The time-dependent two-stream method The time-dependent two-stream method for lidar and radar multiple scatteringfor lidar and radar multiple scattering

Robin Hogan (University of Reading)Robin Hogan (University of Reading)Alessandro Battaglia (University of Bonn)Alessandro Battaglia (University of Bonn)

• To account for multiple scattering in CloudSat and CALIPSO retrievals we need a fast forward model to represent this effect

• Overview:– Examples of multiple scattering

from CloudSat and LITE– The four multiple scattering

regimes– The time-dependent two-stream

approximation– Comparison with Monte-Carlo

calculations for radar and lidar

Examples of multiple scattering• LITE lidar (<r, footprint~1 km)

CloudSat radar (>r)

StratocumulusStratocumulus

Intense thunderstormIntense thunderstorm

Surface echoSurface echoApparent echo from below the surface

• 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: Small-angle multiple scattering

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

– No pulse stretching

• Regime 3: Wide-angle multiple scattering

– Occurs when l ~ x

New radar/lidar forward New radar/lidar forward modelmodel

• CloudSat and CALIPSO record a new profile every 0.1 s– Delanoe and Hogan (JGR 2008) developed a variational radar-lidar

retrieval for ice clouds; intention to extend to liquid clouds and precip.

– It needs a forward model that runs in much less than 0.01 s

• Most widely used existing lidar 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

(Hogan 2006, Applied Optics)– Regime 3: Time-Dependent Two-Stream (TDTS) method (this talk)

• Sum the signal from the relevant methods:– Radar: regime 1 (single scattering) + regime 3 (wide-angle

scattering) – Lidar: regime 2 (small-angle) + regime 3 (wide-angle scattering)

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 modified 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

Spatial derivative Transport of radiation from 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

Hogan and Battaglia (2008, to appear in J. Atmos. Sci.)

Lateral photon Lateral photon transporttransport

• What fraction of photons remain in the receiver field-of-view?

• Calculate lateral standard deviation:

1/ 22 2x y

y x

t

1/ 2t

2

2

41

3n

t

n el

1/ 22 2x y

2

2

4

3t

nl

• Diffusion theory predicts superluminal travel when the mean number of scattering events n = ct/lt is small:

• In ~1920, Ornstein and Fürth independently solved the Langevin equation to obtain the correct description:

1/ 2t

Modelling lateral photon Modelling lateral photon transporttransport

• 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

1 21

1V

V VV V S D

c t r

1 21

1V

V VV V S D

c t r

22V I

Additional source Increasing variance with time is described by Ornstein-Fürth formula

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

Monte Carlo comparison: Monte Carlo comparison: IsotropicIsotropic• I3RC (Intercomparison of 3D radiation codes) lidar case 1

– Isotropic scattering, semi-infinite cloud, optical depth 20

Monte Carlo calculations from Alessandro Battaglia

Monte Carlo comparison: MieMonte Carlo comparison: Mie• I3RC lidar case 5

– Mie phase function, 500-m cloud

Monte Carlo calculations from Alessandro Battaglia

Monte Carlo comparison: Monte Carlo comparison: 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

Ongoing workOngoing work• Apply to “Quickbeam”, the CloudSat simulator (done)• Predict Mie and Rayleigh channels of HSRL lidar (done for PVC)• Implement TDTS in CloudSat/CALIPSO retrieval (PVC 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

Code available from www.met.rdg.ac.uk/clouds/multiscatter

Monte Carlo comparison: H-GMonte Carlo comparison: H-G• I3RC lidar case 3

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

Monte Carlo calculations from Alessandro Battaglia

How How important is important is

multiple multiple scattering scattering

for CALIPSO?for CALIPSO?• Ice clouds:

– FOV such that small-angle scattering almost saturates: satisfactory to use Platt’s approximation with =0.5

• Liquid clouds:– Essential to include wide-

angle scattering for optically thick clouds

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 rayDelanoë and Hogan (JGR 2008)

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 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