**A revised 4D-Var algorithm**

**Outline**

**Used Data**

**ECMWF forecast model
geometry**

**The current operational
4D-Var system**

**A few 4D-Var Characteristics**

**The incremental formulation
of 4D-Var**

**Approximations at inner
iterations**

**The outer iterations**

**Slide 10**

**Slide 11**

**Test of incremental
approximation**

**Evolution of TL model error**

**Impact of TL model
resolution**

**Small scales**

**Impact of TL model
resolution (adiab.)**

**Hessian eigenvector
preconditioning**

**Trajectory (in)consistency**

**The revised 4D-Var
algorithm: Motivation**

**The revised 4D-Var
algorithm: Specification**

**Conjugate Gradients and
Lanczos Algorithms**

**Conjugate Gradients and
Lanczos Algorithms**

**Preconditioning**

**Preconditioning**

**Preconditioning**

**Preconditioning**

**Preconditioning: Spectrum of
Hessian**

**Multi-incremental: RMS of T
analysis increments**

**Conjugate-gradient:
Reduction of Norm of gradient**

**C.G. and Lanczos Summary**

**Slide 31**

**Interpolated trajectory**

**Performance: Jo cost
function**

**Scores**

**Performance: CPU cost**

**Conclusions**