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Keywords:

  • ensemble of assimilations;
  • background error variances;
  • flow dependency;
  • balance operators;
  • ensemble Kalman filter

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental set-up
  5. 3. Diagnostic study of background-error variances of full variables
  6. 4. Impact of an extended specification of flow-dependent errors
  7. 5. Conclusions and perspectives
  8. Acknowledgements
  9. References

A variational ensemble assimilation has been developed at Météo-France to provide errors ‘of the day’ to the operational 4D-Var assimilation of the Arpège model. In the reference Arpège system (operational until April 2010), ensemble-based variances are used for vorticity and for the associated balanced parts of mass and divergence, while variances of the unbalanced parts remain static and horizontally homogeneous, and humidity variances are calculated with an empirical flow-dependent formula. This article presents some diagnostic and impact studies to examine the effects of extending the specification of flow-dependent ensemble-based variances to all variables in the minimization. The extended flow-dependent variances derived from the ensemble are first compared with the reference system values. It is observed that the flow-dependence of local variances of all variables tends to be strengthened with the extended specification. This is particularly noticeable for surface pressure variances in the vicinity of mid-latitude storms and tropical cyclones. The impact of using variances ‘of the day’ for all analysis variables is examined through analysis/forecast experiments over a 1-month period. The global impact of this extended specification is neutral to positive. Results suggest that the new variances are likely to produce localized (in space and time) positive impacts, most likely connected with dynamically active systems. This is supported by a case study of an intense storm that hit the northern part of France on February 2009. It is observed that the extended flow-dependent information improves the 48 h surface pressure forecast, by correcting both position and intensity errors. Copyright © 2011 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental set-up
  5. 3. Diagnostic study of background-error variances of full variables
  6. 4. Impact of an extended specification of flow-dependent errors
  7. 5. Conclusions and perspectives
  8. Acknowledgements
  9. References

Since initial conditions play an important role in the quality of subsequent forecasts, data assimilation schemes aim at making the best possible use of the information provided by the observations and a background state. This requires in particular a precise specification of observation and background error covariances (namely the R and B matrices).

During the last decade, a number of studies have focused on the issue of estimating flow-dependent background-error variances and correlations, either in the context of ensemble Kalman filtering (EnKF) (see Evensen, 2003, for a review on the EnKF) or variational data assimilation (Kucukkaraca and Fisher, 2006; Berre et al., 2007; Pannekoucke et al., 2007; Raynaud et al., 2009). This is still a topic of active and ongoing research.

Several operational centres including Météo-France, ECMWF (European Centre for Medium-range Weather Forecasts), UKMO (UK Met Office), JMA (Japan Meteorological Agency) and CMC (Canadian Meteorological Center) implemented complex and successful 4D-Var assimilation schemes in their global model, with significant efforts devoted to the modelling of the B matrix (Fisher, 2003).

On the other hand, research studies on EnKF show promising results. In particular, the presumed advantage of EnKF is its ability to provide direct flow-dependent estimates of background-error covariances through ensemble covariances, so that the observations and the background are appropriately weighted during the assimilation.

At Météo-France, instead of developing an EnKF in addition to the existing 4D-Var scheme, ideas from EnKF and 4D-Var have been combined, as suggested by Gustafsson (2007). In this perspective, it has been decided to include flow-dependent background-error variances in the 4D-Var B matrix of the operational Arpège global model. This is achieved by implementing a real-time ensemble of perturbed variational assimilations (Berre et al., 2007, 2009), which has been implemented and used operationally at Météo-France since July 2008. A direct estimate of flow-dependent statistics is calculated from this ensemble. These ensemble-based variances are then used in the operational 4D-Var assimilation to provide flow-dependent information. Such an approach is also being tested in the context of the ECMWF 4D-Var system (Kucukkaraca and Fisher, 2006; Isaksen et al., 2007).

Different attempts to include flow-dependent errors in variational data assimilation have been proposed in the past few years. Studies have been conducted on hybrid schemes (Hamill and Snyder, 2000; Lorenc, 2003; Wang et al., 2008) which combine the static 3D-Var B matrix with the flow-dependent EnKF B matrix. Other ideas are also described in works by Purser et al. (2003), Lindskog et al. (2006) and Miyoshi and Kadowaki (2008), for instance. More recently, Buehner et al. (2010a, 2010b) presented promising results from using in a near-operational variational system the background-error covariances that are estimated from the ensemble of background states produced by an EnKF.

In its first operational version, which is used as the reference version in the rest of the paper, Météo-France ensemble assimilation provided flow-dependent estimates only for the specification of vorticity variances. This implies that variances of vorticity and of associated balanced components of divergence and mass were flow dependent, while variances of unbalanced components were kept homogeneous and climatological, and variances of humidity were calculated with an empirical flow-dependent formula. The aim of this paper was to examine the impact of extending the use of flow-dependent variances to all variables in the minimization of the 4D-Var cost function.

The paper is organized as follows. Section 2 describes the background-error covariance model in the Arpège system and the way an ensemble data assimilation can be used to provide flow-dependent information on error variances. A diagnostic study of background-error variances is then presented in section 3, to compare statistics from the reference configuration with the new fully flow-dependent estimates. The impact of specifying extended flow-dependent variances is investigated in section 4, considering forecast scores calculated from a month of analysis-forecast experiments. The particular case of the forecast of an intense storm over France is also examined, to evaluate the impact of flow-dependent variances in an example of intense weather event. Finally, conclusions and perspectives are given in section 5.

2. Experimental set-up

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental set-up
  5. 3. Diagnostic study of background-error variances of full variables
  6. 4. Impact of an extended specification of flow-dependent errors
  7. 5. Conclusions and perspectives
  8. Acknowledgements
  9. References

2.1. The Arpège model

The French operational model Arpège (Courtier et al., 1991), developed in collaboration between Météo-France and the European Centre for Medium-Range Weather Forecasts (ECMWF), is a global spectral forecast model that uses a stretched horizontal resolution emphasizing the European area, with a finer resolution over France (Courtier and Geleyn, 1988). The atmosphere is described on 60 vertical levels from the surface to 0.1 hPa. Model variables are vorticity ζ, divergence η, temperature T, logarithm of surface pressure PS and specific humidity q. The assimilation scheme is based on a multi-incremental 4D-Var (Courtier et al., 1994; Veersé and Thépaut, 1998), with two successive minimizations at spectral resolutions T107 and T224 respectively. There are four daily analyses at the main synoptic hours (00, 06, 12 and 18 UTC).

2.2. Formulation of the background-error covariances

The multivariate description of the background-error covariance matrix in the Arpège model is based on the formulation by Derber and Bouttier (1999). In this formulation, control variable transforms (CVT) are used to represent univariate and multivariate components of the B matrix. Precisely, model variables are partitioned into balanced and unbalanced components (denoted hereafter by the subscripts b and u respectively) by using regressions, and the problem is reformulated in terms of new variables (the control variables): vorticity, unbalanced divergence ηu, unbalanced temperature and logarithm of surface pressure (T,PS)u, and specific humidity. Background errors for these control variables are quasi-independent due to the use of regressions and they are related to the model variables through balance relationships (which represent the mass/wind coupling for example).

Univariate covariance matrices are thus defined for each of the control variables, while multivariate coupling is obtained by applying a balance operator. The background error covariance matrix Bu of the control variables is then assumed to be block-diagonal with no correlation between the parameters:

  • equation image

where C(·) represents the background-error autocovariance matrix for each variable.

The method of CVT in Arpège is represented in Figure 1. With this change of variable, the problem is now easier to handle since the calculation of the multivariate covariance matrix is reduced to the calculation of the univariate blocks only.

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Figure 1. Illustration of the method of control variable transforms. Background-error covariance matrices are represented in terms of model variables (left) and control variables (right). Grey blocks correspond to non-zero elements.

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Each autocovariance matrix in Bu is designed with a hybrid grid point/spectral approach. It combines the grid point space for the representation of local variances, and the spectral space for the representation of spatial correlations. This hybrid formulation is given by Bu = ΣΓΣT, where Σ is the diagonal matrix of grid point standard deviations and Γ is the correlation matrix.

Standard deviations in the matrix Σ are specified for each control variable. For vorticity, variances are space and time varying, and provided by a real-time ensemble variational assimilation, as will be detailed in section 2.3. For the unbalanced parts of divergence, temperature and surface pressure, variances are static in time and horizontally homogeneous. They are provided by an off-line ensemble of perturbed variational assimilations (Fisher, 2003; Belo Pereira and Berre, 2006). For specific humidity, variances are space and time varying, and obtained through an empirical flow-dependent formula, as will be detailed in section 2.5. Moreover, a multiplicative inflation factor equal to 1.8 is applied to standard deviations in Σ, on the basis of a posteriori diagnostics (Desroziers and Ivanov, 2001), in order to represent model error contributions.

The correlation matrix Γ is modelled with a spectral diagonal hypothesis, Γ = S−1D(S−1)T, where D is a block-diagonal matrix in spectral space (with a block for each spectral coefficient and each variable), and S represents the spectral-to-grid transformation. The D matrix is defined such that spectral coefficients are not correlated, but a full vertical autocorrelation matrix is specified for each spectral coefficient. The resulting correlations are non-separable, homogeneous and isotropic in grid point space. The shape of horizontal correlations is determined by the correlation spectra. A climatological estimate of the correlations is obtained from an ensemble of perturbed variational assimilations (Fisher, 2003; Belo Pereira and Berre, 2006), and is specified in Bu.

Finally, the B matrix in terms of model variables is not explicitly defined, but obtained as

  • equation image

where Kb is the balance operator (detailed in section 2.4), which transforms control variables to model variables. The additional elements included in B over those in Bu are the balanced parts of the covariances.

This method for modelling background-error covariances includes a first degree of flow dependence. This arises in particular from the use (within Kb) of linearized versions of nonlinear and omega balance equations (Fisher, 2003) that depend on the background state.

2.3. Calculation of flow-dependent variances

To introduce further flow dependence in the modelled background-error statistics, the ensemble variational assimilation that has been implemented at Météo-France and used operationally since July 2008 is considered. The first operational version of this ensemble (as described in detail in Berre et al., 2007) is based on a lower and unstretched horizontal resolution version (T359C1.0L60) of the Arpège operational version (T538C2.4L60), where C is the stretching factor, and uses 3D-Fgat (first guess at appropriate time) to approximate 4D-Var. In this configuration, it consists of six independent 3D-Fgat assimilation experiments performed in real time in a perfect model framework, with explicit perturbations of observations and implicit perturbations of backgrounds.

Denoting the perturbed backgrounds by equation image and the number of members in the ensemble by N, the ensemble estimate of the background-error covariance is then classically calculated as

  • equation image

Numerous authors have reported and worked on the problems raised by this ensemble formulation, such as the sampling noise that affects finite-size ensemble estimates. Regarding correlations, the key issue of spurious non-zero values at long distances has been widely studied (Houtekamer and Mitchell, 1998; Buehner and Charron, 2007; Pannekoucke et al., 2007; Bishop and Hodyss, 2007; Kepert, 2009). Regarding the sampling noise observed on the estimated variances, the literature is often elusive on this subject. Raynaud et al. (2008, 2009) and Berre and Desroziers (2010) have examined this issue so far, from both experimental and theoretical points of view. The first study gives empirical insights on the spatial properties of the sampling noise and of its appropriate filtering, while the second study, based on these empirical results, proposes an objective and automatic filtering procedure to remove this sampling noise.

In the current study, flow-dependent ensemble-based variances are calculated for all control variables. Estimated variances are then filtered before being used in the assimilation process (i.e. in the matrix Σ). This is done with the filtering approach proposed by Raynaud et al. (2009). Basically, the filtering of the raw estimates is performed with a spectral low-pass filter, defined by

  • equation image(1)

with an objective choice of the truncation Ntrunc for each control variable and each vertical level. This objective truncation is determined according to spectral noise-to-signal ratios of the estimated raw variance fields. Figure 2 presents vertical profiles of objective truncation for vorticity, unbalanced divergence, unbalanced temperature and specific humidity. These profiles display a quite similar behaviour for vorticity, divergence and temperature, with a decrease of the truncation with altitude, apart from a small increase near the tropopause height. This means that the filtering tends to be less scale selective at low levels than at higher levels. It can also be noticed that the filtering truncations for these variables have comparable values above 700 hPa, while they differ markedly at lower levels. The behaviour is a bit more complex for specific humidity, with successive decreases and increases of the truncation.

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Figure 2. Vertical profiles of objective truncation Ntrunc, as defined in Eq. (1), for the filtering of ensemble background-error variances of vorticity (solid), unbalanced divergence (dashed), unbalanced temperature (dotted) and specific humidity (dashed dotted). The left ordinate axis represents the model level, with level 60 being near the surface. The corresponding pressure levels are indicated on the right ordinate axis. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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2.4. Balance relationships

As mentioned in section 2.1, background-error autocovariances in the Arpège model are first calculated for the control variables, and transformed back to the space of model variables with balance relationships. The balance operator Kb is defined so that the covariances for divergence, temperature and surface pressure are related to their unbalanced components and to vorticity through the following equations:

  • equation image

The M, N and P operators define the balance relationships. They are calculated from a combination of multiple linear regressions plus nonlinear and omega balance equations (Derber and Bouttier, 1999; Fisher, 2003).

In the reference system (Table I), flow-dependent vorticity variances in C(ζ) are calculated from the ensemble, while the variances for the unbalanced parts, C(ηu) and C[(T,PS)u], are obtained from climatology. However, the above equations show that vorticity plays an important role since the covariances of balanced parts of divergence, temperature and surface pressure are linear functions of the covariances of vorticity. The introduction of flow dependence in vorticity variances thus enables the variances of associated balanced parts to be flow dependent too. Moreover, the flow dependence of balanced variances is strengthened by the fact that some of the balance operators are also flow dependent (see end of section 2.2).

On the other hand, a full flow-dependent description of background-error variances can be obtained with a flow-dependent specification of the variances for both the balanced and the unbalanced parts. The addition of flow-dependent unbalanced variances is part of the experiments presented in the next sections.

2.5. Calculation of specific humidity variances

In the reference assimilation scheme, variances of specific humidity are not derived from a climatology as is the case for the unbalanced variables. In the case of humidity, variances are calculated with a simple empirical formula as a function of the background temperature and relative humidity (Rabier et al., 1998). Moreover, to avoid a systematic drift of stratospheric humidity, increments are forced to be negligibly small above the tropopause. This is achieved by setting a very low value of 10−8 for humidity background-error standard deviations everywhere the pressure is lower than 70 hPa. The humidity in the stratosphere in this system is then mainly driven by the model, and weakly controlled by observations. The replacement of these empirical variances by ensemble variances is also examined in this study.

2.6. Design of experiments

The main investigation of this study is to extend the use of ensemble-based background-error variances to all variables in the minimization (instead of vorticity only), and to test their impact on the Arpège assimilation/prediction system. For that purpose, variances are calculated at each analysis time from the variational six-member ensemble for all control variables and all vertical levels, and they are specified in the operational 4D-Var as a replacement for the previous operational variances. This means that variances in C(ζ),C(ηu),C[(T,PS)u] and C(q) are all provided by the ensemble calculation.

It may be mentioned that this study deals with flow-dependent variances only. Flow-dependent correlations could also be calculated from the ensemble, but this is beyond the scope of this paper and correlation spectra in this study are kept climatological.

Experiments have been performed over a 1-month period in February–March 2008 with a T538 spectral resolution and a C2.4 stretching factor. This implies a maximum spectral resolution of T1290 over Europe (which corresponds to about 15 km in physical space). A wide range of observation data types are used: surface observations, aircraft data, satellite-derived winds, sea surface observations (e.g. drifting buoys, ship reports), in situ sounding data, wind profiler radar data, geostationary satellite winds (atmospheric motion vectors), Global Positioning System (GPS) ground-based data and radiances from polar-orbiting satellites (e.g. AMSU-A/B, AIRS, SSMI, IASI). Various checks are performed to select a ‘clean’ set of observations. This selection involves quality checks, removal of duplicated observations and thinning of their resolution, in particular. The quality control is based on observation–background departures, with a rejection of the observation if the difference is larger than a given threshold. In this study, ensemble-based background errors ‘of the day’ are specified in the minimization step only, while background-error variances for observation quality control are obtained from quasi-static covariances using a randomization technique (Fisher and Courtier, 1995; Andersson and Fisher, 1999).

A summary of the experiments performed is given in Table I.

Table I. Description of the two systems considered. Each system uses a specific set of background-error standard deviations. The reference system corresponds to the previous operational configuration (until April 2010), while the experimental system, which uses a full set of flow-dependent standard deviations, corresponds to the current operational configuration.
SystemFlow-dependent ensemble SDClimatological or empirical SD
Referenceζηu,(T,PS)u,q
Experimentalζ,ηu,(T,PS)u,q 

3. Diagnostic study of background-error variances of full variables

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental set-up
  5. 3. Diagnostic study of background-error variances of full variables
  6. 4. Impact of an extended specification of flow-dependent errors
  7. 5. Conclusions and perspectives
  8. Acknowledgements
  9. References

Reference and experimental (i.e. with extended flow dependence) background-error standard deviations are compared for the following full variables: vorticity, surface pressure, temperature and specific humidity. Full variables are considered instead of control variables since they are more directly linked to observed variables. The standard deviations were computed in spectral space at truncation T107. Moreover, both maps are filtered according to Eq. (1), with truncation Ntrunc = 42 for reference maps and Ntrunc objectively determined for extended flow-dependent maps (Figure 2). This comparison can be useful to anticipate the expected impacts of the extended errors ‘of the day’.

3.1. Global standard deviations

It has been first verified that global values of ensemble-based standard deviation for all variables are in good agreement with the reference system values at all levels (not shown). This indicates that the ensemble is able to properly capture the variability of the background state.

In the case of humidity, global ensemble variances are close to empirical estimates up to around 70 hPa, while they are much larger above, by approximately a factor between 8 and 15 (not shown). This is related to the artificially very low constant value of 10−8 attributed to humidity variances in the reference system (as discussed in section 2.5). Consequently, the impact of the new humidity variances at high levels will not be linked to flow dependence, but rather to a large systematic increase of amplitude. In order to be coherent with the main purpose of the paper, which is about the impact of flow-dependent background-error variances, and also because more investigations are necessary to clarify the validity of such an increase at high levels, results and discussions are limited to the part of the atmosphere below 70 hPa in the remainder of the paper.

3.2. Local standard deviations

In this section, the geographical distribution of background-error variances is examined, for both the reference and the experimental systems, for the particular case of 03 UTC on 15 February 2008.

Figure 3 shows standard deviation maps of vorticity near the surface. As mentioned before, these two maps are obtained from the same six-member ensemble and only differ in the truncation of the applied filter (which is equal to T42 in the reference map and T110 in the experimental map). As expected, these variance fields are closely linked to the underlying flow, with high values located in the lows and troughs. The larger filtering truncation used in panel (b) tends to enhance small-scale features. The connection with the weather situation tends to be strengthened, with high standard deviation values that follow the shape and the bend of troughs more closely (e.g. Northern Pacific and Atlantic).

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Figure 3. Background-error standard deviations of vorticity near the surface. (a) Reference map and (b) experimental map corresponding to 15 February 2008 at 03 UTC. Unit: 5.10−5 s−1. The mean sea-level surface pressure analysis is overlaid; contour: 10 hPa. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Figure 4 displays standard deviation maps of surface pressure. It can be noticed that the reference map already represents some flow-dependent features, especially in the Southern Hemisphere and in the storm-track regions (e.g. Northern Pacific). This first degree of flow dependence results from the use of both nonlinear balance operators and vorticity variances ‘of the day’ (section 2.4). These geographical variations are confirmed and reinforced in the experimental map. In particular, standard deviations tend to be increased in the lows and in the Tropics. Tropical cyclone Nicholas, along the northwest coast of Australia, has also a stronger signature in background error. These differences between reference and experimental maps partly result from the new flow-dependent specification of unbalanced variances (section 2.4). A more detailed understanding of the role of these unbalanced variances will be given in section 3.3.

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Figure 4. Background-error standard deviations of pressure near the surface. (a) Reference map and (b) experimental map corresponding to 15 February 2008 at 03 UTC. Unit: hPa. The mean sea-level surface pressure analysis is overlaid; contour: 10 hPa. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Standard deviations of temperature at 700 hPa are shown in Figure 5. Positions of highs (storm-track regions) and lows (land and Tropics) of standard deviation appear to be relatively similar in reference and experimental maps. However, local maxima are increased with an extended specification of flow-dependent variances.

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Figure 5. Background-error standard deviations of temperature near 700 hPa. (a) Reference map and (b) experimental map corresponding to 15 February 2008 at 03 UTC. Unit: K. The mean sea-level surface pressure analysis is overlaid; contour: 10 hPa. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Figure 6 presents maps of specific humidity standard deviation near 850 hPa. On the one hand, it turns out that reference and experimental maps are relatively similar in terms of geographical variations (correlation value of 0.9). In particular, they display a common latitudinal structure, with the lowest values in polar regions and the highest values in the Tropics, which is consistent with the temperature dependence of specific humidity. On the other hand, local variance maxima tend to be strengthened in the ensemble map, in particular in the Tropics.

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Figure 6. Background-error standard deviations of specific humidity at 850 hPa. (a) Reference map and (b) experimental map corresponding to 15 February 2008 at 03 UTC. Unit: 0.5 × 10−2 kg/kg. The mean sea-level surface pressure analysis is overlaid; contour: 10 hPa. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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All these comparisons thus show that using ensemble-based variances ‘of the day’ for the unbalanced variables and for specific humidity tends to strengthen the connection between variances and the local weather situation, in particular in the vicinity of intense weather events such as mid-latitude storms and tropical cyclones.

3.3. Diagnostic evaluation of the contribution of flow-dependent variances of unbalanced variables

The respective contributions of the variances of the vorticity-balanced and residual parts to the full variances are investigated in the case of surface pressure. For that purpose total variances represented in Figure 4(b) are decomposed into vorticity-balanced surface pressure variances (term NC(ζ)NT) and into the term PC(ηu)PT + C[(T,PS)u] (section 2.4), which represents the contribution from the new flow-dependent specification of variances of unbalanced components (ηu and (T,PS)u). Figure 7 shows that vorticity-balanced variances already represent a large amount of the total variances, especially in dynamical areas (e.g. Southern Hemisphere and Northern Pacific), where they control up to 80% of the total variances. The contribution of the new flow-dependent unbalanced variances also appears to be linked to the underlying flow and thus tends to reinforce the flow dependence of total variances, through noticeable impacts especially in lows and in the tropical area. Calculations show that these variances explain almost 40% of the total variances on average, with larger relative contributions near extra-tropical lows and in tropical convective areas. This emphasizes the importance of a proper representation of the unbalanced statistics.

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Figure 7. Extended flow-dependent background-error standard deviations of surface pressure corresponding to 15 February 2008 at 03 UTC (Figure 4(b)), decomposed into (a) vorticity-balanced surface pressure variances (NC(ζ)NT) and (b) ηu-balanced plus unbalanced surface pressure variances (PC(ηu)PT + C[(T,PS)u]). Unit: hPa. The mean sea-level surface pressure analysis is overlaid; contour: 10 hPa. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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4. Impact of an extended specification of flow-dependent errors

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental set-up
  5. 3. Diagnostic study of background-error variances of full variables
  6. 4. Impact of an extended specification of flow-dependent errors
  7. 5. Conclusions and perspectives
  8. Acknowledgements
  9. References

In this section, it is examined how analyses and forecasts are influenced by the extended specification of flow-dependent background-error variances in the experimental system.

4.1. Innovation statistics and forecast scores

The global impact of the specification of flow-dependent variances of unbalanced variables and humidity, measured with space and time-averaged innovation statistics and forecast scores (RMS error), appears to be neutral to positive. This is illustrated in Figure 8 with temperature innovation RMS against dropsondes over the globe, and in Figure 9 with RMS error of 4-day geopotential height forecasts against ECMWF analyses over a European Atlantic domain (30–70° N; 10° W to 35° E).

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Figure 8. Innovation RMS for the experimental system (dashed line) and for the reference system (solid line). Black lines represent observation–background statistics and red lines represent observation–analysis statistics. Results are given for temperature from dropsondes over the globe. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Figure 9. Vertical profile of RMS error of 96 h forecasts of geopotential height (m) over a European Atlantic domain for the reference run (solid line) and for the experimental run (dashed line). Forecasts are verified against ECMWF analysis.

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To evaluate the temporal robustness of the positive impact highlighted in Figure 9, the temporal evolution of the RMS forecast error is presented in Figure 10 for the case of 96 h geopotential height and wind forecasts at 250 hPa over Europe. One can point out noticeable reductions of RMS error over the whole period, with a tendency to reduce error peaks.

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Figure 10. Temporal evolution of RMS error of 96 h forecasts of (a) geopotential height (m) and (b) wind (m s−1) at 250 hPa over a European Atlantic domain, for the reference run (solid line) and for the experimental run (dashed line). Forecasts are verified against ECMWF analysis.

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This suggests that the impact of the new variance specification may be relatively localized both in space and time. This was to be expected in view of variance fields modifications (section 3.2), which mostly concern localized dynamically active systems. Consequently, there is not a systematic improvement in forecast scores, but rather a tendency to reduce local errors, possibly connected to intense weather systems. To further investigate this aspect, the case of an intense storm event is examined in more detail in the next section.

4.2. A case study of cyclogenesis

As can be seen on the variance maps presented in section 3, ensemble variances tend to be increased in low-predictability regions, which suggests that observations there should be better used through a larger weight in the analysis. While impact studies in the Arpège model showed that extended flow-dependent variances have a neutral to positive global impact on forecasts (section 4.1), larger improvements may be expected in intense weather events(Isaksen et al., 2007).

On Tuesday 10 February 2009, an intense storm hit the northern part of France. This storm developed during the night of 9/10 February over the Atlantic Ocean and moved along the English Channel coasts. The strongest winds have been localized south of the storm, with violent gusts that first hit the coast, before reaching the eastern part of the country in the early morning. The highest wind gusts have been observed along the coasts, with values up to 40 m s−1.

Figure 11 shows the analysis of surface pressure valid on 10 February 2009 at 00 UTC, along with 48 h forecasts issued from analyses valid on 8 February 2009 at 00 UTC. Forecasts from the reference run and from the experimental run are compared. The experimental run has been initialized on 4 February 2009 at 18 UTC in order to cycle flow-dependent variances a few days before the development of the storm.

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Figure 11. 48h Arpège forecasts of surface pressure valid on 10 February 2009 at 00 UTC: issued from the reference run (in blue) and from the experimental run (in red). The reference analysis is overlaid (in black). Contour: 5 hPa. The 985 hPa isoline is plotted in bold for the three fields.

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While the storm is moved to the southwest in the reference forecast compared to the analysis, the experimental forecast tends to reposition the storm toward the correct position. The central pressure of the storm is also predicted better, as shown by the spatial extension of the 985 hPa isoline. In connection with this, the central pressure of the storm is decreased from 981.7 hPa (in the reference forecast) to 980.5 hPa (in the experimental forecast), which is closer to the analysis (976.6 hPa). Flow-dependent background-error variances thus contribute to correct both position and intensity errors in this example of intense weather event.

To examine the impact of surface pressure variances on this storm forecast, another experiment has been run (not shown), in a configuration similar to the experimental one, except that variances for equation image are static and homogeneous. This can be compared to the previous experimental configuration to obtain the impact of extended flow-dependent surface pressure variances. It is observed that this other experimental forecast locates the centre of the storm north of the reference forecast, but it remains too westerly compared to the analysed storm. The addition and the cycling of flow-dependent surface pressure variances thus enable this remaining position error to be partly corrected.

The results obtained from the impact studies thus support the idea that there is some relevant and robust information to be used routinely in the extended flow-dependent variance maps. In particular, the case study of cyclogenesis suggests that flow-dependent variances should be beneficial to the prediction of intense local weather events. Further case studies will be considered in future studies to assess the statistical significance of these conclusions.

5. Conclusions and perspectives

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental set-up
  5. 3. Diagnostic study of background-error variances of full variables
  6. 4. Impact of an extended specification of flow-dependent errors
  7. 5. Conclusions and perspectives
  8. Acknowledgements
  9. References

Representation of flow-dependent aspects in variational data assimilation is currently the subject of a number of research studies. Among the different approaches proposed, the ensemble variational assimilation implemented at Météo-France offers a suitable framework to estimate background-error covariances ‘of the day’.

In the present paper, this ensemble method has been used to provide flow-dependent background-error variances to the operational Arpège 4D-Var. While ensemble estimates were previously specified for vorticity only (reference configuration), the impact of extending their use to all variables in the minimization has been examined (experimental configuration).

Diagnostic studies first showed that local variance maxima for all variables tend to be strengthened in the experimental configuration. This is particularly noticeable in the vicinity of lows and troughs on variance maps of surface pressure. This suggests that observations could potentially be better used in dynamically active regions.

The impact of an extended specification of flow-dependent variances in the operational Arpège assimilation has then been explored through a series of impact studies. It turned out that the extended use of ensemble variances has a neutral to positive global impact on the forecast scores, which is related to both humidity and unbalanced components variances. This impact is more pronounced for medium-range geopotential height and wind forecasts over Europe.

Examination of a severe winter storm over the northern half of France showed that an improvement in the 48 h forecast of surface pressure –in terms of both position and intensity of the low –can be obtained from the use of flow-dependent variances for all control variables, with a large contribution from surface pressure variances. This supports the idea that localized improvements in dynamically active regions, rather than a general positive impact, are to be expected with the new variance specification.

Since a real-time variational ensemble is already running operationally at Météo-France, the use of ensemble variances for all control variables in the operational Arpège assimilation is straightforward and cost free. Given the results of this study, the extended use of errors ‘of the day’ has been made operational in April 2010 for the unbalanced variables. A further extension to humidity has been made operational in November 2010. It may also be mentioned that another operational change in April 2010 involved the use of a 4D-Var ensemble (instead of 3D-Fgat).

Moreover, this ensemble approach assumes that the forecast model is perfect (apart from the inflation of B). It is likely that this current neglect of model error causes underestimation of the background perturbation amplitudes. Therefore, it is also planned to work on inflation techniques within the ensemble simulation, by using a posteriori diagnostics (Desroziers et al., 2005, 2009; Miyoshi, 2005).

The specification of ensemble variances in limited-area models is another important topic that should be further examined. There have already been successful attempts to estimate the static part of regional error covariances (Desroziers et al., 2007; Brousseau et al., 2011), and an extension to represent space- and time-dependent covariances is also investigated (Monteiro and Berre, 2010). For example, Montroty (2008) showed the first promising results with the ALADIN-Réunion model for the prediction of tropical cyclones (in the context of a 3D-Var assimilation).

Finally, the new variance specification tested in this study is to be considered as a first step in the implementation of flow-dependent covariances in the Arpège model. The following step is to include also flow-dependent correlations, based on the developments by Fisher (2003) and Pannekoucke et al. (2007) on a wavelet representation for instance.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Experimental set-up
  5. 3. Diagnostic study of background-error variances of full variables
  6. 4. Impact of an extended specification of flow-dependent errors
  7. 5. Conclusions and perspectives
  8. Acknowledgements
  9. References
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