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  • View in gallery

    Average variances of 3-h forecasts at level 50 (~550 m), for zonal wind (solid), temperature (dashed), and specific humidity (dotted), as the function of their starting time. The vertical line indicates the time from which the ensemble is considered as mature.

  • View in gallery

    RMSE of the background error variances of (a),(b) vorticity, (c),(d) unbalanced temperature, and (e),(f) unbalanced specific humidity, computed with (left) 6 or (right) 12 members. The reference is computed from an independent ensemble of 78 members, its uncertainty [Eq. (11)] being illustrated with the shaded areas. Tested configurations are raw sample variances (dashed–dotted black) and variances filtered with various kernels optimized to minimize the RMSE knowing the reference (other colored curves).

  • View in gallery

    As in Fig. 2, but for the RAE.

  • View in gallery

    As in Fig. 2, but for other tested configurations: modified spectral filtering of Raynaud et al. (2009) (dashed brown) and filtering with a Gaussian kernel, whose length scale is computed either to minimize the RMSE knowing the reference (solid red), to fit the general criterion (12) (dashed orange), or to fit the Gaussian criterion (14) (dotted green).

  • View in gallery

    Taylor diagrams displaying the correlation and normalized standard deviation of the raw (circles) and filtered (triangles) background error variances of (a),(b) vorticity, (c),(d) unbalanced temperature, and (e),(f) unbalanced specific humidity, computed with (left) 6 or (right) 12 members. The reference is computed from an independent ensemble of 78 members. Colors indicate the averaged pressure level.

  • View in gallery

    (top) Raw sample and (bottom) filtered [general criterion (12)] background error standard deviation of unbalanced specific humidity at level 50 (~550 m), with increasing ensemble sizes (from left to right) 6, 12, and 78.

  • View in gallery

    Filtering spectral truncation found to minimize the RMSE knowing the reference (solid red), to fit the general criterion (12) (dashed orange), or to fit the Gaussian criterion (14) (dotted green), for the background error variances of (a),(b) vorticity, (c),(d) unbalanced temperature, and (e),(f) unbalanced specific humidity, computed with (left) 6 or (right) 12 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

  • View in gallery

    Filtered background error standard deviation of (top) vorticity, (middle) unbalanced temperature, and (bottom) unbalanced specific humidity at level 50 (~550 m) for three kinds of filters. (from left to right) Gaussian filters optimized with the general criterion (12) or with the Gaussian criterion (14) and modified Raynaud et al. (2009) filter.

  • View in gallery

    (top) Filtered background error standard deviation of unbalanced divergence at level 50 (~550 m) and (bottom) corresponding filtering length scale, within increasing number of tiles (from left to right) 1, 64, and 4096 for a 6-member ensemble.

  • View in gallery

    Histograms of the RMSE of the background error variances of (a) unbalanced divergence and (b) unbalanced specific humidity at level 50 (~550 m), computed with 6 (gray) or 12 members (black). The reference is computed from an independent ensemble of 78 members, its uncertainty [Eq. (11)] being illustrated with the light gray area. Variances are heterogeneously filtered to fit the general criterion (12) on local domains (mask technique, see text) using (from left to right) (i.e., homogeneous filtering on the whole domain) to tiles, and finally a precipitating–nonprecipitating mask.

  • View in gallery

    Diagnosed horizontal correlation (solid black) and localization functions (other colored curves) for zonal wind at level 50 (~550 m), computed with Eq. (19) (“1”), Eq. (20) (“2”), and Eq. (21) (“3”) for ensembles of (a) 30 or (b) 90 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

  • View in gallery

    Length-scale profiles of fitted correlation (solid black) and fitted localization functions (other colored curves) computed with Eq. (19) (“1”), Eq. (20) (“2”), and Eq. (21) (“3”) for ensembles of (left) 30 or (right) 90 members. Tested variables are (a),(b) zonal wind, (c),(d) temperature, and (e),(f) specific humidity. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

  • View in gallery

    (left) Raw and (right) localized [with Eq. (21)] sample correlations of specific humidity at level 50 (~550 m), for ensembles of (top) 30 or (bottom) 90 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

  • View in gallery

    Diagnosed vertical correlation (solid black) and localization functions (other colored curves) for zonal wind around 600 hPa, computed with Eq. (19) (“1”), Eq. (20) (“2”), and Eq. (21) (“3”) for ensembles of (a) 30 or (b) 90 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

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Linear Filtering of Sample Covariances for Ensemble-Based Data Assimilation. Part II: Application to a Convective-Scale NWP Model

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  • 1 Centre National de Recherches Météorologiques–Groupe d’étude de l’Atmosphère Météorologique, Météo-France/CNRS, Toulouse, France
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Abstract

In Part I of this two-part study, a new theory for optimal linear filtering of covariances sampled from an ensemble of forecasts was detailed. This method, especially designed for data assimilation (DA) schemes in numerical weather prediction (NWP) systems, has the advantage of using optimality criteria that involve sample estimated quantities and filter output only. In this second part, the theory is tested with real background error covariances computed using a large ensemble data assimilation (EDA) at the convective scale coupled with a large EDA at the global scale, based respectively on the Applications of Research to Operations at Mesoscale (AROME) and ARPEGE operational NWP systems. Background error variances estimated with a subset of this ensemble are filtered and evaluated against values obtained with the remaining members, which are considered as an independent reference. Algorithms presented in Part I show relevant results, with the homogeneous filtering being quasi optimal. Heterogeneous filtering is also successfully tested with different local criteria, yet at a higher computational cost, showing the full generality of the method. As a second application, horizontal and vertical localization functions are diagnosed from the ensemble, providing pertinent localization length scales that consistently depend on the number of members, on the meteorological variables, and on the vertical levels.

Corresponding author address: Benjamin Ménétrier, CNRM-GAME/GMAP 42 avenue G. Coriolis, 31057 Toulouse, France. E-mail: benjamin.menetrier@meteo.fr

Abstract

In Part I of this two-part study, a new theory for optimal linear filtering of covariances sampled from an ensemble of forecasts was detailed. This method, especially designed for data assimilation (DA) schemes in numerical weather prediction (NWP) systems, has the advantage of using optimality criteria that involve sample estimated quantities and filter output only. In this second part, the theory is tested with real background error covariances computed using a large ensemble data assimilation (EDA) at the convective scale coupled with a large EDA at the global scale, based respectively on the Applications of Research to Operations at Mesoscale (AROME) and ARPEGE operational NWP systems. Background error variances estimated with a subset of this ensemble are filtered and evaluated against values obtained with the remaining members, which are considered as an independent reference. Algorithms presented in Part I show relevant results, with the homogeneous filtering being quasi optimal. Heterogeneous filtering is also successfully tested with different local criteria, yet at a higher computational cost, showing the full generality of the method. As a second application, horizontal and vertical localization functions are diagnosed from the ensemble, providing pertinent localization length scales that consistently depend on the number of members, on the meteorological variables, and on the vertical levels.

Corresponding author address: Benjamin Ménétrier, CNRM-GAME/GMAP 42 avenue G. Coriolis, 31057 Toulouse, France. E-mail: benjamin.menetrier@meteo.fr

1. Introduction

Estimation of the background error covariance matrix (hereafter called the matrix) is a key point for data assimilation (DA) schemes of numerical weather prediction (NWP) systems. Advanced algorithms are often based on an ensemble of perturbed forecasts that sample the forecast error distribution. Since ensembles of forecasts are very expensive in terms of computational resources, the affordable number of members lies between O(10) and O(100), and filtering of covariances is needed to alleviate the sampling noise while keeping the signal of interest.

In the first part of this study, this point has been addressed theoretically by presenting a new method based on theories of linear filtering and of sample centered moments estimation. A first major result is the development of an iterative method allowing a relevant estimation of gridpoint background error variances from such small ensembles. Compared to other previous studies, this method has the advantages of involving sample estimated quantities and filter output only, of ensuring the positivity of filtered variances, and of being computationally cheap. This seems to be well adapted for estimating the error variance “of the day” that is used to add some flow dependency in climatological of deterministic four-dimensional variational data assimilation (4DVar), as it is done operationally for global models at Météo-France (Raynaud et al. 2009; Berre and Desroziers 2010) and ECMWF (Bonavita et al. 2012).

A second application of the theory exposed in Ménétrier et al. (2015, hereafter Part I) concerns the estimation of localization functions used in the localization of covariances through a Schur product (Houtekamer and Mitchell 2001), referred to in Part I as Schur filtering. This is of great interest for DA schemes that make use of ensembles to fully or partly compute the background error covariances for ensemble Kalman filter (EnKF) and its relatives, respectively (Evensen 1994; Houtekamer and Mitchell 1998), or for hybrid variational methods such as ensemble–variational (EnVar; Lorenc 2003; Buehner 2005). As a matter of fact, such methods generally tackle the problem of sampling noise by applying a damping coefficient, given by a distance-dependent function (generally monotonically decreasing), which “localizes” the covariances. In EnKF-like methods, these covariances are computed between grid points and observations individually, which allows damping coefficients with complex spatial distribution [e.g., the ensemble correlations raised to a power (ECO-RAP) algorithm of Bishop and Hodyss (2009a)] that can be a function of the set of observations [e.g., the successive covariance localization (SCL) method of Zhang et al. (2009)] but that are deficient for satellite data (Campbell et al. 2010). In EnVar-like methods, the covariances are computed between all pairs of grid points, making it necessary to process the damping coefficients with a simplified model. A Schur filtering of the sample covariances, which retains the positive semidefiniteness of the covariance matrix, is widely used for this purpose.

In all these DA approaches, the crucial point remains the determination of such damping functions. Beyond the option of a manual tuning, which is always expensive and should be redone whenever the DA system is modified, several techniques have been developed over the years to get more objective damping functions. Anderson (2007) proposed to use a series of “hierarchical ensemble filters” to detect the sampling error and minimize it, with a relatively high computational cost, unfortunately. More recently, Anderson (2012) used a linear regression to minimize the root-mean square (RMS) between the updated ensemble mean and a maximum likelihood estimate. In Anderson and Lei (2013), a new method was introduced to compute localization functions using the output of an observing system simulation experiment (OSSE). Another way of getting an adaptive localization function is to specify it as a power of a correlation function. This has been done with the smoothed ensemble correlations raised to a power (SENCORP) method of Bishop and Hodyss (2007), which is rather computationally expensive, and also with the ECO-RAP method of Bishop and Hodyss (2009a,b), which is much faster and can keep long-range relevant covariances. This kind of method can be extended to fit variational DA schemes requirements (Bishop and Hodyss 2011) for ensemble 4DVar. Thus, Clayton et al. (2013) used a power of the vertical correlation to construct their vertical localization. In all these methods, the choice of the power to which the (possibly smoothed) correlation functions should be raised is arbitrary and has to be tuned manually through expensive experiments. Therefore, the new formulae for localization developed in Part I have the significant advantage of being cheaply computable from the ensemble itself, without any kind of external tuning.

Thus, the detailed theory presented in Part I seems to provide interesting possibilities for variance filtering and covariance localization applications. As a preliminary proof of concept, these new methods have been successfully tested in an idealized framework at the end of Part I to filter error variances and to diagnose horizontal localization functions, but extensive tests with a real NWP model are still required to get a reliable validation. For this task, the same ensemble as in Ménétrier et al. (2014, hereafter M2014) has been used. This ensemble of forecasts is deduced from a 90-member ensemble data assimilation (EDA) at convective scale based on the Applications of Research to Operations at Mesoscale (AROME) model at 2.5-km horizontal resolution, coupled with a 90-member EDA based on the global ARPEGE model (50-km resolution). In M2014, strong discrepancies have been shown for background error parameters at these two different scales, mainly because of the resolution differences and the implied different physical parameterizations. At convective scale, for instance, background errors are characterized by much stronger anisotropies of local correlations and stronger gradients of variance. Testing the algorithms described in Part I in this context thus seems particularly challenging.

The second section of this paper details the experimental framework. Section 3 describes and discusses the results of variance filtering, while section 4 shows some diagnosed localization functions and their application. Concluding remarks are given in section 5.

2. Experimental framework

The NWP system used to test the filtering methods has been detailed in M2014. A brief summary is given in this section.

a. The convective event

For our DA experiments, we chose a heavy precipitation event in the time frame of the Hydrological Cycle in Mediterranean Experiment (HyMeX; Drobinski et al. 2014), occurring over France during the night between 3 and 4 November 2011. At that time, and as shown in Fig. 1 of M2014, moist and unstable air was advected from the Mediterranean Sea over southern France, forced by a strong trough over the northeastern Atlantic. Radar reflectivities displayed in Fig. 3 of M2014 show heavy rainfalls over the highlands of the Cévennes region, but thunderstorms also struck along a north–south moving band, while northwestern France was subject to more stratiform precipitation.

b. The AROME model

AROME is a limited-area model (LAM) at convective scale, operational at Météo-France since the end of 2008 (Seity et al. 2011). It integrates nonhydrostatic equations with fully explicit precipitation at a 2.5-km resolution, on a 1875 km × 1800 km domain over western Europe, including most of the Iberian Peninsula, Germany, and England (see Fig. 6). Its stretched 60 hybrid levels, close together in the lowest atmospheric layers, go up to 1.4 hPa. The coupling fields are provided by the global model ARPEGE, with a Davies relaxation on the lateral boundaries and a spectral relaxation at the top.

Its DA system is an incremental 3DVar using a control variable transform (Berre 2000), assimilating a comprehensive set of observations with a rapid update cycle of 3 h (Brousseau et al. 2011). In particular, SEVIRI radiances are used at high resolution for clear sky over sea and above low clouds (Montmerle et al. 2007), and the radar data from the French Application Radar à la Météorologie Infrasynoptique (ARAMIS) network are used with both Doppler winds and reflectivities (Montmerle and Faccani 2009; Wattrelot et al. 2014).

c. Background error covariance modeling

The background error covariance matrix is necessarily modeled for a system of high dimension. AROME is using the same modeling as its hydrostatic predecessor ALADIN (Berre 2000). The modeled covariance matrix is split into a balance operator dealing with the cross covariance between the model variables that are the vorticity , the divergence , the temperature T, and the specific humidity q, and an autocovariance model providing the spatial covariance of unbalanced variables that are the unbalanced divergence , the unbalanced temperature , and the unbalanced specific humidity , with the vorticity being univariate. The main ideas behind this error covariance modeling framework can be found in Derber and Bouttier (1999) and Bannister (2008).

d. Ensemble data assimilation

The methodology of an EDA system is to simulate the forecast error dynamics in a Monte Carlo framework [more details can be found in Berre et al. (2006)]. Thus, a set of implicitly perturbed analyses and forecasts is propagated through DA cycles [Eq. (4) of M2014]. The explicit perturbation of observations and of models allows the forecasts and analyses perturbations dynamics to mimic their respective error dynamics. Looking at the dynamics of the sample covariances computed from the ensemble, we can separate two kinds of errors: 1) systematic errors that arise from approximations in the observation-error and model-error specification and 2) random errors coming from the finite ensemble size (also called sampling errors). An improvement of observation-error and model-error models is the only way to reduce the impact of systematic errors. This is not the goal of this paper, which focuses on the filtering of sampling errors.

In our study, an EDA of 90 members of the global model ARPEGE is run at first. After a spinup period of a few assimilation cycles, these forecasts are then used as perturbed coupling fields for the AROME ensemble. A spinup period is also required, in order to stabilize the averaged variance of the forecasted parameters. This point is illustrated in Fig. 1, where it is shown that four cycles are needed to reach this condition. Thus, only assimilation times after 1800 UTC 3 November are considered in the following.

Fig. 1.
Fig. 1.

Average variances of 3-h forecasts at level 50 (~550 m), for zonal wind (solid), temperature (dashed), and specific humidity (dotted), as the function of their starting time. The vertical line indicates the time from which the ensemble is considered as mature.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

3. Spatial filtering of variances

In Fig. 8 of M2014, a spatial filtering was applied on AROME variances, in order to study the effect of sampling noise, following the formulation of Raynaud et al. (2009) modified to fit convective-scale model requirements. In this section, we are evaluating the iterative filtering methods proposed in Part I, using our large ensemble. It should be noted that in practice, a spectral truncation is applied to ensemble members before computing gridpoint variances, in order to avoid aliasing. This step decreases the resolution of ensemble members by a factor of 2, from 2.5 to 5 km.

For the sake of simplicity, the variances are gathered in column vectors: denotes the asymptotic variances, obtained from an infinite-size ensemble, whereas is actually computed with the finite-size ensemble. The sampling noise is defined as their difference: .

a. Methodology of evaluation

Considering sample variances , and assuming that we have a reference available, we can compute the following diagnostics:

  • the spatial root-mean-square error (RMSE) with respect to the reference:
    e1
    where n is the number of grid points;
  • the spatial relative absolute error (RAE) with respect to the reference:
    e2
    where denotes the spatial average:
    e3
  • the spatial correlation with respect to the reference:
    e4
    where is the spatial spread:
    e5
    and
  • the spatial spread ratio (SR):
    e6

To evaluate the filtering algorithms, the whole ensemble is partitioned into two independent subsets: a test sample of size N and a reference sample of size . Of course, this reference is not perfect, but its uncertainty can be estimated. Assuming that the distribution of the ensemble is Gaussian, variances of the sampling error for both subsets are deduced from Eq. (12) of Part I:
e7a
and
e7b
Hence, the ratio of sampling error variances is
e8

In our experiments, the 90-member ensemble is split into a 12-member test subset and a 78-member reference subset. To assess the impact of the ensemble size, a 6-member subset is drawn from the 12-member subset. The 78-member subset can be used to provide the common reference variance estimate , since the associated sampling noise is much smaller in this case, which implies that variance filtering is less essential. Considering such reference, the ratio of sampling error standard deviations is equal to 3.92 and 2.65 with respect to 6- and 12-member subsets.

Since the test and reference variances are computed from independent subsets, we can consider that their sampling errors are uncorrelated. Using the definition of , we get
e9
Under the spatial ergodicity assumption, the mathematical expectations can be replaced by spatial averages:
e10
where “” is the Schur product, that is, element by element. Thus, the uncertainty of the reference can be estimated as a function of the RMSE:
e11
This uncertainty is added as a shaded area in figures displaying RMSE.

As pointed out in the introduction, one possible use of filtered variances from ensembles is to add some flow dependency in climatological . Instead of using climatological variances in the spatial transform operator , that is, after the application of to total variables (see section 2c), it is possible to use variances of the day. These variances of “unbalanced” variables are computed with a small ensemble, so they need to be filtered before any use to damp the sampling noise out. However, the formalism presented in Part I is fully general and could be used for any kind of variance (e.g., for the EnKF).

b. Homogeneous and isotropic filtering

1) Algorithms

The filtering algorithms developed in the Part I are based on a “trial and update” iterative process. A filtering length scale is selected, then the filter—whose shape is arbitrarily chosen—is applied to the raw variances , in order to get the filtered variances . Both variances are then used to compute a specific criterion aimed at determining if the selected length scale is too small or too large. Once the filtering length scale is updated, the process starts again. The pseudocode of this algorithm is given is Part I. We showed that the criteria to verify for a homogeneous and isotropic filtering are defined by
e12
where
e13
and is the sample fourth-order centered moment. If the distribution of the ensemble is Gaussian, the Wick–Isserlis theorem applies [Eq. (10) of Part I], and can be simplified in
e14
These iterative filters based on the criteria (12) and (14) will be compared with two other possible approaches.

The first one is based on an adjustment of the filtering length scale (for a given kernel shape), such as the RMSE of the filtered variances with respect to the reference 78-member variances is minimized. In practice, the adjustment is achieved by using a classical line search algorithm (Brent’s method). This first alternative approach will be considered as an optimal method, in the sense that it allows the best possible length scale to be found once the filtering kernel shape and the reference variances are known.

The second alternative approach is the spectral filter based on Raynaud et al. (2009) and modified for convective-scale model specificity. In this case, the spectral filter is obtained from estimated signal-to-noise ratios in spectral space. In the considered study, these ratios have been computed by using flow-dependent spectral error covariances for both signal and noise estimates instead of global climatological covariances. Since negative filtered variances may occur with the original spectral fit in , a regularization step is applied to the raw filter, which sets local negative variances to a small but positive value. This substitution has been chosen as corresponding to the smallest positive variance, without significant impact on averaged scores.

2) Choice of a kernel shape

In our algorithms, the shape of the filtering kernel is arbitrarily chosen before optimizing its length scale. Therefore, the first question to address is the influence of the kernel shape on the filtering quality. For this purpose, instead of using the optimality criteria developed in Part I, we employ the first alternative approach (detailed in the previous subsection): the filtering length scales of several kernel shapes are adjusted in order to minimize the RMSE between the filtered variances and the reference variances. This allows the best possible length scale to be found for each kernel shape knowing the reference variances, which allows the different kernel shapes to be objectively evaluated.

Three kernels applied in spectral space are tested: Gaussian, negative exponential, and Lorentzian. We also tried a recursive filter of order 1, as described in Purser et al. (2003a). The results in terms of RMSE, displayed in Fig. 2, show that, on average, all the filters reach the same quality whatever the kernel shape, for all variables and for 6- or 12-member ensembles. In every case, the spatial filtering brings a significant decrease of the RMSE with respect to the raw variances. In terms of RAE, Fig. 3 shows that the improvement lies between 10% and 50%. As expected, the improvement is larger for a smaller ensemble, which is consistent with the analytical conclusions of Fig. 3 in Part I, and for small-scale variables such as vorticity.

Fig. 2.
Fig. 2.

RMSE of the background error variances of (a),(b) vorticity, (c),(d) unbalanced temperature, and (e),(f) unbalanced specific humidity, computed with (left) 6 or (right) 12 members. The reference is computed from an independent ensemble of 78 members, its uncertainty [Eq. (11)] being illustrated with the shaded areas. Tested configurations are raw sample variances (dashed–dotted black) and variances filtered with various kernels optimized to minimize the RMSE knowing the reference (other colored curves).

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

Fig. 3.
Fig. 3.

As in Fig. 2, but for the RAE.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

3) Evaluation of the filtering quality

The second question relates to the ability of criteria (12) and (14) to provide a good approximation of the optimal filtering length scale. For a Gaussian filtering kernel, Fig. 4 points out that the general criterion (12) always supplies accurate quasi-optimal filtered variances. The criterion (14), based on the assumption that the sample distribution is Gaussian, works as well for as for , but is less accurate for ζ (Figs. 4a,b) and (unbalanced divergence, not shown). This is consistent with Gaussianity tests performed on the ensemble, showing that the null hypothesis of a Gaussian distribution is rejected for ζ and , but not for and (R. Legrand and Y. Michel 2014, personal communication).

Fig. 4.
Fig. 4.

As in Fig. 2, but for other tested configurations: modified spectral filtering of Raynaud et al. (2009) (dashed brown) and filtering with a Gaussian kernel, whose length scale is computed either to minimize the RMSE knowing the reference (solid red), to fit the general criterion (12) (dashed orange), or to fit the Gaussian criterion (14) (dotted green).

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

The conclusions given here for a Gaussian filtering kernel are exactly similar for other kernel shapes, as well as for a recursive filtering: the general criterion (12) is the most efficient to optimize the filtering length scale, whatever the filtering kernel shape.

For relatively large-scale variables such as temperature and humidity, and compared to the optimized filter based on the reference 78-member ensemble, the modified filter of Raynaud et al. (2009) provides filtered variances that are close to optimality, in a similar way as for the iterative filters (although slight suboptimalities can be seen for with 6 members). Nevertheless, in the case of smaller-scale variables such as vorticity, the RMSE of such filters is found to be larger. Although this has not been studied in detail, this is likely to point out a rather specific behavior of vorticity, for which approximations in the estimation of the spectral filter seem to be more sensitive.

4) Behavior of the filtered variances

To get a better understanding of effects on variances of a spatial filtering with a Gaussian kernel optimized with the general criterion (12), Fig. 5 proposes a Taylor diagram (Taylor 2001) representing some raw and filtered variance properties. In such a diagram, the radial coordinate is the spatial spread ratio [ and , respectively] and the angular coordinate is the spatial correlation [ and , respectively]. The gray circles indicate the distance to the reference point that is characterized by values of 1 (indicated as REF on the chart). Each level from the ground up to 200 hPa is represented by a colored dot. In addition, and to get a more visual impression of the filtering effects, horizontal cross sections of raw and filtered variances for are plotted in Fig. 6 for different ensemble size at low levels. Several lessons can be drawn from these plots:

  • For all variables, an increase of the ensemble size improves both the spread ratio and the spatial correlation of the raw variances, straight toward the reference point. This is consistent with the first line of Fig. 6, showing the decrease of the sampling noise as the ensemble size increases.
  • The homogeneous and isotropic filtering improves the spatial correlation, but to a lesser extent for larger ensemble sizes, meaning that most of the spatial structures are already well represented for larger ensembles.
  • The spread ratio is always strongly reduced during the filtering, toward values below 1, meaning that the spread of filtered variances is lower than the spread of reference variances. This is not a real issue, considering that the unfiltered reference variances include some sampling noise that artificially increases their spread. Moreover, as mentioned in Part I, a slight overfiltering would not be harmful for the use of variances in modeled background error covariances, since in that case we prefer to lose a bit of signal on variances than to keep sampling noise on them.
  • The accuracy of 12-member filtered variances is slightly higher than their 6-member counterparts, characterized by a displacement of the corresponding points toward REF. This is consistent with an interesting property of our filtering method, already pointed out in Part I and shown in the bottom panels of Fig. 6: the larger the ensemble is, the softer the filtering is and the finer the remaining structures are.

Fig. 5.
Fig. 5.

Taylor diagrams displaying the correlation and normalized standard deviation of the raw (circles) and filtered (triangles) background error variances of (a),(b) vorticity, (c),(d) unbalanced temperature, and (e),(f) unbalanced specific humidity, computed with (left) 6 or (right) 12 members. The reference is computed from an independent ensemble of 78 members. Colors indicate the averaged pressure level.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

Fig. 6.
Fig. 6.

(top) Raw sample and (bottom) filtered [general criterion (12)] background error standard deviation of unbalanced specific humidity at level 50 (~550 m), with increasing ensemble sizes (from left to right) 6, 12, and 78.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

5) Behavior of filtering length scales

Figure 7 shows the filtering spectral truncation, which is inversely proportional to the filtering length scale. Since results obtained for are comparable to those obtained for ζ, only figures for ζ are discussed. As expected, the filtering length scales are shorter for larger ensembles and for smaller-scale fields, such as ζ. Regarding the vertical variability, we notice that the filtering length scale increases in the troposphere for ζ and but is almost constant for .

Fig. 7.
Fig. 7.

Filtering spectral truncation found to minimize the RMSE knowing the reference (solid red), to fit the general criterion (12) (dashed orange), or to fit the Gaussian criterion (14) (dotted green), for the background error variances of (a),(b) vorticity, (c),(d) unbalanced temperature, and (e),(f) unbalanced specific humidity, computed with (left) 6 or (right) 12 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

The filtering length scales obtained from the three different configurations are consistent with the results of Fig. 4 about their respective quality. The general criterion (12) provides quasi-optimal length scales, with a slight tendency to overfiltering (already noted in Fig. 4 of Part I). The Gaussian criterion (14) is accurate for and , but gives far too short filtering length scales for ζ. Figure 8 shows some examples of standard-deviation fields deduced from variances filtered with the general criterion (12), with the Gaussian criterion (14), and with the modified Raynaud et al. (2009) filter. As expected, the general criterion (12) provides smoother filtered variances than the Gaussian criterion (14). The modified filter of Raynaud et al. (2009) smooths least.

Fig. 8.
Fig. 8.

Filtered background error standard deviation of (top) vorticity, (middle) unbalanced temperature, and (bottom) unbalanced specific humidity at level 50 (~550 m) for three kinds of filters. (from left to right) Gaussian filters optimized with the general criterion (12) or with the Gaussian criterion (14) and modified Raynaud et al. (2009) filter.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

Finally, it should be noted that the temporal variability of the filtering length scales between the close assimilation times is small, meaning that optimality criteria give robust results (not shown). Contrary to the case of global models, optimal filters for LAM seem prone to far more temporal modulations. Indeed, for a longer time period, a day to day variability has been observed, consistent with the background error variability documented, for instance, in Brousseau et al. (2012; not shown). Instead of applying a climatological filter to sample variances, we think that it would be better in this case to optimize the filter at each assimilation time, which is affordable since our method is cheap to apply and does not require any tuning.

c. Heterogeneous filtering

In real cases and as pointed out by M2014, the assumption of horizontal homogeneity of the statistical properties of raw variances is not valid. Several sources of heterogeneity can be listed: dynamics (e.g., convective–stratiform areas), topography, observation network (e.g., land–sea contrast), effects of coupling fields near boundaries for LAM, etc. Thus, the filtering of raw variances should ideally be heterogeneous and anisotropic to reach optimality.

1) Geographical masks method

There is a feasible method of filtering that has a reasonable computational cost and a manageable number of parameters to estimate: a homogeneous filter can be applied and optimized independently on distinct subdomains, then a short-range smoothing is applied at the borders of the subdomains to get continuous filtered variances. The underlying idea is based on the geographical masks method, used to model heterogeneous background error covariances and applied for precipitation (Montmerle and Berre 2010) and for fog events (Ménétrier and Montmerle 2011). In our case, the subdomains have to be small enough to get a truly heterogeneous filtering, yet large enough so that the number of filtering length-scale optimizations remain affordable and can be performed from reliable estimations of criteria (12) or (14). Importantly, the positivity of filtered variances is still guaranteed. The partition of the domain can be arbitrary (e.g., regular tiles) or based on a meteorological criterion (e.g., precipitating–nonprecipitating areas). The filtering can be parallelized in a straightforward manner, for each subdomain.

2) Distribution of filtering length scales

Figure 9 shows an example of partition in regular tiles with an increasing resolution for a 6-member ensemble. At its maximum resolution, the filtering length-scale distribution is consistent with its expected heterogeneity sources. For instance, we can see in Fig. 11 of M2014 that background error correlation length scales are very sensitive to the coupling fields and to the topography, since we notice larger values over sea where there is an inflow from the coupling fields (e.g., west of Corsica and Sardinia and southwest of England) and smaller values on steep topography areas (e.g., over the Alps, the Pyrenees, Corsica, and Sardinia). The positive correlation between background error correlation and filtering length scale, detailed in Fig. 2 of Part I, is made clear in Fig. 9, where the filtering length scale is following the same pattern. The filtering length scale is also larger over areas where the asymptotic variances seem rather homogeneous (e.g., over southwestern France), corresponding to the negative correlation between the standard deviation s of (defined in section 8a of Part I) and the filtering length scale . Moreover, the filtered variances reflect the filtering length-scale heterogeneity, while being continuous enough because of the smoothing at the subdomains borders.

Fig. 9.
Fig. 9.

(top) Filtered background error standard deviation of unbalanced divergence at level 50 (~550 m) and (bottom) corresponding filtering length scale, within increasing number of tiles (from left to right) 1, 64, and 4096 for a 6-member ensemble.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

3) Evaluation of the filtering quality

This heterogeneous filtering method can perform better in terms of RMSE than its homogeneous counterpart, as shown in Fig. 10. However, the amplitude of the RMSE reduction as a function of the number of tiles is varying, depending on the field to filter. We see that for a very heterogeneous field such as at level 50 (~550 m), the RMSE decreases when the number of tiles increases and saturates at a plateau for tiles, whatever the ensemble size. For at the same level, a more contrasted behavior is displayed. For the 12-member ensemble, the improvement is growing with the number of tiles with saturation after , whereas the 6-member ensemble shows an increase of RMSE when the number of tiles exceeds . Two explanations for this degradation can be cited: 1) the gain due to an increased heterogeneity is polluted by the artifacts of tiles junction and 2) when the tiles get smaller, the sample on which the optimality criterion is estimated has a lower size, so this estimation is less reliable. Splitting the whole domain into precipitating–nonprecipitating subdomains also shows a positive impact, since its overall quality is about the same as for a cut into regular tiles (see case P/N-P in Figs. 10a,b).

Fig. 10.
Fig. 10.

Histograms of the RMSE of the background error variances of (a) unbalanced divergence and (b) unbalanced specific humidity at level 50 (~550 m), computed with 6 (gray) or 12 members (black). The reference is computed from an independent ensemble of 78 members, its uncertainty [Eq. (11)] being illustrated with the light gray area. Variances are heterogeneously filtered to fit the general criterion (12) on local domains (mask technique, see text) using (from left to right) (i.e., homogeneous filtering on the whole domain) to tiles, and finally a precipitating–nonprecipitating mask.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

The detail of these contrasted improvements with regular tiles is provided in Table 1, giving the spread ratio, the spatial correlation, and the relative impact of filtering in terms of RMSE:
e15
Homogeneously and heterogeneously filtered variances of and at level 50 (~550 m), computed with a 6-member ensemble, are compared. The heterogeneous filtering is performed on regular tiles for and tiles for , which corresponds to their respective optimal splitting of the domain according to Fig. 10. The small improvement of heterogeneous filtering for is due to slightly improved spread ratio and correlation, the former being more important than the latter. Conversely for , the significant improvement of heterogeneous filtering is mainly due to a better correlation, even if the spread ratio also converges toward 1.
Table 1.

Comparison of homogeneous and heterogeneous filters applied on variances of unbalanced divergence and unbalanced specific humidity at level 50 (~550 m), computed with six members. The heterogeneous filtering is performed on regular tiles for and tiles for . Diagnostics are the spread ratio, the spatial correlation, and the relative impact of filtering in terms of RMSE (see text for details).

Table 1.

Thus, we have shown that the theory of homogeneous variance filtering developed in Part I can be applied to a heterogeneous filtering in a straightforward simple manner, yet at a higher but affordable computational cost. The benefits of such heterogeneous filtering compared to homogeneous filtering are clear, but depend on the considered variable. To get the best out of heterogeneous filtering, a trade-off has to be found between the computational cost and the expected improvement of filtered variance accuracy.

4. Covariance localization

The estimation of covariances requires larger ensembles to reach a given accuracy than the estimation of variances. Indeed, if we define the signal-to-noise ratio (SNR) of a sample covariance for a single realization of as
e16
then in a Gaussian framework, we get from Eq. (11) of Part I
e17
where is the asymptotic correlation. From this formula, we can compute that a 201-member ensemble is enough to estimate a variance with an SNR of 10, whereas a 501-member ensemble would be required to get the same accuracy on a covariance whose asymptotic correlation is 0.5.

Thus, our 90 members are not large enough to properly estimate asymptotic covariances at nonzero separation. Therefore, the aim of this section is not to get an objective evaluation of the diagnosed localization functions and of the localized covariances, but to compare several methods derived in Part I to diagnose quasi-optimal localization functions from the ensemble itself. To illustrate the impact of ensemble size, these methods are applied on two ensembles of 30 and 90 members.

a. Horizontal localization functions diagnostic

The sample covariance matrix is localized through a Schur product (i.e., element by element) with a matrix , to get a filtered matrix :
e18
In section 7d of Part I, we showed that, under a spatial ergodicity assumption, several approximations of the optimal localization were available. With decreasing computational costs, these localization functions can be written
  • in the general case:
    e19
  • with a Gaussian sample distribution:
    e20
  • and with a Gaussian sample distribution and using sample correlations:
    e21
    where is the matrix such that is the sampled fourth-order centered moment matrix of the ith and jth elements of background errors, and where is the sample correlation matrix. The variable denotes a spatial and angular average, performed for large systems through random sampling. Obviously, the size of this sampling is directly affecting the noise observed on the diagnosed functions.

The practical implementation of such localization diagnostics may be summarized by the following three steps:

  1. computation of local fourth-order moment and products of variances, covariances, or correlations (depending on the chosen formula), for different separation distances and using a subsample of grid points;
  2. computation of spatial and angular averages of these local terms and of the diagnosed localization from these averages; and
  3. fit of the resulting localization function by an analytical positive-definite function, for localization length scale diagnostic and other applications.

As expected, Fig. 11 shows noise is slightly more important for methods (19) and (20), where full covariances, fourth-order moments, and variance products are subject to the spatial and angular averages. Instead, method (21) is based on the average of sample correlations, which are naturally bounded between −1 and 1, giving a more stable estimation. As we suggested in the analytical illustrations of Part I, the localization functions have a rather flat top compared to the correlation function, and they widen with the ensemble size, as already shown in previous studies such as in Lorenc (2003).

Fig. 11.
Fig. 11.

Diagnosed horizontal correlation (solid black) and localization functions (other colored curves) for zonal wind at level 50 (~550 m), computed with Eq. (19) (“1”), Eq. (20) (“2”), and Eq. (21) (“3”) for ensembles of (a) 30 or (b) 90 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

The optimal localization being obviously linked to the background error covariance that is strongly flow dependent (Brousseau et al. 2012; M2014), some significant spatiotemporal variation of its value is expected in a LAM context.

b. Horizontal localization length-scale diagnostic

Introduced in Part I, the localization length scale is a useful tool to analyze the variations of the localization functions depending on the variable and on the vertical level. To get an accurate estimation of this parameter from functions diagnosed through Eqs. (19), (20), and (21), we fit them with hyper-Gaussian functions, described by Purser et al. (2003b). Only two parameters are required to define a hyper-Gaussian function: a scale parameter and a shape parameter γ. In our case, they are determined to minimize the mean square error between diagnosed and fitted localization functions. Then, we used the following relationship, which is demonstrated in the appendix following Daley (1991):
e22
The profiles of localization length scales of zonal wind, temperature, and specific humidity, as displayed in Fig. 12, show a good agreement between the three formulations given above in Eqs. (19), (20), and (21). The vertical variations of the diagnosed localization length scales make sense: there is a regular increase for wind and temperature in the troposphere, with a strong decrease above the tropopause for temperature. Such decrease is probably due to an increase of sampling error for temperature at these levels, as shown in Fig. 2, consistently with the increase of averaged variance shown in Fig. 6 of M2014. Larger values are also displayed below 400 hPa for temperature compared to zonal wind, reflecting a larger uniformity between members at those levels. For specific humidity, the values are more uniform throughout the troposphere, with a large gradient at the tropopause, above which error structures are much wider. As it is displayed in Fig. 6 of M2014, contrarily to other variables, the specific humidity has the particularity to reach values very close to zero above the troposphere where very few levels are considered in AROME. As a matter of fact, AROME is mainly dedicated in representing tropospheric phenomena like convective systems or fog, which explains why many more vertical levels are considered in lower layers. Furthermore, since only few AMSU-A radiances are assimilated at those levels, the resulting dispersion between members may be very small. As a consequence, sampling noise of a quasi-null field is quasi inexistent, which results in very large diagnosed localization length scale.
Fig. 12.
Fig. 12.

Length-scale profiles of fitted correlation (solid black) and fitted localization functions (other colored curves) computed with Eq. (19) (“1”), Eq. (20) (“2”), and Eq. (21) (“3”) for ensembles of (left) 30 or (right) 90 members. Tested variables are (a),(b) zonal wind, (c),(d) temperature, and (e),(f) specific humidity. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

c. Horizontal localization application

For illustration, the impact of localization on sample correlations computed with ensembles of various sizes is displayed for some sampled points in Fig. 13. The ubiquitous presence of sampling noise in the raw sample correlations is clearly visible, especially for the 30-member ensemble. The application of a fitted localization function, using , damps most of the long-range sampling noise, especially for the 90-member case. The heterogeneity and anisotropy of the correlation functions are mostly maintained despite the localization, even if some relevant large-scale correlations might be reduced, as in the southeast corner of Fig. 13. As already shown in Fig. 11, long-distance correlations are also less damped for the 90-member ensemble since they are more trusted.

Fig. 13.
Fig. 13.

(left) Raw and (right) localized [with Eq. (21)] sample correlations of specific humidity at level 50 (~550 m), for ensembles of (top) 30 or (bottom) 90 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

d. Vertical localization functions diagnostic

As a proof of concept, we show in Fig. 14 that the formalism developed in Part I to get horizontal localization functions can also be applied for the diagnostic of vertical localization functions. The spatial and angular average of Eqs. (19), (20), and (21) is replaced by a horizontal spatial average alone. Thus, we are diagnosing horizontally homogeneous vertical localization functions, which are varying depending on the variable and the vertical level.

Fig. 14.
Fig. 14.

Diagnosed vertical correlation (solid black) and localization functions (other colored curves) for zonal wind around 600 hPa, computed with Eq. (19) (“1”), Eq. (20) (“2”), and Eq. (21) (“3”) for ensembles of (a) 30 or (b) 90 members. Results are obtained from forecasts issued at 1800 UTC 3 Nov 2011.

Citation: Monthly Weather Review 143, 5; 10.1175/MWR-D-14-00156.1

We can notice that the diagnosed vertical localization functions have a flatter top than the correlation function, as for horizontal localizations. Their amplitude and their width increase also with the ensemble size. The three formulae are still providing rather consistent results, which points out the relevance of their different underlying assumption. We can be surprised by the fact that diagnosed vertical localization functions do not exactly converge to zero when vertical distance increases. However, our experiments have shown that the diagnosed localizations converge to zero only if the correlation itself becomes very close to zero, which is verified in the horizontal case, but not in the vertical case.

5. Conclusions

In the first part of this study, a new theory of covariance filtering has been presented and applied in an analytical idealized framework. To confirm the relevance of these previous results, tests with a real NWP model have been undertaken. The use of the convective-scale model AROME, coupled to the global-scale model ARPEGE, seems particularly challenging for this task, since background error parameters present strong gradients and are strongly anisotropic.

The same 90-member EDA based on AROME as in M2014 has been exploited in this context. Such ensemble has proven to be large enough to quantify objectively the accuracy of the proposed filters for variances. It particularly appears that this new method is fully efficient, as well as computationally cheap, in providing the best length scale in a homogeneous and isotropic filtering context, whatever the filtering kernel. It should be noted that the simplified method assuming that the ensemble distribution is Gaussian works well for all variables except vorticity and divergence, for which the general method performs better. A preliminary comparison with the method of Raynaud et al. (2009) has also been conducted, and a more detailed comparative study with such existing filtering methods would be interesting to consider in future studies. Furthermore, spatial and temporal variabilities of the filtering length scales seem robust and consistent. An affordable method of heterogeneous filtering, based on homogeneous subdomains, has been also successfully tested, showing that the theoretical criteria developed in Part I can be applied through various ergodicity assumptions.

As a second application, the ensemble has been used to diagnose horizontal and vertical localization functions applied to background error correlations. The various formulae given in Part I, which are based on gradual approximations, provide consistent outcomes. As in previous studies (e.g., Lorenc 2003), the strong dependency between the diagnosed localization length scales and the number of members has been shown. Variations of such length scales between variables and between the different vertical levels have also been noticed, which argues in favor of the use of nonhomogeneous Schur filtering in EnKF- or EnVar-like methods. Finally, it has been shown that vertical localization can also be diagnosed using the same approach. To our knowledge, it is the first method that allows an objective estimation of the localization from the ensemble itself and that seems efficient with state-of-the-art NWP models.

The study of the impact of such methods on the analyses and forecasts quality is ongoing, the first step being the inclusion of filtered background error variance “of the day” in the deterministic AROME 3DVar.

Acknowledgments

This work has been supported by the École Normale Supérieure of Paris. Benjamin Ménétrier warmly thanks Thomas Auligné and Chris Snyder (NCAR) for the very enriching visit in Boulder during the fall of 2013.

APPENDIX

Length Scale of Hyper-Gaussian Functions

From Purser et al. (2003b), the family of hyper-Gaussian functions can be defined in 1D as
ea1
where is a scale parameter, γ is a shape parameter, and s here is a dummy variable. Therefore, is given by
ea2
and the second-order derivative at the origin is given by
ea3
Knowing that for a positive a
ea4
the ratio of Eqs. (A2) and (A3) can be simplified in
ea5
so that the length scale defined as in Daley (1991) is given by
ea6

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