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**Variational cloud retrievals from radar, lidar and radiometers**

Robin Hogan Julien Delanoë Nicola Pounder University of Reading

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Introduction 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

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**Ingredients developed before In progress Not yet developed**

Retrieval framework 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 and mean drop diameter 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 Either 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 Ingredients developed before In progress Not yet developed

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**Target 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

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**Example from the AMF in Niamey**

94-GHz radar reflectivity Observations 532-nm lidar backscatter Forward model at final iteration 94-GHz radar reflectivity 532-nm lidar backscatter

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**Results: radar+lidar only**

Retrievals in regions where radar or lidar detects the cloud Retrieved visible extinction coefficient Retrieved effective radius Large error where only one instrument detects the cloud Retrieval error in ln(extinction)

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**Results: radar, lidar, SEVERI radiances**

Delanoe & Hogan (JGR 2008) Retrieved visible extinction coefficient TOA radiances increase retrieved optical depth and decrease particle size near cloud top Retrieved effective radius Cloud-top error greatly reduced Retrieval error in ln(extinction)

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**Unified algorithm: state variables**

Proposed list of retrieved variables held in the state vector x 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 Droplet number concentration One value per liquid layer Rain Rain rate Cubic spline basis functions with flatness constraint Normalized number conc. Nw One value per profile Dependent on whether from melting ice or coallescence Melting-layer thickness scaling factor Aerosols Extinction coefficient 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?

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**The cost function + Smoothness constraints**

The essence of the method is to find the state vector x that minimizes a cost function: 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 + Smoothness constraints

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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

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**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 Typical convergence behaviour

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**Forward model components**

From state vector x to forward modelled observations H(x)... Ice & snow Liquid cloud Rain Aerosol x Jacobian matrix H=y/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 Computationally expensive matrix-matrix multiplications: the most expensive part of the entire algorithm Radar scattering profile Lidar scattering profile Radiometer scattering profile Sum the contributions from each constituent 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 Radar forward modelled obs Lidar forward modelled obs Radiometer forward modelled obs H(x) Radiative transfer models

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**Equivalent adjoint method**

From state vector x to forward modelled observations H(x)... Ice & snow Liquid cloud Rain Aerosol x Gradient of cost function (vector) J/x=HTy* 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 Vector-matrix multiplications: around the same cost as the original forward operations Radar scattering profile Lidar scattering profile Radiometer scattering profile Sum the contributions from each constituent Radar forward modelled obs Lidar forward modelled obs Radiometer forward modelled obs H(x) Radiative transfer models Adjoint of radar model (vector) Adjoint of lidar model (vector) Adjoint of radiometer model Adjoint of radiative transfer models

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Scattering 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... Particle type Radar (3.2 mm) Radar Doppler Thermal IR, Solar, UV Aerosol Aerosol not detected by radar Mie theory, Highwood refractive index Liquid droplets Mie theory Beard (1976) Rain drops T-matrix: Brandes et al. (2002) shapes Ice cloud particles T-matrix (Hogan et al. 2010) Westbrook & Heymsfield Baran (2004) Graupel and hail TBD Melting ice Wu & Wang (1991)

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**Radiative transfer forward models**

Computational cost can scale with number of points describing vertical profile N; we can cope with an N2 dependence but not N3 Radar/lidar model Applications Speed Jacobian Adjoint Single scattering: b’=b exp(-2t) Radar & lidar, no multiple scattering N N2 Platt’s approximation b’=b exp(-2ht) Lidar, ice only, crude multiple scattering Photon Variance-Covariance (PVC) method (Hogan 2006, 2008) Lidar, ice only, small-angle multiple scattering N or N2 Time-Dependent Two-Stream (TDTS) method (Hogan and Battaglia 2008) Lidar & radar, wide-angle multiple scattering N3 Depolarization capability for TDTS Lidar & radar depol with multiple scattering 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 Radiometer model Applications Speed Jacobian Adjoint RTTOV (used at ECMWF & Met Office) Infrared and microwave radiances N Two-stream source function technique (e.g. Delanoe & Hogan 2008) Infrared radiances N2 LIDORT Solar radiances Infrared will probably use RTTOV, solar radiances will use LIDORT Both currently being tested by Julien Delanoe

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Gradient constraint A. Slingo, S. Nichols and J. Schmetz, Q. J. R. Met. Soc. 1982 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:

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**Example 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 Two FOVs: very good performance One 10-m footprint: saturates at optical depth=5 One 100-m footprint: multiple scattering helps! One 10-m footprint with gradient constraint: can extrapolate downwards successfully Optical depth=30 Footprint=100m Footprint=600m

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Progress 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…

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