Implementation of a 3D Variational Data Assimilation System at the ...

Implementation of a 3D Variational Data Assimilation System at the Canadian Meteorological Centre. Part I: The Global Analysis P. Gauthier, C. Charette, L. Fillion, P. Koclas* and S. Laroche Data Assimilation and Satellite Meteorology Division Atmospheric Environment Service 2121 Trans-Canada Hwy. Dorval, P.Q. H9P 1J3

[Original manuscript received 9 September 1998; in revised form 26 November 1998]

abstract In this paper, the operational 3D variational data assimilation system (3D-var) of the Canadian Meteorological Centre (CMC) is described and its performance is compared to that of the previously operational statistical interpolation analysis. Deliberately configuring the 3D-var to be as close as possible to the statistical interpolation system permits an evaluation of the impact of data selection on both the analysis and the resulting forecasts. The current implementation of the 3D-var is incremental in the horizontal and the vertical since the analysis increments are constructed at a lower horizontal resolution on prescribed pressure levels. They are subsequently interpolated vertically to the s levels of the model. The results show that although there could be significant differences in the single analysis increments, the impact on the resulting forecasts is neutral. The 3D-var implements a multivariate covariance model implicitly through changes of variables. It is shown that the implicit covariances resulting from using the linear balance relationship leads to correlations that are not compactly supported. Using a “local” balance relationship has enabled us to avoid this problem when background-error variances vary fully in physical space. There are also indications that the 3D-var analysis increments are better balanced than those of the statistical interpolation. This implies that the 3D-var analysis may not need to undergo a nonlinear normal mode initialization. A brief description is given of the work going on at the Atmospheric Environment Service to improve the 3D-var making possible the direct assimilation of new types of data from a variety of instruments. résumé Cet article décrit un nouveau système variationnel d’assimilation de données, le 3Dvar, maintenant opérationnel au Centre météorologique canadien. Sa performance est comparée à celle du système opérationnel précédent basé sur l’interpolation statistique. L’implémentation actuelle du 3D-var est incrémentale à l’horizontale et à la verticale puisque les *

Canadian Meteorological Centre, Atmospheric Environment Service, Dorval, Québec, Canada.

ATMOSPHERE-OCEAN 37 (2) 1999, 103–156 0705-5900/99/0000-0103$1.25/0 © Canadian Meteorological and Oceanographic Society

104 / P. Gauthier et al. incréments d’analyse sont construits à une résolution horizontale plus basse que celle du modèle et sur des niveaux de pression prescrits. Ils sont ensuite interpolés verticalement aux niveaux s du modèle. En configurant délibérément le 3D-var pour qu’il soit aussi semblable que possible au système d’interpolation statistique, il nous a été possible d’évaluer l’impact de la sélection de données autant sur les analyses que sur les prévisions qui en résultent. Les résultats montrent que même s’il peut exister des différences importantes dans les incréments d’analyse, l’impact sur les prévisions est plutôt neutre. Le 3D-var implémente un modèle de covariance multivarié défini implicitement par l’intermédiaire de changements de variables. On montre que les covariances implicites obtenues en utilisant la relation de balance linéaire conduit à des corrélations qui n’ont pas un support compact. Par contre, en utilisant une relation de balance dite “locale”, il nous a été possible d’éviter ce problème lorsque les variances d’erreur de prévision varient librement dans l’espace physique. Plusieurs éléments nous portent à croire que les incréments d’analyse du 3D-var sont mieux balancés que ceux de l’interpolation statistique. Ceci implique qu’il n’est peut-être pas nécessaire d’appliquer une initialisation aux modes normaux à l’analyse du 3D-var. Une brève description est donnée des travaux ayant cours au Service de l’environnement atmosphérique pour améliorer le 3D-var qui permet une assimilation directe de nouveaux types de données provenant de plusieurs instruments différents.

1 Introduction Data assimilation is an essential part of any numerical weather prediction (NWP) system. Its purpose is to blend information contained in observations originating from different sources with information contained in a prior estimate of the state of the atmosphere referred to as the background state. In practice, current operational assimilation systems use a short term forecast from an NWP model as the background state. The model carries the information gained from past observations forward in time. To assimilate the data properly, forward models are needed for each type of observation to describe the link between the model state and the observations. However, it is just as important to have a good estimate of the accuracy of both the observations and the background state. The quality of the analysis is controlled largely by the covariances of these errors. Since 1993, the main effort of the Data Assimilation and Satellite Meteorology Division of the Atmospheric Environment Service (AES) has been centred on the development of the 3D variational data assimilation system (3D-var) for the operational global spectral model (Ritchie and Beaudoin, 1994) of the Canadian Meteorological Centre (CMC). This 3D-var analysis is similar to those that have been developed at the National Centers for Environmental Prediction (NCEP) (Parrish and Derber, 1992) and the European Centre for Medium-Range Forecasts (ECMWF) (Pailleux, 1990; Courtier et al., 1998). In da Silva et al. (1995) and Cohn et al. (1998), a variant of 3D-var is proposed in which the minimization problem is solved in observation space and the analysis is obtained by transferring the results to model space. In Courtier (1997), it is shown that there is a dual relationship between the two formulations that makes them equivalent. In particular, it is shown in that paper that the two minimization problems have exactly the same conditioning. As

Implementation of a 3D Variational Data Assimilation System / 105

pointed out in Lorenc (1986), the 3D-var is based on the same statistical estimation principles as those of statistical interpolation (also called optimal interpolation (OI)): the two only differ in their way of solving the problem. Statistical interpolation has been used operationally for over twenty years at CMC (Rutherford, 1972; Mitchell et al., 1996, MCCHL96 hereafter). There are several reasons why it was considered necessary to switch to 3D-var, the original motivation being that a variational framework is more suited to the assimilation of observations indirectly linked (even nonlinearly) to the variables of the forecast model. For instance, as shown by Eyre (1989a, b), the direct assimilation of radiance measurements can be achieved once a direct model and its adjoint are available. Andersson et al. (1994) showed that there are clear advantages in assimilating radiances directly instead of the retrieved temperature and humidity proþles. With the advent of an increasing number of new instruments, it has become necessary to have an assimilation system that can assimilate data that are as close as possible to the raw measurements. Another advantage of 3D-var is that it can relax some of the assumptions which are necessary in OI. The most important of these is data selection. In MCCHL96, the maximum number of data used to produce each analyzed value was set to 24 predictors (increased later to 96); implicitly the 3D-var uses all data. As will be shown later, this can create signiþcant differences in the analysis. Finally, the 3D-var makes it possible to consider new types of covariance models. Rabier et al. (1998) have shown that with 3D-var, it is possible to use non-separable correlation models which could be introduced in OI but not as easily and at a higher computational cost. These are beneþcial for the assimilation of winds and temperatures. Bouttier et al. (1997) introduced a representation for the divergence error that has a signiþcant positive impact on the analysis and resulting forecasts. In Lorenc (1997) and Parrish et al. (1997), covariance models formulated in physical space are considered to relax the homogeneity and isotropy assumptions usually made on the correlations. For a þrst implementation, we have deliberately set a more modest goal, to reproduce as closely as possible the results of our OI system. This was considered to be more prudent in that it allowed us to identify some choices in the operational suite that were appropriate for OI but were seen to be detrimental to a 3D-var. It also allowed us to assess the impact of data selection on the analysis. The background-error statistics are therefore similar, but not identical, to those used in the operational OI system (Mitchell et al., 1990, MCCB90 hereafter). The paper is organized as follows. The general formulation of the 3D-var is described in Section 2 and the multivariate background-error covariance representation is explained in Section 3. Section 4 discusses the background-error statistics and presents a few single observation experiments to illustrate some differences that exist between the OI and 3D-var. The nature of the observations used operationally (and their error characteristics) is given in Section 5. In Section 6, the analysis increments that result from using all the data are compared to examine the global response of the two systems: in these experiments both systems use the same background þeld. Finally, in Section 7, the results from several assimilation

106 / P. Gauthier et al.

cycles are presented to study the impact of the analyses on the resulting forecasts. We conclude in Section 8. The variational analysis became operational for the global analysis on 18 June 1997 and was followed on 24 September 1997 by the implementation of a regional variational assimilation for the operational variable resolution model of C^ote et al. (1998). A description of the regional variational assimilation system will be presented in a companion paper (Laroche et al., 1999). 2 Formulation of the 3D-var As presented in Lorenc (1988), the basis of 3D-var is the same as statistical interpolation: to correct a background þeld given a set of M observations both having known error statistics. For Gaussian probability distributions, the best linear unbiased estimate is obtained by þnding the model state X that minimizes the analysis error variance. It can then be shown that this amounts to minimizing the functional J (X) 12 (X with

Xb )T B 1 (X Jb

y)T O 1 (H(X) Jo

Xb ) + 12 (H(X) +

y)

(2.1)

y : observation vector for the \M "observations; X : model state variable (dimension N ), Xb : background þeld (or þrst guess), H : forward interpolation (linear or nonlinear) (RN ! RM ); O : observation-error covariance matrix; B : background-error covariance matrixØ

When H(X) HX is linear, the matrix equation HT O 1 (HX

y) + B 1 (X

Xb ) 0

is obtained by setting the gradient =J to zero and the solution X Xa is Xa Xb + K(y

H(Xb ))Ù

(2.2)

K BHT (O + HBHT ) 1 Ø

(2.3)

with K the gain (or weight) matrix

This establishes the equivalence between OI and 3D-var. Instead of trying to solve the equation =J 0 exactly, the variational analysis proceeds iteratively by using descent algorithms such as conjugate gradient or quasiNewton algorithms (Navon and Legler, 1987; Gilbert and LeMarechal, 1989). What has to be provided are the means to compute J (X) and its gradient. This requires: a. the forward model H and its adjoint, b. a representation of the background-error covariance B,


Fig. 1

Schematic of the sequence of operations included in the forward model.

c. the observation values and their observation-error covariances O. a The forward model and its adjoint The variational analysis is being developed for the Spectral Finite Element (SFE) global model (Ritchie and Beaudoin, 1994) for which the model state can be deþned ~ ), the in terms of the spectral components (noted by tildes) of the streamfunction (Ψ ~) velocity potential (χ~ ) and a number of other scalar þelds such as geopotential (Φ and speciþc humidity (~q). The forward interpolation encompasses all operations required to obtain a model equivalent of the observations. These are: Inverse spectral transform (S 1 ) to obtain the model state on the Gaussian grid at all levels. This also includes a transformation to obtain wind components instead of the streamfunction and velocity potential: this is part of this transform. II Horizontal interpolation to the observation locations (h). This operation yields the model state in vertical columns located at the observation points. At this stage the problem has been reduced to a similar setting as in 1D-var in which the model state is reduced to vertical proþles instead of 3D þelds. III Generalized vertical interpolation (V). For observations that correspond to the model state (e.g., u, v, Φ and q), it sufþces to interpolate vertically to the observation level. For observations that are indirectly related to the model state, a more complex transformation would be needed. For example, the assimilation of radiances would require a radiative transfer model (Eyre, 1989a; Garand et al., 1998).

I

Therefore H VhS 1 and it can be summarized as shown in Fig. 1. These operations have been described in detail in Appendix A of Parrish and Derber (1992). The evaluation of the functional J (X) only requires the forward model but its adjoint is necessary to obtain the gradient required by the descent algorithm used for the minimization. In Talagrand and Courtier (1987), it is shown how the adjoint of these operators can be used to obtain the gradient. The deþnition of adjoint is the following. Deþnition 2.1. If E and F are Euclidean spaces with respective inner products hx Ù y iE and hx Ù y iF , the adjoint L of a linear operator L : E ! F is deþned by the property


hx Ù Ly iF hL x Ù y iE Ø The adjoint of L is deþned with respect to a given inner product that can be chosen arbitrarily to suit the needs of the problem. In the case where hx Ù y iF x T y and hx Ù y iE x T y, it follows that L LT . The following property follows immediately from the deþnition. Property 2.2. If E, F and G are Euclidean spaces, and if L : E ! F, M : F ! G and N : E ! G with N ML, then N L M and N : G ! E. Let X 2 E and y 2 O where E, is the Euclidean space of model states and O, that of the observation. If their respective inner products are hx1 Ù x2 iE xT1 x2 and hy1 y2 iO yT1 y2 , then J (X) 12 hX

Xb Ù B 1 (X

Xb )iE + 12 hH(X)

yÙ O 1 (H(X)

y)iO Ø

(2.4)

The change of J due to a change δX is found to be δJ(X) hδXÙ B 1 (X

Xb )iE + hH0 (X)δXÙ O 1 (H(X) 0

hδXÙ B (X

Xb )iE + hδXÙ H (X) O (H(X)

hδXÙ B 1 (X

Xb ) + H0 (X) O 1 (H(X)

1

1

y)iE

y)iO y)iE

(2.5)

hδXÙ =X Jb + =X Jo iE Ø

where H(X + δX) H(X) (∂HÛ∂X)δX H0 (X)δX is the so-called tangent linear approximation to the forward interpolation. In the present case, since both h and S 1 are linear, property (2.2) implies that

H0 (X) S

1

h V0 (X) H0 (X)T

(2.6)

and only V has to be linearized if necessary. More details on the construction of adjoints are given in Appendix A. b The incremental formulation The resolution of the analysis increments is determined to a great extent by the horizontal correlation lengths used in the background-error covariance model. Referring to Hollingsworth and Lonnberg (1986) and MCCB90, these scales are found to be of the order of a few hundreds of kilometres. These rather large correlation scales result from the þltering of the small scales due to the fact that here, B is representative of the forecast error averaged in time over a period of at least a month and in space over large areas (e.g., North America). This results in analysis increments with a lower resolution than the background þeld that has developed smaller scales. However, an accurate computation of the innovations (y H(Xb )) requires the full resolution of the background þeld. This point will be further discussed in the companion paper by Laroche et al. (1999) in the context of the regional analysis.


In Courtier et al. (1994), an incremental formulation of the variational problem was introduced and applied to both 3D and 4D-var. We outline their derivation here to stress a few points about our implementation of the incremental 3D-var. By writing X Xref + ∆X, the tangent linear approximation of the forward model yields

H(Xref + ∆X) H(Xref ) + H0 (Xref )∆XØ

(2.7)

so that (2.1) can be approximated as J (∆X) 12 (∆X

X0 )T B 1 (∆X +

X0 ) 0 1 2 [H (Xref )∆ X

y0 ]T O 1 [H0 (Xref )∆X

y0 ]

(2Ø8)

where X0 Xb Xref and y0 y H(Xref ). What makes (2.8) interesting is that it has an identical form to the original problem except that the observation value is replaced by y0 , the departure between the observation and the background state, while the background state is replaced by (Xb Xref ). This can be further simpliþed by introducing δx ∆X X0 so that (2.8) reduces to J (δx) 12 δxT B 1 δx + 12 [H0 (Xref )δx

y00 ]T O 1 [H0 (Xref )δx

y00 ]

(2.9)

where

y00 y0

H0 (Xref )X0 y

H(Xb )Ø

(2.10)

The increment is now written as δx [δxTL Ù δxTS ]T , expressed in terms of its large- and small-scale components. Let Π be the projection operator such that δxL Πδx allowing us to rewrite (2.9) as follows: J (δxL ) 12 δxTL B 1 δxL + 12 [H0 (Xref )ΠT δxL

y00 ]T O 1 [H0 (Xref )ΠT δxL

y00 ]Ø

(2.11)

Again, we want to emphasize that the \innovations" y00 are computed with respect to the model state at its full resolution. Usually, one would start with Xref Xb which could be updated through outer iterations in which the tangent linear approximation of the forward models would be deþned with respect to a different point in phase space. In the experiments to be presented later, no outer iterations are needed since all observations used have a linear forward model. In the current implementation of the 3D-var, the model has a horizontal resolution of T199 while a resolution of T108 is used for the analysis increments. For the global analysis, the coarser analysis increments are interpolated spectrally to the high horizontal resolution grid and added to the background þeld to obtain the analysis. In Part II of this paper, it is shown how the incremental formulation makes it possible to compute the innovations with respect to a forecast from a variable resolution model having a high resolution over North America. The increments are

110 / P. Gauthier et al. þrst computed on a global uniform resolution grid identical to the one used here and then horizontally interpolated to the high resolution grid. In the previous operational OI system used at CMC, an incremental analysis has also been used for several years (Mitchell et al., 1993). Since the increments were obtained on a prescribed set of pressure levels, that also made it incremental in the vertical. All observations were preprocessed so that they were located exactly on these levels and no vertical interpolation to the observations was then necessary. The forward model then consisted of þrst vertically interpolating the 6-hr forecast from the σ-levels to the pressure levels and then horizontally interpolating to the observation locations. The net result was an analysis increment deþned on pressure levels that needed to be vertically interpolated back to the σ-levels of the model. This approach has been kept in the current implementation of the 3D-var at CMC in order not to alter the preprocessing of the observations (including quality control). In the next section, the representation of the background-error covariances is discussed. 3 A multivariate formulation of background-error covariances based on a local balance relationship Considering for the moment the model state to be deþned as X (ΨÙ χÙ Φ)T in physical space, the background-error covariances can be represented as 0 1 hΨΨT i hΨχT i hΨΦT i BΨÙχÙΦ @ hχΨT i hχχT i hχΦT i A Ù (3.1) T hΦΨT i hΦχ i hΦΦT i

where h. . .i represents an ensemble average. Following Parrish and Derber (1992), the geopotential is split in a geostrophic component ΦB and an ageostrophic component Φ0 with ΦB being obtained through a balance constraint. For instance, the geostrophic component ΦB can be obtained through the linear balance equation =2 ΦB = f =Ψ

so that the departure from geostrophy of the geopotential is then Φ0 Φ Φ

ΦB ∆ 1 (= f =Ψ) Φ

B ΨÙ

(3.2)

where ∆ 1 is the inverse Laplacian operator. Assuming there is no background-error correlation between Ψ, χ and Φ0 , the covariances are now written as 1 0 BΨΨ 0 0 BΨÙχÙΦ0 @ 0 (3.3) Bχχ 0 AÙ

0

0

BΦ0 Φ0

Implementation of a 3D Variational Data Assimilation System / 111 0

with BΨΨ hΨΨT iÙ Bχχ hχχT i and BΦ0 Φ0 hΦ0 Φ T i. It is through the change of variables 2 3 0 12 3 2 3 I 0 0 Ψ Ψ Ψ 4 χ 5 @ 0 I 0A4 χ 5 M 4 χ 5 (3.4) Φ

B

0

Φ0

I

Φ0

induced by the balance relationship that the covariances are implicitly deþned from (3.3) as 0 1 BΨΨ 0 BΨΨ B T AØ (3.5) BΨÙχÙΦ M BΨÙχÙΦ0 M T @ 0 Bχχ 0

B BΨΨ

0

[B BΨΨ B T + BΦ0 Φ0 ]

The only cross-terms appearing in (3.5) are due to the geostrophic coupling between the geopotential and the streamfunction. In Bouttier et al. (1997), a further coupling is introduced between the divergent component and the streamfunction to describe the balanced component of the divergence. This has not been considered in the present formulation. The covariances BΨÙχÙΦ can be factorized as BΨÙχÙΦ0 DCD with

0

σΨ D@ 0

0

0 σχ 0

0

0

1

0 A

(3.6)

CΨ

0

C@ 0

Cχ

0

0

σΦ0

0

1

0 A

CΦ0

representing respectively the variances and correlations for the analysis variables. a Representation of the variances Referring to Daley (1985, 1991), the background-error variances for Ψ and χ can be related to those of the winds σ2u through the relationships σ2Ψ (1

v 2 )σ2u L2Ψ Ù

σ2χ v 2 σ2u L2χ Ù

(3.7-a) (3.7-b)

where LΨ and Lχ are the local characteristic lengths of the correlations of Ψ and χ respectively. The parameter v 2 stands for the ratio of the divergent component of the kinetic energy to the total kinetic energy of the wind error expressed as v2

σ2χ Ø σ2u L2χ

(3.8)


It can be estimated by using a lagged forecast approach as in Rabier et al. (1998). Being deþned in physical space, this parameter could vary spatially to re ect the increased importance of divergence in the tropics or near the surface. Referring to (3.5), we have hΦΦT i hΦ0 Φ0T i + h(B Ψ)(B Ψ)T iØ

(3.9)

At any particular point, the linear balance is approximated by expressing it on a local f -plane so that B Ψ f (ϕ)Ψ. An expression for the variances is obtained by restricting (3.9) to its diagonal elements in physical space so that at any point (λÙ ϕÙ p), we have σ2Φ (λÙ ϕÙ p) σ2Φ0 (λÙ ϕÙ p) + f 2 (ϕ)σ2Ψ (λÙ ϕÙ p)Ø

In summary, the variances for the analysis variables are given by (3.7) and σ2Φ0 σ2Φ

f 2 L2Ψ (1

v 2 )σ2u Ø

(3.10)

With the Coriolis parameter going to zero at the equator, (3.10) shows that the analysis gradually becomes univariate as we approach the equator. b Non-separable homogeneous and isotropic correlations When background-error correlations are assumed to be homogeneous and isotropic, the correlation C(i Ù j) between two points is expressed as C(i Ù j) f (r Ù ηi Ù ηj ) with r being the horizontal distance between the two points located respectively at levels ηi and ηj . Moreover, if the correlations are also taken to be separable, then f (r Ù ηi Ù ηj ) h(r)V (ηi Ù ηj ). The non-separable representation has been discussed in Bartello and Mitchell (1992) on an f -plane and by Rabier et al. (1998) and Bouttier et al. (1997) on the sphere. On the sphere, the correlations can be expanded as f (r Ù ηi Ù ηj )

N X

f~n (ηi Ù ηj )Yn0 (µ)Ù

(3.11)

n0

where Yn0 (µ) are the zonal spherical harmonics (which reduce to Legendre polynomials). Referring to Boer (1983), Gauthier et al. (1993) and Courtier et al. (1998), one can use the addition theorem for spherical harmonics to show that the correlation between the spectral components (mÙ nÙ ηp ), and (m0 Ù n0 Ù ηq ) is then 0

~ mÙÙm0 hε~ m (ηp )ε~ m00 (ηj )i αm f~n (ηi Ù ηj )δnn0 δmm0 Ù C n nn n

6 0 and α0 1Û2. with ε standing for any of the analysis variables and αm 1 if m Here, δnn0 is the Kronecker delta function. The associated spectral representation ~ is then a block-diagonal matrix, the entries of which do of the correlations C not depend on the zonal wavenumber m (except for a constant when m 0). Consequently, a compact representation of the correlations is

Implementation of a 3D Variational Data Assimilation System / 113 1 T ~ C S 1 C(S )

where S 1 is the inverse spectral transform. Introducing this form into the background term leads to 1

~ (SD 1 x)Ù Jb (x) 12 (SD 1 x)T C

(3.12)

with x δxL X Xb standing here for the analysis variables Ψ, χ, Φ0 and q in physical space. Introducing the new variable j such that x G(j ) DS 1 j , the full functional can now be written 1

~ j + 1 [H0 G(j ) J (j ) 12 j T C 2

y00 ]T O 1 [H0 G(j )

y00 ]Ù

(3.13)

where the notation H0 H0 (Xref )ΠT has been used. One þnal remark should be made to state that the conversion Φ0 ! Φ Φ0 + B Ψ

is also part of G(j ). c Preconditioning of the minimization ~ ϕT L ϕ, the As discussed in Courtier et al. (1998), by using the normal form C variable k L 1Û2 ϕξ further reduces the functional to J (k ) 12 k T k + 12 [H0 G0 (k )

y00 ]T O 1 [H0 G0 (k )

y00 ]Ù

(3.14)

with G0 (k ) G(ϕT L 1Û2 k ). This additional transformation amounts to projecting onto the eigenmodes of the vertical correlation matrices. This form introduces a perfect preconditioning of the minimization with respect to the Hessian of the background term. There is a signiþcant difference in behaviour between (3.13) and (3.14) with respect to convergence even in the presence of a large number of observations which makes the Hessian deviate from the identity. When k is used as the control variable, the minimization requires a number of iterations that corresponds to 35% of the number of iterations necessary when j is used (60 iterations vs. 170). In both cases, the convergence criterion is to have the norm of the gradient reduced by 3 orders of magnitude. This result is surprising given that ~ 1 that is block-diagonal, each block being using j leads to a representation of C only Nk Nk , Nk being the number of vertical levels (16 in the present case). More importantly, when the convergence criterion required a reduction of 6 orders of magnitude (the limit imposed by the numerical accuracy), no signiþcant changes were observed when k is used but some small changes were observed when using j , particularly for extreme values of winds at the jet level. This is a reminder that it is important to verify the impact of using a more stringent convergence criterion for the minimization. These experiments were done after the 3D-var had been implemented and there-

114 / P. Gauthier et al. fore all the results presented here have been obtained by using j as the control variable. The 3D-var managed then to do the analysis in about the same CPU time as the OI and computational efþciency was not an issue. Since then, improving the preconditioning as described above and some work on the optimization of the spectral transforms has brought down the cost of the 3D-var to 30% of that of our previous OI analysis. d Implicit covariances induced by the geostrophic balance The background-error covariances B BΨΨ B T associated with the balanced component of the geopotential are only deþned implicitly through the balance relationship used. When the linear balance relationship (3.2) is used, the constant term is arbitrary and is usually set to zero. On the other hand, the correlations for Ψ are represented using a series of zonal harmonics that includes a component for n 0. The net result of using the linear balance in this manner is to yield correlations that do not vanish at inþnity but instead lead to a plateau of very small amplitude. For a correlation length of 500 km with a Gaussian proþle (exp( r 2 Û2L2 )), the amplitude of the plateau is of the order of 10 3 of the maximum value. One way to circumvent this difþculty is to use the local balance relationship ΦB f (ϕ)Ψ. The associated covariances are then

B BΨΨB T f (ϕp )f (ϕq )σΨ(p)σΨ(q)CΨΨ (jp

qj)

This allows for a complete modulation of the variances in physical space without altering the correlation model introduced for Ψ. A scale analysis shows that the local balance approximates reasonably well the linear balance relationship for length scales corresponding to the characteristic lengths of the streamfunction (1000 km). In response to a single height observation located at a grid point and for which y HXb 1, the analysis increments represent the covariances hΦΦT i implicitly used by the 3D-var. Figure 2 shows a comparison of these increments using the linear and local balance. As expected, to a good approximation, the two formulations are very similar. Magnifying the scale however, the lower panel of Fig. 2 shows that with the linear balance, this single observation has an impact at the antipode. For the case where the variances are constant, the linear balance will create an analysis increment that has a zero global average which is achieved by generating a small plateau of the opposite sign of that of the increment in the neighbourhood of the observation. When there is a bias in the forecast error, the analysis increments are of the opposite sign to that of the bias. Therefore, the plateaus accumulate to create a signiþcant component that has the same sign as the bias. This is illustrated in Fig. 3 showing the zonal average of the geopotential analysis increments in response to the radiosonde network of the Southern Hemisphere only. The impact over the Northern Hemisphere is far from being negligible. 4 Background-error statistics The background-error statistics used in the previous OI analysis were estimated


Fig. 2 Geopotential analysis increment in response to a single observation of Φ located at 45 N. The bottom panel is a magniþed view of the same increment shown on the top panel. The two curves are extended to complete the great circle going through the observation point: these continuation curves are represented with short dashes. Geopotential units (m2 s 2 ) are used on the vertical axis and the latitude is expressed in degrees.


Fig. 3

Zonally averaged geopotential height analysis increments (m) resulting from radiosonde height observations in the southern hemisphere only. Multivariate covariances are based on the linear (sold line) and the local balance (dotted line).

by direct comparison with radiosonde observations. Estimates were obtained for the correlation lengths of geopotential, variances and vertical correlations for wind and geopotential. The reader is referred to MCCB90 for more details on the modelling and estimation of the background-error statistics which were later reestimated (MCCHL96). Here we only very brie y summarize some of their results. a Horizontal correlations For geopotential height, MCCB90 used a third-order autoregressive (TOAR) model F(cÙ r) (1 + α) 1 [f (cÙ r) + αf (cÛN Ù r)]

(4.1)

where

f (cÙ r) (1 + cr + c2 r 2 Û3)e 31Û2 2

cr

Ù

6 3(1 + α) 7 1 Lc 4 α 5 c Ø 1+ 2 N The parameter values α 0Ø 2 and N 3 were determined by þtting wind and height residuals. The parameter c is seen to vary as 1ÛLc , Lc being the correlation length. Under the geostrophy assumption, (4.1) induces the horizontal wind correlation model

F 0 (cÙ r) (1 + αÛN 2 ) 1 [f 0 (cÙ r) + αÛN 2 f 0 (cÛNÙ r)]Ù

(4.2)


Fig. 4 Analysis increments for a) geopotential height (m) and b) zonal wind (m s 1 ) in response to a single geopotential observation when Gaussian (dashed line) and Third Order Autoregressive (solid line) correlations are used.

where f 0 (cÙ r) (1 +cr)e cr is the \normalized logarithmic derivative". The second term of (4.1) is a large scale horizontal component that has a reduced impact on the winds. Figure 4 shows the analysis increments in response to a single height observation when Gaussian and TOAR correlations are used for height. Through the multivariate coupling, the impact on the winds (Fig. 4b) is signiþcantly larger when Gaussian correlations are used. Note also that the TOAR correlations lead to a much broader height increment (Fig. 4a). It is important also to remember that a TOAR model was used by MCCB90 because it led to a better þt to the data. The characteristic lengths LΨ used in the 3D-var were those estimated for the Northern Hemisphere in MCCHL96 based on wind observation residuals. These were taken to vary with height as shown in Fig. 5. This þgure also shows the characteristic lengths used for the horizontal correlations of Φ0 and humidity in terms of the dew-point depression (T Td ). The characteristic lengths of Φ0 are


Fig. 5 Variation of characteristic lengths (km) with pressure for ΨÙ Φ0 and dewpoint depression (T Td ).

based on the estimates of MCCHL96 for the tropics. For these last two variables (dew-point depression and Φ0 ), the analysis is univariate and Gaussian correlations were used because they lead to sharper horizontal increments. b Variances The background-error variances of MCCHL96 resulted in estimates of wind and height variances averaged over three latitudinal bands: extra-tropics in the Northern (Southern) Hemisphere with 30 Ú ϕ Ú 90 ( 90 Ú ϕ Ú 30 ) and the tropics deþned as 30 ϕ 30 . Table 1 summarizes the results as a function of the 16 analysis pressure levels. From these estimates, estimates for σ2Φ0 and σ2Ψ were obtained by imposing the geostrophy constraint such that σ2Φ0 vanishes outside the tropical band. In the extra-tropics, geostropic variances for Φ were obtained from the wind variance estimate using σ2Φ jgeostr. f 2 (ϕ)L2Ψ σ2u . These differ signiþcantly from the independently estimated geopotential variances. To attenuate this difference, an average between the geostrophic estimate of the wind variances σ2u jgeostr. σ2Φ Ûf 2 (ϕ)L2Ψ and the observed wind variance was used to create the geopotential variances. The characteristic length of the Northern Hemisphere has been used. For the tropical band between 0 and 30 N, the variances are taken to vary linearly with latitude as

Implementation of a 3D Variational Data Assimilation System / 119 Table 1. Standard deviation of background-error used for wind vector intensity (m s–1) and geopotential height (m) for the extratropics of the winter and summer hemispheres and for the Tropics. Winter Level (hPa) 10 20 30 50 70 100 150 200 250 300 400 500 700 850 925 1000

Summer

Tropics

σu

σΦ

σu

σΦ

σu

σΦ

7.3 5.2 4.4 3.5 3.6 3.7 4.1 5.1 5.9 6.0 5.2 4.4 3.3 3.2 3.2 3.2

53.6 43.3 37.7 32.1 28.3 24.3 21.0 19.8 19.2 17.5 13.6 10.9 7.6 7.1 6.5 7.2

3.4 2.9 2.5 2.3 2.5 2.7 3.2 4.6 5.7 5.7 4.4 3.5 3.1 2.8 2.7 2.6

28.3 23.2 22.5 21.6 19.5 17.2 15.4 15.8 15.6 14.8 11.2 9.2 6.7 5.9 5.5 6.1

7.3 6.4 6.0 4.5 5.4 6.3 7.5 7.5 7.0 6.7 5.4 4.6 3.6 3.3 3.3 3.3

38.2 32.3 30.1 27.4 24.7 21.7 18.7 16.8 15.3 13.7 10.7 9.0 6.8 6.5 6.5 7.1

σ2uÙΦ (ϕ)

(30 ϕ) 2 ϕ σuÙΦ jTrop + σ2uÙΦ jNorth 30 30

and the ageostrophic component of the geopotential derived from (3.10). These were computed for the Northern Hemisphere in winter and summer. In the Southern Hemisphere, the scarcity of observations introduces much of uncertainty in these statistics. To circumvent this, the global summer (winter) statistics used the winter (summer) statistics of the Northern Hemisphere for the Southern Hemisphere. The variation with latitude and height of the resulting variances is presented for the total geopotential, its ageostrophic component, the wind and dewpoint depression for summer (Fig. 6) and winter (Fig. 7). The analysis increments are therefore forced to satisfy the geostrophic constraint in the extra-tropics. Finally, due to a lack of observations above 300 hPa, the analysis increments of the dew-point depression are forced to vanish above 300 hPa by setting the covariances to zero. c Vertical correlations As discussed in MCCB90, the geopotential vertical correlations are not used in their raw form but þtted to an analytic form. Due to the geostrophic coupling, only these height correlations are needed in the extratropics. There are no signiþcant variations in the geopotential vertical correlations when estimated over different latitudinal bands or from season to season. Therefore, the same vertical correlations were used throughout the year for both the streamfunction and its ageostrophic component. These vertical correlations are shown in Fig. 8 for 30, 100, 250, 500 and 850 hPa.


Fig. 6 Background-error statistics used for summer. Standard deviations of the background-error as a function of latitude and height for a) the geopotential height (m), b) the ageostrophic geopotential height (m), c) the winds (m s 1 ) and d) the dew-point depression (T Td ).

These are broad and lead to increments that have a considerable vertical extent. For instance, an observation at 30 hPa would have an impact that is still signiþcant at the jet level (250 hPa). The semi-separable correlations implicitly deþne sharper horizontal correlations for winds because these are obtained from a horizontal differentiation. The vertical correlations however remain very similar to those of the streamfunction. The opposite occurs for the covariances of the balanced component of temperature. Being obtained through a vertical differentiation, they are seen to be sharper in the vertical but with a horizontal structure that is very similar to those of the streamfunction. It must be said however that the variation with height


Fig. 7 As in Fig. 6 but for winter.

of the horizontal correlation lengths LΨ (p) of the streamfunction induces different horizontal structures for temperature but only at those levels where LΨ (p) varies signiþcantly. Finally, vertical correlations for the dew-point depression (not shown) are very sharp so that the analysis increment produced by one observation is mostly conþned between the levels above and below the observation. d Analysis increments produced from data of one radiosonde The 3D-var covariance model uses non-separable correlations of the form (3.11). As a special case, it can represent the \semi-separable" model of the Canadian OI system. By semi-separable, we mean that the horizontal correlation scale varies with height and can be represented as


Fig. 8 Composite of the vertical correlations used for Ψ and Φ0 at levels 30, 100, 250, 500 and 850 hPa.

h(r; p)

N X

h~ n (p)Yn0 (µ)Ù

(4.3)

n0

where r a cos 1 µ and a stands for the radius of the Earth. The notation h(r; p) is to emphasize that the horizontal correlations vary with height. Introducing f~n (pi Ù pj ) h~ n1Û2 (pi )h~ n1Û2 (pj )V (pi Ù pj ) in (3.11) yields the semi-separable form F(r Ù pi Ù pj )

N X n0

h~ 1nÛ2 (pi )h~ 1nÛ2 (pj )V (pi Ù pj )Yn0 (µ)

(4.4)

Implementation of a 3D Variational Data Assimilation System / 123 which has the property that F(0Ù pi Ù pj ) V (pi Ù pj ) and F(r Ù pi Ù pi ) h(r; pi ). Figure 9a compares the analysis increments obtained with the 3D-var (solid line) and OI (dashed line) in response to a single geopotential height observation. This þgure shows that there are still a few differences arising from the way OI is implemented. When the analyzed value is produced at level pi , the horizontal correlation is taken to be that of this level for all points. This leads to slight differences at the levels where the horizontal lengths scale differs the most from that at level pi . There is an important difference between OI and 3D-var: data selection. To produce Fig. 9a, the data selection algorithm could be deactivated because of the small number of observations involved. Operationally, the search limited itself to two levels above and below and the total number of predictors was limited to 96, selected as the most representative. To show the impact of data selection, we repeated the experiment with the limited vertical search used in the OI system. Consequently, an observation can only in uence 5 levels thereby leading to effective covariances that have narrower vertical correlations than those speciþed. The OI increment when data selection is activated is compared in Fig. 9b to the analysis increment of the 3D-var (identical to that of Fig. 9a). Finally, all geopotential observations from a single radiosonde located at Sable Island (44 N, 60 W) were assimilated and Fig. 10 compares the OI and 3D-var geopotential analysis increments. The two systems exhibit some differences especially in the upper levels. In the OI, the horizontal data selection is limited within a search radius of 2000 km. With only one radiosonde, this causes the OI increment to vanish beyond this distance. However, the horizontal extent of the 3D-var increment remains signiþcant at distances beyond 2000 km at 10 and 20 hPa where the horizontal correlation length is larger than 1000 km. As will be seen later, the bias of the forecast error being more important at levels above 100 hPa, the innovations are bigger and this is spread vertically according to the vertical correlations (see Fig. 8) to create important differences even at the level of the tropopause (250 hPa). In the OI, since the in uence of an observation is limited in the vertical by data selection, this conþnes the impact of these large innovations to the upper levels. In the next section, the nature of the observations used at the CMC is described together with their error characteristics. 5 Observations The observations used in the 3D-var are in exactly the same form as in the OI. This means that they have all gone through several quality control procedures (including a statistical interpolation check) and all data have been interpolated to the analysis levels. No changes have been made to the observational errors except for the SATEM data (temperature soundings retrieved from the TIROS-N Operational Vertical Sounding instrument commonly known as TOVS). In this last instance, the variances had to be reduced in the OI system to augment their impact on the analysis (MCCHL96). In the 3D-var, all available data are used and these


Fig. 9 Geopotential analysis increments in response to a single geopotential observation located at 250 hPa. For the OI, the results have been obtained a) without data selection, b) with data selection restricting the vertical search to 2 levels above and below the analysis level. Solid and dashed lines depict the 3D-var and OI increments respectively.


Fig. 10 Analysis increments obtained in response to geopotential observations from a complete radiosonde (data at all 16 levels) located at Sable Island (44 N, 60 W). Solid and dashed lines depict the 3D-var and OI increments respectively. The OI increments were obtained using the same data selection as in Fig. 9b.

reduced variances resulted in analyses that were drawing too strongly in uenced by the SATEMs thereby increasing the departures from radiosonde data. Reverting to the original estimates corrected the problem. This is consistent with observations made by Andersson et al. (1998) who also report that in the 3D-var, the adjustment of the observation-errors is less important and the original estimates are retained. The observation-error characteristics are summarized in Appendix B. Horizontal correlations of observational errors were taken into account in the OI for SATEM and HUMSAT data (Garand and Halle, 1997). In the 3D-var, these horizontal correlations were neglected for two reasons. Firstly, as noted in MCCHL96, these have a negligible impact for SATEMs when they are assimilated at their current resolution of 500 km. Secondly, for HUMSAT data, they can be divided into two categories according to whether they are within a dry or moist area and there is no error correlation between the two groups. Consequently, in the 3D-var, each category was independently thinned to a 300-km resolution which made them less horizontally correlated. Their combination, however, resulted in data often being 100 km apart in regions of sharp gradients in the humidity þeld. This led to humidity analyses having sharper gradients, a feature that was noted as being positive by the operational forecasters during the parallel run.


Vertical correlations are necessary for the SATEM thicknesses and radiosonde height data and were included in the 3D-var. The vertical correlations are those used in the OI system. When introduced as is, these produced changes to the analysis increments which were small in amplitude but had vertical proþles which exhibited short-scale variations. This was related to the eigenvectors of O associated with small eigenvalues. To understand the problem, the O matrix is diagonalized so that the Jo term can be rewritten as L1 1 0 Jo (x) 12 (Hx y)T ET E(Hx y) (5.1) 1L 1 0 ε 2 where E is the matrix having the eigenvectors of O as rows. To emphasize the different order of magnitudes between the two groups of eigenvalues, we have factorized out the scale ε 1 of the smallest ones so that L1 and L2 are then of the same order of magnitude. Consequently, Jo is expressed as Jo (x)

1 (Hx 2

y)T P1T L1 1 P1 (Hx

y) +

1 (Hx 2ε

y)T P2T L2 1 P2 (Hx

y)

(5.2)

with P1 and P2 standing for the projectors onto the eigenmodes associated with L1 and L2 respectively. The second term in (5.2) acts as a penalty term which requires that the components along the eigenvectors with small eigenvalues be set to zero. If O were a perfect estimate of the observation error, small eigenvalues mean that we have very accurate observations that should be þtted very closely. However, one must remember that O has been estimated and consequently, there is some uncertainty about the estimate itself. These components with small eigenvalues may be indicative of the fact that it was not possible to estimate them because these components are difþcult to observe. If one retains only those well observed components, one would be left only with the þrst term on the right-hand-side of (5.2) which means that we should be using the generalized inverse of O deþned as L1 1 0 1 T EØ (5.3) O E 0 0 Figure 11 shows the height analysis increments and their difference when O is used as is and when O is þltered by neglecting all eigenvalues below 0.2 (dashed line). Filtering the correlations resulted in slight differences in the increments. In general, þltering the small eigenvalues tends to smooth the increment in the vertical. The corresponding increment for OI (not shown) is closer to the increment obtained with þltered correlations even though the unþltered correlations were used. This was related to the approximate way the problem is solved within OI. When more accuracy was requested in the solution of the problem, the OI increment converged towards the 3D-var increment obtained with unþltered correlations. Figure 12 de-


Fig. 11 Impact on the analysis increments of þltering the eigenmodes associated with small eigenvalues (Ú 0Ø2) of the vertical correlation of radiosonde observations (RAOB) geopotential height error. a) Vertical proþles of geopotential height increments at the station's location, b) difference between these two increments. The data used are all geopotential data from a radiosonde located at 50 N. In a), the solid (dashed) line corresponds to the increment obtained by using unþltered (þltered) correlations.


Fig. 12 Representation of the vertical correlations of RAOB geopotential height error when all eigenvectors associated with eigenvalues lower than 0.2 are þltered out (dashed line). This retains over 95% of the total \variance". The solid line corresponds to the unþltered correlations.

picts the vertical correlation before and after the application of the þlter: it shows that the þltering does not signiþcantly change the shape of the correlations and that these changes are well within the estimation error of these correlations. This is consistent with the fact that the þltering retains over 95% of the total variance of the correlation matrix. Finally, we would like to point out that even for radiosonde height data, the correlation matrix O 1 may differ from one sounding to another because of missing level data. This would require one matrix inversion per radiosonde and per iteration of the 3D-var. On the other hand, if OF stands for the full correlation matrix

Implementation of a 3D Variational Data Assimilation System / 129 deþned for the 16 prescribed levels and if h stands for the vertical interpolation to those levels where data are available, then O hOF hT . The computation of w O 1 (Hx y) is then obtained by minimizing J (w) 12 wT (hOF hT )w

wT (Hx

y)

(5.4)

with a conjugate gradient algorithm (Gollub and van Loan, 1983) which converges rapidly. The costly part of the computation is the matrix multiplication of hOF hT which is achieved in three steps:

I Obtain full proþles through the application of hT ; II Multiply by OF ; III Interpolate the results to those levels having data. This procedure in three steps can be applied to all proþles and since the multiplication by OF is done on identical vectors, a substantial gain in efþciency is obtained on vector computers at the matrix multiplication step. In the next section, the results of the evaluation of the parallel run will be presented. 6 Single analysis using all the operational data In preparation for its implementation, the 3D-var went through several comparative assimilation experiments against the previously-operational OI system. Table 2 summarizes the characteristics of the two systems. The data selection used in the OI analysis is limited to 96 predictors: to produce the analyzed value at a given point, data are selected by keeping the highest correlated observations to analyze one variable. A consequence of this is that the analysis of geopotential and winds may not be obtained from exactly the same set of observations. The experiments presented in this section use all the operational data to examine the global response of the two systems to a complete set of observations with the same background þeld. In the next section, the results of assimilation cycles will be presented to study the impact of the analyses on the resulting forecasts. By using the same set of observations and the same background þeld, the OI and 3D-var see the same innovation values because all observations are linearly related to the model variables. Differences in the global analysis increments provide information on how the two systems differ in the way they combine the observations. Referring to Table 2, one of the important differences between OI and 3D-var is the fact that the OI analysis allows the characteristic lengths of the backgrounderror correlation to vary latitudinally. In practice, the variation is small except for the tropical regions where the correlation lengths for geopotential are signiþcantly larger in the troposphere (800 km in the tropics and 400 km in the extratropics). This particular aspect is taken into account in the 3D-var through the speciþcation of different background-error statistics for Φ0 that totally describe the geopotential statistics at the equator. The broadening of the increments due to the

130 / P. Gauthier et al. Table 2. Characteristics of OI vs. 3D-var In both systems, the analysis increments are constructed on the same 16 pressure levels: 10, 20, 30, 50, 70, 100, 150, 200, 250, 300, 400, 500, 700, 850, 925 and 1000 hPa. OI

3D-var

Data selection

96 predictors, 3D data selection limited to 2 levels above and below the analysis level

No data selection

Characteristic lengths for horizontal correlations

Background-error correlations with latitudinally varying characteristic length

Background-error correlations with uniform characteristic length

Horizontal resolution of the analysis

Local increments produced in physical Spectral increments produced at T108 space on a 240 × 120 Gaussian grid. and transformed to a 240 × 120 Gaussian grid.

Geostrophic coupling in the extratropics

Geostrophic variances at mid-latitudes Geostrophic variances at mid-latitudes (local f-plane approximation is used) (local balance)

Geostrophic coupling in the tropics (–30°S < ϕ < 30°N)

Geopotential analysis gradually becomes univariate as the equator is approached.

Wind increments

Purely rotational wind increments are Purely rotational wind increments take derived by assuming f to be locally into account the variation of f with constant. latitude (local balance).

Background-error correlation representation

‘‘Semi-separables'' correlations are defined as C(r,Zi,Zj) = h(r;Zi)V(Zi,Zj) where Zi is the analysis level.

Geopotential analysis gradually becomes univariate as the equator is approached.

‘‘Semi-separables'' correlations are defined as C(r,Zi,Zj) = h(r;Zi)1/2h(r;Zj)1/2V(Zi,Zj).

increased characteristic lengths is illustrated in Fig. 13 which shows a composite of the meridional cross-section of geopotential analysis increments in response to single observations located at 0 and 52 N respectively, both observations being at the 500-hPa level. The background-error covariance model has been deþned so that the wind increments are purely rotational everywhere and geostrophically balanced in the extratropics. Rotational wind increments imply that the zonal average of the meridional wind component should vanish identically. This is exactly what the 3D-var produces to numerical accuracy but not the OI as shown in Fig. 14. Although small, it is important to remember that this is a spurious component resulting from the approximate way the problem is solved in the OI due mostly to either data selection or the local f -plane approximation. As was the case for the OI, the CMC's 3D-var produces its wind and geopotential increments on pressure levels: these need to be interpolated to the model's σ levels, where σ pÛps , before being added to the background þeld to obtain the analysis. This vertical interpolation is detrimental in that it destroys some of


Fig. 13 Composite of two analysis increments of geopotential height obtained with the 3D-var in response to a single geopotential height observation located at 0 and 52 N respectively, both being at 500 hPa. Both increments are produced form an innovation value (y Hxb ) of 10 m.

the constraints imposed by the analysis through the covariance model. For instance, the 3D-var wind increments were purely rotational in pressure but not after this vertical interpolation. This is illustrated in Fig. 15 which shows that after the vertical interpolation, the zonal average of the meridional wind component increment of the 3D-var does not vanish anymore. The covariance model also implies that the analysis increments should be locally geostrophic in the extratropics. The local balance used by both systems is an approximation to the linear balance which is itself an approximation to the nonlinear balance as deþned in nonlinear normal mode initialization (NNMI). The latter usually being performed under adiabatic conditions, it also differs from the true balance as the model (and the atmosphere) sees it. Figure 16 shows the geopotential and wind increments of the OI and 3D-var at 50 hPa over a region of dense data coverage. It illustrates that the OI increments are not always geostrophic. The reason for this is that in regions where data are plentiful and characteristic lengths are broad, the data selection algorithm results in signiþcantly different sets of data being used for the geopotential and wind analyses. The analysis increments are therefore not constructed from the same data and situations, such as those shown in Fig. 16, can be found quite often in the OI at upper levels where the characteristic lengths are broad. A close examination of the OI wind increments reveals that they follow high pressure systems that can only be seen in the 3D-var increment. The latter starts to show some ageostrophy only as we get closer to the equator where the geostrophy constraint is gradually relaxed. Parrish and Derber (1992) had observed that their 3D-var analyses were better balanced than their OI counterparts. Their analysis increments are produced directly on the levels of the model just as in the 3D-var of ECMWF (Courtier et al., 1998). In our case, the increments are vertically interpolated to obtain ∆Xa (σ), the analysis increments on the σ-levels. Temperature and surface pressure increments


Fig. 14 Zonal mean of the meridional wind component of the OI analysis increments on pressure levels.

Fig. 15 Zonal mean of the meridional wind component of the 3D-var analysis increments after the vertical interpolation to the model's σ levels.


Fig. 16 Geopotential height and wind vector analysis increments at 50 hPa from a) OI, b) 3D-var.

134 / P. Gauthier et al. are obtained from those of geopotential using the hydrostatic relationship and the fact that at the surface, the geopotential is constant and prescribed. The increments are then added to the background þeld Xb (σ) to obtain the analysis Xa (σ). If N stands for the normal mode initialization, the initialized increment is deþned as

∆1 X(σ) N (Xa (σ)) N (Xb (σ)) ∂N (Xb (σ)) ∆Xa (σ) ∂X N 0 (Xb (σ))∆Xa (σ)

(6Ø 1)

where N 0 stands for the tangent linear approximation to the NNMI valid in the neighbourhood of Xb . The changes due to the initialization of the 3D-var and OI analyses are depicted in Fig. 17 which shows the initialization increments ∆I X(σ) ∆Xa (σ) for the zonal wind component at the level σ 0Ø245. The 3D-var analysis increments are clearly less altered by the initialization even though both went through the vertical interpolation process which upsets the balance imposed via the covariance model. To answer the question as to whether or not the 3D-var analyses could do without NNMI, forecasts were made from the OI and 3D-var analyses with and without NNMI. As a complement, the initialized increment as deþned by (6.1) was added to the background þeld to assess the impact of initializing only the increment instead of the complete analyzed þelds. Figure 18 shows time series of surface pressure over a point located west of Hudson's Bay at (53 N, 93 W). The results clearly show that the 3D-var analysis is better balanced than that of the OI. This is also observed in similar time series that were examined at several other points. The initialization of the increments does even better at þltering out spurious gravity wave oscillations than the full initialization scheme. This is true both for the 3D-var and the OI. Ideally, with the current 3D-var system, it would have been appropriate to initialize only the analysis increments. However, as actually implemented, the full analysis undergoes NNMI just as before. To change this in the operational implementation would have required a more thorough evaluation than has been done here. It was decided to postpone this study to the next version of the 3D-var in which analysis increments will be obtained directly on the model's levels. In the next section, the impact of the 3D-var analysis on the forecast will be discussed. 7 Impact of the 3D-var global analysis in the CMC operational data assimilation cycle The CMC global model is a semi-Lagrangian spectral model using σ pÛps as a vertical coordinate. As described in Ritchie and Beaudoin (1994), it has 21 vertical σ-levels but since then, the horizontal resolution has been increased from T119 to T199 (100 km). The global analysis is obtained from a data assimilation cycle that comprises the following sequence of operations:


Fig. 17 Changes to the zonal wind analysis increments at σ 0Ø245 produced by the nonlinear normal mode initialization. a) for the OI analysis, b) for the 3D-var analysis. Contour intervals of 0.2 m s 1 have been used in both cases. The increments vary from 1Ø1 to 1.3 m s 1 in the case of the OI analysis and from 1Ø1 to 0.8 m s 1 for the 3D-var analysis.


Fig. 18 Time series of surface pressure over a 24-hr period. a) 3D-var analysis, b) OI analysis. The impact of initializing the complete analysis is compared with that of initializing only the analysis increment. The trace is for a point located at (53 N, 93 W).

i.

Integrate the model from the previous initialized analysis for 6-h to produce the background þeld Xb on σ levels. ii. From the high resolution forecast, compute the innovations (y HXb ). In the current system, all observations are preprocessed to be exactly on 16 prescribed pressure levels. The innovations are computed by þrst performing a vertical interpolation of the forecast from σ to pressure levels followed by a horizontal interpolation to the observations' locations to obtain vertical proþles of model states. The observation operators of the 3D-var are then applied to obtain the innovations. Quality control is also performed to identify erroneous data. iii. The incremental 3D-var (based on (2.9)) is used to produce the analysis incre-


ments ∆Xa (p) at a lower horizontal resolution (T108) on prescribed analysis pressure levels. iv. The analysis increments ∆Xa (p) are interpolated horizontally to the higher resolution grid. This step is done spectrally. v. A vertical interpolation of ∆Xa (p) to the σ-levels of the model is done and the resulting increment ∆xa (σ) is added to the background þeld Xb (σ) to obtain the analysis Xa (σ) Xb (σ) + ∆xa (σ). The hydrostatic relationship and the fact that ∆Φ(σ 1) 0 are used to obtain the analysis increments for surface pressure and temperature. vi. Nonlinear normal mode initialization is applied to the analysis to þlter out spurious gravity waves (XI (σ) N (Xa (σ))). The OI procedure differs from that of the 3D-var mostly at steps iii and iv. Slight differences also exist in the way each system computes the innovations because the 3D-var uses a linear horizontal interpolation while the OI uses a cubic interpolation. Since these are computed from the high resolution grid, this difference in interpolation does not lead to signiþcant differences. To test the system, several assimilation cycles were run over a period of two weeks to a month during January 1996 and August 1996. These experiments were done under different conditions to compare the reaction of the 3D-var with that of the OI. It was one of these experiments that revealed the bias problem caused by using the linear balance. The problem appeared in one of the diagnostics used to judge the quality of the analyses and forecasts: comparison against radiosonde observations over particular regions and time periods to obtain estimates of the r.m.s. error and the bias. Figure 19 shows the bias for geopotential analyses and forecasts over the Southern Hemisphere (30 {90 S). An important bias appears in the 6-h forecasts of the 3D-var but is seen to be substantially smaller in the analyses. We have seen in Fig. 3 that the linear balance can create small plateaus that accumulate to create signiþcant increments over regions where there are no observations. The veriþcation of the analysis could not detect this because there are no observations in these regions. Where the analysis can be veriþed, the veriþcation uses observations that were included in the analysis. To detect the problem in the analysis would have required veriþcation against independent observations. This error over data voids propagates to regions where observations are present, which explains why the bias problem appears in the forecast but not in the analysis. This problem completely disappeared when the local balance formulation of the multivariate analysis was introduced. This rendered the error level of the 3D-var system comparable to that of the OI system. Experiments were conducted to make the þnal adjustments to the 3D-var system. For example, it was beneþcial to change the horizontal correlation model from the Gaussian used at þrst, to the TOAR correlations described in Section 4. As shown in Fig. 4, when Gaussian correlations are used, geopotential observations produce more important wind increments at some distance from the observation's location.


Fig. 19 Mean error (bias) made on the analysis (}) and 6-h forecast ( ) for OI (solid lines) and 3Dvar using the linear balance (dashed lines). These estimates were obtained by verifying with radiosonde data and are based on 12 winter cases taken from an assimilation cycle started on 16 January 1996.

In regions where there is good data coverage, this results in a broadening of the wind increments which tends to smooth the gradients of the wind increments. Using the TOAR correlations resulted in stronger gradients which translated into a slight improvement in the 6-h forecasts. The analysis and 6-h forecast error estimates were calculated for the period 23 April to 18 June 1997. This includes 112 cases taken from the parallel run preceding the implementation of the 3D-var (on 18 June 1997). These error statistics have been computed for the Northern Hemisphere (Fig. 20), Southern Hemisphere (Fig. 21) and the Tropics (Fig. 22). They compare the 3D-var (dashed curve) against the then operational OI system (solid line). Over the Northern Hemisphere, the r.m.s. errors for winds are comparable for both systems but those of the geopotential are slightly worse for the 3D-var, particularly for the analysis above 300 hPa. This could be explained by two things. Firstly, data selection in the OI þlters the data by choosing those that are the best correlated with the analyzed variable. This tends


Fig. 20 Veriþcation against radiosonde data of analyses and 6-h forecast errors over the extratropical northern hemisphere (30 N{90 N) for the period 23 April to 18 June 1997 (112 cases). The left (right) panels show the r.m.s. error (bias) for the zonal wind component (top) and geopotential (bottom) as a function of pressure (expressed in hPa). The analysis and 6-h forecasts for OI and 3D-var are distinguished by the following symbols: analysis (}), 6-h forecast ( ), OI (solid line) and 3D-var based on local balance (dashed line).


Fig. 21

As in Fig. 20 but for the extratropical southern hemisphere (30 S{90 S).


Fig. 22 As in Fig. 20 but for the Tropics (30 S{30 N). Note that the scale for the bias error of u has been increased by a factor of 4 due to the large bias in this region.

142 / P. Gauthier et al. to retain data closer to the analysis point making the analysis more closely þt the data used for the veriþcation. Moreover, the increasing characteristic lengths at these levels result in observations further away having more impact in the 3D-var on a given analysis point which deviates more from the local data. Secondly, as discussed in Section 6, there are good indications that the 3D-var analysis is better balanced than that of the OI which can locally þt the observations more closely by not imposing the balance constraint as stringently as the 3D-var does. However, after the NNMI, the OI analysis exhibits r.m.s. errors that are comparable to those of 3D-var (results not shown). In the Southern Hemisphere, there are less radiosonde data available for the veriþcation and at 10 hPa, these were not found to be reliable. Consequently, Fig. 21 shows the error only up to 20 hPa. The 3D-var analyses are better but this advantage is not retained in the resulting 6-h forecasts. It is also interesting to notice that the bias in geopotential forecasts is similar to the one that arose when the linear balance is used (see Fig. 19). The results shown then were obtained from a short assimilation cycle comprising only 12 cases. Figure 21 shows that over a longer period, the bias in the forecast error is genuine and is observed by both systems. Finally, the results in the Tropics (Fig. 22) reveal that there is a signiþcant bias in the wind error which contributes to the r.m.s. error as much as the variance. This could be explained by the low vertical resolution of the model which cannot resolve the sharp transition from easterly to westerly winds occurring between 10 and 50 hPa associated with the quasi-biennial oscillation (Andrews et al., 1987). This region is well within the sponge layer of the model (Ritchie and Beaudoin, 1994). For all three regions, the r.m.s. errors of our 3D-var analyses and 6-h forecasts compare with those of ECMWF (Fig. 7 of Andersson et al., 1998). In 3D-var, the impact on the 5-day forecast was found to be mostly neutral over all regions and at all levels. Based on a comparison with radiosonde data, Fig. 23 shows time series of r.m.s. forecast errors of wind vector intensity (UV) and geopotential height (GZ) at 250, 500 and 850 hPa over North America for a series of forecasts based on analyses from an assimilation cycle of two weeks beginning on 14 August 1996. Overall, the maximum difference is less than 0.5 m s 1 for winds and 5 m for height. The beneþt of the 3D-var is not considered to be statistically signiþcant because the results can vary from one assimilation cycle to another. Figure 24 shows similar results but for a period of two weeks starting on 16 January 1996. In this case, the OI has a slight advantage over the 3D-var but again these differences cannot be considered signiþcant. Finally, Figs 25 and 26 show the corresponding results over the Southern Hemisphere for the same periods. In this case, the 3Dvar leads to slightly positive results for the two periods, the August case having a stronger signal. In the Southern Hemisphere, radiosonde coverage is sparse and the analysis relies more on SATEM data. These results may be an indication that they are better assimilated in the 3D-var. However, the number of cases is too low to make a þrm statement.


Fig. 23 Time series of r.m.s. error of the wind vector (UV, left panels) and geopotential (GZ, right panels) of 5-day forecasts at 250, 500 and 850 hPa over North America. These forecasts were initiated from analyses of a two-week assimilation cycle starting on 14 August 1996: a total of 25 cases is included. As before, the dashed line is for the 3D-var and the solid line for the OI. The error estimates were obtained by comparing against radiosonde data. Notice that the range of values used on the vertical axis varies with level in order to make the differences perceptible.


Fig. 24 As in Fig. 23 but the forecasts are based on analyses from a two-week assimilation cycle starting on 16 January 1996. A total of 25 cases is included. The range of values used on the vertical axis varies with level in order to make the differences perceptible.


Fig. 25 As for Fig. 23 but for the southern hemisphere.


Fig. 26 As for Fig. 24 but for the southern hemisphere.


The overall conclusion to be drawn is that under the same hypotheses, OI and 3D-var yield very similar forecasts. The r.m.s. error statistics have about the same level of error for both systems. The only change that had to be made to the observational error statistics was to reset the SATEM errors from the lowered values used in the OI to their estimates provided by the National Environmental Satellite, Data and Information Service (NESDIS). As was discussed earlier, the most signiþcant difference between OI and 3D-var is the data selection mechanism which creates a different response when only a few observations are used. In the current context, these differences only seem to have a marginal impact on the resulting forecasts. 8 Discussion and conclusions In this paper, the operational 3D-var analysis of the CMC has been described and its performance has been compared to that of the previously-operational OI analysis (MCCHL96). Deliberately conþguring the 3D-var to be as close as possible to the OI system has permitted an evaluation of the impact of data selection on both the analysis and the resulting forecasts. Our results have shown that although there can be signiþcant differences in single analysis increments, the impact on the forecasts is neutral. The merit of this system is that whatever can be done in the OI system could be reproduced in 3D-var. In Section 3, it was shown that the 3D-var implements a multivariate covariance model implicitly through changes of variables similar to those use in Parrish and Derber (1992). It has been shown that the implicit covariances resulting from using the linear balance relationship yield correlations that are not compactly supported. Using a \local" balance relationship resolves this problem when background-error variances vary fully in physical space. In Section 4, it was shown that the 3D-var analysis increments are better balanced than those of the OI and that the 3D-var would likely be better off by not applying NNMI at all or applying it only to the analysis increments. This has not been included in our current implementation of the 3D-var but will be in the next version. The purpose of developing the variational analysis was certainly not to have a system giving similar results to the previous system but was rather to introduce an analysis system that will permit improvements in the short and long terms. In terms of improvements to the analysis itself, work is being done to implement a new model of the background-error statistics similar to that described in Bouttier et al. (1997) by introducing non-separable error correlations and a divergent component in the wind analysis increments. Preliminary results of this work are presented in Gauthier et al. (1998) in which the analysis increments are also produced directly on the model's own vertical coordinate. That study is based on forecasts obtained from the Global Environmental Multiscale (GEM) model of C^ote et al. (1998), a variable resolution model used operationally to produce 48-h regional forecasts. The companion paper by Laroche et al. (1999) describes how the incremental formulation of the 3D-var has permitted it to be used in an intermittent regional data assimilation cycle. As a proposed replacement for the global spectral model,

148 / P. Gauthier et al. GEM is currently being tested in its uniform resolution conþguration in global data assimilation cycles using the 3D-var as described here. Work is also being carried out on the variational assimilation of satellite data. Experiments have been conducted by Deblonde (1999) on the assimilation of measurements of total precipitable water from a Special Sensor Microwave Imager (SSM/I) using the 3D-var in a similar conþguration to that presented here. Work has also been in progress over the last few years on the assimilation of cloudcleared TOVS radiances (Chouinard and Halle, 1997). In Garand et al. (1999), the direct assimilation of radiances has been done by using a physically based radiative transfer model as a forward model to relate temperature and humidity to observed radiance data from the TOVS instrument. However, the experiments on the assimilation of satellite data have led us to conclude that these data would have a more signiþcant impact if the temperature and speciþc humidity were used as analysis variables instead of geopotential and dewpoint depression. For this reason, it was thought best to defer the assimilation of both SSM/I and TOVS data in the operational system until after the implementation of a 3D-var based on new statistics deþned in the model's own vertical coordinate. 9 Acknowledgments The þrst author would like to thank Philippe Courtier for many discussions that occurred over the years on the subject of variational data assimilation. All of us beneþted from his theoretical and practical input during his stay with us in July 1996. Because the 3D-var is built on the knowledge contained in the previous OI analysis system, we would like to acknowledge the contribution of all who worked on its development and in particular that of H.L. Mitchell who led this group for many years. We thank him also for his careful review of our manuscript which contributed to the improvement of the þnal version. Real Sarrazin from the CMC did the experiments on the impact of nonlinear normal mode initialization reported in Section 6. We would also like to thank Erik Andersson and two anonymous reviewers whose comments helped to improve the þnal version of this paper. The CMC operational 3D-var uses the M1QN3 minimization code provided to us by Jean-Charles Gilbert of the Institut National de Recherche en Informatique et en Automatique (INRIA). Appendix A Forward models and their adjoints Over the last ten years, the derivation of adjoint operators has been the subject of numerous papers. The purpose of this appendix is to give some details about the forward model used in our implementation of 3D-var and to mention a few points on how the adjoint of some of its key elements was obtained. The reader is referred to Talagrand and Courtier (1987) and Parrish and Derber (1992) for a more detailed discussion on the derivation of the forward model and its adjoint. a Forward model Referring to Sections 2 and 3, the forward model establishes the link between the


control variable k used in the variational analysis and the observations. The control variable is related to components along the eigenvectors of the correlation matrix C which, for homogeneous and isotropic correlation functions, is more conveniently represented in spectral space. To relate these components to the actual atmospheric variables, the sequence of operations is as follows: I

Transform from the eigenvector space of C to normalized departures in spectral space: 0 ~ 1 δΨÛσΨ B δχ~ Ûσχ C 1Û2 C δx~ B @ ~ 0 Ûσ 0 A ϕΛ k δΦ Φ δq~ Ûσq

II Perform an inverse spectral transform to obtain the normal departures in physical space and then multiply through by the background-error standard deviation expressed in terms of the analysis variables Ψ, χ, Φ0 and q: 0 1 δΨ B δχ C 1 C (A.1) δx B @ 0 A DS δx~ Ø δΦ δq III Convert δΦ0 to geopotential by adding the ageostrophic and geostrophic components. 0 δΨ 1 0 1 δΨ B δχ C B C δχ C B C δx2 B @ δΦ A @ δΦ f δΨ + δΦ0 A M δxØ δq δq

IV Go from a (ΨÙ χ) to a (uÙ v) representation by using the relationships: p 1 1 µ2 ∂ ∂ χ p δu δ δΨ U (u) (δΨÙ δχ)Ù a ∂µ a 1 µ2 ∂λ p 1 1 µ2 ∂ χ ∂ δΨ + δ U (v) (δΨÙ δχ)Ø δv p 2 a ∂λ ∂µ a 1 µ

(A.2)

(A.3a)

(A.3b)

The short-hand notation δX U δx2 is used to describe this step. V Horizontal interpolation (h) of model states (in terms of u, v, Φ and q) to the observation locations (δXobs hδX).


VI Apply the observation operators to obtain the model equivalent of each observation. This deþnition of the forward model has two interesting characteristics. Firstly, in the incremental formulation, the innovation y0 y HXb is computed directly from the model state expressed in terms of u, v, Φ and q and therefore only requires the last two operators, h and V. Interpolating horizontally from the high resolution deþnition of the background state, proþles of model states are obtained at each observation location and are kept to deþne the tangent linear approximation V0 needed for nonlinear observation operators. It is also worth noticing that only multiplication by the variances appears above so that starting from k 0, no changes would be brought by the analysis to those components of the background state that have vanishing variances. The same applies to the small eigenvalues that may appear in Λ1Û2 so that the inversion of the background-error covariances is perfectly consistent with a generalized inverse deþnition.

b The adjoint of the forward model With the deþnitions given above, the forward model is now expressed as H0 k (V0 hIG UB DS 1 IS ϕT Λ1Û2 )k where IS and IG stand for the identity transformations IS : S ! S and IG : G ! G in spectral and physical spaces respectively. As discussed in Talagrand and Courtier (1987) and Gauthier et al. (1993), it is convenient to change the deþnition of the norms used at intermediate stages to make the spectral transform and its inverse unitary transformations such that hSf Ù SgiS hf Ù giG . It can then easily be shown that (S 1 ) S. The functional (3.14) is J (k ) 12 k T k + 12 [H0 k

y00 ]T O 1 [H0 k

y00 ]Ù

where the notation H0 H0 G0 has been used. The gradient of J is then 0 0 1 =κ J k + (H ) O [H k

y00 ]

0 h V oband requires the adjoint operator (H0 ) Λ1Û2 ϕIS [(S 1 ) DM U ]IG tained by composition of the individual adjoint operators. The þrst two, Λ1Û2 and ϕT , are straightforward matrix multiplications. The inner product haÙ bi1 aT b being used in step I, the adjoint operator corresponds to the transpose of these matrices. The same applies to the observation operators V0 and h so that h hT and V0 V0 T.

1 ADJOINT OF U The inner product induced by IS in physical space is the Euclidean norm


hf (λÙ µ)Ù g(λÙ µ)iG2

2π Z+1 1 Z d λ d µf (λÙ µ)g(λÙ µ)Ø 4π

(A.4)

1

0

Using integration by parts, it can be shown that δ Ψ U δ u δχ δv with δ Ψ

δ χ

1 1 ∂ p δv 2 a ∂λ 1 µ 1 1 ∂ p δ u+ 2 a ∂λ 1 µ

∂ p ( 1 ∂µ ∂ p ( 1 ∂µ

(A.5)

µ2 δ u) Ù µ2 δ v)

Ù

where the superscript designates adjoint variables and a is the radius of the earth. 2 ADJOINT OF M The adjoint of M is δ Ψ M δ Ψ δ Ψ + f δ Φ Ø 0 δΦ δΦ δΦ

(A.6)

3 ADJOINT OF IG The identity transformation IG : G1 ! G2 is such that in G1 the discretized form of the inner product (A.4) X 1 Z hx Ù y iG1 fgdS xi yi ωi Ù (A.7) 4π S i is used, ωi being the weights of the numerical quadrature. In G2 , the usual Euclidean inner product X hx Ù y iG2 x T y xi yi Ù i

is used, the sum being carried out over all grid points. It follows that the adjoint of the identity IG is (I x)i

xi Ø ωi

(A.8)

4 ADJOINT OF IS As for the identity transform IS : S1 ! S2 in spectral space, it changes the norm from

152 / P. Gauthier et al. ~ Gi ~ S F~ T G ~ hFÙ 1

N 2N +1 X X

~ lm F~ lm G

(A.9)

m0 l 2m

to ~ Gi ~ S hFÙ 2

N X

~ n0 + 2 F~ l0 G

l 0

N 2N +1 X X

~ lm F~ lm G

m1 l 2m

(A.10)

~ F~ T AG with A, a real diagonal matrix. The latter inner product is a rewriting of h f~ Ù gi ~ S2

N +N X X

f~nm (g~ nm ) Ù

m N njmj

in which the complex spectral coefþcients are mapped into a real vector f~nm ! F~ lm m m Reff~nm g and F~ 2n+1 Imff~nm g when m Ü 0 and F~ n0 Reff~n0 g. Therefore, with F~ 2n ~ IS F~ AFØ Appendix B Characterization of the observations used in the CMC variational analysis This appendix summarizes the standard deviations of the observational errors used for the different types of observations assimilated in the 3D-var. For radiosondes, one distinguishes between a TEMP which provides observations of geopotential height, winds and dew-point depression and a PILOT which provides only measurements of the wind components. In the Northern Hemisphere, different statistics are used for winter and summer but for the Tropics and the Southern Hemisphere, the same statistics are used throughout the year. Wind error is given for the wind vector intensity and a distinction is made between TEMP and PILOT for the wind error statistics. Table 3 gives these standard deviations. Table 4 gives the statistics used for SATOB and AIREP wind data. Different constant values are used depending on whether the data are above or below 500 hPa. Table 5 gives the statistics used for SATEM data, i.e., retrieved geopotential thicknesses provided by NESDIS. Table 6 gives the error statistics for the surface data used in the CMC 3D-var. Except for the SATEMs, the observation-error statistics are identical to those previously used in the OI. Finally, the HUMSAT data and their error estimates come from a retrieval process described in Garand and Halle (1997).

Implementation of a 3D Variational Data Assimilation System / 153 Table 3. Standard deviation of observational error for wind vector intensity (UV) in m s–1, geopotential height (GZ) in m and dewpoint depression (T – Td) in K used for radiosonde data (TEMP and PILOT) in the operational analysis. Different statistics for summer and winter are used in the northern extratropics while they are kept constant throughout the year for the Tropics and the southern extratropics. Observation-errors (extratropics north winter) Pressure hPa

TEMP UV m s–1

1000 925 850 700 500 400 300 250 200 150 100 70 50 30 20 10

3.1 3.1 3.1 3.1 3.3 3.5 3.8 3.8 3.7 3.7 3.5 3.5 3.5 3.5 3.8 5.1

PILOT TEMP UV GZ m s–1 m 3.1 3.1 3.1 3.1 3.3 3.5 3.8 3.8 3.7 3.7 3.5 3.5 3.5 3.5 3.8 5.1

5.9 5.3 5.9 6.7 8.8 10.8 13.5 14.8 16.0 17.4 21.0 23.9 26.8 29.8 33.9 40.2

TEMP T – Td K

Pressure hPa

TEMP UV m s–1

1.7 1.8 3.0 3.6 3.7 3.3 2.8 — — — — — — — — —

1000 925 850 700 500 400 300 250 200 150 100 70 50 30 20 10

2.3 2.4 2.4 2.4 2.4 2.5 2.9 3.0 2.6 2.2 2.2 2.0 1.9 1.9 2.4 2.8

TEMP UV m s–1

1000 925 850 700 500 400 300 250 200 150 100 70 50 30 20 10

3.0 3.0 3.0 3.0 2.8 2.8 2.8 3.5 3.5 3.7 3.5 3.5 3.5 3.5 3.8 5.1


5.9 5.3 5.9 6.7 8.8 10.8 13.5 14.8 16.0 17.4 21.0 23.9 26.8 29.8 33.9 40.2


5.0 4.3 4.4 5.0 6.9 8.4 11.1 12.0 12.3 12.3 14.1 15.9 17.7 17.6 21.8 28.3

TEMP T – Td K 1.7 1.8 3.0 3.6 3.7 3.3 2.8 — — — — — — — — —

Observation-errors (extratropics south)

Observation-errors (Tropics) Pressure hPa

Observation-errors (extratropics north summer)

TEMP T – Td K

Pressure hPa

TEMP UV m s–1

2.3 2.2 3.2 4.2 4.3 3.9 4.0 — — — — — — — — —

1000 925 850 700 500 400 300 250 200 150 100 70 50 30 20 10

3.0 3.0 3.0 3.3 4.0 4.2 4.2 4.2 4.2 4.0 3.3 2.5 2.3 2.0 2.4 3.0


6.6 5.0 4.4 4.8 7.4 9.6 11.1 11.8 13.1 12.9 14.0 15.4 16.7 18.6 21.8 30.2

TEMP T – Td K 1.7 1.8 3.0 3.6 3.7 3.3 2.8 — — — — — — — — —

154 / P. Gauthier et al. Table 4. Standard deviation of observational error for wind vector intensity (UV) in m s–1 used for SATOB and AIREP wind data. These statistics are kept constant throughout the year. Pressure hPa

SATOB UV m s–1

AIREP UV m s–1

1000 925 850 700 500 400 300 250 200 150 100 70 50 30 20 10

5.4 5.4 5.4 5.4 5.4 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9 10.9

5.4 5.4 5.4 5.4 5.4 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2

Table 5. Error for geopotential height thicknesses (m) used for the 7 layer measurements obtained from SATEM. A distinction is made between clear and cloudy conditions. These statistics are kept constant throughout the year. Thickness hPa

clear m

p. cloudy cloudy m

1000–700 700–500 500–300 300–100 100–50 50–30 30–10

27.0 17.0 25.0 53.0 33.0 36.0 74.0

33.0 20.0 31.0 56.0 33.0 36.0 74.0

Table 6. Standard deviation of observational error for wind vector intensity (UV) in m s–1, geopotential height (Z) in m and dewpoint depression (T – Td) in K used for surface data (SYNOP, SHIP, fixed buoys and drifting buoys DRIBU) in the operational analysis. Dewpoint depression measurements are only provided by SYNOP and SHIP while the DRIBU provide data only for geopotential height. These statistics are kept constant throughout the year. SYNOP UV m s–1 5.1

SYNOP Z m

SYNOP T – Td K

SHIP UV m s–1

SHIP Z m

SHIP T – Td K

DRIBU Z m

FIXED BUOY UV m s–1

FIXED BUOY Z m

6.0

3.0

5.1

6.0

3.0

6.0

5.1

6.0


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