NOTES AND CORRESPONDENCE. Mesoscale Background Error Covariances: Recent Results Obtained with the Limited-Area Model ALADIN over Morocco

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1 3927 NOTES AND CORRESPONDENCE Mesoscale Background Error Covariances: Recent Results Obtained with the Limited-Area Model ALADIN over Morocco WAFAA SADIKI Direction de la Météorologie Nationale, Casablanca, Morocco CLAUDE FISCHER AND JEAN-FRANÇOIS GELEYN Météo-France/CNRM/GMAP, Toulouse, France 20 October 1999 and 31 March 2000 ABSTRACT This note presents recent results on the properties of background error covariances computed using the National Meteorological Center (now known as the National Centers for Environmental Prediction) method and applied to a mesoscale limited-area model. The error covariances are computed over a winter period for the Moroccan version of the ALADIN forecast system. The overall results compare well with previous outputs obtained with the same forecast model but on a typical midlatitude domain (viz., ALADIN-France). Thus, the climatological error statistics can be used with an equivalent efficiency in the frame of a 3DVAR analysis system over a wide range of low-latitude and midlatitude limited-area domains. Furthermore, the error statistics between the true forecast fields and the coupling fields for the same domain are compared. The coupling fields are provided by interpolation from the global ARPÈGE model. The comparison gives insight into the range of scales over which ALADIN is building its own dynamics and errors. In particular, the study shows at what scale the limited-area model dynamics depart from the global model forced solution. 1. Introduction The formulation of variational data assimilation requires the knowledge of model forecast error statistics for the computation of the background cost function. Several methods have been developed to estimate forecast error statistics. One approach consists of collecting error statistics from differences between model forecasts and an observation network. Hollingsworth and Lönnberg (1986) and Lönnberg and Hollingsworth (1986) use radiosonde data to construct the forecast error covariances. Pairs of observation points are sampled into bins for a given spatial separation, then the empirical covariances are fit to the modeled covariance functions by a least square method. The background error variances are retrieved by continuation of the empirical covariances for zero spatial separation, and the observation error variances are then the remaining part of the total empirical error variances. One shortcoming of this method comes from the inhomogeneous observation network that makes computations over data-sparse areas subject Corresponding author address: Dr. Claude Fischer, Météo-France/ CNRM/GMAP, 42 Avenue Gustave Coriolis, Toulouse, France. claude.fischer@meteo.fr to caution. Simple covariance models are used, assuming the separability of the horizontal and vertical structures. The technique also relies on a heuristic separation between background and observation error variances, since it only provides the total sum of the two variances. Other approaches use randomization techniques or adaptive methods. Wahba et al. (1995) use one single random perturbation, applied to the observations, along with the original data, to estimate the minimizer of a prescribed predictive mean-square error. They obtain estimates for some so-called observable parameters that enter into the formulation of the forecast error statistics. In particular, a scalar parameter representing the relative weight of background to observation error variance is estimated for an idealized variational analysis of 500- hpa geopotential height. The European Centre for Medium-Range Weather Forecasts (ECMWF) uses randomly perturbed observational data to produce a second analysis, in parallel to the control, unperturbed one. New forecasts are then carried out, and the difference with the original forecasts is found to produce statistics that are close to the background error statistics (with twice the background error variances and with the very same error correlations as the background; M. Fisher 1999, personal communication) American Meteorological Society

2 3928 MONTHLY WEATHER REVIEW VOLUME 128 An adaptive approach is proposed by Dee (1995). The example of an adaptive Kalman filter is used to estimate parameters that enter the formulation of either model or observation or background errors. A maximum-likelihood function is maximized in order to estimate the covariance parameters that will make the covariance model match the observed one. Adaptive methods therefore rely on an online, time-dependent adjustment of the statistical model toward the actual performances of the analysis. It is worth noting that both randomization and adaptive techniques demand the a priori formulation of a covariance model. It is likely that a physically sound covariance model will allow for a reduced number of adjustable parameters if any such method has to be implemented in an operational environment. Regardless, the determination of the proper estimatable covariance parameters remains a crucial problem in these methods. As an alternative, the National Meteorological Center (NMC, now known as the National Centers for Environment Prediction) method provides an approximation of model forecast error covariances by the computation of statistics on a set of differences of model forecasts for the same validating time (Parrish and Derber 1992). This information gives insight into the dynamics of the numerical model itself. Shortcomings of the NMC method have been noticed by several authors. The method gives a strong weight to the model data and might not reveal all the structures and the intensity of the background errors. Indeed, observational data only enter indirectly into the computations, via the 24-h extra data assimilation in the lagged forecast. Thus, model spinup and data rejection in the analysis can lead to background error misspecification. Furthermore, van Tuyl and Daley (1999) compute forecast minus observation statistics using North Pacific Experiment dropsonde data and compare the results with those from the NMC method. They show that the latter method does not retrieve the proper correlation structures in (usually) data-sparse areas such as the northern Pacific. Also, the correlation length scales can be overestimated because of the 24-h lag that is generally used, while 3DVAR or 4DVAR with a 6-h window would require sharper, less dynamically evolved, structures (M. Fisher 1999, personal communication). One advantage of the NMC method is that it provides background error statistics that are easy to implement in a variational scheme. The background variances usually are then weighted by a scalar in order to tune the strength of the feedback toward the observations. Statistics of background error covariances with the NMC method are now used on a regular basis in the ECMWF variational data assimilation system (Bouttier et al. 1997) along with statistical balance relationships. The empirical balance coefficients are obtained from multiple regression over the set of forecast differences and provide a set of uncorrelated variables used as predictors in the analysis instead of the model variables (Parrish et al. 1997). The technique is also implemented in the Météo-France operational 3DVAR system. It was applied to the limited-area mesoscale model ALADIN by Berre (2000) with an additional balance relation for humidity. Berre s study shows that the mesoscale error statistics do provide useful extra information for the multivariate analysis of humidity, even in the range of scales below 60 km. More generally, his study shows a separation of scales in the balance relationships, with dominating synoptic-scale error structures for mass and vorticity and dominating small-scale structures when divergence or unbalanced temperature is considered. In this study, we briefly review the general formulation of the statistical approach and we show results for the error structures associated with temperature and specific humidity. We then compare the error statistics computed with true forecast data and those errors computed with coupling data provided by the global model ARPÈGE. The latter data are used to provide the lateral boundary forcing for the operational ALADIN-Morocco model using the Davies formulation (Davies 1976). The ALADIN model is based on the spectral technique (Orszag 1970): horizontal diffusion, semi-implicit corrections, and horizontal derivatives are computed in the space of bi-fourier coefficients (Machenhauer and Haugen 1987; Haugen and Machenhauer 1993). The biperiodization is achieved by an elaborated fit in gridpoint space over a so-called extension zone (for details, see Bubnova et al. 1993; Radnoti et al. 1995). Let Q(x, y, z) be any such periodic gridpoint solution, then the spectral coefficients read J 1 K 1 1 l Q Q(x, y, z ) exp{ im[(2 /L )x ]} mn j k l x j JK j 0 k 0 exp{ in[(2 /L y)y k]} (1) with x j (j/j)l x and y k (k/k)l y. We consider in our formulation a regular gridpoint lattice (x j, y k ) with J K points. The total length of the domain is L x along x and L y along y. The vertical coordinate is simply represented as z l, with l the vertical index. In bi-fourier space, each wave is represented by a couple of zonal and meridional wavenumbers (m, n) and the total effective wavenumber k* is obtained by an elliptic relationship: 2 2 m n k* N (2) 2 2 M N with M and N the maximal zonal and meridional wavenumbers in the elliptic truncation. l l The spectral covariances QmnQ * m n are computed under the assumptions of horizontal homogeneity (invariance by translation) and isotropy (invariance with respect to the wave direction). A thorough development of these assumptions and their impact on the spectral covariance formulation can be found in Berre (2000). With similar

3 3929 TABLE 1. Main geometrical parameters for the ALADIN-Morocco model. Latitude and longitude of the corner points are given in degrees. Parameter J K M N L x L y x y NW corner SE corner Operational value km km N, W N, 9.83 E notations to Berre s study, the covariances between two spectral coefficients only depend on the total horizontal wavenumber and finally read l l m n ll QmnQ m n * m n I k*, (3) where the overbar stands for an ensemble mean and 2 1 ll l l k* mn m n I Q Q * d. (4) 2 0 The right-hand-side integral represents the sum over all contributing wave couples (m, n) along the slice of ellipse associated to the total wavenumber k*. In practice, the forecast error statistics are obtained by computing the ensemble mean over a series of model forecast differences for the same validating time. The final covariances, as functions of k* only, are then computed by summation based on Eq. (4). Furthermore, linear regression coefficients are computed in the same way as presented in Bouttier et al. (1997) and Derber and Bouttier (1999). The statistical balance relationships start with the computation of a linearized hydrostatic geopotential height P t : l t i i ref s i l(surface) P (l) RT logp RT log(p ), (5) where T i is temperature, p i is pressure, and is the difference between two model levels i and i 1. Here, T ref stands for a reference temperature (270 K) and p s for the surface pressure. Then a horizontal regression of P t with the total spectral vorticity is performed, assuming a geostrophic type of balance: P b H. (6) Here, H is a horizontal diagonal operator with respect to k*. The balanced mass P b is then used to derive multiple regressions for the other meteorological fields, including the divergence, the mass field represented by the temperature T, and the surface pressure p s and the specific humidity q (Berre 2000): M MH b (T, p ) N P NH P s b u u q Q R S(T, p ) b u s u QH R u S(T, p s) u. (7) In the latter equations, the subscripts b and u stand for balanced and unbalanced (total minus balanced) fields, respectively. The multiple regression operators for vorticity have the specific algebraic structure of a product between a vertical operator (M, N or Q) and the horizontal geostrophic balance operator H. Note that the term balance is used here in a different context than in atmospheric dynamics. Balanced stands for derived by a linear regression with a series of iteratively prescribed predictors. Thus, our balances imply in general more than one predictor and no active filtering of rapid waves. This situation is quite different from dynamical balances such as the barotropic nondivergent one or the potential vorticity inversion formulations (see Hoskins et al. 1985; Bishop and Thorpe 1994). These dynamical balances usually are nonlinear and implicitly filtered formulations, while our statistical balance is by definition linear and unfiltered. However, the results shown in Derber and Bouttier (1999), Berre (2000), or in this paper suggest that, due to the physical considerations prior to the choice of the predictors, the linear regressions recover part of the general equilibria of the atmosphere. The unbalanced predictors are assumed to provide an independent set of variables suitable for the 3DVAR analysis. Thus, the background error covariance matrix B u in the predictor space is taken as block diagonal. Each block consists of the error covariances for one of the predictors, and these blocks are themselves block diagonal with one subblock of vertical covariances for each total wavenumber: C C u C(T,p ) 0 s u B. (8) u C The total background error covariance matrix B in the space of the actual analysis variables is retrieved as with T B KBuK (9) I M I 0 0 K. (10) N P I 0 Q R S I The superscript T stands for the transpose and I is the identity. The regression coefficients provide also as a by-product the percentages of explained variances of each total field by its predictors. The explained variance gives insight into the strength of the statistical link. In general, the vertical covariances can be related to the physical processes at work. q u

4 3930 MONTHLY WEATHER REVIEW VOLUME Error statistics over the Morocco domain Table 1 gives the values of the main geometrical parameters that define the ALADIN-Morocco domain. The model dynamics are hydrostatic. The gridpoint domain is square with a quadratic grid to avoid spurious aliasing, which means that the number of grid points J is related to the spectral truncation M by J 3M 1. At the time of this study, the model used 27 levels on the vertical. Model forecasts are run with a two-time-level semi-lagrangian advection scheme and a complete package of physical parameterizations. The physics are the same as in ARPÈGE, but with some different settings to take into account the higher resolution. We present statistics computed over a winter period from 1 October 1997 until 16 December 1997 and from 3 until 20 January 1998 (i.e., a 95-day period on the whole). Forecast differences are taken between 12- and 36-h forecasts, for the same validating time. We compare our error covariance structures for a few significant fields with those presented in Berre (2000). He worked with ALADIN-France data and over a different winter period (1 Dec Feb 1997). Figures 1a and 1b show the vertical covariances between temperature and balanced mass and unbalanced divergence, respectively. The statistical link with balanced mass is mostly explained at scales larger than 100 km. This characteristic is common to all balances with the linearized mass field. The positive correlation throughout the troposphere indicates that an overestimated forecast in height is likely to be due to an overestimated temperature of the air mass. Thus, the errors reproduce the Laplace relationship. The deep vertical penetration of this correlation is a consequence of the large-scale hydrostatic equilibrium. At the tropopause, the correlation is reversed in accordance with the change in vertical stability characteristics. The covariances between temperature and unbalanced divergence (Fig. 1b) are built up mostly at mesoscales (below 100 km, not shown). Different patterns are visible depending on the vertical levels. In the planetary boundary layer (PBL), negative covariances show that low-level overestimated convergence is linked to a positive temperature error, which can be explained by the low-level warm advection in the active systems: an overestimation of a convergent pattern is linked to an overestimation of warm temperatures, and to an overestimation of the strength of an active mesoscale system in general. By considering mass conservation, underestimated convergence (overestimated divergence) is believed to be above the boundary layer, which explains the positive covariances between a PBL positive temperature error and low-tropospheric positive divergence error (at about 850 hpa). Throughout the mid- and upper troposphere, positive covariances flanked by negative values are obtained. The positive and negative patches are swapped from one to the other when crossing the tropopause. Figures 1a and 1b compare well with Berre s results for the FIG. 1. (a) Vertical error covariances between balanced mass and temperature. Positive values start at 5 K J kg 1 (solid). Negative values start at 5 KJkg 1 (dashed). Contours are plotted every 10 KJkg 1. (b) Vertical error covariances between temperature and unbalanced divergence. Positive values start at Ks 1 (solid). Negative values start at Ks 1 (dashed). Contours are plotted every 10 6 Ks 1. ALADIN-France data. We obtain very similar structures. However, the extrema of the covariances are smaller with the ALADIN-Morocco data. R For covariances between temperature and balanced mass, the extrema located at about 285 hpa are 50%

5 % smaller with the ALADIN-Morocco data than in Berre s study ( 48 and 49 K J kg 1 in our case, as compared with 88 and 78 K J kg 1 in Berre s). R For covariances between temperature and divergence, the PBL minimum is about Ks 1 and the maximum ahead is Ks 1, as compared with Ks 1 and about Ks 1 for ALADIN-France cases. Figures 2a and 2b show the vertical covariances between specific humidity and unbalanced divergence or unbalanced temperature, respectively. The vertical covariances between humidity and unbalanced divergence show a uniformly negative signal in the planetary boundary layer that remains present throughout part of the troposphere (though, in the midtroposphere, negative covariances are only associated with specific humidity at a level lower than the level considered for divergence; see Fig. 2a). A positive signal appears between divergence and specific humidity at higher levels. Thus, Fig. 2a shows the mesoscale coupling between a low-level positive error on convergence and an overestimation of specific humidity, explainable by the supply of humidity in convergent systems. An overestimation of the activity of a convective or frontal event is likely to produce both too strong PBL convergence and too strong humidity and precipitation patterns. In the midtroposphere, any area of convergence has its diverging counterpart at a higher level, which is retrieved from the error statistics. The error covariances between humidity and unbalanced temperature show a strong negative correlation along the diagonal. Furthermore, negative covariance structures are evident between low-level temperature and midtropospheric humidity (up to about 480 hpa in Fig. 2b). These negative values are explained physically by the coupling between low-level mesoscale subsidence (the compensating downward movement to convective or frontal updrafts) and loss of humidity by condensation and precipitation in the active systems. In this sense, the small positive kernel between upper-tropospheric temperature and humidity indicates the supply of specific humidity below the tropopause in subsiding areas. It is worth mentioning that the negative signal is built up for ⅔ by large-scale structures (with length scales above 100 km) and for ⅓ by mesoscale information (scales below 100 km). The positive signal is almost completely built up by large scales (not shown). These scale considerations are consistent with the physical explanations. The structures in Fig. 2 are very similar to those obtained by Berre (2000). Also, the extrema are close in both studies: for humidity versus divergence covariances, the maxima and minima are slightly smaller in our study than in Berre s, by about 10%; and for humidity versus temperature, the PBL signal is very clear in our data, and the minimum in the midtroposphere is also stronger than in the ALADIN-France study. FIG. 2. (a) Vertical error covariances between specific humidity and unbalanced divergence. Positive values start at s 1 gkg 1 (solid). Negative values start at s 1 gkg 1 (dashed). Contours are plotted every (b) Vertical error covariances between specific humidity and unbalanced temperature. Positive values start at Kgkg 1 (solid) and negative values start at Kgkg 1 (dashed). Contours are plotted every Forecast fields versus coupling fields We now concentrate on the comparison between the statistical balance relationships as obtained from the coupling data and from the forecast data. The figures

6 3932 MONTHLY WEATHER REVIEW VOLUME 128 FIG. 3. Vertically averaged percentage of explained variance of specific humidity as a function of the horizontal scale and predictor: sum over the three predictors (solid), balanced mass (long dashed), unbalanced divergence (short dashed), and unbalanced temperature surface pressure (dashed dotted): (a) as obtained from the coupling data (interpolated global model data), and (b) as obtained from the model forecast data. will be studied either as functions of the total horizontal wavenumber k* or as functions of the horizontal length scale, which is defined by convention as one-fourth of the corresponding resolved wavelength : L y /k* and /4. Figures 3a and 3b show the variance of specific humidity explained by the predictors in the coupling data and in the forecast data, respectively. At large scales, the curves are almost identical (scales above 150 km), while the two sets of curves differ at smaller scales. Below 100 km, the coupling data ratio of explained variance presents two peaks: a soft one at about 70 km and a sharp one at about 30 km. The whole shape is mainly controlled by unbalanced mass. The forecast data variances exhibit flatter curves than the coupling data variances between 100 and 40 km. Between 40 and 25 km, the explained variance drops quickly in the forecast data, and it increases strongly below 25 km (Fig. 3b). It is important here to note that the error covariances obtained from the coupling data contain no physical signal at all below length scales of about 25 km. Indeed, below this value, all the structures are purely mathematical and come from the horizontal interpolations of the global model data onto the Moroccan domain. Actually, two steps of interpolations (of the same mathematical nature) are performed, because the global model data are first projected onto an intermediate resolution, ARPÈGE-like domain (suitable for transmitting the files from Toulouse to Casablanca), then a second interpolation is performed to obtain high-resolution coupling fields in Casablanca. Thus, the structures of explained variance in Fig. 3a, below 25 km, correspond to the response of the horizontal interpolation operators to the multiple regressions. Physically, the only meaningful information in the interpolation is the assumption of a hydrostatic balance on the vertical and thus the obtained variances do not represent any effective balance relationship. In a similar way, trivial structures are noticed on plots of vertical covariances of coupling data, below the 25-km threshold (not shown). The same restriction holds for the curves in Fig. 4a. Finally, it is worth noting that the ratio of explained variance increases strongly at very small scales, in a more pronounced way than in the ALADIN-France data of Berre (2000). This behavior might be due to a stronger signal of convective activity at lower latitudes, which would be in agreement with the fact that the explained variance comes both from unbalanced mass and unbalanced divergence. Figures 4a and 4b show the ratio of explained variance for temperature. As noticed previously, the very small scales in the coupling data are physically irrelevant. They show simply in this case that the horizontal interpolations produce no statistical link between mass and either balanced mass (i.e., vorticity) or unbalanced divergence. To be explicit, the interpolations do not seek a geostrophic type of equilibrium, or any refined boundary layer equilibrium (Ekman layer). Comparing Figs. 4a and 4b, it can be seen that the curves match only for scales above 200 km. These large scales are controlled by the global model and forced via the Davies coupling in ALADIN. Between 200 and 50 km, the limited-area solution is clearly different, with a softly decreasing negative slope. At first glance, this difference is rather due to the large-scale balance between temperature and the balanced mass. Below 50 km, the forecast error statistics bring a significant new amount of information, which is also useful for the statistical balances since the ratio of explained variance increases between 50 and 30 km. At these small scales, the physics are controlled by the coupling between temperature and unbalanced divergence. Again, this finding

7 3933 FIG. 4. Vertically averaged percentage of explained variance of temperature as a function of the horizontal scale and predictor: sum over the two predictors (solid), balanced mass (long dashed), and unbalanced divergence (short dashed): (a) as obtained from the coupling data (interpolated global model data), and (b) as obtained from the model forecast data. is consistent with a dominating role played by convective or frontal convergence. While the separation of the physically useful signal at small scales is rather simple, based on considerations on the production of the coupling data, the separation at large scale is less obvious. To shed more light on the scale at which the ALADIN solution truly departs from the large-scale forced solution, we compare plots of horizontal power spectra of error covariances, at different vertical levels l. The power spectra are defined following Berre (2000): l ll 2 k* I, where ll I k* k* k* (11) is the sum of vertical autocovariances over FIG. 5. (a) Horizontal covariance spectra for divergence at model level 4 in s 2. (b) Same in (a) but at model level 20. the slice of the ellipse associated with the total wavenumber k*, deduced from Eqs. (3) and (4). Figures 5a and 5b show the horizontal spectra for total divergence (balanced plus unbalanced) and at model levels 4 and 20 (stratosphere and midtroposphere, respectively). Again, Fig. 5a exhibits the usual irrelevant signal at small scales (below wavenumber 20) due to the interpolations of the coupling data. At large and medium scales, there is an increase of the absolute values of the spectra in the forecast fields, which shows that the signal is always richer in the dynamically adapted data than in the interpolated coupling data. Moreover, the curves start to depart significantly at about wavenumbers 7 (model level 4) and 10 (model level 20). These values correspond to resolved wavelengths of about 400 and 300 km, respectively. Figures 5a and 5b show that there is a relatively small amount of energy contained in the mesoscale spectra of the coupling fields. On the con-

8 3934 MONTHLY WEATHER REVIEW VOLUME 128 FIG. 6. (a) Horizontal covariance spectra for specific humidity at model level 4 in (g kg 1 ) 2. (b) Same as in (a) but at model level 20. trary, the forecast builds up true, dynamically consistent energy spectra and it brings new, additional information for the error variances. Identical behavior is noticed in the plots of horizontal power spectra of specific humidity error covariances (Figs. 6a and 6b). The limited-area model builds up its own structures below a wavenumber of about 7 10 depending on the model level. 4. Conclusions The error covariances computed over a 95-day winter period of ALADIN-Morocco forecasts are in good agreement with previous results obtained for ALADIN- France forecasts. The vertical error covariances and the main scale selections from the multiple regressions are stable results from one limited-area domain to another. Differences are encountered in some of the extremum values for covariances and some of the ratios of explained error variances. In particular, vertical error covariances that involve temperature are smaller for ALA- DIN-Morocco (Figs. 1a and 1b). This finding might be due to less intense cyclonic and convective activity over Morocco in the 1997/98 winter than in the French domain in the previous wintertime. An alternative or complementary explanation would be that Berre s statistics were computed for a higher-resolution dataset: the smallest resolved wavelength is min 39 km in Berre s study versus min 50.5 km here. A more regular use of the NMC statistics in the coming years could improve our knowledge on the dependence of the error covariances and variances on the domain and the season. The sharp increase of the ratio of humidity variance explained by unbalanced temperature and divergence at the highest wavenumbers (Fig. 6b) might be an indication that convective processes play a bigger role in the Moroccan data than in the French data. This fact can also be tackled by a comparative study over a summer period. The statistics underline some of the basic large- and mesoscale processes at work in the atmosphere, such as hydrostatism and PBL convergence. Thus, they provide reliable data for a later use in a refined 3DVAR system, in the spirit of Derber and Bouttier (1999) for example. Furthermore, despite the southern extent of the Moroccan domain below 20 N, the covariances appear to be fully informative and there is no restriction for the horizontal balance operator (H) on this domain. Probably, the signal of latitudes above 30 N is dominating and enables us to build midlatitude -type statistics. In our study, we have shown that the limited-area dynamics only become effective beyond a critical wavenumber. For smaller wavenumbers, as expected, the solution is controlled completely by the global model. In the ALADIN-Morocco case, the critical wavenumber is in the range [7,10], which corresponds to wavelengths of about km. At all levels, the mesoscale model produces its own solution for scales smaller than 300 km, whatever field is considered. This result gives a lower bound to the range of wavenumbers that should be analyzed in a mesoscale 3DVAR system, for example. Indeed, one does not wish to reanalyze the whole spectrum, since then the large scales would be analyzed twice, which seems unnecessary. Moreover, the reanalyzed large scales can be in contradiction with the large scales provided in subsequent coupling files and produce a disrupture in the large-scale coupling information used for the limited-area forecast. The statistical balance formulation along with the NMC method is intended to be used in future for a 3DVAR data assimilation system in the ALADIN software. The present study therefore gives useful insight into the potential use of this formulation on a limitedarea domain of significant extent but located at lower latitudes than the usual ALADIN operational domains, which are over Europe. A further use of the NMC sta-

9 3935 tistics would be for the comparison of the forecast fields with analysis data, directly retrieved from the global model data assimilation and simply projected onto the limited-area domain. This work would give information on the model error at large and medium scales, where the analysis is taken as the truth. Also, it would show at what scales the dynamical adaptation in ALADIN brings more information than the analyzed state of the global model. Acknowledgments. This study benefited from pertinent remarks by Jean-Noël Thépaut, Chris Snyder, and an anonymous reviewer. The authors are also grateful to Loïk Berre and Maria Monteiro for their preparative work, without which this study would not have been possible. Wafaa Sadiki got a leave of absence from the Moroccan Direction de la Météorologie Nationale for a short-term period. Her stay was made possible thanks to French Embassy funding. REFERENCES Berre, L., 2000: Estimation of synoptic and mesoscale forecast error covariances in a limited area model. Mon. Wea. Rev., 128, Bishop, C. H., and A. J. Thorpe, 1994: Potential vorticity and the electrostatics analogy: Quasi-geostrophic theory. Quart. J. Roy. Meteor. Soc., 120, Bouttier, F., J. Derber, and M. Fisher, 1997: The 1997 revision of the J b term in 3D/4D-var. ECMWF Research Dept. Tech. Memo., 238, 54 pp. Bubnova, R., A. Horanyi, and S. Malardel, 1993: International project ARPEGE-ALADIN. EWGLAM Newsletter, No. 22, Institut Royal Météorologique de Belgique, Davies, H. C., 1976: A lateral boundary formulation for multi-level prediction models. Quart. J. Roy. Meteor. Soc., 102, Dee, D. P., 1995: On-line estimation of error covariance parameters for atmospheric data assimilation. Mon. Wea. Rev., 123, Derber, J., and F. Bouttier, 1999: A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus, 51A, Haugen, J. E., and B. Machenhauer, 1993: A spectral limited-area model formulation with time-dependent boundary conditions applied to the shallow-water equations. Mon. Wea. Rev., 121, Hollingsworth, A., and P. Lönnberg, 1986: The statistical structure of short-range forecast errors as determined from radiosonde data. Part I: The wind field. Tellus, 38A, Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, Lönnberg, P., and A. Hollingsworth, 1986: The statistical structure of short-range forecast errors as determined from radiosonde data. Part II: The covariance of height and wind errors. Tellus, 38A, Machenhauer, B., and J. E. Haugen, 1987: Test of a spectral limited area shallow water model with time-dependent lateral boundary conditions and combined normal mode/semi-lagrangian time integration schemes. ECMWF Workshop Proceedings: Techniques for Horizontal Discretisation in Numerical Weather Prediction Models, Reading, United Kingdom, ECMWF, Orszag, S. A., 1970: Transform method for the calculation of vectorcoupled sums: Application to the spectral form of the vorticity equation. J. Atmos. Sci., 27, Parrish, D., and J. Derber, 1992: The National Meteorological Center s spectral statistical interpolation analysis system. Mon. Wea. Rev., 120, ,, R. J. Purser, W.-S. Wu, and Z.-X. Pu, 1997: The NCEP global analysis system: Recent improvements and future plans. J. Meteor. Soc. Japan, 75, Radnoti, G., and Coauthors, 1995: The spectral limited area model ARPEGE-ALADIN. PWPR Rep. Series, No. 7, WMO TD No.- 699, van Tuyl, A. H., and R. Daley, 1999: Estimation of forecast error covariances over an oceanic region using NORPEX data. Preprints, 13th Conf. on Numerical Weather Prediction, Denver, CO, Amer. Meteor. Soc., Wahba, G., D. R. Johnson, F. Gao, and J. Gong, 1995: Adaptive tuning of numerical weather prediction models: Randomized GCV in three- and four-dimensional data assimilation. Mon. Wea. Rev., 123,