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  • View in gallery

    Vertical background error covariances involving the specific humidity (y axis): covariances with the unbalanced divergence (x axis; units: 10−8 s−1 kg−1 kg, with negative values plotted as dashed lines) in (a) rainy areas and (b) in the climatological operational configuration; covariances with the unbalanced temperature (x axis; units: 10−5 K kg−1 kg) in (c) rainy areas and (d) in the operational configuration.

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    The 0900 UTC 15 Jun 2010 case over France: (a) radar mosaic and (b) gridpoint mask composed of 0 (white) to 1 (red) values deduced from this mosaic. This mask defines the areas where rainy background error covariances are applied. The black rectangle corresponds to the subdomain that is considered in the following horizontal cross sections.

  • View in gallery

    Relative humidity (%) in observation space deduced from the 1D Bayesian inversion of reflectivities from the Collobrière radar (located by the star) at 3.6° elevation, 0900 UTC 15 Jun 2010.

  • View in gallery

    Horizontal cross sections of specific humidity increments (g kg−1) at 600 hPa for (left) EXP and (right) CNTRL at 0900 UTC 15 Jun 2010. This figure is a zoom within the subdomain plotted in Fig. 2b. For EXP, the area where the “rainy” forecast errors are applied is dotted.

  • View in gallery

    As in Fig. 4, but for the divergence (10−5 s−1) at (left) 800 and (right) 400 hPa for (top) EXP and (bottom) CNTRL.

  • View in gallery

    Horizontal cross sections of the analyses difference between EXP and CNTRL for temperature T (K) at 0900 UTC 15 Jun 2010 at (a) 950 and (b) 600 hPa. The dotted area denotes where the “rainy” forecast errors are applied in EXP.

  • View in gallery

    Mean total surface pressure tendencies (hPa h−1) between 6 and 18 Jun 2010 for EXP (dashed line) and CNTRL (solid line).

  • View in gallery

    Scatterplot of the mean total surface pressure tendency for EXP averaged during the first 3 h of forecast vs the percentage of rainy pixels within the computational domain for each assimilation time between 6 and 18 Jun 2010.

  • View in gallery

    Time series of the liquid cloud ql (blue), rain qr (black), ice cloud qi (red), snow flakes qs (green), and graupel qg (orange) averaged in the SE of France and produced by EXP (solid lines) and CNTRL (dashed lines) at 0900 UTC 15 Jun 2010 (g kg−1).

  • View in gallery

    Horizontal cross sections of (a) the observed radar mosaic of reflectivity compared to the reflectivity at 1500 m simulated by (b) CNTRL and (c) EXP after 1 h of integration starting at the 0900 UTC 15 Jun 2010 analysis (dBZ). These figures are zoomed within the subdomain plotted in Fig. 2b.

  • View in gallery

    The 3-h cumulated rainfall scores vs. rain gauges measurements for different precipitation thresholds between 6 and 18 Jun 2010 for EXP (solid gray) and CNTRL (solid black): (a) probability of detection and (b) false alarm rate. The dashed histograms, related to the right y axis, indicate the number of forecasts for which >10 observations have been taken into account in the computation.

  • View in gallery

    The 24-h cumulated rainfall scores against rain gauges measurements for different precipitation thresholds between 6 and 18 Jun 2010 for EXP (black) and CNTRL (gray): (a) bias and (b) Brier skill score. The closer to 1 these scores are, the better is the forecast.

  • View in gallery

    Time series between 6 and 18 Jun 2010 of rms errors (solid lines) and biases (dashed lines) against radiosoundings for (a) temperature and (b) humidity for EXP (black) and CNTRL (gray) at 850 hPa after 12 h of forecast starting from the 0000 UTC analysis time.

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Optimization of the Assimilation of Radar Data at the Convective Scale Using Specific Background Error Covariances in Precipitation

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  • 1 Centre National de Recherches Météorologiques-Groupe d’étude de l’Atmosphère Metéorologique, Toulouse, France
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Abstract

This study focuses on the impact of using specific background error covariances in precipitating areas in the Application of Research to Operations at Mesoscale (AROME-France) numerical weather prediction (NWP) system that considers reflectivities and radial velocities in its assimilation system. Such error covariances are deduced from the application of geographical masks on forecast differences generated from an ensemble assimilation of various precipitating cases. The retrieved forecast error covariances are then applied in an incremental three-dimensional variational data assimilation (3D-Var) specifically in rainy areas, in addition to the operational climatological background error covariances that are used elsewhere. Such heterogeneous formulation gives better balanced and more realistic analysis increments, as retrieved from the assimilation of radar data. For instance, midlevel humidification allows for the reinforcement of the low-level cooling and convergence, the warming in clouds, and high-level divergence. Smaller forecast error horizontal lengths explain the smaller-scale structures of the increments and render possible the increase of data densities in rainy areas. Larger error variances for the dynamical variables give more weight to wind observations such as radial winds. A reduction of the spinup is also shown and is positively correlated to the size of the area where rainy forecast error covariances are applied. Positive forecast scores on cumulated rain and on low-level temperature and humidity are finally displayed.

Corresponding author address: Thibaut Montmerle, Météo-France/CNRM-GAME/GMAP, 42 av. G. Coriolis, 31057, Toulouse, France. E-mail: thibaut.montmerle@meteo.fr

Abstract

This study focuses on the impact of using specific background error covariances in precipitating areas in the Application of Research to Operations at Mesoscale (AROME-France) numerical weather prediction (NWP) system that considers reflectivities and radial velocities in its assimilation system. Such error covariances are deduced from the application of geographical masks on forecast differences generated from an ensemble assimilation of various precipitating cases. The retrieved forecast error covariances are then applied in an incremental three-dimensional variational data assimilation (3D-Var) specifically in rainy areas, in addition to the operational climatological background error covariances that are used elsewhere. Such heterogeneous formulation gives better balanced and more realistic analysis increments, as retrieved from the assimilation of radar data. For instance, midlevel humidification allows for the reinforcement of the low-level cooling and convergence, the warming in clouds, and high-level divergence. Smaller forecast error horizontal lengths explain the smaller-scale structures of the increments and render possible the increase of data densities in rainy areas. Larger error variances for the dynamical variables give more weight to wind observations such as radial winds. A reduction of the spinup is also shown and is positively correlated to the size of the area where rainy forecast error covariances are applied. Positive forecast scores on cumulated rain and on low-level temperature and humidity are finally displayed.

Corresponding author address: Thibaut Montmerle, Météo-France/CNRM-GAME/GMAP, 42 av. G. Coriolis, 31057, Toulouse, France. E-mail: thibaut.montmerle@meteo.fr

1. Introduction

Variational methods, that are used for data assimilation (DA) purposes, are largely based on the specification of the background (or forecast) error covariances matrix, classically defined as the matrix. As pointed out by Daley (1991), the new model state (or analysis), that results from the minimization of a cost function, closely depends on the matrix specification, which determines (i) the weight that is given to the a priori state, (ii) the amplitudes of the information spreading and filtering from the observation points, and (iii) the balances of the initial fields that are necessary to limit the excitation of fast gravity wave noise in the early stage of the forecast. Unfortunately, as it is impossible to measure the error by lack of a “true state” and since cannot be determined at full rank because of its size, the forecast error covariances have to be modeled. Practically, rank reduction can be obtained by making assumptions in the representation of point-by-point spatial covariances (such as homogeneity and isotropy), and balance relationships can be deduced analytically or by using linear regressions that are considered to model the multivariate parts of the covariances. These different entities are usually deduced from statistics performed on ensembles of forecast differences. A complete review of the different methods that have been developed in the main operational numerical weather prediction (NWP) centers to calibrate and to model can be found in Bannister (2008a) and Bannister (2008b), respectively.

The assumptions that are made in the computation of partly explain why the use of observations in DA schemes is still suboptimal: for different weather regimes and for some local meteorological phenomena, climatological balance relationships, background error length scales, or variances are likely to be inadequate. Using an ensemble assimilation, Buehner (2005) and Pereira and Berre (2006) have for instance shown strong geographical variations of background error length scales over the globe. Caron and Fillion (2010) have demonstrated that the deviation from geostrophic balance is proportional to the intensity of precipitation at the regional scale. Furthermore, using a cloud-resolving model and a computation method based on geographical masks that will be detailed in the sequel, Montmerle and Berre (2010, hereafter MB10) and Ménétrier and Montmerle (2011) have shown forecast errors respectively in precipitating areas and in fog that strongly differ from errors computed over the whole computational domain because of diabatic processes. For instance, for fog, the lowest layers of the troposphere have been found to be totally decorrelated with the levels located above the inversion. Such behavior has a direct consequence on the analysis by avoiding information from the surface to spread nonrealistically above the fog layer.

In recent years, much effort had been made to add flow dependency to the spatial covariances of forecast errors in variational DA systems. This flow dependency can be obtained by setting up an ensemble of forecasts coming either from an ensemble assimilation (Houtekamer et al. 1996; Fisher 2003) or an ensemble Kalman filter (Buehner 2005), and by filtering raw horizontal correlations and/or error standard deviations, by using algorithms based on, for example, spectral or wavelet decompositions [a review on this topic can be found in Berre and Desroziers (2010)]. In gridpoint space, successive covariance localization can also be applied on an ensemble of forecasts, as in Zhang et al. (2009), to account for the different scales that are present in the forecast errors. Some specific balance operators, such as the nonlinear geostrophic and the quasigeostrophic omega balance equations (Barker et al. 2004; Fisher 2003), can also bring some degree of flow dependency, but the assumptions in their formulation often make them inadequate at meso- to convective scales where it is of essence to take into account diabatic processes (Caron and Fillion 2010; MB10). At those scales, prognostic rather than diagnostic balance equations may be more appropriate (Pagé et al. 2007), but are technically difficult to implement.

Recently, MB10 proposed an original way to add a dependency to meteorological phenomena, rather than a flow dependency, to forecast errors using a heterogeneous formulation of the matrix. This dependency is obtained through the simultaneous use of different matrices in a variational system, each matrix being specifically computed in some geographical areas. The idea came from the statement that observations obtained in precipitating areas (especially radar reflectivities and radial velocities) are not exploited as they should be in the operational Application of Research to Operations at Mesoscale (AROME)-France NWP system at convective scales (Seity et al. 2011), mainly because of the very weak signal of diabatic processes in its three-dimensional variational data assimilation (3D-Var) system. This results from the very spatially and temporally localized nature of precipitation, which explains their under-representation in the ensemble used to calibrate the climatological matrix. By comparison, this heterogeneous formulation of forecast errors allows us to consider error covariances and balance relationships that are much more representative of the considered meteorological phenomenon, which allow a more optimal use of observations in areas of interest.

As MB10 only presented a draft concept of the method in the frame of a typical one observation experiment, this paper stresses its impact in real-case experiments in the framework of radar data assimilation at convective scales. Real-case experiments have been already tested for the analysis and the forecast of fog events in Ménétrier and Montmerle (2011) who showed that the global impact is closely related to the quality of the fog mask that is used to determine the position of the “foggy” background error covariances. Section 2 introduces the main features of forecast errors in precipitation and how they are computed. The impact of using such diabatic forecast errors, specifically in precipitation, will then be discussed in terms of increment structures, spinup reduction, and forecast scores in ensuing sections.

2. Forecast errors in precipitation

a. Modeling of forecast errors in the CVT framework

As in most operational centers, the AROME 3D-Var is based on the control variable transform (CVT) formalism. This consists in replacing the analysis increment δx by a control variable χ in the 3D-Var written in an incremental way (Courtier et al. (1994)), such as
e1
This shift in variable generally improves the conditioning of the minimization, which results in a faster convergence. In the total cost function, this methodology leads to a trivial formulation of the background term:
e2

Here the second term represents the observation contribution measuring the distance between the innovation vector d = y, which is the difference between the observation y and its simulated counterpart computed by applying the nonlinear observation operator H to the background xb, and the increment written in the observation space. Here stands for the linearized version of H and stands for the observation error covariance matrix.

In the 3D variational assimilation system used by the AROME model, the following control variables, written in the spectral space, are stored in χ: vorticity ζ, divergence η, temperature T, surface pressure Ps, and specific humidity q. The background error covariances are decomposed as a sequence of two operators. The first one is the balance operator, which aims to link forecast errors of each control variable to the unbalanced part of the forecast errors of the other control variables. This is achieved using regression operators that adjust coupling with scales, following the formulation proposed by Berre (2000), which is an extension of the formulation of Parrish et al. (1997) and Derber and Bouttier (1999) for limited-area models, that takes into account a multivariate relationship for q. We will show in the following that this additional coupling is of great interest at convective scale, mainly by allowing balance between q and the divergence η. The second operator is the spatial transform, which is a block-diagonal matrix containing the autocovariances of the unbalanced part of each control variables. Each of those unbalanced terms are thus uncorrelated in spectral space. In this former operator, the spatial correlations are constructed through an empirical orthogonal decomposition in the vertical and spectral diagonal assumptions in the horizontal, which produce homogeneous and isotropic increment structure.

Practically, these covariances and the regression coefficients that are used in the balance operator are deduced from statistics performed on 3-h forecast differences. These forecasts are generated from an ensemble assimilation, as explained in Berre et al. (2006), which consists in cycling forecast–analysis steps making use of explicitly perturbed observations (consistently with their error covariances) and implicitly perturbed background fields (through the cycling). Each member of the ensemble is furthermore coupled with the operational ensemble assimilation at the global scale based on the Action de Recherche Petite Echelle Grande Echelle (ARPEGE) model (AEARP; Desroziers et al. 2008) in order to add perturbed lateral boundary conditions. Following this procedure, a climatological version of the matrix has been computed by Brousseau et al. (2011) and is used operationally to generate new AROME analyses 8 times per day (Seity et al. 2011).

b. Computation of forecast errors representative of precipitating areas

Somehow, these climatological, or static, background error covariances summarize the different meteorological situations that are represented in the ensemble over the domain covered by the AROME-France model. However, Brousseau et al. (2012) have shown a significant day-to-day variability of these covariances that are linked to meteorological conditions over France (e.g., to anticyclonic or to perturbed weather regimes). Moreover, some meteorological phenomena, such as convective systems or fog, are likely to be under-represented in the ensemble because they are spatially and temporally very localized. As a consequence, observations performed in those conditions may be suboptimally used through DA.

Because of technical issues due to an increase in the number of vertical levels in the operational configuration of AROME, forecast errors computed in MB10, specifically in precipitating and in nonprecipitating areas, have unfortunately had to be recalculated. The same approach as in MB10 has thus been followed by setting up an ensemble assimilation composed exclusively of precipitating events that occurred over France, but for the summer 2009 instead. For each events, four forecast–assimilation steps have been performed prior to the last 3-h forecasts used to calibrate the statistics in order to better take into account errors from the boundaries. Binary geographical masks, deduced from a threshold applied on vertically integrated rain computed for each background perturbations, have then been applied to the differences between pairs of forecasts. To alleviate subsampling effects, an inflation deduced from the size of the mean rainy area of the ensemble was applied to the error variances. Such methodology has also been followed recently by Ménétrier and Montmerle (2011) to calibrate forecast errors representative of fog (the geographical mask being based on low-level nebulosity in that particular case) and by Michel et al. (2011) to compute forecast errors of various hydrometeors in the Weather Research and Forecasting (WRF) model.

c. Main behaviors of forecast errors in precipitation

The matrices that have been computed share the same behaviors as in MB10. The main discrepancies between the “rainy” and the operational climatological background error covariances are as follows:

  • shorter horizontal correlation lengths for T and q, reflecting the finescale structures of relative humidity in precipitation;
  • larger spectrally averaged variances for η and ζ due to more intense dynamical activities in convective areas;
  • larger vertical autocorrelations, especially for q, generated by updrafts and downdrafts within precipitating systems;
  • much larger coupling between forecast errors of q and the unbalanced divergence ηu, which is coherent with processes linked to convection development and maintenance (Figs. 1a,b). As a matter of fact, the bipolar structure of the vertical covariances between these two variables that has been found in rain shows that, depending on the level of humidification, convergence and divergence will be enhanced, respectively, below and above this level. On the contrary, forecast errors of q are almost univariate in the operational configuration;
  • different coupling between forecast error of q and the unbalanced temperature Tu (Figs. 1c,d). For rainy cases, such coupling signifies a diabatic effect: midlevel humidification will result in low-level cooling and higher-level warming, reflecting rain evaporation and latent heat release, respectively. In the operational configuration, a positive increment of q will mainly result in a local cooling.
Fig. 1.
Fig. 1.

Vertical background error covariances involving the specific humidity (y axis): covariances with the unbalanced divergence (x axis; units: 10−8 s−1 kg−1 kg, with negative values plotted as dashed lines) in (a) rainy areas and (b) in the climatological operational configuration; covariances with the unbalanced temperature (x axis; units: 10−5 K kg−1 kg) in (c) rainy areas and (d) in the operational configuration.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

Considering such forecast error covariances in the minimization process would obviously optimize the use of observations linked to precipitation such as radar data or in situ measurements. For instance, one can expect such observations to have more weight in the analyses, to produce midtropospheric humidity increments that are more spread vertically, and to impact the low-level convergence structure. An example of the typical impacts of using such error covariances in precipitating areas will be discussed in section 4 in the framework of radar data assimilation. On the opposite, it must be noted that, in the current operational configuration, the very weak coupling between humidity and divergence is somehow alleviated in the case of a simultaneous use of radial velocities and reflectivities from Doppler radar.

3. Use of a heterogeneous matrix in a 3DVar

The purpose the heterogeneous formalism is to allow a simultaneous use of different matrices that correspond to different geographical locations in a DA system. Its detailed formulation is already given in MB10, so only the main principles are recalled here. The basic concept is to express the matrix as a linear combination of N terms:
e3
The matrices reflect the forecast errors of some particular meteorological phenomena (e.g., precipitation, fog, …), the operators defining the subdomains where these matrices are applied. These operators are written in the spectral space and are deduced from the application of direct and inverse Fourier transforms and −1 to the binary gridpoint geographical masks , whose sum is the identity matrix:
eq1
In the CVT framework, the increment of Eq. (1) becomes
e4
Thus, this method implies that the length of control variable χ must be multiplied by N. For instance, in cases when N = 2 and when and correspond to precipitating and nonprecipitating areas, this expression is written as
e5

Here, the gridpoint mask can be deduced from a threshold applied to a radar mosaic or satellite imagery. As explain in detail by Ménétrier and Montmerle (2011), a convolution with a normalized Gaussian kernel is furthermore performed to to avoid the sharp transitions between 0 and 1. The size of the convolution kernel has been chosen in order to get a smooth transition between variance and horizontal correlation length scale values of the two homogeneous matrices. A 4 × 4 gridpoint box has been found to be the best choice in our case. Such a procedure allows the covariance functions to propagate without losing their finescale heterogeneity, while adding anisotropy in the mask border, which is particularly interesting around clouds where strong gradients of atmospheric variables often occur.

4. Impact on the analysis of convective systems considering radar data

To illustrate the impact of using the heterogeneous formulation on analyses, two experiments are set up:

  • CNTRL is the control experiment that mimics the operational NWP system AROME: a 3D-Var is applied at 2.5-km horizontal resolution in order to assimilate a comprehensive set of observations including conventional and aircraft measurements, data from ground based GPS and scatterometers, radiances from several radiometers [the Spinning Enhanced Visible and InfraRed Imager (SEVIRI), the Advanced Television and Infrared Observation Satellite (TIROS) Operational Vertical Sounder (ATOVS), and the Infrared Atmospheric Sounding Interferometer (IAISI), among others], the GPS radio occultation technique (GPSRO; for more details refer to Seity et al. 2011). In addition to this set of observations, volumic scans of radial velocities (Montmerle and Faccani 2009) and reflectivities (Wattrelot et al. 2008) are also considered. The background error covariances are the climatological operational ones (hereafter named ).
  • EXP is based on the CNTRL configuration but its 3D-Var uses the heterogeneous formulation of Eq. (5): is the rainy covariance matrix described in section 2, is equal to , and is deduced from the application of a Gaussian kernel to a binary mask inferred from the observed radar mosaic thresholded to 12 dBZ.

At first, these two configurations have been launched for one single assimilation time on 15 June 2010, which coincided with strong convection occurring in southeast France. Such an event classically appends in this area when moist and unstable meteorological conditions from the Mediterranean Sea encounter the French coastline in the southern part of the Alps and the Massif Central. The corresponding radar mosaic, which is displayed in Fig. 2a, allows us to retrieve the gridpoint mask shown in Fig. 2b. To assess the impact of the heterogeneous formulation on the analyses of the different control variables, the same background fields have been considered in both experiments. For clarity sake, analysis increments will be discussed in the subdomain centered over the southeast of France shown in Fig. 2b. This subdomain being mainly covered by clouds and since only clear-sky satellite radiances are assimilated, the main source of information about the midtropospheric meteorological state comes from the Collobrière radar. As displayed in Fig. 3 for the 3.6° elevation, the 1D Bayesian inversion of radar reflectivities (so-called 1D+3DVar method), as described in Caumont et al. (2010), allows to generate pseudo-observations of relative humidity (RH) that are drier and moister west and east of the radar, respectively. Such a layout is the consequence of a displacement error of the forecasted reflectivity pattern which is obviously located too much on the west.

Fig. 2.
Fig. 2.

The 0900 UTC 15 Jun 2010 case over France: (a) radar mosaic and (b) gridpoint mask composed of 0 (white) to 1 (red) values deduced from this mosaic. This mask defines the areas where rainy background error covariances are applied. The black rectangle corresponds to the subdomain that is considered in the following horizontal cross sections.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

Fig. 3.
Fig. 3.

Relative humidity (%) in observation space deduced from the 1D Bayesian inversion of reflectivities from the Collobrière radar (located by the star) at 3.6° elevation, 0900 UTC 15 Jun 2010.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

In this particular domain, the assimilation of these RH pseudo-observations leads to very different increment structures between EXP and CNTRL, radial velocities being unfortunately unavailable at the time of this study for this particular radar. At first, q increments (as well as T increments, not shown) are more localized spatially in EXP, thanks to the shorter horizontal correlation length, as shown at 600 hPa in Fig. 4. Note that, keeping in mind the geometry of a radar scan, areas of moistening (drying) coincide with increased (decreased) values of RH measurements as expected. These shorter length scales would reduce the size of the thinning boxes used to avoid observation error correlations between adjacent pixels [e.g., to exploit radar data with a higher spatial resolution; see Montmerle and Faccani (2009) for more details]. Fig. 5 displays the result of the much stronger coupling between q and ηn in rainy conditions, as shown in Fig. 1a: humidification at midlevel, as displayed in Fig. 4, implies a low-level convergence below and a divergence above, on top of convective systems. In CNTRL, low-level increments of η result from the assimilation of surface and aircrafts measurements, Doppler winds being unavailable. Furthermore, the comparison between the two analyses retrieved in EXP and CNTRL displayed in Fig. 6 reveals that such midtropospheric humidification has a cooling and a warming effect in EXP on the lower and the higher levels, respectively, thanks to the dipolar structure of the vertical covariances between q and Tu, as shown in Fig. 1c. As discussed earlier, these structures reflect diabatic effects in precipitating clouds (e.g., the cooling is due to the rain evaporation and the warming to the latent heat release). Stronger forecast error variances for the dynamical variable η and ζ in precipitation (not shown) give also more weight in EXP to measurements of the air circulation in these areas, which is of great interest to increase the weight of radial winds in analyses.

Fig. 4.
Fig. 4.

Horizontal cross sections of specific humidity increments (g kg−1) at 600 hPa for (left) EXP and (right) CNTRL at 0900 UTC 15 Jun 2010. This figure is a zoom within the subdomain plotted in Fig. 2b. For EXP, the area where the “rainy” forecast errors are applied is dotted.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

Fig. 5.
Fig. 5.

As in Fig. 4, but for the divergence (10−5 s−1) at (left) 800 and (right) 400 hPa for (top) EXP and (bottom) CNTRL.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

Fig. 6.
Fig. 6.

Horizontal cross sections of the analyses difference between EXP and CNTRL for temperature T (K) at 0900 UTC 15 Jun 2010 at (a) 950 and (b) 600 hPa. The dotted area denotes where the “rainy” forecast errors are applied in EXP.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

In nonprecipitating regions, since the same background error covariances are applied, the two experiments show logically the same increment structures (see e.g., Fig. 5, outside the dashed area). The dedicated matrix for rain, purposely calibrated to represent forecast error in precipitation, allows us to better exploit observations in pertaining areas in a variational DA system. The heterogeneous formulation strengthens the main characteristics of the convection, in particular in the event of a positive innovation of the specific humidity that occurs when the model either fails to predict precipitation, which has been otherwise observed, or fails to show an intensification of rain.

5. Effect of the diabatic balances in the matrix on spinup

Furthermore, the use of in precipitation allows us to reduce the model spinup, which generally designates the time lapse necessary for the model to adjust the initial fields regarding its equations. For instance, a diabatic adjustment consists in balancing water vapor, the microphysical variables, and their physical tendencies with the temperature and the dynamic. To quantify a possible spinup reduction, EXP and CNTRL ran continuously for 12 days, using a 3-h assimilation–forecast cycle as in operation, from 6 to 18 June 2010.

On average, the heterogeneous formulation reduces almost twofold the dynamical adjustment amplitude in the first time steps (not shown), which demonstrates the ability of this method to limit the propagation of spurious numerical waves (e.g., acoustic or gravity waves in a nonhydrostatic model such as AROME). This improvement in the balance of the analyzed fields is illustrated in Fig. 7, which shows the time evolution of the surface pressure tendency averaged on the whole computational domain for this 12-day period. Compared to CNTRL, EXP displays a reduction of this tendency by about 0.7 hPa h−1 during the first hour of forecast. This reduction varies between the different assimilation times and is quite well correlated with the percentage of grid points that are considered as rainy (e.g., where is applied). Figure 8 shows indeed a correlation of 0.64 for the whole time period. This result seems to indicate that, in the operational configuration of AROME, the larger the precipitating area (which implies great number of assimilated reflectivity profiles), the more spurious waves tend to appear in rain because of a nonoptimal balancing of the analysis increments.

Fig. 7.
Fig. 7.

Mean total surface pressure tendencies (hPa h−1) between 6 and 18 Jun 2010 for EXP (dashed line) and CNTRL (solid line).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

Fig. 8.
Fig. 8.

Scatterplot of the mean total surface pressure tendency for EXP averaged during the first 3 h of forecast vs the percentage of rainy pixels within the computational domain for each assimilation time between 6 and 18 Jun 2010.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

6. Impact on forecasts

These balance differences in the first time steps between the two experiments have a direct impact on the forecasted hydrometeor quantities. Figure 9 shows, for instance, time series of such quantities for one assimilation time, starting from scratch in order to avoid the effect of cycling. For this particular case, considering specific forecast errors in rain explains why EXP produces more liquid cloud, ice water cloud, and snow contents than CNTRL during the first hour of forecast. Those larger values explain why forecasted contents of precipitating species such as rain and graupel are contrarily smaller for EXP. For this situation, EXP seems thus to favor stratiform cloud development. Such a result is unfortunately difficult to validate, no quantitative observations of hydrometeors being available.

Fig. 9.
Fig. 9.

Time series of the liquid cloud ql (blue), rain qr (black), ice cloud qi (red), snow flakes qs (green), and graupel qg (orange) averaged in the SE of France and produced by EXP (solid lines) and CNTRL (dashed lines) at 0900 UTC 15 Jun 2010 (g kg−1).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

The 1-h forecasts starting at 0900 UTC 15 June 2010 analyses, which have been compared in section 4, are displayed in Fig. 10. For this particular case and compared to what is displayed for CNTRL, EXP shows a more realistic precipitating pattern characterized by a more narrow global structure, by a better-defined alignment of convective cells along a north–south axis in its southern part and by more intense stratiform precipitation in its northern part. EXP displays also a larger number of small precipitating cells over the Mediterranean Sea, which is difficult to validate because they are located too far from the radars to be sampled and because this area is particularly overcast, as shown by IR images from the geostationary satellite Meteorological Satellite-9 (Meteosat-9, not shown).

Fig. 10.
Fig. 10.

Horizontal cross sections of (a) the observed radar mosaic of reflectivity compared to the reflectivity at 1500 m simulated by (b) CNTRL and (c) EXP after 1 h of integration starting at the 0900 UTC 15 Jun 2010 analysis (dBZ). These figures are zoomed within the subdomain plotted in Fig. 2b.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

Focusing now on the time period from 6 to 18 June 2010, the heterogeneous formulation used in EXP has a positive impact on quantitative precipitation forecast (QPF) scores for the first 3 h compared to rain gauges (Fig. 11). The probability of detection is improved for all the considered precipitation thresholds, the false alarm rate being unmodified, except for thresholds above 10 mm h−1 that are nonsignificant because of the low number of samples that have been taken into account in the computations. This tendency is confirmed after 24 h of forecast, as displayed in Fig. 12, which shows smaller bias and higher Brier skill scores for thresholds smaller than 5 mm h−1. Here again, scores for large thresholds are not statistically reliable for lack of samples.

Fig. 11.
Fig. 11.

The 3-h cumulated rainfall scores vs. rain gauges measurements for different precipitation thresholds between 6 and 18 Jun 2010 for EXP (solid gray) and CNTRL (solid black): (a) probability of detection and (b) false alarm rate. The dashed histograms, related to the right y axis, indicate the number of forecasts for which >10 observations have been taken into account in the computation.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

Fig. 12.
Fig. 12.

The 24-h cumulated rainfall scores against rain gauges measurements for different precipitation thresholds between 6 and 18 Jun 2010 for EXP (black) and CNTRL (gray): (a) bias and (b) Brier skill score. The closer to 1 these scores are, the better is the forecast.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

When averaged for the whole time period, forecast scores against radiosoundings and the European Centre for Medium-Range Weather Forecasts (ECMWF) analyses are neutral to slightly positive up to the 12-h range. Note that, since the heterogeneous formulation is activated only if precipitation occur, such a result is not surprising. Positive impact can, however, be found in the lower troposphere, especially for temperature and humidity, as shown in Fig. 13. Rms errors and biases are indeed constantly reduced in EXP compared to CNTRL after the first two days when only isolated precipitating cells have been observed. When compared to wind observations from profilers, radiosoundings or Doppler radar, smaller rms errors are also found for the analyses (not shown), which seems to indicate that observation error variances are currently overestimated in operations for such observations since the background error variances are much larger in rain as discussed in section 2.

Fig. 13.
Fig. 13.

Time series between 6 and 18 Jun 2010 of rms errors (solid lines) and biases (dashed lines) against radiosoundings for (a) temperature and (b) humidity for EXP (black) and CNTRL (gray) at 850 hPa after 12 h of forecast starting from the 0000 UTC analysis time.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-12-00008.1

7. Conclusions

This paper presents an application of the heterogeneous background error covariances that was formulated in MB10 for the assimilation of observations in precipitating areas (especially radar data) at the convective scale. The AROME-France NWP system, which is based on an incremental 3D-Var assimilation system and a continuous 3-h cycle of assimilation–forecast steps, was used in this context. Forecast errors in rain were at first computed using geographical masks applied on forecast differences coming from an ensemble assimilation of precipitating events. The estimated forecast error covariances were then exploited specifically in rainy areas following the heterogeneous formulation of the incremental 3D-Var and compared to the operational configuration that uses climatological background error covariances. The impact of such methodology on structures of analysis increments has been first highlighted in the context of the 1D+3DVar assimilation of radar reflectivities that are used in operation. In precipitating areas, (i) shorter background error horizontal correlation lengths explain the much smaller-scale increment structures, (ii) larger error variances for the dynamical variables give more weight to wind observation such as radial winds, (iii) the strong coupling between the specific humidity and the divergence allows to enhance low-level convergence and high-level divergence in the case of a midlevel humidification, and (iv) vertical covariances between the temperature and the specific humidity allow the reinforcement of the low-level cold pool and of the warming above (also in the case of a midlevel humidification). Using more adequate background error covariances in rainy areas furthermore allows the spinup time of the model to be reduced, as the amplitude of this reduction is correlated with the size of these rainy areas. This clearly shows that the initial fields are better balanced. Finally, positive forecast scores have been found up to 12 h for the temperature and the humidity at low levels, and up to 24 h for precipitation. Fits to wind observation have also been improved for the analyses.

Those results are encouraging and validate the concept of using separate climatological forecast errors for specific meteorological phenomena such as precipitation. However, since the “rainy” forecast error covariances are static and representative of the precipitating systems that have been sampled in the ensemble, one can reasonably expect these results to be improved if those covariances would have been more suitable to the time period where they have been applied. Such flow dependency could be obtained for AROME by mimicking what is performed at the global scale since 2008 at Météo-France (Berre et al. 2007) and since 2011 at ECMWF (Bonavita et al. 2011) by setting up a daily ensemble assimilation at the convective scale, as described in Brousseau et al. (2012). Based on such an ensemble, work is currently under way to evaluate the methodology presented in this paper and methods based on the filtering of background error variances or horizontal correlations deduced from this ensemble. Spectral (Raynaud et al. 2009) or wavelet-based (Deckmyn and Berre 2005) filtering will be tested in this context. Note that, in the latter, no flow dependency is considered for the balance operator, whereas it may be, when using the heterogeneous formulation. Tests will be mainly performed in the Hydrological cycle in the Mediterranean Experiment (HyMeX) framework, which aims at a better understanding, quantification, and modeling of the hydrological cycle in the Mediterranean.

In parallel and equivalent to what has been presented by Michel et al. (2011), the methodology presented herein has been extended to estimate forecast error covariances of different hydrometeors that are considered in AROME. Coupling relationships linking temperature and humidity to the latter variables have been used to this purpose. This work is a first step to the direct assimilation of homogeneous liquid and ice clouds observed by cloudy radiances in the infrared (Martinet et al. 2012) or of hydrometeor quantities observed by polarimetric radars.

Acknowledgments

The author is very grateful to Jean Maziejewski for his careful reading of this manuscript.

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