The Evolution of Dispersion Spectra and the Evaluation of Model Differences in an Ensemble Estimation of Error Statistics for a Limited-Area Analysis

Simona Ecaterina Ştefănescu National Meteorological Administration, LMDCC, Bucharest, Romania

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Loïk Berre Météo-France, CNRM/GAME-GMAP, Toulouse, France

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Margarida Belo Pereira Instituto de Meteorologia, DOV/CMAR, Lisbon, Portugal

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Abstract

An ensemble of limited-area forecasts has been obtained by integrating the Aire Limitée Adaptation Dynamique Développement International (ALADIN) limited-area model, in cold-starting mode, from an ensemble of Action de Recherche Petite Echelle Grande Echelle (ARPEGE) global analyses and forecasts. This permits error covariances of the ALADIN 6-h forecast and of the ARPEGE analysis to be estimated. These two fields may be combined in a future ALADIN analysis.

The evolution of dispersion spectra is first studied in a perfect model framework. The ARPEGE analysis reduces the large-scale dispersion of the ARPEGE background by extracting some information from observations. Then, the digital filter initialization reduces the small-scale dispersion by removing the noise caused by interpolation of the ARPEGE analysis onto the ALADIN grid. Finally, the ALADIN 6-h forecast strongly increases the small-scale dispersion, in accordance with its ability to represent small-scale processes.

Some model error contributions are then studied. The variances of the differences between the ALADIN and ARPEGE forecasts, which are started from the same ARPEGE analysis, are of smaller scale than are the ALADIN and ARPEGE perfect model dispersions. The small-scale part of these ARPEGE–ALADIN model differences is shown to correspond to structures that are represented by ALADIN and not by ARPEGE. Therefore, this part may be added to the ARPEGE analysis dispersion. The residual large-scale part is more ambiguous, but it may be added to the ALADIN dispersion; this may reflect some effects of the coupling inaccuracies, and strengthen (in a future ALADIN analysis) the use of the large-scale information from the ARPEGE analysis.

Corresponding author address: Simona Ecaterina Ştefănescu, National Meteorological Administration, LMDCC, 97 Şos. Bucureşti-Ploieşti, Sector 1, 013686 Bucharest, Romania. Email: simona.stefanescu@meteo.inmh.ro

Abstract

An ensemble of limited-area forecasts has been obtained by integrating the Aire Limitée Adaptation Dynamique Développement International (ALADIN) limited-area model, in cold-starting mode, from an ensemble of Action de Recherche Petite Echelle Grande Echelle (ARPEGE) global analyses and forecasts. This permits error covariances of the ALADIN 6-h forecast and of the ARPEGE analysis to be estimated. These two fields may be combined in a future ALADIN analysis.

The evolution of dispersion spectra is first studied in a perfect model framework. The ARPEGE analysis reduces the large-scale dispersion of the ARPEGE background by extracting some information from observations. Then, the digital filter initialization reduces the small-scale dispersion by removing the noise caused by interpolation of the ARPEGE analysis onto the ALADIN grid. Finally, the ALADIN 6-h forecast strongly increases the small-scale dispersion, in accordance with its ability to represent small-scale processes.

Some model error contributions are then studied. The variances of the differences between the ALADIN and ARPEGE forecasts, which are started from the same ARPEGE analysis, are of smaller scale than are the ALADIN and ARPEGE perfect model dispersions. The small-scale part of these ARPEGE–ALADIN model differences is shown to correspond to structures that are represented by ALADIN and not by ARPEGE. Therefore, this part may be added to the ARPEGE analysis dispersion. The residual large-scale part is more ambiguous, but it may be added to the ALADIN dispersion; this may reflect some effects of the coupling inaccuracies, and strengthen (in a future ALADIN analysis) the use of the large-scale information from the ARPEGE analysis.

Corresponding author address: Simona Ecaterina Ştefănescu, National Meteorological Administration, LMDCC, 97 Şos. Bucureşti-Ploieşti, Sector 1, 013686 Bucharest, Romania. Email: simona.stefanescu@meteo.inmh.ro

1. Introduction

In data assimilation, a short-range forecast (known as background) is often used as a source of information in addition to observations. The typical range of this forecast is 6 h, or even 1 h for some regional assimilation schemes. Moreover, the background error covariances are used to filter and propagate the observed information. However, estimating the forecast errors is not trivial, because the true atmospheric state is never exactly known. An ensemble method has been proposed by Houtekamer et al. (1996). This method relies on the time evolution of some perturbations that are constructed to be consistent with the involved error contributions. There are three basic steps or components that can be involved in the time evolution. First, the analysis scheme is applied to some perturbed observations and to a perturbed background, which provides a perturbed analysis: this simulates the effect of the two information errors (namely, the observation errors and the background errors) and of the analysis equation on the initial state uncertainties. Second, the perturbed analysis leads to a perturbed forecast, by using the forecast model integration: this aims to reproduce the effects of the atmospheric processes on the evolution of analysis errors into forecast errors. Third, some model perturbations may be added, to reflect the effect of the model errors: this can be done either explicitly at the end of the forecast (or even during the forecast), or implicitly by using different model versions for the different members of the ensemble.

This kind of approach has also been applied at the European Centre for Medium-Range Weather Forecasts (ECMWF) by Fisher (2003), using the first two components only, that is, without adding some model perturbations (partly due to the lack of knowledge about model errors); this version of the method has also been tested at Météo-France (Belo Pereira and Berre 2006), to derive statistics for the global model Action de Recherche Petite Echelle Grande Echelle (ARPEGE; Courtier et al. 1991). A similar approach is also presented in Buehner (2005).

While this ensemble method has been used previously for global models, we propose to consider its possible applications in the context of a limited-area model. The involved limited-area model is Aire Limitée Adaptation Dynamique Développement International (ALADIN) [Bubnová et al. (1993) or Radnóti et al. (1995)], which is coupled to the Météo-France global model ARPEGE. The ALADIN model is integrated to provide a dynamical adaptation of large-scale fields to topography and to other surface characteristics at high resolution. The integration is currently based on a cold-starting mode. This means that its initial condition is basically supplied by the ARPEGE analysis. An ensemble of ALADIN forecasts can thus be produced, with initial conditions and boundary conditions provided by an ARPEGE ensemble.

There are three main aspects of interest in the experimental results of this ensemble method. The idea is first to study how the ensemble dispersion evolves, through the different steps of the ARPEGE–ALADIN integration. The first step of an ALADIN integration is a global ARPEGE analysis, which is itself a combination of an ARPEGE 6-h forecast with some recent observations. This analysis is then interpolated onto the ALADIN grid. This is followed by a digital filter initialization (DFI; Lynch and Huang 1992) to remove imbalances introduced by the interpolation (Lynch et al. 1997). Finally, a forecast is obtained by integrating the ALADIN model, with boundary conditions provided by the ARPEGE model. To understand the statistical features of the ALADIN forecast dispersion, it is natural to examine how the dispersion changes through the successive aforementioned processes.

A second point of interest is to study the differences between the ALADIN forecasts and the ARPEGE forecasts when they are started from the same ARPEGE analysis field. This is interesting in itself and also when considering some possible model error representations. Indeed, as mentioned previously, model errors are sometimes simulated by integrating different numerical weather prediction (NWP) models or different versions of a NWP model (e.g., Errico et al. 2001). Thus, it seems interesting to examine the differences between the two NWP models that are involved here, namely the global (low resolution) coupling model (ARPEGE) and the limited-area (high resolution) coupled model (ALADIN).

A third issue is the possibility, offered by the ensemble approach, of estimating and comparing the error statistics of the ARPEGE analysis (over the ALADIN domain), with the error statistics of the ALADIN forecast. Generally, the global model analysis is thought to be particularly accurate in the large scales, while the limited-area forecast tends to be considered to be especially useful in providing some small-scale information. This general idea is evoked, for example, when trying to design a limited-area analysis that accounts for both a limited-area model forecast and an available global analysis (Gustafsson et al. 1997; Brožkova et al. 2001; Bouttier 2002).

The paper is organized as follows. Section 2 gives a short description of the dataset and of notations used herein. In section 3, the evolution of the dispersion spectra is studied, through the successive basic steps of an ALADIN integration. Section 4 deals with the evaluation and decomposition of the ARPEGE–ALADIN model differences. In section 5, the implications for the specification of the error statistics of the ALADIN forecast and of the ARPEGE analysis are discussed. Conclusions and perspectives are summarized in section 6.

2. Description of the dataset

The ensemble statistics were computed over a 48-day period, from 4 February 2002 until 23 March 2002. A two-member ALADIN ensemble was obtained as follows. First, a two-member ensemble of ARPEGE assimilation cycles was produced. Each of these two cycles was then used as a source of initial and boundary conditions for the corresponding ALADIN experiments. Therefore, the ARPEGE ensemble is first briefly described, and then the features of the ALADIN integrations are presented.

a. The ARPEGE ensemble experiment

The ARPEGE perturbed analysis and first guess fields were taken from two independent assimilation cycles (Belo Pereira et al. 2002). The basic steps of the ARPEGE ensemble experiment simulate the time evolution of the ARPEGE uncertainties, during the data assimilation cycle and in a perfect model framework. Starting from a perturbed analysis that is valid at time t0, a perturbed 6-h forecast (valid at time t1 = t0 + 6 h) is obtained by integrating the ARPEGE model. A perturbed analysis that is valid at time t1 is then obtained by applying the analysis scheme to the perturbed background and to the perturbed observations. The observation perturbations are constructed as random values, which have a Gaussian distribution with a mean equal to zero and a variance equal to the assumed variance of the observation errors. The analysis perturbations will thus result from the background and observation perturbations and from the analysis equation. Each perturbed analysis is then integrated in time to provide a perturbed 6-h forecast that is valid at time t2 = t1 + 6 h. These basic steps are repeated during successive analysis–forecast cycles.

The very first initial state of the two ARPEGE experiments is the operational analysis that is valid on ≈31 January 2002 at 1800 UTC: it is therefore the same for the two members, which means that there are no background differences on 1 February 2002 at 0000 UTC, at the beginning of the ensemble experiments. The amplitudes of the background differences then grow during the first days, but typically stabilize after 3 days. Therefore, the period of the dataset over which the statistics are calculated starts after this preliminary 3-day period.

The configuration of the involved ARPEGE system is based on the operational version of the ARPEGE model. The corresponding grid is not uniform. It is stretched by a factor c, in order to obtain a higher resolution over the European area than over the rest of the globe (Courtier and Geleyn 1988). The stretching factor is equal to c = 3.5, and the spectral truncation is T298, which leads to a grid length of about 19 km over the ALADIN–France area. There are 41 vertical levels, and the model top is at 1 hPa. The analysis is provided by a four-dimensional variational data assimilation (4DVAR) scheme (Veersé and Thépaut 1998). The assimilated observations are surface pressure from SYNOP; temperature and wind from aircrafts; temperature, wind, and humidity from soundings (TEMP radiosondes and PILOT messages); Advanced Television Infrared Observation Satellite (TIROS) Operational Vertical Sounder (ATOVS) Advanced Microwave Sounding Unit-A and High Resolution Infrared Sounder cloud-cleared radiances, and atmospheric motion vectors from SATOB. Marginal (in number) data also come from drifting buoys and 10-m wind observations over sea.

b. The ALADIN integrations and data

To obtain an ALADIN ensemble, one ALADIN 6-h integration has been performed in cold-starting mode, from each of the two members of the ARPEGE ensemble, and for each date of the 48-day period, at 0600 and at 1200 UTC. The results are similar at 0600 and 1200 UTC, and they will be shown for fields valid at 1200 UTC mostly.

Two additional ALADIN members have also been generated (from two other ARPEGE members), and it has been verified that their time-averaged statistics are quite similar to those of the first two members. This test confirms that the statistics are already stable with the first two members, whose statistical features will be shown in this paper.

The basic steps of an ALADIN integration are the following. The ARPEGE analysis is first interpolated onto the ALADIN grid: the ARPEGE fields are interpolated horizontally (by a 12-point quadratic interpolation; the interpolation points are the 12 nearest points of a 4 × 4 stencil, which means that the four corner points are not used), and then vertically (in a linear way as a function of pressure) to account for changes in surface pressure (which are induced by changes in the orography). A DFI is applied afterward. This provides the limited-area initial state. Then numerical integrations of the ALADIN model are forced by lateral boundary conditions, which are interpolated ARPEGE forecasts (obtained simply from the ARPEGE integrations, after the analysis).

These steps were applied to derive a two-member ALADIN ensemble from a two-member ARPEGE ensemble. The interpolation and initialization procedures were also applied to the ARPEGE 6-h forecast fields, for diagnostic purposes. The ALADIN model was integrated over the ALADIN–France integration domain. The main geometrical characteristics for this integration domain are the following: there are 41 vertical levels as in ARPEGE (with the model top at 1 hPa), the domain is a square with sides of length Lx = Ly = 2850 km, and the number of grid points in each direction is J = K = 300; the grid resolution is thus δx = δy = 9.5 km and the spectral truncation corresponds to M = N = 149 (with M, N being the maximum wavenumbers in each direction). The basic physics and dynamics are the same as in ARPEGE.

The time-averaged variance spectra of four variables (divergence, vorticity, temperature, and surface pressure), at four model levels (13, 22, 29, and 41), were studied for several types of state differences which are listed in section 2c. The humidity statistics have not been examined, but they will be the subject of a future study. The ranges that have been studied are 0 and 6 h. Model level 13 is located around the 225-hPa pressure level, model level 22 corresponds to approximately the 535-hPa pressure level, model level 29 is located around 775 hPa, and model level 41 is located near the surface. The main results will generally be illustrated by the variance spectra of a small-scale variable, namely vorticity at level 29, and of a large-scale variable, namely the logarithm of surface pressure (in the remainder of the text it will be referred to as the surface pressure). In addition, the vertical profiles of the standard deviation for temperature and divergence will be studied in section 3, and also partly in section 4 for the ARPEGE–ALADIN model differences.

c. Notations and terminology

Section 3 presents results of the ARPEGE–ALADIN assimilation ensemble experiment in a perfect model framework. This allows the contributions of initial condition errors to be simulated under the assumption that the models are perfect. Section 4 deals with a study of 6-h accumulated ARPEGE–ALADIN model differences. The corresponding contributions to model errors are discussed further in section 5. Initial condition and model error contributions are added in section 5 to provide total error variance estimates, for the ALADIN background and also for the ARPEGE analysis. The corresponding notations will be summarized here.

1) Contributions of initial condition errors (in a perfect model framework)

In section 3, the time-averaged variances of differences between the two members k and l will be shown at different stages of the ALADIN ensemble experiment, such as for the initial state
i1520-0493-134-11-3456-e1
where for member k, for example, the initial state xarpia(k) is the ARPEGE analysis, interpolated onto the ALADIN grid, and initialized by DFI,
i1520-0493-134-11-3456-e2
where Xarpa(k) is the ARPEGE analysis field over the globe (for member k), 𝗣 is the interpolation operator onto the ALADIN grid, and 𝗗 is DFI. Here, xarpia will be referred to as the initialized ARPEGE analysis. Similarly, the variances of the ALADIN 6-h forecast differences correspond to
i1520-0493-134-11-3456-e3
where 𝗠ald is the ALADIN 6-h forecast operator. The variances of the differences between the ALADIN 6-h forecasts will be compared in particular to the variances of the differences between the initialized fields of the (interpolated) ARPEGE 6-h forecast,
i1520-0493-134-11-3456-e4
where 𝗠arp is the ARPEGE 6-h forecast operator. Here, xarpib will be referred to as the initialized ARPEGE 6-h forecast (or background). Note that the initialization of the ARPEGE field is indicated by the letter i in the superscript arpi. This is introduced to distinguish these fields from the variances of the uninitialized ARPEGE fields (which will also be studied at the beginning of section 3); namely,
i1520-0493-134-11-3456-e5
i1520-0493-134-11-3456-e6

2) Contributions of model errors

In section 4, the following time-averaged variances of the differences between the ALADIN and (initialized) ARPEGE 6-h forecasts (noted by εald−arpim), which are started from the same ARPEGE analysis field (i.e., differences between the ALADIN and ARPEGE forecasts that belong to the same member), will be studied:
i1520-0493-134-11-3456-e7
For simplicity, all operators have been assumed to be linear. The model difference expression 𝗠ald𝗗𝗣 − 𝗗𝗣𝗠arp indicates that εald−arpim corresponds to 6-h accumulated differences between the ALADIN and ARPEGE models. The examination of such model differences is similar to the approach in Errico et al. (2001), for instance.

Cohn and Dee [1988, p. 592, Eq. (2.6c)] and Dee [1995, p. 1131, Eq. (5)] also discuss the estimation of model error by using model differences. While a projection operator of a low-resolution to a high-resolution representation is used in our study, these two references present a similar approach in which a projection in the opposite direction [i.e., a truncation operator transforms a high- (or infinite) resolution representation to a lower-resolution representation] is performed.

In practice, the respective model difference statistics for members k and l are very similar to each other, as expected (since they describe the basic differences between the ARPEGE and ALADIN integrations, independently from the details of their initial conditions). They were averaged to formally preserve an equal statistical weight for both members in the examined statistics.

The variance V(εald−arpim) will also be compared with V(xaldb) and with V(xarpib), which are variances of the full forecast fields. As for V(εald−arpim), they are calculated as an average of the respective results for members k and l. The detailed expression of, for example, V(xaldb) is thus the following:
i1520-0493-134-11-3456-e8
The variances V(xaldb) and V(xarpib) describe the atmospheric variability, as seen by the ALADIN and (initialized) ARPEGE forecast fields, respectively.

In section 5a, the respective model error variances of ARPEGE and ALADIN will be estimated from V(εald−arpim). They will be denoted V(εarpim) and V(εaldm), respectively.

3) Total error variance estimates

A total error variance V(εaldbm) will be defined as follows for the ALADIN 6-h forecast:
i1520-0493-134-11-3456-e9
This amounts to summing up the following two contributions to the ALADIN 6-h forecast error variance: the errors induced by initial condition uncertainties (under the assumption that the models are perfect), and the ALADIN model errors. This is analogous to the Kalman filter equations such as described in Daley [1991, Eqs. (13.3.5) on p. 376 and (13.3.7) p. 377].
Similarly, a total error variance V(εarpiam) for the (initialized) ARPEGE analysis will be defined as follows, to account for the two contributions to the corresponding uncertainties:
i1520-0493-134-11-3456-e10

3. Contributions to the evolution of dispersion spectra

The dispersion of the ALADIN 6-h forecasts results from a rather complex evolution: the contributions to this evolution are studied in the current section, by examining the spectral and vertical variations of the dispersion, after the successive steps of the ALADIN integration.

a. The effect of the ARPEGE analysis

The effect of the ARPEGE analysis can be diagnosed by comparing the respective dispersions of the ARPEGE analysis and of the ARPEGE background (both being first projected onto the ALADIN grid). The corresponding horizontal variance spectra for surface pressure and vorticity at level 29 and the vertical profiles of standard deviations for temperature and divergence are presented in Fig. 1.

The analysis dispersion appears to be smaller than the background dispersion. This is consistent with the expected reduction of uncertainty, when assimilating observations to correct the ARPEGE background.

This reduction of dispersion is somewhat stronger in the large scales than in the small scales, which is particularly visible for surface pressure. This is consistent with the large amplitude of background errors in the large scales (compared to observation errors) for the main observed variables, such as surface pressure, temperature, radiances, and wind. The analysis is expected to correct more efficiently background components whose error amplitude (compared to the corresponding observation error amplitude) is large, as discussed also in Daley (1991, the final paragraph of section 4.5 on p. 130 and Fig. 5.9 on p. 174) and Daley and Ménard (1993, Fig. 2 on p. 1558).

Similarly, the reduction of dispersion is also more pronounced in the midtroposphere than near the surface, which may be linked with the greater importance of large-scale features at these middle levels. The analysis effect is smaller at the highest levels, probably due to the lower observation density there.

The resolution of the ARPEGE model over the ALADIN–France integration domain is about 19 km, which corresponds roughly to wavenumber 40. The shape of the dispersion spectra for wavenumbers larger than 40 is therefore completely determined by the interpolation itself (there is no information about these small scales from the ARPEGE model itself). The peak of the small-scale variance at wavenumbers 70–90, which is particularly visible for vorticity in Fig. 1, is therefore likely to correspond to some artificial noise that is introduced by the interpolation. Such interpolation effects are consistent with those observed in previous studies relying on the National Meteorological Center (NMC) method (Sadiki et al. 2000). As discussed in this latter reference, they are related to the fact that the interpolation is essentially mathematical (rather than physical), compared for instance to what a high-resolution NWP model can provide. In particular, physically, the only meaningful information in the interpolation is the assumption of hydrostatic balance in the vertical.

b. The effect of the DFI

The effect of the DFI (applied on the ALADIN grid) can be examined by comparing the respective dispersions of the (uninitialized) ARPEGE analysis and of the initialized ARPEGE analysis (Fig. 1). The dispersion of the initialized fields is smaller than for the uninitialized fields. The comparison of the dispersion spectra indicates moreover that the reduction of dispersion is rather small in the large scales, and that it increases toward the small scales.

It can be observed in particular that the artificial peak of the variance for vorticity in the small scales, which was present in the uninitialized analysis, has been removed by the DFI. The DFI thus appears to remove the noise that is introduced by the interpolation. The reduction of the dispersion is particularly large for divergence in the midtroposphere. At these levels, the artificial peak of the variance was strong.

For diagnostic purposes, it is interesting to apply DFI to the ARPEGE 6-h forecast field as well: it allows us to visualize the ARPEGE analysis effect, without the contribution of the interpolation noise itself (Fig. 2). Note that the ARPEGE 6-h forecast is the field used as the background for the ARPEGE analysis, and therefore the ARPEGE analyses and forecasts are valid at the same time. A comparison of Figs. 1 and 2 indicates that the analysis effect on the initialized background is similar to the analysis effect on the uninitialized background. The dispersion variance is indeed reduced by the analysis, as expected.

c. The effect of the ALADIN 6-h forecast

Similarly, in order to visualize the effects of the atmospheric evolution during the ALADIN 6-h forecast, it is convenient to compare the dispersion statistics of the ALADIN 6-h forecast with those of the ARPEGE initialized fields. It is better to use the ARPEGE initialized fields (instead of the ARPEGE uninitialized ones), as DFI is able to remove the main part of the interpolation noise.

A first way to evaluate the effect of the 6-h integration of ALADIN is thus to compare the respective dispersion statistics of the ALADIN initial state (which is an ARPEGE analysis initialized by DFI) and of the ALADIN 6-h forecast (Fig. 3). Note that the latter field is valid 6 h later than the former. A general increase of the variance can be observed. The large-scale increase of variance likely reflects the increase of the dispersion induced by some atmospheric instabilities (such as baroclinic developments for instance). Moreover, probably the increase of the small-scale variance is induced by the higher resolution of the ALADIN model.

To confirm this, it is useful to also compare the respective dispersion statistics of the ARPEGE (initialized) 6-h forecast and those of the ALADIN 6-h forecast (Fig. 4). It has been noticed previously that the DFI step implies essentially the removal of some smallscale interpolation noise. In contrast, as can be seen especially from the vorticity variance spectrum, the ALADIN 6-h forecast provides a relatively large amount of additional small-scale energy. This is consistent with the representation of small-scale dynamical effects in the ALADIN model, as discussed in Sadiki et al. (2000). The increase of small-scale energy is somewhat more pronounced at low levels, which may be related to the higher-resolution topography and, more generally, to the larger importance of small-scale processes at low levels.

The increase of the small-scale dispersion by the ALADIN 6-h forecast implies an increase of the total variance. As expected, this effect is much stronger for a small-scale variable such as divergence than it is for a large-scale variable such as temperature (see bottom panels in Figs. 3 and 4).

The smaller amplitude of the small-scale variance in the ARPEGE 6-h (initialized) forecast field is expected to some extent, due to the lower resolution of the ARPEGE model. The latter implies naturally that the contributions of some small-scale processes, which exist on the ALADIN grid, are dissipated by the ARPEGE numerical diffusion, or not even represented explicitly in ARPEGE. On the other hand, this result may also be seen as unrealistic, knowing that the dispersion is supposed to reflect the involved uncertainties: one could consider in fact that the smaller variances of the ARPEGE forecasts suggest that the ARPEGE forecast is more accurate than the ALADIN forecast in the small scales, which would be rather unexpected knowing that the ARPEGE model has a lower resolution. To study these issues more deeply, differences between the ARPEGE and ALADIN forecasts will be examined in the next section.

4. Model differences evaluation and decomposition

In this section, differences between ALADIN and (initialized) ARPEGE 6-h forecasts, which are started from the same ARPEGE analysis field, will be studied. As mentioned in section 2c, this amounts to examining the influence of the 6-h accumulated model differences between ARPEGE and ALADIN. First, the nature of the involved model differences will be summarized. Second, the resulting forecast differences will be examined and compared to the ARPEGE and ALADIN perfect model dispersions (which were studied in section 3). Third, the respective variance spectra of the ARPEGE and ALADIN full forecast fields and their difference will be studied. This will allow us to highlight the strong contribution of the associated forecast amplitude differences in the small scales. Finally, this will lead us to consider a scale decomposition of the variance of the ARPEGE–ALADIN forecast differences.

a. The nature of the ARPEGE–ALADIN model differences

Three basic features are expected to contribute to the differences between the ARPEGE and ALADIN forecast fields on the ALADIN grid, when they are started from the same ARPEGE analysis.

First, the difference of horizontal resolution means that some small-scale structures are either not represented at all by ARPEGE or are dissipated by ARPEGE numerical diffusion.

Second, ARPEGE is a global model, while ALADIN is a limited-area model. The ALADIN coupling is based on a Davies–Kålberg relaxation scheme (Davies 1976), with a 3-h coupling frequency. Any coupling technique is imperfect, which means that the large-scale information that is provided by ARPEGE may be distorted.

Third, ARPEGE fields are interpolated onto the ALADIN grid, in order to obtain fields that can be used by ALADIN. As evidenced in section 3a, this interpolation is not perfect and likely produces some unrealistic small-scale features beyond the ARPEGE resolution.

The interpolation noise was also shown to be reduced by DFI in section 3b. Moreover, in the remainder of the paper, we will concentrate on initialized ARPEGE fields. This means that the horizontal resolution and the coupling are the main two factors contributing to the ARPEGE–ALADIN forecast differences.

The basic physics and dynamics are the same in both models; consequently, the ARPEGE–ALADIN differences are not expected to provide a full model error simulation. On the other hand, this allows us to focus on the aforementioned specific differences and on their implications for the estimation of error statistics. In other words, this is a first natural step in order to go beyond the perfect model approach.

b. Comparison of model differences with the ARPEGE and ALADIN perfect model dispersions

Figure 4 presents the spectral and vertical distribution of the ARPEGE–ALADIN forecast differences, together with the respective perfect model dispersions of the ARPEGE and ALADIN 6-h forecasts (which were studied in section 3). The examination of the vorticity variance spectrum indicates that the dispersion of the ARPEGE–ALADIN differences is shifted more toward the small scales than the ARPEGE and ALADIN dispersions, with a maximum around wavenumbers 30–40 (instead of 10–20 for the two other curves). This is expected, given that the ARPEGE and ALADIN models are based on similar equations and that the model differences result to a large extent from the resolution differences. The vertical profiles presented in Fig. 4 (bottom panels) show larger standard deviation values of the ARPEGE–ALADIN differences at low levels (as compared to the values for the mid- and high troposphere): this is consistent with the effect of the higher-resolution topography and, more generally, with the larger importance of small-scale processes at low levels.

On the other hand, it may be noticed that the nonzero variance values of the ARPEGE–ALADIN differences do not concern only the scales that are resolved by ALADIN and not by ARPEGE. The ALADIN-specific scales correspond indeed to wavenumbers that are larger than 40. This is visible in particular in the variance spectra of surface pressure: the variance of the ARPEGE–ALADIN differences appears to be maximum in the largest scales, rather than in the small or intermediate scales. It may therefore be interesting to identify more specifically the part of the ARPEGE–ALADIN differences that corresponds to the resolution differences. This is the objective of the next section.

c. Decomposition of the ARPEGE–ALADIN differences

The difference in resolution implies that some small-scale structures will be represented by ALADIN and not by ARPEGE (either not at all, or dissipated by the ARPEGE numerical diffusion).

To estimate the contribution of these small-scale structures, one may first express the variance of the ARPEGE–ALADIN difference field, as a function of the variances of the two full forecast fields:
i1520-0493-134-11-3456-e11
where V(xb) = is the variance of the full forecast field xb and cov(xaldb, xarpib) = is the covariance between the two forecast fields. As mentioned in connection with Eq. (8), note that V(xb) describes the full atmospheric variability, as seen by the forecast field xb.
It is possible to rewrite Eq. (11) as
i1520-0493-134-11-3456-e12
where β = [V(xarpib)/V(xaldb)] is the ratio between the ARPEGE variance and the ALADIN variance (which can also be seen as the percentage of the ALADIN variance that is represented by ARPEGE), and ρ is the correlation between the ALADIN and ARPEGE forecast fields. The parameter β describes the influence of the average amplitude differences, while the parameter ρ corresponds to the influence of the average phase differences. For instance, β is likely to be close to zero for small-scale structures that are dissipated by ARPEGE, and ρ is likely to be close to zero if the ARPEGE and ALADIN waves are in quadrature. These two contributions may be partly distinguished, by rearranging the Eq. (12) as the sum of the following two terms:
i1520-0493-134-11-3456-e13
where the first term of the right-hand side can be easily shown to correspond to the difference between the full forecast variances of ALADIN and of ARPEGE, namely V(xaldb)(1 − β) = V(xaldb) − V(xarpib).

This term has been calculated and is plotted in Fig. 5, where it can be compared with the variances of the ARPEGE and ALADIN full forecast fields: V(xarpib) and V(xaldb) (left panel). The ARPEGE and ALADIN variances are very similar in the large scales, while the ALADIN variance is much larger than the ARPEGE variance in the small scales, as expected (see the beginning of the last paragraph in section 3c). This implies that the difference between these two variance spectra will correspond to a spectrum that has its maximum in the small scales.

This difference between the ARPEGE and ALADIN variances can also be compared with the variance of the ARPEGE–ALADIN differences (right panel in Fig. 5). The two curves look very similar in the small scales (for wavenumbers larger than 10), which means
i1520-0493-134-11-3456-e14
In other words, the (energy) amount of the ARPEGE–ALADIN forecast differences tends to be the same as the (energy) amount of ALADIN structures that are not represented by ARPEGE. This is consistent with the expectation that the small-scale structures that are represented by ALADIN and missed by ARPEGE predominate in the ARPEGE–ALADIN forecast differences. Therefore, one may conclude that the small-scale ARPEGE–ALADIN forecast differences correspond essentially to some resolution-induced ARPEGE model errors with respect to the ALADIN grid.

It should also be mentioned that, in general, some possible small-scale noise of the interpolation (of ARPEGE fields onto the ALADIN grid) is another potential contribution to the variance of the ARPEGE– ALADIN forecast differences. Nevertheless, there are two results that indicate that this is not a major contribution in the present case, which deals with initialized ARPEGE fields. First, while some spurious noise is visible in the small-scale spectrum of uninitialized ARPEGE fields, the DFI appears to remove most of it (top panels in Fig. 1). Second, if the possible remaining interpolation noise had been a major small-scale contribution to the ARPEGE field xarpib and to the associated ARPEGE–ALADIN forecast differences εald−arpim, one would expect a strong statistical link between the structures of xarpib and those of εald−arpim. This can be measured by the amplitude of the corresponding cross covariance cov(xarpib, εald−arpim) [which can be compared to V(εald−arpim) and cov(xaldb, εald−arpim) for instance]. In fact (see appendix A), the result V(εald−arpim) ≈ V(xaldb) − V(xarpib) (right panel in Fig. 5) indicates that cov(xarpib, εald−arpim) ≈ 0 ≪ cov(xaldb, εald−arpim) ∼ V(εald−arpim). Therefore, the small-scale structures of εald−arpim appear to be much more linked to the structures of xaldb. This is consistent with the idea that εald−arpim corresponds essentially to some (resolution induced) structures that are represented by ALADIN and that are missed by ARPEGE.

The sharper decrease of V(xaldb) − V(xarpib) toward the large scales [compared with the corresponding decrease of V(εald−arpim)] suggests moreover that there may be some large-scale components in the ARPEGE–ALADIN differences that do not correspond to the former resolution-induced structures and differences in the variance amplitudes. Therefore, one may decompose the variance of the ARPEGE–ALADIN differences as the sum of two components:
i1520-0493-134-11-3456-e15
where α is a parameter that varies from 0 in the large scales to 1 in the small scales. Thus, V(εald−arpim,ss) corresponds to a small-scale part of the ARPEGE–ALADIN differences, which are induced by the resolution differences. In contrast, V(εald−arpim,ls) = V(xaldbxarpib) − V(εald−arpim,ss) is a residual, which corresponds to a large-scale part of the ARPEGE–ALADIN differences.

The parameter α may be modeled by a simple hyperbolic tangent function, as a function of the two-dimensional (2D) horizontal wavenumber k*: α(k*) = a{tanh[(k* − b)/c] + d}, with a = 0.5, b = 10, c = 3, and d = 1 (Fig. 6). These values were chosen in order to have variations between 0 and 1, and in order to ensure that V(εald−arpim,ss) = αV(xaldbxarpib) has a similar shape as V(xaldb) − V(xarpib) (right panel in Fig. 5).

The results of the decomposition are shown in Fig. 7, which illustrates how the two parts contribute differently to the variance of the ARPEGE–ALADIN differences as a function of scale.

5. Implications for the specification of the ARPEGE and ALADIN error statistics

a. Implied changes in the variance spectra

The possibility of estimating some model error contributions, V(εarpim) and V(εaldm), will now be considered, on the basis of the variance of the ARPEGE–ALADIN differences and of its decomposition (as discussed in section 4). The resulting total error variances V(εarpiam) and V(εaldbm) of the ARPEGE (initialized) analysis and of the ALADIN 6-h forecast respectively will also be examined, following Eqs. (9) and (10) in section 2c. The former field is the actual initial state of the ALADIN forecast, when ALADIN is integrated in cold-starting mode; the latter field is a possible background for an ALADIN three-dimensional variational data assimilation (3DVAR). These two fields may also be combined, either within a variational analysis or through a blending technique (Brožkova et al. 2001).

The small-scale part of the model differences variance, V(εald−arpim,ss), appeared to be closely related to the small-scale structures that are represented by ALADIN, and are missed by ARPEGE. Therefore, it can be considered that these small-scale variances correspond to some ARPEGE model errors, with respect to the truth on the ALADIN grid:
i1520-0493-134-11-3456-e16
It may be mentioned that the contribution of the small-scale model differences, namely 2V(εald−arpim,ss), can be considered to be applicable equally to the ARPEGE 6-h forecast and to the ARPEGE analysis: this is due to the same low resolution of both ARPEGE fields. Another point of note is that the factor of 2 in Eq. (16) yields a proper weight for this model error contribution, compared to the contribution of the dispersion arising from the perturbations of the ARPEGE assimilation cycle. The detailed equations are presented in appendix B, and the explanation can be described briefly as follows. The assimilation dispersion corresponds to the variance of the difference between two perturbed fields, involving two independent realizations of the perturbations: the resulting variance can easily be shown to be twice the variance of the basic perturbations. In contrast, the small-scale part of an ARPEGE–ALADIN model difference field may be seen as a single realization of the associated model error probability distribution function (pdf); therefore, a factor of 2 is to be used, in order to recover a contribution that is twice the variance of the involved model error.

The implied changes in the variance spectra are shown in the top panels in Fig. 8 [where V(εarpia) and V(εarpiam) can be compared]. While the large-scale variance is practically unchanged, the small-scale variance is significantly increased, as expected.

The large-scale part of the model differences variance, V(εald−arpim,ls), is more difficult to interpret. It is likely to correspond to a mixture of several effects. One may consider in particular two contributions to these large-scale model differences.

A first contribution may come from the interactions between the small scales and the large scales. This corresponds to the nonlinear effects of resolving smaller-scale features on the simulation of the large-scale solution. One may consider, in particular, that a more realistic simulation of small-scale processes may also contribute to the improvement of the simulation of the large-scale phenomena, if the involved processes are nonlinear.

A second contribution may come from the coupling inaccuracies. Any coupling technique is imperfect, which means that the large-scale information that is provided by the ARPEGE model may be distorted.

By themselves, the ARPEGE–ALADIN model differences, and the proposed scale decomposition, are insufficient to distinguish the two aforementioned possible contributions. To go further, one may consider some additional experiments in the future. For instance, one could integrate some ALADIN forecasts that have the same (low) resolution as the ARPEGE forecasts. Statistics of the differences between the high- and low-resolution ALADIN forecasts may thus be calculated: this would provide an estimate of the contribution of the ARPEGE–ALADIN resolution differences to the ARPEGE–ALADIN forecast differences variance. Statistics of the differences between the low-resolution ALADIN forecasts and the ARPEGE forecasts may also be calculated: this would provide an estimate of the contribution of the coupling inaccuracies to the ARPEGE–ALADIN forecast differences variance.

Due to these two possible contributions, three different scenarios will be considered in the remainder of this paper. A first possible scenario is to consider that the large-scale model differences provide an (upper bound) estimate of the contribution of the ALADIN coupling inaccuracies:
i1520-0493-134-11-3456-e17
The implied changes of the ALADIN variance spectra are shown in the bottom panels in Fig. 8 [where V(εaldb) and V(εaldbm) can be compared]. As expected, the small-scale variance is essentially the same, while the large-scale variance is significantly increased.

The implications of this first scenario are illustrated in Fig. 9, which represents the comparison between the variance spectra of the ARPEGE analysis and of the ALADIN forecast. When the contributions of the model differences are not considered at all (top panels in Fig. 9), the ALADIN variances appear to be larger than the ARPEGE variances, for all considered scales. In the large scales, this is consistent with the reduction of error due to the ARPEGE analysis. But it does not look realistic in the small scales, when considering that the dispersion is supposed to describe the involved uncertainties, and knowing that ALADIN has higher resolution than ARPEGE.

When the contributions of the model differences are added to the ARPEGE and ALADIN dispersions, as described above, the situation changes. The small-scale variances of ARPEGE are now larger than those of ALADIN: this is consistent with the lower ARPEGE resolution. Moreover, the representation of the larger uncertainty of the ALADIN forecast in the large scales has been strengthened.

The second scenario is similar to the first one, except that the variance of the large-scale model differences is no longer added to the ALADIN dispersion (or to the ARPEGE dispersion): V(εaldm) = 0 and therefore V(εaldbm) = V(εaldb). Compared to the previous scenario, this implies essentially that the uncertainty of the ALADIN forecast in the large scales is reduced (top panels in Fig. 10), although it remains larger than the ARPEGE analysis uncertainty.

A third scenario is to consider that the large-scale model difference variance (in addition to the variance of the small-scale ones) should be added to the ARPEGE dispersion (by considering, e.g., that they correspond rather to the beneficial effects of the ALADIN’s higher resolution on the ALADIN large-scale simulation): V(εarpim) = 2V(εald−arpim,ls) + 2V(εald−arpim,ss). Compared to the other two scenarios, this would imply for vorticity (bottom-right panel in Fig. 10) that the ARPEGE variance in the large scales would be much closer to (although still smaller than) the ALADIN variance. For surface pressure (bottom-left panel in Fig. 10), this would make even the large-scale ARPEGE variance larger than the large-scale ALADIN variance.

The implications of these different scenarios on a possible ALADIN analysis are discussed in the next section.

b. Implied weights in an ALADIN analysis

To determine the initial state of ALADIN, one may consider using the ARPEGE analysis, the ALADIN background, and some observations. These three sources of information can be combined through a variational formalism. The latter approach may be seen as a generalized formulation of 3DVAR, in which the ARPEGE analysis is introduced as an additional source of information (Gustafsson et al. 1997; Bouttier 2002). The first two sources of information may also be combined, by using a preliminary blending technique, before performing a classical 3DVAR analysis with the observations.

In this section, we will examine the implications of the ensemble variance spectra on the information extraction in this generalized 3DVAR analysis. This will be done by considering the case where one tries to combine the ARPEGE analysis and the ALADIN background in a manner similar to that in the blending technique, namely without using observations.

It may be mentioned that there are other issues involved in this generalized 3DVAR that will not be studied in the present paper. For instance, ideally, when also using observations, the ARPEGE analysis errors and the observation errors should be uncorrelated, in order to make the analysis procedure less complicated. This means that it may be advantageous to assimilate a large amount of observations that have not been used by the ARPEGE analysis. Some examples of such observations (whose assimilation is planned for the ALADIN analysis) are surface screen–level data, Meteosat-8 infrared radiances from the Spinning Enhanced Visible and Infrared Imager (SEVIRI), and radar data.

There may be also some correlations between the ARPEGE analysis errors and the ALADIN background errors, as the basic dynamics and physics are very similar in the two models. On the other hand, there are some model differences that arise from the ALADIN model’s high resolution and from its coupling scheme. These correlations may also depend on the amount and weight of assimilated observations in the ARPEGE analysis, as observation errors will contribute to the ARPEGE analysis errors, and as they tend to be uncorrelated with the ALADIN background errors. In the remainder of the paper, we will only consider the simple case where the two involved errors are assumed to be uncorrelated.

The ARPEGE analysis field considered here is the initialized ARPEGE analysis, which is the initial state used in cold-starting mode. This implies that this ARPEGE field is on the same grid as the ALADIN background. In other words, the equivalent of the observation operator for the ARPEGE field (when seeing the ARPEGE values as some observation values) is simply the identity matrix.

The equation that would correspond, for example, to a 3DVAR analysis xalda (without observations) is thus the following, if the two involved errors are uncorrelated [i.e., = 0]:
i1520-0493-134-11-3456-e18
where 𝗪arpia = 𝗕ald(𝗔arpi + 𝗕ald)−1 and 𝗪aldb = 𝗔arpi (𝗔arpi + 𝗕ald)−1 are the respective weight matrices of the two fields, where 𝗔arpi and 𝗕ald are the error spatial covariance matrices of the ARPEGE analysis and of the ALADIN background, 𝗔arpi = and 𝗕ald = .
While the previous equations are general, in order to simplify, one can consider a univariate and 2D horizontal framework, and the case where the two covariances are homogeneous and isotropic. In this case, the two weight matrices will be diagonal in spectral space. The corresponding diagonal (scalar) values, Warpia and Waldb, will only depend on the 2D horizontal wavenumber k*, and their expressions become simple functions of the ratio between the ARPEGE and ALADIN error variances:
i1520-0493-134-11-3456-e19
i1520-0493-134-11-3456-e20
The spectral variations of these weights are represented in Fig. 11 for the three different scenarios. In the small scales, the weight of the ALADIN forecast is largest: its value varies from 60% (for vorticity) to 80% (for surface pressure). When attributing the large-scale model differences to the ALADIN coupling inaccuracies (top panels in Fig. 11), it is the ARPEGE analysis that has the largest weight in the large scales: its maximum weight varies from around 70% (for vorticity) to around 85% (for surface pressure). The transition between these two different regimes occurs for wavenumbers around 11 and 12 (which corresponds to horizontal scales of about 60–65 km), for which the two involved weights are both close to 50%.

This larger weight of ARPEGE in the large scales is reduced, but still present, if the term 2V(εald−arpim,ls) is not added to either of the two involved variances (middle panels in Fig. 11): the maximum weight is around 60% for vorticity, and around 70% for surface pressure.

In the third scenario, the term 2V(εald−arpim,ls) is added to the ARPEGE variance (bottom panels in Fig. 11): this implies a maximum ARPEGE weight that is slightly larger than 50% for vorticity, and around 40% for surface pressure.

These different figure panels thus illustrate the implications of the ensemble statistics on the information extraction in such an ALADIN 3DVAR scenario. For instance, if one of the goals (of this ALADIN analysis) is to rely on the ARPEGE analysis for the large scales, and on the ALADIN background for the small scales, then the first two scenarios (and in particular the first one) are more appropriate for the specification of the error statistics.

6. Conclusions and perspectives

An ensemble of ALADIN fields has been obtained by integrating the ALADIN limited-area model, in cold-starting mode, from an ensemble of global ARPEGE analyses and forecasts. The latter global ensemble was itself obtained by integrating two independent perturbed assimilation cycles in a perfect model framework (involving perturbed observations, and perturbed backgrounds that are provided by the evolution of the previous perturbed analyses).

The evolution of the perfect model dispersion spectra has been studied, in order to examine the effects of the successive steps of an ALADIN integration. The ARPEGE analysis reduces the large-scale dispersion of the ARPEGE background, by using some recent observational information. The DFI reduces the small-scale dispersion, as it removes some noise that is induced by the interpolation of the ARPEGE analysis onto the ALADIN grid. Finally, the ALADIN 6-h forecast implies a strong increase of the small-scale dispersion, in accordance with its representation of small-scale dynamical and physical processes.

Then, the variances of the differences between the ALADIN and ARPEGE 6-h forecasts, which are produced from the same ARPEGE analysis field, were examined. These model differences are of smaller scale than are those of the ARPEGE and ALADIN perfect model dispersions. Moreover, it has been found that, in the small scales, the corresponding variances are very close to the difference between the variances of the ARPEGE and ALADIN full forecast fields. This indicates that the small-scale part of the model differences corresponds to some structures that are produced by the ALADIN high-resolution dynamics and physics, and which are not represented in ARPEGE (due to its lower resolution). This has allowed us to propose a decomposition of the model differences variance as the sum of a small-scale part, which is induced directly by the resolution differences, and of a residual large-scale part.

Finally, the possible implications of these model differences have been studied, for the specification of the error statistics of the ARPEGE analysis and of the ALADIN 6-h forecast. This is linked with the idea of combining these two fields (and observations) in order to provide an ALADIN analysis. The variance spectra of the ARPEGE and ALADIN dispersions indicate that the large-scale information from the ALADIN analysis would preferably be extracted from the ARPEGE analysis, which benefits from the use of some recent observations. The small-scale part of the model difference variance may be added to the ARPEGE dispersion: this allows for the representation of the larger uncertainty of the ARPEGE analysis with respect to the small-scale features. This implies that the small-scale information from the ALADIN analysis would tend to be extracted from the ALADIN background. The large-scale part of the model difference variance is more ambiguous. It may be added to the ALADIN dispersion, if it is seen as corresponding to the effects of the ALADIN coupling inaccuracies for instance: this would emphasize the extraction of the large-scale information from the ARPEGE analysis. Another possibility would be to add this large-scale part to the ARPEGE dispersion, by considering that it reflects the beneficial (nonlinear) effects of the ALADIN model’s high resolution on the simulation of large scales. However, this approach would significantly reduce the amount of large-scale information extracted from the ARPEGE analysis.

There are many natural extensions to this kind of study. Some other statistical features may be examined, such as the multivariate and three-dimensional aspects (which involve the vertical auto- and cross covariances). Gridpoint and wavelet statistics could be computed, in order to diagnose the local (space dependent) features of the spatial structures of the ensemble dispersion. It would also be interesting to calculate and study the cross correlations between the ARPEGE and ALADIN errors, and time-dependent features may be studied as well.

Moreover, an ALADIN 3DVAR analysis is being developed, including a term measuring the distance to the ARPEGE analysis (Bouttier 2002): this will allow for the possibility of integrating an ALADIN ensemble, in data assimilation mode. A posteriori diagnostics can then be calculated also, which may allow for the comparison of the ensemble-based statistics with the estimates retrieved from the observations (Sadiki and Fischer 2005). The study of the model differences may be pursued, in particular by calculating differences between two models that differ either by their resolution, or by the geometry of their integration domain (global versus limited area). Another natural perspective is to assess the impact of, for example, model error modifications of the statistics on ARPEGE–ALADIN forecasts.

Acknowledgments

The authors thank François Bouttier, François Bouyssel, Claude Fischer, and Alex Deckmyn for carefully reading the early versions of the manuscript. We would like also to thank the two reviewers for their constructive comments. Simona Ecaterina Ştefănescu and Margarida Belo Pereira are grateful to Météo-France for supporting their stays at CNRM/GMAP.

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APPENDIX A

i1520-0493-134-11-3456-eqa1

The covariance cov(xarpib, εald−arpim)

By definition of εald−arpim, the ALADIN forecast field xaldb can be written as xaldb = xarpib + εald−arpim, which implies the following expression for the variance of the full ALADIN forecast field: V(xaldb) = V(xarpib) + V(εald−arpim) + 2 cov(xarpib, εald−arpim). This is equivalent to
i1520-0493-134-11-3456-eqa2
Therefore, the result V(εald−arpim) ≈ V(xaldb) − V(xarpib) implies that
i1520-0493-134-11-3456-eqa3

The covariance cov(xaldb, εald−arpim)

By definition of εald−arpim, the ARPEGE forecast field xarpib can be written as xarpib = xaldbεald−arpim, which implies the following expression for the variance of the full ARPEGE forecast field: V(xarpib) = V(xaldb) + V(εald−arpim) − 2 cov(xaldb, εald−arpim). This is equivalent to
i1520-0493-134-11-3456-eqa4
Therefore, the result V(εald−arpim) ≈ V(xaldb) − V(xarpib) implies that
i1520-0493-134-11-3456-eqa5
which tends to be much larger than cov(xarpib, εald−arpim) ≈ 0.

APPENDIX B

The Covariance of Perturbation Differences and of Single Perturbations

The covariance of perturbation differences

In the two-member ensemble experiment, two independent sets of observation perturbations, δo(k) and δo(l), are added to the unperturbed observation set y, for members k and l, respectively: y(k) = y + δo(k) and y(l) = y + δo(l). In practice, the two sets of observation perturbations are obtained as random realizations of the Gaussian probability distribution function whose mean is zero, and whose covariance matrix 𝗥δ corresponds to the specified observation error covariance matrix 𝗥: = = 𝗥δ with 𝗥δ = 𝗥. These two random realizations are uncorrelated, which implies that the covariance matrix of the observation difference εo = δo(k) − δo(l) is equal to twice the covariance matrix of the observation perturbations: 𝗥ε = = + = 2𝗥δ. Similarly, the analysis difference εarpia can be seen as the difference between two independent analysis perturbations, δarpia(k) and δarpia(l): εarpia = δarpia(k) − δarpia(l). This implies that the covariance matrix 𝗔ε of the analysis difference will be equal to twice the covariance matrix 𝗔δ of the analysis perturbations: 𝗔ε = 2 Aδ.

The covariance of single perturbations

In sections 4 and 5, εald−arpim are the differences between an ALADIN forecast and an ARPEGE forecast that are started from the same ARPEGE analysis. These differences arise from errors that are present in one model and not in the other one, such as the ALADIN coupling errors, or the lack of representation of some small-scale structures by the ARPEGE model (due to its low resolution). Said differently, either the ARPEGE model can be seen as a perturbation of the ALADIN model (e.g., with an excessive dissipation of small-scale structures), or the ALADIN model can be seen as a perturbation of the ARPEGE model (e.g., with some distortions of ARPEGE large-scale fields by the ALADIN coupling procedure). In both cases, the difference εald−arpim is to be seen as a single realization of the probability distribution function of a model perturbation δm (related either to ARPEGE or to ALADIN): εald−arpim = δm, and thus V(εald−arpim) = V(δm) instead of 2V(δm). This implies that a factor of 2 has to be applied to V(εald−arpim) (or to its subparts), when adding the initial condition and model error variance contributions. This allows us to recover a weight similar to that for the initial condition error variance.

Fig. 1.
Fig. 1.

The respective effects of the ARPEGE analysis (seen on the ARPEGE uninitialized fields) and of the DFI: the solid, dotted, and dashed–dotted lines correspond to the dispersion statistics of the ARPEGE uninitialized background, uninitialized analysis, and initialized analysis, respectively.

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 2.
Fig. 2.

The effect of the ARPEGE analysis (seen on initialized fields): the solid and dotted lines correspond to the dispersion statistics of the ARPEGE (initialized) background and (initialized) analysis, respectively.

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 3.
Fig. 3.

The effect of the ALADIN 6-h forecast, with respect to the evolution of the ARPEGE initialized analysisdispersion (valid at 1200 UTC, solid lines) into the ALADIN 6-h forecast dispersion (valid at 1800 UTC, dotted lines).

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 4.
Fig. 4.

The perfect model dispersions of the ARPEGE initialized 6-h forecast (solid lines) and of the ALADIN 6-h forecast (dotted lines); the dashed–dotted lines represent the statistics of the differences between the ALADIN 6-h forecasts and the ARPEGE (initialized) 6-h forecasts, which correspond to the same ensemble member.

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 5.
Fig. 5.

The difference between the ARPEGE and ALADIN variances (dashed–dotted lines). (left) Comparison with the variances of the ARPEGE (dotted line) and ALADIN (solid line) forecast fields. (right) Comparison with the variance of the ARPEGE–ALADIN differences (solid line). Missing values in the largest scale range of the curve V(xaldb) − V(xarpib) correspond to negative values. In the right panel, the small-scale part of the variance of the ARPEGE–ALADIN differences is also shown (dotted line).

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 6.
Fig. 6.

The hyperbolic tangent function α(k*) = 0.5{tanh[(k* − 10)/3] + 1}.

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 7.
Fig. 7.

The decomposition of the variance of the ARPEGE–ALADIN differences (solid lines), as a function of a small-scale part (dotted lines) that is induced by resolution differences, and of a residual large-scale part (dashed–dotted lines).

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 8.
Fig. 8.

(top) The variance spectra of the ARPEGE analysis, before (solid lines) and after (dotted lines) adding the contribution of the small-scale part of the model differences. (bottom) The variance spectra of the ALADIN 6-h forecast, before (solid lines) and after (dotted lines) adding the contribution of the large-scale part of the model differences.

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 9.
Fig. 9.

(top) The variance spectra of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines), before the addition of the contributions of the model differences. (bottom) The “total” variance spectra of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines) (i.e., after adding the model difference contributions, according to the first scenario).

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 10.
Fig. 10.

The “total” variance spectra of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines), according to the other two scenarios: (top) when the contribution of the large-scale part of the model differences is added neither to the ALADIN dispersion, nor to the ARPEGE dispersion and (bottom) when the contribution of the large-scale part of the model differences is added to the ARPEGE dispersion.

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

Fig. 11.
Fig. 11.

The respective weights of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines), resulting from their total variance spectra: (top) when the contribution of the large-scale part of the model differences is added to the ALADIN dispersion, (middle) when the contribution of the large-scale part of the model differences is not accounted for, and (bottom) when this large-scale part is added to the ARPEGE dispersion.

Citation: Monthly Weather Review 134, 11; 10.1175/MWR3230.1

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  • Fig. 1.

    The respective effects of the ARPEGE analysis (seen on the ARPEGE uninitialized fields) and of the DFI: the solid, dotted, and dashed–dotted lines correspond to the dispersion statistics of the ARPEGE uninitialized background, uninitialized analysis, and initialized analysis, respectively.

  • Fig. 2.

    The effect of the ARPEGE analysis (seen on initialized fields): the solid and dotted lines correspond to the dispersion statistics of the ARPEGE (initialized) background and (initialized) analysis, respectively.

  • Fig. 3.

    The effect of the ALADIN 6-h forecast, with respect to the evolution of the ARPEGE initialized analysisdispersion (valid at 1200 UTC, solid lines) into the ALADIN 6-h forecast dispersion (valid at 1800 UTC, dotted lines).

  • Fig. 4.

    The perfect model dispersions of the ARPEGE initialized 6-h forecast (solid lines) and of the ALADIN 6-h forecast (dotted lines); the dashed–dotted lines represent the statistics of the differences between the ALADIN 6-h forecasts and the ARPEGE (initialized) 6-h forecasts, which correspond to the same ensemble member.

  • Fig. 5.

    The difference between the ARPEGE and ALADIN variances (dashed–dotted lines). (left) Comparison with the variances of the ARPEGE (dotted line) and ALADIN (solid line) forecast fields. (right) Comparison with the variance of the ARPEGE–ALADIN differences (solid line). Missing values in the largest scale range of the curve V(xaldb) − V(xarpib) correspond to negative values. In the right panel, the small-scale part of the variance of the ARPEGE–ALADIN differences is also shown (dotted line).

  • Fig. 6.

    The hyperbolic tangent function α(k*) = 0.5{tanh[(k* − 10)/3] + 1}.

  • Fig. 7.

    The decomposition of the variance of the ARPEGE–ALADIN differences (solid lines), as a function of a small-scale part (dotted lines) that is induced by resolution differences, and of a residual large-scale part (dashed–dotted lines).

  • Fig. 8.

    (top) The variance spectra of the ARPEGE analysis, before (solid lines) and after (dotted lines) adding the contribution of the small-scale part of the model differences. (bottom) The variance spectra of the ALADIN 6-h forecast, before (solid lines) and after (dotted lines) adding the contribution of the large-scale part of the model differences.

  • Fig. 9.

    (top) The variance spectra of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines), before the addition of the contributions of the model differences. (bottom) The “total” variance spectra of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines) (i.e., after adding the model difference contributions, according to the first scenario).

  • Fig. 10.

    The “total” variance spectra of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines), according to the other two scenarios: (top) when the contribution of the large-scale part of the model differences is added neither to the ALADIN dispersion, nor to the ARPEGE dispersion and (bottom) when the contribution of the large-scale part of the model differences is added to the ARPEGE dispersion.

  • Fig. 11.

    The respective weights of the ARPEGE analysis (solid lines) and of the ALADIN 6-h forecast (dotted lines), resulting from their total variance spectra: (top) when the contribution of the large-scale part of the model differences is added to the ALADIN dispersion, (middle) when the contribution of the large-scale part of the model differences is not accounted for, and (bottom) when this large-scale part is added to the ARPEGE dispersion.

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