Modelling radar and lidar multiple scattering Modelling radar and lidar multiple scattering Robin Hogan The CloudSat radar and the Calipso lidar were launched on 28 th 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

Eastern RussiaJapan Sea of JapanEast China Sea Calipso lidar (<r) CloudSat radar (>r) Molecular scattering Aerosol from China? Cirrus Mixed-phase altocumulus Drizzling stratocumulus Non-drizzling stratocumulus Rain 7 June km

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!

1D variational method New ray of data First guess of x 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 Gauss-Newton iteration step Predict new state vector: x i+1 = x i +A -1 {H T R -1 [y-H(x i )] +B -1 (b-x i )} where the Hessian is A=H T R -1 H+B -1 Calculate error in retrieval The solution error covariance matrix is S=A -1 No Yes Proceed to next ray –In this problem, the observation vector y and state vector x are:

Examples of multiple scattering Examples of multiple scattering 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 (sea) Stratocumulus 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 Examples of multiple scattering Examples of multiple scattering LITE lidar (<r) CloudSat radar (>r)Stratocumulus Intense thunderstorm Surface echo Apparent echo from below the surface!

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

Intense convection over the Amazon Multiple scattering

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

Regime 1: Single scattering –Apparent backscatter is easy to calculate from optical depth along range r : (r)= r exp[- (r)] Scattering regimes Footprint x Regime 2: Quasi-small-angle multiple scattering – Only for wavelength much less than particle size, leading to strong forward scattering – Fast models exist (e.g. Hogan, Applied Optics 2006) Regime 3: Wide-angle multiple scattering – Large instrument footprint – How can this be modelled?

Lidar forward model: multiple scattering 90-m footprint of Calipso means that multiple scattering is a problem Elorantas (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) New 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) Hogan (Applied Optics, 2006). Code: Ice cloud Molecules Liquid cloud Aerosol Narrow field-of-view: forward scattered photons escape Wide field-of- view: forward scattered photons may be returned

New algorithm Use the Hogan (Applied Optics 2006) algorithm for the quasi- direct return (contribution from regimes 1 and 2) How should we model the diffuse radiation responsible for wide-angle multiple scattering (regime 3)? Space-time diagram

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°

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 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 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

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 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 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)

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

Future work Modify the numerics so that discretizations can be used where the optical depth is large within one layer Add the capability to have a partially reflecting surface 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