robin hogan julien delanoë nicola pounder university of reading variational cloud retrievals from...
TRANSCRIPT
Robin HoganJulien DelanoëNicola Pounder
University of Reading
Variational cloud Variational cloud retrievals from radar, retrievals from radar, lidar and radiometerslidar and radiometers
IntroductionIntroduction• Best estimate of the atmospheric state from instrument synergy
– Use a variational framework / optimal estimation theory• Some important measurements are integral constraints
– E.g. microwave, infrared and visible radiances– Affected by all cloud types in profile, plus aerosol and precipitation– Hence need to retrieve different particle types simultaneously
• Funded by ESA and NERC to develop unified retrieval algorithm– For application to EarthCARE– Will be tested on ground-based, airborne and A-train data
• Algorithm components– Target classification input– State variables– Minimization techniques: Gauss-Newton vs. Gradient-Descent– Status of forward models and their adjoints
• Progress with individual target types– Ice clouds– Liquid clouds
Retrieval Retrieval frameworframewor
kkIngredients developed before
In progressNot yet developed
1. New ray of data: define state vector
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 and mean drop diameterAerosol: extinction coefficient, particle size and lidar ratio
3a. Radar model
Including surface return and multiple scattering
3b. Lidar model
Including HSRL channels and multiple scattering
3c. Radiance model
Solar and IR channels
4. Compare to observations
Check for convergence
6. Iteration method
Derive a new state vectorEither Gauss-Newton or quasi-Newton scheme
3. Forward model
Not converged
Converged
Proceed to next ray of data
2. Convert state vector to radar-lidar resolution
Often the state vector will contain a low resolution description of the profile
5. Convert Jacobian/adjoint to state-vector resolution
Initially will be at the radar-lidar resolution
7. Calculate retrieval error
Error covariances and averaging kernel
Target classificationTarget classification• In Cloudnet we used radar and lidar to provide a detailed
discrimination of target types (Illingworth et al. 2007):
• A similar approach has been used by Julien Delanoe on CloudSat and Calipso using the one-instrument products as a starting point:
• More detailed classifications could distinguish “warm” and “cold” rain (implying different size distributions) and different aerosol types
Example from the AMF in Niamey
94-GHz radar reflectivity
532-nm lidar backscatter
Forward model at
final iteration
94-GHz radar reflectivity
532-nm lidar backscatter
Observations
Retrievals in regions where radar or lidar detects the cloud
Retrieved visible extinction coefficient
Retrieved effective radius
Results: radar+lidar only
Large error where only one instrument detects the cloud
Retrieval error in ln(extinction)
TOA radiances increase retrieved optical depth and decrease particle size near cloud top
Cloud-top error greatly reduced
Retrieval error in ln(extinction)
Retrieved visible extinction coefficient
Retrieved effective radius
Results: radar, lidar, SEVERI radiances
Delanoe & Hogan (JGR 2008)
Unified algorithm: state variables
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
The cost function• The essence of the method is to find the state vector x that minimizes
a cost function:
2 2
2 21 1
( )y x
i i
n ni i i
i iy b
y H x bJ
x+ Smoothness
constraints
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 an a priori estimate
This term penalizes curvature in the
extinction profile
Gauss-Newton method• See Rodgers’ book (p85): write the cost function in matrix form:
• Define its gradient (a vector):
• …and its second derivative (a matrix):
• Approximate J as quadratic and apply this:
– Advantage: rapid convergence (instant convergence for linear problems)
– Another advantage: get the error covariance of the solution “for free”– Disadvantage: need the Jacobian matrix of every forward model: can be
expensive for larger problems
112 BHRHxTJ
axBaxxyRxy 11
2
1)()(
2
1 TT HHJ
axBxyRHx 11 )(HJ T
JJii xxxx
12
1
Gradient descent methods• Just use gradient information:
– Advantage: we don’t need to calculate the Jacobian so forward model is cheaper!
– Disadvantage: more iterations needed since we don’t know curvature of J(x)– Use a quasi-Newton method to get the search direction, such as BFGS used
by ECMWF: builds up an approximate form of the second derivative to get improved convergence
– Scales well for large x– Disadvantage: poorer estimate of the error at the end
• Why don’t we need the Jacobian H?
– The “adjoint” of a forward model takes as input the vector {.} and outputs the vector Jobs without needing to calculate the matrix H on the way
– Adjoint can be coded to be only ~3 times slower than original forward model– Tricky coding for newcomers, although some automatic code generators exist
Jii xxx 1
)(1 xyRH HJ Tobs
Typical convergence behaviour
Forward model components• 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
Jacobian matrix
H=y/x
Computationally expensive matrix-matrix multiplications: the most
expensive part of the entire algorithm
x
Radar forward modelled obs
Lidar forward modelled obs
Radiometer forward modelled obs
H(x)Radiative transfer models
Gradient of radar measurements with respect to radar inputs
Gradient of lidar measurements with respect to lidar inputs
Gradient of radiometer measurements with respect to radiometer inputs
Jacobian part of radiative transfer models
Equivalent adjoint method• 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
Adjoint of radar model (vector)
Adjoint of lidar model (vector)
Adjoint of radiometer model
Gradient of cost function (vector)
J/x=HTy*
Vector-matrix multiplications: around the same cost as the original forward
operations
x
Radar forward modelled obs
Lidar forward modelled obs
Radiometer forward modelled obs
H(x)Radiative transfer models
Adjoint of radiative transfer models
• 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
• Graupel and melting ice still uncertain; but normal ice is decided...
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
• 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 models
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
Gradient constraintGradient constraint
• We have a good constraint on the gradient of the state variables with height for:– LWC in stratocu (adiabatic profile, particularly near cloud base)– Rain rate (fast falling so little variation with height expected)
• Not suitable for the usual “a priori” constraint• Solution: add an extra term to the cost function to penalize
deviations from gradient c:
A. Slingo, S. Nichols and J. Schmetz, Q. J. R. Met. Soc. 1982
2
i
i
dxJ c
dz
Example in liquid cloudsExample in liquid clouds• Using simulated observations:
– Triangular cloud observed by a 1- or 2-field-of-view lidar– Retrieval uses Levenberg-Marquardt minimization with Hogan and
Battaglia (2008) model for lidar multiple scattering
Optical depth=30Footprint=100mFootprint=600m
Two FOVs: very good performanceOne 10-m footprint: saturates at optical depth=5One 100-m footprint: multiple scattering helps!One 10-m footprint with gradient constraint: can
extrapolate downwards successfully
ProgressProgress• Done:
– 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
– Code to generate particle scattering libraries in NetCDF files– Adjoint of radar and lidar forward models with multiple scattering
and HSRL/Raman support– Radar/lidar model interfaced and cost function can be calculated
• In progress / future work:– Interface to BFGS algorithm (e.g. in GNU Scientific Library)– Implement ice, liquid, aerosol and rain constituents– Interface to radiance models– Test on a range of ground-based and spaceborne instruments– Test using ECSIM observational simulator– Apply to large datasets of ground-based observations…