4DEnVar: link with 4D state formulation of variational assimilation and different possible implementations

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1 QuarterlyJournalof theoyalmeteorologicalsociety Q J Meteorol Soc 4: 97 October 4 A DOI:/qj35 4DEnVar: lin with 4D state formulation of variational assimilation and different possible implementations Gérald Desroziers* Jean-Thomas Camino and Loï Berre CNM-GAME Météo-France and CNS Toulouse France *Correspondence to: G Desroziers CNM-GAME Météo-France and CNS 4 avenue G Coriolis 357 Toulouse Cedex France GeraldDesroziers@meteofr The four-dimensional ensemble variational (4DEnVar) formulation is receiving increasing interest especially in numerical weather prediction centres which until now have mostly relied on the four-dimensional variational (4D-Var) formalism It may indeed combine some of the best features of variational and ensemble methods In this article it is shown that the 4DEnVar formulation is lined with the 4D state formulation of variational assimilation and that the 4DEnVar is relatively easy to precondition in addition of being parallelizable Practical implementations of the 4DEnVar are also investigated and two new preconditioned algorithms are proposed The hybrid formulation of 4DEnVar combining static and ensemble bacground-error covariances is discussed for the different possible algorithms An application of the proposed implementations of 4DEnVar is shown with the Burgers model and compared to the use of 4D-Var Key Words: variational assimilation; ensemble assimilation; minimization algorithms; preconditioning eceived 6 April 3; evised 7 December 3; Accepted January 4; Published online in Wiley Online Library 5 April 4 Introduction Most numerical weather prediction (NWP) centres rely on the four-dimensional variational (4D-Var) formalism for their data assimilation system (Lewis and Derber 985; Le Dimet and Talagrand 986; Courtier 987; abier et al ) A nice property of 4D-Var is that it allows bacground-error covariances to evolve implicitly over the assimilation period (Thépaut et al 996) However they are often simplified at the beginning of the assimilation period with respect to space and time variations Moreover linearized and adjoint models are difficult to develop and to maintain and their low resolution and sequential use in 4D-Var raise parallelization issues (Isasen ) The Ensemble Kalman Filter (EnKF) formalism (Evensen 994; Houteamer et al 996) has received great attention during recent years and has initiated a lot of wor on the use of ensembles in data assimilation Indeed it allows bacground-error covariances to evolve in space and time Other square-root forms of the EnKF are possible with no perturbation of observations in the ensemble (Bishop et al a; Whitaer and Hamill ; Tippett et al 3; Hunt et al 7) A difficulty raised by the EnKF is the necessary localization of ensemble covariances in observation space which is questionable especially for satellite data (Campbell et al ) The idea of EnKF based on an explicit perturbation of observations and an implicit perturbation of bacground can be transposed to an ensemble of perturbed variational assimilations (Fisher 3; Kucuaraca and Fisher 6; Berre et al 6) Such an ensemble was implemented first at Météo-France (Berre et al 7; Desroziers et al 9; Berre and Desroziers ) and subsequently at the European Centre for Medium-range Weather Forecasts (ECMWF; Isasen et al ) It is used at both centres to partly initialize ensemble prediction systems but also to allow bacground-error variances to vary in space and time in the variational minimization The homogeneity assumption made in 4D-Var on bacground-error correlations can also be relaxed by representing them in wavelet space (Fisher 3) Such a more realistic model for bacground covariances can be calibrated by using bacground perturbations given by an ensemble of variational assimilations carried out over a few wees or even on shorter periods (Panneouce et al 7; Varella et al ; Buehner ) Another possibility to utilize the benefit of ensemble perturbations is to use them in order to diagnose and introduce an objective deformation of a bacground correlation model (Michel 3) or to use them in the combination of static covariances and ensemble covariances (Hamill and Snyder ; Wang et al 7a 8; Clayton et al 3; Wang et al 3) The 4DEnVar formulation (Lorenc 3 3; Buehner 5; Buehner et al ) has also received considerable attention in the very recent years The interest in this formulation relies on different nice properties: it allows flow-dependent bacgrounderror covariances; in contrast to the EnKF it avoids the localization of bacground-error covariances in observation space; it is potentially highly parallel on new computer architectures; and it also gets rid of the development maintenance and cost of tangentlinear and adjoint models A comparison of 4D-Var with 4DEnVar using a toy model has been made by Fairbairn et al (4) who in particular showed the effect of simple spatial localization on the spatial temporal cross-covariances in 4DEnVar The aim of this article is to point out the strong formal similarity between the 4D state formulation of variational assimilation and its 4DEnVar counterpart It also aims to show that the c 4 oyal Meteorological Society

2 98 G Desroziers et al implementations of 4DEnVar proposed so far implicitly rely on a preconditioning of the variational problem using the square root of the ensemble covariance matrix Two other possible formulations are proposed relying rather on a preconditioner given by the complete ensemble covariance matrix The next section recalls the formulation of 4D-Var in its most general form including the representation of model errors within the assimilation window Different formulations of the incremental 4D-Var are shown Section 3 presents the general formalism of the 4DEnVar and also introduces the different ways to apply the localization of ensemble covariances The practical implementation of 4DEnVar is presented in section 4 Hybrid formulations are discussed in section 5 Finally an application of the 4DEnVar formulation with the Burgers model is shown in section 6 Conclusions and future prospects are given in section 7 4D variational assimilation General formulation The principle of variational data assimilation is to see an analysis that fits the observations and a previous forecast In its weaconstraint form this includes the representation of model errors (Courtier 997; Trémolet 6; Fisher et al ) and assuming that these model errors are uncorrelated in time it consists of minimizing the cost function J(x x K ) = ( x x b ) T ( B x x b ) + K q T Q q = + K { H (x ) y o T { } H (x ) y o } = where x x K is the set of analyzed states at the different times t of the assimilation window [t t K ] x and B respectively stand for the bacground state and error covariance matrix at initial time t q = x M (x ) is the model error term at time t and Q is the corresponding model error covariance matrix y o is the vector of observations at time t H is the possibly nonlinear associated observation operator and is the observation-error covariance matrix Incremental formulation The minimization of the above cost function can be achieved by a sequence of minimizations of quadratic cost functions with observation operators linearized around updated trajectories These cost functions have the following form (Fisher et al ): at the previous outer loop; H is the linear observation operator associated with observations y o and linearized around xt ;and d ot = y o H (x t ) is the difference between the values of observations at time t and the corresponding values of the previous trajectory using the nonlinear observation operator H 3 Alternative expressions Following Fisher et al () a more compact form of the weaconstraint 4D-Var cost function can be obtained by introducing the following set of 4D vectors: δx = δx δx δx K δp = δx δq δq K where δx = M δx + δq Here the underline notation denotes the extra time dimension as in Fairbairn et al (4) In matrix form δx and δp are lined by the expression with δx = L δp I M I L = M M I M K M K M K K I and M = M M M M = I Matrix L is invertible and its inverse is given by I M I L = M K I Defining H H H = ( δx d bt) T B ( δx d bt) H K J(δx δx K ) = + + K ( = K = δq d qt ( H δx d ot ) T ( Q ) T δq d qt ) ( H δx d ot ) () d ot = d ot d ot d ot K = K where δx δx K is the set of increments to be added to the previous high-resolution trajectory x t xt K ; dbt = x b xt is the difference between the bacground and the previous trajectory at the beginning of the assimilation window; δq = δx M δx is the model error term for the increments; d qt = M (x t ) xt = M (x t + δx ) (xt + δx ) where x and δx respectively stand for the trajectory and the increment and d t = d bt d qt d qt K D = B Q Q K c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

3 4DEnVar: State Formulation of 4D-Var and Possible Implementations 99 it is easy to chec that the cost function defined by expression () can be simply rewritten as J(δx) = ( L δx d t) T D ( L δx d t) + ot ( ) T ( ot) () The cost function () can be further rewritten J(δx) = ( L δx L L d t) T D ( L δx L L d t) + ( ot ) T ( ot) = ( δx L d t ) T( L D L T ) ( δx L d t ) + ( ot ) T ( ot) Using the expressions of L and D it follows that each line with K of matrix L D corresponding to a given time t with t t t K has the generic form ( M B M Q M Q Q with + non-null blocs Therefore L D L T is a full 4D matrix that can be written as B B B K L D L T B B B K = B K B KK where each bloc B has the generic form B = M B M min( T ) + = and M = M = M = I The expression L d t can be developed as M Q T M I M I L d t = M M I M K M K M K K I x b xt M (x t ) xt M (x t ) xt M K (xk t ) xt K Maing the approximations M (xb xt ) M (xb ) M (xt ) M [M (xt ) xt ] M (xt ) M (xt ) with M = M {(M )} each line with K of matrix L d t can be written M (xb ) M (xt ) + M (xt ) M (xt ) + + M (xt ) M (xt ) + M (x t ) xt ) which simplifies to M (xb ) xt and therefore x b xt x b xt L d t M = (xb ) xt = x b xt = dbt M K(xb ) xt K xk b xt K where d bt denotes the vector of the differences between the elements of the 4D bacground trajectory x b and of the updated 4D trajectory x t Using the previous expressions for δx d bt H D and the additional notation B = L D L T the wea-constraint 4D-Var cost function () is finally equivalent to the following cost function: J(δx) = ( δx d bt ) T B ( δx d bt) + ( ot ) T ( ot) (3) This is an expression consisting only of 4D vectors and matrices In particular B is a full 4D covariance matrix of the 4D bacground errors including additive model errors consistent with matrices Q = K 3 4D ensemble variational assimilation 3 General formulation Following Fairbairn et al (4) the 4DEnVar cost function is given by J(x) = ( x x b )T Be ( x x b ) + { H(x) y o} T { H(x) y o} (4) All the elements appearing in expression (4) have a fourdimensional trait with x the 4D vector of the states x at the different times t of the assimilation period x b the 4D vector of the bacground states x b Be the 4D covariance matrix of bacground errors given by an ensemble y o the 4D vector of observations H the corresponding 4D observation operator possibly nonlinear the 4D covariance matrix of observation errors possibly non-diagonal which maes it more general than the definition given in section The covariance matrix Be is deduced from an ensemble of 4D perturbations If L is the size of the ensemble each perturbation x b l is given by a scaled difference between the nonlinear forecast x b l and the ensemble mean xb : x b l = (x b l L xb ) (5) Defining the rectangular matrix ( ) X b = x b xb L containing the L vectors of 4D perturbations matrix Be is given by B e = X b X b T Since the perturbations x b are 4D matrix Be has the following structure: B e B e B e K B e B e = B e B e K B e K B e KK c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

4 G Desroziers et al Each bloc B e of Be contains the covariance of perturbations at time t with perturbations at time t : L B e = Xb Xb T = l= xl b T xb l where X b is the rectangular matrix containing the L subcolumns of X b corresponding to the L ensemble perturbations {xl b l = L} at time t 3 Incremental formulation An incremental formulation of the 4DEnVar cost function is also possible under the form J(δx) = ( δx d bt )T Be ( δx d bt ) + ( ot ) T ( ot) (6) with the same notation as in the expression (3) of the weaconstraint 4D-Var formulation and where d bt = x b x t still denotes the 4D vector containing the difference between the 4D bacground x b and the previous 4D trajectory x t However note that the two forms () and (3) of the weaconstraint 4D-Var cost function are equivalent only under the linearity hypothesis used to derive the approximation L d t = d bt in section 3 In the 4DEnVar formulation (6) the bacground error covariance matrix Be is explicitly derived from an ensemble of nonlinear integrations 33 epresentation of model error The ensemble perturbations x b are assumed to sample the actual bacground errors If these errors follow a linear evolution within the assimilation period and if they also include a model error component with uncorrelated and additive errors in time then each one of these 4D perturbations can be seen as a random realization given by x b = L D η L where η is a 4D Gaussian random vector where all elements have a zero mean and a variance equal to and where matrices L and D have the same meanings as in the wea-constraint 4D-Var formulation presented in section With such ensemble perturbations the expression X b X b T will approximate the matrix B = L D L T that appears in the formulation (3) of the wea-constraint 4D-Var assuming that the ensemble B e matrix and B matrix correspond at the beginning of the assimilation period Therefore there is a close connection between the 4D state formulation of 4D-Var and the 4DEnVar formulation because both allow the same representation of additive model errors From that point of view the 4DEnVar formulation appears even more general than the particular form of wea-constraint 4D-Var presented in section since it can easily allow different representations of model errors such as errors correlated in time additive or multiplicative model perturbations corresponding to inflation techniques (eg aynaud et al ) stochastic physics (Buizza et al 999) stochastic inetic energy bacscatter (SKEB; Shutts 5) or different versions of the model physics 34 Localization of raw ensemble covariances 34 General formulation of the localization The main drawbac of the raw ensemble covariance matrix B e is that it has a very limited ran L and is affected by sampling error A way to alleviate this problem proposed by Houteamer and Mitchell () is to replace this raw matrix by B e = Be C where C is a valid correlation matrix with the same size as B e and denotes the element-by-element product of the two matrices also called the Hadamard-Schur product Since Be isa4draw covariance matrix C is also a 4D covariance matrix and B e is given by B e B e B e K B e B e B e K B e = B e K B e KK C C C K C C C K C K C KK where the C are possibly different localization correlation matrices for the different covariance matrices B e where and stand for two time indices inside the assimilation period If M denotes the number of model fields and assuming here for simplicity that each one of these model fields has a 3D structure with size N then the blocs B e and C also are bloc matrices with C C C M C C = C C M (8) C M C MM where each matrix C mm is the localization correlation applied to cross-covariances between two times and and two variables m and m With such a localization of the raw covariances the incremental formulation (6) of the 4DEnVar cost function can then be rewritten as J(δx) = ( δx d bt ) T B e ( δx d bt) + ( ot ) T ( ot) (9) with the expression (7) for B e 34 Unique localization matrix C A particular but common case is the one where the same localization correlation C is used for all auto- and crosscovariances for the different variables and times as in the implementation of the 4DEnVar described by Buehner et al () In this case it is easy to chec that the full 4D correlation matrix C is defined by C C C C C C C = = N CT N C C where I N I Ṇ N = I N (7) c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

5 4DEnVar: State Formulation of 4D-Var and Possible Implementations is the [(K + )MN] N matrix where each one of the (K + )M blocs I N is the N N identity matrix and C also is a N N matrix lie any bloc C mm in expression (8) 343 Low-resolution representation of localization matrices where Ĉ = C C C () A spectral representation of the localization correlation C can be written: C = S C s (S ) T where S is the inverse spectral transform and C s is the matrix of correlation in spectral space Furthermore if the localization correlation is assumed to be homogeneous in the horizontal C s will be diagonal Because the correlations contained in matrices C mm are rather broad C can be represented at a reduced resolution N e : C T e C e (T e ) T where T e is the interpolation from a reduced grid with size N e to the perturbation grid C e can be represented in spectral space with C e = (S e ) C es { (S e ) } T where (S e ) is the inverse spectral transform in low-resolution space and C es is the spectral representation of C e Then a possible representation for C is C T e (S e ) C es { (S e ) } T (T e ) T 4 Practical implementation of 4DEnVar 4 Algorithm : Change of variable for increments 4 General formulation of the change of variable It is well nown that the standard 4D-Var cost function is difficult to minimize This is due to the ill-conditioning of the Hessian of the cost function which is itself related to the ill-conditioning of the bacground error covariance matrix A classical way to improve the conditioning of the Hessian of the cost function and then the minimization is to apply a change of variable involving the square-root of the bacground error covariance matrix Following Lorenc (3) and Buehner (5) it can indeed be shown that the square root B e of B e exists and is given by where B e = (D C D L C ) D l D l D l = DlK is the diagonal matrix containing the elements of the vector of the 4D perturbations x b l along the diagonal (each diagonal matrix D l contains the perturbations x b l at time t ) Defining the matrix D = (D D L ) matrix B e can be simply written B e = D Ĉ is the bloc diagonal matrix where the L blocs contain the same matrix C A preconditioning of the 4DEnVar cost function (6) can then be obtained by defining the new variable χ with δx = B e χ + d bt = D Ĉ χ + d bt With such a change of variable the cost function (9) becomes J(χ) = χ T χ + ( ot ) T ( ot) () and the gradient of this new cost function with respect to the control variable χ is J(χ) = χ + Ĉ T D T H T ( ot) The associated Conjugate Gradient (CG) algorithm to minimize this preconditioned cost function is: Algorithm d = β = χ = g = Ĉ T D T H T d ot for i = :I d i = g i + β i d i f i = (I + Ĉ T D T H T H D Ĉ ) d i α i = (g Tg i i )/(dt i f i ) χ i+ = χ i + α i d i g i+ = g i α i f i β i = (g T g i+ i+ )/(gtg ) i i end δx I = D Ĉ χ I where I is the maximum number of CG iterations In this algorithm all the vectors d i f i χ i g i have the same dimension which can be potentially as large as L(K + )MN 4 Simplified matrix C The control variable χ potentially is a L(K + )MN vector where M is the number of model variables and N their size in threedimensional (3D) space assuming here for simplicity that every variable has a 3D structure However as mentioned above a particular case is the one where the same localization correlation C is used for all auto- and cross-covariances for the different variables and times as in the implementation of the 4DEnVar described by Buehner et al () In this case it is easy to chec that each bloc C of the bloc diagonal matrix Ĉ becomes C = N C c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

6 G Desroziers et al This shows that the B e = D Ĉ matrix used for the change of variable now is a L N rectangular matrix with such a simplification for the expression of C In this case the control variable χ becomes a single set of L 3D fields with dimension N and can be simply noted χ Furthermore if a low-resolution spectral representation of C is used C can be rewritten as C = N T e (S e ) (C es ) which shows that the dimension of the control variable χ is now limited to LN e For any application x = B e χ of B e to a vector χifxm and xlm b respectively stand for the parts of x and xb l corresponding to a particular time t and a particular variable m x m is given by x m = L l= x b lm { T e (S e ) (C es ) χ } () where χ l is the part of χ associated to the ensemble perturbation l Similarly for any application χ = B e T x of B e T to an adjoint variable vector x with the dimension (K + )MN of the model 4D state the sub-part χ l of χ is given by χ l = K = m = M (C es ) { (S e ) } T (T e ) T (x b l m x m ) (3) With such a simplification of the matrix C the minimization algorithm is similar to Algorithm but with a size of the vectors d i f i χ i g i reduced to LN e dropping here the underline notation since the time variation of these vectors vanishes 4 Algorithm : C-preconditioned CG in weight space 4 General formulation with perturbation weights Another formulation of the 4DEnVar is possible by introducing a set of ensemble perturbation weights w l l = L (Lorenc 3; Liu et al 8) Indeed a new cost function can be defined as J(w) = wt Ĉ w + ( ot ) T ( ot) (4) where Ĉ has the same definition as in expression () w is the L(K + )MN vector that is the concatenation of the L fields w l with size (K + )MN and δx = D w + d bt = L l= x b l w l + d bt The above expressions show that the spatial localization of the covariances is caused by the fact that the weights applied to each ensemble member are allowed to vary spatially according to the localization correlation matrix 4 Equivalence with the change of variable for increments Wang et al (7b) showed that after a linear transformation of the control variable w the above cost function is equivalent to expression () which is used in Buehner (5) or Buehner et al () and which also relies implicitly on a change of variable as shown in section 4 Indeed defining a new variable χ w such as w = Ĉ χ w the cost function (4) can be rewritten J(χ w ) = χ wt χ w + ( ot ) T ( ot) (5) with δx = D w + d bt = D Ĉ χ w + d bt (6) It appears that the cost function (5) is exactly the same as the cost function () which shows that the two approaches are completely equivalent 43 Use of a Ĉ-preconditioned CG It is easy to chec that the Hessian of the cost function (4) is given by Ĉ + D T H T H D (7) The use of a CG iterative algorithm to minimize the cost function (5) is equivalent to solving the linear system (Ĉ + D T H T H D) w = H T d ot Due to the ill-conditioning of matrix Ĉ the Hessian matrix (Eq (7)) will be ill-conditioned As shown by Wang () a Ĉ-preconditioned formulation of the minimization can be defined in order to implicitly invert the Ĉ-preconditioned Hessian matrix ) Ĉ(Ĉ + D T H T H D = I + Ĉ D T H T H D which should have a far better conditioning than the original one The Ĉ-preconditioned CG algorithm is: Algorithm d = e = β = w = g = D T H T d ot h = Ĉ g for i = :I d i = h i + β i d i e i = g i + β i e i f i = e i + D T H T H D d i α i = (g T i h i )/(dt i f i ) w i+ = w i + α i d i g i+ = g i α i f i h i+ = Ĉ g i+ β i = (g T i+ h i+ )/(gt i h i ) end δx I = D w I In this algorithm all the vectors d i e i f i w i g i h i have the same dimension which can be potentially as large as L(K + )MN c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

7 4DEnVar: State Formulation of 4D-Var and Possible Implementations 3 However if the same localization correlation C is used for all auto- and cross-covariances for the different variables and times then w will simplify to a set of L 3D fields with a size LN e if a low-resolution representation of C is also used The minimization algorithm will be similar to Algorithm but with a size of the vectors d i f i χ i g i reduced to LN e dropping here again the underline notation since the time variation of these vectors vanishes 43 Algorithm 3: B-preconditioned CG in primal space Starting from the incremental formulation (9) of the 4DEnVar with a full 4D control variable δx the Hessian of this cost function is B e + H T H (8) It is again ill-conditioned due to the presence of the inverse of B e The minimization of the cost function can be largely improved by using a CG in model space with the full 4D δx vector as a control variable preconditioned by the B e matrix This gives the following new iterative algorithm Algorithm 3 d = e = β = δx = g = H T d ot h = B e g for i = :I d i = h i + β i d i e i = g i + β i e i f i = e i + H T H d i α i = (g T i h i )/(dt i f i ) δx i+ = δx i + α i d i g i+ = g i α i f i h i+ = B e g i+ β i = (g T i+ h i+ )/(gt i h i ) end In this primal space algorithm all the vectors d i e i f i δx i g i have the same dimension (K + )MN as the bacground vector x b In this case the square root of B e is no longer required However a simplification of B e is still possible by relying on a single localization correlation C for all covariances between variables and times and also on a low-resolution representation C T e C e (T e ) T of C For any application h = B e g of B e to a vector gifg m and h m respectively stand for the parts of g and h corresponding to a particular time t and a particular variable m h m is given by L K M { h m = x b lm T e C e (T e ) T (x b l m g m } ) (9) l= = m = This expression is equivalent to the application of Eq () followed by the application of Eq (3) in Algorithm However it does not require the storage of the LN e vector resulting from the application of B e T in expression (3) Note that this expression does not explicitly rely on a squareroot form of matrix C e However the need to use a valid positive semi-definite correlation matrix still implies to rely on a squareroot form of C e 44 Algorithm 4: B-preconditioned CG in dual space As shown for example by Courtier (997) a formulation of the assimilation problem is also possible in observation or dual space Indeed following Gürol et al (4) the solution of the minimization of the incremental 4DEnVar cost function (6) is δx = ( B e +H T H ) ( B e d bt + H T d ot) Using the Sherman Morrison Woodbury formula and after some reordering δx canalsobeexpressedas δx = d bt + B e H T (H B e H T + ) (d ot H d bt ) = d bt + B e H T δy where δy is the solution of the linear system ( H B e H T + ) δy = d ot H d bt It is easy to chec that δy also is the solution of the following new cost function in observation space: J(δy)= δyt( H B e H T + ) δy δy T( d ot H d bt) By analogy with the use of B e to precondition the primal problem in model space a preconditioning by was proposed by Courtier (997) First implementations of a dual-space assimilation employed such a preconditioning via the use of the square root of (Courtier 997; El Araoui et al 8; El Araoui and Gauthier ) However it was shown that this preconditioning combined with the use of a CG algorithm was characterized by a poor convergence of the cost function in primal space (El Araoui and Gauthier ) ecently Gürol et al (4) proposed a new minimization algorithm BCG (educed B Conjugate Gradient) based on the use of the bacground-error covariance matrix as a preconditioner but with a control variable in dual space They showed that the minimization has theoretically exactly the same convergence as the minimization performed in primal space and using the same preconditioning This strategy can be transposed to the 4DEnVar problem which gives the following additional iterative algorithm Algorithm 4 d = e = β = δy = g = d ot h = H B e H T g for i = :I d i = g i + β i d i e i = h i + β i e i f i = e i + d i α i = (g T i h i )/(et i f i ) δy i+ = δy i + α i d i g i+ = g i α i f i h i+ = H B e H T g i+ β i = (g T i+ h i+ )/(gt i h i ) end δx I = d bt + B e H T δy I In this dual-space algorithm all the vectors d i e i f i δy i g i have the same dimension P of the vector y o of observations c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

8 4 G Desroziers et al 5 Hybrid formulation 5 General formulation In practice it might be desirable to combine a static covariance matrix B s as may be used in the 4D-Var formulation presented in section and a localized ensemble covariance matrix B e to form an hybrid covariance matrix B h with B h = γ s B s + γ e B e where γ s and γ e are two scalars such as γ s + γ e = Matrix B s can possibly be a full 4D matrix with the form B s = L D L T as shown in section This form expresses the evolution of covariances along the assimilation period with the tangent-linear model M However if the tangent-linear model is avoided or in other words if the tangent-linear model is replaced by the identity model it is easy to chec that the B s matrix will be a particular matrix where all the covariances between any times are assumed to be the same This can be written in a matrix form as B s B s B s B s B s B s B s = B s B s = MN B s T MN where = MN B s B s T T MN I MN I MN MN = I MN isa[(k + )MN] MN matrix where each one of the K + blocs I MN is the MN MN identity matrix and B s is the square root of matrix B s It is important to point out that matrix B s is not necessarily asquaremn MN matrix and that it can even have a number of columns larger than MN by a factor that will be denoted by S This is in particular the case with the wavelet formulation of the B s matrix introduced by Fisher (3) at ECMWF and also used at Météo-France (Varella et al ) which allows a better representation of bacground-error correlation heterogeneity but at the price of an increase of the leading dimension of matrix B s of the order of S = 3 Note that for other formulations of the static B s matrix the factor S may on the contrary be less than one for example when the analysis increment is on an (unreduced) lat-long grid and the control vector is in spectral space It was shown in section 4 that if the same localization correlation C is used for all auto- and cross-covariances for the different variables and times then the representation of the localized covariance matrix B e can be simplified The nature and the size of the control vector depend on the choice of one of the formulations described in section 4 to solve the hybrid assimilation problem 5 Hybrid formulation of Algorithm Algorithm presented in section 4 relies on a change of variable using the square root of the localized ensemble covariance matrix In the hybrid formulation it involves the square root of the hybrid covariance matrix which is defined by ( ) B h = γ s B s γ e B e This means that the control variable χ for this hybrid formulation will be an augmented control vector with a first part χ s with dimension MN associated with the static covariance matrix B s and a second part χ e with dimension LN e associated with the ensemble covariance matrix B e In this case the cost function to minimize is with and J(χ h ) = χ ht χ h + ( ot ) T ( ot) () χ h = χ s χ e δx = γ s B s χ s + γ e B e χ e + d bt = γ s MN B s χ s + γ e D Ĉ χ e + d bt = MN δx s + δx e + d bt The previous expression shows that in this case the hybrid increment is the combination of a static increment δx s with an evolutive ensemble increment δx e As in Bishop et al (b) the following values can be assumed to give an approximation of the size of χ s and χ e : K + = (times) M = (variables) N = (levels) 6 (grid points) = 8 L = (members) If an increase by a factor S = 3 of the size of the static part of the control vector is assumed due to the square-root representation of B s dim(χ s ) = SMN = 3 ( 6 ) = 3 9 Assuming that the low-resolution representation of C induces a reduction = both within the vertical and the horizontal it follows that dim(χ e ) = L (N/ ) = ( 8 / ) = 8 In this case the dimension of the whole control vector χ is then dominated by the dimension of χ s As mentioned above the situation would be different in the case where the factor S is less than one 53 Hybrid formulation of Algorithm Following Wang () the hybrid formulation of Algorithm corresponds to the minimization of the cost function with J(δx s w) = γ s δxst B s δx s + γ e wt Ĉ w + ( ot ) T ( ot) δx = MN δx s + D w + d bt = MN δx s + δx e + d bt () This cost function can be minimized with a preconditioned CG algorithm using as a preconditioner the bloc diagonal matrix γ s B s γ e Ĉ In this case because the square root of B s is not needed dim(δx s ) = MN = ( 6 ) = 9 and dim(w) = L(N/ ) = ( 8 / ) = 8 The dimension of the whole control vector is diminished when compared to Algorithm but still dominated by the dimension of δx s Again the situation would be different in the case where the B s form induces a factor S less than one c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

9 4DEnVar: State Formulation of 4D-Var and Possible Implementations 5 54 Hybrid formulation of Algorithm 3 The hybrid formulation of Algorithm 3 simply corresponds to the minimization of the incremental formulation (6) of the 4DEnVar cost but with the substitution of B e by B h This cost function can be minimized with a preconditioned CG algorithm using B h as a preconditioner Each computation of the type B h g will thus require the computation of γ s B s g and γ e B e g followed by the summation of these two contributions In this case a single 4D control vector δx is used with the dimension dim(δx) = (K + )MN = ( 6 ) = 55 Hybrid formulation of Algorithm 4 The hybrid formulation of Algorithm 4 corresponds to the minimization of the previous cost function but in dual space In the iterative algorithm it requires the replacement of the application of B e by B h The size of the control vector δy is simply given by the number P of observations If we assume that P = 7 in a global assimilation system then dim(δy) = P = 7 56 Discussion The previous fully ensemble-based or hybrid formulations are different ways to obtain the solution of the same assimilation problem They all involve the same elements H H T B e B s but applied in a different order None of these algorithms requires the use of the inverse of B e or B s The four classes of algorithms should lead mathematically to the same iterates and then to the same final analysis increment One of the main differences between the different algorithms is the size of the control vector For the first two algorithms the size of the control vector is related to the size of the ensemble and to the dimension necessary to represent the localization correlation assuming that the same localization correlation can be applied to all auto- and cross-covariances for the different variables and times In the third algorithm the control vector size corresponds to the full 4D state but is independent of the ensemble size and does not require a single low-resolution localization correlation to be applied Finally the fourth algorithm relies on a control vector that has simply the size of the 4D observation vector It does not require either a single low-resolution localization correlation to be applied The different iterative algorithms require many scalar products of vectors that all have the size of the control vector Their cost will then depend partly on this size related to the real size application of the assimilation problem Furthermore although all algorithms should lead to the same solution at each iteration round-off errors will inevitably lead to a loss of orthogonality between successive gradients in the application of CG methods with large size problems This problem can be alleviated by a re-orthogonalization of successive gradients Such a re-orthogonalization requires to eep in memory the successive gradients With this respect the last algorithm relying on a control vector with a size reduced to the number of observations can be appealing in a real size problem where the number of observations is generally smaller than the full 4D state within the assimilation window In any case all of the previous 4DEnVar algorithms require the initial input of the ensemble of perturbed bacgrounds x b and the computation and storage of the corresponding perturbations x b and these are significant costs for 4DEnVar Figure True wind field x true (solid blac line) simulated bacground x b (dashed green line) and simulated observations y o (blac circles) all at time t 6 Application to a simplified problem The different 4DEnVar algorithms described in previous sections have been tested in the Burgers toy model and compared to the classical 4D-Var approach 6 The Burgers model The Burgers model describes the evolution of the wind u on a circular domain The evolution of u is given by the nonlinear advection diffusion equation u t + u u s = ν u s () where ν is the diffusion coefficient and s the coordinate of the D periodic domain s πa with a = 637 m In spectral space this partial differential equation leads to an equation of evolution for each spectral coefficient This simple model is an adaptation of the model developed by Liu and abier (3) The integration is performed with a semi-implicit leapfrog scheme initialized by a forward Euler step Three forms of the model are used here: the nonlinear tangentlinear and adjoint models The last two forms are only used for the comparison of the 4DEnVar with the 4D-Var The model is run here with spectral modes which corresponds to N = + grid points and spectral coefficients The true initial condition x true (solid blac line in Figure ) has a simple sinusoidal form ( s x true = Usin a) with U = m s The value of the diffusion coefficient is ν = m With such values a localized sharp gradient is obtained after 7 h of integration and has already started to form at 48 h (solid blac line in Figure ) 6 Comparison of 4D-Var and 4DEnVar In the present comparison between 4D-Var and 4DEnVar assimilation experiments are performed on a single assimilation period This means that in this first set of experiments there is no cycling along successive assimilation periods of bacground-error covariances although this potential evolution of bacgrounderror covariances between successive windows is of course one of the main advantages of the 4DEnVar formulation This aspect will be investigated in future wor either in the framewor of a toy assimilation system or in the real-size assimilations performed at c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

10 6 G Desroziers et al Figure True wind field x true (solid blac line) and simulated bacground x b (dashed green line) at time t f = 48 h Météo-France for the global and limited-area applications The aim of the present experiments is to understand how the 4DEnVar wors and how it compares to 4D-Var inside a single assimilation period The idea of the present experiments is to impose reliance of both 4DEnVar and 4D-Var on the same bacground-error covariance B s at the beginning of the assimilation period and also of course the use of the same bacground trajectory inside this assimilation window A bacground x b is simulated with xb = xtrue + B s/ η b where η b is a random vector with normal distribution The bacground trajectory x b is obtained by the integration of x b with the nonlinear model given by Eq () Matrix B s is built with a Gaussian correlation function and a length-scale L b = 5 m and a uniform error square root σ b = 5ms Similarly simulated observations distributed inside the assimilation period are obtained with y o = H x true + / η o where η o is a random vector with normal distribution H relies on a simple cubic interpolation using the four closest points surrounding observation locations and at the right time of the x true trajectory Matrix relies on an uncorrelated model with an uniform error square root σ o = 5 m s Figure shows the shape of the true signal x true at the initial time t (solid blac line) as well as the simulated bacground x b (dashed green line) and three simulated observations (blac circles) Figure shows xf true and xf b at time t f = 48 h when the sharp wind gradient has started to form in the eastern part of the domain In order to implement the 4DEnVar algorithm an ensemble of L = perturbed bacgrounds x b l at initial time t is built with x b l = xb + Bs/ η b l whereηb l is a random vector with normal distribution The L perturbed bacground trajectories are obtained by the integration of these L states x b l with the nonlinear Burgers model The 4D ensemble perturbations x b l are computed as the differences of these trajectories with the 4D ensemble mean x b as in expression (5) In a first set of experiments the length of the assimilation period is T a = 48 h and both 4D-Var and 4DEnVar formulations are implemented with Algorithm 3 which relies on a CG algorithm preconditioned by the 3D B s bacground-error covariance matrix for the 4D-Var formulation and the 4D ensemble covariance matrix B e for the 4DEnVar case Figure 3 shows the 4D-Var increment (solid blue line) obtained at the initial time t of the assimilation period with three single observations placed at this initial time The increments are consistent with the ratio (σ b /σ o ) = (5/5) = 4 and also with the length-scale of the correlation L b = 5 m specified Figure 3 Square root at time t of bacground error deduced from the ensemble (upper dashed red line) innovations d ot (blac circles) 4D-Var increment (solid blue line) and 4DEnVar increments with L c = 5 m (dashed red line) and L c = 45 m (dotted green line) There are three observations at t (circles) with σ o = 5 m s The length-scale of the bacground-error correlation is L b = 5 m and σ b = 5ms Figure 4 As Figure 3 but for the increments produced at t f = 48 h in matrix B s The bacground-error variance deduced from the ensemble is logically close to the value σ b = 5 m s specified in matrix B s at the beginning of the assimilation period but is slightly varying due to the finite size L = of the ensemble The 4DEnVar increment (dashed red line) obtained with a lengthscale of the localization correlation L c = 5 m= 3 L b is very close to the 4D-Var increment A length-scale of the localization correlation L c = 45 m induces an increment which is noisier far from the three observations (dotted green line) and consistently the 4DEnVar increment obtained with no localization (not shown in the figure) is even less realistic Figure 4 shows the same increments produced by the three single observations at t but at the end of the assimilation period t f = 48 h It appears that the 4D-Var increments have a smaller amplitude and also are broader outside the frontal area (Figure ) On the contrary the increment due to the observation placed in the eastern part of the domain has a larger amplitude and has a slightly smaller horizontal extension The increment due to the observation placed in the middle of the domain is also shifted westerly relative to its position at the initial time This can be understood by the comparison of the structures of the signal at t and t f (Figures and ) The amplitudes of the increments in the different parts of the domain can be directly related to the bacground-error variance deduced from the ensemble which is no more spatially homogeneous at final time; it has slightly c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

11 4DEnVar: State Formulation of 4D-Var and Possible Implementations Figure 5 As Figure 3 but with different sizes of the ensemble: L = (dashed red line) L = (dotted green line which nearly coincides with the solid blue line corresponding to the 4D-Var increment) Figure 6 Decrease of the cost function J for 4D-Var (solid blue line) and 4DEnVar (dashed red line) There are observations at t and t f = 6hThesize of the ensemble is L = decreased outside the frontal zone and on the contrary is much increased in the vicinity of the forming front The 4DEnVar increments at t f obtained with L c = 5 m (dashed red line) are rather close to the 4D-Var increments (solid blue line) for the western and eastern observations but are very bad for the increment due to the observation placed in the middle of the domain This is due to the fact that as mentioned above this increment is shifted to the west between the initial and final times and that the spatial localization applied for the 4DEnVar case is completely cancelling the increment at t f due to the observation at t This is indeed a limitation of the localization process in the 4DEnVar formulation Of course the increment obtained at t f with no localization (not shown in the figure) is able to represent the impact of the observation but at the price of unrealistic oscillations The choice of spatial localization with L c = 45 m allows us to compromise between an increment closer to the 4D-Var increment and not too much noise (dotted green line) 63 Sensitivity to ensemble size The sensitivity of the realism of the 4DEnVar increments to the ensemble size L can also be easily investigated in the Burgers toy problem Consistently eeping the length-scale of the localization correlation constant (L c = 5 m) the 4DEnVar increment becomes relatively poor if the size of the ensemble is reduced to L = (dashed red line in Figure 5) and on the contrary cannot be distinguished from the 4D-Var increment if L becomes as large as (dotted green line) This is also evident on the realism of the bacground-error square-root fields (upper curves in Figure 5) Note that the aim of the experiments conducted here with the simple Burgers model is mainly to chec the validity of the two new algorithms and also to show the expected sensitivity of the 4DEnVar results to parameters such as the ensemble size and localization length The 48 h length of the assimilation period was particularly chosen here to allow a sufficient evolution of bacground-error covariances It is clear that the assimilation period and the results would be different in a real-size problem although published articles on real-size experiments showed that an ensemble size of about already brings a large amount of information on flow-dependent covariances 64 Convergence properties The different minimization algorithms presented in previous sections have been applied to the Burgers toy problem with the 4DEnVar formulation It appears that they give exactly the same Figure 7 Square root of bacground error deduced from the ensemble (upper dashed red line) 4D-Var increment (solid blue line) 4DEnVar increment (dashed red line nearly coincident with the solid blue line) innovations d ot (blac circles) at time t There are observations at t and t f = 6 h The length-scale of the localization correlation is L c = 5 m iterates for such a small problem where the round-off errors are ept small This is also the case for the 4D-Var case It also appears that the convergences of 4DEnVar and 4D-Var are very similar and sufficient with the previous preconditioned algorithms Figure 6 shows such a comparison between the two formulations in a case with a larger number of observations ( observations at t and t f ) and a smaller assimilation window (t f = 6 h) It appears that the decreases of the 4D-Var and 4DEnVar cost functions are very similar and that the same final values are obtained after a limited number of iterations Of course this is a case where the above-mentioned localization problem is less serious due to the limitation of the assimilation window In accordance with the behaviour of the two cost functions the 4D-Var and 4DEnVar increments cannot be distinguished at t (Figure 7) and t f (not shown) 65 Introduction of additive model error in 4DEnVar As explained in section 3 the 4DEnVar formulation formulation offers the possibility to tae into account a large panel of representations of model error Here a simple application of this possibility is investigated with an arbitrary additive representation of model error at the end of the assimilation period Consistently c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)

12 8 G Desroziers et al Figure 8 True wind field x true (solid blac line) simulated bacground x b (thin dashed green line) simulated bacground with additive error x b (thic dashed green line) and simulated observations y o (circles) at time t f Figure 9 Square root of bacground error at time t f deduced from the ensemble without (upper thin dashed red line) and with (upper thic dashed red line) additive model error innovations d ot with model error in the bacground (blac circles) 4D-Var increment without (thin solid blue line) and with (thic solid blue line) model error representation and 4DEnVar increment with model error representation (thic dashed red line) There are three observations at t f only with σ o = 5 m s The length-scale of the bacground-error correlation is L b = 5 m and σ b = 5ms The length-scale of the model error correlation is L q = 5 m and σ q = 5ms the bacground trajectory with model error x b is built with x b = x b except at time t f where x b f = x b f + Q / η q whereη q is a random vector with normal distribution Matrix Q is built with a Gaussian correlation function and a length-scale L q = 5 m and a uniform error square root σ q = 5 m s Figure8 allows us to compare the simulated bacgrounds without (thin dashed green line) and with (thic dashed green line) additive model error at final time t f and Figure 9 shows the geographical variation of σ b without (thin dashed red line) and with (thic dashed red line) the addition of model errors in the ensemble at t f Figure 9 also shows the 4D-Var increment with no model error representation (thin solid blue line) obtained at the final time t f of the assimilation period with three single observations placed at this final time The amplitude of the increment due to the observation placed in the western part of the domain is consistent with the fact that the value of the square root of bacground error is close to the value of σ o at this position (σ o = 5 m s ) On the contrary the fit to the observation placed in the frontal area is much larger because bacground error has increased in this area As in the previous case with observations placed at time t it appears that at time t f the increment in Figure As Figure 9 but for the increments produced at time t the western part of the domain is broader than in the frontal area The increments produced at time t (Figure ) by the three observations placed at time t f show the smoothing property of 4D-Var At time t the amplitude of the increments is changed due to the fact that the square root of bacground error represented by the ensemble is homogeneous (σ b = 5 m s ) The horizontal extension of the increments also becomes homogeneous and consistent with the length-scale L b = 5 m used to build the ensemble perturbations at initial time The introduction of a consistent representation of additive model error in the 4D-Var formulation produces different increments at final time t f Indeed since the amplitude of the variance of the bacground error is augmented over all the domain (upper thic dashed red line in Figure 9) the fit to observations at time t f is larger in the assimilation with a model error representation in 4D-Var It can also be noted that the increment due to the observation placed in the western part of the domain has a reduced horizontal extension consistent with the model error covariance matrix Q with L q = 5 m The smoothing effect of the 4D-Var appears in the increments produced at time t by the observations placed at time t f Interestingly the amplitude of the increments is largely reduced at time t relative to time t f This is consistent with the fact that the model error perturbations added at time t f are here uncorrelated with those of bacground errors at time t The increments produced at time t f and t by the 4DEnVar (thic dashed red line in Figures 9 and ) appear to be similar to the wea-constraint 4D-Var increments (except again for the incorrect increment produced at t in the middle of the domain due to a too fast advection of the signal in this area) This good agreement between 4D-Var and 4DEnVar in the presence of model error is obtained here without introducing a particular time localization of the model error covariances Nevertheless it can be thought that the introduction of a model error representation in 4DEnVar should correspond to the introduction of a specific time localization of covariances in the case where model error is assumed to be uncorrelated in time In the current simple experimental case an explicit temporal localization is possible by avoiding the computation of the covariances between the additive model error perturbations at t f with other perturbations In this simple framewor this brings no additional improvement (not shown) However the good results obtained here with a 4DEnVar formulation with additive model errors in the perturbations and no particular temporal localization of the covariances might be due to the large size of the ensemble compared to the degrees of freedom being sampled Anyhow there is a more general need for developing more sophisticated spatio-temporal localization techniques of ensemble covariances to improve the performance of 4DEnVar The development of c 4 oyal Meteorological Society Q J Meteorol Soc 4: 97 (4)