We previously performed local ensemble transform Kalman filter (LETKF) experiments with up to 10 240 ensemble members using an intermediate atmospheric general circulation model (AGCM). While the previous study focused on the impact of localization on the analysis accuracy, the present study focuses on the probability density functions (PDFs) represented by the 10 240-member ensemble. The 10 240-member ensemble can resolve the detailed structures of the PDFs and indicates that non-Gaussianity is caused in those PDFs by multimodality and outliers. The results show that the spatial patterns of the analysis errors are similar to those of non-Gaussianity. While the outliers appear randomly, large multimodality corresponds well with large analysis error, mainly in the tropical regions and storm track regions where highly nonlinear processes appear frequently. Therefore, we further investigate the life cycle of multimodal PDFs, and show that they are mainly generated by the on–off switch of convective parameterization in the tropical regions and by the instability associated with advection in the storm track regions. Sensitivity to the ensemble size suggests that approximately 1000 ensemble members are necessary in the intermediate AGCM-LETKF system to represent the detailed structures of non-Gaussian PDFs such as skewness and kurtosis; the higher-order non-Gaussian statistics are more vulnerable to the sampling errors due to a smaller ensemble size.

Data assimilation is a statistical approach to estimate a posterior probability density function (PDF) using information from a prior PDF and observations. Based on the posterior PDF estimate, the optimal initial state is given for numerical weather prediction (NWP). The ensemble Kalman filter (EnKF; Evensen, 1994) is an ensemble data assimilation method based on the Kalman filter (Kalman, 1960) and approximates the background error covariance matrix by an ensemble of forecasts. The EnKF can explicitly represent the PDF of the model state, where the ensemble size is essential because the sampling error contaminates the PDF represented by the ensemble. Although the sampling error is reduced by increasing the ensemble size, the EnKF is usually performed with a limited ensemble size up to O(100) due to the high computational cost of ensemble model runs. Recently, EnKF experiments with a large ensemble have been performed using powerful supercomputers. Miyoshi et al. (2014; hereafter MKI14) implemented a 10 240-member EnKF with an intermediate atmospheric general circulation model (AGCM) known as the Simplified Parameterizations, Primitive Equation Dynamics model (SPEEDY; Molteni, 2003), and found meaningful long-range error correlations. In addition, they reported that sampling errors in the error correlation were reduced by increasing the ensemble size. Further, Miyoshi et al. (2015) assimilated real atmospheric observations with a realistic model known as the Nonhydrostatic Icosahedral Atmospheric Model (NICAM; Tomita and Satoh, 2004; Satoh et al., 2008, 2014) using an EnKF with 10 240 members. Kondo and Miyoshi (2016; hereafter KM16) investigated the impact of covariance localization on the accuracy of analysis using a modified version of the MKI14 system.

MKI14 also focused on the PDF and reported strong non-Gaussianity, such as a
bimodal PDF. Previous studies investigated the impact of non-Gaussianity on
the EnKF. Anderson (2010) reported that an

Using the precious dataset of KM16 with 10 240 ensemble members, we can carry out various investigations such as non-Gaussian statistics and sampling errors in the background error covariance. Here we focus on the non-Gaussian statistics in this study. As the Gaussian assumption makes the minimum variance estimator of the EnKF coincide with the maximum likelihood estimator, the non-Gaussian PDF may have some negative impacts on the LETKF analysis. KM16 showed that the improvement in the tropics was relatively small by increasing the ensemble size up to 10 240, and suggested that the small improvement was related to the convectively dominated tropical dynamics. This study aims to investigate the non-Gaussian statistics of the atmospheric dynamics in more detail to explore the relationship between the analysis error and the non-Gaussian PDF, as well as the behavior and life cycle of the non-Gaussian PDF. To the best of the authors' knowledge, this is the first study investigating the non-Gaussian PDF using a 10 240-member ensemble of an intermediate AGCM. This study also discusses how many ensemble members are necessary to represent a non-Gaussian PDF without contamination due to the sampling error, as higher-order non-Gaussian statistics are generally more vulnerable to the sampling error due to a limited ensemble size. This paper is organized as follows: Section 2 describes measures for the non-Gaussian PDF; Section 3 describes experimental settings; Sect. 4 presents the results; and a summary and discussions are provided in Sect. 5.

Sample skewness

Ensemble-based histograms with 10 240 ensemble members when the
Kullback–Leibler (KL) divergence

A non-Gaussian PDF can also be caused by outliers. Although detailed results
are shown in Sect. 4, one or several ensemble members are detached from
the main cluster; this also results in the large KL divergence

Histograms of background temperature (K) at the fourth model level
(

In the SD method, the ensemble members beyond a prescribed threshold in the
unit of SD are defined as outliers. If we make 10 240 independent random
draws from a Gaussian PDF, statistically 27.6, 0.65, and 0.0059 samples
(0.270, 0.00633, and 0.0000573 %) are expected beyond the

Unlike the SD method, the LOF method is based on the local density, not on
the distance from the sample mean. For a given two-dimensional dataset

Schematic diagrams of reach-dist

The statistics of the KL divergence, and the SD and LOF methods with 10 240 samples
are evaluated numerically with 1 million trials of 10 240 random draws from
the standard normal distribution by the Box–Muller method (Box and Muller,
1958). The results show that the expected value of KL divergence

We use the 10 240-member global atmospheric analysis data from an idealized
LETKF experiment of KM16. That is, the experiment was performed with the
SPEEDY-LETKF system (Miyoshi, 2005) consisting of the SPEEDY model (Molteni,
2003) and the LETKF (Hunt et al., 2007; Miyoshi and Yamane, 2007). The SPEEDY
model is an intermediate AGCM based on the primitive equations at T30/L7
resolution, which corresponds to

The LETKF applies the ETKF (Bishop et al., 2001) algorithm to the local
ensemble Kalman filter (LEKF; Ott et al., 2004). The LETKF can assimilate
observations at every grid point independently, which is particularly
advantageous in high-performance computation. In fact, Miyoshi and Yamane (2007) showed that the parallelization ratio reached 99.99 % on the
Japanese Earth Simulator supercomputer, and KM16 performed 10 240-member
SPEEDY-LETKF experiments within 5 min for one execution of LETKF, not
including the forecast part on 4608 nodes of the Japanese K supercomputer.
The LETKF is computed as follows. Let

KM16 performed a perfect-model twin experiment for 60 d from 00:00 UTC on 1 January in the second year of the nature run, which was initiated at 00:00 UTC on 1 January from the standard atmosphere at rest (zero wind). The first
year of the nature run was discarded as spin-up. To resolve detailed PDF
structures, the ensemble size was fixed to 10 240. No localization was
applied, yielding the best analysis accuracy as shown by KM16, who compared
five 10 240-member experiments with different choices of localization: step
functions with 2000, 4000, and 7303 km localization radii, a Gaussian
function with a 7303 km localization radius, and no localization. The
observations for horizontal wind components (

The non-Gaussian measures, skewness

Figure 4 shows the spatial distributions of the analysis absolute error,
ensemble spread, background skewness

Spatial distributions of

Scatter diagrams of the local outlier factor (LOF) method versus
distance from the ensemble mean for all ensemble members for background
temperature at the fourth model level (

Spatial distributions of the number of outliers for background
temperature at the fourth model level (

Figure 7 shows the spatial distributions of the time-mean analysis RMSE,
the ensemble spread, the background absolute skewness

Spatial distributions of the time-mean

Spatial distributions of the frequency of the non-Gaussian PDF with high
KL divergence

Similar to Fig. 8, but showing the frequency of identifying at least one outlier with high a LOF > 8.0 on a 10 240-member ensemble.

To investigate how the non-Gaussian PDF is generated, we plot the forecast
and analysis update processes at 1.9

Life cycle of non-Gaussianity at 1.9

Scatter diagram of 06:00 UTC versus 12:00 UTC on 9 February for the
background temperature at the fourth model level (

Similar to Fig. 11, but for 06:00 UTC versus 12:00 UTC on 9 February for background precipitation.

In the extratropics, non-Gaussian PDF is generated differently. To
investigate the genesis of non-Gaussian PDF in the extratropics, we focus on
a case around an extratropical cyclone over the Atlantic Ocean. A
non-Gaussian PDF appears at 06:00 UTC on 15 February at 42.7

Similar to Fig. 11, but for 06:00 UTC versus 12:00 UTC on 9 February
for background zonal wind at the fourth model level (

Spatial distributions of the KL divergence for background
temperature at the fourth model level (

Scatter diagram of background specific humidity at the second
model level (

Similar to Fig. 14, but for background specific humidity versus
meridional wind background at the second level (

The non-Gaussian measures are sensitive to the ensemble size due to sampling
errors. Figure 17 shows the spatial distributions of the skewness

Spatial distributions of (

Similar to Fig. 5b, but for the ensemble sizes

Rank histograms verified against truth for background specific
humidity at the lowest model level (

Spatial distributions of background KL divergence for the SPEEDY
and NICAM models. Upper panels show

We saw good agreement between the RMSE and ensemble spread (Fig. 7a, b),
but it is useful to further evaluate the 10 240-member ensemble using ranked
probability scores. The rank histogram (Hamill and Collucci, 1997; Talagrand
et al., 1999; Anderson, 1996; Hamill, 2001) evaluates the reliability of
ensemble statistically. Figure 19 shows almost flat rank histograms at all
grid points and the grid points with non-Gaussian PDF. The truth is known in
this study and used as a verifying analysis. The flat rank histograms
correspond to healthy background ensemble distributions. The continuous
ranked probability score (CRPS, Hersbach, 2000) is another method to evaluate
ensemble distributions, decomposed into reliability, resolution and
uncertainty as

CRPS and its three components (reliability, resolution, and
uncertainty) for background specific humidity at the lowest model level
(

Kalman filters provide the minimum variance linear estimator, which coincides with the maximum likelihood estimator if the PDFs are Gaussian. This study investigated the non-Gaussian PDF and its behavior using the SPEEDY-LETKF system with 10 240 members. Non-Gaussian PDFs appear frequently in the areas where the RMSE and ensemble spread are larger. Moreover, an ensemble size of about 1000 is necessary to identify the possible non-Gaussianity of PDFs, which may be difficult to detect in the presence of sampling error.

The non-Gaussian PDF frequently appears in the tropics and the storm track regions over the Pacific and Atlantic oceans, particularly for temperature and specific humidity, but not for winds. With the SPEEDY model, the genesis of non-Gaussian PDF in the tropics is mainly associated with the convective instability. These results suggest that the non-Gaussianity is mainly caused by precipitation processes such those associated with cumulus convection, but much less by dynamic processes. Generally, the atmosphere in the tropics tends to become unstable, and the convective instability is mitigated by vertical convection with precipitation. In the SPEEDY model, a simplified mass-flux scheme developed by Tiedtke (1993) is applied. Convection occurs when either the specific or relative humidity exceeds a prescribed threshold (Molteni, 2003). The members that hit the threshold have precipitation, and this process mitigates their own convective instability resulting in a temperature rise and humidity decrease. In contrast, the members with no or little precipitation enhance or cannot mitigate their own convective instability. Therefore, convective instability is a key to non-Gaussianity genesis in the tropics in the SPEEDY model.

In the extratropics, non-Gaussianity is generally weak and seldom appears except in the storm track regions, where the genesis of the non-Gaussian PDF is also associated with instabilities, but with different processes from the tropics. This study focused on a case near the extratropical cyclone in the North Atlantic, and the results showed that the instability was associated with the horizontal advections. The members with reduced instabilities had lower humidity in the lower troposphere and higher temperature in the mid troposphere by meridional advections. In contrast, the members with higher humidity in the lower troposphere and lower temperature in the mid troposphere enhanced their instability. Moreover, the precipitation process through the cumulus parameterization did not explain the non-Gaussian PDF. Precipitation associated with extratropical cyclones is usually caused by synoptic-scale baroclinic instabilities and does not mitigate the local instability completely.

As mentioned in Sect. 4, generalizing the process of non-Gaussianity genesis in the extratropics is not simple. The non-Gaussianity genesis is generally associated with instability from various processes such as the convection, advection, and larger-scale atmospheric phenomena, so that it is very difficult to find general mechanisms of the non-Gaussianity genesis in the extratropics even for the simple SPEEDY model. Furthermore, if we use more realistic models with complex physics schemes, the process of non-Gaussianity genesis would be much more diverse and complicated. This is partly why we did not go into details to investigate different cases of non-Gaussianity genesis with the SPEEDY model.

The non-Gaussianity is less frequent in the wind components not only on the
timescale of 1 month but also for the snapshot, although the dynamic
process of the atmosphere is a nonlinear system. Moreover, the non-Gaussian
PDFs of temperature and specific humidity seldom affect the PDFs of the wind
components. We hypothesize that the model complexity may be a reason for
this. The SPEEDY model could not resolve some local interactions between
wind components and other variables due to its coarse resolution and
simplified processes. With more realistic models, physical processes are
much more complex, and the local interactions can also be represented.
Indeed, we obtained widely distributed non-Gaussianity with a 10 240-member
NICAM-LETKF system with a 112 km horizontal resolution assimilating real
observations from the National Centers for Environmental Prediction (NCEP)
known as PREPBUFR from 00:00 UTC on 1 November to 00:00 UTC on 8 November (Miyoshi
et al., 2015). Figure 20 shows the spatial distributions of background KL
divergence of zonal wind and temperature at the second model level
(

The outliers appear almost randomly regardless of locations, levels, and variables, and the lifetime is about a few analysis steps. When the outliers appear, the number of outliers is basically one per grid point, but sometimes the number is more than one. Anderson (2010) also reported similar results using a low-order dry atmospheric model. These results seem not to be consistent with Amezcua et al. (2012), who reported that just one outlier appeared with the ensemble square root filters in low-dimensional models and that the outlier did not rejoin the cluster easily. These properties of their outlier and our outliers in the SPEEDY model are somewhat different. In the low-dimensional models, a certain ensemble member tends to become an outlier at all grid points and all variables. In contrast, the outliers in the SPEEDY model appear at just some grid points but not all grid points and do not appear in all variables simultaneously. In addition, the negative influence of outliers on the analysis accuracy may be quite small in high-dimensional models due to the randomness and short longevity of outliers. In fact, the results showed no clear correspondence between the outlier frequency and analysis accuracy. These are the results from the simple SPEEDY model. How the outliers behave with a more realistic model and real observations remains the subject of future research.

As measures of non-Gaussianity, skewness, kurtosis, and KL divergence for the non-Gaussianity, and the SD and LOF methods for outliers, are introduced and compared with each other. The KL divergence is a more suitable measure because it measures the direct difference between the ensemble-based histogram and the fitted Gaussian function. The LOF method is better than the SD method because it can detect the outliers depending on the density of objects. Although it is easy to detect the outliers using the SD method, misdetection of outliers is possible because this method categorizes a small cluster far from the main cluster into outliers. The small cluster may be generated via physical processes and have physical significance; therefore, such cases should not be treated as outliers. The measures of non-Gaussianity are evaluated in the univariate field in this study. An extension to multivariate fields with multivariate analysis remains a subject for future research.

Non-Gaussian measures tend to be more sensitive to the sampling error due to the limited ensemble size (see Figs. 17, 18). When the ensemble size is small, it is difficult to determine whether a split member is a real outlier or a sample from a small cluster. Amezcua et al. (2012) discussed the outliers by skewness using the 20-member SPEEDY-LETKF and reported that the skewness is clearly large in the tropics and the Southern Hemisphere for the temperature and humidity fields. These results were not consistent with those of the present study because the outliers appear randomly. However, this inconsistency may have been due to the small ensemble size. The large skewness of Amezcua et al. (2012) could possibly indicate the non-Gaussianity rather than the outliers with a large ensemble size. Having a sufficient ensemble size, suggested to be about 1000 according to this study, would be essential when discussing about non-Gaussianity and outliers.

All data and source code are archived in RIKEN Center for Computational
Science and are available upon request from the corresponding authors under
the license of the original providers. The original source code of the
SPEEDY-LETKF is available at

KK performed the experiments and analyzed data. TM is the PI and directed the research. Both authors wrote the paper.

The authors declare that they have no conflict of interest.

The authors are grateful to the members of the Data Assimilation Research
Team, RIKEN R-CCS, and the Meteorological Research Institute for fruitful
discussions. The SPEEDY-LETKF code is publicly available at

This research has been supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (grant no. JP16K17806), the JST CREST (grant no. JPMJCR1312), the JST AIP (grant no. JPMJCR19U2), and the Japan Aerospace Exploration Agency (JAXA) Precipitation Measuring Mission (PMM).

This paper was edited by Olivier Talagrand and reviewed by three anonymous referees.