Robin HoganRobin Hogan

A variational scheme A variational scheme for retrieving rainfall for retrieving rainfall rate and hail intensityrate and hail intensity

OutlineOutline• Rain-rate estimated by Z=aRb is 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 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

Using Using ZZdrdr and and dpdp for for rainrain

• Useful at low and high R• Differential attenuation

allows accurate attenuation correction but difficult to implement

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

• Need accurate calibration

• Too noisy at each gate• Degraded by hail

dp

Variational methodVariational 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 dp

– Include all the relevant physics, such as attenuation etc.

• Compare observations with forward-model values, and refine a by minimizing a cost function:

2

2

2

2

,,

12

2

,,

apdpdr a

api

fwdidpidp

n

i Z

fwdidridr aaZZ

J

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

• Observations

• Retrieval

Forward-model values at final iteration are essentially least-squares fits to the observations, but without instrument noise

ChilboltoChilbolton n

example example

A ray of dataA ray of data

• Zdr 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

What if we What if we only use only use

only only ZZdrdr or or dp dp ? ?

Very similar retrievals: in moderate rain rates, much more useful information obtained from Zdr than dp

Zdr

only

dp

only

Zdr

and

dp

Retrieved a Retrieval 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

Nominal Zdr error of ±0.2 dB Additional random error of ±1 dB

Response to observational Response to observational errorserrors

• Observations

• Retrieval

Difficult case: differential attenuation of 1 dB and differential phase shift of 80º!

Heavy Heavy rain andrain and

hailhail

How is hail How is hail retrieved?retrieved?

• Hail is nearly spherical– High Z but much lower Zdr than

would get for rain– Forward model cannot match

both Zdr and dp

• First pass of the algorithm– Increase error on Zdr so that rain

information comes from dp

– Hail is where Zdrfwd-Zdr

> 1.5 dB

• Second pass of algorithm– Use original Zdr 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 Zdr and dp

Distribution of Distribution of hailhail

– Retrieved rain rate much lower in hail regions: high Z no longer attributed to rain

– Can avoid false-alarm flood warnings

Retrieved a Retrieval error Retrieved hail fraction

SummarySummary• New scheme achieves a seamless transition between

the following separate algorithms:– Drizzle. Zdr and dp are both zero: use a-priori a coefficient

– Light rain. Useful information in Zdr 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 Zdr and dp information according to their errors

– Weak attenuation. Use dp 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 dp measurements (Smyth et al. 1999)

– Rain coexisting with hail. Estimate rain-rate in hail regions from dp alone (Sachidananda and Zrnic 1987)