A revised 4D-Var algorithm Yannick Trémolet ECMWF IMA W/S 29 April, 2002 Mike Fisher, Lars Isaksen and Erik Andersson For increased efficiency and improved accuracy.

2 Outline o Introduction o Current 4D-Var with 12-hourly cycling Main characteristics, used data types. The incremental formulation - and associated approximations and limitations. o The revised algorithm Quadratic inner iterations - using Conjugate Gradient. Hessian eigen-vector preconditioning. Trajectory interpolation from T511 to T42/T95/T159. Evaluation. o Prospects and Conclusions

3 Used Data o SYNOP Surf.Press, Wind-10m, RH-2m o AIREP Wind, Temperature o SATOB Cloud drift winds o DRIBU Surf.Press, Wind-10m o TEMP Wind, Temp, Humidity profiles o DROPSONDE Wind and Temp profiles o PILOT/Am+Eu Profilers Wind profiles o PAOB Surface pressure proxy o ATOVS HIRS, MSU and AMSU-A radiances o SSM/I TCWV, Wind speed o METEOSAT Water Vapour channel o QuikScat Ambiguous winds o SBUV Layer ozone o GOME Total ozone Conventional Satellite

4 ECMWF forecast model geometry Vertical resolution 60 levels levels below 850 hPa Horizontal resolution T L 511 ~ 40 km Observations are compared against a short-range 3-15 hour forecast

5 The current operational 4D-Var system Forecast model at T511 (40 km) resolution o Observation minus Background departures are computed using the full model at full resolution at the observed time. o Analysis increments are computed at coarser T159 resolution (125 km), using a tangent linear forecast model and its adjoint. All observations are analysed simultaneously. o 12 hours worth of global obser- vations are used in one go. o Around data are used, in total, per 12-hour cycle. o Satellite radiances are the most numerous data source

6 o 4D-Var finds the 12- hour forecast evolution that best fits the available observations o It does so by adjusting 1) surface pressure, and the upper-air fields of 2) temperature, 3) wind, 4) specific humidity and 5) ozone A few 4D-Var Characteristics All data within a 12-hour period are used simultaneously, in one global (iterative) estimation problem

7 o The i-summation is over 1h or ½h-long sub-divisions (or time slots) of the 12- hour assimilation period. The incremental formulation of 4D-Var In the incremental formulation (Courtier et al. 1994) the cost function is expressed in terms of increments with respect to the background state, with and linearized around.  The innovations are calculated using the non-linear operators, and : This ensures the highest possible accuracy for the calculation of the innovations, which are the primary input to the assimilation!

8 Approximations at inner iterations 1) The tangent-linear approximation: and 2) Approximations to reduce the cost: this involves degrading the tangent-linear (and its adjoint) with respect to the full model.  Lower resolution (T159 instead of T511),  Simplified physics (some processes ignored),  Simpler dynamics (e.g. spectral instead of grid-point humidity). This results in a shorter control vector, and cheaper TL and AD model during the minimisation - i.e. the inner iterations.

9 The outer iterations After each minimisation at inner level;  is updated:,  and are re-linearized around. o Innovations are re-calculated using the full non-linear model : o Superscript represents the outer iterations. o The full model remains at T511 throughout.

10 TL testing o Test of linear model based on Taylor series: o Valid for any perturbation (in practice a set of random vectors). o TL and NL run with the same setup:  Resolution  Physics  Time step  Simpler dynamics  Configuration (IFS)

11 Test of incremental approximation o In 4D-VAR the perturbation is not any vector, it is an analysis increment. It is not random and it is the result of a algorithm which involves the linear model. o The linear and non-linear models are used at different resolutions (T511/T159). o The non-linear model uses more physics. o Humidity is represented in spectral space in the linear model, in grid point space in the non-linear model. o Relative error: vs.

12 Test of incremental approximation o Compare TL output with finite difference in 4D-VAR setup (resolution, physics, …). o All the necessary information is present during the minimisation: o All the components are used exactly in their 4D-VAR configuration. High resolution non-linear update Low resolution non-linear trajectory Low resolution TL (cost function) Minimisation (TL) Low resolution TL (diagnostic) High resolution non-linear update

13 Evolution of TL model error Operational configuration. The error is large. It grows very rapidly in the first hours. It is not the case in the adiabatic test.

14 Impact of TL model resolution T511 outer loop. Varying inner loop resolution. The resolution of the inner loop may have reached a limit.

15 Small scales TL at T255, 12h forecast, Spectral norms

16 Impact of TL model resolution (adiab.) Adiabatic test. Better linear physics is needed. It is expensive both in development work and CPU.

17 Hessian eigenvector preconditioning The optimal pre-conditioner for the 4D-Var minimisation problem is the Hessian of the cost function,. The full 4D-Var Hessian is not known. So far has been used as an approximate preconditioner, neglecting the observation term. The consequence is that patches of very dense or particularly accurate observations may deteriorate the conditioning and slow down the rate-of- convergence.

18 Trajectory (in)consistency T319-T63T511-T159 The discrepancies between the high resolution non-linear update and the low resolution non-linear trajectory runs can be important.

19 The revised 4D-Var algorithm: Motivation o Improved efficiency  To offset some the cost of planned 1) higher resolution, 2) improved TL physics and 3) increased numbers of satellite data. o Increased TL accuracy  Discrepancies between and can introduce errors which grow quickly over the 12-hour assimilation window, especially affecting the analysis of small-scale phenomena and humidity. o Preparation for new high density satellite data  Coping with large numbers of observational data without deterioration of the rate-of-convergence. o Preparation for cloud and rain assimilation  Requires more extensive use of TL physics, and a good agreement between at inner iterations and at outer.

20 The revised 4D-Var algorithm: Specification o Quadratic inner iterations. Variational quality control and SCAT ambiguity removal moved to outer level. o Conjugate Gradient minimisation. With objective stopping- criterion based on the gradient-norm reduction. o Hessian eigenvector pre-conditioning. Updated after each inner minimisation. o Multi-Incremental, T42/T95/T159. With some tests at T255. o Interpolation of the trajectory. From T511 to T42/T95/T159. o TL physics. Used during all inner iterations.

21 o The conjugate gradient algorithm minimizes a quadratic function with a symmetric positive-definite Hessian: Conjugate Gradients and Lanczos Algorithms o The algorithm is: step to the line minimum recalculate the gradient calculate a new direction where: o Eliminate to get the 3-term recurrence (Lanczos):

22 Conjugate Gradients and Lanczos Algorithms o The gradient vectors in conjugate gradients are orthogonal. o Let be the matrix whose columns are. o Then where is tri-diagonal and o The residual term becomes small during the minimization as the gradient decreases. o After iterations, we get. o i.e. has the same eigenvalues as. o Intermediate matrices have interleaving eigenvalues: o Even for, some eigenvalues are well approximated.

23 Preconditioning o Write the analysis cost function as: o Preconditioning replaces by: o The Hessian of this new function is o The trick is to choose so that has a small condition number. o Eigenvector preconditioning sets: o Writing, gives: o If we choose so that, then the condition number of is. with

24 Preconditioning Eigenvalue N =1 1 = = Preconditioning reduces the condition number k= 1 / N from to

25 Preconditioning Variational Quality Control

26 Preconditioning Convergence is roughly twice as fast with Hessian preconditioning.

27 Preconditioning: Spectrum of Hessian o The leading eigenvectors of the Hessian are large-scale. o It is very effective and cost-efficient to calculate them at low resolution (T42/T95), o They can be used as pre-conditioner to reduce the number of iterations at higher resolutions (T95/T159 or T255). o This naturally leads to a multi-incremental setup.

28 Multi-incremental: RMS of T analysis increments Most of the total An-increment is formed at T42. There is a clear scale-separation between successive minimisation. The rapid decrease beyond ~T100 is due to the filtering properties of Jb, and the lack of observational information on smallest scales.

29 Conjugate-gradient: Reduction of Norm of gradient With C.G. minimisation the gradient norm reduces nearly monotonically with iteration. It is therefore possible to introduce an objective stopping-criterion based on its ratio. We have chosen a value =

30 C.G. and Lanczos Summary o The close connection between conjugate gradients and the Lanczos algorithm allows us to simultaneously:  Minimize the cost function.  Calculate the eigenvectors and eigenvalues of the Hessian. o The extra computational effort required to calculate the eigenpairs is negligible. o The connection can be exploited to improve the minimization. o The consequence is a more efficient and more robust 4D-Var minimisation – w. r. t. observation amounts and distribution.

31 Interpolated Trajectory o The increments which are appropriate for the low resolution situation are not always suitable for the high resolution situation. o The trajectory can be interpolated: o This algorithm could not be tested with the traditional tangent linear test because of the two resolutions involved. High resolution non-linear update Low resolution non-linear trajectory Low resolution TL (cost function) Minimisation (TL) Low resolution TL (diagnostic) High resolution non-linear update Interpolation

32 Interpolated trajectory

33 Performance: Jo cost function Current 4D-Var Revised 4D-Var

34 Scores

35 Performance: CPU cost o Operational setup: 1h20min. o Revised algorithm: 1h13min. o On Fujitsu VPP5000, 16 processors. o Elapsed time, I/O not fully optimised.

36 Conclusions ECMWF’s 4D-Var has been improved: o Conj. Gradient minimisation o Hessian pre-conditioning o Inner/outer iteration algorithm o Improved TL approximations o Multi-incremental T42/T95/T159 These developments will help facilitate: o Use of higher density data o Higher resolution o Enhanced use of (relatively costly) TL physics o Cloud and rain assimilation Prospects o The minimisation of the cost function has been improved. o More work is needed to improve the representation of the small scales in the inner loop. o Efficiency gains will pay for improved inner loop physics and resolution.