Review Doppler Radar (Fig. 3.1) A simplified block diagram 10/29-11/11/2013METR 50041

Complex plane (Phasor diagram) A jQ(t) I(t) ψeψe 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 f d. 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 Echo phase at the output of the detectors and filters

10/29-11/11/2013METR 50043

Pulsed Radar Principle cτcτ λ c = speed of microwaves = c h for H and = c v for V waves τ = pulse length λ = wavelength = λ h for H and λ v for V waves τ s = time delay between transmission of a pulse and reception of an echo. r=c τ s /2 10/29-11/11/2013METR 50044

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 Super refraction (trapping) (anomalous propagation: Unusually cold moist air near the ground) Sub refraction 10/29-11/11/2013METR Free space Normal atmosphere: The ray’s radius of curvature R c ≈ constant)

Angular Beam Formation ( the transition from a circular beam of constant diameter to an angular beam of constant angular width) 6 Fresnel zone 10/29-11/11/2013METR 5004 Far field region θ 1 = 1.27 λ/D (radians)

Eq. (3.4) Antenna (directive) Gain g t The defining equation: (W m -2 ) = power density incident on scatterer r = range to measurement = radiation pattern (= 1 on beam axis) = transmitted power (W) 10/29-11/11/2013METR 50047

Wavenumbers for H, V Waves Horizontal polarization: Vertical polarization: where k = free space wavenumber = 3.6x10 6 (deg./km) for λ = 10 cm (e.g., for R= 100 mm h -1, = 24.4 o km -1, = 20.7 o km -1 ) Therefore: c h k v Specific differential phase: (for R = 100 mm h -1 ) (K DP : an important polarimetric variable related to rainrate) 10/29-11/11/2013METR 50048

Specific Differential Phase 10/29-11/11/2013METR Echo phase of H =  h = 2k h r (Fig.6.17) Eq. (6.60)

Backscattering Cross Section, σ b for a Spherical Particle 10/29-11/11/2013METR

Backscattered Power Density Incident on Receiving Antenna 10/29-11/11/2013METR

Echo Power P r Received A e is the effective area of the receiving antenna for radiation from the θ,φ direction. It is shown that: (3.20) (3.21) If the transmitting antenna is the same as the receiving antenna then: 10/29-11/11/2013METR

The Radar Equation (point scatterer/discrete object) 10/29-11/11/2013METR

Unambiguous Range r a If targets are located beyond r a = cT s /2, their echoes from the n th 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 T s = PRT Unambiguous range: r a = cT s /2 – Echoes from scatterers between 0 and r a are called 1 st trip echoes, – Echoes from scatterers between r a and 2r a are called 2 nd trip echoes, Echoes from scatterers between 2r a and 3r a are called 3 rd trip echoes, etc time True delay > T s ( n +1)th pulse n th pulse TsTs Apparent delay < T s rara 10/29-11/11/2013METR

Ambiguous Doppler Shifted Echoes (Fig. 3.14) 10/29-11/11/2013METR

10/29-11/11/2013METR Δr = v r T s is the change in range of the scatterer between successive transmitted pulses

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 10/29-11/11/2013 METR

18 Signal Coherency How large a T s 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 v a = λ/4T s ) Increasing T s decreases correlation exponentially – also increases exponentially! Pick a threshold: – – Violation of this condition results in very large errors of estimates! 10/29-11/11/2013METR 5004

Spectrum width σ v 19 Signal Coherency and Ambiguities Range and velocity dilemma: r a v a =c Signal coherency:  v < v a /  r a constraint: Eq. (7.2c) – This is a more basic constraint on radar parameters than the first equation above Then,  v and not v a imposes a basic limitation on Doppler weather radars – Example: Severe storms have a median  v ~ 4 m/s and 10% of the time  v > 8 m/s. If we want accurate Doppler estimates 90% of the time with a 10-cm radar ( = 10 cm); then, r a ≤ 150 km. This will often result in range ambiguities 10/29-11/11/2013METR 5004 Unambiguous range r a 150 km 8 m s -1 Fig. 7.5

10/24-11/11/2013METR 5004 Echoes (I or Q) from Distributed Scatterers (Fig. 4.1) Weather signals (echoes) mT s 20  c (  s ) ≈  t (  t = transmitted pulse width) tt

Weather Echo Statistics (Fig. 4.4) 10/24-11/11/2013METR

Spectrum of a transmitted rectangular pulse 10/24-11/11/2013METR 5004 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 L r. For and ideal matched filter L r = 1.8 dB (ℓ r = 1.5). 22 f t f

Reflectivity Factor Z (Spherical scatterers; Rayleigh condition: D ≤ λ/16) 10/24-11/11/2013METR

Differential Reflectivity in dB units: Z DR (dB) = Z h (dBZ) - Z v (dBZ) in linear units: Z dr = Z h (mm 6 m -3 )/Z v (mm 6 m -3 ) - is independent of drop concentration N 0 - depends on the shape of scatterers 10/24-11/11/2013METR

10/24-11/11/2013METR Shapes of raindrops falling in still air and experiencing drag force deformation. D e is the equivalent diameter of a spherical drop. Z DR (dB) is the differential reflectivity in decibels (Rayleigh condition is assumed). Adapted from Pruppacher and Beard (1970)

The Weather Radar Equation 10/24-11/11/2013METR