The%20time-dependent%20two-stream%20method%20for%20lidar%20and%20radar%20multiple%20scattering%20Robin%20Hogan%20(University%20of%20Reading)%20Alessandro%20Battaglia%20(University%20of%20Bonn)

1
The time-dependent two-stream method for lidar
and radar multiple scatteringRobin Hogan
(University of Reading)Alessandro Battaglia
(University of Bonn)

• To account for multiple scattering in CloudSat
  and CALIPSO retrievals we need a fast forward
  model to represent this effect
• Overview
• Examples of multiple scattering from CloudSat and
  LITE
• The four multiple scattering regimes
• The time-dependent two-stream approximation
• Comparison with Monte-Carlo calculations for
  radar and lidar

2
Examples of multiple scattering

• LITE lidar (lltr, footprint1 km)
• CloudSat radar (lgtr)

3
Scattering regimes

• Regime 0 No attenuation
• Optical depth d ltlt 1
• Regime 1 Single scattering
• Apparent backscatter b is easy to calculate from
  d at range r
• b(r) b(r) exp-2d(r)

Footprint x
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New radar/lidar forward model

• CloudSat and CALIPSO record a new profile every
  0.1 s
• Delanoe and Hogan (JGR 2008) developed a
  variational radar-lidar retrieval for ice clouds
  intention to extend to liquid clouds and precip.
• It needs a forward model that runs in much less
  than 0.01 s
• Most widely used existing lidar methods
• Regime 2 Eloranta (1998) too slow
• Regime 3 Monte Carlo much too slow!
• Two fast new methods
• Regime 2 Photon Variance-Covariance (PVC)
  method (Hogan 2006, Applied Optics)
• Regime 3 Time-Dependent Two-Stream (TDTS) method
  (this talk)
• Sum the signal from the relevant methods
• Radar regime 1 (single scattering) regime 3
  (wide-angle scattering)
• Lidar regime 2 (small-angle) regime 3
  (wide-angle scattering)

5
Regime 3 Wide-angle multiple scattering
Space-time diagram
I(t,r)
60

• Make some approximations in modelling the diffuse
  radiation
• 1-D represent lateral transport as modified
  diffusion
• 2-stream represent only two propagation
  directions

60
I(t,r)
60
r
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Time-dependent 2-stream approx.

• Describe diffuse flux in terms of outgoing stream
  I and incoming stream I, and numerically
  integrate the following coupled PDEs
• These can be discretized quite simply in time and
  space (no implicit methods or matrix inversion
  required)

Source Scattering from the quasi-direct beam
into each of the streams
Time derivative Remove this and we have the
time-independent two-stream approximation
Gain by scattering Radiation scattered from
the other stream
Loss by absorption or scattering Some of lost
radiation will enter the other stream
Spatial derivative Transport of radiation
from upstream
Hogan and Battaglia (2008, to appear in J. Atmos.
Sci.)
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Lateral photon transport
x
y

• What fraction of photons remain in the receiver
  field-of-view?
• Calculate lateral standard deviation

• Diffusion theory predicts superluminal travel
  when the mean number of scattering events n
  ct/lt is small

• In 1920, Ornstein and Fürth independently solved
  the Langevin equation to obtain the correct
  description

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Modelling lateral photon transport

• Model the lateral variance of photon position,
  , using the following equations (where
  )
• Then assume the lateral photon distribution is
  Gaussian to predict what fraction of it lies
  within the field-of-view
• Resulting method is O(N2) efficient

Additional source Increasing variance with time
is described by Ornstein-Fürth formula
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Simulation of 3D photon transport

• Animation of scalar flux (II)
• 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

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Monte Carlo comparison Isotropic

• I3RC (Intercomparison of 3D radiation codes)
  lidar case 1
• Isotropic scattering, semi-infinite cloud,
  optical depth 20

Monte Carlo calculations from Alessandro Battaglia
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Monte Carlo comparison Mie

• I3RC lidar case 5
• Mie phase function, 500-m cloud

Monte Carlo calculations from Alessandro Battaglia
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Monte Carlo comparison Radar

• Mie phase functions, CloudSat reciever
  field-of-view

Monte Carlo calculations from Alessandro Battaglia
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Comparison of algorithm speeds
Model Time Relative to PVC
50-point profile, 1-GHz Pentium 50-point profile, 1-GHz Pentium 50-point profile, 1-GHz Pentium
PVC 0.56 ms 1
TDTS 2.5 ms 5
Eloranta 3rd order 6.6 ms 11
Eloranta 4th order 88 ms 150
Eloranta 5th order 1 s 1700
Eloranta 6th order 8.6 s 15000
28 million photons, 3-GHz Pentium 28 million photons, 3-GHz Pentium 28 million photons, 3-GHz Pentium
Monte Carlo with polarization 5 hours (0.6 ms per photon) 3x107
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Ongoing work

• Apply to Quickbeam, the CloudSat simulator
  (done)
• Predict Mie and Rayleigh channels of HSRL lidar
  (done for PVC)
• Implement TDTS in CloudSat/CALIPSO retrieval (PVC
  already implemented for lidar)
• More confidence in lidar retrievals of liquid
  water clouds
• Can interpret CloudSat returns in deep convection
• But need to find a fast way to estimate the
  Jacobian of TDTS
• Add the capability to have a partially reflecting
  surface
• Apply to multiple field-of-view lidars
• The difference in backscatter for two different
  fields of view enables the multiple scattering to
  be interpreted in terms of cloud properties
• Predict the polarization of the returned signal
• Difficult but useful for both radar and lidar

Code available from www.met.rdg.ac.uk/clouds/multi
scatter
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Monte Carlo comparison H-G

• I3RC lidar case 3
• Henyey-Greenstein phase function, semi-infinite
  cloud, absorption

Monte Carlo calculations from Alessandro Battaglia
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How important is multiple scattering for CALIPSO?

• Ice clouds
• FOV such that small-angle scattering almost
  saturates satisfactory to use Platts
  approximation with h0.5
• Liquid clouds
• Essential to include wide-angle scattering for
  optically thick clouds

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The basics of a variational retrieval scheme
New ray of data First guess of profile of
cloud/aerosol properties (IWC, LWC, re )
Forward model Predict radar and lidar
measurements (Z, b ) and Jacobian (dZ/dIWC )
Gauss-Newton iteration step Clever mathematics to
produce a better estimate of the state of the
atmosphere
Compare to the measurements Are they close enough?
No
Yes
Calculate error in retrieval
Proceed to next ray

• Delanoë and Hogan (JGR 2008)

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Phase functions

• Radar cloud droplet
• l gtgt D
• Rayleigh scattering
• g 0
• Radar rain drop
• l D
• Mie scattering
• g 0.5
• Lidar cloud droplet
• l ltlt D
• Mie scattering
• g 0.85

Asymmetry factor
Q
q
20
Regime 2
Forward scattering events
s
r

• Elorantas (1998) method
• Estimate photon distribution at range r,
  considering all possible locations of scattering
  on the way up to scattering order m
• Result is O(N m/m !) efficient for an N -point
  profile
• Should use at least 5th order for spaceborne
  lidar too slow

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Comparison of Eloranta PVC methods
Wide field-of-view forward scattered
photons may be returned
Narrow field-of-view forward scattered
photons escape

• For Calipso geometry (90-m field-of-view)
• PVC method is as accurate as Elorantas method
  taken to 5th-6th order

Ice cloud
Molecules
Liquid cloud
Aerosol
Download code from www.met.rdg.ac.uk/clouds
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