METR 3223, Physical Meteorology II: Radar Doppler Velocity Estimation Mark Askelson Adapted from: Doviak and Zrnić, 1993: Doppler radar and weather observations. 2nd Ed. Academic Press, 562 pp. I. Essentials--Wave Propagation A. General equation for a traveling wave 1. E = the value at a time and place 2. A = amplitude 3. j = 1 4. ω = circular (angular) frequency a. ω = 2πf b. f = frequency 5. t = time, x = distance 6. k = wavenumber 2π a. k = λ b. λ = wavelength E = A exp[ j( ω t kx) ] (1) 7. Here, E and A are vectors because we are considering electric fields, a vector quantity. B. For the electromagnetic waves utilized by weather radars: 1. x r, where r is range 2. λ = c f 3. (1) becomes: E 2π ( ) r A exp j + + = + ( + ) ω r j ϕt ϕb A exp j2πf t j ϕt ϕ c c. (2) f = b
a. ϕ t = transmitter phase 1. Constant for coherent receivers. 2. Random for incoherent receivers. This does not permit velocity estimation! Why? Might there be ways around this? 3. We will consider a coherent receiver. b. ϕ b = backscattering phase shift. 1. Generally not of consequence for Rayleigh scattering a) Definition: Scatterer diameter is much smaller than the wavelength of the radar. b) Strictly, D λ, but in practice this approximation can be pushed further. 16 2. Will be considered to be constant in this discussion. 4. The actual evolution of E is given by the real part of (2). An imaginary representation, however, is useful because from this representation one can obtain r the amplitude A and the phase ϕ t = 2 πf t + ( ϕt + ϕb ). See the phasor c diagram below: [from Doviak and Zrnić (1993, p. 12)] 2
II. Velocity Determination A. Stationary scatterer. 1. An electromagnetic (EM) wave is transmitted by the radar and impinges upon a scatterer (a rain drop for example). 2. The electric field in the wave excites an alternating dipole in the scatterer that radiates an EM wave back to the radar (backscattering). 3. Since the scatterer dipole oscillates with the frequency of the transmitted wave, the frequency of the backscattered EM wave received at the radar is that of the wave that was transmitted. A. Still Hydrometeor B. Moving at Radar + + Head-on view Head-on view Radar Radar Side view Side view B. Scatterer moving towards radar. 1. Same as above except the oscillation of the dipole in the scatterer will be faster because the scatterer is experiencing a higher frequency EM wave due to its motion into the wave. 2. The backscattered wave received at the radar has a higher frequency than the wave that was transmitted. This is totally a result of the motion of the scatterer towards (or away) from the radar and is known as the Doppler shift. 3. The Doppler shift can be measured using the phase of the returned EM waves. Consider 3
dϕt d = + ( + ) 2r 2 πf t ϕt ϕb dt dt c 4π dr 4π = 2πf = ωc + ωd = ωc vd. (3) λ dt λ a. ω c = "carrier" angular frequency b. ω d = Doppler angular frequency c. v d = Doppler velocity d. It is assumed that ϕ t & ϕb are constant (not true for incoherent radars). e. Note that here r 2r because we are considering now the phase at the receiver as opposed to the phase at the scatterer. 4. Coherent weather radars are equipped with thingamajiggers like STALOs and synchronous detectors that filter out the carrier wave and determine the Is and Qs [the real and imaginary parts, respectively (see phasor diagram)] of the left-over wave that is oscillating with the Doppler angular frequency. C. A highly simplified view of velocity determination. 1. From two successive I and Q estimates for a scatterer (or collection of scatterers) at range r, one can obtain two estimates of total phase ϕ t1 and ϕ t 2. 2. With the oscillations owing to the carrier wave removed, the difference in these phases is due to the Doppler velocity of the scatterer(s): ϕ ϕ 2 ϕ1 4π = = v t T λ s d, (4) where T s is the sampling time, also known as the pulse repetition time (PRT). 4
III. Ambiguities and the Doppler dilemma A. Radar measurements of phase are digital 1. Data are digital when you do not have continuous information concerning a phenomenon. If, for instance, you knew that the temperature varied linearly from one hour to the next you would have continuous information concerning the temperature and the data set would be analog. If, on the other hand, you had measurements of temperature every ten minutes during that hour, your data would be digital. B. Because radar measurements of phase are digital, they can suffer aliasing. 1. Fundamental to digital data 2. Can be difficult to handle 3. Example: A car is rolling forward. The "spokes" on the hubcaps of the car, however, give the appearance that the car is rolling backward. What happens is the "spokes" move more than 1/2 the way around the tire as your optical system samples the motion of the spokes. Since it is a lesser distance if the spokes had actually rotated backwards, you interpret the information as if the tire was rotating backwards! Seen Actual Motion 4. Occurs with weather-radar Doppler velocity estimation as well. Because we do not know the sign of the velocity (inbound or outbound), the largest phase shift we can measure is ± π. 5. Plugging this value into (4) gives the equation for the Nyquist velocity v dn λ = ±. (5) 4T s 5
C. Doppler dilemma 1. The maximum range of a scatterer that can be determined unambiguously is cts r a = ±. (6) 2 2. Note that r a is directly proportional to T s while v dn is inversely proportional to T s. Thus, if we try to decrease range ambiguities by increasing T s we exacerbate the velocity ambiguity problem and vice-versa. This is known as the Doppler dilemma. 6