lecture 12: curvature and refraction radar equation for...
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MET4410RemoteSensing:RadarandSatelliteMeteorologyMET5412RemoteSensinginMeteorology
Lecture12:CurvatureandRefractionRadarEquationforPointTargets
(RinehartCh3-4)
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Radar Wave Propagation� In the clear air without rain or cloud:
Reflection & refractionØ We can treat that the air as homogenous medium
stratified into horizontal layers. The the radar EM waves are incident on a planar (flat) boundary between different layers. In this case, we can use reflection/refraction to understand how the radar wave is propagating in the atmosphere.
� In rain or cloud: we need to consider absorption & scattering by particles
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Radar Wave Propagation� In rain or cloud: scattering by particles
ØNegligible scattering regime: when particle size is much smaller than the wavelength. We only need to consider absorption
ØOptics regime: when particle size is much larger than the wavelength, we can use the laws of reflection & refraction
Ø Rayleigh or Mie regime : when the wavelength is comparable to the particle size. For microwave band radar detecting cloud drops or raindrops, it falls into this regime, therefore, we have to consider scattering.
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Radar wave propagation in the air: Curvature and Refraction
� Recall complex refractive index:
� The real part nr controls the phase speed of the EM wave through the medium. nr is defined as the ratio of the speed of light in vacuum c to the speed of EM waves through the medium c1: 𝑛" =
$$%
◦ For all real substance, nr >1.◦ nr air ≈ 1.0003 to 1.0004 at visible band, at sea level
r iN n n i= −
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The real part of refractive index in the air
� 𝑛" is between 1.0003 to 1.0004� The EM radiation travels about 0.03% to
0.04% slower in the air than in a vacuum� 𝑛" decreases from 1.0003 near surface of
the Earth to 1.0000 at the top of the atmosphere: usually it’s a gradual decrease, but there could be some abrupt changes.
� 𝑛" is important to understand the radar wave propagation in the atmosphere!
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Refractivity
� Refractivity: � When 𝑛"=1.0003, Nr=300, which is a more
convenient number� Refractivity of the atmosphere is dependent
upon pressure, temperature, and humidity.
6( 1) 10r rN n= − ×
� Radar wave propagation is more dependent upon the gradient of refractivity rather than the absolute value of refractivity at any point.
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-39
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Snell’s Law� Under normal atmospheric conditions,
refractivity is largest near the ground and decreases with height. Therefore, radar waves will travel faster aloft than near the surface. This bends the waves in a downward direction relative to the horizontal. Why?
� Snell’s Law: sin ( )sin ( )
i ir r
r r i r
unn u
θθ
= =
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� Earth’scurvatureeffect: ifthere isnoatmosphere onEarth, radarrayswouldtravel instraightlines.Thecurvature oftheEarthbeneath theraywouldgraduallycausetheraytobehigherandhigherabove theEarthfartherandfarther fromtheradar.RelativetotheEarth, theradarwavewouldappear bendupward!
More Dense Medium N larger
Less Dense Medium N smaller
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Radar Wave Propagation: Curvature Effect� Curvature is the angular rate of change
necessary to follow a curved path.� The curvature C of a circle is thus the
angular distance traversed divided by the linear distance traversed:
Where R is the radius of the circle.
2 12
CS R Rδθ πδ π
= = =
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Curvature� Therefore, the Earth’s curvature is
� The curvature due to refraction in the atmosphere is dependent on the refractive index change with height:
where is the real part of the refractive index, and H is the height. For standard refraction condition:
rnH
δδ
1 , 6374R kmR
=
6 139 10rn kmH
δδ
− −= − ×
rn
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Curvature� Then the total curvature of radar rays due to
both Earth curvature and standard atmospheric refraction is:
Which is equivalent to:
� The effective Earth radius is a fictitious Earth’s radius. Under standard refraction conditions, the radar rays travel along the curvature of this fictitious Earth’s surface.
8483 1.33 (4 / 3)R km R Rʹ = ≅ ≅
6 41 1 1 39 10 1.179 106374
rnR R H km km km
δδ
− −× ×= + = − =ʹ
Rʹ
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Standard Refraction (Radar book Fig 3.7)
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Super Refraction/Ducting (Trapping) � Super refraction happens when the downward bending of radar
waves is stronger than normal. � Super refraction occurs when temperature increases with height
(inversion) � Ducting (trapping) is exceptional super refraction when the radius
of curvature for the wave becomes smaller than Earth’s and the radar waves are trapped in a layer of the atmosphere.
� Ducting increases the radar detection range, but it also increases ground clutter.
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Radar Equation for Point Targets
� Motivation: Beside locating storms, radars can also measure the strength of the returned power, therefore estimating rain rate and convective intensity, etc. Therefore, we must know how to use radar quantitatively.
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Radarequation:Solvingtheradarreceivedpowerfromtransmittedpower
� Panela:Radarpower(unit:W)transmittedbyanisotropicantenna
� Panelb:usingarealantenna,thepoweratapointonthebeamaxisisincreasedbyafactoroftheantennagain.
� Panelc:thepowerinterceptedbyatargetwithareaAisreradiatedisotropically inalldirections,withsomeofitreceivedbackattheradar.
Fig.4.1
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Radar Equation for Point Targets� Assume the radar power is Pt at radar’s original
location. For an isotropic antenna, the radar power density S (W/m2) at range r from the radar is:
� Then for a real antenna, it should be:
24tPSrπ
=
24tPgSrπ
=
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Radar Equation for Point Targets� Then the power intercepted by the target at
range r from the radar is:
where Aσ is the cross section area of the target.
24tPgAPrσ
σ π=
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Radar Equation for Point Targets� Assume that power intercepted by the target is
reradiated isotropically back into space, then the energy amount received back at the radar will be:
where Ae is the effective area of the receiving antenna:
2 2 24 (4 )e t e
rP A PgA APr r
σ σ
π π= =
2
4egA λπ
=
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Radar Equation for Point Targets� Substitute and get the radar equation for point
target:
� Wait a minute, is this correct?-- No. In fact, because the physical size of the target is not necessarily the size the target interacts with (appears ) to the radar. Instead, we should use the backscattering cross-section σ toreplaceAσ. So the final radar equation is:
2 2
3 464t
rPg AP
rσλ
π=
2 2
3 464t
rPgP
rλ σπ
=
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BackScatteringCrossSectionforSphericalTargets
� Opticalregion:validforsphereslargerthanwavelength(x=πD/λ>10,Disthediameter,aistheradius):
� Rayleighregime:whenthesizeofasphereissmallcomparedtothewavelength(x=πD/λ<0.1)
where
N2 is the complex index of refraction of the sphere (target).
2aσ π=
25 6
4
K Dπσ
λ=
22
221
1 ,2
NmK m Nm N
−= = ≈
+
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HowtoderiveBackScatteringCrossSection(PettyCH12)
� RecallRayleighsolution(lec 10)scatteringefficiency:
𝑄' =()𝑥+ 𝐾 -
� BackscatteringefficiencyQb isQs atbackwarddirection𝜃 = 𝜋.Thenforrayleigh phasefunction,
𝑄2 = 4𝑥+ 𝐾 -
� Backscatteringcrosssection:
𝜎2 = 𝑄2𝜋𝑟- = 4𝑥+ 𝐾 -𝜋𝑟- = 67 8 9:;
𝝺=
where diameter D=2r , 𝜎 = 𝜎225 6
4
K Dπσ
λ=
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Back Scattering Cross Section: Mie Solution
Y-axis: backscattering efficiency
X-axis: size Parameter x
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Back Scattering Cross Section
� In Rayleigh regime, the backscattering cross section is proportional to D6
� Mie regime (0.1< π D/λ<10): the backscattering cross section σ candecreaseastheparticlesizeincreasesforcertainsizeparameters
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Summary� The combination effect of Earth curvature and
atmosphere refraction causes the radar rays bend upward relative to the Earth surface under standard refraction condition.
� Super refraction may cause ducting.� Rayleigh regime: σ ~D6 ; Optical regime: σ=geometricarea;Mieregime:complicated