1. Fast lidar & radar multiple-scattering models for cloud retrievals Robin Hogan (University of Reading) Alessandro Battaglia (University of Bonn)
How can we account for radar and lidar multiple scattering in CloudSat/Calipso/EarthCARE retrievals?
Overview of talk:
Examples of multiple scattering from CloudSat and LITE
“Variational” retrievals and forward modelling of radar/lidar signals
The four scattering regimes
Fast modelling of “quasi-small-angle” multiple scattering using the photon variance-covariance method
Fast modelling of “wide-angle” multiple scattering using 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=300 m)
CloudSat radar (l>r)
Stratocumulus
Intense thunderstorm
LITE lidar on the space shuttle in 1994
Large detector footprint (300 m) means that photons may be scattered many times and still remain within the field-of-view
The long path-length means that those detected appear to have been scattered back from below cloud base (or below the surface)
Need a sophisticated lidar forward model to represent this
Surface echo
Apparent echo from below the surface

3. 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
e.g. Delanoë and Hogan, in preparation

4. 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

5. 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: Quasi-small-angle multiple scattering
Occurs when Ql ~ x
Only for wavelength much less than particle size, e.g. lidar & ice clouds
Footprint x
Regime 3: Wide-angle multiple scattering
Occurs when l ~ x

6. New radar/lidar forward model
CloudSat/Calipso record a new profile every 0.1 s
An “operational” forward model clearly must run in much less than 0.01 s!
Most widely used existing 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
Regime 3: Time-Dependent Two-Stream (TDTS) method
Sum the signal from the relevant methods:
Radar: regime 1 (single scattering) plus regime 3
Lidar: regime 2 plus regime 3

7. 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

8. 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:

9. 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 a diffusion
2-stream: represent only two propagation directions
60°
60°
I+(t,r)
60° r

10. 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 used in weather models
Source Scattering from the quasi-direct beam into each of the streams
Gain by scattering Radiation scattered from the other stream
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

11. Lateral photon spreading
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 a diffusivity D

12. 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

13. Comparison with Monte Carlo: Lidar
I3RC (Intercomparison of 3D radiation codes) case 1
Isotropic scattering, 500-m cloud, optical depth 20
Monte Carlo calculations from Alessandro Battaglia

14. Comparison with Monte Carlo: Lidar
I3RC case 3
Henyey-Greenstein phase function, semi-infinite cloud, absorption
Monte Carlo calculations from Alessandro Battaglia

15. Comparison with Monte Carlo: Lidar
I3RC case 5
Mie phase function, 500-m cloud
Monte Carlo calculations from Alessandro Battaglia

16. Comparison with Monte Carlo: Radar
Mie phase functions, CloudSat reciever field-of-view
Monte Carlo calculations from Alessandro Battaglia

17. 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

18. Future work
Implement TDTS in CloudSat/Calipso retrieval (PVC is 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
It would be simple to predict the HSRL channel of EarthCARE

20. Interpretation of radar and lidar
We want to know the profile of the important cloud properties:
Liquid or ice water content (g m-3)
The mean size of the droplets or ice particles
In principle these properties can be derived utilizing the very different scattering mechanisms of radar and lidar
We have developed a variational algorithm to interpret the combined measurements (“1D-Var” in data assimilation):
Make a first guess of the cloud profile
Use forward models to simulate the corresponding observations
Compare the forward model values with the actual observations
Use Gauss-Newton iteration to refine the cloud profile to achived a better fit with the observations in a least-squares sense
We need accurate radar and lidar forward models, but multiple scattering can make life difficult!

21. 7 June 2006 Calipso lidar (l<r) CloudSat radar (l>r)
Molecular scattering
Aerosol from China?
Cirrus
Mixed-phase altocumulus
Drizzling stratocumulus
Non-drizzling stratocumulus
Rain
5500 km
Japan
Eastern Russia
East China Sea
Sea of Japan

22. CloudSat radar MODIS infrared image Multiple scattering Bangladesh
Bay of Bengal
MODIS infrared image
Bangladesh
Himalayas

23. Intense convection over the Amazon
Multiple scattering

25. The 3D radiative transfer equation
Also known as the “Boltzmann transport equation”, this describes the evolution of the radiative intensity I as a function of time t, position x and direction W:
Can use Monte Carlo but very expensive
Spatial derivative representing how much radiation is upstream
Loss by absorption or scattering
Source
Gain by scattering Radiation scattered from all other directions
Time derivative r
I–(t,r)
I+(t,r)
Must make some approximations:
1-D: represent lateral transport as a diffusion
2-stream: represent only two propagation directions
60°

26. Effect on integrated backscatter
Need to be careful in applying the calibration technique for wide field-of-view lidars; may no longer asymptote
It is possible that integrated backscatter could provide information on optical depth