Review Doppler Radar (Fig. 3.1) A simplified block diagram

1. ReviewDoppler Radar (Fig. 3.1)A simplified block diagram METR 5004

2. Electric field incident on scatterer Reflected electric field incident on antenna Voltage input to the synchronous detectors; This pair of detectors shifts the frequency f to 0 jQ(t) A Echo phase at the output of the detectors and filters ψe I(t) If the range r of the scatterer is fixed, phasor A is fixed. But if scatterer has a radial velocity, Phasor A rotates about the origin at the Doppler frequency fd. Complex plane (Phasor diagram)

3. 1 μ s Range-Time 2 3 1 0 4 5 4 3 5 0 2 1 Stationary Moving (A) scatterers scatterer (B) METR 5004

4. r=cτs/2 cτ λ Pulsed Radar Principle c = speed of microwaves = ch for H and = cv for V waves τ = pulse length λ = wavelength = λh for H and λv for V waves τs = time delay between transmission of a pulse and reception of an echo. METR 5004

5. For typical atmospheric conditions (i.e., normal) the propagation path is a straight line if the earth has a radius 4/3rds times its true radius. Normal and Anomalous Propagation Sub refraction Free space Normal atmosphere: The ray’s radius of curvature Rc≈ constant) Super refraction (trapping) (anomalous propagation: Unusually cold moist air near the ground) METR 5004

6. Angular Beam Formation(the transition from a circular beam of constant diameter to an angular beam of constant angular width) Fresnel zone Far field region θ1= 1.27 λ/D (radians) METR 5004

7. Antenna (directive) Gaingt The defining equation: Eq. (3.4) (W m-2) = power density incident on scatterer r = range to measurement = radiation pattern (= 1 on beam axis) = transmitted power (W) METR 5004

8. Wavenumbers for H, V Waves Horizontal polarization: Vertical polarization: where k = free space wavenumber = 3.6x106 (deg./km) for λ = 10 cm (e.g., for R= 100 mm h-1, = 24.4okm-1, = 20.7okm-1) Therefore: ch< cv ; λh < λv ; kh>kv Specific differential phase: (for R = 100 mm h-1) (KDP:an important polarimetric variable related to rainrate) METR 5004

9. Specific Differential Phase (Fig.6.17) Echo phase of H = h = 2khr Eq. (6.60) METR 5004

10. Backscattering Cross Section, σbfor a Spherical Particle METR 5004

11. Backscattered Power Density Incident on Receiving Antenna METR 5004

12. Echo Power PrReceived (3.20) Aeis the effective area of the receiving antenna for radiation from the θ,φ direction. It is shown that: (3.21) If the transmitting antenna is the same as the receiving antenna then: METR 5004

13. The Radar Equation(point scatterer/discrete object) METR 5004

14. nth pulse (n+1)th pulse Ts time Apparent delay < Ts True delay > Ts Unambiguous Range ra • If targets are located beyond ra = cTs/2, their echoes from the nth transmitted pulse are received after the (n+1)th pulse is transmitted. Thus, they appear to be closer to the radar than they really are! • This is known as range folding • Ts = PRT • Unambiguous range: ra = cTs/2 • Echoes from scatterers between 0 and raare called 1st trip echoes, • Echoes from scatterers between ra and 2ra are called 2nd trip echoes, Echoes from scatterers between 2ra and 3ra are called 3rd trip echoes, etc ra METR 5004

15. Ambiguous Doppler Shifted Echoes(Fig. 3.14) METR 5004

16. Δr= vrTs is the change in range of the scatterer between successive transmitted pulses METR 5004

17. Another PRT Trade-Off • Correlation of pairs: • This is a measure of signal coherency • Accurate measurement of power requires long PRTs • More independent samples (low coherency) • But accurate measurement of velocity requires short PRTs • High correlation between sample pairs (high coherency) • Yet a large number of independent sample pairs are required METR 5004

18. Signal Coherency • How large a Ts can we pick? • Correlation between m = 1 pairs of echo samples is: • Correlated pairs: (i.e., Spectrum width must be much smaller than unambiguous velocity va = λ/4Ts) • Increasing Ts decreases correlation exponentially • also increases exponentially! • Pick a threshold: • Violation of this condition results in very large errors of estimates! METR 5004

19. 8 m s-1 Signal Coherency and Ambiguities • Range and velocity dilemma: rava=cl/8 • Signal coherency: sv<va /p • raconstraint: Eq. (7.2c) • This is a more basic constraint on radar parameters than the first equation above • Then, sv and not va imposes a basiclimitation on Doppler weather radars • Example: Severe storms have a median sv ~ 4 m/s and 10% of the time sv > 8 m/s. If we want accurate Dopplerestimates 90% of the time with a 10-cmradar (l = 10 cm); then, ra ≤ 150 km. This will often result in range ambiguities Fig. 7.5 150 km Spectrum width σv METR 5004 Unambiguous range ra

20. Echoes (I or Q) from Distributed Scatterers (Fig. 4.1) t c(s)≈ t (t = transmitted pulse width) mTs Weather signals (echoes) METR 5004

21. Weather Echo Statistics (Fig. 4.4) METR 5004

22. Spectrum of a transmitted rectangular pulse ftf If receiver frequency response is matched to the spectrum of the transmitted pulse (an ideal matched filter receiver), some echo power will be lost. This is called the finite bandwidth receiver loss Lr. For and ideal matched filter Lr = 1.8 dB (ℓr= 1.5). METR 5004

23. Reflectivity Factor Z(Spherical scatterers; Rayleigh condition: D≤ λ/16) METR 5004

24. Differential Reflectivity in dB units: ZDR(dB) = Zh(dBZ) - Zv(dBZ) in linear units: Zdr = Zh(mm6m-3)/Zv(mm6m-3) - is independent of drop concentration N0 - depends on the shape of scatterers METR 5004

25. Shapes of raindrops falling in still air and experiencing drag force deformation. De is the equivalent diameter of a spherical drop.ZDR (dB) is the differential reflectivity in decibels (Rayleigh condition is assumed). Adapted from Pruppacher and Beard (1970) METR 5004

26. The Weather Radar Equation METR 5004