Transcript WEATHER RADAR SYSTEMS Lecture 3 (Incomplete Draft)

WEATHER RADAR
SYSTEMS
Draft of Lecture 3
MEASUREMENT OF RADAR REFLECTIVITY
• Reflectivity is the most important measurement performed by weather
radar
• Reflectivity is correlated with precipitation activity
• A reflectivity map is closest to a weather ‘picture’
• Early weather radars had only reflectivity measuring capability
• Reflectivity of precipitation is caused by the scattering of radar energy
by particles suspended in air
• Scatterers have unknown sizes, shape, orientation positions,
velocities, composition
• The important task is to measure and relate the echo power to
precipitation activity
• This is done under certain assumptions, depending on the capability
of the radar and complexity of the measurement models
SHAPE OF RAINDROPS
D = 8 mm
7.35 mm
5.80 mm
5.30 mm
3.45 mm
2.70 mm
• Drop shape depends on a balance of surface tension, gravity and
drag forces
• In typical rain, smaller drops are more numerous
• Smaller droplets tend to have spherical shape
• Spherical shape is a good approximation for raindrops
SCATTERING BY SMALL PARTICLES
 /  r2
10.0
RAYLEIGH
MIE
OPTICAL
1.0
0.1
0.01
0.001
0.1
1.0
10.0
2r/
NORMALIZED RADAR CROSS SECTION OF SPHERE vs SIZE
SCATTERING BY SINGLE RAINDROP
• Raindrops are Rayleigh scatterers (D/ << 1) for practical radar
wavelengths
• Scattering depends strongly on drop diameter D and wavelength 
• Backscattering cross section b of a small (D  /16) spherical drop of
water is (Rayleigh scattering approximation):
b  k
D6

k   Km
5
For a given radar  is constant
4
 b  D6
2
m2  1
Km  2
m 2
m = complex refractive index of scatterer = n – jk)
At =10 cm:
For water: n ≈ 9, k = 0.63 to 1.47 (20 to 0° C)
For ice: n ≈ 1.78, k = 2.410–3 to 5.510–4 (0 to –20° C)
|Km|2 = 0.91 to 0.93 for  = 1 to 10 cm, constant with temperature
REFLECTIVITY AND REFLECTIVITY FACTOR
• Energy backscattered from all particles in a resolution volume will be
received at a given instant (with appropriate weighting)
• Resolution volume contains numerous drops of varying sizes
• Important parameter is backscattering cross section per unit volume of
space
Reflectivity
k 
   4 Z
 
Z = ‘reflectivity factor’ =
 1 
6

  Di
 V  i
Di is the diameter of ith raindrop in the volume element V
Summation is over the drops in the volume V
V should be large enough to represent the drop statistics, but
small enough to ensure homogeneity
In case of radar V may represent the resolution volume
REFLECTIVITY FACTOR
• Z cannot be directly evaluated from particle sizes, which are
unknown
• Evaluated statistically for large ensembles of scatteres
• Reflectivity factor has SI units m6/m3 or m6 m–3 (dimensionally m3)
• A more practical unit is mm6/m3 or mm6 m–3 (differs from SI by a
factor of 1018)
• In practice Z varies over wide range
• To avoid deling with large values, Z is most often expressed in dB:
dBZ = 10 log10 Z , with Z expressed in mm6/m3
• Examples:
Clouds: ~0 dBZ
Drizzle: ~25 dBZ
Very heavy rain with hail: >60 dBZ
NWS* REFLECTIVITY LEVELS
Level
1
Reflectivity Rainfall category
interval (dBZ)
18-30
Light (Mist)
2
30-41
Moderate
3
41-46
Heavy
4
46-50
Very heavy
5
50-57
Intense
6
>57
Extreme (with hail)
* National Weather Service, USA
WEATHER RADAR RANGE EQUATION
resolution
volume
R
c /2
2

 1   rb   c
Pr  Pt G  

 
2 
 4r    2   2
Effective
power
along
beam
Spherical
spread
Volume of
resolution
volume
2



 1  G 
   

2
4

r
4

 


Reflec- Reverse Antenna
tivity spherical collecting
spread
area
η = reflectivity = backscattering cross section of droplets per unit volume
WEATHER RADAR RANGE EQUATION
   Km
5
2
1
 4 Z e
 
Equivalent reflectivity factor
Units:  (m), Ze (m6/m3), η (m2/m3)
 Pt G  c km Z e
Pr 
29 (2 ln 2)2 r 2
3
2
2
b
2
2ln2 is a shape factor due to nonideal antenna – calculated for halfpower (3-dB) beamwidth of Gaussian-shaped beam pattern
 Pt G  c km Z e
Pr 
210 (ln 2)2 r 2
3
2
2
b
2
EFFECT OF LOSSES
System Loss Facor (Ls): Denotes fraction of energy lost between
points where Pt is specified and where Pr is measured. Accounts for
losses that are not included in any other variable in the radar
equation.
Atmospheric Loss actor (La one-way, La2 2-way): Loss of radar
signals due to weather and other factors. Includes lens effect of ~1
dB (2-way, worst case 0° elevation at 450 km range)
Receiver Filtering Loss Factor (Lf): Accounts for spectral
components of transmitted signal that do not pass through the finite
bandwidth of the receiver filter
Signal power at the output of radar receiver:
 Pt G G  c k m Z e
Pro  10
2
2 2
2 (ln 2) Ls La L f  r
3
2
2
s b
2
Gs = Receiver power gain, usually measured with a continuous wave
or monotone signal
RANGE EQUATION: PRACTICAL FORM
Using more practical units,
Pt G G  c k m Z e
2
 25
Pro  4.37 10
Where the units are:
2
s b
2
s a f
2
L L L 2 r 2
Pro (mW)
Pt
(W)

b
(μs)
Ze
(mm6/m3)

(cm)
r
(km)
(°)
The range equation is used to determine the reflectivity Ze. All other
quantities are known.
RAIN RATE ESTIMATION
• Radar reflectivity can be estimated from range equation
• Reflectivity is a good indicator of rainfall intensity (rain rate)
• However, quantitative rain estimate is more difficult
• The relation between reflectivity and rain rate is complex (not definite
/ unique / exact)
• Rain rate may vary by a factor of 3 or more for a given reflectivity
factor
Rain liquid water content  amount of suspended liquid water per
unit volume


6
3
D
 i over unit volume
i
Rain rate = Volume of water passing through unit horizontal area per
unit time, depends on drop fall speeds in addition to
diameters and numerical density of drops
RAIN RATE ESTIMATION
Drop terminal velocity wt(D) ≈386.6 D0.67 m/s where D is in m
Rain rate R  D3.67 for given drop diameter D
But reflectivity Z  D6
Hence Z-R relation is not unique, depends on drop size distribution
N(D) is the number of drops / unit volume,
having diameter between D and D + D
[Unit: number/(m3 mm) or m-3 mm-1]
N
N  D   lim
D 0 D
Then rain rate R 


3
D
 N Dwt D dD
60
and
Reflectivity factor Z 


6
D
 N D  dD
60
If drop size distribution N(D) is known then Z-R relation is unique
MARSHALL-PALMER DISTRIBUTION
N D  N0 eD
N0 = 8000 m-3 mm-1
D = Drop diameter in mm
 = 4.1 R-0.21
R = rain rate in mm h-1
Actual drop size distribution and Z-R relation may vary substantially
REFLECTIVITY-RAIN RATE RELATION
Z-R relationship is often expressed directly
Typical form: Z = a Rb
where a,b are constants, and
R and Z are expressed in mm h–1 and mm6 m–3 respectively
Examples: for stratiform rain
a = 200, b = 1.6 (Marshall-Palmer distribution)
a = 400, b = 1.4 (Laws and Parsons)
a = 300, b = 1.5 (Joss and Waldvogel)
REFLECTIVITY-RAIN RATE RELATION
Multiplicity of Z-R
relationships