Modelling radar and lidar Modelling radar and lidar multiple scatteringmultiple scattering

Robin Hogan

• The CloudSat radar and the Calipso lidar were launched on 28th 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 RussiaJapanSea of JapanEast China Sea

• Calipso lidar (<r)

• CloudSat radar (>r)

Molecular scattering

Aerosol from China?

CirrusMixed-phase

altocumulus

Drizzling stratocumulus

Non-drizzling stratocumulus

Rain

7 June 2006

5500 km

1D variational method1D variational methodNew ray of dataFirst guess of x

Forward modelPredict measurements y and Jacobian H from state vector x using forward model H(x)

Compare measurements to forward modelHas the solution converged?2 convergence test Gauss-Newton iteration step

Predict new state vector: xi+1= xi+A-1{HTR-1[y-H(xi)]

+B-1(b-xi)}where the Hessian is

A=HTR-1H+B-1

Calculate error in retrievalThe 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:

n

n

N

N

ln

ln

ln

ln

1

1

x

m

mZ

Z

ln

ln 1

1

y

Examples of multiple Examples of multiple scatteringscattering

• 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)Surface echo (sea)StratocumulusStratocumulus

Apparent echo from below the sea surface!

BangladeshBay of Bengal

Bangladesh Himalayas

Multiple scattering

Multiple scattering

CloudSat radar

MODIS infrared image

Intense Intense convectioconvection over the n over the

Amazon Amazon

Multiple scattering

Multiple scattering so strong it corrupts next ray, appearing above the cloud!

Phase functionsPhase 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 cosg

• Regime 0: No attenuation– Optical depth << 1– mm-wave radar & ice cloud– Lidar & optically thin aerosol

• Regime 1: Single scattering– mm-wave radar & liquid cloud– Lidar & optically thick aerosol

Scattering Scattering regimesregimes

Footprint x

Mean free path l

•Regime 2: Quasi-small-angle multiple scattering

– l ~ x– Only for wavelength much less than particle size– Lidar & ice clouds

•Regime 3: Full multiple scattering

– l ~ x

New algorithmNew algorithm• Use the Hogan (Applied Optics, 2006) algorithm for the “quasi-

direct” return (contribution from regimes 1 and 2)• Use the “time-dependent two-stream” approximation for wide-

angle multiple scattering (regime 3)

Space-time diagram

Quasi-small-angle multi-Quasi-small-angle multi-scatteringscattering

• To calculate the “quasi-direct” component:

• Eloranta’s (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)

• Hogan (2006) 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: www.met.rdg.ac.uk/clouds

Ice cloud

Molecules

Liquid cloud

Aerosol

Narrow field-of-view:

forward scattered

photons escape

Wide field-of-view:

forward scattered

photons may be returned

Time-dependent two-stream Time-dependent two-stream approx.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

SII

r

I

t

I

c 211

1

SII

r

I

t

I

c 211

1

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 spreadingLateral photon spreading• Two-stream equations are inherently 1D, so can’t predict the

lateral spreading of photons that is necessary to determine the fraction of them remaining within the instrument field-of-view

• Solution: 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

DISVV

r

V

t

V

c V211

1

DISVV

r

V

t

V

c V211

1

2s2sIV

Additional source Increasing variance with time is described by a diffusivity D

Simulation of 3D photon Simulation of 3D photon transporttransport

• Scalar (actinic) flux is shown (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 CarloComparison 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 Effect on integrated backscatterbackscatter

• Need to be careful in applying the O’Connor et al. (2004) 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 workFuture work• Modify the numerics so that discretizations can be used where

the optical depth is large within one layer• 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