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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

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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 Examples of multiple scattering Examples of multiple scattering LITE lidar (

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The basics of a variational retrieval scheme New ray of data First guess of profile of cloud/aerosol properties (IWC, LWC, r e …) Forward model Predict radar and lidar measurements (Z,…) and Jacobian (dZ/dIWC …) Compare to the measurements Are they close enough? Gauss-Newton iteration step Clever mathematics to produce a better estimate of the state of the atmosphere Calculate error in retrieval No Yes Proceed to next ray e.g. Delanoë and Hogan, in preparation We need a fast forward model that includes the effects of multiple scattering for both radar and lidar

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Phase functions Radar & cloud droplet – >> D –Rayleigh scattering –g ~ 0 Radar & rain drop – ~ D –Mie scattering –g ~ 0.5 Lidar & cloud droplet – << D –Mie scattering –g ~ 0.85 Asymmetry factor

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Regime 0: No attenuation –Optical depth << 1 Regime 1: Single scattering –Apparent backscatter is easy to calculate from at range r : (r) = (r) exp[-2 (r)] Scattering regimes Footprint x Mean free path l Regime 2: Quasi-small-angle multiple scattering – Occurs when l ~ x – Only for wavelength much less than particle size, e.g. lidar & ice clouds Regime 3: Wide-angle multiple scattering –Occurs when l ~ x

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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

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Regime 2 Elorantas (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 5 th order for spaceborne lidar: too slow r s Forward scattering events 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 Equivalent medium theorem: use lidar FOV to determine the fraction of distribution that is detectable (we can neglect the return journey)

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Comparison of Eloranta & PVC methods For Calipso geometry (90-m field-of-view): –PVC method is as accurate as Elorantas method taken to 5 th -6 th order Download code from: Ice cloud Molecules Liquid cloud Aerosol Narrow field-of-view: forward scattered photons escape Wide field-of- view: forward scattered photons may be returned

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Regime 3: Wide-angle multiple scattering Make some approximations in modelling the diffuse radiation: –1-D: represent lateral transport as a diffusion –2-stream: represent only two propagation directions Space-time diagram r I – (t,r) I + (t,r) 60°

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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 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

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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(N 2 ) efficient Additional source Increasing variance with time is described by a diffusivity D

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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

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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

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

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

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

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Comparison of algorithm speeds ModelTimeRelative to PVC 50-point profile, 1-GHz Pentium: PVC0.56 ms1 TDTS2.5 ms5 Eloranta 3 rd order6.6 ms11 Eloranta 4 th order88 ms150 Eloranta 5 th order1 s1700 Eloranta 6 th order8.6 s million photons, 3-GHz Pentium: Monte Carlo with polarization 5 hours (0.6 ms per photon) 3x10 7

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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

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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!

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Eastern RussiaJapan Sea of JapanEast China Sea Calipso lidar (

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

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Intense convection over the Amazon Multiple scattering

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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 : Can use Monte Carlo but very expensive Time derivative Spatial derivative representing how much radiation is upstream Loss by absorption or scattering Source Gain by scattering Radiation scattered from all other directions 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°

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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

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