**How to distinguish rain from hail using radar:A** cunning,variational method Robin Hogan Last Minute Productions Inc.

**Outline** • Increasingly in active remote sensing (radar and lidar), many instruments are being deployed together, and individual instruments may measure many variables • We want to retrieve an “optimum” estimate of the state of the atmosphere that is consistent with all the measurements • But most algorithms use at most only two instruments/variables and don’t take proper account of instrumental errors • The “variational” approach (a.k.a. optimal estimation theory) is standard in data assimilation and passive sounding, but has only recently been applied to radar retrieval problems • It is mathematically rigorous and takes full account of errors • Straightforward to add extra constraints and extra instruments • In this talk, it will be applied to polarization radar measurements of rain rate and hail intensity • Met Office recently commissioned new polarization radar • A variational retrieval is a very useful step towards assimilation of polarization data

**Active sensing** Passive sensing No attenuation With attenuation Isolated weighting functions (or Jacobians) so don’t need to bother with variational methods? With attenuation (e.g. spaceborne lidar) weighting functions are broader: variational method required • Radiance at a particular wavelength has contributions from large range of heights • A variational method is used to retrieve the temperature profile

**Chilbolton 3GHz radar: Z** • We need to retrieve rain rate for accurate flood forecasts • Conventional radar estimates rain-rate R from radar reflectivity factorZ using Z=aRb • Around a factor of 2 error in retrievals due to variations in raindrop size and number concentration • Attenuation through heavy rain must be corrected for, but gate-by-gate methods are intrinsically unstable • Hail contamination can lead to large overestimates in rain rate

**Chilbolton 3GHz radar: Zdr** • Differential reflectivity Zdr is a measure of drop shape, and hence drop size: Zdr= 10log10(ZH/ZV) • In principle allows rain rate to be retrieved to 25% • Can assist in correction for attenuation • But • Too noisy to use at each range-gate • Needs to be accurately calibrated • Degraded by hail Drop 1 mm ZV 3 mm ZH 4.5 mm ZDR = 0 dB (ZH = ZV) Drop shape is directly related to drop size: larger drops are less spherical Hence the combination of Z and ZDR can provide rain rate to ~25%. ZDR = 1.5 dB (ZH > ZV) ZDR = 3 dB (ZH >> ZV)

**Chilbolton 3GHz radar: fdp** phase shift • Differential phase shift fdp is a propagation effect caused by the difference in speed of the H and V waves through oblate drops • Can use to estimate attenuation • Calibration not required • Low sensitivity to hail • But • Need high rain rate • Low resolution information: need to take derivative but far too noisy to use at each gate: derivative can be negative! • How can we make the best use of the Zdr and fdp information?

**Simple Zdr method** Retrieval Noisy or no retrieval Rainrate Lookup table Observations • Use Zdr at each gate to infer a in Z=aR1.5 • Measurement noise feeds through to retrieval • Noise much worse in operational radars Noisy or Negative Zdr

**Variational method** • Start with a first guess of coefficient a in Z=aR1.5 • Z/R implies a drop size: use this in a forward model to predict the observations of Zdr and fdp • Include all the relevant physics, such as attenuation etc. • Compare observations with forward-model values, and refine a by minimizing a cost function: + Smoothness constraints Observational errors are explicitly included, and the solution is weighted accordingly For a sensible solution at low rainrate, add an a priori constraint on coefficient a

**Finding the solution** New ray of data First guess of x • In this problem, the observation vector y and state vector x are: Forward model Predict measurements y and Jacobian H from state vector x using forward modelH(x) Compare measurements to forward model Has the solution converged? 2 convergence test No Gauss-Newton iteration step Predict new state vector: xi+1= xi+A-1{HTR-1[y-H(xi)] +B-1(b-xi)} where the Hessian is A=HTR-1H+B-1 Yes Calculate error in retrieval The solution error covariance matrix is S=A-1 Proceed to next ray

**First guess of a** • Use difference between the observations and forward model to predict new state vector (i.e. values of a), and iterate First guess: a =200 everywhere Rainrate

**Final iteration** • Zdr and fdp are well fitted by forward model at final iteration of minimization of cost function Rainrate • Retrieved coefficient a is forced to vary smoothly • Prevents random noise in measurements feeding through into retrieval (which occurs in the simple Zdr method)

**Enforcing smoothness** • In range: cubic-spline basis functions • Rather than state vector x containing “a” at every range gate, it is the amplitude of smaller number of basis functions • Cubic splines solution is continuous in itself, its first and second derivatives • Fewer elements in x more efficient! Representing a 50-point function by 10 control points • In azimuth: Two-pass Kalman smoother • First pass: use one ray as a constraint on the retrieval at the next (a bit like an a priori) • Second pass: repeat in the reverse direction, constraining each ray both by the retrieval at the previous ray, and by the first-pass retrieval from the ray on the other side

**Full scan from Chilbolton** • Observations • Retrieval • Note: validation required! Forward-model values at final iteration are essentially least-squares fits to the observations, but without instrument noise

**Response to observational errors** Nominal Zdr error of ±0.2 dB Additional random error of ±1 dB

**What if we use only Zdr or fdp ? ** Retrieved a Retrieval error Zdr and fdp Very similar retrievals: in moderate rain rates, much more useful information obtained from Zdr than fdp Zdr only Where observations provide no information, retrieval tends to a priori value (and its error) fdp only fdp only useful where there is appreciable gradient with range

**Heavy rain andhail** Difficult case: differential attenuation of 1 dB and differential phase shift of 80º • Observations • Retrieval

**How is hail retrieved?** • Hail is nearly spherical • High Z but much lower Zdrthan would get for rain • Forward model cannot match both Zdr andfdp • First pass of the algorithm • Increase error on Zdrso that rain information comes from fdp • Hail is where Zdrfwd-Zdr> 1.5 dB and Z > 35 dBZ • Second pass of algorithm • Use original Zdrerror • At each hail gate, retrieve the fraction of the measured Z that is due to hail, as well as a. • Now the retrieval can match both Zdr andfdp

**Distribution of hail** Retrieved a Retrieval error Retrieved hail fraction • Retrieved rain rate much lower in hail regions: high Z no longer attributed to rain • Can avoid false-alarm flood warnings • Use Twomey method for smoothness of hail retrieval

**Summary** • New scheme achieves a seamless transition between the following separate algorithms: • Drizzle.Zdr andfdp are both zero: use a-prioria coefficient • Light rain. Useful information in Zdr only: retrieve a smoothly varying a field (Illingworth and Thompson 2005) • Heavy rain. Use fdp as well (e.g. Testud et al. 2000), but weight the Zdr and fdp information according to their errors • Weak attenuation. Use fdp to estimate attenuation (Holt 1988) • Strong attenuation. Use differential attenuation, measured by negative Zdr at far end of ray (Smyth and Illingworth 1998) • Hail occurrence. Identify by inconsistency between Zdr and fdp measurements (Smyth et al. 1999) • Rain coexisting with hail. Estimate rain-rate in hail regions from fdp alone (Sachidananda and Zrnic 1987) • Could be applied to new Met Office polarization radars • Testing required: higher frequency higher attenuation! Hogan (2007, J. Appl. Meteorol. Climatology)

**Conclusions and ongoing work** Lake district Isle of Wight France England Scotland • Variational methods have been described for retrieving cloud, rain and hail, from combined active and passive sensors • Appropriate choice of state vector and smoothness constraints ensures the retrievals are accurate and efficient • Could provide the basis for cloud/rain data assimilation • Ongoing work: cloud • Test radiance part of cloud retrieval using geostationary-satellite radiances from Meteosat/SEVIRI above ground-based radar & lidar • Retrieve properties of liquid-water layers, drizzle and aerosol • Incorporate microwave radiances for deep precipitating clouds • Apply to A-train data and validate using in-situ underflights • Use to evaluate forecast/climate models • Quantify radiative errors in representation of different sorts of cloud • Ongoing work: rain • Validate the retrieved drop-size information, e.g. using a distrometer • Apply to operational C-band (5.6 GHz) radars: more attenuation! • Apply to other radar problems, e.g. the radar refractivity method