Transcript Probabilistic Structure of the Ocean Surface

A Prelude
to
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; 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.
Historic events
• Instruments proposed for Earth system studies:
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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.
Seasat-I
1978
Applications of Satellite Altimetry
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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 jointdistribution 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
Real Ocean Surface
Surface Probability Model
If  1  a cos  ;  2  a sin are Gaussian distributed RV.
Z1 

a
a2 / 2

1/ 2
cos  ; Z 2 

a
a2 / 2

1/ 2
1
 1 2
2 
p( Z 1 , Z 2 ) =
exp    Z 1 + Z 2  
2
 2

sin 
Surface Probability Model
1 2
3 3 2
2
Now ,  = a k + a cos  + a k cos 2   a k cos 3   ...
2
8
9 3 3
 x = -ak sin  - a k sin 2   a k sin 3   ...
8
2
2
x x
 
Define:  =
and  =
 ( )
 ( x )
Surface Probability Model
Z Z 12
3 Z 13  3Z 1 Z 22 2 2
   2 k 
 k  
3
N N
8
N
Z 2 2Z 1 Z 2
9  3Z 12 Z 2  Z 23  2 2
 

k  
 k
2
M
MN
8
MN

N   1  8 S
2


S
2
0
2

1
2
; M   1  16 S
2
2

1
2
1/ 2
, Significant Slope ;
 = wave length at spectral peak.
0
Surface Probability Model
2

Z1
13
9
M


2
3
2
2 2
     1   k     2  


 k
2
N
 8 N
 8

 21 2
Z2
9 M2 3 2 2
   2 k      2 
  k
2
M
8 N
8

Surface Probability Model
Fundamental Theorem of Probability
governing transform of random variables
 Z1 , Z 2
P  Z 1 , Z 2  dZ 1dZ 2  p  Z 1  ,  , Z 2  ,  J 
  ,

 dd 

Surface Probability Model
NM
P  ,  
2

 23 2

1

4

k



4


 2



9 M 2 2  2 2 

  k 
2
2 N




 1 2 2
 exp  
N   M 2 2  2  2  1  N 2  4 2 M 2  k
 2
2
  17 4
9
M
2
2 2
2
     6  1 


N

2
4
4
N


2
 2 2  
 37 2 2
9
M
2
4 
2
     4 
  M  k  
2
4 N
 
 4


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.