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  • View in gallery

    Time series for the nature run (averaged over five realizations): domain-average of the composite reflectivity in dBZ (solid with diamonds), average horizontal (dotted), and maximum (dashed) size of the rain objects in the composite reflectivity thresholded to >10 dBZ, percentage of rainy and, therefore, observed volume grid points with a volume reflectivity >5 dBZ. The assimilation window is between 1400 and 1700 UTC (shaded dark gray) and the forecast window is between 1700 and 2000 UTC (shaded light gray). Variation of the object sizes can be due to merger of anvils of separate convective systems.

  • View in gallery

    Snapshots of composite reflectivity from nature run 03.

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    Composite reflectivity of nature run 03 (cf. Fig. 2 at 1700 UTC) and of the corresponding analysis ensemble means of the different scale experiments at the last analysis time 1700 UTC.

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    As in Fig. 3, but showing the temperature T at z = 150 m.

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    As in Fig. 3, but showing the vertical velocity W at z = 3500 m.

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    RMSE and spread of the ensemble mean of L8 through the assimilation (1400–1700 UTC) and forecast (1700–2000 UTC) phases of the experiment, with gray shading indicating the forecast phase. The gray lines show the RMSE and spread of the mean of the free control ensemble without any assimilation. All grid points are evaluated. The error values are averaged over five repetitions of the experiment.

  • View in gallery

    Composite reflectivity of nature run 03 (cutout of the domain in Fig. 2), analysis ensemble members 01, 15, 35, and 50, and analysis ensemble mean of L8 at the last assimilation time 1700 UTC.

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    Relative frequencies of model values within the analysis ensemble-members of L8 (light gray), L32SOCG (gray), and L32SOCG20 (black) at the last analysis time of 1700 UTC (cf. Figs. 3, 5, 7, and 13), averaged over the five repetitions of the experiments. Distributions are computed for regions inside storms defined by (a) grid points with updrafts where W of the nature run is 5 ± 0.5 m s−1 and (b) where Reflnature = 25 ± 0.5 dBZ, both illustrated by a gray bar in the background. The bin width for the ensemble values is 3 times their bar width.

  • View in gallery

    As in Fig. 6, but only the subset of grid points is evaluated where the volume reflectivity of the nature run exceeds the observation threshold of 5 dBZ.

  • View in gallery

    As in Fig. 6, but now showing RMSE of the ensemble means for all experiments (spread not shown). All grid points are evaluated. The error values are averaged over five repetitions of the experiments.

  • View in gallery

    As in Fig. 7, but for L8SO.

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    As in Fig. 7, but for L32SOCG.

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    As in Fig. 7, but for L32SOCG20.

  • View in gallery

    As in Fig. 10, but showing the mean RMSE of the individual ensemble members of all experiments [see (7)].

  • View in gallery

    As in Fig. 3, but showing composite reflectivity of nature run 03 (cf. Fig. 2 at 2000 UTC), and the forecast ensemble means of the different scale experiments at 2000 UTC, after 3 h of ensemble forecast.

  • View in gallery

    As in Fig. 15, but showing the temperature T at z = 150 m.

  • View in gallery

    As in Fig. 15, but showing the vertical velocity W at z = 3500 m.

  • View in gallery

    As in Fig. 8, but for the forecast ensemble-members at 2000 UTC after 3 h of ensemble forecast (cf. Figs. 15 and 17).

  • View in gallery

    Composite reflectivity of nature run 03 (cutout of the domain), forecast ensemble members 01, 15, 35, and 50, and forecast ensemble mean of L8 at 1800 UTC after 1 h of forecast.

  • View in gallery

    As in Fig. 19, but for L32SOCG20.

  • View in gallery

    DIS component of the DAS score applied to the composite reflectivity field thresholded by 10 dBZ. Displayed is the mean DIS score of the ensemble members of all experiments, with the nature run as reference. The score is averaged over the five random repetitions of all experiments. (A value of DIS = 0 is a perfect match.)

  • View in gallery

    SAL score components S, A, and L for all experiments, with the nature run as reference (see text). The scored field is the composite reflectivity, the scores are averaged over the five random repetitions of all experiments.

  • View in gallery

    Surface pressure tendencies dps/dt (domainwide maximum) of the first member (averaged over five repetitions of the experiments) within the first 20 min after starting the 3-h ensemble forecast. Shown are L8, L32SOCG, and L32SOCG20.

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The Impact of Data Assimilation Length Scales on Analysis and Prediction of Convective Storms

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  • 1 Hans Ertel Centre for Weather Research, Data Assimilation Branch, Ludwig-Maximilians-Universität München, Munich, Germany
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Abstract

An idealized convective test bed for the local ensemble transform Kalman filter (LETKF) is set up to perform storm-scale data assimilation of simulated Doppler radar observations. Convective systems with lifetimes exceeding 6 h are triggered in a doubly periodic domain. Perfect-model experiments are used to investigate the limited predictability in precipitation forecasts by comparing analysis schemes that resolve different length scales. Starting from a high-resolution reference scheme with 8-km covariance localization and observations with 2-km resolution on a 5-min cycle, an experimental hierarchy is set up by successively choosing a larger covariance localization radius of 32 km, observations that are horizontally averaged by a factor of 4, a coarser resolution in the calculation of the analysis weights, and a cycling interval of 20 min. After 3 h of assimilation, the high-resolution analysis scheme is clearly superior to the configurations with coarser scales in terms of RMS error and field-oriented measures. The difference is associated with the observation resolution and a larger localization radius required for filter convergence with coarse observations. The high-resolution analysis leads to better forecasts for the first hour, but after 3 hours, the forecast quality of the schemes is indistinguishable. The more rapid error growth in forecasts from the high-resolution analysis appears to be associated with a limited predictability of the small scales, but also with gravity wave noise and spurious convective cells. The latter suggests that the field is in some sense less balanced, or less consistent with the model dynamics, than in the coarser-resolution analysis.

Corresponding author address: Heiner Lange, Meteorological Institute Munich, Theresienstrasse 37, 80333 Muenchen, Germany. E-mail: heiner.lange@lmu.de

Abstract

An idealized convective test bed for the local ensemble transform Kalman filter (LETKF) is set up to perform storm-scale data assimilation of simulated Doppler radar observations. Convective systems with lifetimes exceeding 6 h are triggered in a doubly periodic domain. Perfect-model experiments are used to investigate the limited predictability in precipitation forecasts by comparing analysis schemes that resolve different length scales. Starting from a high-resolution reference scheme with 8-km covariance localization and observations with 2-km resolution on a 5-min cycle, an experimental hierarchy is set up by successively choosing a larger covariance localization radius of 32 km, observations that are horizontally averaged by a factor of 4, a coarser resolution in the calculation of the analysis weights, and a cycling interval of 20 min. After 3 h of assimilation, the high-resolution analysis scheme is clearly superior to the configurations with coarser scales in terms of RMS error and field-oriented measures. The difference is associated with the observation resolution and a larger localization radius required for filter convergence with coarse observations. The high-resolution analysis leads to better forecasts for the first hour, but after 3 hours, the forecast quality of the schemes is indistinguishable. The more rapid error growth in forecasts from the high-resolution analysis appears to be associated with a limited predictability of the small scales, but also with gravity wave noise and spurious convective cells. The latter suggests that the field is in some sense less balanced, or less consistent with the model dynamics, than in the coarser-resolution analysis.

Corresponding author address: Heiner Lange, Meteorological Institute Munich, Theresienstrasse 37, 80333 Muenchen, Germany. E-mail: heiner.lange@lmu.de

1. Introduction

Over the last decade, data assimilation (DA) with an ensemble Kalman filter (EnKF) (Evensen 1994; Houtekamer and Mitchell 1998) for Doppler radar observations has been demonstrated to be a feasible method to obtain suitable initial states of convective storms for very short-range ensemble forecasts in studies with both simulated (Snyder and Zhang 2003; Tong and Xue 2005) and real observations (Dowell et al. 2004; Dowell and Wicker 2009; Aksoy et al. 2009, 2010). As Stensrud et al. (2009) note, a goal of convective DA is to be competitive to nowcast-warning systems that have generally been superior to model forecasts without data assimilation for at least the first three hours of lead time (Kober et al. 2012; Scheufele et al. 2014).

Previous studies that used radar data in an EnKF have focused on storm-scale analyses, applied relatively small covariance localization lengths in the range of ~10 km (Sobash and Stensrud 2013) and converged the analysis ensemble closely toward the observations, pursuing the goal of obtaining initial states with small errors as the basis of their ensemble forecasts.

This study tries to assess the benefits of such close convergence, because even a very precise analysis with small errors may be of limited value when used as an initial state for a forecast. The reason for this is the short predictability of convection in the chaotic atmospheric system where small-scale errors grow rapidly (Lorenz 1969) because the small-scale error spectrum saturates faster than the larger-scale error spectrum. By adding random perturbations to convection in twin experiments, Zhang et al. (2003, 2007) observed a saturation of small-scale error growth within a few hours—a limitation that directly affects approaches of highly resolved ensemble data assimilation on the convective scale.

Taking the limited predictability of convection into account, the benefits of a high-resolution approach in convective EnKF system may not justify the cost: an analysis ensemble not constrained to accurately reproduce the smallest scales in the observations might provide comparably good quantitative precipitation forecasts (QPFs) for lead times where the small-scale error growth has had enough time to saturate.

a. Limited predictability in convective-scale data assimilation

Lilly (1990) and Skamarock (2004) estimated the predictability of mesoscale convective systems to be in the range of tens of minutes to 1 h before the upscale error growth taints the forecast completely. Zhang et al. (2003), Hohenegger and Schär (2007), and Done et al. (2012) compared randomly perturbed forecasts of organized convection to unperturbed reference runs. They found small-scale perturbations to grow very quickly and nonlinearly, saturating within 3–6 h. The specific predictability limit in these studies depended on the presence of a large-scale forcing that determined the type of convection and the spatial position of the cells. Craig et al. (2012) used perturbations from a latent heat nudging assimilation scheme for convective storm forecasts and concluded a lower predictability for convection in regimes with weak synoptic forcing in which the storm properties are not strongly constrained by the large-scale forcing.

To assess the error growth processes in the framework of data assimilation (Kuhl et al. 2007), this study performs experiments with the local ensemble transform Kalman filter (LETKF; Hunt et al. 2007) coupled to an idealized setup of the nonhydrostatic Consortium for Small-Scale Modeling (COSMO) model (Baldauf et al. 2011) containing severe and long-lived convection. Observing system simulation experiments (OSSEs) of cycled DA are performed where synthetic radar observations are drawn from a nature run. A perfect-model approach and a horizontally homogeneous environment without large-scale forcing are applied to focus on the intrinsic predictability of convective storms. While model error is usually the largest contributor to forecast errors of convection, this study uses a perfect model to investigate the properties and results of the data assimilation cycling alone.

b. Assimilation experiments with different length scales

First, a reference experiment is devised in order to reproduce the results of previous studies on convective EnKF DA. For this, finescale observations of radar data are drawn from the nature run at the full model resolution and are assimilated with a suitably small covariance localization length in order to converge the analysis ensemble closely toward the nature run.

Analysis schemes with different spatial and temporal resolution of the observations in combination with different covariance localization radii are then constructed to provide EnKF-generated perturbations with errors at different scales. Successively, (i) the horizontal localization radius of the covariances is increased (Sobash and Stensrud 2013), (ii) the scale of the observations is coarsened by averaging them into superobservations (Alpert and Kumar 2007), (iii) the computation of the LETKF analysis is done on a reduced horizontal grid (Yang et al. 2009), and (iv) the temporal assimilation interval is extended to provide the analysis with observations less frequently.

The cycled assimilation covers a time span of 3 h, followed by 3 h of ensemble forecast. Different combinations of the experiments given by (i)–(iv) are evaluated using the root-mean-square error (RMSE) of the states together with object- and field-based forecast skill scores to see how quickly the advantage of a fine analysis state is lost because of changes in the precision and scale of the analysis errors. It is also discussed how a tight fit to the observations might cause problems for the dynamics of the forecast model.

The LETKF experiments of this study try to focus on basic properties of such an assimilation system and do not make use of some recent innovations that can help to improve EnKF analyses, such as adaptive covariance inflation (Anderson 2009), additive inflation (Dowell and Wicker 2009), “running in place” (Kalnay and Yang 2010; Wang et al. 2013), or Gaussian anamorphosis of precipitation observations (Lien et al. 2013).

For all experiments, the EnKF will be initiated with a convective ensemble that is spun up from random initial white noise and therefore lacks any prior knowledge about the position of the observed storms in the nature run. This “bad background” leads to a longer period for the initial convergence of the ensemble, but avoids the possibly beneficial influence, for example, of arbitrary convective triggers that are “manually” introduced at predetermined locations (Tong and Xue 2005; Aksoy et al. 2009, 2010) to help the background ensemble resemble the observations even in the first assimilation step.

2. Model configuration and experimental design

This section first describes the data assimilation setup, consisting of nature run, synthetic observations, convective ensemble, and LETKF algorithm, followed by the implementation of the scale-varying experiments. The kilometer-scale ensemble data assimilation (KENDA) system (Reich et al. 2011) is being developed at the Deutscher Wetterdienst (DWD). It couples an LETKF implementation with an ensemble of the COSMO model simulations in the domain over Germany (COSMO-DE; Baldauf et al. 2011).

COSMO solves the full nonhydrostatic and compressible Navier–Stokes equations using a time-splitting Runge–Kutta approach for fast and slow tendencies in the prognostic wind variables U, V, and W and the deviations of temperature T and pressure perturbation PP from a stationary hydrostatic base state. The moist physics uses a single-moment bulk microphysics scheme with six state variables: water vapor QV, cloud water QC, cloud ice QI, rain QR, snow QS, and graupel QG. A radiation scheme for long- and shortwave radiation is applied. Surface fluxes of latent and sensible heat are parameterized and constrained throughout the simulation by a constant surface temperature and a constant surface specific humidity in both the nature run and the ensembles.

a. Nature run

This study uses the test bed setup of COSMO with idealized initial state, periodic boundary conditions, and a homogeneous flat landscape at an elevation of 500 m as the lower boundary. A convection-permitting horizontal resolution of 2 km and 50 vertical levels is set up in a domain of 396 km × 396 km × 22 km extent. The vertical resolution ranges from 800 m at the model top to 100 m at the surface. The initial profile of all model runs is horizontally homogeneous and based on the sounding of Payerne (CH, Radiosonde 06610) at 1200 UTC 30 July 2007, a day with strong convective storms and mesoscale convective systems, favored by a high CAPE value of 2200 J kg−1 together with a vertical wind shear that allows organized convection with heavy precipitation and propagating gust fronts (Bischof 2011).

Instead of initializing convection with predefined warm bubbles (Aksoy et al. 2009) or targeted noise (Dowell et al. 2004; Tong and Xue 2005) with amplitudes that directly trigger thermals, uncorrelated gridpoint noise is added at the initial time t0 to the temperature field T and the vertical wind speed W in the boundary layer with amplitudes of 0.02 K and 0.02 m s−1, respectively. The model runs start at 0600 UTC and quickly develop a convective boundary layer. Instability increases prior to the outbreak of convection due to radiative cooling of the upper troposphere while the surface temperature is held constant.

Small showers initialize at random locations at 0800 UTC, grow until 1000 UTC and mostly die off by 1200 UTC (Fig. 1). The surviving systems grow into intense storms and mesoscale convective systems by 1400 UTC and propagate with the mean wind in a northeastern direction through the domain with lifetimes ≥6 h (Fig. 2). Horizontally contiguous rain areas with a composite reflectivity >30 dBZ extend over distances from 30 to 150 km. Surface fluxes of sensible and latent heat lead to gradual decay of the cold pools in the wake of the storms. The periodic boundary conditions allow the storms to spin up in a way that is “natural” for the model physics for the given sounding despite the modest domain size. The time window between 1400 and 2000 UTC is chosen for the experiments because the storm properties such as size and organization change little during this period (Fig. 1). For 3 h, cycled assimilation is performed every 5 (or 20) min from 1400 to 1700 UTC, followed by ensemble forecasts with 3-h lead time from 1700 to 2000 UTC.

Fig. 1.
Fig. 1.

Time series for the nature run (averaged over five realizations): domain-average of the composite reflectivity in dBZ (solid with diamonds), average horizontal (dotted), and maximum (dashed) size of the rain objects in the composite reflectivity thresholded to >10 dBZ, percentage of rainy and, therefore, observed volume grid points with a volume reflectivity >5 dBZ. The assimilation window is between 1400 and 1700 UTC (shaded dark gray) and the forecast window is between 1700 and 2000 UTC (shaded light gray). Variation of the object sizes can be due to merger of anvils of separate convective systems.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 2.
Fig. 2.

Snapshots of composite reflectivity from nature run 03.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

All experiments in this study are repeated 5 times. Each repetition uses a different seed for the random noise field to initialize its nature run (not shown). These five repetitions represent a variety of different storm positions and shapes that are possible given the initial sounding. When generating the nature runs, one random case was excluded in which the preliminary showers (seen in all nature runs, e.g., in Fig. 1 at 1200 UTC) died out and no larger storm grew. An examination of the full set of ensemble members, which were also randomly initialized, showed that this was a rare event, occurring for of the members, and therefore no such case was included in the five nature runs.

b. Synthetic observations

Synthetic radar observations of reflectivity Z and the component of horizontal wind in the x direction U are generated from the nature runs. To mimic a region of good radar coverage, the observations are taken at every model grid point in the horizontal direction with the grid spacing of Δxmodel = Δxobs = 2 km. In the vertical, every third grid point between 500 m and 13 km above the model surface is observed.

Reflectivity Z is computed from mixing ratios of graupel (QG), rain (QR), and snow (QS) using the simple formula of Done et al. (2004): Zfac,Q = AQ(ρQ)1.75 where AQR = 3.63 × 109 for Q = QR, AQS = 2.19 × 109, and AQG = 1.03 × 109, where ρ is the air density. The reflectivity is given by Z = 10 log10(Zfac,QR + Zfac,QS + Zfac,QG) in dBZ. These values were designed for the Weather Research and Forecasting (WRF) Model but also give reasonable values of dBZ with COSMO, mimicking the behavior of real logarithmic reflectivity observations.

As in other OSSE studies (Tong and Xue 2005), Gaussian noise with a standard deviation of σrefl = 5 dBZ is added to simulate measurement errors. The reflectivity observations Z are masked to regions where Z > 5 dBZ. Below this threshold, they are assimilated as observations of no reflectivity (Tong and Xue 2005; Aksoy et al. 2009; Weygandt et al. 2008; Benjamin et al. 2012).

In reality, most elevation angles of a radar volume scan are shallow, so the radial wind mainly contains information about the horizontal wind. Observations of the horizontal wind component U, masked to Z > 5 dBZ and with an added error of σU = 1 m s−1, are therefore used as a proxy for radial wind observations. As the storms move in a northeastern direction, U contains information of both storm propagation and horizontal divergence patterns. The regular observation geometry and the usage of U ensures that the observed information coverage is uniform and all storms are equally well observed.

c. Initial ensemble

The synthetic radar observations are assimilated using an ensemble of k = 50 members that differ from the nature run and among themselves only in the random seed for the initial noise. The spinup time between 0600 and 1400 UTC enables the members to contain storms with similar characteristics but completely uncorrelated horizontal positions. Snapshots of the reflectivity field at different times throughout one of the nature runs are shown in Fig. 2. Other runs appear similar, but with random displacement and shape variations of the intense storm systems. This initialization method was chosen to deprive the ensemble of any prior knowledge about the state of the nature run when the assimilation starts, as would have been provided by a “manual” positioning of warm bubbles in the members or a confinement of the initial noise to regions of observed reflectivity (Tong and Xue 2005; Dowell and Wicker 2009).

d. Implementation of the LETKF

To produce an analysis ensemble, the LETKF algorithm (described fully by Hunt et al. 2007) determines the vector that minimizes the cost function as follows:
e1
where the k-dimensional vector w defines the optimal linear combination of ensemble member states that minimizes J*. Here yo is the vector of observations and is the observation error covariance matrix, which is treated as diagonal here. Here and bw are given by approximating the observation operator H to be linear about the m-dimensional background ensemble mean state :
e2
where b is a m × k matrix whose columns are given by the deviations of the single forecast members from their mean and
e3

The minimum of the cost function in (1) is computed locally for every analysis grid point to determine the best local linear combination of forecast members in the weighting vectors wa(i). However, these single analyses do not necessarily need to be computed at the full model resolution: the spatial field of the local wa(i) is usually quite smooth in case of localization (Janjić et al. 2011), so a coarser analysis grid can be chosen horizontally and vertically on which the local analysis weights wa(i) are computed before being interpolated onto the model grid (Yang et al. 2009).

For the local analysis, only nearby observations are taken into account by localizing the observation error covariance matrix with a Gaussian-like correlation function (Gaspari and Cohn 1999) that is zero where the distance r of the single observations is larger than the “cutoff length” rLoc of the localization radius. Consistent with the doubly periodic lateral boundary conditions of the model used in this study, the synthetic observations are periodically replicated around the original domain and the filter algorithm of KENDA is configured to take also observations into account that are nominally outside of the domain while still within the horizontal localization radius of the outermost grid points. This leads to a fully periodic LETKF analysis.

For all experiments, the analysis weights wa(i) are multiplied by a constant covariance inflation factor of ρ = 1.05 (Aksoy et al. 2009) in order to enhance the span of the analysis state space. All prognostic variables of the model are updated in the analysis computations (zonal wind U, meridional wind V, vertical wind W, temperature T, pressure perturbation PP, water vapor mixing ratio QV, cloud water mixing ratio QC, cloud ice mixing ratio QI, rainwater mixing ratio QR, snow mixing ratio QS, and graupel mixing ratio QG).

e. Reference configuration

To reproduce the typical behavior of an EnKF DA system with the assimilation of simulated radar observations, a reference setup with the name L8 was used and is described here. In L8, an assimilation interval of Δtass = 5 min represents the typical availability of volume observations from a scanning Doppler radar network (Lu and Xu 2009).

L8 uses a horizontal localization cutoff length of rLoc,h = 8 km, so the ensemble covariances contain storm-internal structures, while the overall structure of the observed storms has to be recovered by assembling the overlapping local analyses from neighboring analysis grid points. In L8, the horizontal resolution of this analysis grid coincides with the full model grid (Δxana = Δxmodel = 2 km). The analysis grid has 20 vertical levels with a spacing that varies with the logarithm of the reference pressure. This is similar to the vertical grid structure of the model’s 50 layers, but with a vertical spacing twice as large, varying from 1600 m at the model top to 250 m at the surface.

The vertical localization radius rLoc,υ varies with height, so that observations close to the surface have a vertical influence of ~1 km, while observations taken a height of 12 km have a vertical influence of ~6 km. The combination of the vertical analysis grid structure, the vertical localization radius, and the resolution of the synthetic observations at every third grid point vertically should provide a sufficient overlap for vertically consistent analysis computations. Although the vertical covariance structures in deep convection can extend up to lengths of 10 km when sampled by an EnKF background ensemble (Tong and Xue 2005), a shallow vertical localization appears to be sufficient here due to the good data coverage, and has an advantage for computational efficiency as less observations have to be taken into account for each local analysis computation.

In the L8 configuration, the positions of the storms in the analysis should closely coincide with those of the observed storms. Within the storm cores, the analysis states are expected to be detailed with a low error and small variance, while spurious convection is suppressed outside of them by assimilating volume observations of no reflectivity. These are the requirements for a “converged” analysis ensemble as defined in the introduction.

f. Configurations with different length scales

All experiments are performed with the full model resolution of Δxmodel = 2 km for both nature run and ensemble members. To produce analyses with varying scales, rLoc,h, Δxobs, Δxana, and Δtass are varied to create the experimental hierarchy shown in Table 1:

  1. Horizontal localization (L8, L32): the horizontal localization is increased from of rLoc,h = 8 to 32 km. The vertical localization is not varied.
  2. Superobservations (SO): The observation spacing Δxobs is increased from 2 to 8 km in experiments L8SO and L32SO, by horizontally averaging the values and positions of 4 × 4 blocks of the original observations into one central SO. If the original observations do not cover the full 4 × 4 block because of their reflectivity threshold, like at the edge of an anvil, the SO is the average of the available original observations in the partially observed block. This procedure is preferable to data thinning and reduces the information to the desired coarse scale (Alpert and Kumar 2007; Salonen et al. 2009; Seko et al. 2004; Xu 2011). The entries of the observation error covariance matrix are kept the same for one SO as for one original observation. The vertical resolution of the observations is not varied.
  3. Coarse analysis grid (CG): The analysis grid spacing Δxana is increased from 2 to 8 km in experiments L8SOCG and L32SOCG by computing the local analysis weights wa(i) at every fourth model grid point (Δxmodel = 2 km) and then linearly interpolating onto the full model grid (Yang et al. 2009). After the interpolation, the transformation from ensemble space into model space is performed in the LETKF as a linear combination of background members as before. The vertical resolution of the analysis grid is not varied.
  4. Assimilation interval (20): The cycling interval Δtass is increased from 5 to 20 min in experiments L8SOCG20 and L32SOCG20. A comparable amount of observations is provided at every intermittent analysis time for the analogous 5- and 20-min experiments, as the generation of synthetic observations from the nature run is not changed but only performed less often. By increasing the cycling interval, the amount of assimilated information within the 3-h assimilation period is therefore reduced and the model dynamics of the ensemble are given a longer time to adapt to the updated analysis states.
Table 1.

Length scales used in the assimilation experiments, as described in section 2: rLoc,h is the cutoff length of the horizontal covariance localization function, Δxobs is the horizontal resolution of observations, Δxana is the horizontal resolution of the analysis grid, Δtass is the assimilation interval between two subsequent analyses, nobs,ana = (rLoc,hxobs)2 is the number of horizontal observation points per one local analysis within an square, and nobs,total = nobs,ana × 180/Δtass is the total number of assimilated local observations in the 3-h assimilation period.

Table 1.

These parameter combinations were selected first (i) to use observation and background information about larger-scale correlations for the analysis, (ii) to reduce the horizontal resolution of the observation information, (iii) to let the filter compute the solution only directly on this coarsened scale before the full scales are updated, and (iv) to further temporally reduce the amount of information provided to the analysis.

The factor of 4 in the parameters between (i) and (iv) implies that all experiments with “L32SO” have the same number of observations per local analysis as the reference experiment L8, while the pure L32 has to fit the analysis ensemble to more local observations (see nobs,ana in Table 1).

The analysis field of local linear ensemble member combinations wa(i) in L32 is therefore computed with a larger horizontal overlap (rLoc = 32 km), so the “assembly” of members in L32 is performed less locally than with the small localization of rLoc = 8 km in L8. It might be dynamically favorable to have less variation in the linear combination of ensemble members especially in regions with large horizontal gradients such as up- or downdraft cores because a linear combination of nonlinear dynamics such as convection is not necessarily a dynamically consistent state for every member.

g. RMSE, spread, and consistency ratio

The accuracy of analysis and forecast states x is measured by the RMSE of the ensemble mean , computed in model space for the different model variables:
e4
where m is the number of grid points. The corresponding variance is the spread of the ensemble xi around its mean , given by
e5
where k is the number of ensemble members. To fulfill the Gaussian assumption of the filter, the ensemble spread should represent the actual error of the analysis, so the consistency ratio CR
e6
should be close to CR = 1. In addition to (4), the mean RMSE of the single members with respect to the nature run is computed for some comparisons:
e7

As in previous studies, RMSE and spread are evaluated for rainy model grid points above the detection threshold of 5 dBZ where the up- and downdraft dynamics are most active and the error reduction most significant. Additionally, the present study evaluates RMSE and spread for all grid points of the domain. This results in a generally lower error level because many small error values of clear-air regions contribute to the mean error, but is of interest since temperature and wind errors can occur outside of precipitating regions, especially in the boundary layer. Analysis and forecast errors are compared to a reference error level computed from a free-running control ensemble that has the same initial conditions as the background ensemble of the LETKF but does not undergo assimilation and runs parallel to the assimilation experiments.

In section 4, additional feature-based scores are described and applied to analyses and forecasts.

3. Assimilation results

First the reference experiment L8 is evaluated during the assimilation window 1400–1700 UTC in section 3a, then the assimilation results of the scale-varying experiments L8S0-L32SOCG20 (Table 1) are compared to L8 in section 3b. The ensemble forecast parts of the experiments are discussed in section 4.

An example of the assimilation results for one realization of the nature run is given in Figs. 3, 4, and 5, which compare snapshots of the nature run 03 to the analysis ensemble means at 1700 UTC. The composite reflectivity is the vertical maximum of the 3D-reflectivity field (Fig. 3). Here T at the height of z = 150 m shows the cold pools (Fig. 4) and W at z = 3500 m shows regions of up- and downdrafts (Fig. 5). Visually, the L8 analysis closely matches the nature run, particularly in reflectivity and vertical velocity.

Fig. 3.
Fig. 3.

Composite reflectivity of nature run 03 (cf. Fig. 2 at 1700 UTC) and of the corresponding analysis ensemble means of the different scale experiments at the last analysis time 1700 UTC.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 4.
Fig. 4.

As in Fig. 3, but showing the temperature T at z = 150 m.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 5.
Fig. 5.

As in Fig. 3, but showing the vertical velocity W at z = 3500 m.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

a. Performance of the reference scheme L8

Figure 6 shows the RMSE of the ensemble mean and the spread of L8 during the assimilation window and during the forecast window from 1700 to 2000 UTC for each of the prognostic model variables and the derived reflectivity (Refl), computed at all model grid points. To illustrate the relative error reduction, the RMSE and spread of the free-running ensemble (which has not undergone any assimilation) are also depicted. An “error reduction” is therefore a reduction of RMSE with respect to the free error level given by the control ensemble, which changes throughout the diurnal cycle due to an increase in domain-wide storm coverage (cf. Fig. 1). The two observed variables are U and Refl, all other variables are updated only through covariances provided by the background ensemble.

Fig. 6.
Fig. 6.

RMSE and spread of the ensemble mean of L8 through the assimilation (1400–1700 UTC) and forecast (1700–2000 UTC) phases of the experiment, with gray shading indicating the forecast phase. The gray lines show the RMSE and spread of the mean of the free control ensemble without any assimilation. All grid points are evaluated. The error values are averaged over five repetitions of the experiment.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

The reduction of RMSE for all model variables shows the effectiveness of the LETKF cycling. The error of the meridional wind U decreases during the assimilation window. At the same time the spread adjusts toward a good consistency ratio by 1700 UTC, showing that the chosen inflation factor is appropriate. The zonal wind V, although only updated through covariances, behaves similarly to U. This is probably due to the strong coupling of U and V in the domain sounding with a southwesterly background wind.

The pressure field PP, the humidity QV, and the temperature T (illustrated by cold pool structures in Fig. 4) also benefit from the LETKF-cycling updates through ensemble covariances. Note that the filter update of T is slightly detrimental at analysis times before 1600 UTC, showing a “reversed” sawtooth pattern (visible in a close examination of Fig. 6), while the error decreases during the 5 min of forecast intervals due to the dynamical convergence of the ensemble members toward a physical meaningful state. This indicates that the direct ensemble covariances of T with the observed U and Refl are probably not well sampled by the background ensemble. The deficient T analysis in the EnKF assimilation of radar data has been noticed in previous studies (Zhang et al. 2004; Dong et al. 2011).

Information about vertical motion of W (up- and downdrafts in Fig. 5) is provided by observations of Refl [cf. Fig. 5 of Tong and Xue (2005)] because the observed reflectivity field is confined to vertically active regions and by horizontal convergence patterns of U that enclose the up- and downdrafts [cf. Fig. 8 of Snyder and Zhang (2003)].

The horizontally intermittent Refl field is well captured by the analysis ensemble (cf. Fig. 7), and the RMSEs of the precipitation variables QR, QS, and QG are reduced. QR, QS, and QG are used to compute observations and first guesses of Refl in the observation operator and therefore are well constrained by the ensemble covariances. The unobserved cloud variables QC and QI also benefit from the LETKF update. On the other hand, the spread of the precipitation variables is strongly reduced and does not recover during the assimilation window. This probably happens due to the non-Gaussian and multimodal climatological distribution of the clouds prior to the assimilation, which is converged toward a Gaussian solution in the analyses. The variance of a multimodal distribution with rain and no-rain present is larger than the variance of Gaussian distributions around rain and no rain (Dance 2004; Craig and Würsch 2013). The localized Gaussian analysis solutions of the filter therefore exhibit a smaller spread than the climatology, which is the free error level, displayed for example in Fig. 6. This convergence toward a Gaussian analysis solution is illustrated in Fig. 8, which shows histograms of W and Refl of the L8 analysis ensemble, computed from the gridpoint values of all analysis members at locations inside the storm in the nature run (i.e., points satisfying Wnature = 5 ± 0.5 m s−1 or Reflnature = 25 ± 0.5 dBZ in left and right panels, respectively). These show an approximately Gaussian distribution around the observed (Refl) or covariance-derived (W) values from the nature run.

Fig. 7.
Fig. 7.

Composite reflectivity of nature run 03 (cutout of the domain in Fig. 2), analysis ensemble members 01, 15, 35, and 50, and analysis ensemble mean of L8 at the last assimilation time 1700 UTC.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 8.
Fig. 8.

Relative frequencies of model values within the analysis ensemble-members of L8 (light gray), L32SOCG (gray), and L32SOCG20 (black) at the last analysis time of 1700 UTC (cf. Figs. 3, 5, 7, and 13), averaged over the five repetitions of the experiments. Distributions are computed for regions inside storms defined by (a) grid points with updrafts where W of the nature run is 5 ± 0.5 m s−1 and (b) where Reflnature = 25 ± 0.5 dBZ, both illustrated by a gray bar in the background. The bin width for the ensemble values is 3 times their bar width.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Figure 9 shows the RMSE and spread of L8 as in Fig. 6, but now computed only at those grid points where Refl of the nature run exceeds the observation threshold of 5 dBZ. The overall variance of all variables is larger in these convective regions, as is the relative amount of error reduction. This is because observations of U and Refl are available for the whole subset volume where RMSE and spread are evaluated. The main difference between Figs. 6 and 9 during the assimilation period from 1400 to 1700 UTC appears in U, V, T, and QV with a stronger error reduction inside the thresholded subset, but also a smaller spread when compared to the evaluation including all grid points. During the forecast period from 1700 to 2000 UTC, the spread of such variables, which are strongly associated with convective updrafts (W, Refl, QR, QG, and QI), quickly exceeds the RMSE of the forecast ensemble mean if all grid points are considered (Fig. 6). This increase in variance is due to the development of spurious updrafts in the L8 forecast ensemble members. The grid points of these spurious cells are not contained in the thresholded subset of the observed storms (Fig. 9). They will be discussed in the forecast results in section 4a.

Fig. 9.
Fig. 9.

As in Fig. 6, but only the subset of grid points is evaluated where the volume reflectivity of the nature run exceeds the observation threshold of 5 dBZ.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

As the relative differences between the two choices of grid point sets for RMSE computations are small during the assimilation period, and the full-domain RMSEs of Fig. 6 contain information about the whole three-dimensional fields, further discussion of RMSEs in this paper will refer to the full domain. Furthermore, since the RMSE behavior of many variables are similar to each other, only plots of four representative variables U, W, T, and QR will be considered. The U will serve as a proxy for V, T as a proxy for PP and QV, and QR as a proxy for QC, QI, QR, QS, QG, and Refl (Fig. 10).

Fig. 10.
Fig. 10.

As in Fig. 6, but now showing RMSE of the ensemble means for all experiments (spread not shown). All grid points are evaluated. The error values are averaged over five repetitions of the experiments.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Summarizing the results for the high-resolution assimilation, the L8 scheme appears to produce LETKF analyses that are comparable in quality to the previous convective EnKF studies with radar data mentioned earlier. The mean of this strongly converged ensemble is representative of the best possible solution, with little variance among the analysis members inside the observed storms.

b. Influence of length scales on assimilation results

In section 2f and Table 1, the different length scales employed in the schemes L8SO–L32SOCG20 are specified. Here, the assimilation results of the various experiments are evaluated in comparison to the reference experiment L8 (section 3a).

First, the effect of increasing only the localization radius is considered [step (i) in the experimental hierarchy of section 2f], by comparing L8 to L32, where the horizontal localization rLoc,h is changed from 8 to 32 km. In L32, the storms in the analysis ensemble mean do not converge well onto the observations. This is clearly visible in the plots of the last analysis ensemble mean in Figs. 3, 4, and 5. The mean reflectivity field is much weaker than the nature run, as is the cold pool intensity. These results are typical of the five repetitions of the experiment although the fields differ in detail. The RMSE of the L32 ensemble mean (Fig. 10) shows very poor performance for U, W, and QR. For T, the ensemble mean of L32 is strongly degraded by every analysis cycle and is even worse than the free error level. These deficiencies indicate that the number of 50 ensemble members is too small in L32: for every local analysis, 4 times the number of full-scale (Δxobs = 2 km) observations must be fitted by the LETKF, in comparison to L8. As a result, the spread of L32 decreases, leading to a very poor consistency ratio for all variables (not shown).

Second, the change in horizontal observation resolution Δxobs from 2 to 8 km [step (ii) superobservations (SO)] is evaluated by comparing L8SO to L8 and L32SO to L32. In L8SO, the analysis field of reflectivity appears more spotty (Fig. 3) than in L8. In the single members of L8SO in Fig. 11, the inner storm core is somewhat broken up and spurious convective cells exist in many locations. The superobservations are obtained by coarse graining horizontally onto 4 × 4 blocks, so that finescale errors are obscured and are not penalized appropriately in the analysis. This is evident in the worse RMSE of W and QR of L8SO compared to L8 (Fig. 10). For U and T, L8SO does not perform much worse than L8, indicating that less horizontal information is needed for a reasonable analysis of the horizontally smoother variables U and T. It is perhaps surprising that in L8SO the filter cycling does not completely fail although there is very little horizontal overlap in the solution of adjacent local analyses when the observation resolution of Δxobs = 8 km matches the localization radius of rLoc,h = 8 km, resulting in a very nonsmooth analysis solution.

Fig. 11.
Fig. 11.

As in Fig. 7, but for L8SO.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

In L32SO, the same horizontal number of observations per local analysis is available as in L8, but with a coarser observation resolution of 8 km. Indeed, the analysis ensemble of L32SO is now able to converge toward the superobservations, in contrast to L32 (Fig. 10), albeit with less precision in the small scales than L8. This can be seen comparing Figs. 3 and 12, which shows L32SOCG as a proxy for L32SO.

Fig. 12.
Fig. 12.

As in Fig. 7, but for L32SOCG.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

In the third set of experiments, the LETKF computations are performed on an analysis grid with a horizontal resolution decreased from Δxana = 2 to 8 km [step (iii) coarse analysis grid (CG)]. The resulting field of local analysis weights wa(i) is then interpolated onto the model grid. Using Δxobs = 8 km (SO) in combination with Δxana = 8 km (SOCG), the analyses of the SOCG experiments are computed at the same horizontal scale where the observations are available. The full-resolution covariances in model space sampled from the background ensemble are still used for the analysis ensemble. Comparing the ensemble mean field plots of L8SOCG to L8SO and L32SOCG to L32SO, the change from SO to SOCG appears insignificant. The RMSE shown in Fig. 10 shows little influence of the coarse grid method in both L32SO(CG) and L8SO(CG). This indicates that the fields of wa(i) are smooth enough to be accurately represented on the coarse grid (although the setup of L8SO and L8SOCG could not be recommended for actual usage because of the lack of horizontal overlap between local analysis regions).

Finally, the interval between the cycling steps is increased from 5 to 20 min [step (iv) 20 min]. In L8SOCG20, this less frequent introduction of observation information leads to strong deterioration in the analysis mean solution of storm positions and cold pools (Figs. 3 and 4) and the RMSEs are significantly worse than for L8SOCG (Fig. 10).

In L32SOCG, the lowest spatial resolution of observations and analysis is reached, and L32SOCG20 additionally lowers the temporal resolution. As for the L8SOCG20 experiment, the precision of the analysis mean is degraded (Fig. 10). In Fig. 12, the analysis members of L32SOCG show a much larger spatial variability of the storm field than L8 (Fig. 7). This variability is even larger with L32SOCG20 (Fig. 13). Figure 8 compares the distributions of W and Refl of the analysis ensembles of L32SOCG and L32SOCG20 to analysis ensemble L8, at subsets of points inside the nature run’s storms. Figure 8b shows that the analysis distribution of L32SOCG is broader than L8 and includes values of zero reflectivity. This is even more evident for L32SOCG20 where many nonprecipitating points are present, and the values of precipitating points are distributed broadly around the observed value. This behavior is closely coupled to the nonobserved W updrafts in Fig. 8a where L32SOCG and L32SOCG20 show a more climatological distribution than the closely converged L8, where climatology is defined here by the frequency distribution of Refl or W at the rainy grid points in the nature runs (cf. Fig. 2).

Fig. 13.
Fig. 13.

As in Fig. 7, but for L32SOCG20.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

The RMSE of the L32SOCG20 analysis mean of W (Fig. 10) is larger than that of the free ensemble and it shows that the ensemble mean of L32SOCG20 has converged toward a solution where the mean updrafts cores are displaced with respect to the nature run (cf. Figs. 3 and 13). This large RMSE of W is therefore associated with a double penalty, in contrast to the error of the free ensemble, where the randomly placed convective updrafts contribute no strong features to the ensemble mean. Despite the larger RMSE, the consistency ratio of L32SOCG20 and L8 in W at 1700 UTC is CG ≈ 0.2 inside the storms (cf. W in Fig. 10), so both the finest and the coarsest experiment have converged to a comparable degree on the updrafts of the nature run.

Before the forecast results are presented in the next section, we look briefly in Fig. 14 at the mean RMSE of the individual ensemble members with respect to the nature run, as defined in (7). Although the single members are not themselves the solution of the LETKF (which is the ensemble mean and analysis perturbations from the mean), can tell us about the deviations of the single members from the truth. It is remarkable that for U and T is the lowest in L32SOCG20, and for W and QR it is well within the range of other experiments, compared to Fig. 10. For U and T, this property indicates that horizontally smooth fields such as U and T are sensitive to the introduction of noise by the filter increments (Greybush et al. 2011; Holland and Wang, 2013), especially for T, where the error of the “low information” experiment L8SOCG20 is also smaller than for other experiments.

Fig. 14.
Fig. 14.

As in Fig. 10, but showing the mean RMSE of the individual ensemble members of all experiments [see (7)].

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

This introduction of noise is also evident in the vertical wind field in Fig. 5 in the form of spurious gravity waves, where the W field of L32SOCG20 in locations of updrafts appears smoother than in all other experiments, and outside the storms is much less tainted by gravity wave noise (cf. the nature run), especially those with Δtass = 5 min.

4. Ensemble forecasts from analyses with different length scales

The main goal of this paper is to evaluate the influence of the spatial scale of ensemble perturbations on the quality of convective ensemble forecasts, and to assess how the benefits of a high-resolution analysis are limited by the predictability of the atmospheric system.

As described before, after the 3 h of cycled LETKF assimilation, ensemble forecasts with lead times up to 3 h are run from 1700 to 2000 UTC for all experiments. This section will focus on the forecast results of L8, L32SOCG, and L32SOCG20, which exemplify the differences between fine and coarse resolution LETKF analyses.

a. Forecast fields and RMS error

For U and T, the RMSE of the forecast ensemble mean (Fig. 10) grows similarly slowly for all experiments, aside from the badly converged L32. The RMSE of the forecast ensemble mean is generally larger for the L32 experiments than for the L8 experiments, but appears to converge toward the end of the forecast window.

For W and QR, the RMSE of the L8 forecast ensemble mean increases from a low error to the free error level within the 3 h from 1700 to 2000 UTC (Fig. 10). The RMSEs of L32SOCG and L32SOCG20 also converge toward the free error level, but from a larger (and for L32SOCG20 doubly penalized) initial error at 1700 UTC. As large values of W and QR are present mainly within convective cell cores, the convergence to the uncorrelated free ensemble error level in Fig. 10 is consistent with W and QR, which smooth out in the forecast ensemble mean, becoming similar to the very smooth free ensemble mean fields of W and QR due to the random position of the storms in the free control ensemble. This smoothing can be seen directly in Fig. 15 for the mean forecast composite reflectivity in Fig. 16 for T and in Fig. 17 for W. The forecast ensemble mean reflectivity field, in particular, resembles a smoothed probability map of the nature run’s storm position. For W and Refl, the analysis ensembles of L8, L32SOCG, and L32SOCG20 had different initial distributions of values (Fig. 8) inside the nature run’s storm locations, with the values for L8 narrowly distributed around the observation. However, after 3 h (Fig. 18), the distributions of W and Refl from the three forecast ensembles are indistinguishable and similar to the climatology of the simulated convective regime.

Fig. 15.
Fig. 15.

As in Fig. 3, but showing composite reflectivity of nature run 03 (cf. Fig. 2 at 2000 UTC), and the forecast ensemble means of the different scale experiments at 2000 UTC, after 3 h of ensemble forecast.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 16.
Fig. 16.

As in Fig. 15, but showing the temperature T at z = 150 m.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 17.
Fig. 17.

As in Fig. 15, but showing the vertical velocity W at z = 3500 m.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 18.
Fig. 18.

As in Fig. 8, but for the forecast ensemble-members at 2000 UTC after 3 h of ensemble forecast (cf. Figs. 15 and 17).

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

The values of the members in Fig. 14 grow similarly slowly for U and T, compared to the RMSE of the ensemble mean in Fig. 10 for all experiments. However, the of W and QR overshoots the free error level for all experiments except L32SOCG20. The free error level here, in contrast to error of the ensemble mean in Fig. 10, is strongly penalized due to the random storm positions in the single free control members. The overshooting of the forecasts of L8 compared to L32SOCG indicates a strong double penalty for the forecast updrafts. In Fig. 19, a snapshot of a 1-h forecast of L8 is displayed. It can be seen that spurious convection arises in the L8 forecast members outside of the true storm position of the nature run. Within the region of stratiform precipitation, the positions of the active parts of the updrafts have diverged very strongly. Such spurious development was found in almost all experiments (L8SO–L32SOCG, not shown) and has been observed in previous studies [cf. Fig. 3 of Aksoy et al. (2010)]. This is also evident through the rapid growth of spread for storm-core variables like W, Refl, QR, QG, and QI in Fig. 6 after 1700 UTC. In L32SOCG20, however, almost no spurious convection is seen in the 1-h forecast outside of the organized convective systems (cf. Fig. 20) and the respective spread of W, Refl, QR, QG, and QI grows more slowly and stays below the RMSE of the forecast mean (not shown), as opposed to L8. This indicates that the analysis states of L32SOCG20 are internally more consistent although the members had not converged closely to the observations, whereas the strongly converged analysis of L8 is not well handled by the model dynamics and is probably unbalanced in some sense.

Fig. 19.
Fig. 19.

Composite reflectivity of nature run 03 (cutout of the domain), forecast ensemble members 01, 15, 35, and 50, and forecast ensemble mean of L8 at 1800 UTC after 1 h of forecast.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

Fig. 20.
Fig. 20.

As in Fig. 19, but for L32SOCG20.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

b. Spatial forecast verification methods

To supplement the visible and the RMSE forecast evaluation of the volume fields, two spatial verification measures for QPFs are chosen to compare the forecast rain fields to the nature run. The composite reflectivity field is used here because high reflectivity is usually accompanied with strong precipitation and winds—the essential threats to be predicted by a convective storm forecast. For this purpose, the composite reflectivity fields of both the nature run and the forecasts are masked to values above a threshold of 10 dBZ to separate the storms and overlapping anvils for object identification.

The displacement and amplitude score (DAS; Keil and Craig 2009) uses a pyramidal matching algorithm to compare two fields by an optical flow technique. A vector field is computed that morphs the forecast onto the reference field and vice versa, using a maximum search radius of 45 km here. The average magnitude of the displacement vector field, normalized by the maximum search radius, defines the displacement (DIS) component of the DAS score, displayed in Fig. 21.

Fig. 21.
Fig. 21.

DIS component of the DAS score applied to the composite reflectivity field thresholded by 10 dBZ. Displayed is the mean DIS score of the ensemble members of all experiments, with the nature run as reference. The score is averaged over the five random repetitions of all experiments. (A value of DIS = 0 is a perfect match.)

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

The SAL score (Wernli et al. 2008) compares statistical properties of thresholded rain objects. The structure or S component indicates whether the forecast objects are smoother and broader than the observations (S > 0) or spikier (S < 0), with S = 0 indicating the correct structure. Only average object properties are compared rather than matching individual objects, so SAL-S is independent of location errors and biases (Fig. 22a). The amplitude or SAL-A component (Fig. 22b) compares the domain-wide total reflectivity (not thresholded) between forecast and observation and thus displays the overall bias. The SAL-L component (Fig. 22c) determines the location error by measuring the horizontal deviation of the centroid of the forecast reflectivity field from the centroid of the observations. Without matching objects, the curves of the location error SAL-L are therefore less continuous than the DIS score. This is a consequence of the small sample of five random cases that is used in this study.

Fig. 22.
Fig. 22.

SAL score components S, A, and L for all experiments, with the nature run as reference (see text). The scored field is the composite reflectivity, the scores are averaged over the five random repetitions of all experiments.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

c. Spatial forecast verification results

The DIS score in Fig. 21 shows that after the 3 h of assimilation, the reference experiment L8 has the lowest position error for the reflectivity field. The L8 experiments with superobservations suffer from the low SO resolution and the small overlap, so the L8SOx experiments are disregarded here. The L32 experiments, however, appear comparable. In the first 30 min of the ensemble forecast, the DIS displacement error of the L8 members grows rapidly and exceeds L32SOCG20, then saturates. The DIS of the L32SOCG20 members grows slowly and converges with L8, L32SO, and L32SOCG toward the end of the forecast window. This finding corroborates the results of the the of QR and W in Fig. 14, which showed the influence of spurious storms in the forecast (cf. Figs. 19 and 20).

The rapid displacement error growth of the L8 members is also displayed by the SAL score: in the first 30 min of the ensemble forecast, the bias SAL-A of L8 increases, the location error SAL-L of L8 grows strongly, and the structure SAL-S of L8 decreases to negative values associated with spikier objects. This means that during this first half hour additional small and mislocated convective cells are triggered in L8: additional cells increase the bias SAL-A, small cells decrease the structure SAL-S, and mislocated cells increase the location error SAL-L (DIS). In contrast, none of this appears to happen in the experiment L32SOCG20, which has the coarsest analysis properties.

d. “Balance” of initial states

The development of spurious updraft cores in L8–L32SOCG suggests that the states of the analysis ensemble members are in some way inconsistent with the forecast model dynamics. In Fig. 5, it is apparent that all analyses except that of L32SOCG20 have finescale gravity wave noise present in the W variable that is not found in the nature run. While it is not expected that convective forecasts are close to geostrophic balance, the apparent excess of free gravity waves suggests the initial states for the ensemble forecasts may suffer from a similar “dynamical imbalance,” caused by noisy analysis increments of the LETKF. In Fig. 23, the surface pressure tendencies of L8, L32SOCG, and L32SOCG20 during the first 20 min after 1700 UTC are plotted as an indicator of possible imbalance (Greybush et al. 2011). The initial pressure tendencies in L8 are substantially larger than for the large-scale analyses until time step 25 (=5 min), which is also the cycling interval of all experiments except L8SOCG20 and L32SOCG20.

Fig. 23.
Fig. 23.

Surface pressure tendencies dps/dt (domainwide maximum) of the first member (averaged over five repetitions of the experiments) within the first 20 min after starting the 3-h ensemble forecast. Shown are L8, L32SOCG, and L32SOCG20.

Citation: Monthly Weather Review 142, 10; 10.1175/MWR-D-13-00304.1

5. Summary of results

The aim of this study on convective-scale data assimilation was to assess how errors grow in 3-h ensemble forecasts from analysis ensembles computed with differing length scales. Data assimilation of long-lived and organized convective systems was performed under the assumption of a perfect model in order to focus on the intrinsic predictability of the storm systems. An LETKF system with 50 members was used in an idealized OSSE test bed with simulated Doppler radar observations of reflectivity and radial wind. The reference and ensemble storms were triggered randomly in the convection-permitting COSMO model with Δxmodel = 2 km, using radiative forcing and initial small-amplitude random noise in a horizontally homogeneous environment with periodic boundary conditions. The nature run and the ensemble all used the same model with the same horizontal resolution and the same initial sounding.

a. Assimilation schemes with different length scales

A reference experiment L8 was set up to reproduce the typical behavior of a convective EnKF DA system in which the exact observation error covariance matrix was known and used by the LETKF. Assimilation schemes were devised in which the localization radius of was increased from 8 to 32 km (L8 and L32), the observation resolution was coarsened from 2 to 8 km as superobservations (SO), the analysis grid resolution of the LETKF was coarsened (CG) from 2 to 8 km, and the assimilation interval was increased from 5 to 20 min.

The background and analysis ensembles of each of the experiments succeeded in converging toward the observations, with the exception of L32, which suffered from undersampling due to the large localization radius in combination with full-resolution observations. This undersampling of the background error covariance in L32 resulted in an overconfidence of the filter that converged toward a wrong solution, which missed some convective cells. The analyses of L8 had the lowest RMSE and displacement errors after 3 h of cycling. The introduction of superobservations made the analyses of the L8SO experiments less precise than L8 while enabling the L32SO experiments to converge properly. Using a coarser analysis grid (CG experiments) in the SO experiments made no significant difference, because the analysis computation was merely performed on the same coarse horizontal scale as the observations. An increased cycling interval of 20 min degraded the precision of the analyses, but resulted in less spurious gravity wave noise in the members through less-frequent introduction of analysis increments. It was found that the spatially and temporally coarsened observation information in L32SOCG and L32SOCG20 rendered their filter solutions much less Gaussian than L8. The point-value distributions of these solutions were closer to the climatology of the simulated convective regime.

b. Ensemble forecasts from analyses with different scales

In the ensemble quantitative precipitation forecasts (QPFs), the displacement error of the storm’s reflectivity field was measured by the object-based and field-based scores DAS and SAL. During the first hour, forecasts from the reference experiment L8 were clearly superior to those of the coarser schemes in terms of storm positions and internal structure. This advantage was lost in as little as half an hour of forecast lead time through the rapid error growth of small perturbations and the emergence of spurious convective cells. The coarsest scheme L32SOCG20 with large perturbations had much slower error growth and fewer spurious cells.

c. Imbalance and limited predictability

Inspection of the forecast fields suggested that rapid growth of spurious convective cells occurred in LETKF configurations where the analysis showed small-scale gravity wave noise. Surface pressure tendencies at the start of the ensemble forecast were, therefore, evaluated for indications of dynamical imbalance introduced by the analysis increments. The smaller initial pressure tendencies in the L32SOCG20 experiment in comparison with L8 suggest less imbalance, although the analysis errors are larger in the coarse resolution experiment. The L32SOCG20 experiment also featured slower initial growth rates of RMSE. This was not caused by a need to spin up small-scale motions since the analysis ensemble was always constructed from the full-resolution background ensemble. Rather, the rapid error growth in L8 was associated with the appearance of small-scale spurious convective cells that may have been triggered by spurious gravity wave noise from the high-resolution analysis increments as noted previously. As this noise was also present in clear-air regions around the observed storms, the possibility of triggering spurious convection by interfering gravity waves (Hohenegger and Schär 2007) cannot be ruled out. The lower noise level of the experiments with a 20-min cycling interval agrees with experiences in operational convective data assimilation (Seity et al. 2011). A hypothetical better data assimilation scheme that did not introduce spurious imbalance might have slower initial error growth than the L8 results here, but it is unlikely that the results would be better than those of a coarser resolution analysis after 1–2 h since some rapid development of new convective cores is seen in all configurations and the perturbations of the L32SOCG20 updraft positions from the analysis are already larger than the coarse observation scale of 8 km within the first forecast hour.

6. Discussion and outlook

a. On the methods used in this study

The forecast results displayed the limited predictability of the dynamics in large convective systems. In previous twin-experiment studies on error growth (Zhang et al. 2003, 2007; Hohenegger and Schär 2007; Done et al. 2012), uncorrelated noise with very small amplitude was used, which does not disrupt the model dynamics as the LETKF ensemble perturbations do. This study should, therefore, be seen in the tradition of predictability studies that used perturbations created by data assimilation systems (Kuhl et al. 2007; Aksoy et al. 2010; Craig et al. 2012).

One feature of the experiments of this paper is that the linear combinations of ensemble members wa(i) are computed by the LETKF on different scales only in the transformed ensemble perturbation space, but the final analysis increments are added in the physical space and, therefore, contain also the smallest scales, as they are sampled from the full-resolution background ensemble members. This means that the analysis states of the members are full scale for all experiments, regardless of the reduction of the physical observation resolution in the SO experiments or the posteriori downscaled coarse grid analysis in the CG experiments. An alternative approach would be to spatially coarsen not only the input parameters of the transforming LETKF solver but also the final physical solution of the algorithm, in order to update larger physical scales only and leave the fine scales untouched. For example, Gao and Xue (2008) computed ensemble covariances from a background ensemble with 4-km horizontal resolution in an ensemble square root filter to update an analysis state of a 1-km resolution model. To mimic this for the present LETKF study, one could (i) coarse grain the background ensemble for the LETKF, (ii) compute the analysis weights on this coarse scale, and (iii) apply the analysis update in physical space on the coarse scales only (C. Snyder 2012, personal communication). This might also reduce the imbalances that are introduced by the filter, and take into account that the effective resolution of numerical models is much lower than the grid spacing implies (Skamarock 2004). Damping the analysis noise by the means of more diffusion on the small scales could be an instrument to keep the model state stable during a Rapid Update Cycle. Incremental analysis update (IAU; Bloom et al. 1996) is often used to prevent insertion shocks by distributing the increments of intermittent analyses throughout the forecast window. Also, a digital filter initialization (DFI; Lynch and Huang 1992) could be helpful to reduce the initial gravity wave noise of the ensemble forecasts (Whitaker et al. 2008), although DFI might be problematic for a nonhydrostatic model wherein the analysis states contain gravity wave motion explicitly, so essential parts of the dynamic signal could be filtered out erroneously.

As a general remark, the spatial and temporal predictability depends on the model resolution, the model physics and the type of long-lived storm that is simulated. A resolution of 2 km is not sufficient to simulate storm-internal variance on the scale of single plumes. These become addressable with horizontal resolutions of 250 m and less (Bryan and Morisson 2012; Craig and Dörnbrack 2008). Using such a model that is able to resolve three-dimensional turbulence, an even finer assimilation scheme could be applied to further investigate the limits of predictability. In addition, using a different environmental sounding could result in different convective modes, such as mesocyclones, multicell storms, and linear squall lines, and the predictability limit will probably be influenced by the degree of storm-internal organization (Lilly 1990; Aksoy et al. 2010). The quantitative time limit found in this study is specific to the atmospheric model configuration and the data assimilation algorithm used here and should not be generalized.

b. Idealized versus operational convective data assimilation

The present study addressed the limitation of intrinsic convective predictability, assuming a perfect model. In the current convective DA systems (e.g., KENDA), model error may well be the largest factor limiting the practical predictability in convective QPF. In real-world experiments, it may also happen that the radar DA tries to converge the analysis members toward convective modes that are not supported by the model physics and the predicted sounding (Stensrud and Gao 2010; Aksoy et al. 2010). On the other hand, in an operational model the predictability of convective system may be enhanced by effects of synoptic and orographic forcing (Hohenegger and Schär 2007; Craig et al. 2012).

c. Outlook

The results of this study showed that the impact of high-resolution information in convective-scale data assimilation is limited by the intrinsic predictability of the flow to a time interval of a few hours at most, and probably also by the balance and noise properties of the initial states generated by the LETKF. These could potentially be mitigated through a priori constraints (Janjić et al. 2014) or postprocessing like DFI. The observation error covariance matrix (Desroziers et al. 2005) is also likely to affect convergence and balance properties of the solution—choosing a larger observation error than actually added to the synthetic observations may result in less convergence but also in more compatible model states, as preliminary experiments (not shown) indicated. Another related issue is the initial ensemble generation for the filter. The random storms used here lead to large analysis increments in the first couple of cycles, which is likely to have detrimental effects on the dynamics (Lien et al. 2013; Kalnay and Yang 2010). In the present LETKF framework with intermittent and instantaneous state updates consisting of linear combinations of the member states, any noise that is not dissipated between subsequent analyses is always fed back into the system and can possibly amplify or resonate. Methods that accelerate the convergence of the filter, such as running in place, could lead to more consistent states by the end of analysis period, if their increments are smooth enough. However, it is not clear how successful these methods would be for convective-scale data assimilation, where the notions of “balanced” or “consistent” states cannot yet be precisely defined or measured.

A further issue is the difficulty of assimilating observations of precipitation due to their non-Gaussian climatological distribution, making the suppression of spurious convection insufficient because (i) the background distribution of convection is typically intermittent and therefore non-Gaussian and (ii) there is no “negative rain” to assimilate in unaltered observations (Craig and Würsch 2013). A Gaussian anamorphosis of precipitation observations could help here to fulfill the Gaussian assumptions of the EnKF (Simon and Bertino 2009; Bocquet et al. 2010; Lien et al. 2013), also on the convective scales.

Acknowledgments

The authors thank Daniel Leuenberger for providing the template of the COSMO setup, Ulrich Blahak for the support with the idealized COSMO code, and Hendrik Reich and Andreas Rhodin for providing code and support of the KENDA-LETKF. We are grateful to an anonymous reviewer for particularly detailed and constructive advice. This study was carried out in the Hans Ertel Centre for Weather Research. This research network of universities, research institutes, and the Deutscher Wetterdienst is funded by the BMVI (Federal Ministry of Transport and Digital Infrastructure).

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