Robin Hogan, Julien Delanoë, Nicola Pounder,Robin Hogan, Julien Delanoë, Nicola Pounder,

Nicky Chalmers, Thorwald Stein, Anthony IllingworthNicky Chalmers, Thorwald Stein, Anthony Illingworth

University of ReadingUniversity of Reading

Thanks to Alessandro Battaglia and Richard ForbesThanks to Alessandro Battaglia and Richard Forbes

Towards “unified” retrievals of Towards “unified” retrievals of cloud, precipitation and aerosol from cloud, precipitation and aerosol from

combined radar, lidar and combined radar, lidar and radiometer observationsradiometer observations

Clouds in climate modelsClouds in climate models

14 global models (AMIP)

90N 80 60 40 20 0 -20 -40 -60 -80 90S

0.05

0.10

0.15

0.20

0.25

Latitude

Ver

tical

ly in

tegr

ated

clo

ud w

ater

(kg

m-2) But all

models tuned to give about the same top-of-atmosphere radiation

The properties of ice clouds

are particularly uncertain

• Via their interaction with solar and terrestrial radiation, clouds are one of the greatest sources of uncertainty in climate forecasts

• But cloud water content in models varies by a factor of 10• Need instrument with high vertical resolution…

Stephens et al. (2002)

Vertical structure of liquid water Vertical structure of liquid water contentcontent

• Cloudnet: several years of retrievals from 3 European ground-based sites• Observations in grey (with range indicating uncertainty)

• How do these models perform globally?

– ECMWF has far too great an occurrence of low LWC values

0-3 km

– Supercooled liquid water content from seven forecast models spans a factor of 20

Illingworth, Hogan et al. (2007)

Spaceborne radar, lidar and Spaceborne radar, lidar and radiometersradiometers

The A-Train– NASA– 700-km orbit– CloudSat 94-GHz radar (launch 2006)– Calipso 532/1064-nm depol. lidar– MODIS multi-wavelength radiometer– CERES broad-band radiometer– AMSR-E microwave radiometer

EarthCARE: launch 2012– ESA+JAXA– 400-km orbit: more

sensitive– 94-GHz Doppler radar– 355-nm HSRL/depol. lidar– Multispectral imager– Broad-band radiometer– Heart-warming name

EarthCare

2013201420152016201720182019

OverviewOverview• What do spaceborne radar and lidar see?

– Classification of targets from radar and lidar– Global distribution of supercooled clouds from the LITE lidar

• Towards a “unified” retrieval of cloud, precipitation and aerosol – Variational retrieval framework

• Results from CloudSat-Calipso ice-cloud retrieval– Consistency with top-of-atmosphere radiative fluxes– Evaluation and improvement of models

• Challenges and opportunities from multiple scattering– Fast forward model– Multiple field-of-view lidar retrieval

• EarthCARE• First results from prototype unified retrieval• Outlook for model evaluation and improvement

What do CloudSat and Calipso What do CloudSat and Calipso see?see?

Cloudsat radar

CALIPSO lidar

Target classificationInsectsAerosolRainSupercooled liquid cloudWarm liquid cloudIce and supercooled liquidIceClearNo ice/rain but possibly liquidGround

Delanoe and Hogan (2008, 2010)

• Radar: ~D6, detects whole profile, surface echo provides integral constraint

• Lidar: ~D2, more sensitive to thin cirrus and liquid but attenuated

• Radar-lidar ratio provides size D

CloudSat and Calipso CloudSat and Calipso sensitivitysensitivity

7

• In July 2006, cloud occurrence in the subzero troposphere was 13.3%

• The fraction observed by radar was 65.9%

• The fraction observed by lidar was 65.0%

• The fraction observed by both was 31.0%

Distribution versus temperature & Distribution versus temperature & latitudelatitude

• No supercooled water colder than –40°C (as expected)• Supercooled water more frequent in southern hemisphere storm track

Hogan et al. (2004)

Ingredients of a variational Ingredients of a variational retrievalretrieval

• Aim: to retrieve an optimal estimate of the properties of clouds, aerosols and precipitation from combining these measurements– To make use of integral constraints must retrieve components

together• For each ray of data, define observation vector y:

– Radar reflectivity values– Lidar backscatter values– Infrared radiances– Shortwave radiances (not yet implemented)– Surface radar echo providing two-way attenuation (ditto)

• Define state vector x of properties to be retrieved:– Ice cloud extinction, number concentration and lidar-ratio profile– Liquid water content profile and number concentration– Rain rate profile and number concentration– Aerosol extinction coefficient profile and lidar ratio

• Forward model H(x) to predict the observations from the state vector– Microphysical component: particle scattering properties– Radiative transfer component

The cost functionThe cost function• The essence of the method is to find the state vector x that minimizes

a cost function:

TxxδxBδxδyRδy

x

T1T1T

1

2

211

12

2

12

2

2

1

2

1

2

1

22

1

2

1)(

2

1

n

iiiii

n

i bi

iim

i yi

ii xxxbxHy

J

Each observation yi is weighted by the inverse of

its error variance

The forward model H(x) predicts the observations from the state vector x

Some elements of x are constrained by a

prior estimate

This term can be used to penalize curvature in the retrieved profile

Unified Unified retrievalretrieval

Ingredients developedWork in progress

1. New ray of data: define state vector x

Use classification to specify variables describing each species at each gateIce: extinction coefficient , N0’, lidar extinction-to-backscatter ratio

Liquid: extinction coefficient and number concentrationRain: rain rate, drop diameter and melting iceAerosol: extinction coefficient, particle size and lidar ratio

2a. Radar model

Including surface return and multiple scattering

2b. Lidar model

Including HSRL channels and multiple scattering

2c. Radiance model

Solar and IR channels

3. Compare to observations

Check for convergence

4. Iteration method

Derive a new state vectorAdjoint of full forward modelQuasi-Newton or Gauss-Newton scheme

2. Forward model

Not converged

Converged

Proceed to next ray of data5. Calculate retrieval error

Error covariances and averaging kernel

Unified retrieval: Forward Unified retrieval: Forward modelmodel

• From state vector x to forward modelled observations H(x)...

Ice & snow Liquid cloud Rain Aerosol

Ice/radar

Liquid/radar

Rain/radar

Ice/lidar

Liquid/lidar

Rain/lidar

Aerosol/lidar

Ice/radiometer

Liquid/radiometer

Rain/radiometer

Aerosol/radiometer

Radar scattering profile

Lidar scattering profile

Radiometer scattering profile

Lookup tables to obtain profiles of extinction, scattering & backscatter coefficients, asymmetry factor

Sum the contributions from each constituent

x

Radar forward modelled obs

Lidar forward modelled obs

Radiometer fwd modelled obs

H(x)Radiative transfer models

Adjoint of radar model (vector)

Adjoint of lidar model (vector)

Adjoint of radiometer model

Gradient of cost function (vector)

xJ=HTR-1[y–H(x)]

Vector-matrix multiplications: around the same cost as the original forward

operations

Adjoint of radiative transfer models

yJ=R-1[y–H(x)]

Gauss-Newton method

• Requires the curvature 2J/x2

– A matrix– More expensive to calculate

• Faster convergence– Assume J is quadratic and

jump to the minimum• Limited to smaller retrieval

problems

J

x

x1

J/x2J/x2

Minimization methods - in 1DMinimization methods - in 1DQuasi-Newton method (e.g. L-BFGS)

• Rolling a ball down a hill– Intelligent choice of direction

in multi-dimensions helps convergence

• Requires the gradient J/x– A vector (efficient to store)– Efficient to calculate using

adjoint method• Used in data assimilation

J

x

x2x3x4x5x6x7x8

x1

J/x

x2x3

x4x5

Ice retrievalIce retrieval• First challenge:

– How do we model radar scattering from complex ice particles?

• In Rayleigh regime (»D), backscatter proportional to mass squared– Particle shape irrelevant– Use a mass-D relationship

• But Rayleigh assumption often not valid at 94 GHz (=3 mm)– Most papers assume homogeneous ice-air spheres with a diameter

equal to the maximum dimension D of the particle, and apply Mie theory

• Problems with this approach:– Non-Rayleigh behaviour depends on the vertical dimension of the

particle– Most are irregular aggregates with an average axial ratio of 0.6– They fall with their maximum dimension horizontal

• Alternative approach:– Approximate as horizontally aligned oblate spheroids

Spheres versus spheroidsSpheres versus spheroids

SpheroidSphere

Transmitted wave

Sphere: returns from

opposite sides of particle out

of phase: cancellation

Spheroid: returns from

opposite sides not out of

phase: higherb

Hogan et al. (2011)

– Sphere produces ~5 dB error (factor of 3)– Spheroid approximation matches Rayleigh reflectivity (mass is about right)

and non-Rayleigh reflectivity (shape is about right)

Test with dual-wavelength Test with dual-wavelength aircraft dataaircraft data

Hogan et al. (2011)

Test with 3-GHz differential Test with 3-GHz differential reflectivityreflectivity

Hogan et al. (2011)

• Horizontal polarization Zh

– Differential reflectivity Zdr is larger for more extreme axial ratios

– Agreement with Z and Zdr confirms the Brown & Francis (1995) mass-D relationship and axial ratio = 0.6

• Zdr = 10log10(Zh/Zv)

Ice cloud: non-variational Ice cloud: non-variational retrievalretrieval

• Donovan et al. (2000) algorithm can only be applied where both lidar and radar have signal

Observations

State variables

Derived variables

Retrieval is accurate but not perfectly stable where lidar loses signal

Aircraft-simulated profiles with noise (from Hogan et al. 2006)

Delanoe and Hogan (2008)

Variational radar/lidar Variational radar/lidar retrievalretrieval

• Noise in lidar backscatter feeds through to retrieved extinction

Observations

State variables

Derived variables

Lidar noise matched by retrieval

Noise feeds through to other variables

Delanoe and Hogan (2008)

……add smoothness constraintadd smoothness constraint

• Smoothness constraint: add a term to cost function to penalize curvature in the solution (J’ = id2i/dz2)

Observations

State variables

Derived variables

Retrieval reverts to a-priori N0

Extinction and IWC too low in radar-only region

Delanoe and Hogan (2008)

……add a-priori error add a-priori error correlationcorrelation

• Use B (the a priori error covariance matrix) to smooth the N0 information in the vertical

Observations

State variables

Derived variables

Vertical correlation of error in N0

Extinction and IWC now more accurate

Delanoe and Hogan (2008)

Lidar observations

Radar observations

Visible extinction

Ice water content

Effective radius

Lidar forward model

Radar forward model

Example ice Example ice cloud cloud

retrievalsretrievalsDelanoe and Hogan (2010)

Evaluation using CERES TOA Evaluation using CERES TOA fluxesfluxes

• Radar-lidar retrieved profiles containing only ice used with Edwards-Slingo radiation code to predict CERES fluxes

• Small biases but large random shortwave error: 3D effects?

Chalmers (2011)

ShortwaveBias 4 W m-2, RMSE 71 W m-2

LongwaveBias 0.3 W m-2, RMSE 14 W m-2

CERES versus a radar-only CERES versus a radar-only retrievalretrieval

• How does this compare with radar-only empirical IWC(Z, T) retrieval of Hogan et al. (2006) using effective radius parameterization from Kristjansson et al. (1999)?

Bias 10 W m-2

RMS 47 W m-2

ShortwaveBias 48 W m-2, RMSE 110 W m-2

LongwaveBias –10 W m-2, RMSE 47 W m-2

Chalmers (2011)

How important is lidar?How important is lidar?• Remove lidar-only pixels from radar-lidar retrieval• Change to fluxes is only ~5 W m-2 but lidar still acts to improve

retrieval in radar-lidar region of the cloud

ShortwaveBias –5 W m-2, RMSE 17 W m-2

LongwaveBias 4 W m-2, RMSE 9 W m-2

Chalmers (2011)

A-Train A-Train versus versus

modelsmodels• Ice water

content• 14 July 2006• Half an orbit• 150°

longitude at equator

Delanoe et al. (2011)

• Both models lack high thin cirrus• ECMWF lacks high IWC values; using this work, ECMWF have

developed a new prognostic snow scheme that performs better• Met Office has too narrow a distribution of in-cloud IWC

Evaluation of gridbox-mean ice water Evaluation of gridbox-mean ice water contentcontent

In-cloud mean ice water In-cloud mean ice water contentcontent

Radiative transfer forward Radiative transfer forward modelsmodels

• Infrared radiances– Delanoe and Hogan (2008) model– Currently testing RTTOV (widely used, can do microwave, has

adjoint)• Solar radiances

– Currently testing LIDORT• Radar and lidar

– Simplest model is single scattering with attenuation: ’= exp(-2)– Problem from space is multiple scattering: contains extra

information on cloud properties (particularly optical depth) but no-one has previously been able to rigorously make use of data subject to pulse stretching

– Use combination of fast “Photon Variance-Covariance” method and “Time-Dependent Two-Stream” methods

– Adjoints for these models recently coded– Forward model for lidar depolarization is in progress

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 (pulse stretching)

– Occurs when l ~ x

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 (J. Atmos. Sci., 2008)

Fast multiple scattering forward Fast multiple scattering forward modelmodel

CloudSat-like example

• New method uses the time-dependent two-stream approximation

• Agrees with Monte Carlo but ~107 times faster (~3 ms)

Hogan and Battaglia (J. Atmos. Sci. 2008)

CALIPSO-like example

Multiple field-of-view lidar Multiple field-of-view lidar retrievalretrieval

• To test multiple scattering model in a retrieval, and its adjoint, consider a multiple field-of-view lidar observing a liquid cloud

• Wide fields of view provide information deeper into the cloud

• The NASA airborne “THOR” lidar is an example with 8 fields of view

• Simple retrieval implemented with state vector consisting of profile of extinction coefficient

• Different solution methods implemented, e.g. Gauss-Newton, Levenberg-Marquardt and Quasi-Newton (L-BFGS)

lidar

Cloud top

600 m100 m

10 m

Results for a sine profileResults for a sine profile

• Simulated test with 200-m sinusoidal structure in extinction

• With one FOV, only retrieve first 2 optical depths

• With three FOVs, retrieve structure of extinction profile down to 6 optical depths

• Beyond that the information is smeared out

Pounder, Hogan et al. (2011)

Optical depth from multiple Optical depth from multiple FOV lidarFOV lidar• Despite vertical

smearing of information, the total optical depth can be retrieved to ~30 optical depths

• Limit is closer to 3 for one narrow field-of-view lidar

• Useful optical depth information from one 100-m-footprint lidar (e.g. Calipso)!

Pounder, Hogan et al. (2011)

THOTHOR R

lidarlidar

EarthCAREEarthCARE• The ESA/JAXA “EarthCARE”

satellite is designed with synergy in mind

• We are currently developing synergy algorithms for its instrument specification

EarthCARE lidarEarthCARE lidar• High Spectral Resolution

capability enables direct retrieval of extinction profile

EarthCARE radar: Doppler EarthCARE radar: Doppler capabilitycapability

94-GHz radar

10-GHz radar

• Example from NASA airborne cloud radar demonstrates– Can estimate ice fall-speed globally: important for radiation budget– Can identify strong updrafts in convective cores

94-GHz reflectivity in convection disappears very quickly: multiple scattering from CloudSat may be giving us a false impression of how far we are penetrating

Unified algorithm: progressUnified algorithm: progress• Bringing all the aspects of this talk together…• Done:

– Functioning algorithm framework exists– C++: object orientation allows code to be completely flexible:

observations can be added and removed without needing to keep track of indices to matrices, so same code can be applied to different observing systems

– Preliminary retrieval of ice, liquid, rain and aerosol– Adjoint of radar and lidar forward models with multiple scattering

and HSRL/Raman support– Interface to L-BFGS quasi-Newton algorithm in GNU Scientific

Library• In progress / future work:

– Estimate and report error in solution and averaging kernel – Interface to radiance models– Test on a range of ground-based, airborne and spaceborne

instruments

Observations vs forward Observations vs forward modelsmodels

– Radar and lidar backscatter are successfully forward modelled (at final iteration) in most situations

– Can also forward model Doppler velocity (what EarthCARE would see)

• Radar reflectivity factor• Lidar backscatter

Three retrieved componentsThree retrieved components• Liquid water content

• Ice extinction coefficient

• Rain rate

OutlookOutlook• Use of radiances in retrieval should make retrieved profiles consistent with

broadband fluxes (can test this with A-Train and EarthCARE)• EarthCARE will take this a step further

– Use imager to construct 3D cloud field 10-20 km wide beneath satellite – Use 3D radiative transfer to test consistency with broadband radiances

looking at the cloud field in 3 directions (overcome earlier 3D problem)• How can we use these retrievals to improve weather forecasts?

– Assimilate cloud products, or radar and lidar observations directly?– Assimilation experiments being carried out by ECMWF– Still an open problem as to how to ensure clouds are assimilated such

that the dynamics and thermodynamics of the model are modified so as to be consistent with the presence of the cloud

• How can we use these retrievals to improve climate models?– We will have retrieved global cloud fields consistent with radiation– So can diagnose in detail not only what aspects of clouds are wrong in

models, but the radiative error associated with each error in the representation of clouds

• First part of a forward model is the scattering and fall-speed model– Same methods typically used for all radiometer and lidar channels– Radar and Doppler model uses another set of methods

Scattering modelsScattering models

Particle type Radar (3.2 mm) Radar Doppler Thermal IR, Solar, UVAerosol Aerosol not

detected by radarAerosol not detected by radar

Mie theory, Highwood refractive index

Liquid droplets Mie theory Beard (1976) Mie theoryRain drops T-matrix: Brandes

et al. (2002) shapes

Beard (1976) Mie theory

Ice cloud particles

T-matrix (Hogan et al. 2010)

Westbrook & Heymsfield

Baran (2004)

Graupel and hail Mie theory TBD Mie theoryMelting ice Wu & Wang

(1991)TBD Mie theory

Unified algorithm: state Unified algorithm: state variablesvariables

State variable Representation with height / constraint A-priori

Ice clouds and snow

Visible extinction coefficient One variable per pixel with smoothness constraint None

Number conc. parameter Cubic spline basis functions with vertical correlation Temperature dependent

Lidar extinction-to-backscatter ratio Cubic spline basis functions 20 sr

Riming factor Likely a single value per profile 1

Liquid clouds

Liquid water content One variable per pixel but with gradient constraint None

Droplet number concentration One value per liquid layer Temperature dependent

Rain

Rain rate Cubic spline basis functions with flatness constraint None

Normalized number conc. Nw One value per profile Dependent on whether from melting ice or coallescence

Melting-layer thickness scaling factor One value per profile 1

Aerosols

Extinction coefficient One variable per pixel with smoothness constraint None

Lidar extinction-to-backscatter ratio One value per aerosol layer identified Climatological type depending on region

Ice clouds follows Delanoe & Hogan (2008); Snow & riming in convective clouds needs to be added

Liquid clouds currently being tackled

Basic rain to be added shortly; Full representation later

Basic aerosols to be added shortly; Full representation via collaboration?

• Proposed list of retrieved variables held in the state vector x

• Computational cost can scale with number of points describing vertical profile N; we can cope with an N2 dependence but not N3

Radiative transfer forward Radiative transfer forward modelsmodels

Radar/lidar model Applications Speed Jacobian Adjoint

Single scattering: ’= exp(-2) Radar & lidar, no multiple scattering N N2 N

Platt’s approximation ’= exp(-2) Lidar, ice only, crude multiple scattering N N2 N

Photon Variance-Covariance (PVC) method (Hogan 2006, 2008)

Lidar, ice only, small-angle multiple scattering

N or N2 N2 N

Time-Dependent Two-Stream (TDTS) method (Hogan and Battaglia 2008)

Lidar & radar, wide-angle multiple scattering

N2 N3 N2

Depolarization capability for TDTS Lidar & radar depol with multiple scattering N2 N2

Radiometer model Applications Speed Jacobian Adjoint

RTTOV (used at ECMWF & Met Office) Infrared and microwave radiances N N

Two-stream source function technique (e.g. Delanoe & Hogan 2008)

Infrared radiances N N2

LIDORT Solar radiances N N2 N

• Infrared will probably use RTTOV, solar radiances will use LIDORT• Both currently being tested by Julien Delanoe

• Lidar uses PVC+TDTS (N2), radar uses single-scattering+TDTS (N2)• Jacobian of TDTS is too expensive: N3

• We have recently coded adjoint of multiple scattering models• Future work: depolarization forward model with multiple scattering

and 2nd derivative (the Hessian matrix):

Gradient Descent methods

– Fast adjoint method to calculate xJ means don’t need to calculate Jacobian

– Disadvantage: more iterations needed since we don’t know curvature of J(x)

– Quasi-Newton method to get the search direction (e.g. L-BFGS used by ECMWF): builds up an approximate inverse Hessian A for improved convergence

– Scales well for large x– Poorer estimate of the error at the

end

Minimizing the cost functionMinimizing the cost function

Gradient of cost function (a vector)

Gauss-Newton method

– Rapid convergence (instant for linear problems)

– Get solution error covariance “for free” at the end

– Levenberg-Marquardt is a small modification to ensure convergence

– Need the Jacobian matrix H of every forward model: can be expensive for larger problems as forward model may need to be rerun with each element of the state vector perturbed

112 BHRHxTJ

axBaxxyRxy 11

2

1)()(

2

1 TT HHJ

axBxyRHx 11 )(HJ T

JJii xxxx

12

1 Jii xAxx 1

Comparison of convergence Comparison of convergence ratesrates

• Solution is identical• Gauss-Newton method converges in < 10 iterations• L-BFGS Gradient Descent method converges in < 100 iterations• Conjugate Gradient method converges a little slower than L-BFGS• Each L-BFGS iteration >> 10x faster than each Gauss-Newton one!• Gauss-Newton method requires the Jacobian matrix, which must be

calculated by rerunning multiple scattering model multiple times

Unified algorithm: first results for Unified algorithm: first results for ice+liquidice+liquid

Ob

serv

ati

on

s

Retr

ieval

But lidar noise degrades retrieval

TruthRetrieval

First guessIterations

ObservationsForward modelled retrievalForward modelled first guess

Convergence!

Add smoothness constraintAdd smoothness constraintO

bserv

ati

on

s

Retr

ieval

TruthRetrieval

First guessIterations

ObservationsForward modelled retrievalForward modelled first guess

Smoother retrieval but slower convergence