Chapter 4: Radar equation and Attenuation

η(r) ≡ ∫0∞ σb (D) N(D, r) dD (1), 稱之回波

(the reflectivity), 或是在雷利散射假設下

η = (π5/λ4)|Km|2 Z (2) , 其中

Z ≡ ∫0∞ { N(D, r) D6 }dD (3), 稱之回波因子

(Reflectivity factor) , 當雷利散射假設不滿足時

Ze = η / {(π5/λ4)|Km|2} (4), 稱之相當回波因子

(equivalent reflectivity factor) 使用單位：dBZ = 10 x log [ Ze / (1 mm6 m-3)]

Reflectivity η, reflectivity factor Z, and equivalent reflectivity factor Ze

How does parallel beam become conical form?

• 微波脈衝離開天線時主要是一群平行電磁波束，由天線反射器(其直徑為Da)射出

• 由於繞射(diffraction)之故，在距離天線 r= Da

2/λ處(第一繞射點)，此平行電磁波束開始成圓錐狀發射，其角寬大約為 Δθ~ 104λ/Da。

• 定義(電磁波)束寬(beam width)，當能量僅達其最大能量一半之角寬稱之，一般又稱之3dB束寬。

Diffraction and radiation pattern

• 輻射型態(radiation pattern)描述電磁波束由天線射出時能量分布狀況，受工藝技術影響實務上無法將能量完全集中在一個窄束內，部分能量會落在主瓣(main lobe)外，稱之副瓣(side lobe)。一般而言副瓣能量大都不超過主瓣能量的百分之一，所有副瓣內的總能量一般僅僅是主瓣能量的百分之幾大小。副瓣總能量大小常常是天氣雷達設計時的重要考量，避免雜訊比太高。

• 微波脈衝發射可以設想成厚度為cτ球殼以光速c往外膨脹，雖然在任一球殼內之發射的脈衝能量Pt’為一常數，但是電磁波能量密度Si(θ,φ)與距離r2

成反比。 • 在每一脈衝延時內發射之能量並不一致，因此脈衝能量定義為在脈衝延時平均之發射能量。

• 由於能量在天線,傳輸線,天線罩中都會損失，因此送往天線輸入口脈衝能量Pt將較實際發射之脈衝能量Pt’為大。

Radar equation for point target The maximum directional gain of the antenna

gt’ ≡ Sp / [ Pt’/4πr2 ] (R1)

Pt’ is difficult to measure, so usually Pt (the power delivered to the antenna’s input port)is measured. Sp is the peak power and measured at distance far ( r >2Da

2/λ) from the antenna.

Thus, the computed gain gt accounts for losses of energy associated with the antenna system (radome and waveguide).

The incident radiation power density at range r without considering attenuation is given by

Si(θ, φ) = Pt gt ƒ2(θ, φ) / 4πr2 (R2)

ƒ2(θ, φ) is the normalized [i. e., ƒ(θ, φ) = 1 at θ0, φ0] power gain pattern and gt is the antenna power gain along the beam axis.

一般天線-反射器成拋物圓形, 由焦點（喇叭口）發射出波束,反射天線上能量分佈並不均勻,能流密度(輻射強度)隨天線軸心距離 (ρ) 成 [1-4(ρ/Da)2] 2的關係。在此情形下, 常態化能流密度形態(或稱為輻射形態函數radiation pattern)與偏角的關係可寫成：

ƒ2(θ) = S(θ)/S(θ0) = {8 J2[(πDasinθ)/λ] / [(πDasinθ)/λ]2 }2 (R3)

其中θ為離天線軸心之偏角大小, λ為雷達發射電磁波波長, Da為天線直徑, J2 是Bessel function of second order。

雷達發射之微波脈衝可視為厚度為cτ (脈衝寬) 的球形殼以光速往外傳播並擴張。因此, 在一封閉球體內所發射之脈衝能量（Pt’)為常數, 但是散射物所接收到的能流密度大小與距離平方成反比。此外由於天線本身,傳輸線,天線罩等造成的電磁波能量減損, 進入天線之脈衝能量 Pt 要大於實際發射出的 Pt’ 。

**一般Pt較Pt’好量度,因此將能量損失加入天線增益中表達

The backscattering cross section 反散射截面 The cross section σ of a scatterer (hydrometeor) is an apparent area

that intercepts a power σ Si which, if radiated isotropically, produces at the receiver a power density

Sr = Si σ (θ,φ) /4πr2 (R4) equal to that scattered by the actual hydrometeor.

注意: 目標物的散射截面並不等於目標物的幾何截面

The backscattering cross section σb of a spherical water drop of diameter D small compared to λ (I.e., Rayleigh scattering or small drop scattering approximation, D≤ λ/16) is well approximated by

σb ≈ ( π5/λ4 )Km2 D6 (R5), Where Km = (m2-1)/(m2+2) and m = n-jnκ is the complex refractive

index of water, n is refractive index, κ is the attenuation index. Some authors define the absorption coefficient k = nκ (a function of wavelength and temperature)

Complex refractive index: ice and water

λ = 10 cm and longer,

For water, n = 9 (independent of T), k = 0.63 ~1.47 for 20~0℃ For ice, n = 1.78 (also independent of T), k = 2.4x10-3~5.5x10-4 for 0~-20 ℃

For water drops, Km2 = 0.91~0.93, λ =1-10cm, indep. of T

For ice spheres, Km2 = 018, ρ=0.917gcm-3, indep. of T and λ

Note, when D~∞ , σb / [πD2/4] = (m-1)2/(m+1)2

If D<<λ σ b (ice) = 0.20 σ b (water) (-6.989dB)

If D ~∞ σ b (ice) = 0.12 σ b (water) (-9.2dB)

衰減 Si(r2) = Si(r1) exp {-∫r1r2 k dr} , 其中

衰減係數 k = ∫0 ∞ N(D, r) σe (D) dD N(D, r)為粒徑譜(DSD), σe (D) 為消光截面, k的單位為km-1。

衰減係數 k 也可寫成以dB km-1為單位的K，

K = d/dr2 {10 log[S(r1)/S(r2)]} = 4.34 x 103 k dB km-1

Then, the echo power density at the radar antenna is

Sr (r, θ, φ) = Pt gt ƒ2(θ, φ) σb Ǎ2 / (4πr2 )2 (R6)

其中 Ǎ = exp { - ∫0r (kg + k) dr} one-way transmission loss due to hydrometeors as well as that caused by atmospheric gases.

Receiving aspects • The echo power collected by the antenna system from a

wave scattered by a hydrometeor at r, θ, φ is Pr = Sr (r, θ, φ) Ae (θ, φ) (R7) where Ae is the effective collection area of the antenna for

radiation from direction (θ, φ) • Ae (θ, φ) = grλ2ƒ2(θ, φ) /4π

where gr is the gain of the receiving antenna, equal to gt if the transmitting antenna is used for echo reception and Pr is measured at the same location in the antenna system as Pt.

Ae=42m2, but Ap=65.7m2

Radar equation for point target

The radar equation for a point scatterer having backscatter cross section σb

Pr (r,θ,φ) = Pt g2λ2ƒ4(θ,φ)σbǍ2 / [(4π)3 r4]

(R8)

where gt=gr=g.

Pr is in dBm (milli watts)

Radar equation in terms of Ze The radar equation in terms of Ze: P0(r) = {[π3Pt g2gscτ θ1

2]/ [210 ln2 λ2 ϑ2]} x {|Km|2 Ze/ r2} To the meteorologists, the radar equation is P0(mW) =[π510-17Pt(w)g2gs τ(µs)θ1

2(deg)] /[6.75 x 1014 ln2 λ2(cm) ϑ2

] x {|Km|2 Ze(mm6m-3)/ r2(km)} • mW=milli-Watts (dBm)

The scattering phase function P(θ)= (λ2 /8π2) (π D/λ)6Km2 (1+cos2θ)Fo

θis the angle between the incident beam and the scattered beam.

Mie and Rayleigh Scattering 米散射和雷利散射

米散射 Mie scattering

Attenuation:衰減 Attenuation: A wave suffers power loss both from energy

absorption and scattering.

Each drop absorbs an amount of power PL = σaSi, where σa is the absorption cross section, an apparent area that intercepts from the incident radiation a power equal to the power dissipated as heat in the drop. (a strong function of wavelength of the incident wave)

For small spheres, total scattering cross section (which is proportional to the integral of scattered power density over a sphere enclosing the particles) σs= 2 σb/3。(forward + backward + lateral scattering)

The attenuation (extinction) cross section, σe = σa + σs

For non-spherical drop, the cross sections are rather complicated and numerical evaluations are needed.

當電磁波傳播經體積為∆V (r) 內一短距離∆r, 其能流密度的改變 ∆Si 可寫成∆Si = - [∆r/ ∆V] Σn{(σan + σsn) Si}, (n=1,N) 負號代表能量損失，N代表∆V體積內之總粒子數。當∆r→0, 則 dSi / dr = - k Si，k為specific attenuation (比衰減)或稱衰減係數。積分可得 Si(r2) = Si(r1) exp {-∫r1

r2 k dr} , 其中

衰減係數 k = lim ∆r→0 Σ{(σan + σsn)/ ∆V}

= ∫0 ∞ N(D, r) σe (D) dD

N(D, r)為粒徑譜(DSD), σe (D) 為消光截面, k的單位為km-1。衷減係數 k 也可寫成以dB km-1為單位的K，

K = d/dr2 {10 log[S(r1)/S(r2)]} = 4.34 x 103 k dB km-1

Then, the echo power density at the radar antenna is

Sr (r, θ, φ) = Pt gt ƒ2(θ, φ) σb η2 / (4πr2 )2

其中 η = exp { - ∫0r (kg + k) dr} one-way transmission loss due to hydrometeors as well as that caused by atmospheric gases.

雨的衰減 Mie 散射公式顯示在小粒子

散射假設吸收截面和散射截面可寫成

σa ≈ (π2 D3 / λ) Im(-Km) σs ≈ (2π5 D6 / 3λ4) |Km|2 Note:雨的衰減在微波波段主

要以吸收為主, 即使是10公分波長, 吸收所造成的衰減亦相當可觀. 僅有在低降雨率對波長1公分之電磁波,散射才較吸收為大.

波長0.86公分電磁波在雨中的衰減截面和D/λ呈線性關係

One-way specific attenuation, calculated by function of rain rates (in mm/hour) for different wavelength and at a temperature of 18℃ by using Laws and Parsons (1943) drop size data.

Note that rainwater on radome significantly attenuates weather signals.

Rain attenuation K = 4.34x103 ∫0∞ N(D, r)σe(D) dD dB km-1

= K (absorption) + K (scattering)

Rain attenuation K = 4.34 x 103 ∫0∞ N(D, r)σe(D) dD dBkm-1,

= K (absorption) + K (scatter)

Rain attenuation in terms of Z

Raindrop size distributions Marshall-Palmer distribution N(D) = N0 exp (-ΛD) Λ = 4.1 R-0.21 mm-1

N0 = 8 x 103 m-3mm-1

Laws and Parsons distribution

(measured)

Cloud attenuation, kc ≈ π2/λ Im(-Km) ∫0∞ N(D, r) D3 dD

Liquid water density, M = ρwπ/6 ∫0∞ N(D, r) D3 dD

Specific attenuation kc ≈ 6πM / [ρwλ] Im(-Km)

For weather radar wavelength 3.3-10 cm, clouds at 0C, Im(-Km) ≈ 1.1 x 10-3 / λ , with an accuracy of ~3%.

Attenuation by dry snow

ks = 3.5 x 10-2 (R2/λ4) + 2.2 x 10-3 (R/λ ) dB km-1

Wet snow and water snowflakes, because of the irregular shapes, attenuates more than dry snow. No simple formula are available for it.

Attenuation in gases kg, mainly water vapor and oxygen, and absorption >>scattering; is appreciable when storms are far away and beam elevation is low.