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

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

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

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

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

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

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

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: www.met.rdg.ac.uk/clouds

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

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

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

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

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

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

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

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

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

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

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!

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

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

Intense convection over the Amazon Multiple scattering

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°

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