Remote Sensing of Ocean-Atmosphere - UPRM

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					  Scattering from Hydrometeors:
        Clouds, Snow, Rain
Microwave Remote Sensing INEL 6069
Sandra Cruz Pol
Professor, Dept. of Electrical & Computer Engineering,
UPRM, Mayagüez, PR


                                                         1
 Outline: Clouds & Rain
1. Single sphere (Mie vs. Rayleigh)
2. Sphere of rain, snow, & ice (Hydrometeors)
        Find their ec, nc, sb
3.   Many spheres together : Clouds, Rain, Snow
       a. Drop size distribution
       b. Volume Extinction= Scattering+ Absorption
       c. Volume Backscattering

4. Radar Equation for Meteorology
5. TB Brightness by Clouds & Rain


                                                      2
Clouds Types on our Atmosphere




                         3
    Cirrus Clouds
    Composition
    70
    60
                         hexagonal
    50                   plates
    40                   bullet rosettes
%
    30                   dendrites
    20
                         others
    10
     0
          Ice Crystals



                                           4
    EM interaction with
    Single Spherical Particles
   Absorption                       Si
     – Cross-Section, Qa =Pa /Si
     – Efficiency, xa= Qa /pr2
   Scattered
     – Power, Ps
     – Cross-section , Qs =Ps /Si
     – Efficiency, xs= Qs /pr2
   Total power removed by sphere from the
    incident EM wave, xe = xs+ xa

   Backscatter, Ss(p) = Sisb/4pR2        5
Mie Scattering: general solution to EM
scattered, absorbed by dielectric sphere.

   Uses 2 parameters (Mie parameters)
    – Size wrt. l :
                           2pr
                      
                           lb
    – Speed ratio on both media:

           np
      n
           nb
                                        6
Mie Solution

   Mie solution
                                  
                         2
         x s (n,  ) 
                         2
                                  (2m  1)(| am |2  | bm |2 )
                                 m 1
                                  
                         2
         x a (n,  ) 
                            2    (2m  1) Re{a
                                 m 1
                                                    m   bm }


   Where am & bm are the Mie coefficients given
    by eqs 5.62 to 5.70 in the textbook.
                                                         7
 Mie coefficients
      Am m 
      n    Re{Wm }  Re{Wm 1}
            
am         
          Am m                            2pr 2πr
          n   Wm  Wm 1
                 
                                                 ec
                                         lp    λo


            m
     
      nAm   Re{Wm }  Re{Wm 1}
             
bm                                 np   e pc        (   j )
                m                n           ec 
           nAm  Wm  Wm 1
                                    nb   e bc            ko
                

where      Wo  sin   j cos 
n  n '  jn"                                     8
 Non-absorbing
 sphere or drop
 (n”=0 for
 a perfect dielectric,
 which is a
 non-absorbing sphere)
      Re call
                     (   j )
      n  n' jn " 
                         ko        =.06
      k o    oe o

Rayleigh region |n|<<1
                                           9
Conducting (absorbing) sphere




           =2.4

                          10
    Plots of Mie xe versus 

                                            Four Cases of sphere in air :
                                            n=1.29 (lossless non-absorbing sphere)
                                            n=1.29-j0.47 (low loss sphere)
                                            n=1.28-j1.37 (lossy dielectric sphere)
                                            n=  perfectly conducting metal sphere



   As n’’ increases, so does the absorption (xa), and less is the
    oscillatory behavior.
   Optical limit (r >>l) is xe =2.
   Crossover for
     – Hi conducting sphere at  =2.4
     – Weakly conducting sphere is at  =.06
                                                              11
Rayleigh Approximation |n|<<1
   Scattering efficiency
                         8 4
                    x s   | K |2 ...
                         3

   Extinction efficiency
                              8 4
          x e  4  Im{ K }   | K |2 ...
                              3

   where K is the dielectric factor
                n2 1 e c 1
              K 2   
                n  2 ec  2                   12
Absorption efficiency in Rayleigh
region



x a  x e  x s  4  Im{  K }  x e

     i.e. scattering can be neglected in Rayleigh region
     (small particles with respect to wavelength)
     |n|<<1

                                                13
Scattering from Hydrometeors

   Rayleigh Scattering         Mie Scattering




   l >> particle size    l comparable to particle size
                         --when rain or ice crystals
                         are present.      14
Single Particle Cross-sections vs.

                                       For small drops, almost
   Scattering cross section           no scattering, i.e. no
                  2l2 6
             Qs      | K |2 [m 2 ]   bouncing from drop since
                  3p                   it’s so small.
   Absorption cross section
                 l2 3
             Qa   Im{ K } [m 2 ]
                 p
In the Rayleigh region (n<<1) =>Qa is
  larger, so much more of the signal is
                            x s  x a
  absorbed than scattered. Therefore
                                            15
16
Rayleigh-Mie-GeometricOptics
   Along with absorption, scattering is a major cause of the
    attenuation of radiation by the atmosphere for visible.
   Scattering varies as a function of the ratio of the particle
    diameter to the wavelength (d/l) of the radiation.
    When this ratio is less than about one-tenth (d/l1/10),
    Rayleigh scattering occurs in which the scattering
    coefficient varies inversely as the fourth power of the
    wavelength.
   At larger values of the ratio of particle diameter to
    wavelength, the scattering varies in a complex fashion
    described by the Mie theory;
   at a ratio of the order of 10 (d/l>10), the laws of geometric
    optics begin to apply.
                                                   17
Mie Scattering (d/l1),

   Mie theory : A complete mathematical-physical theory of the
    scattering of electromagnetic radiation by spherical particles,
    developed by G. Mie in 1908.
   In contrast to Rayleigh scattering, the Mie theory embraces all
    possible ratios of diameter to wavelength. The Mie theory is very
    important in meteorological optics, where diameter-to-
    wavelength ratios of the order of unity and larger are
    characteristic of many problems regarding haze and cloud
    scattering.
   When d/l  1 neither Rayleigh or Geometric Optics Theory
    applies. Need to use Mie.
   Scattering of radar energy by raindrops constitutes another
    significant application of the Mie theory.
                                                        18
Backscattering Cross-section
   From Mie solution, the backscattered
    field by a spherical particle is
                                            2
                      
                                                 sb
      x b (n,  )  2   1 (2m  1)(am bm )  2
                  1          m

                    m1                         pr

Observe that
  perfect dielectric
 (nonabsorbent) sphere
exhibits large
oscillations for >1.
  Hi absorbing and perfect
conducting spheres show
regularly damped oscillations.
                                                      19
Backscattering from metal sphere
   Rayleigh Region defined as x  4  4 | K |2
                                b

                                  for n  0.5



                             where,
                                      K




   For conducting sphere (|n|=  )    x20  9  4
                                        b
Scattering by Hydrometeors
Hydrometeors (water particles)
 In the case of water, the index of
  refraction is a function of T & f. (fig 5.16)
 @T=20C
                   9  j.25 @ 1 GHz
                 
nw  n' jn' '   4.2  j 2.5 @ 30 GHz
                 2.4  j.47 @ 300 GHz
                 
 For ice. n'i  1.78
   For snow, it’s a mixture of both above.
                                      21
Liquid water refractivity, n’




                                22
Sphere pol signature


Co-pol




Cross-pol




                       23
Sizes for cloud and rain drops




                            24
                    Snowflakes

           Snow is mixture of ice crystals and air
           
        i  0.916 g/cm  a  0 0.05   s  0.3g/cm
                       3                             3


         The relative permittivity of dry snow
                      e ds  1  s  e ds  1 
                        '                 '
                                    '           
                       3e ds
                           '                   ' 
                                 i  e i  2e ds 
                                                                  ei 1
                                                              Ki 
            The Kds factor for dry snow                           ei  2
K ds
       
           1.1K i
                     0.5
                                                p 5 D6              p 5 D6
                            s bs  x bprs 2           | K ds |2          | K i |2
 ds           i                                l4 o                4l4 o
                                                                 25
Volume Scattering

   Two assumptions:
    – particles randomly distributed in volume--
      incoherent scattering theory.
    – Concentration is small-- ignore shadowing.
   Volume Scattering coefficient is the total
    scattering cross section per unit volume.
       bsb   p(rD)bs (b ( Dr d D
          N ( )s s r )d )
                       Q                 [Np/m]

x s  Qs / pr 2   x a  Qa / pr 2   x b  s b / pr 2
                                          26
Total number of drops per unit volume



     Nv   p(r )dr   N ( D)d D   in units of mm-3


                   / 
       p(r )  arc e
                         D / Do
       N ( D)  N o e

                                     27
Volume Scattering

                                               lo
Using...   2pr / lo , x s  Qs / pr and dr     d
                                                  2

                                               2p
 It’s also expressed as

                       3 
                    l
     s , e ,b                     p(  )x s ,e,b (  )d 
                       o          2                                            [Np/m]
                   8p   2
                          0
           [s,e,b stand for scattering, extinction and backscattering.]
   or in dB/km units,
                                        
 dB   b  4.34 103  N ( D)s b ( D)d D                                 [dB/km]
                                        0                                 28
    For Rayleigh approximation

   Substitute eqs. 71, 74 and 79 into definitions of
    the cross sectional areas of a scatterer.
                        2p 5 D 6
         Qs  x spr 2           | K w |2
                          3l4
                         p D
                           2   3
         Qa  x apr 2
                                 Im( K w )
                              l
                           p 5 D6
         s b  x bpr 2           | K w |2
                             l4
    D=2r =diameter                            29
Noise in Stratus cloud image
-scanning Ka-band radar




                               30
    Volume extinction from clouds
   Total attenuation is due to gases,cloud, and rain
                 a   g   e c   ep
   cloud volume extinction is (eq.5.98)
                        p   2
        e c   Qa dD  Im{ K w} D3dD
                        lo
   Liquid Water Content LWC or mv )
               4p 3          6 p
     mv   w  r dr  10  D 3dD
                3              6
  w  water density = 106 g/m3
                                               31
Relation with Cloud water content

   This means extinction increases with
    cloud water content.
        e c   1mv

where
               6p
     1  .434    Im(  K )       1   1
                              [dBkm g m ]   3

               lo

and wavelength is in cm.               32
Raindrops symmetry




                     33
Volume backscattering from Clouds

 Many applications require the modeling of
  the radar return.
 For a single drop
                               p 5 D6
               s b  x bpr 2         | K w |2
                                 l4
For many drops (cloud) 5
                            p
  s vc   s b N ( D )dD  4 | K w |2    N(D)D 6 dD 
                            l
   p5
  4 | K w |2 Z
   l                                        34
  Reflectivity Factor, Z
     Is defined as
                                  p5
Z   D6 N ( D)dD so that s vc    4 | K w |2 Z
                                  lo

     and sometimes expressed in dBZ to cover a
      wider dynamic range of weather conditions.

      dBZ  10 log Z
     Z is also used for rain and ice measurements.
                                         35
Reflectivity in other references…


                p5
   10   12             2
                    | Kw | Z
                lo4




 where is in cm and   -1

                            6   3
 Z is expressedin mm /m
                                    36
                         Reflectivity & Reflectivity Factor
                                                                       Z (in dB)
Reflectivity,  [cm-1]




                                                        dBZ for 1g/m3



                          Reflectivity and reflectivity factor produced by 1g/m3 liquid water
                          Divided into drops of same diameter. (from Lhermitte, 2002).
                                                                             37
Cloud detection vs. frequency




                           38
Rain drops




             39
Precipitation (Rain)

   Volume extinction
               3 
               l
        er              p(  )x e (  )d    1 R
               o        2                            b

              8p2                                    r      [dB/km]

                  0
                                         Mie coefficients
 where Rr is rain rate in mm/hr
  1 [dB/km] and b are given in Table 5.7
 can depend on polarization since large
  drops are not spherical but ~oblong.
                                    40
W-band UMass CPRS radar




                          41
Rain Rate [mm/hr]
        If know the rain drop size distribution, each
         drop has a liquid water mass of m  p D 3 
                                                     w
                                               6
        total mass per unit area and time
           

           
           0
             N ( D)m( D)dD dAdt  (  w p / 6)  D 3 N ( D)vt dD

        rainfall rate is depth of water per unit time
               Rr  p / 6 vt ( D) N ( D) D dD    3




                                                         
        a useful formula
                                       (-6.8D2  4.88D)
                vt ( D)  9.251 - e                42
Volume Backscattering for Rain

   For many drops in a volume, if we use
    Rayleigh approximation
                    p5                    p5
s vr     s br dD  4 | K w |2  D 6 dD  4 | K w |2 Z
                     l                    l
   Marshall and Palmer developed
            Z  200 Rr .6
                     1


   but need Mie for f>10GHz.
                   p  5
           s vr    4 | K w |2 Z e
                   l                           43
Rain retrieval Algorithms
Several types of algorithms used to retrieve rainfall
  rate with polarimetric radars; mainly
 R(Zh),
 R(Zh, Zdr)          ˆ ( K )  11.62K 0.937 for S band
                     R dp
 R(Kdp)                                 dp

 R(Kdp, Zdr)         ˆ
                     R( K dp )  40.5K dp85 for X band
                                       0.
where
R is rain rate,
Zh is the horizontal co-polar radar reflectivity factor,
Zdr is the differential reflectivity
Kdp is the differential specific phase shift a.k.a.
  differential propagation phase, defined as
                           dp (r2 )  dp (r1 )
                  K dp 
                               2(r2  r1 )         44
Snow extinction coefficient

   Both scattering and absorption
    ( for f < 20GHz --Rayleigh)

       e s  4.34 10
                     3
                          Q dD   Q dD
                            a        s

 for snowfall rates in the range of a few
  mm/hr, the scattering is negligible.
 At higher frequencies,the Mie formulation
  should be used.
 The  e s is smaller that rain for the same R,
  but is higher for melting snow.
                                         45
  Snow Volume Backscattering

      Similar to rain
        p5                    p5
s vs    4 | K ds |2  D6 dD  4 | K ds |2 Z s
        l                     lo

                              1                           1
Z s   D s N ( D )dD                D           dD 
              6                                6
                                                                  Zi
                                 2
                                  s
                                           i
                                                             2
                                                              s


                                                     46
   Radar equation for Meteorology
      For weather applications
         Pt G l 2 2
                                         g   e c   ep dr
                                          R
                    2
   Pr          se
                o o

        4p  R 3 4
                                          o


                                              R          c p 
                                                       2
      for a volume s  s vV             V p            
                                                            2   
                                              2                
                          2
       Pt G l  c p e
          2 2   2

Pr 
          o o
                                sv
          324pR 
                      2
                                                  47
   Radar Equation

       For power
        distribution in the       Pt Go2 l2 oo c p Lr 2 s v
                            Pr 
                                          o
        main lobe                                       L 2
                                     1024 p ln 2
                                              2
                                                           R
        assumed to be
        Gaussian function. where,
                            s v    radar reflectivity
                                 and Lr  receiver losses

And the two - way atmospheric losses are defined here as
   L2  e  2                                     48

				
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