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Examples of multiple scattering

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:

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Examples of multiple scattering

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  1. Fast lidar & radar multiple-scattering models for cloud retrievalsRobin 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 Stratocumulus Surface echo Apparent echo from below the surface Intense thunderstorm • LITE lidar (l<r, footprint=300 m) CloudSat radar (l>r) • 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

  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 e.g. Delanoë and Hogan, in preparation New ray of data First guess of profile of cloud/aerosol properties (IWC, LWC, re …) Forward model Predict radar and lidar measurements (Z, b…) andJacobian (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

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

  5. Scattering regimes • Regime 2: Quasi-small-angle multiple scattering • Occurs when Ql~ x • Only for wavelength much less than particle size, e.g. lidar & ice clouds • Regime 3: Wide-angle multiple scattering • Occurs when l~ x Mean free path l • 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)] Footprint 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. Regime 2 Equivalent medium theorem: use lidar FOV to determine the fraction of distribution that is detectable (we can neglect the return journey) Calculate at each gate: • Total energy P • Position variance • Direction variance • Covariance z s r • 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!) • 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 Forward scattering events s r

  8. Wide field-of-view: forward scattered photons may be returned Narrow field-of-view: forward scattered photons escape 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: www.met.rdg.ac.uk/clouds

  9. Regime 3: Wide-angle multiple scattering Space-time diagram • Make some approximations in modelling the diffuse radiation: • 1-D: represent lateral transport as a diffusion • 2-stream: represent only two propagation directions I–(t,r) 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) Source Scattering from the quasi-direct beam into each of the streams Timederivative Remove this and we have the time-independent two-stream approximation used in weather models 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

  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

  19. 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!

  20. 7 June 2006 Aerosol from China? Molecular scattering Non-drizzling stratocumulus Cirrus Mixed-phase altocumulus Drizzling stratocumulus Rain • Calipso lidar (l<r) • CloudSat radar (l>r) 5500 km Japan Eastern Russia East China Sea Sea of Japan

  21. The 3D radiative transfer equation Loss by absorption or scattering Source Gain by scattering Radiation scattered from all other directions Spatial derivative representing how much radiation is upstream I–(t,r) 60° • Must make some approximations: • 1-D: represent lateral transport as a diffusion • 2-stream: represent only two propagation directions 60° I+(t,r) 60° r • 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 Time derivative

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