1. Robin Hogan A variational scheme for retrieving rainfall rate and hail intensity

2. Outline Rain-rate estimated by Z=aR b is at best accurate to a factor of 2 due to: –Variations in drop size and number concentration –Attenuation and hail contamination In principle, Z dr and dp can overcome these problems but tricky to implement operationally: –Need to take derivative of already noisy dp field to get dp –Errors in observations mean we must cope with negative values –Difficult to ensure attenuation-correction algorithms are stable The variational approach is standard in data assimilation and satellite retrievals, but has not yet been applied to polarization radar: –It is mathematically rigorous and takes full account of errors –Straightforward to add extra constraints

3. Using Z dr and dp for rain Useful at low and high R Differential attenuation allows accurate attenuation correction but difficult to implement Z dr Calibration not required Low sensitivity to hail Stable but inaccurate attenuation correction Need high R to use Must take derivative: far too noisy at each gate Need accurate calibration Too noisy at each gate Degraded by hail dp

4. Variational method Start with a first guess of coefficient a in Z=aR 1.5 Z/a implies a drop size: use this in a forward model to predict the observations of Z dr and dp –Include all the relevant physics, such as attenuation etc. Compare observations with forward-model values, and refine a by minimizing a cost function: 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 + Smoothness constraints

5. Observations Retrieval Forward-model values at final iteration are essentially least- squares fits to the observations, but without instrument noise Chilbolton example

6. A ray of data Z dr and dp are well fitted by the forward model at the final iteration of the minimization of the cost function Retrieved coefficient a is forced to vary smoothly –Represented by cubic spline basis functions Scheme also reports error in the retrieved values

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

8. Nominal Z dr error of ±0.2 dBAdditional random error of ±1 dB Response to observational errors

9. Observations Retrieval Difficult case: differential attenuation of 1 dB and differential phase shift of 80º! Heavy rain and hail

10. How is hail retrieved? Hail is nearly spherical –High Z but much lower Z dr than would get for rain –Forward model cannot match both Z dr and dp First pass of the algorithm –Increase error on Z dr so that rain information comes from dp –Hail is where Z dr fwd -Z dr > 1.5 dB Second pass of algorithm –Use original Z dr error –At each hail gate, retrieve the fraction of the measured Z that is due to hail, as well as a. –Now can match both Z dr and dp

11. Distribution of hail –Retrieved rain rate much lower in hail regions: high Z no longer attributed to rain –Can avoid false-alarm flood warnings Retrieved aRetrieval errorRetrieved hail fraction

12. Summary New scheme achieves a seamless transition between the following separate algorithms: –Drizzle. Z dr and dp are both zero: use a-priori a coefficient –Light rain. Useful information in Z dr only: retrieve a smoothly varying a field (Illingworth and Thompson 2005) –Heavy rain. Use dp as well (e.g. Testud et al. 2000), but weight the Z dr and dp information according to their errors –Weak attenuation. Use dp to estimate attenuation (Holt 1988) –Strong attenuation. Use differential attenuation, measured by negative Z dr at far end of ray (Smyth and Illingworth 1998) –Hail occurrence. Identify by inconsistency between Z dr and dp measurements (Smyth et al. 1999) –Rain coexisting with hail. Estimate rain-rate in hail regions from dp alone (Sachidananda and Zrnic 1987)