1. Robin Hogan, Chris Westbrook University of Reading, UK Alessandro Battaglia University of Leicester, UK Fast forward modelling of radar and lidar depolarization subject to multiple scattering

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

3. Depolarization induced by multiple scattering Can we model effect of multiple scattering on depolarization? Potentially very useful information on extinction (e.g. Sassen & Petrilla 1986) Radar: Battaglia et al. (2007) Lidar: Observations at Chilbolton

4. Overview Lidar observations in liquid clouds difficult to interpret quantitatively –Difficult to correct for strong attenuation Radar & lidar multiple scattering contains useful info on extinction –Information mostly in the tail - see Nicola Pounders talk on Friday Depolarization induced by multiple scattering contains more info –Information content first noted by Sassen and Petrilla (1986) Provides a range-resolved index of multiple scattering –Useful for spaceborne radar: is apparent echo from low in a storm actually from radiation that has just bounced around at cloud top? –Useful for spaceborne lidar: can we retrieve the extinction profile in optically thick liquid clouds? Challenge: write a fast forward model to use in radar & lidar retrievals –May need to be quite heuristic…

5. Regime 1: Single scattering –Apparent backscatter is easy to calculate –Zero depolarization from droplets Scattering regimes Footprint x Regime 2: Small-angle multiple scattering –Only for wavelength much less than particle size, e.g. lidar & ice clouds –Fast Photon Variance-Covariance (PVC) model of Hogan (2008) –Depolarization due to backscatter slightly away from 180 degrees Regime 3: Wide-angle multiple scattering –Fast Time Dependent Two Stream (TDTS) method of Hogan & Battaglia –Depolarization increases with number of scattering events

6. A typical Mie phase function for a distribution of droplets Fraction of cross-polar rather than co-polar scattered radiation Forward scattering is unpolarized The glory is polarized

7. Calculate at each gate: Total energy P Position variance Direction variance Covariance r s Equivalent medium theorem (Katsev et al. 1997): Use double optical depth on outward journey and zero on return Apparent backscatter is fraction of photon distribution in FOV Photon Variance-Covariance method Hogan (JAS 2008): small-angle lidar scattering Backscatter co-angle variance Construct distribution of backscatter co-angles Convolve with either total, co- or cross-polar phase function Infrastructure to do this already present in Hogan (2008) model to account for shape of total phase function

8. Time-dependent 2-stream approximation Hogan and Battaglia (JAS 2008): wide-angle scattering Describe diffuse flux in terms of outgoing stream I + and incoming stream I –, and numerically integrate the following coupled PDEs: I + and I – are used to calculate total (unpolarized) backscatter tot = || + T Time derivative Remove this and we have the time- independent two- stream approximation used in weather models 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 Gain by scattering Radiation scattered from the other stream Source Scattering from the quasi-direct beam into each of the streams

9. ...with depolarization Define co-polar weighted streams K + and K – and use them to calculate the co-polar backscatter co = || – T : Evolution of these streams governed by the same equations but with a loss term related to the rate at which scattering is taking place, since every scattering event randomizes the polarization and hence reduces the memory of the original polarization: Where is the fraction of the original polarization that is retained every scattering event (to be determined by comparison with Monte Carlo calculations provided by Alessandro Battaglia) Depolarization ratio is then calculated from

10. Interlude for gratuitous animation Animation of unpolarized 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

11. Evaluation in isotropic conditions Compare with Alessandro Battaglias rigorous Monte Carlo model I3RC case 1 isotropic scattering: use single scattering + TDTS model (no forward lobe so no need to treat small-angle scattering) Best fit to depolarization with = 0.75, independent of field of view

12. Evaluation in Mie scattering conditions I3RC case 5: PVC for small-angle and TDTS for wide-angle scattering Small-angle scattering: excellent fit at close range Wide-angle scattering: best fit achieved for = 0.6f + 0.85(1–f), where f is the fraction of energy remaining in the field-of-view of the lidar New model appears to perform well for different fields of view Small-angle scattering dominates Wide-angle scattering dominates

13. We have developed a fast forward model for the depolarization effects of multiple scattering –Applicable to lidar in liquid clouds, yet to be tested for radar –Less useful for lidar in ice clouds because of the uncertain single- scattering depolarization Next steps –Refine coefficients for different droplet size distributions –Incorporate into retrieval scheme (talk on Friday for possible framework) Non-polarized multiple scattering code freely available from http://www.met.rdg.ac.uk/clouds/multiscatter –Combines two fast multiple scattering models, PVC & TDTS –Includes C & Fortran-90 interfaces, adjoint model, HSRL capability... –For lidar, much more accurate than Platts approximation with –Can be used in retrievals and in instrument simulators –Fast: One profile can cost the same as a single Monte Carlo photon!Outlook