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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)