4dVar assimilation experiments into highly non linear models with adjointless technique Gleb Panteleev (IARC) Max Yaremchuk, (NRL), Dmitri Nechaev (USM) Monthly Weather Review, 137, 2966-2978, 2009 AOMIP, October 20-23, 2009

Outline: Linear background 1.1 4Dvar 1.2 Krylov space methods: GMRES, CG, … 1.3 EOF analysis of model solutions 1.4 Adjointless 4Dvar 2. Comparison with the adjoint 4dVar: twin-data experiment 2.0 Setting of the twin-data experiments 2.1 Instability of the TL and adjoint models and validity of the TL approximation 2.2 Impact of spatial resolution 2.3 Impact of noise 3. Twin-data experiment with ice model: preliminary results 4. Summary & possible directions of further application

Conventional 4Dvar: state space formulation Model: Cost. Fun: Important: There is a necessity to build forward and adjoint models to solve problem Model spectrum Data spectrum 2) The model spectrum is always broader than data spectrum

Krylov subspace methods: GMRES, CG, BiCG.. Krylov basis Krylov subspace methods

EOF analysis of model solution The largest eigenvalues of the sample covariance matrix correspond to the largest projections of the initial condition on the least damped eigen-modes of the dynamical operator, i.e. EOF analysis selectively retrieves those components of the initial state that are projected on the 4d data locations with the best accuracy.

Motivation: 1. Adjoint models are hard to develop and keep updated in correspondence with continuously developing forward code. 2. Community OGCMs with 4Dvar capabilities (ROMS, OPA, MIT ) use approximation to their tangent linear and adjoint codes because of a) non-differentiable processes that cannot be rigorously linearized b) breakdown of the TL approximation in the presence nonlinear instabilities of the flow 3. Automatic adjoint compilers generate less computationally efficient code, which may require up to 4-10 times more CPU time than the forward code. 4. Universal tool for quantitative model inter-comparison: model-data fusion or OSSE.

The method External loop Internal loop 7

The method External iterations A-spectrum of the solution first guess Krylov subspace dimension 8

QG model: setting and reference solution L = 480 km Rd = 25 km dx = 15 km T = 45 days f = 10-4 s-1 b = 2 10-13 s-1 n = 50 (500) m2/s tdiss = 52 (5.2) days curlt = 10-7 s-1 sin(4px/L)cos((4px/L)

QG model: setting of the twin-data experiments

QG: Instability of the adjoint model Cost function gradient after 30 iterations Validity of the TL approximation: [

QG model: setting of the twin-data experiments 12

RESULTS: Experiments with n= 50 (unstable adjoint) En4d Adjoint

RESULTS: Experiments with n= 500 (stable adjoint)

Ice concentration and velocity Wind, 15m/s RESULTS: Ice model - zonal wind forcing 2 days, dt=2 hours, 30 steps, 10 “layers” of data Ice concentration and velocity Wind, 15m/s Ice model: Complementary to J.F. Lemieux

Ice concentration and velocity Wind, 15m/s RESULTS: Ice model - zonal wind forcing ~10 days, dt=2 hours, 130 steps, 10 “layers” of data Ice concentration and velocity Wind, 15m/s Ice model: Complementary to J.F. Lemieux

Ice concentration and velocity Wind, 15m/s RESULTS: Ice model - zonal wind forcing ~20 days, dt=2 hours, 230 steps, 10 “layers” of data Ice concentration and velocity Wind, 15m/s Ice model: Complementary to J.F. Lemieux

Ice concentration and velocity Wind, 15m/s RESULTS: Ice model - zonal wind forcing Dt=2 hours, 230 steps, 10 “layers” of data Ice concentration and velocity Wind, 15m/s Ice model: Complementary to J.F. Lemieux

Summary (1) 1. A version of the R4dVar is proposed. The algorithm employs a sequence of low- dimensional Krylov subspaces that are iteratively updated in the process of finding a minimum of the cost function 2. Compared to conventional 4dVar, the method provides similar or better reduction of the cost function after several updates of the search subspace. 3. In terms of the computational cost R4dVar performs similarly with 4dVar if the latter is terminated after more than 2N - 4N iterations, where N is the (fixed) dimension of the Krylov subspaces. 4. The method gains substantial advantage over 4dVar when the dynamical constraints have strong non-linear instabilities which cause the breakdown of the tangent linear approximation. Compared to 4dVar, the method gains extra efficiency when observations become more sparse and/or noisy Error analysis of optimal state is straightforward in the reduced space LIMITATION: in contrast to the adjoint-based 4dVar, performance of R4dVar depends on the spectrum of the first guess (in the linear case). 19

Summary (2) 6. R4dVar works well with strongly non-linear ice model, but ice model solution reveal limited controllability relatively initial conditions. Wind forcing and ice parameters should be included into the control vector. 7. There is a possibility possible to build R4Dvar tool that can be applied to the arbitrary ICE-OCEAN model. This tool can be used: a) For data assimilation into any model b) Analysis of the model skill to reproduce the data c) For model errors and bias analysis ( ~ Zupasnky et al., 2006) For OSSE (or twin data) experiments and model inter-comparison: Quantitative matrix of model “skill”.

POSSIBILITIES OF FURTHER RESEARCH 1. Extend the method to include forcing control (atmosphere, open lateral boundaries) – employ (time-lagged) correlations between the model interior and the boundaries. Try other approaches to the generation of the Krylov subspaces: a) replace background error covariance-based projection by a “non- linear approximation” of the adjoint b) include eigenfunctions of the background error covariance in the Krylov matrix c) Breeding ? d) Lyapunov exponents? Apply the method to a AO OGCM (multivariate state vector): community data assimilation tool (OSSE, model-data fusion, ext.) 21

THANKS