Modelling radar and lidar multiple scattering Robin Hogan

Modelling radar and lidar multiple scattering Robin Hogan • The Cloud. Sat radar and the Calipso lidar were launched on 28 th April 2006 as part of the Atrain 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

• Calipso lidar (l<r) Mixed-phase altocumulus 7 June 2006 Aerosol from China? Drizzling stratocumulus • Cloud. Sat radar (l>r) Rain East China Sea Cirrus Japan Molecular scattering Non-drizzling stratocumulus Sea of Japan Eastern Russia 5500 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 (“ 1 D-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!

Examples of multiple scattering • LITE lidar (l<r) Stratocumulus Surface echo Apparent echo from below the surface! Intense thunderstorm Cloud. Sat radar (l>r)

Scattering regimes • Regime 1: Single scattering – Apparent backscatter b’ is easy to calculate from optical depth d along range r: b’(r)=b(r) exp[-d(r)] • Regime 2: Quasi-small-angle multiple scattering Footprint x – 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?

The 3 D 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: Loss by absorption Time derivative or scattering Gain by scattering Spatial derivative Radiation scattered from representing how much radiation is upstream Source all other directions • Can use Monte Carlo but very expensive • Must make some approximations: – 1 -D: represent lateral transport as a diffusion – 2 -stream: represent only two propagation directions I–(t, r) 60° + 60° I (t, r) 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: Source Time derivative Scattering from the quasi-direct beam into each of the streams Remove this and we have the timeindependent twostream approximation used in weather models Gain by scattering 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 Radiation scattered from the other stream • These can be discretized using simple schemes in time and space, provided that the optical depth of each layer is small

Lateral photon spreading • Model the lateral variance of photon position, following equations (where ): , using the Additional source Increasing variance with time is described by a diffusivity D • Then assume the lateral photon distribution is Gaussian to predict what fraction of it lies within the field-of-view

Simulation of 3 D 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 I 3 RC case (Intercomparison of 3 D Radiation Codes)

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 Cloud. Sat/Calipso retrieval scheme – More confidence in lidar retrievals in liquid water clouds – Can interpret Cloud. Sat 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