• Navigation• Prediction of seafloor depth• Plate tectonic• Geoid• Sea level changing, global warming• Ocean circulation• Wave and sea state monitoring
Geoid
Equi-potential surface. Local fluctuation could reach 0.1 M/ KM.
ECGM96 Geoid (30’x30’): the equi-potential surface
Local Geoid from ERS-1
Geoid
1. Ocean 2. Ellipsoid 3. Local plumb 4. Continent 5. Geoid (important military applications)
Sea Level Changes
A global warning related topic
Sea Level Change measured by Altimetry
Ocean Circulation
Geo-strophic balance
Mean absolute sea surface height estimated from
5 years of T/P data relative to the EGM-96 geoid.
TOPEX : 4 year mean
Principle of Altimetry
Pulse-limited radar
Specular Reflection
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!!
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.
Ocean Surface
Ocean Surface
The basic law of the seaway is the apparent lack of any law
-- Lord Rayleigh
Surface Probability Model
1 2
1 21 / 2 1 / 22 2
2 21 2 1 2
If a cos ; a sin are Gaussian distributed RV.
a aZ cos ; Z sin
a / 2 a / 2
1 1p( Z , Z ) = exp Z + Z
2 2
Surface Probability Model
2 2 3 2
2 2 3 3x
x x
x
1 3Now, = a k + a cos + a k cos 2 a k cos 3 ...
2 8
9 = -ak sin - a k sin 2 a k sin 3 ...
8
Define: = and = ( ) ( )
Surface Probability Model
2 3 22 21 1 1 2
2 3
2 32 22 1 2 1 2 2
2
Z Z 3Z ZZ 3k k
N N 8 N
Z 2Z Z 3Z Z Z9k k
M MN 8 MN
1 1
2 2 2 22 2N 1 8 S ; M 1 16 S
0
1 / 22
0
= wave length at spectral peakS , Significant Slope ; .
Surface Probability Model
2
2 3 2 2 212
22 3 2 22
2
Z 13 9 M1 k 2 k
N 8 8 N
Z 21 9 M2 k 2 k
M 8 8 N
Surface Probability Model
Fundamental Theorem of Probability governing transform of random variables
1 21 2 1 2 1 2
Z ,ZP Z ,Z dZ dZ p Z , ,Z , J d d
,
Surface Probability Model
2
22 2 2
2
2 2 2 2 2 2 2 2
24 2 2 2 2
2
22 2 2 4 2 2 2
2
NM 23P , 1 4 k 4
2 2
9 Mk
2 N
1exp N M 2 1 N 4 M k
2
17 9 M6 1 N
4 4 N
37 9 M4 M k
4 4 N
Non-Gaussian Elevation Distribution
Verifications
Laboratory studies in
NASA Wind-Wave Experimental Facility
NASA Wind-Wave Experimental Facility
Non-Gaussian Elevation Distribution
Non-Gaussian Elevation Distribution
Joint Distributions
Elevation-Slope
Verification
Conditional Distributions
Joint Elevation-Zero-Slope
Sea Surface Bias
Verification
Theoretic and Observed
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.
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.
