# 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

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

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
 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

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
 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
   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

   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