Fast lidar & radar multiple-Fast lidar & radar multiple-scattering models for cloud scattering models for cloud

retrievalsretrievals

Robin Hogan (University of Reading)Robin Hogan (University of Reading)Alessandro Battaglia (University of Bonn)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 Examples of multiple scatteringscattering• LITE lidar (<r, footprint=300 m)

CloudSat radar (>r)

StratocumulusStratocumulus

Intense thunderstormIntense thunderstorm

Surface echoSurface echoApparent echo from below the surface

The basics of a variational retrieval The basics of a variational retrieval schemescheme

New ray of dataFirst guess of profile of cloud/aerosol

properties (IWC, LWC, re …)

Forward modelPredict radar and lidar measurements (Z, …) and Jacobian (dZ/dIWC …)

Compare to the measurementsAre they close enough?

Gauss-Newton iteration stepClever mathematics to produce a

better estimate of the state of the atmosphere

Calculate error in retrieval

No

Yes

Proceed to next raye.g. Delanoë and Hogan, in preparation

We need a fast forward model that includes the effects

of multiple scattering for both

radar and lidar

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

• Regime 1: Single scattering– Apparent backscatter ’ is easy to

calculate from at range r : ’(r) = (r) exp[-2(r)]

Scattering Scattering regimesregimes

Footprint x

Mean free path l

• Regime 2: Quasi-small-angle multiple scattering

– Occurs when l ~ x– Only for wavelength much less than particle size, e.g. lidar & ice clouds

• Regime 3: Wide-angle multiple scattering

– Occurs when l ~ x

New radar/lidar forward New radar/lidar forward modelmodel

• 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

Regime 2Regime 2

• 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

r s

Forward scattering events

2ζ

• 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

ζs

2s

r s

Equivalent medium theorem: use lidar FOV to determine the fraction of

distribution that is detectable (we can neglect

the return journey)

Comparison of Eloranta & PVC Comparison of Eloranta & PVC methodsmethods

• For Calipso geometry (90-m field-of-view):– PVC method is as accurate as Eloranta’s method taken to 5th-6th

order

Download code from: www.met.rdg.ac.uk/clouds

Ice cloud

Molecules

Liquid cloud

Aerosol

Regime 3: Wide-angle multiple Regime 3: Wide-angle multiple scatteringscattering

• Make some approximations in modelling the diffuse radiation:– 1-D: represent lateral transport as a diffusion– 2-stream: represent only two propagation directions

Space-time diagram

r

I–(t,r)

I+(t,r)

60°60°

60°

Time-dependent 2-stream Time-dependent 2-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 quite simply in time and space (no implicit methods or matrix inversion required)

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 scatteringSome 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

• 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

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

• 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: Comparison with Monte Carlo: LidarLidar• 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: Comparison with Monte Carlo: LidarLidar• I3RC case 3

– Henyey-Greenstein phase function, semi-infinite cloud, absorption

Monte Carlo calculations from Alessandro Battaglia

Comparison with Monte Carlo: Comparison with Monte Carlo: LidarLidar• I3RC case 5

– Mie phase function, 500-m cloud

Monte Carlo calculations from Alessandro Battaglia

Comparison with Monte Carlo: Comparison with Monte Carlo: RadarRadar– Mie phase functions, CloudSat reciever field-of-view

Monte Carlo calculations from Alessandro Battaglia

Comparison of algorithm Comparison of algorithm speedsspeeds

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 workFuture 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 Interpretation of radar and lidarlidar

• 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!

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

The 3D radiative transfer The 3D radiative transfer equationequation

• 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

ΩxΩΩxΩΩxxΩ

,,,,,)()(1

tSdtIpIIt

I

c

Time derivative Spatial derivative

representing how much radiation is upstream

Loss by absorption or scattering

SourceGain 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°60°

60°