Transcript Document

Fast forward modelling of
radar and lidar depolarization
subject to multiple scattering
Robin Hogan, Chris Westbrook University of Reading, UK
Alessandro Battaglia University of Leicester, UK
Examples of multiple scattering
• LITE lidar (l<r, footprint~1 km)
Stratocumulus
Apparent echo from
below the surface
Surface echo
Intense thunderstorm
CloudSat radar (l>r)
Depolarization induced
by multiple scattering
Radar: Battaglia et al. (2007)
Lidar: Observations at Chilbolton
Overview
• Lidar observations in liquid clouds difficult to interpret quantitatively
– Difficult to correct for strong attenuation
• Radar & lidar multiple scattering contains useful info on extinction
– Information mostly in the tail - see Nicola Pounder’s talk on Friday
• Depolarization induced by multiple scattering contains more info
– Information content first noted by Sassen and Petrilla (1986)
• Provides a range-resolved index of multiple scattering
– Useful for spaceborne radar: is apparent echo from low in a storm
actually from radiation that has just bounced around at cloud top?
– Useful for spaceborne lidar: can we retrieve the extinction profile in
optically thick liquid clouds?
• Challenge: write a fast forward model to use in radar & lidar retrievals
– May need to be quite heuristic…
Scattering regimes
• Regime 1: Single scattering
– Apparent backscatter b’ is easy to
calculate
– Zero depolarization from droplets
• Regime 2: Small-angle
multiple scattering
Footprint x
– Only for wavelength much less than
particle size, e.g. lidar & ice clouds
– Fast Photon Variance-Covariance
(PVC) model of Hogan (2008)
– Depolarization due to backscatter
slightly away from 180 degrees
• Regime 3: Wide-angle multiple
scattering
– Fast Time Dependent Two Stream
(TDTS) method of Hogan & Battaglia
– Depolarization increases with
number of scattering events
A typical Mie
phase function
for a distribution
of droplets
Fraction of cross-polar
rather than co-polar
scattered radiation
Forward
scattering is
unpolarized
The glory is
polarized
Photon Variance-Covariance method
Hogan (JAS 2008): small-angle lidar scattering
Equivalent medium theorem (Katsev et al. 1997):
• Use double optical depth on outward journey and zero on return
• Apparent backscatter is fraction of photon distribution in FOV
Calculate at each gate:
z
g
r
•
•
•
•
•
s
Total energy P
Position variance s 2
Direction variance ζ 2
Covariance sζ
Backscatter co-angle variance
• Construct distribution of
backscatter co-angles
• Convolve with either total, co- or
cross-polar phase function
• Infrastructure to do this
already present in Hogan (2008)
model to account for shape of
total phase function
g2
Time-dependent 2-stream approximation
Hogan and Battaglia (JAS 2008): wide-angle scattering
•
Describe diffuse flux in terms of outgoing stream I+ and incoming stream I–, and
numerically integrate the following coupled PDEs:
Time derivative
Remove this and
we have the timeindependent twostream
approximation
used in weather
models


1 I
I

  g1 I   g 2 I   S 
1c t
r


1 I  I 

  g1 I   g 2 I   S 
1c t
r


Source
Scattering from
the quasi-direct
beam into each of
the streams
Gain by scattering
Spatial derivative
A bit like an
advection term,
representing how
much radiation is
upstream
•
Loss by absorption
or scattering
Some of lost radiation
will enter the other
stream
Radiation scattered
from the other
stream
I+ and I– are used to calculate total (unpolarized) backscatter btot = b|| + bT
...with depolarization
•
•
Define “co-polar weighted” streams K+ and K– and use them to calculate the
co-polar backscatter bco = b|| – bT:
Evolution of these streams governed by the same equations but with a loss
term related to the rate at which scattering is taking place, since every
scattering event randomizes the polarization and hence reduces the memory
of the original polarization:
1 K 
K 
 




  g 1K  g 2 K   S   K
1c t
r
1
•
•
Where  is the “fraction of the original polarization that is retained” every
scattering event (to be determined by comparison with Monte Carlo
calculations provided by Alessandro Battaglia)
Depolarization ratio is then calculated from

bT btot  bco

b|| btot  bco
Interlude for gratuitous animation
• Animation of unpolarized scalar
flux (I++I–)
– Colour scale is logarithmic
– Represents 5 orders of
magnitude
• Domain properties:
– 500-m thick
– 2-km wide
– Optical depth of 20
– No absorption
• In this simulation the lateral
distribution is Gaussian at each
height and each time
Evaluation in isotropic conditions
• Compare with Alessandro Battaglia’s rigorous Monte Carlo model
• I3RC case 1 isotropic scattering: use single scattering + TDTS model
(no forward lobe so no need to treat small-angle scattering)
• Best fit to depolarization with  = 0.75, independent of field of view
Evaluation in Mie scattering conditions
Wide-angle
scattering dominates
Small-angle
scattering
dominates
• I3RC case 5: PVC for small-angle and TDTS for wide-angle scattering
• Small-angle scattering: excellent fit at close range
• Wide-angle scattering: best fit achieved for  = 0.6f + 0.85(1–f),
where f is the fraction of energy remaining in the field-of-view of the
lidar
• New model appears to perform well for different fields of view
Outlook
• We have developed a fast forward model for the depolarization effects of
multiple scattering
– Applicable to lidar in liquid clouds, yet to be tested for radar
– Less useful for lidar in ice clouds because of the uncertain singlescattering depolarization
• Next steps
– Refine coefficients for different droplet size distributions
– Incorporate into retrieval scheme (talk on Friday for possible framework)
• Non-polarized multiple scattering code freely available from
http://www.met.rdg.ac.uk/clouds/multiscatter
– Combines two fast multiple scattering models, PVC & TDTS
– Includes C & Fortran-90 interfaces, adjoint model, HSRL capability...
– For lidar, much more accurate than Platt’s approximation with h  0.7
– Can be used in retrievals and in instrument simulators
– Fast: One profile can cost the same as a single Monte Carlo photon!