A variational scheme for retrieving rainfall rate and hail intensity

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

2. Outline • Rain-rate estimated by Z=aRbis at best accurate to a factor of 2 due to: • Variations in drop size and number concentration • Attenuation and hail contamination • In principle, Zdr and fdp can overcome these problems but tricky to implement operationally: • Need to take derivative of already noisy fdp field to get Kdp • 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 Zdr and fdpfor rain • Useful at low and high R • Differential attenuation allows accurate attenuation correction but difficult to implement • Need accurate calibration • Too noisy at each gate • Degraded by hail Zdr • 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 fdp

4. Variational method • Start with a first guess of coefficient a in Z=aR1.5 • Z/a 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 For a sensible solution at low rainrate, add an a priori constraint on coefficient a Observational errors are explicitly included, and the solution is weighted accordingly

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

6. A ray of data • Zdr and fdp 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 Zdr or fdp ? Where observations provide no information, retrieval tends to a priori value (and its error) fdp only useful where there is appreciable gradient with range 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 fdp only

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

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

10. 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 • 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 can match both Zdr andfdp

11. 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

12. 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)