A Prelude to HHT Analysis

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A PreludetoHHT Analysis

• My work led to, but before, the HHT

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Probabilistic Structure of the Ocean Surface

• Justification for my NASA job

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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.

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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 therefore, the oceans determine the
  environmental condition.
• We have to understand what the backscattering
  radar signals from the ocean meant. Or, how the
  ocean surface influence the radar backscattering.

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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.

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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.

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Seasat-I 1978
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Applications of Satellite Altimetry

• Navigation
• Prediction of seafloor depth
• Plate tectonic
• Geoid
• Sea level changing, global warming
• Ocean circulation
• Wave and sea state monitoring

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Geoid

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

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ECGM96 Geoid (30x30) the equi-potential surface
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Local Geoid from ERS-1
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Geoid
1. Ocean 2. Ellipsoid 3. Local plumb 4.
Continent 5. Geoid (important military
applications)
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Sea Level Changes

• A global warning related topic

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Sea Level Change measured by Altimetry
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Ocean Circulation

• Geo-strophic balance

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Mean absolute sea surface height estimated from 5
years of T/P data relative to the EGM-96 geoid. 
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TOPEX 4 year mean
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Principle of Altimetry

• Pulse-limited radar

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Specular Reflection
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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!!

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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.

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Ocean Surface
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Ocean Surface
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The basic law of the seaway is the apparent lack
of any law

• -- Lord Rayleigh

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Real Ocean Surface
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Surface Probability Model
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Surface Probability Model
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Surface Probability Model
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Surface Probability Model
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Surface Probability Model
Fundamental Theorem of Probability governing
transform of random variables
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Surface Probability Model
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Non-Gaussian Elevation Distribution
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Verifications

• Laboratory studies in
• NASA Wind-Wave Experimental Facility

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NASA Wind-Wave Experimental Facility
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Non-Gaussian Elevation Distribution
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Non-Gaussian Elevation Distribution
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Joint Distributions

• Elevation-Slope

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Verification
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Conditional Distributions

• Joint Elevation-Zero-Slope

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Sea Surface Bias

• Verification

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Theoretic and Observed
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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.

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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.