1. 4dVar assimilation experiments into highly non linear models with adjointless technique
Gleb Panteleev (IARC)
Max Yaremchuk, (NRL), Dmitri Nechaev (USM)
Monthly Weather Review, 137, , 2009
AOMIP, October 20-23, 2009

2. 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

3. 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

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

5. 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.

6. 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.

7. The method
External loop
Internal loop 7

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

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

10. QG model: setting of the twin-data experiments

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

12. QG model: setting of the twin-data experiments12

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

14. RESULTS: Experiments with n= 500 (stable adjoint)

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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”.

21. 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

22. THANKS