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

Complex plane (Phasor diagram) Ao Ao jQ(t) I(t) ψeψe If the range r of the scatterer is fixed, the phasor (A o, ψ e ) is fixed (i.e., no change in A o nor ψ e. But if scatterer has a radial velocity, phasor (A o, ψ e ) 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 voltage V o at the output of the detectors and filters. The echo amplitude is A o and phase is

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

Angular Beam Formation ( the transition from a circular beam of constant diameter to an angular beam of constant angular width) 5 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 a scatterer r = range to measurement (m) = radiation pattern = 1 on beam axis = transmitted power (W) 10/29-11/11/2013METR 50046

Backscattering Cross Section, σ b for a Spherical Particle 10/29-11/11/2013METR 50047

Backscattered Power Density Incident on Receiving Antenna 10/29-11/11/2013METR 50048

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 50049

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

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

14 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 15 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 16  c (  s ) ≈  t (  t = transmitted pulse width) tt

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

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