1. 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

2. Examples of multiple scattering
LITE lidar (l<r, footprint~1 km)
CloudSat radar (l>r)
Stratocumulus
Intense thunderstorm
Surface echo
Apparent echo from below the surface

3. 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)]
Mean free path l
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
Footprint x
Regime 3: Wide-angle multiple scattering
Occurs when l ~ x

4. 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)
Make some approximations in modelling the diffuse radiation:
1-D: represent lateral transport as modified diffusion
2-stream: represent only two propagation directions
60°
60°
I+(t,r)
60° r

6. 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)
Time derivative Remove this and we have the time-independent two-stream approximation
Source Scattering from the quasi-direct beam into each of the streams
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.)

7. Lateral photon transportx 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:

8. 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

9. 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

10. 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

11. Monte Carlo comparison: Mie
I3RC lidar case 5
Mie phase function, 500-m cloud
Monte Carlo calculations from Alessandro Battaglia

12. Monte Carlo comparison: Radar
Mie phase functions, CloudSat reciever field-of-view
Monte Carlo calculations from Alessandro Battaglia

13. Comparison of algorithm speeds
Model
Time
Relative to PVC
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:
Monte Carlo with polarization
5 hours
(0.6 ms per photon)
3x107

14. 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

16. Monte Carlo comparison: H-G
I3RC lidar case 3
Henyey-Greenstein phase function, semi-infinite cloud, absorption
Monte Carlo calculations from Alessandro Battaglia

17. 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 wide-angle scattering for optically thick clouds

18. 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 …)
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)

19. Phase functions Asymmetry factor Q q Radar & cloud droplet
l >> D
Rayleigh scattering
g ~ 0
Radar & rain drop
l ~ D
Mie scattering
g ~ 0.5
Lidar & cloud droplet
l << D
g ~ 0.85
Asymmetry factor Q q

20. Forward scattering events
Regime 2
Equivalent medium theorem: use lidar FOV to determine the fraction of distribution that is detectable (we can neglect the return journey) z
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!)
Calculate at each gate:
Total energy P
Position variance
Direction variance
Covariance r s
Forward scattering events s r
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

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