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**Retrieving consistent profiles of clouds and rain from instrument synergy**

Robin Hogan, Nicola Pounder, Chris Westbrook University of Reading, UK Julien Delanoë LATMOS, France Alessandro Battaglia University of Leicester, UK

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**Overview Why a “unified” algorithm? Some new components**

New fast model for depolarization due to multiple scattering Automatic adjoints Ice, rain and melting-ice retrieval Testing on simulated profiles Demonstration on A-Train data Skill of global cloud forecasts Outlook

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**Why a “unified” algorithm?**

Combine all measurements available (radar, lidar, radiometers) Forms the observation vector y Retrieve cloud, precipitation and aerosol properties simultaneously Ensures integral measurements can be used when affected by more than one species (e.g. radiances affected by ice and liquid clouds) Forms the state vector x Variational approach Minimizing a cost function J(x) allows for rigorous treatment of errors Aim to be completely flexible Applicable to ground-based, airborne and space-borne platforms Behaviour should tend towards existing two-instrument synergy algos Radar+lidar for ice clouds: Donovan et al. (2001), Delanoe & H (2008) 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 algorithm will provide one of the standard EarthCARE products

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**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 a new state vector Adjoint of full forward model 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

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Forward models Observation Model Speed Status Radar reflectivity factor Multiscatter: single scattering option N OK Radar reflectivity factor in deep convection Multiscatter: single scattering plus TDTS MS model (Hogan and Battaglia 2008) N2 Radar Doppler velocity Single scattering OK if no NUBF; fast MS model with Doppler does not exist Not available for MS HSRL lidar in ice and aerosol Multiscatter: PVC model (Hogan 2008) HSRL lidar in liquid cloud Multiscatter: PVC plus TDTS models Lidar depolarization Multiscatter: under development In progress Infrared radiances Delanoe and Hogan (2008) two-stream source function method Needs to be plugged in Solar radiances LIDORT (Robert Spurr) Testing Multiscatter combines two fast multiple scattering models, PVC & TDTS Includes a Fortran-90 interface, adjoint model, HSRL capability... For lidar, much more accurate than Platt’s approximation with mu=0.7 Can be used in retrievals and in instrument simulators Fast: One profile can cost the same as a single Monte Carlo photon! Freely available from

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Depolarization! Can we model effect of multiple scattering on depolarization? Potentially very useful information on extinction (e.g. Sassen & Petrilla 1986) Battaglia et al. (2007)

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**Scattering regimes Regime 1: Single scattering**

Apparent backscatter b’ is easy to calculate Zero depolarization from droplets Regime 2: Small-angle multiple scattering Only for wavelength much less than particle size, e.g. lidar & ice clouds Fast Photon Variance-Covariance (PVC) model of Hogan (2008) Depolarization due to backscatter slightly away from 180 degrees Footprint x Regime 3: Wide-angle multiple scattering Fast Time Dependent Two Stream (TDTS) method of Hogan & Battaglia Depolarization increases with number of scattering events

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**Fraction of cross-polar rather than co-polar scattered radiation **

Forward scattering is unpolarized A typical Mie phase function for a distribution of droplets The glory is polarized

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**Compare new model to Monte Carlo using I3RC case**

Small-angle scattering dominates Wide-angle scattering dominates Small-angle scattering: convolve cross-polar phase function with modelled distribution of near-backscatter scattering angles Wide-angle scattering: assume that each scattering event randomizes the polarization by a certain fraction = 0.6f (1–f), where f is the fraction of energy remaining in the field-of-view of the lidar (coefficients derived by comparing to Alessandro’s Monte Carlo) New model appears to perform well for different fields of view

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**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

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**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?

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**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; }

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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

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Ice retrieval: status Same as Delanoe & Hogan (2008, 2010) ice-only retrieval State variables Extinction coefficient in geometric optics approximation Normalized number concentration N0* Lidar ratio Notable aspects Molecular signal beyond cloud is used when detectable Oblate spheroids with aspect ratio 0.6 for radar: good match to aircraft data (Hogan et al. 2012) Doppler model using Heymsfield & Westbrook (2010) fall-speeds Remaining steps needed Add density as a retrieved variable to exploit Doppler in riming graupel conditions

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**Liquid cloud retrieval: status**

Discussed in Nicola Pounder’s talk State variables Liquid water content (actually ln LWC) Total number concentration (one number per liquid layer) Notable aspects Wide-angle lidar multiple scattering is included and provides useful constraint on optical depth “One-sided gradient constraint” ensures retrieval near cloud base tends towards the known adiabatic profile Remaining steps needed Forward model shortwave radiances and radar PIA

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**Rain and melting-layer: status**

State variables Rain rate Number concentration param (NL; currently fixed at a-priori value) Notable aspects Radar multiple-scattering included “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 Remaining steps needed Use radar PIA information over the ocean Do we need a more complex melting-layer model?

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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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A mixed-phase case Observations Retrievals

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**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 primitive 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

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**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

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**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)

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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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**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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