Doppler Radar (Fig. 3.1) A simplified block diagram 10/29-11/11/2013 METR

Transcription

1 Review Dopple Rada (Fig. 3.1) A simplified block diagam 10/9-11/11/013 METR

2 A ( θϕ, ) exp ψt c i Ei = j π f t + j E A ( θϕ, ) = exp jπ f t + jψt c 4π Vi = Aiexp jπ ft j + jψt λ jq(t) ψ e Complex plane (Phaso diagam) I(t) Electic field incident on sca>ee Reflected electic field incident on antenna Voltage input to the synchonous detectos; This pai of detectos shiqs the fequency f to 0 [ ] V exp 4 / o = I + jq Ao j π λ + jψt Echo phase at the output of the detectos and filtes ψe = ( 4 π / λ) + ψt If the ange of the sca>ee is fixed, phaso A is fixed. But if sca>ee has a adial velocity, Phaso A otates about the oigin at the Dopple fequency f d.

3 Range-Time 1 µ s (A) Stationay scattees Moving scattee (B) 10/9-11/11/013 METR

4 Pulsed Rada Pinciple =cτ s / λ cτ c = speed of micowaves = c h fo H and = c v fo V waves τ = pulse length λ = wavelength = λ h fo H and λ v fo V waves τ s = time delay between tansmission of a pulse and eception of an echo. 10/9-11/11/013 METR

5 Nomal and Anomalous PopagaTon n = 1+ C P /T + C P /T + C P /T d d w1 w w w efactive index n= c/ v Sub efaction Fee space Nomal atmosphee: The ay s adius of cuvatue R c constant) Supe efaction (tapping) (anomalous popagation: Unusually cold moist ai nea the gound) Fo typical atmospheic conditons (i.e., nomal) the popagaton path is a staight line if the eath has a adius 4/3ds Tmes its tue adius. 10/9-11/11/013 METR

6 Angula Beam FomaTon (the tansiton fom a cicula beam of constant diamete to an angula beam of constant angula width) Fesnel zone D / λ Fa field egion E φ E θ D / λ ; 1.5 km; WSR-88D: D ; 8.53 m; λ=10 cm θ 1 = 1.7 λ/d (adians) 10/9-11/11/013 METR

7 Antenna (diective) Gain g t The defining equation: S i = Pt g f ( θ, φ) 4π t Eq. (3.4) S i (W m - ) = powe density incident on scattee = ange to measuement f ( θ, φ) = adiation patten (= 1 on beam axis) P t = tansmitted powe (W) 10/9-11/11/013 METR

8 Wavenumbes fo H, V Waves Hoizontal polaization: Vetical polaization: kv = ( k+ kʹ v)= πλ / v whee k = fee space wavenumbe = 3.6x10 6 (deg./km) fo λ = 10 cm (e.g., fo R= 100 mm h -1, kʹ = 4.4 o km -1, kʹ = 0.7 o km -1 ) Theefoe: c h < c v ; λ h < λ v ; k h > k v Specific diffeential phase: K kʹ k ʹ. 1 DP = h v = 3 7(deg. km ) h k = ( k+ kʹ )= πλ / h h h (fo R = 100 mm h -1 ) (K DP : an impotant polaimetic vaiable elated to ainate) v 10/9-11/11/013 METR

9 Specific Diffeential Phase (Fig.6.17) Echo phase of H = φ h = k h Eq. (6.60) 10/9-11/11/013 METR

10 Backsca>eing Coss SecTon, σ b fo a Spheical PaTcle Rayleigh condition on a spheical paticle of diamete D: D < λ/ 16; λ wavelength 5 π σ b = K 4 m D λ 6 ; m 1 K ; Dielectic Facto of the medium filling the sphee; Eq.3.6 m m + m= n jnκ = the complex index of efaction K K = fo wate, and m w K K = fo ice (density = g m -3 ) m i 10/9-11/11/013 METR

11 Backsca>eed Powe Density Incident on Receiving Antenna S i Pg t t f ( θφ, ) 1 S(, θφ, ) = σ ( 3.13a) b 4π l 4π l whee l is the loss facto (due to attenuation) l = exp ( kg + k) d ( 3.13b) 0 10/9-11/11/013 METR

12 π λ φ θ 4 / ), ( e f g A = Echo Powe P Received A e is the effective aea of the eceiving antenna fo adiation fom the θ,φ diection. It is shown that: (3.0) (3.1) If the tansmitting antenna is the same as the eceiving antenna then: ), ( ),, ( φ θ φ θ e A P = S ), ( ), ( ), ( φ θ φ θ φ θ gf f g f g t t = 10/9-11/11/013 METR

13 The Rada EquaTon (point sca>ee/discete object) Pgf t ( θ,φ) σb gλ f (θ,φ) P = ( 3. 4 ) 4π l 4π l 4π Example: λ =. ; = ( ; P( = ( t m 0 km x10 m) min) 10 W); P = ( g = x ; = ( W; peak); 3 10 l 1 no path loss) Calculating the minimum detectable backscatteing σ b (min) = x10 m 7 = σ b fo a 6.3 mm dop! σ : b 10/9-11/11/013 METR

14 Unambiguous Range a If tagets ae located beyond a = ct s /, thei echoes fom the n th tansmi>ed pulse ae eceived aqe the (n+1) th pulse is tansmi>ed. Thus, they appea to be close to the ada than they eally ae! This is known as ange folding nth pulse (n+1)th pulse T s = PRT a T s Tue delay > T s time Appaent delay < T s Unambiguous ange: a = ct s / Echoes fom sca>ees between 0 and a ae called 1 st tip echoes, Echoes fom sca>ees between a and a ae called nd tip echoes, Echoes fom sca>ees between a and 3 a ae called 3 d tip echoes, etc 10/9-11/11/013 METR

15 Ambiguous Dopple ShiQed Echoes (Fig. 3.14) 10/9-11/11/013 METR

16 Unambiguous Velocity l A pulsed Dopple ada measues adial Dopple velocity by keeping tack of phase changes between samples that ae T s (pulse epetition time) apat l Recall that echo phase shift is ψ e = 4π/λ. Then, the phase change fom pulse to pulse is Δψ e = 4πΔ/λ = 4πv T s /λ l Note that only phase changes between π and π can be unambiguously esolved l Theefoe, the unambiguous velocity is: l l 4πv a T s /λ = π ð v a = λ/4t s This is elated to the Nyquist sampling theoem: Dopple velocities outside the ±v a inteval will be aliased! Δ = v T s is the change in ange of the scattee between successive tansmitted pulses 10/9-11/11/013 METR

17 Anothe PRT Tade- Off CoelaTon of pais: This is a measue of signal coheency Accuate measuement of powe equies long PRTs Moe independent samples (low coheency) But accuate measuement of velocity equies shot PRTs T lim ρ( T ) = 0 s s lim ρ( T ) = 1 T 0 s s ρ( T ) = exp 8 ( πσ / λ ) s vts High coelaton between sample pais (high coheency) Yet a lage numbe of independent sample pais ae equied 10/9-11/11/013 METR

18 Signal Coheency How lage a T s can we pick? CoelaTon between m = 1 pais of echo samples is: ρ( T )= exp T s 8( π σv s/ λ ) πσ vts λ Coelated pais: ρ( Ts) 1 << 1 >> σv λ πts (i.e., Spectum width σ v must be much smalle than unambiguous velocity v a = λ/4t s ) Inceasing T s deceases coelaton exponentally Va[ vˆ] and Va[ σˆ ] also inceases exponentally! v Pick a theshold: 0.5 ρ( T ) e 8 πσ T / λ = 0.5 σ v / π ( ) s v s v a ViolaTon of this conditon esults in vey lage eos of estmates! 18 10/9-11/11/013 METR 5004

19 Signal Coheency and AmbiguiTes Range and velocity dilemma: a v a =cλ/8 Signal coheency: σ v < v a /π a constaint: a cλ 8πσ Eq. (7.c) This is a moe basic constaint on ada paametes than the fist equaton above Then, σ v and not v a imposes a basic limitaton on Dopple weathe adas Example: Sevee stoms have a median σ v ~ 4 m/s and 10% of the >me σ v > 8 m/s. If we want accuate Dopple estmates 90% of the Tme with a 10- cm ada (λ = 10 cm); then, a 150 km. This will oqen esult in ange ambiguites v Spectum width σ v 8 m s -1 Fig km 19 10/9-11/11/013 METR 5004 Unambiguous ange a

20 Echoes (I o Q) fom Distibuted Scattees (Fig. 4.1) τ t τ c (τ s ) τ t (τ t = tansmitted pulse width) Weathe signals (echoes) 10/4-11/11/013 METR

21 Weathe Echo Statistics (Fig. 4.4) 10/4-11/11/013 METR

22 Spectum of a tansmi>ed ectangula pulse δ f = 1 τ t f t f If eceive fequency esponse is matched to the spectum of the tansmitted pulse (an ideal matched filte eceive), some echo powe will be lost. This is called the finite bandwidth eceive loss L. Fo and ideal matched filte L = 1.8 db (l = 1.5). 10/4-11/11/013 METR 5004

23 ReflecTvity Facto Z (Spheical sca>ees; Rayleigh conditon: D λ/16) η whee η π λ 5 ( ) = K 4 m Z( ) (4.31) ( ) i = (, ) (4.3) ΔV i 0 Z D N D D dd π λ 5 ( ) = K 4 w Ze( ) (4.33) o fo wate dops : Kw 0.93 independent of T( C); fo ice paticles : Ki 0.16 dependent on T and icedensity. 10/4-11/11/013 METR

24 Diffeential Reflectivity in db units: Z DR (db) = Z h (dbz) - Z v (dbz) in linea units: Z d = Z h (mm 6 m -3 )/Z v (mm 6 m -3 ) - is independent of dop concentation N 0 - depends on the shape of scattees 10/4-11/11/013 METR

25 Shapes of aindops falling in still ai and expeiencing dag foce defomation. D e is the equivalent diamete of a spheical dop. Z DR (db) is the diffeential eflectivity in decibels (Rayleigh condition is assumed). Adapted fom Puppache and Bead (1970) 10/4-11/11/013 METR

26 The Weathe Rada EquaTon A fom of the weathe ada equation fo echo powe fom ain is: E π 10 P t (W) g g s τ ( µ s) θ 1 (deg.) K w Z (mm m ) w ( 0 ) (mw) = (ln ) 0 (km) λ (cm) l l [ ] E P [ P ] ( ) Expected peak weathe signal powe in milliwatts; 0 (4.35) P t g s Peak tansmitted pulse powe (typically 500 kw) net powe gain of the echo in going fom the antenna to the ada output. τ = pulse width θ 1 one-way half-powe beamwidth; K w dielectic facto of wate Z eflectivity facto fo wate sphees; w 0 ange (in km) to the cente of the esolution volume V 6 l one-way loss facto (a numbe 1) incued fo popagation though a ain filled atmosphee. l loss facto due to the finite bandwidth of the eceive; λ wavelength of the tansmitted adiation 10/4-11/11/013 METR