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Keywords:

  • AROME;
  • B matrix;
  • precipitation

Abstract

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References

This study focuses on diagnosing variations of background-error covariances between precipitating and non-precipitating areas, and on presenting a heterogeneous covariance formulation to represent these variations in a variational framework. The context of this work is the assimilation of observations linked to precipitation (radar data especially) in the AROME model, which has been running operationally at Météo-France since December 2008 over French territory with a 2.5 km horizontal resolution. This system uses multivariate background-error covariances deduced from an ensemble-based method. At first, such statistics have been computed for 17 precipitating cases using an ensemble of AROME forecasts coupled with an ALADIN ensemble assimilation. Results, obtained from 3 h forecast differences performed separately for non-precipitating and precipitating columns, display large discrepancies in error variances, correlation lengths and the correlations between humidity, temperature and divergence errors.

These results argue in favour of including heterogeneous background-error covariances in AROME incremental 3D-Var, allowing different covariances to be used in regions with different meteorological patterns. Such a method enables us to get increments more adequately structured in those regions, and thus potentially to make better use of observations in a data assimilation system. The implementation consists of expressing the analysis increment as the sum of two terms, one for precipitating areas and the other for non-precipitating areas, making use of a mask that defines rainy regions. This implies a doubling in the size of the control variable and of the gradient of the cost function. The feasibility of this method is shown through experiments with four isolated observations. Copyright © 2010 Royal Meteorological Society


1.  Introduction

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References

In order to provide information in areas with no observations and to supply a realistic reference state for use in observation operators, variational data assimilation needs an a priori (or background) meteorological state. The background state is not perfect and its errors are taken into account in the variational system through the use of the background-error covariance matrix, denoted B. As pointed out by Daley (1991), B has a profound impact on the analysis, by (i) weighting the importance of the a priori state, (ii) smoothing and spreading information from observation points, and (iii) imposing balance between the model control variables. Accurate knowledge of background-error statistics is thus important to the success of the assimilation process. To estimate the background error, two main practical difficulties occur. First, the ‘true’ state is unknown. This can be addressed by using differences between forecasts as a proxy for background errors (see reviews by Berre et al., 2006; Bannister, 2008). Second, because of its size, B can be neither estimated at full rank nor stored explicitly. This is commonly addressed by simplifying the covariance model as the product of a balance operator, which takes multivariate couplings into account, and of a spatial transform, which often uses assumptions such as homogeneity and isotropy.

The AROME numerical weather prediction (NWP) system, which provides operational forecasts of tropospheric phenomena with a 2.5 km horizontal resolution over France, uses a climatological multivariate background-error covariance matrix deduced from an ensemble-based method (Brousseau, et al.., 2008). The ensemble of forecasts built for that purpose gathers summer and winter cases in order to compute background-error covariances representative of a wide range of forecast errors. However it is well known that the structure functions that compose the covariances strongly depend on the weather regime. For instance, large differences between midlatitude and tropical error covariances have been shown at synoptic (Daley 1991; Derber and Bouttier 1999) and regional (Montmerle, et al.., 2006) scales. More recently, Caron and Fillion (2010) have shown that, compared to dry areas, forecast errors drift away from linear geostrophic balance over precipitation areas and that this deviation is proportional to the intensity of precipitation. In variational data assimilation, such discrepancies have a direct impact on the structures of analysis increments and on multivariate couplings through balance relationships. As a consequence, climatological covariances often produce sub-optimal increment structures in regions characterized by strong gradients such as precipitating fronts or the top of the boundary layer. This is particularly true when assimilating observations linked to precipitation such as radar data, because rainy areas are likely to be under-represented in the ensemble.

As pointed out by Auligné, et al.. (2010), one of the main issues to analyze clouds and precipitations at the mesoscale is to develop methods that aim to bring more flow-dependency to the background-error covariances. By applying specific diabatic balance operators in the linear balance operator proposed by Derber and Bouttier (1999), Caron and Fillion (2010) show that some improvements in the coupling between mass and rotational wind increments could be obtained over precipitating areas. However, the horizontal variation of this type of diabatic balance to be used during the minimization is not addressed in this latter study. One way to get some flow dependency is to consider nonlinear balance relationships in the balance operator, such as the nonlinear geostrophic balance equation (Barker, et al.., 2004; Fisher 2003) and the quasi-geostrophic omega balance (Fisher 2003). The latter approach is properly justified for synoptic scales, but for mesoscales prognostic rather than diagnostic balance equations (where moist physical processes play a dominant role to the leading order of approximation) may be more appropriate (e.g. Pagé, et al.., 2007. One could also use 4D-Var instead of 3D-Var, the evolution of B in 4D-Var corresponding to the evolution of the error covariances in a Kalman filter (Fisher, et al.., 2005). However, in this case, flow-dependency is somehow limited because B is replaced by a static estimate at the beginning of each assimilation cycle. Main efforts are nowadays given to the use of ensemble-based flow-dependent background-error covariances, which consists of including (partially or totally) into 3D-Var or 4D-Var a flow-dependent B matrix, which is computed from daily runs of an ensemble assimilation system (e.g. Kucukkaraca and Fisher, 2006; Berre, et al.., 2007). Although very attractive (weather-dependent covariances, sharper correlations), this method is computationally very expensive, especially for cloud-resolving models (CRMs). A solution is to consider an ensemble with few members and use optimized filtering techniques to reduce sampling error in B. Two main approaches are currently used. The first one considers control variable transform in ensemble sub-space (En3D-Var: Lorenc et al., 2003; En4D-Var: Liu et al., 2008) using localizations with Schur operators to reduce sampling noise effects. The second approach, which is developed and used operationally at Météo-France (Desroziers, et al.., 2008), is based on spectral filtering of error standard deviations (Raynaud, et al.., 2009). Flow-dependent ensemble-based correlations may be represented and filtered using a wavelet approach (Fisher 2003; Pannekoucke, et al.., 2007).

We propose here an alternative approach allowing specifically computed background-error covariances to be applied in regions characterized by different meteorological situations. In this paper, these different regions are precipitating and non-precipitating areas. The goal of such a method is to make better use of observations through the application of more adequate multivariate relationships and through a better localization of increments, avoiding the computation of a daily ensemble of forecast. This idea has been briefly mentioned in Courtier et al. (1998; their Eq. (31)), who propose such a formulation in order to use different length-scales depending on geographical location, allowing for broader length-scales in data-sparse areas than in data-rich areas. A similar idea has been considered at Environnement Canada (M. Buehner, personal communication) for representing latitude-dependent covariances.

By applying geographical masks to differences of AROME forecasts valid at the same time, section 2 presents to what extent modelled covariances can differ in precipitating and non-precipitating areas over France for a CRM that explicitely represents convective processes. The large differences that have been found lead us to include a heterogeneous background-error covariance matrix in AROME incremental 3D-Var, allowing different covariances to be used in precipitating and non-precipitating areas. The theoretical aspects of such a method are discussed in section 3, followed by a proof of concept based on a simple four-observation experiment in section 4.

2.  Background-error covariances for rainy and non-rainy areas

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References

2.1.  The multivariate formulation

Berre (2000) has proposed a multivariate formalism for the ALADIN limited-area model adapted from Parrish et al. (1997) and Derber and Bouttier (1999) for global NWP systems. This formalism uses linear regressions between errors of different physical quantities, to represent multivariate couplings such as geostrophy. Following the expression used in (e.g.) Derber and Bouttier (1999), these regressions are referred to as balance operators. These couplings are inferred from statistical regressions with an extra multivariate relationship for specific humidity. Compared to a univariate approach, the multivariate relationship for humidity allows (e.g.) misplaced low-level moisture convergence to be shifted within convective systems as observed by Doppler radars (Montmerle and Faccani 2009). Positive impacts of similar multivariate approaches are also reported in Berre and Vignes (2002) and Liu, et al.. (2009), for instance. The use of spectral regressions allows one to obtain scale-dependent balance relationships that are representative of the area of interest, which explains why the formulation has been used for the operational AROME model which provides analyses and forecasts over France with a 2.5 km horizontal resolution up to 1.35 hPa. The statistical relationships are

  • equation image(1)

where (ζ,η,(T,Ps),q) are respectively forecast errors of vorticity, divergence, temperature and surface pressure, and specific humidity, which are the model control variables that are analyzed on model vertical levels; the subscript u stands for unbalanced (total minus balanced) fields. ℳ,��,��,��,ℛ and �� are linear regression operators relating the spectral vertical profiles of predictors and those of the predictands. As explained in the next section, the vertical balance operators are deduced from homogeneous and isotropic auto-covariance and cross-covariance matrices computed for several days, using geographical masks in order to compute separately statistics for rainy and non-rainy regions.

ℋ is a horizontal balance operator that transforms the spectral coefficients of vorticity ζ into those of the balanced geopotential Pb (i.e Pb = ℋζ), using an f-plane assumption. Balanced geopotential is supposed to be the balanced part of Pt, which is the linearized mass variable deduced from (T,Ps) by the linearized hydrostatic relationship (Parrish, et al.., 1997).

These regressions also give an insight into the percentage of explained variances of each total field by its predictors, and thus indicate the strength of the statistical links. Separating predictors into balanced and unbalanced components provides a set of independent variables better suited for the 3D-Var minimization. The background-error covariance matrix Bu is taken as block diagonal, each block consisting of the vertical error covariances for each residual variable and for each wave number:

  • equation image

The total background-error covariance matrix B is then retrieved as follows:

  • equation image

with

  • equation image

where the superscript T stands for the transpose and I is the identity.

2.2.  The AROME ensemble dataset

In the operational configuration of AROME, background-error covariances are computed using a method based on an ensemble of perturbed assimilations (Desroziers, et al.., 2008), whose formalism can be found in Berre, et al.., (2006). In this technique (Houtekamer, et al.., 1996; Fisher, 2003), observation and background perturbations are added (either explicitly or implicitly) to the unperturbed assimilation system, in order to simulate associated error contributions and their effects on the error cycling of the assimilation system.

Here the construction of the dataset follows the same approach: 6-member ensembles are considered and computed with the ALADIN and AROME models at 10 km and 2.5 km horizontal resolution respectively, nested into the real-time ARPEGE ensemble assimilation that has run operationally at Météo-France since July 2008 (Desroziers, et al.., 2008). Each member has been computed for 17 cases, chosen from April to July 2008, that are characterized by strong convective activity. The ALADIN ensemble includes a perturbed 3D-Var step (with perturbed backgrounds and perturbed observations), while the AROME ensemble has been conducted in spin-up mode, using the ALADIN ensemble as initial and lateral boundary conditions. Since the AROME RUC (Rapid Update Cycle) is based on cycled assimilation/forecast steps every 3 h (Brousseau, et al.., 2008), statistics on 3 h forecast differences equation image between members (k,l) have been calculated. Forecasts valid at 2100 UTC have been used; this time corresponds approximately to the maximum of convective activity for the chosen dates.

In this study, statistics have been computed separately for precipitating and non-precipitating areas without distinguishing between precipitation rates. As only members characterized with strong convection have been chosen to compute the error covariances, the following results should be representative of convective precipitation. For separating rainy and non-rainy areas, a mask has been applied to the forecast differences equation image in model space (i.e. for ζ,η,(T,Ps) and q) in order to take into account only those profiles whose vertically integrated mixing ratio of simulated precipitating rain exceeds 0.1 g kg−1 in both forecasts. In order to smooth the spatial structures of masked fields, a Gaussian blur has been applied in addition to the mask operators. This filtering has been found useful by allowing the retrieved correlation length-scales to be increased by 20% in precipitation areas that were unrealistically lower due to the sharp transitions between 0 and 1 at the edge of the mask. This Gaussian blur is based on a convolution with a Gaussian function in two dimensions:

  • equation image

where x and y are the distances from the origin on the zonal and meridional axes respectively, and σ is the standard deviation of the Gaussian distribution. The convolution has been performed within a 5 × 5 kernel with σ = 1.4dx, where dx is the horizontal grid spacing.

Applying such masks implies that the masked field will contain many zero values. As a consequence, standard deviations of the masked fields averaged over the whole horizontal domain are underestimated. Thus, in practice, they must be rescaled with respect to the proportion of the gridpoints within each area of interest (averaged over the time period), in order to recover accurate values.

2.3.  Comparisons of statistics obtained for non-precipitating and precipitating regions

2.3.1.  Auto-covariances

Figure 1 shows vertical profiles of standard deviations σb of forecast errors, as used in the AROME operational suite (with a horizontal average over the whole domain), and as deduced from the ensemble of precipitating cases.

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Figure 1. Vertical profiles of standard deviations of (a) humidity, (b) temperature, (c) divergence and (d) vorticity forecast errors for AROME over France for the operational version (full line), and deduced from an ensemble of precipitating cases: total domain (dotted line), precipitating (dashed line) and non-precipitating (dot-dashed line) areas.

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The humidity error standard deviation is smaller in rainy areas. This could be explained by the fact that, in this case, only saturated profiles characterized by small spread of q have been considered in the computation. Two maxima are displayed, in the boundary layer and around 600 hPa, which can be associated with humidity entrainment and detrainment respectively. This corresponds to larger vertical gradients of moisture near these levels, which implies that (e.g.) a given phase difference with respect to these structures is likely to produce relatively large error contributions. For the rainy subdomain, errors are also much larger at all altitudes for ζ and η, and to a lesser extent for T. For dynamical variables, this standard deviation increase is not induced by the mask itself; it reflects the important small-scale dynamical circulation within precipitating clouds. Variability in the low-level convergence and in the vertical extension of the clouds are in particular shown by the η profile. Statistics that have been obtained considering only non-rainy profiles are close to what is used operationally, whereas for the whole domain with the ensemble of precipitating cases, σb values are larger for dynamical variables that are representative of small scales, which reflects the higher convective activity in this ensemble.

To compare horizontal correlations, horizontal length-scales as defined by Daley (1991, section 4.3) have been computed and the results are plotted in Figure 2. This type of diagnostic indicates how fast the correlation function decreases away from the origin, and it is sensitive to the small-scale part of the variance spectrum in particular (Berre 2000). Smaller length-scales (5 km versus 10 km approximately) are obtained over rainy areas than over non-rainy areas for q and T, denoting more spatial variability for those variables in these areas. Furthermore, for these variables, the length-scales are nearly constant in the vertical, contrary to what is used in AROME operationally, where length-scales increase with height. This is probably due to the fact that the same bi-dimensional mask is applied in the entire column. Very similar profiles are obtained for ζ and η however (Figures 2(c) and (d)). These results indicate that much more localized increments can be obtained in precipitating areas using specific background-error statistics. This is of great interest for high-density observation networks like radar data or satellite radiances, since the representative scales of the resulting analyses should become much smaller (obviously by also paying attention to correlations between observation errors) and therefore more realistic.

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Figure 2. Vertical profiles of horizontal length-scales (km) of (a) humidity, (b) temperature, (c) divergence and (d) vorticity forecast errors for AROME over France for the operational version (full line), and deduced from an ensemble of precipitating cases: total domain (dotted line), precipitating (dashed line) and non-precipitating (dot-dashed line) areas.

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Signatures of vertical cloud development can be seen mainly in the mean vertical auto-correlations for q errors computed over rainy areas (Figure 3(a)), these mean correlations being a spectral average of the scale-dependent vertical correlations. Compared to what has been obtained for non-precipitating areas (Figure 3(b)), which is comparable to what is used operationally (not shown), much broader correlations are displayed for this quantity at mid-level (and to a lesser extent for T (Figures 3(c) and (d)) and ζ (Figures 4(c) and (d)). This point denotes stronger vertical mixing within clouds performed by the explicitly resolved convection. Broader vertical auto-correlations are also displayed for rainy areas in the lower troposphere because of mixing processes in the boundary layer. This is consistent with Figure 10(e) of Caron and Fillion (2010), in which such broadness of the analysis increment increases when the precipitation rate is larger.

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Figure 3. Average vertical auto-correlations computed for (a, c) precipitating and (b, d) non-precipitating areas of (a, b) humidity and (c, d) temperature forecast errors for AROME over France deduced from an ensemble of precipitating cases. Negative values are plotted as dashed lines.

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Figure 4. As Figure 3, but for (a, b) divergence and (c, d) vorticity forecast errors.

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2.3.2.  Cross-covariances

Variance ratios are used to compute the relative importance of balanced and unbalanced terms following the multivariate approach defined in Eq. (1). They are given by the ratio of the variance of each balanced term divided by the variance of the full field. They indicate the amount of increment (e.g. the analyzed minus a priori values) for a given variable that will be balanced with increments of other variables.

Figure 5 shows the vertical distributions of these ratios obtained over rainy and non-rainy areas for the specific humidity q. The differences in behaviour between these two areas are striking: in the rainy case, the total explained variance of q varies between 25% and 50% in the troposphere, and it is mainly controlled by unbalanced divergence ηu; in contrast, in the non-precipitating case, the total explained variance of q varies between 10% and 30% in the troposphere, and it is mainly driven by the unbalanced mass field (T,Ps)u. Figure 6 gives an insight into the scales that are involved in those couplings: the total explained variance of q is predominantly controlled by the unbalanced mass field (T,Ps)u at all scales for non-precipitating areas (with a maximum of 30% around 100 km, Figure 6(d)), and by the unbalanced divergence ηu mainly for mesoscales below 30 km (Figure 6(a)) for rainy regions. Note that this relatively strong coupling between moisture and convergence in precipitating areas is consistent with the analysis of mesoscale balance coupling in Pagé et al. (2007, section 4c). For the two regions, the coupling with balanced geopotential Pb (which is linked to vorticity by the horizontal balance operator ℋ, as seen in section 2) is almost non-existent. The statistics that are used operationally are computed without distinction between precipitating and non-precipitating areas. As expected, they display intermediate values with contributions of (T,Ps)u and ηu being predominant below and above 600 hPa respectively (not shown). It can be mentioned that the importance of q–(T,Ps)u couplings at low levels in this context is also consistent with the sensitivity of heating rates to low-level moisture perturbations as described in Fillion and Bellair (2004).

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Figure 5. Spectral averages of the percentages of explained specific humidity q error variances, as a function of height for AROME, computed over (a) precipitating and (b) non-precipitating areas. Pb is the so-called balanced mass, divu is the unbalanced divergence, and (T,Ps)u is the unbalanced temperature and surface pressure forecast error (see text for details).

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Figure 6. Fraction of specific humidity q error variance explained by (a, b) the unbalanced divergence (divu) forecast error and by (c, d) the unbalanced temperature and surface pressure (T,Ps)u forecast error as a function of scale for (a, c) precipitating and (b, d) non-precipitating areas. Contours are plotted every 5%, except for (b) where 1% has been used.

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For rainy and non-rainy areas, the main predictors for the computation of the specific humidity forecast-error covariances are thus respectively the unbalanced divergence ηu and the unbalanced temperature and surface pressure (T,Ps)u covariances. To show the main characteristics of the coupling between humidity and divergence, which has a key role for the assimilation of precipitating observations such as Doppler radar data (e.g. reflectivities and radial winds), spectrally averaged covariances between q and the total divergence η are plotted in Figure 7. For rainy areas, Figure 7(a) shows the mesoscale coupling between a low-level positive error on convergence and an over-estimation of specific humidity, explainable by the supply of humidity in a convergent system. A positive increment of humidity at 800 hPa will result from a strong convergence below and divergence above. On the contrary, and in accordance with Figure 7(b), such a humidification has much less impact on the wind field in non-rainy conditions, in the form of a weak local divergence. The signature of convection that has been found for rainy areas is also visible in the error covariances that are used operationally, but with a much smaller vertical extent (not shown). This indicates that, in the operational suite, increments are often balanced in a suboptimal way, either in precipitating areas (where the qη coupling is too weak) or in non-precipitating areas (where the qη coupling is too strong).

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Figure 7. Vertical covariances between specific humidity (y-axis) and divergence (x-axis) forecast errors for (a) precipitating and (b) non-precipitating areas deduced from an ensemble of precipitating cases. Contour interval is 5 × 10−10s−1kg−1kg, with negative values plotted as dashed lines.

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Couplings that explain the total variance of temperature and surface pressure errors (T,Ps) look quite similar over both domains (Figure 8): the balanced part of such variance is small (around 10% and 20% up to 300 hPa for non-precipitating and precipitating areas respectively), except around the tropopause where almost half of the variance is balanced and explained by ηu, mostly for scales below 100 km in both cases. This feature shows the variability of the interaction between strong winds at these levels and the tropopause height.

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Figure 8. (a,b) Spectral averages of the percentages of explained variances of temperature and surface pressure (T, Ps), as a function of height for AROME computed over (a) precipitating and (b) non-precipitating areas. As in Figure 5, Pb stands for the so-called balanced mass and divu for the unbalanced divergence.

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The strongly different behaviours that have been shown in this section, which are directly linked to convection processes that are resolved explicitly in AROME, confirm the interest of using different statistics in regions with different meteorological patterns. Such a method would indeed allow increments to be more adequately balanced and structured in those regions, and thus would make better use of observations in a data assimilation system.

3.  Implementation of a heterogeneous B matrix in 3D-Var

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References

3.1.  Theoretical aspects

The AROME system uses an incremental 3D-Var, directely derived from the one used by the regional model ALADIN (Fischer, et al.., 2005). The principle of this incremental formulation (Courtier, et al.., 1994) is to seek the increment δx to be added to the background xb, so that the analysis given by xa = xb + δx minimizes the cost function

  • equation image(2)

The background term Jb measures the distance between the analysis xa and the background xb, with B the background-error covariance matrix. The observation term Jo represents the distance between the innovation vector (yH[xb]) and the increment written in the observation space through the linearized observation operator H.

Following an approach suggested by Courtier et al. (1998, their Eq. (31)), we propose here to express the B matrix as a linear combination of two terms, characterizing non-precipitating (np subscript) and precipitating (p subscript) areas respectively:

  • equation image(3)

which can also be expressed as:

  • equation image(4)

The operators F and G define the areas where the non-precipitating and the precipitating statistics are applied respectively. These operators are based on 2D grid point masks that can be deduced from radar observations, as detailed in section 3.3.2. Theoretically, other terms could be added to this expression, each of these additional terms being applied exclusively to another part of the domain where the analysis is performed. One can imagine for example partitioning precipitating areas into convective and stratiform parts, or clear-air areas into stable and unstable parts. However, for the sake of simplicity (and of memory saving), we have chosen here to focus on only two terms.

The increment δx is written as the control variable χ renormalized by B1/2 :

  • equation image

From Eq. (4), the increment is a linear combination of two terms:

  • equation image(5)

Thus, this method doubles the size of the control vector. In the space of this renormalized control variable, the Jb cost function and its gradient are simply:

  • equation image(6)
  • equation image(7)

In the same space and following Eq. (2), the Jo cost function and its gradient are:

  • equation image(8)
  • equation image(9)

The size of the cost function gradient is doubled just as the size of the control variable (or multiplied by the number of different B matrices used in Eq. (4), if more than two matrices are used).

3.2.  Set-up of the masks

The complementary F and G operators define the spatial locations where Bnp and Bp are applied. The minimization being performed in spectral space, these operators are written as:

  • equation image(10)

Here S and S−1 are the Fourier and inverse Fourier, I is the identity matrix whose dimension is the number of grid points of the AROME domain times the number of control variables, and D is a diagonal matrix of the same dimension whose diagonal components are composed of 1 and 0 for rainy and non-rainy areas respectively. For the academic single-observation experiments that are presented in the next section, the computational domain is simply split into two parts: the northern part (e.g. north of 46.5°N) is considered as rainy, the southern as non-rainy. In a real-case configuration, D can for instance be deduced through the interpolation of reflectivity observations (coming from a dense radar network) to the model grid. A proper threshold on reflectivity value would then allow one to apply exclusively the rainy covariances displayed in section 2, representative of strong convection, within areas characterized by convective precipitation.

4.  Results of assimilation with four observations

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References

To ensure the reliability of the new formulation of the variational system described in the previous section, three different experiments have been performed:

  • CNTRL-Bnp aims at controlling the impact on analysis of the non-precipitating Bnp matrix, using the standard formulation of the variational system (e.g. considering only one B matrix). For that purpose, four pseudo-observations, whose locations are 48°N, 4.5°E and 42.5°N, 4.5°E at 800 and 500 hPa, are assimilated. These pseudo-observations have been generated by considering–30% relative humidity innovations (e.g. observation minus background) at those locations.

  • CNTRL-Bp is the equivalent of CNTRL-Bnp but using the precipitating Bp matrix.

  • Bnp-Bp uses the hybrid formulation of Eq. (4) and other ingredients listed in the previous section in the variational system, considering that the northern (i.e. north of 46.5°N) and southern halves of the domain are precipitating and non-precipitating areas respectively.

The large differences of correlation lengths between the two B matrices displayed in Figure 2 directly impact on the structure of q and T increments, much tighter increments being obtained in CNTRL-Bp than for CNTRL-Bnp (Figures 10 and 9 respectively). A more pronounced impact on the wind field can be seen for CNTRL-Bp, due to differences in error balances. The Bnp-Bp experiment displays what was expected: in the ‘rainy’ northern part and in the ‘non-precipitating’ southern part of the domain, Figure 11 shows exactly the same increment structures as CNTRL-Bp and CNTRL-Bnp respectively. This is a proof of concept that increments with very different behaviours, in terms of intensities and shapes, can be obtained simultaneously using the heterogeneous B matrix formulation, and that different multivariate relationships can be used over different areas in a more adaptive way.

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Figure 9. Vertical north–south cross-sections of the increments of (a) temperature, (b) specific humidity and (c) wind (contours at intervals 5 × 10−2 K, 0.1 g kg−1 and 5 × 10−2m s−1 respectively), generated by four pseudo-observations located at 48°N, 4.5°E and 42.5°N, 4.5°E, at 800 and 500 hPa, characterized by–30% innovations of relative humidity. These increments have been obtained by considering the non-precipitating Bnp matrix in the standard formulation of AROME data assimilation system. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Figure 10. As Figure 9, but considering the precipitating Bp matrix in the standard formulation of the AROME data assimilation system. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Figure 11. As Figure 9, but after having applied the heterogeneous B matrix, composed of Bnp and Bp as in Eq. (4), and considering the northern half of the domain as rainy. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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To check how the variational system behaves near the precipitating/non-precipitating border, one extra experiment that considers four pseudo-observations localized near 46.6°N, 4.5°E and 46.3°N, 4.5°E has been performed. Strongly anisotropic vertical structures of increments are displayed in Figure 12 for that case. This shows that, using the heterogeneous background-error covariance formalism, the information brought by observations of precipitation such as radar reflectivities or radial velocities could be localized more realistically and used more optimally in precipitating areas, without spreading too much towards non-precipitating regions, thanks to the masking and the use of smaller correlation lengths. To smooth increment structures in the transition zone however, recent studies have shown that the use of a gaussian transition between masks in D seems well suited.

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Figure 12. As Figure 11, but after having assimilated the pseudo-observations near the precipitating/non-precipitating border, and expanded between 46°N and 48.5°N. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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5.  Conclusions

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References

When assimilating observations at the mesoscale, such as satellite radiances in clear-air conditions or radar data in precipitating areas, one can often notice unrealistic increments. Typically, an isolated observation spreads information too much spatially, due to the background-error covariance matrix that propagates the innovation (observation minus background) with no regard to the surrounding meteorological conditions. In the French operational model at the mesoscale (AROME), such a matrix contains climatological values computed from an ensemble assimilation that gathers winter and summer situations, precipitating and non-precipitating cases. Thus, it poorly represents strong spatial and temporal meteorological variations, such as the overpass of a narrow rainband or the presence of strongly convective conditions. In order to quantify this misrepresentation, background-error covariances have been firstly diagnosed for precipitating and non-precipitating areas. For that purpose, an ensemble of six AROME forecasts, nested in an ensemble of six ALADIN forecasts initialized from perturbed analyses and coupled to perturbed ARPEGE forecasts, has been built for 17 convective situations. Statistics have been computed from forecast differences, and considering separately rainy and non-rainy profiles in those forecasts. Compared to non-rainy areas, precipitating areas are mainly characterized by:

  • (i)
    larger error standard deviations for η and ζ (and for T, to a lesser extent), which denotes a more intense small-scale dynamical activity,
  • (ii)
    smaller error standard deviations for q,
  • (iii)
    50% smaller correlation lengths for q and T,
  • (iv)
    larger vertical auto-correlations in the mid-troposphere reflecting the stronger vertical mixing within clouds performed by the explicitly resolved convection, and
  • (v)
    very different contributions, in scale and in intensity, to the explained q error variances due to different effects of precipitating clouds such as the presence of low-level cold pool, low-level convergence, latent heat release, and cloud-top divergence.

Thanks to smaller correlation lengths, using such statistics exclusively in precipitating regions would allow the number of observations such as radar data to be increased, permitting more realistic analysis of precipitating structures. Furthermore, the impact of such observations would be optimized through the different auto-covariances and couplings between variables which favour convective activity. This has motivated us to find a heterogeneous covariance formulation to represent different background-error covariances over different regions.

An original method, which consists of expressing the increment as a linear combination of two terms, characterizing precipitating and non-precipitating areas respectively, has been developed in this way. This method implies the doubling of the size of the control variable and of the gradient of the cost function. A mask, which specifies where each of the two matrices should be applied, must also be defined. The feasibility of such a method has been shown by performing academic experiments using four observations and by considering basic masks. The experiment that uses the heterogeneous B matrix formulation displays increments with strongly different shapes in pseudo-non-precipitating and pseudo-precipitating areas. Thus, this approach addresses some of the main issues that have been pointed out by Caron and Fillion (2010), namely the use of different horizontally varying scale-dependent correlations between mass, humidity, vorticity and divergence increments. On real-case experiments however, one important issue may be the conservation of the subsequent structures of the analysis increments in the forecast, considering the spin-up of precipitation and the possible generation of spurious gravity waves.

Tests on such real cases are ongoing using the operational AROME 3D-Var and by considering both reflectivities (as in Wattrelot, et al.., 2008) and Doppler velocities (as in Montmerle and Faccani, 2009) from the French ARAMIS radar network as additional data in the variational process. For these experiments, the mask is deduced from an interpolation of reflectivities from the first elevations on the AROME grid. Other applications will be addressed at the mesoscale, especially the use of heterogeneous background-error covariances devoted to the analysis of fog.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References

This work has been supported by the ANR (Agence Nationale pour la Recherche) through the ADDISA (Assimilation de Données Distribuées et Images SAtellite) project. The authors would like to thank Claude Fischer for his valuable technical advice on the variational system.

References

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Background-error covariances for rainy and non-rainy areas
  5. 3.  Implementation of a heterogeneous B matrix in 3D-Var
  6. 4.  Results of assimilation with four observations
  7. 5.  Conclusions
  8. Acknowledgements
  9. References
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