Transcript Document

The time-dependent two-stream method
for lidar and radar multiple scattering
Robin 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
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)
Scattering regimes
• Regime 0: No attenuation
– Optical depth d << 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
• Regime 2: Small-angle
multiple scattering
– Occurs when Ql ~ x
– Only for wavelength much less than
particle size, e.g. lidar & ice clouds
– No pulse stretching
Mean free path l
• Regime 3: Wide-angle multiple
scattering
– Occurs when l ~ x
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)
Regime 3: Wide-angle multiple scattering
Space-time diagram
• Make some approximations in modelling the
diffuse radiation:
– 1-D: represent lateral transport as modified diffusion
– 2-stream: represent only two propagation directions
I–(t,r)
60°
60°
+
60°
I (t,r)
r
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:
Time derivative
Remove this and
we have the timeindependent twostream
approximation


1 I
I

  1 I    2 I   S 
1c t
r


1 I  I 

  1 I    2 I   S 
1c t
r

Spatial derivative
Transport of
radiation from
upstream

Loss by absorption
or scattering
Some of lost radiation
will enter the other
stream
Source
Scattering from
the quasi-direct
beam into each of
the streams
Gain by scattering
Radiation scattered
from the other
stream
• These can be discretized quite simply in time and space (no implicit
methods or matrix inversion required)
Hogan and Battaglia (2008, to appear in J. Atmos. Sci.)
Lateral photon transport
x
y
• What fraction of photons
remain in the receiver field-ofview?
• Calculate lateral standard
deviation:
  x2  y 2
1/ 2
• Diffusion theory predicts
superluminal travel when the
mean number of scattering
events n = ct/lt is small:
2
4

n
2
lt
3
  t1/ 2
  x2  y 2
1/ 2
  t1/ 2
 t
• In ~1920, Ornstein and Fürth
independently solved the
Langevin equation to obtain the
correct description:
2
lt2

4
n  e n  1

3
Modelling lateral photon transport
• Model the lateral variance of photon position,  , using the
following equations (where V  I   2 ):
2
1 V  V 

   1V    2V    SV  D
1c t
r
1 V  V 

   1V    2V    SV  D
1c t
r
Additional source
Increasing variance
with time is described
by Ornstein-Fürth
formula
• 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
Simulation of 3D photon transport
• Animation of 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
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
Monte Carlo comparison: Mie
• I3RC lidar case 5
– Mie phase function, 500-m cloud
Monte Carlo calculations from Alessandro Battaglia
Monte Carlo comparison: Radar
– Mie phase functions, CloudSat reciever field-of-view
Monte Carlo calculations from Alessandro Battaglia
Comparison of algorithm speeds
Model
Time
Relative to 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
1s
1700
Eloranta 6th order
8.6 s
15000
5 hours
(0.6 ms per photon)
3x107
50-point profile, 1-GHz Pentium:
PVC
28 million photons, 3-GHz Pentium:
Monte Carlo with
polarization
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/multiscatter
Monte Carlo comparison: H-G
• I3RC lidar case 3
– Henyey-Greenstein phase function, semi-infinite cloud, absorption
Monte Carlo calculations from Alessandro Battaglia
How important
is multiple
scattering for
CALIPSO?
• Ice clouds:
– FOV such that small-angle
scattering almost
saturates: satisfactory to
use Platt’s approximation
with h=0.5
• Liquid clouds:
– Essential to include wideangle scattering for
optically thick clouds
The basics of a variational retrieval scheme
We need a fast
forward model that
includes the effects of
multiple scattering for
both radar and lidar
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 …)
Compare to the measurements
Are they close enough?
Yes
No
Gauss-Newton iteration step
Clever mathematics to produce a
better estimate of the state of
the atmosphere
Calculate error in retrieval
Proceed to next ray
Delanoë and Hogan (JGR 2008)
Phase functions
Asymmetry factor
Q
g  cos
• Radar & cloud droplet
– l >> D
– Rayleigh scattering
– g~0
• Radar & rain drop
– l~D
– Mie scattering
– g ~ 0.5
• Lidar & cloud droplet
– l << D
– Mie scattering
– g ~ 0.85

Regime 2
Equivalent medium theorem:
use lidar FOV to determine
the fraction of distribution
that is detectable (we can
neglect the return journey)
Calculate at each gate:
•
•
•
•
Forward
scattering
events
Total energy P
Position variance s 2
Direction variance ζ 2
Covariance sζ
z
r
s
• Eloranta’s (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
r
s
• Photon variance-covariance
(PVC) method
– Photon distribution is estimated
considering all orders of scattering
with O(N 2) efficiency (Hogan
2006, Appl. Opt.)
– O(N ) efficiency is possible but
slightly less accurate (work in
progress!)
Comparison of Eloranta & PVC methods
• For Calipso geometry (90-m field-of-view):
– PVC method is as accurate as Eloranta’s method taken to 5th-6th order
Ice cloud
Molecules
Liquid cloud
Aerosol
Download code from: www.met.rdg.ac.uk/clouds