Tbilisi, GGSWBS'14 Optimization for inverse modelling Ketevan Kasradze 1 Hendrik Elbern 1,2 and the Chemical Data Assimilation group of RIU 1 Rhenish Institute for Environmental Research at the University of Cologne, Germany 2 Institute for Energy and Climate Research -Troposphere, Germany

Tbilisi, GGSWBS'14 Atmospheric layers 3/18

Tbilisi, GGSWBS'14 Atmospheric layers 3/18 SACADA

Tbilisi, GGSWBS'14 SACADA assimilation-system Background Meteorological ECMWF analyses Trace gas observations Analysis SACADA PREP DWD GME CTM CTMad Diffusion L-BFGS

Tbilisi, GGSWBS'14 Horizontal GME Grid 9/18 ~147km between the grid points grid points per Model layer

Tbilisi, GGSWBS'14 Additional refinement troposphere/lower stratosphere SACADA Vertical Grid 54 layer CRISTA-NF MLS

Tbilisi, GGSWBS'14 HN O ~137hPa 12 Uhr UTC MLS 15

Tbilisi, GGSWBS'14 SCOUT-O3 campaign Stratospheric-Climate Links with Emphasis on the UTLS - O3 November-December 2005 AMMA- campaign African Monsoon Multidisciplinary Analyses /18

Tbilisi, GGSWBS'14 Cost function Vector of observations Observation error covariance matrix Projection operator Background Model operator SACADA assimilation-system 4D-Var Background error covariance matrix BECM ~ ~ 80 Terrabyte

Tbilisi, GGSWBS'14 Gradient Adjoint Model SACADA assimilation-system 4D-Var

Tbilisi, GGSWBS'14 Quasi-Newton method L-BFGS SACADA assimilation-system 4D-Var

Tbilisi, GGSWBS'14 Quasi-Newton method L-BFGS Background error covariance matrix BECM ~ ~ 80 Terrabyte SACADA assimilation-system 4D-Var

Tbilisi, GGSWBS'14 Radius of Influence ((de-)correlation length): Extending the information from an observation location Textbook: horizontal influence radius L around a measurement site, to be based on a priori statistical assessments L vertical cut L Horizontal structure function, to be stored as a column of the forecast error covariance matrix diffusion operator construction For atmospheric chemistry covariance modelling the diffusion approach is advocated: localisation intrinsically performed sharp gradients easily feasible matrix square roots for preconditioning straightforward to calculate; no inversion needed Background error covariance matrix formulation

Tbilisi, GGSWBS'14 Isopleths of the cost function and transformed cost function and minimisation steps Minimisation by mere gradients, quasi-Newon method L-BFGS (Large dimensional Broyden Fletcher Goldfarb Shanno), and preconditioned (transformed) L-BFGS application concentration species 1 transformed species 1 concentration species 2 transformed species 2

Tbilisi, GGSWBS'14 Solution: Diffusion Approach Transformation of cost-function: => Inverse of B and B -1/2 are not needed, if x b = 1. guess. 2 outstanding problems: 1.With linear estimation: How to treat the background error covariance matrix B (O(10 12 ))? 2.How can this be treated for preconditioning? (need B -1, B 1/2, B -1/2 ) With variational methods: minimisation procedure Background error covariance matrix formulation

Tbilisi, GGSWBS'14 Background error covariance matrix formulation Background

Tbilisi, GGSWBS'14 Background error covariance matrix formulation Background Observation: 3 ppm Ozone

Tbilisi, GGSWBS'14 Analysis (B diagonal) Background error covariance matrix formulation Background Observation: 3 ppm Ozone

Tbilisi, GGSWBS'14 Background error covariance matrix formulation Background Observation: 3 ppm Ozone

Tbilisi, GGSWBS'14 Background error covariance matrix formulation Observation: 3 ppm Ozone Analysis increment isotropic correlation The increment in initial values is spread out to neighbouring grid-points depending on the correlations that are known / assumed. Background

Tbilisi, GGSWBS'14 Assumption: Strong correlation along isolines of Potential Vorticity  Enhancement of diffusion  flow-dependent BECM Diffusion can be generalised to account for inhomogeneous and anisotropic correlations: Stratospheric case Background error covariance matrix formulation use PV field for anisotropic correlation modelling

Tbilisi, GGSWBS'14 Background Observation: 3 ppm Ozone Background error covariance matrix formulation

Tbilisi, GGSWBS'14 Background Observation: 3 ppm Ozone Analysis increment Background error covariance matrix formulation

Tbilisi, GGSWBS'14 Quasi-Newton method L-BFGS Adjoint Model SACADA assimilation-system 4D-Var

Tbilisi, GGSWBS'14 Construction of the adjoint code (3 different possible pathways) forward model (forward differential equation) algorithm (solver) code backward model (backward differential equation) adjoint algorithm (adjoint solver) adjoint code

Tbilisi, GGSWBS'14 Adjoint model A numerical model integration over a time interval [t 0 ; t i ] Accordingly, the tangent linear of this sequence of model operators is given by Thus, the adjoint model operator Mi propagates the gradient of the cost function with respect to xi backwards in time, to deliver the gradient of the cost function with respect to x0.

Tbilisi, GGSWBS'14 Adjoint model example

Tbilisi, GGSWBS'14 Adjoint model example

Tbilisi, GGSWBS'14 Quasi-Newton method L-BFGS Limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm SACADA assimilation-system 4D-Var

Tbilisi, GGSWBS'14 Gradient of the cost function h Hessian of the cost function

Tbilisi, GGSWBS'14 BFGS algorithm (2) From an initial guess x 0 a nd an approximate Hessian matrix H 0 the following steps are repeated as x k converges to the solution. 1.Obtain a direction s k by solving: 2.Perform a line search to find an acceptable step size in the direction found in the first step, then update 3.Set Convergence can be checked by observing the norm of the gradient,.

Tbilisi, GGSWBS'14 BFGS example with MATLAB

Tbilisi, GGSWBS'14 BFGS example with MATLAB

Tbilisi, GGSWBS'14 BFGS example with MATLAB

Tbilisi, GGSWBS'14 BFGS example with MATLAB

Tbilisi, GGSWBS'14 BFGS example with MATLAB

Tbilisi, GGSWBS'14 BFGS example with MATLAB

Tbilisi, GGSWBS'14 BFGS example with MATLAB

Tbilisi, GGSWBS'14 BFGS example with MATLAB it= 40 f= e-13 ||g||= e-05 sig=1.200 step=BFGS it= 41 f= e-15 ||g||= e-06 Successful termination with ||g||< e-08*max(1,||g0||):

Tbilisi, GGSWBS'14 Thank you for your attention! გმადლობთ ყურადღებისათვის ! Vielen Dank für Ihre Aufmerksamkeit!