1. Modelling radar and lidar multiple scattering Robin Hogan
The CloudSat radar and the Calipso lidar were launched on 28th April 2006 as part of the A-train of satellites
They represent an opportunity to retrieve the vertical profile of cloud properties globally for the first time: important for climate
But multiple scattering presents a problem in interpreting both the radar and the lidar signals

3. 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

4. xi+1= xi+A-1{HTR-1[y-H(xi)]
1D variational method
New ray of data
First guess of x
In this problem, the observation vector y and state vector x are:
Forward model
Predict measurements y and Jacobian H from state vector x using forward model H(x)
Compare measurements to forward model
Has the solution converged?
2 convergence test No
Gauss-Newton iteration step
Predict new state vector:
xi+1= xi+A-1{HTR-1[y-H(xi)]
+B-1(b-xi)}
where the Hessian is
A=HTR-1H+B-1
Yes
Calculate error in retrieval
The solution error covariance matrix is S=A-1
Proceed to next ray

5. Examples of multiple scattering
Stratocumulus
Surface echo (sea)
Apparent echo from below the sea surface!
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

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

7. Intense convection over the Amazon
Multiple scattering

8. Multiple scattering so strong it corrupts next ray, appearing above the cloud!

10. 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

11. Scattering regimes Regime 0: No attenuation
Optical depth << 1
mm-wave radar & ice cloud
Lidar & optically thin aerosol
Regime 1: Single scattering
mm-wave radar & liquid cloud
Lidar & optically thick aerosol
Mean free path l
Regime 2: Quasi-small-angle multiple scattering
Ql ~ x
Only for wavelength much less than particle size
Lidar & ice clouds
Footprint x
Regime 3: Full multiple scattering
l ~ x

12. New algorithm
Use the Hogan (Applied Optics, 2006) algorithm for the “quasi-direct” return (contribution from regimes 1 and 2)
Use the “time-dependent two-stream” approximation for wide-angle multiple scattering (regime 3)
Space-time diagram

13. Quasi-small-angle multi-scattering
To calculate the “quasi-direct” component:
Eloranta’s (1998) model
O (N m/m !) efficient for N points in profile and m-order scattering
Too expensive to take to more than 3rd or 4th order in retrieval (not enough)
Hogan (2006) method treats third and higher orders together
O (N 2) efficient
As accurate as Eloranta when taken to ~6th order
3-4 orders of magnitude faster for N =50 (~ 0.1 ms)
Narrow field-of-view: forward scattered photons escape
Wide field-of-view: forward scattered photons may be returned
Ice cloud
Molecules
Liquid cloud
Aerosol
Hogan (Applied Optics, 2006). Code:

15. Time-dependent two-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 using simple schemes in time and space, provided that the optical depth of each layer is small
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

16. Lateral photon spreading
Two-stream equations are inherently 1D, so can’t predict the lateral spreading of photons that is necessary to determine the fraction of them remaining within the instrument field-of-view
Solution: 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
Additional source Increasing variance with time is described by a diffusivity D

17. Simulation of 3D photon transport
Scalar (actinic) flux is shown (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

18. Comparison with Monte Carlo
Very good agreement found with Monte Carlo (much slower!) for simple cloud case and a wide range of fields-of-view
Monte Carlo calculation courtesy of Tamas Varnai (NASA) for an I3RC case (Intercomparison of 3D Radiation Codes)

19. Effect on integrated backscatter
Need to be careful in applying the O’Connor et al. (2004) calibration technique for wide field-of-view lidars; may no longer asymptote
It is possible that integrated backscatter could provide information on optical depth

20. Future work
Modify the numerics so that discretizations can be used where the optical depth is large within one layer
Find a way to estimate the Jacobian so that the new forward model can be applied in a variational retrieval scheme
Implement in the CloudSat/Calipso retrieval scheme
More confidence in lidar retrievals in liquid water clouds
Can interpret CloudSat returns in deep convection
Apply to multiple field-of-view lidars
The difference in backscatter for two different fields of view enables the multiple scattering to be quantified and interpreted in terms of cloud properties
Predict the polarization of the returned signal
Difficult, but useful for lidar because multiple scattering depolarizes the return in liquid water clouds which would otherwise not depolarize