Robin Hogan

1
A variational scheme for retrieving rainfall rate
and hail intensity

• Robin Hogan

2
Outline

• Rain-rate estimated by ZaRb 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 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 fdp for 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
  ZaR1.5
• 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 ?
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 Zdr than would get for rain
• Forward model cannot match both Zdr and fdp
• First pass of the algorithm
• Increase error on Zdr so that rain information
  comes from fdp
• Hail is where Zdrfwd-Zdr gt 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 fdp

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 and fdp 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 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)