1. Retrieving consistent profiles of clouds and rain from instrument synergy
Robin Hogan, Nicola Pounder University of Reading, UK
Julien Delanoë LATMOS, France

2. Overview
The synergy of instruments in the A-Train and EarthCARE offer more accurate retrievals of cloud & light precipitation than ever before
But to exploit integral measurements (radar PIA, radiances) we must retrieve cloud and precipitation simultaneously
Variational approach offers rigorous treatment of errors
Behaviour should tend towards existing two-instrument synergy algos
Radar+lidar for ice clouds: Donovan et al. (2001), Delanoe & H (2008)
DARDAR available at:
CloudSat+MODIS for liquid clouds: Austin & Stephens (2001)
Calipso+MODIS for aerosol: Kaufman et al. (2003)
CloudSat surface return for rainfall: L’Ecuyer & Stephens (2002)
This talk presents progress towards “unified algorithm” for EarthCARE
Retrieval framework
Automatic adjoints
Ice, rain and melting-ice retrieval: testing on simulated profiles
Demonstration on A-Train data
Future work

3. Ingredients developed Implement previous work Not yet developed
Unified retrieval
1. New ray of data: define state vector
Use classification to specify variables describing each species at each gate
Ice: extinction coefficient, N0’, lidar extinction-to-backscatter ratio
Liquid: extinction coefficient and number concentration
Rain: rain rate, drop diameter and melting ice
Aerosol: 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 new state vector using 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
7. Calculate retrieval error
Error covariances and averaging kernel
Ingredients developed
Implement previous work
Not yet developed

4. Unified retrieval: forward model
From state vector x to forward modelled observations H(x)...
Ice & snow
Liquid cloud
Rain
Aerosol x
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)]
Ice/radar
Liquid/radar
Rain/radar
Ice/lidar
Liquid/lidar
Rain/lidar
Aerosol/lidar
Ice/radiometer
Liquid/radiometer
Rain/radiometer
Aerosol/radiometer
Lookup tables to obtain profiles of extinction, scattering & backscatter coefficients, asymmetry factor
Radar scattering profile
Lidar scattering profile
Radiometer scattering profile
Sum the contributions from each constituent
Radar forward modelled obs
Lidar forward modelled obs
Radiometer fwd modelled obs
H(x)
Radiative transfer models

5. Minimization methods - in 1D
Quasi-Newton method (e.g. L-BFGS)
Rolling a ball down a hill
Smart choice of direction in many dimensions aids convergence
Requires the gradient J/x
A vector (efficient to store)
Efficient to calculate using adjoint method
Used in data assimilation
Gauss-Newton method
Requires the curvature 2J/x2
A matrix
More expensive to calculate
Fewer iterations to converge
Assume J is quadratic and jump to the minimum
Limited to smaller retrieval problems J x x1
J/x
2J/x2 J x1
J/x x2 x3 x4 x5 x6 x7 x8 x2 x3 x4 x5 x
Coding up adjoints and Jacobian matrices is very time consuming and error prone – is there a better way?

6. Automatic adjoints Algorithm y(x) in C++:
double algorithm_AD(const double x[2],
double y_AD[1], double x_AD[2]) {
double y = 4.0;
double s = 2.0*x[0] + 3.0*x[1]*x[1];
y *= sin(s);
/* Adjoint part: */
double s_AD = 0.0;
y_AD[0] += sin(s) * y_AD[0];
s_AD += y * cos(s) * y_AD[0];
x_AD[0] += 3.0 * s_AD;
x_AD[1] += 6.0 * x[0] * s_AD;
s_AD = 0.0;
y_AD[0] = 0.0;
return y; }
Quite fiddly and error-prone to code-up dJ/dx given dJ/dy
adouble algorithm(const adouble x[2]) {
adouble y = 4.0;
adouble s = 2.0*x[0] + 3.0*x[1]*x[1];
y *= sin(s);
return y; }
// Main code
adept::Stack stack;
y = algorithm(x);
stack.compute_adjoint(); // Do the hard work
This can be done automatically in C++ using a special active type “adouble” and overloading mathematical operators and functions
Existing libraries CppAD and ADOL-C provide this capability but they’re quite slow
New library “Adept” uses expression templates in C++ to do this much faster
Freely available at
Hogan (Mathematics & Computers in Simulation, in review)
double algorithm(const double x[2]) {
double y = 4.0;
double s = 2.0*x[0] + 3.0*x[1]*x[1];
y *= sin(s);
return y; }

7. Benchmark results
Tested PVC and TDTS multiple scattering algorithms (here for PVC)
Time relative to original code for profile with N=50 cloudy points:
Adjoint calculation is:
Only 5-20% slower than hand-coded adjoint
5-9 times faster than leading alternative libraries ADOL-C and CppAD
Jacobian calculation is:
4-20 times faster than ADOL-C/CppAD for a matrix of size 50x350
Now used for entire unified algorithm
Sorry, it won’t work for Fortran until Fortran has template capability!
Adjoint
Jacobian (50x350)
Hand-coded
3.0
New C++ library: Adept
3.5 20
ADOL-C 25 83
CppAD 29
352

8. Retrieved (state) variables
Ice clouds and snow (extension of Delanoe and Hogan 2008, 2010)
[1] log extinction coefficient, [2] log lidar ratio
[3] log number conc parameter N0* (good a priori from temperature)
[4] “Riming factor” to allow Doppler information to be assimilated
Lidar molecular signal is used beyond cloud, when detectable
Liquid clouds (see Nicola Pounder’s talk)
[1] log LWC, [2] total number concentration (one value per liquid layer)
Extinction info from lidar multiple scattering, radar PIA, SW radiances
Rain and melting ice
[1] log rain rate, [2] number conc parameter Nw* (one value per profile)
“Flatness constraint” on rain rate: extra terms in cost function penalize deviations from a constant rain rate with height
Different a-priori values for stratocu drizzle and rain from melting ice
Melting layer radar attenuation uses Matrosov’s empirical relationship: 2-way attenuation [dB] = 2.2 x rain rate [mm h-1]
Additional term in cost function penalizes difference in snow flux above melting layer and rain rate below
Use radar PIA information over the ocean

9. Idealized nimbostratus retrieval
Constraint of constant flux across melting layer and smooth rain profile, but we have the classic ill-constrained attenuation retrieval problem

10. Idealized nimbostratus retrieval
Much better behaviour with flatness constraint on rain rate

11. A-Train case
Forward modelled radar and lidar match observations well, indicating good convergence
Can also simulate the Doppler velocity that would be observed by EarthCARE
Currently omits multiple scattering and air motion effects on Doppler

12. Retrievals
Ice cloud properties retrieved similarly to Delanoe and Hogan (2008, 2010) algorithm
Water flux is approximately conserved across the melting layer
Rain rate is relatively constant with range

14. Further work Forward models and observations
Implement LIDORT solar radiance model (has adjoint/Jacobian)
Implement Delanoe & Hogan infrared radiance code
Implement multiple scattering model with depolarization (but are measurements too noisy?)
Incorporate radar path integrated attenuation
Incorporate Doppler velocity
Retrieved quantities
Add “riming” factor
Refine aerosol retrieval
Verification
Consistency of different sources of information using A-Train data
Aircraft data with in-situ sampling from NASA ER-2 and French aircraft
EarthCARE simulator (ECSIM) scenes using EarthCARE specification

16. Minimizing 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
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

17. Ice fall speeds Terminal fall-speed (m s-1)
In convective clouds, intend to add a multiplication factor (or similar) to allow denser particles (e.g. rimed aggregates, graupel and hail) to be retrieved using the Doppler measurements
Heymsfield & Westbrook (2010) expression predicts fall speed as a function of particle mass, maximum dimension and “area ratio”
Currently we assume Brown and Francis (1995) mass-size relationship, so fall speed is a function of size alone
Brown & Francis (1995)

18. Simple melting-layer model
Minimalist approach:
2-way radar attenuation in dB is 2.2 times rain rate (Matrosov 2008)
No effect on other variables
Add term to cost function penalizing difference between ice flux above and rain flux below melting layer
Matrosov (IEEE Trans. Geosci. Rem. Sens. 2008)

19. A mixed-phase case
Observations
Retrievals

21. Model skill Use “DARDAR” CloudSat-CALIPSO cloud mask
How well is mean cloud fraction modelled?
Tend to underestimate mid & low cloud fraction
How good are models at forecasting cloud at right time? (SEDI skill score)
Winter mid to upper troposphere: excellent
Tropical mid to upper troposphere: fair
Tropical and sub-tropical boundary layer: virtually no skill!
Hogan, Stein, Garcon & Delanoe (in prep)