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

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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: LEcuyer & 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

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Unified retrieval Ingredients developed Implement previous work Not yet developed 1. New ray of data: define state vector Use classification to specify variables describing each species at each gate Ice: extinction coefficient, N 0, 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

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Unified retrieval: forward model From state vector x to forward modelled observations H(x)... Ice & snowLiquid cloudRainAerosol 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)H(x) Radiative transfer models Adjoint of radar model (vector) Adjoint of lidar model (vector) Adjoint of radiometer model Gradient of cost function (vector) x J=H T R -1 [y–H(x)] Vector-matrix multiplications: around the same cost as the original forward operations Adjoint of radiative transfer models y J=R -1 [y–H(x)]

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Gauss-Newton method Requires the curvature 2 J/x 2 –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 x1x1 J/x 2 J/x 2 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 J x x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x1x1 J/x x2x2 x3x3 x4x4 x5x5 Coding up adjoints and Jacobian matrices is very time consuming and error prone – is there a better way?

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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 theyre 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) Automatic adjoints Algorithm y(x) in C++: 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; } 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

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Benchmark results AdjointJacobian (50x350) Hand-coded3.0 New C++ library: Adept ADOL-C25 83 CppAD 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 wont work for Fortran until Fortran has template capability!

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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 N 0 * (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 Pounders 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 N w * (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 Matrosovs 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

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

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

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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

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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

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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 Further work

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and 2 nd derivative (the Hessian matrix): Gradient Descent methods –Fast adjoint method to calculate x J means dont need to calculate Jacobian –Disadvantage: more iterations needed since we dont 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 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

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Ice fall speeds 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 Terminal fall-speed (m s -1 ) Brown & Francis (1995) 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

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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)

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A mixed-phase case ObservationsRetrievals

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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)

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