Transcript Linear and Nonlinear Representations of Wave Fields

Linear and nonlinear representations of
wave fields and their application to
processing of radio occultations
M. E. Gorbunov, A. V. Shmakov
Obukhov Institute for Atmospheric Physics, Moscow
K. B. Lauritsen
Danish Meteorological Institute, Copenhagen
S. S. Leroy
Harvard University, Cambridge MA, USA
Contents
• Radio occultation sounding of the Earth's atmosphere
• The problem of the interpretation of measured wave
fields
• Linear representations of wave fields and the
reconstruction of the ray manifold
• Approach to the reconstruction of generic structure of ray
manifold: definition of 2D density in the phase space
• Winger distribution function (WDF): definition and
properties
• Application for radio occultations
Radio occultations
GPS signals:
L1: 1.57542 GHz, 19.04 cm
L2: 1.22760 GHz, 24.43 cm
1. Measurements of refraction of radio signals of GNSS (GPS, GLONASS,
Galileo) in the Earth's atmosphere on limb paths.
2. Retrieval of vertical profiles of temperature, pressure and humidity.
Radio occultations
Bending angle ε
Impact parameter p
Linear
transformation
( p) 
 ln n( x)
Refractivity n
Distance from the Earth's center r
Refractive radius x

d ln n(r )
dr
( p)  2 p 
dr
n( r ) 2 r 2  p 2
r0
 1  ( p)dp
n( x)  exp  
  p p2  x2

x
x  n( r ) r ; r ( x ) 
n( x )




Multipath propagation
For the retrieval of ray manifold structure in
multipath zones, we used approaches
introduced by V.A.Fock, V.P.Masolv,
Yu.V.Egorov, and L. Hörmander and
developed Canonical Transform method.
Ray manifold and its projections
Ray manifold and its projections
Canonical Transforms
Wave field in t- and p-representations:
u  t    Ai  p  exp i  i  t  dt 


uˆ  p    Ai  p  exp i  i  p  dp 


Linear transformation between the representations
uˆ  p    K  p, t  u  t  dt
corresponds to canonical transform
t,    p,  
which conserves volume element:
dt  d   dp  d 
 dp   dt  dS  p, t 
 2 S  p, t 
K  p, t     p, t 
exp  iS  p, t  
p t
Density in phase space
• Linear representations of wave field are linked to different
projections of the ray manifold.
• For the correct retrieval of the ray manifold structure it is necessary
to find a single-valued projection. Such a projection may be
unknown in advance. For radio occultations, projection to the impact
parameter axis is single-valued for a spherically-symmetric
atmosphere, but this property may not take place in presence of
horizontal gradients.
• We will now define an universal energy density in 2D phase space,
which is not linked to any specific coordinate choice.
• An example of such density is spectrogram (radio holographic
sliding spectra). Its disadvantage is the limited resolution due to the
uncertainty relation.
Radio holographic analysis
y
LEO t 1
GPS
LEO t 2
x
pE
The manifold of all virtual rays that can be received in the given point form a line
in the ray coordinate plane (ε,p). The intersections of this line with bending angle
profile correspond to actual rays being received. Red color marks the reflected
ray and the fragment of the bending angle profile corresponding to reflected rays.
Radio holographic analysis
At every moment of time we compute a sliding spectrum of the signal in the
aperture from t-Δt/2 to t+Δt/2. Each Doppler frequency correspond to some ray
direction. Maxima of the spectrum correspond to observed rays.
Uncertainty relation: Δt Δω ~ 2π
Tomography
x
A
h
B
y
Dy
C
a
Dx
ua  y  Dy 
2

DEi
x
Tomographic equation
A, B ,C ,...
DE  W  x, x  Dx Dx
Energy distribution W
ua  y    K a  y, x  u  x  dx
uα(y) is a linear representation of u(x)
Tomography
W  x, x  
1
s 
s

u
x

u
x


 
 exp  ixs  ds

2 
2 
2
The solution of the tomographic equation is
Wigner distribution function (quantum density).
M. E. Gorbunov and K. B. Lauritsen, Analysis of wave fields by Fourier
Integral Operators and its application for radio occultations, Radio
Science, 2004, 39(4), RS4010, doi:10. 1029/2003RS002971.
M. E. Gorbunov, K. B. Lauritsen, and S. S. Leroy, Application of
Wigner distribution function for analysis of radio occultations, Radio
Science , 2010, 2010RS004388, in press.
Properties of WDF
1. WDF is real
2. WDF is not necessary positive, but its integral over the phase plane equals the
full energy.
3. WDF consists of two components: positive component in the vicinity of ray
manifold (tends to the micro-canonical distribution) and oscillation interference
component (quantum oscillations, cross-terms).
4. WDF contains full information about the wave field. Wave field can be retrieved
from WDF up to a constant phase shift.
5. Accurate determination of the linear trend of frequency:

u  x   exp iax  ibx 2 / 2

x  a  bx
W  x, x     x   a  bx  
Properties of WDF
6. Composition of two fields:
u  x   exp  ia1x   exp  ia2 x 
W  x, x     x  a1     x  a2   2 cos   a1  a2  x 
DxDa  2
The sum of two waves, the two ray manifold (two δ-functions) in the phase
space are accurately reproduced, but interference terms emerge. The
period of the interference oscillations is a function of the momentum
difference as follows from the uncertainty relation.
7. WDF can be used for processing arbitrary oscillating signals consisting of
multiple sub-signals with gliding frequencies. It is possible to consider a
cross-density of two signals:
W12  x, x  
1
s 
s

u
x

u
x

1
 2
 exp  ixs  ds

2
2 
2

Processing of radio occultations
rG  t  , rL  t 
A t 
S0  t   rG  t  - rL  t 
S t 
   k  S0  S 
SM  t 

coordinates of transmitter (GPS) and receiver (LEO)
amplitude
vacuum phase delay
atmospheric phase excess
Doppler frequency
phase model (predicting Doppler frequency within 10–15 Hz)
u  t   A  t  exp ik  S  t   SM  t  

wave field with down-converted frequency
WDF and Weighted WDF
k
 s  s
W t, x 
exp  ik xs  u  t   u  t   ds

2
 2  2
W t, x 


k
 s  s
2
2 2
exp
ik
x
s
u
t

u
t

exp

k
Dx
s ds



 


2
 2  2
Processing of radio occultations
LEO
GPS
x G uG
VG

p

Земля
S0  t   S M  t   x  VLu L  VG uG 
p
LEO
xL
uL
VL
rL  t  2
rG  t  2
2
2
  t  p t, x 
rL  t   p  t , x  
rG  t   p  t , x 
rL  t 
rG  t 
p t, x
p t, x
  t , x     t   arccos
 arccos
rL  t 
rG  t 
W  t , x   W  p,  
WDF can be looked at as a function of impact parameter
and bending angle
Processing artificial data
Left: WDF; right: spectrogram
Processing artificial data
Processing artificial data
Dependence of WWDF from the window width. From left to right: increase of the
window (4, 8, 16 s).
Processing COSMIC data
Processing COSMIC data
Processing COSMIC data
Processing COSMIC data
Processing COSMIC data
Processing COSMIC data
Processing artificial data
Left: artificial event without noise. Right: the same event with model of turbulent
fluctuations superimposed (power spectrum, anisotropic turbulence).
Conclusions
1. We suggested the application of Wigner Distribution
Function for the retrieval of ray manifold structure from
radio occultation measurements of wave fields.
2. The only practical problem is what to do now.
3. We are going to consider the application of WDF for the
retrieval of bending angles, ionospheric correction, noise
filtering and the investigation of the structural uncertainty of
radio occultation data.