Characteristics of Initial Perturbations in the Ensemble

Characteristics of Initial Perturbations in the Ensemble Prediction Systemof the Korea Meteorological Administration

JUN KYUNG KAY AND HYUN MEE KIM

Atmospheric Predictability and Data Assimilation Laboratory, Department of Atmospheric Sciences,

Yonsei University, Seoul, South Korea

(Manuscript received 7 August 2013, in final form 20 January 2014)

ABSTRACT

In this study, the initial ensemble perturbation characteristics of the new Korea Meteorological Admin-

istration (KMA) ensemble prediction system (EPS), a version of the Met Office Global and Regional En-

semble Prediction System, were analyzed over two periods: from 1 June to 31 August 2011, and from

1 December 2011 to 29 February 2012. The KMAEPS generated the initial perturbations using the ensemble

transform Kalman filter (ETKF). The observation effect was reflected in both the transform matrix and the

inflation factor of the ETKF; it reduced (increased) uncertainties in the initial perturbations in regions with

dense observations via the transform matrix (inflation factor). The reduction in uncertainties is generally

governed by the transformmatrix but locallymodulated by the inflation factor. The sea ice significantly affects

the initial perturbations near the lower boundary layer. The large perturbation energy in the lower strato-

sphere of the tropics was related to the dominant zonal wind, whereas the perturbation energy in the upper

stratosphere of the winter hemispheres was related to the dominant polar night jet. In the early time-

integration stage, the initial perturbations decayed in the lower troposphere but grew rapidly in the mid- to

upper troposphere. In the meridional direction, the initial perturbations grew greatest in the northern polar

region and smallest in the tropics. The initial perturbations maintained a hydrostatic balance, especially

during the summer in both hemispheres and during both the summer andwinter in the tropics, associatedwith

the smallest growth rates of the initial perturbations. The initial perturbations in the KMAEPS appropriately

describe the uncertainties associated with atmospheric features.

1. Introduction

Because the atmosphere is a chaotic system, initial

condition errors and imperfect numerical forecastmodels

limit predictions of future states of the atmosphere. If

information regarding the distribution of forecast errors

is provided, then the imperfections of deterministic pre-

dictions can be mitigated by obtaining more information

on the probabilistic forecast (Palmer 1999). Ensemble

prediction (EP) estimates forecast uncertainties by in-

tegrating multiple initial conditions sampled from the

error probability of the initial state. To predict the un-

certainties associated with the forecast under the perfect

model assumption, initial conditions for EP should esti-

mate the probability density function (PDF) associated

with the error of the initial state (Ehrendorfer 1994).

Forecast uncertainties depend on the growth of the initial

state errors that are differences between initial ensemble

members and their mean field; therefore, initial ensemble

perturbations should have realistic growth rates that re-

flect the dynamic property of error growth in the atmo-

sphere (Palmer 1993; Magnusson et al. 2008).

The numerical weather prediction (NWP) centers have

developed and implemented various initial perturbation

generation methods for EP: the total energy norm sin-

gular vector (TESV) methods (Buizza and Palmer 1995;

Molteni et al. 1996) of the European Centre for Medium-

Range Weather Forecasts (ECMWF), the breeding

method (Toth and Kalnay 1993) of the National Centers

for Environmental Prediction (NCEP), the ensemble

transform (ET) technique (McLay et al. 2008; Wei et al.

2008) that is a new method of the NCEP, and the en-

semble transformKalman filter (ETKF)method (Bowler

et al. 2008, 2009) of the Met Office (UKMO). The TESV

method generates the perturbations with the greatest

linear growth over the optimization time interval (Buizza

et al. 2005; Kim and Morgan 2002) and the breeding

Corresponding author address: Hyun Mee Kim, Dept. of At-

mospheric Sciences, YonseiUniversity, 50Yonsei-ro, Seodaemun-gu,

Seoul 120-749, South Korea.

E-mail: [email protected]

JUNE 2014 KAY AND K IM 563

DOI: 10.1175/WAF-D-13-00097.1

� 2014 American Meteorological Society

mailto:[email protected]

method calculates the leadingLyapunov vectors ofmodel

solutions associated with the error of the greatest non-

linear growth (Toth and Kalnay 1993). Both the TESV

and breeding method consider the dynamical growth of

the initial perturbations, but may not draw the initial

ensembles from the PDF of the analysis error.

The ETKF method is based on the ensemble square

root filter (EnSRF) data assimilation scheme (Tippett

et al. 2003). Using the information of forecast and ob-

servational uncertainty, the ETKF generates initial en-

semble perturbations based on Kalman filter (KF)

analysis error covariance statistics. Because the calcu-

lations associated with the ETKF can be performed

rapidly, this method effectively generates ensemble

initial conditions. To estimate the analysis and forecast

error covariance appropriately using a finite number of

ensembles, each ensemble member should be main-

tained independently (i.e., orthogonally) so that the

finite number of ensembles could represent most of

the analysis and forecast uncertainties. A concentrated

variance to a specific subset of ensemble members

among the entire set of ensemble members cannot

represent forecast uncertainties appropriately. There-

fore, the even distribution of the variance of the initial

ensemble perturbations to the orthogonal ensemble di-

rections is a desirable property when generating the

initial ensemble perturbations. Wang and Bishop (2003)

first implemented the ETKF without localization as an

ensemble generationmethod and compared this method

with a masked-breeding method using a global climate

model [Community Climate Model (CCM3)] with a

fixed artificial observation network. They found that the

variance of the initial ensemble perturbations from the

ETKF reflected the spatial variability of observational

density and that this variance was more evenly distrib-

uted along the orthogonal ensemble direction for the

ETKF method than for the masked-breeding method.

Wei et al. (2006) applied the methodology of Wang and

Bishop (2003) to the NCEP operational ensemble pre-

diction system (EPS) using their operational observa-

tions. In contrast to Wang and Bishop (2003), Wei et al.

(2006) showed that the effect of the real observations on

the initial ensemble variance was not large on the global

scale due to small ensemble sizes (i.e., 10 members).

Miyoshi and Yamane (2007) used a local ETKF ap-

proach (LETKF; an ETKF method with localization as

a data assimilation scheme) in an AGCM model [the

Atmospheric General Circulation Model for the Earth

Simulator (AFES)] and performed observing system

simulation experiments (OSSEs) using a real observa-

tion network; they showed that the initial ensemble

spread from the LETKF was able to represent the anal-

ysis error. Bowler et al. (2008) described the overall

characteristics and performance of theMetOfficeGlobal

andRegionalEnsemble Prediction System (MOGREPS)

and demonstrated that the horizontal structure of the

initial ensemble spread from the ETKF was related to

the distributions of dynamically active areas and ob-

servational densities in themidtroposphere. Bowler et al.

(2009) adopted localization of the ETKF in MOGREPS

and verified that the localization made the latitudinal

distributions of the initial ensemble spread appropriately

reflect the ensemble-mean error distribution. Neverthe-

less, most ETKF studies have focused on the general

features of the initial perturbations, but have not dis-

cussed the characteristics of the ETKF components

composing the initial perturbations in detail.

Since 2011, the Korea Meteorological Administration

(KMA) has operationally implemented theUnifiedModel

(UM) and its related pre-/postprocessing systems, impor-

ted from theMet Office (Kay et al. 2013). The newEPS of

the KMA is based on MOGREPS and uses ETKF with

localization as the initial ensemble perturbation genera-

tion method. To verify that the ETKF method generates

operational ensembles appropriately, it is necessary to

understand the characteristics of the initial ensemble per-

turbations over the globe for statistically significant time

periods. Therefore, the present study analyzes the global

characteristics of the ETKF components andETKF-based

initial ensemble perturbations in the KMA EPS from

1 June to 31 August 2011 (hereafter JJA) and from

1 December 2011 to 29 February 2012 (hereafter DJF).

In particular, the role of each ETKF component in the

generation of initial ensemble perturbations is inves-

tigated, and the ability of the initial ensemble pertur-

bations to represent the initial state uncertainties is

examined from various dynamic viewpoints.

Section 2 presents the details of the system configu-

ration, the theoretical description of the ETKF, and the

experimental framework. Section 3 describes the char-

acteristics of the transform matrix and the inflation

factor of the ETKF. Section 4 presents the structure and

property of the initial ensemble perturbations. Section 5

provides a summary and discussion.

2. Methodology

a. The KMA EPS

The KMA EPS used in the present study is fundamen-

tally the same as the original MOGREPS. MOGREPS

consists of a global and regional ensemble system

(Bowler et al. 2008). The global ensemble system used in

the present study has a horizontal resolution of ap-

proximately 40 km and 70 vertical levels (N320L70) and

24 members (23 perturbed forecasts and 1 unperturbed

control forecast). The local ETKF generates initial

564 WEATHER AND FORECAST ING VOLUME 29

perturbations (see Bowler et al. 2009). The 23 initial

perturbationswere added to the analysis derived from the

four-dimensional variational data assimilation (4DVAR)

system of the KMAUM; the sum of the analysis and the

perturbations are then integrated forward for 10 days

using the KMAUMwith the same resolution used in the

KMA EPS. The global ensemble run is performed twice

daily (0000 and 1200 UTC). MOGREPS considered

model uncertainties using stochastic-physics schemes

that consist of ‘‘random parameters’’ and ‘‘stochastic

convective vorticity’’ schemes (Bowler et al. 2008). The

random parameters scheme (Bright and Mullen 2002)

introduces uncertainties into the empirical parameters of

the physical parameterization schemes. The stochastic

convective vorticity scheme (Gray and Shutts 2002) ad-

dresses a potential vorticity (PV) anomaly dipole similar

to the one typically associated with a mesoscale convec-

tive system (MCS). The horizontal components of wind u0

and y0, potential temperature u0, Exner pressure p 0, andspecific humidity q0 perturbations were variables used in

MOGREPS.

b. ETKF formulation

ETKF is a type of ensemble square root filter that

estimates forecast and analysis error covariance with

ensemble perturbations of size N based on the optimal

Kalman filter theory. If the size of the ensemble pertur-

bations is N, then the forecast and analysis perturbation

vectors z fi 5 x fi 2 xf and zai 5 xa

i 2 xa (i5 1, 2, . . . ,N) are

used to compose the matrix:

Zf 51ffiffiffiffiffiffiffiffiffiffiffiffi

N2 1p (z

f1 , z

f2 , . . . , z

fN) and

Za51ffiffiffiffiffiffiffiffiffiffiffiffi

N2 1p (za1 , z

a2 , . . . , z

aN) , (1)

where x fi and xa

i are the ith ensemble forecast and

analysis fields, respectively; xf and xa are the ensemble-

mean forecast and analysis from N members, re-

spectively; and the overbar represents the expectation

operator. The forecast and analysis covariance matrices

are expressed as

P f 5Zf (Zf )T and

Pa5Za(Za)T , (2)

where T indicates the matrix transpose. The analysis

ensemble perturbation matrix Za is calculated by solving

the Kalman filter update equation:

Pa5P f 2P fHT(HP fHT 1R)21HP f , (3)

where R is the observational error covariance matrix that

includes the errors of representativeness and observation

operator. In this system, R is defined as a diagonal

matrix and H is a linearized form of the observation

operator H.

If the analysis ensemble perturbations are derived by

transforming the forecast ensemble perturbations as

Za5ZfT , (4)

then the analysis error covariance matrix in Eq. (2) can

be rewritten as Pa 5ZfT(ZfT)T. By substituting this

analysis error covariance matrix into Eq. (3), the trans-

form matrix T is derived, as described by Wang et al.

(2004):

T5C(G1 I)21/2CT , (5)

where G is an (N 2 1) 3 (N 2 1) diagonal matrix con-

taining all of the eigenvalues of

(R21/2HZf )T(R21/2HZf ) , (6)

C is the eigenvectormatrix of Eq. (6), and I is the identity

matrix. InMOGREPSand theKMAEPS,H(xf )2H(x f )

is used instead of HZf in Eq. (6). The observational in-

formation matrix R and the formH are used in Eq. (6) to

incorporate the observation effects.

If the ensemble size is smaller than the degrees of

freedom of the model state, then the analysis error co-

variance is underestimated because the forecast error

covariance is not fully estimated by the ensemble mem-

bers. The inflation factor P is applied to the initial en-

semble perturbations in Eq. (4) to solve

Pn 5Pn21

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrace(dnd

Tn )2 trace(R)

trace(HPfnH

T)

s, (7)

where n represents the time steps and dn 5 yn 2H(x fn) is

an innovation vector that represents the difference be-

tween observations y and the 12-h ensemble-mean

forecast in observational space verified at the time of

observation. The inflation factor ensures that the sum of

the spreads of the ensemble 12-h forecast matches the

sum of the variations in the error of the ensemble-

mean 12-h forecast in the observational space over the

verification region (Bowler et al. 2008, 2009; Kay et al.

2013).

The two perturbation matrices Zf and Za are square

roots of the forecast and analysis covariance matrices

and have the same rank as their covariance matrices.

The initial ensemble perturbations are orthogonal to

each other in the observational space normalized by the

square root of the observation error (Wang and Bishop


2003). If the ensemble mean becomes the best estimate

of the true state (i.e., analysis), the sum of the ensemble

perturbations should be zero (Wang et al. 2004; Wei

et al. 2006; McLay et al. 2008). By applying the spherical

simplex method (Wang et al. 2004) that postmultiplies

the initial analysis perturbations by the transpose of C

[see Eq. (5)], the initial ensemble perturbations become

unbiased and the sum of the initial ensemble perturba-

tions becomes zero.

The ensemble-based data assimilation system uses

a covariance localization to decrease the long-range

spurious correlations due to the limited ensemble size

(Kay et al. 2013). The basic idea of covariance locali-

zation is to reduce the elements of the forecast error

covariance to remove the correlation between distant

grid points (Houtekamer and Mitchell 2001; Hamill

et al. 2001). The aforementioned covariance localization

cannot be implemented in ETKF because the forecast

error covariance is not explicitly calculated. Rather, the

horizontal localization of ETKF in MOGREPS is re-

alized by dividing the globe into 92 centers of approxi-

mately equal distance; the ETKF is then calculated

using the observations taken from a specific radius of

influence of each center to improve the likelihood that

the spread of EP is represented as a function of latitude

(Bowler et al. 2009). The KMA EPS does not currently

apply a vertical localization.

c. Experimental design

The characteristics of the initial ensemble perturba-

tions of the KMA EPS were investigated for JJA and

DJF. The observations used to calculate the initial en-

semble perturbations in the ETKF included sonde, surface

observation, aircraft, satellite wind (Satwind), Advanced

Television Infrared Observation Satellite (TIROS) Op-

erationalVertical Sounder (ATOVS), scatterometerwind

(Scatwind), Atmospheric Infrared Sounder (AIRS), GPS

radio occultation (GPS RO), and Infrared Atmospheric

Sounding Interferometer (IASI), all of which are con-

ventional observations collected for theKMAoperational

data assimilation system (Table 1). The observational

information corresponding to the observations in Table 1

is used in Eq. (6) to calculate the transform matrix. The

number of observations used is shown in Fig. 1. The in-

flation factors in Eq. (7) are calculated using sonde and

ATOVS observations with 12-h ensemble forecasts

because the error estimates of these observations are

relatively easily obtained in the KMA data assimilation

system (Kay et al. 2013). The inflation factors were cal-

culated vertically for four layers: the lower troposphere

(from the surface to 2 km), themid- to upper troposphere

(from approximately 2 to 15 km), the stratosphere (from

approximately 16 to 56 km), and the mesosphere [from

approximately 62 km to the top of the model (76 km)].

The detailed method of calculating the inflation factor is

described in Kay et al. (2013).

3. ETKF component characteristics

a. The transform matrix and the inflation factor

When the initial ensemble perturbations are generated

from Eqs. (4), (5), and (7), the 12-h ensemble forecast

perturbations are rotated by the eigenvectormatrix of the

forecast covariance matrix C and rescaled by (G1 I)21/2.

Therefore, the initial ensemble perturbations normalized

by the square root of R have the same direction as the

eigenvector of the 12-h forecast covariance matrix in

observational space (Wang and Bishop 2003). As men-

tioned in section 1, even distribution of the variance of the

initial ensemble perturbations to the orthogonal ensem-

ble directions (i.e., flatter eigenvalue spectrum) is one of

the desirable properties when generating the initial en-

semble perturbations. As the eigenvalue spectrum of the

forecast covariance matrix is flatter, the forecast ensem-

ble is equally rescaled in the eigenvector direction by

(G1 I)21/2. Thus, the eigenvalue spectrum of the forecast

covariance matrix provides information regarding how

well the initial ensemble perturbations include equally

filtered information in the direction of the eigenvectors.

Figure 2a shows the global eigenvalue spectrum of the

forecast covariance matrix; in other words, all observa-

tions on the globe are used to calculate the transform

matrix. Figure 2b shows the average of the 92 eigenvalue

spectra calculated from all localization centers. The ei-

genvalue spectra in Figs. 2a and 2b are normalized by

their leading eigenvalues. The flatter slope of the ei-

genvalue spectra implies that a large number of eigen-

vectors is necessary to explain the overall forecast

uncertainties. In contrast, the steeper slope of the

spectra implies that a relatively smaller number of

eigenvectors can explain much of the forecast un-

certainties (Szunyogh et al. 2007; Kim and Jung 2009).

The eigenvalue spectra of JJA are slightly flatter than

those of DJF in the global domain (Fig. 2a), which im-

plies that the forecast uncertainties are governed by

a large number of atmospheric systems during JJA

compared with DJF. The slopes of the eigenvalue

spectra of JJA and DJF do not significantly differ for

localized domains (Fig. 2b). The average of the 92 eigen-

value spectra has a steeper slope than the global eigen-

value spectra; thus, the average variance in the different

eigenvector directions of the forecast covariance vector

decreases via localization. The localization in the co-

variance matrix increases the rank of the forecast error

covariance (Hamill et al. 2000; Oke et al. 2007), and

averaged eigenvalue spectra over increasing ranks show


increasing flatness because the increased rank of the

forecast error covariance helps the ensembles to get more

effective degrees of freedom to fit the observations

(Ehrendorfer 2007). However, the localization removes

and consequently ignores the information associated with

the trailing modes of the eigenvalue spectrum (Hacker

et al. 2007). Although the localization of this system im-

proves the relationship between the ensemble spread and

the forecast error of the ensemble mean as a function of

latitude (Bowler et al. 2009), localization produces ei-

genvalueswith less variability than the global eigenvalues.

To investigate the spatial distribution of the variance

of the uncertainty spanned by the initial ensemble per-

turbations following Patil et al. (2001), the ensemble

dimension at each localization center was calculated as

C(s1,s2, . . . ,sN21)5

�N21

i51

si

!2

�N21

i51

s2i

, (8)

where si is the ith singular value of the 12-h forecast

covariancematrix in the observational space normalized

by the square root of R. The largest value of the en-

semble dimension is N 2 1, which is possible when the

uncertainty variance is evenly distributed across all en-

semble perturbations (Oczkowski et al. 2005).

Figure 3 shows the spatial distribution of the ensemble

dimensions. For both JJA and DJF, these dimensions

are large over East Asia and America in the tropical

region (TR) and greater in the extratropical Northern

Hemisphere (NH) compared with the extratropical

Southern Hemisphere (SH). On average, continents

have greater ensemble dimensions than oceans over the

same latitudes. This distribution of the ensemble di-

mension is similar to that in Kuhl et al. (2007), which

investigated the ensemble dimension using the NCEP

global model and LETKF. Two factors determine the

ensemble dimension distributions. First, because small

ensemble dimensions are related to large forecast error

growth in local regions (Oczkowski et al. 2005; Szunyogh

et al. 2005; Kuhl et al. 2007), large ensemble dimensions

in the TR are caused by small forecast error growth

during JJA andDJF (see Reynolds et al. 1994; Straus and

Paolino 2009). The large ensemble dimension in the TR

TABLE 1. Descriptions of the various observations used in the present study.

Observation type Description

Sonde Radiosonde observations (on land, ship, and mobile platforms), dropsonde observations, and wind profiler

observations

Surface Surface-based observations at or near the earth’s surface

Aircraft Aircraft-based observations reported by the Aircraft Meteorological Data Reporting (AMDAR) system

and aircraft reports (AIREPs)

Satwind Atmospheric wind observations derived from successive geostationary satellite observations

ATOVS Advanced Television Infrared Observation Satellite (TIROS) Operational Vertical Sounder system

observations

Scatwind Sea wind observations obtained from the Quick Scatterometer (QuikSCAT)

AIRS Atmospheric Infrared Sounder observations

GPS RO Global positioning system radio occultation observations

IASI Infrared Atmospheric Sounding Interferometer observations

FIG. 1. Observation numbers (contour) used to calculate the trans-

form matrix at each local center averaged for (a) JJA and (b) DJF.

The observation locations are interpolated into model grids.


indicates that the ensemble perturbations are highly

independent; as a result, the variability of the un-

certainty along different directions is maintained. In

contrast to the forecast error in the midlatitude, the

forecast error in the TR is not explained by several

major systems, but associated with many small systems

of complicated dynamics. Because of this feature, the

ensemble dimensions are relatively large in the TR.

Therefore, increasing the ensemble size might result in

more accurate estimations of the forecast error structure

in the TR. Second, because the transform matrix in

Eq. (5) is calculated in the regions where the observa-

tions are made, the observation density affects the en-

semble dimension. A high density of observations over

the NH continents (Fig. 1) leads to smaller forecast er-

rors in the NH; therefore, the ensemble dimension in the

NH is generally greater than that in the SH for both JJA

and DJF. In contrast, the ensemble dimension over the

ocean is relatively small, which is associated with large

forecast errors due to sparse observations. The ensem-

ble dimensions in the northern polar (NP) region differ

from those in the southern polar (SP) region. In the NP

region, the ensemble dimensions are large during JJA

but small during DJF. The seasonal change of the en-

semble dimension in the NP region is associated with

a small error growth (i.e., a large ensemble dimension)

in the summer and a large error growth (i.e., a small

ensemble dimension) in the winter due to baroclinic

instability. Therefore, the ensemble dimension in theNP

region depends on the atmospheric dynamic features.

However, these seasonal changes are not observed in

the SP region because this region has fewer observations

(Fig. 1); as a result, the observational density exerts

more influence on the seasonal change of the ensemble

dimension than the dynamic atmospheric features in the

SP region.

The inflation factor in Eq. (7) increases the spread of

the initial ensemble perturbations to match their mean

error. Figure 4 shows the average inflation factors ap-

plied to the troposphere. The distribution of the sonde

and ATOVS observations used to calculate the inflation

factor is similar to the distribution of the inflation factor

itself. The regions of dense observations have greater

inflation factors because more observations are used to

calculate the inflation factor. In contrast, the regions

over the SH and the ocean have smaller inflation factors.

b. The combined effect of the transform matrixand the inflation factor on the initial ensembleperturbations

Observational information reduces the forecast un-

certainty in the ETKF method by applying the trans-

form matrix to the forecast ensemble perturbations in

the Kalman filter update equation. To investigate the

observation effect on the initial ensemble perturbations,

the ratio of the spread of the initial ensemble per-

turbations with regard to that of the 12-h ensemble

forecasts was calculated (Fig. 5). This ratio effectively

identifies the spatial distributions of rescaling factors

(i.e., the transform matrix) that reduce the forecast

spread to the initial ensemble spread using the obser-

vational information and the inflation factor (Wang and

Bishop 2003; Wei et al. 2006). According to Wang and

Bishop (2003), the ETKF, the optimal data assimilation

scheme, attenuates the forecast uncertainty represented

by the forecast ensemble spread to generate the initial

ensemble spread via observational information; therefore,

the ratio is expected to be less than 1. Wang and Bishop

(2003) also showed that this ratio is lower in regions with

dense observations compared with those with sparse ob-

servations due to the filtering effect of the ETKF (here-

after called the transform effect). In contrast, the value of

FIG. 2. An eigenvalue spectrum of (R21/2HZf )T(R21/2HZf ) (a) over the global domain and (b) averaged over 92 local

centers. The outlined and filled bars represent JJA and DJF, respectively.


the inflation factor increases in regions with dense ob-

servations. Therefore, the ratio of the spread of the

initial ensemble perturbations with regard to the spread

of the 12-h ensemble forecast in Fig. 5 shows the com-

bined effect of both the transform matrix and the in-

flation factor.

The ratio is greater in the TR than at higher latitudes

and is generally higher in the SH than in the NH because

there are more observations in the NH (Fig. 5), which

implies that the transform effect governs the general

features of the ratio in the latitudinal direction. The

ratios do not change significantly with model variables;

however, they do change with seasons. In the NH

midlatitudes, the ratio over the continents is relatively

greater than that over the ocean (except in North

America during DJF). This finding implies that the in-

flation factor effect predominates in these regions. The

ratios exhibit less seasonal variation in the SH compared

with the NH, and they are relatively greater over land

regions with a large amount of satellite observation data

(cf. Fig. 5 with Fig. 4). This finding implies that the in-

flation factor effect also predominates over the land in

the SH. Therefore, the transform effect governs the

general features of the ratio along the latitudinal di-

rection, but the inflation factor effect governs the local

features of the ratio along the longitudinal direction,

FIG. 3. Ensemble dimension averaged for (a) JJA and (b) DJF.


which implies that the ratio generally follows the trans-

form effect and is locally modulated by the inflation

factor. The distribution of the ratios in the present study is

similar to that of the ensemble transform rescaled to

consider the climatologically derived analysis error vari-

ance [cf. Fig. 5 with Fig. 2a in Wei et al. (2008)], which

implies that the initial ensemble perturbations of the

ETKF in the KMA EPS are consistent with the clima-

tologically derived analysis error variance.

To investigate the vertical structure change of the

ensemble perturbations that result from the application

of the transform matrix and the inflation factor, two

areas were chosen: one is the northern midlatitude area

(208–608N) where the average ratio is less than 1, and

the other is the area in which the ratio is greater than 1

(denoted A in Fig. 5a). Figure 6 shows the vertical

profiles of the initial ensemble and the 12-h forecast

ensemble spreads over the northern midlatitude area

and area A. In the northern midlatitude, the spread of

the initial ensemble perturbations was generally smaller

than that of the forecast ensemble perturbations dur-

ing JJA and DJF, and distinctly smaller approximately

FIG. 4. Number of sonde and ATOVS observations (contours) used to calculate the inflation

factor at each local center and the averaged inflation factors in the troposphere (shaded) for

(a) JJA and (b) DJF. The observation locations and inflation factors at 92 local centers are

interpolated into model grids.


below 50 and above 10hPa (Figs. 6a,c). This finding im-

plies that the transform effect generally dominates the

northernmidlatitude area. In areaAabove 1hPa (i.e., the

midstratosphere), the initial ensemble spread is smaller

than the forecast ensemble spread (Figs. 6b,d), which

implies that the inflation factor suppresses the trans-

form effect that usually decreases forecast uncertainty

throughout most of the atmosphere below 1 hPa.

Overall, the transform effect and inflation factor op-

positely affect the initial uncertainties. The transform

effect reduces the size of the initial perturbations, but

the inflation factor increases that of the initial pertur-

bations. The transform effect affects the general fea-

tures of the initial perturbations, but the inflation factor

affects the local features of the initial perturbations.

4. Initial ensemble perturbation characteristics

a. The structure of the initial ensemble perturbations

The total energy (TE) norm of the initial ensemble

perturbations was calculated as described by Bannon

(1995), Ehrendorfer and Errico (1997), Rabier et al.

(1996), and Wang and Bishop (2003):

TE51

2(u021 y02)1

1

2

R

kTref

u 0211

2

RTref

gp2p 02 , (9)

where u0, y0, u0, and p0 are variables in the KMA EPS;

R is the gas constant for dry air; k (’0.286) is the ratio of

R to cp (the specific heat of dry air at a constant pres-

sure); g (’1.4) is the ratio of cp to cy (the specific heat of

dry air at a constant volume); Tref is a reference tem-

perature of 300K; andp is taken from the analysis of the

KMA UM. The first term on the right-hand side of Eq.

(9) represents the kinetic energy (KE), whereas the

second and third terms together represent the potential

energy (PE) (Eckart 1960).

Figure 7 shows the TE of the initial ensemble per-

turbations averaged during JJA and DJF at several

vertical layers, as well as the vertical profiles of the TE

for six regions: the SP region from 908 to 708S, the SH

from 708 to 208S, the TR from 208S to 208N, the NH from

208 to 708N, the NP region from 708 to 908N, and the

entire globe. At the lower boundary layer, the TE during

JJA is large along the border between Antarctica and

the ocean in the SH and large over the Arctic Ocean in

the NH (Fig. 7a); conversely, the TE during DJF is large

over the Arctic Ocean (Fig. 7f), which implies that sea

ice constitutes an important source of uncertainty re-

garding the initial ensemble perturbations near the

surface. In the troposphere, a large amount of TE is

found in the midlatitude during JJA and DJF, isolated

over Africa, the eastern Pacific region, and central Asia

in the NH; the area of large TE in the SH is located over

the ocean and Antarctica (Figs. 7b,g). In the lower- to

midstratosphere, the TE is concentrated in the TR from

approximately 20 to 30 km (from 50 to 10 hPa) and

forms a zonal band during JJA and DJF (Figs. 7c,h). As

FIG. 5. The average ratios of the square root of the initial ensemble variance with regard to the square root of the

12-h ensemble forecast variation for (a),(c) zonal wind and (b),(d) temperature during (left) JJA and (right) DJF.

Box A represents the area with large ratios.


the altitude increases above 10 hPa, the TEs during JJA

and DJF are located broadly in the winter hemispheres

(i.e., SH, Fig. 7d; NH, Fig. 7i). The TE is greater during

DJF (Fig. 7i) compared with JJA (Fig. 7d) near the top

of themodel. As the altitude increases, the TE decreases

to its minimum in the stratosphere and then increases at

the top of the model (Figs. 7e,j).

Figure 8 shows the vertical cross sections of the zon-

ally averaged TE of the initial ensemble perturbations

superimposed on the zonal wind of the KMA UM

analysis. The initial ensemble perturbations have the

greatest TE above the upper stratosphere of the winter

hemisphere where the polar night jets are strong. Over

the TR, the large TE of the initial ensemble perturba-

tions was concentrated in the lower stratosphere where

the dominant zonal winds have large spatial variations

and directional changes with altitude. Kawatani and

Hamilton (2013) demonstrated that the descending

easterly wind was predominant during JJA 2011, and

this finding was confirmed by the KMA UM analysis

(Fig. 8a). During DJF, the easterly wind regime was

predominant in regions above 40 hPa, and the westerly

regime located below the easterly regime is weaker than

the easterly regime (Fig. 8b). As Figs. 7c and 7h show,

the zonally averaged TE of the initial ensemble pertur-

bations over the TR in the stratosphere is greater along

FIG. 6. The vertical profiles of the zonally averaged initial ensemble spread (solid line) and 12-h forecast ensemble

spread (dashed line) for zonal wind averaged over (top) the NH (208–608N) and (bottom) area A during (a),(b) JJA

and (c),(d) DJF.


FIG. 7. The vertically averaged TE of the initial ensemble perturbations (shaded; JKg21) at

each layer during (left) JJA and (right) DJF in the (a),(f) lower boundary; (b),(g) tropo-

sphere; (c),(h) lower to midstratosphere; and (d),(i) top of the model. (e),(f) The vertical

profiles for horizontally averaged TE (contour; JKg21) over the NP (gray), NH (purple), TR

(blue), SH (green), and SP (red) regions, as well as the entire globe (black) for (left) JJA and

(right) DJF.


the dominant easterly wind during DJF compared with

JJA (cf. Fig. 8b with Fig. 8a). From the surface to the

lower troposphere, the initial ensemble perturbation TE

is greater in the winter polar regions, which is due to the

uncertainty associated with the sea ice (see Figs. 7a,f).

Figure 9 presents the vertically and zonally averaged

ensemble-mean TE, KE, and PE of the initial ensemble

perturbations and the standard deviation of the TE for

each vertical layer identified in section 2c. The TE, KE,

and PE are vertically integrated considering the density

in each layer and normalized by the sum of the density of

all vertical layers at each model grid. During JJA and

DJF, the magnitude of KE was greater than that of PE,

which is consistent with the results of Bowler et al.

(2008), who used MOGREPS without localization. In

the lower troposphere, the KE and PE were large and

showed large variations in the ensemble members in the

NP and SP regions. The TE was especially large in the

winter polar regions and relatively weak in the TR (Figs.

9a,e); Bowler et al. (2009) reported similar findings,

showing underestimated ensemble spreads in the TR.

The PE was relatively large in the NP and SP regions;

however, the magnitude of the PE was approximately

one order of magnitude lower than that of the KE.

FIG. 8. Vertical cross sections of the zonally averaged TE of the initial ensemble perturba-

tions (shaded; J kg21) and the zonal wind of the KMA UM analysis (contour; m s21) during

(a) JJA and (b) DJF.


Vertically and zonally averaged zonal and meridional

winds contribute equally to the KE in all regions except

the TR where the zonal wind perturbation energy is

approximately 2 times greater than the meridional wind

perturbation energy. In the mid- to upper troposphere,

the TE during JJA is large in the SP region (Fig. 9b), and

the TE during JJA and DJF is large in the TR (Figs.

9b,f). In the stratosphere, the TE during JJA and DJF

was large and had a high standard deviation in the

TR (Figs. 9c,g). In the stratosphere, the dominant

FIG. 9. The vertically and zonally averaged ensemble-mean TE (black; J kg21), KE (red; J kg21), and PE (blue;

J kg21), as well as the std dev of the TE (green shading; J kg21) of the initial ensemble perturbations during (left)

JJA and (right) DJF, for the (a),(e) lower troposphere from the surface to 2 km; (b),(f) mid- to upper troposphere

from 2 to 15 km; (c),(g) stratosphere from 16 to 56 km; and (d),(h) mesosphere from 62 km to the top of the model

(76 km).


component of the TE in the TR is the zonal wind, which

is primarily affected by the various waves generated by

convective activities and other forcings in the TR, as

well as by ozone concentrations (Holton 2004). These

factors might affect the direction and magnitude of

background zonal wind and the initial ensemble per-

turbations. Although the magnitude of the TE in the

mesosphere was largest in the vertical distribution (Figs.

7e,j), few vertical model layers in the mesosphere ren-

dered the vertically integrated ensemble-mean TE

(normalized by the sum of the density of all vertical

layers) relatively small (Figs. 9d,h). The TE was large in

the winter hemispheres and reached maximum values at

the midlatitudes (Figs. 9d,h). Enomoto et al. (2010)

showed that the large TE in the midlatitudes at the up-

per layer is associated with a temporal change in zonal

wind and temperature.

b. The growth rate of the initial ensembleperturbations

The perturbation growth rates are important factors in

estimating the quality of the initial ensemble perturba-

tions (Magnusson et al. 2008). To reflect the instability

in the atmosphere, the initial ensemble perturbations

should grow appropriately to span the subspace of the

forecast error throughout the model integration. This

property is a critical constraint needed to identify the ini-

tial ensemble perturbations because randomly sampled

perturbations from analysis uncertainty tend to decay

during the early stages of integration and consequently

degrade the quality of the EPS. Wang and Bishop (2003)

showed that the growth rate of the initial ensemble per-

turbations from the ETKF was greater than those of sin-

gular vectors and breeding methods. In this study, to

estimate the growth rate of the initial ensemble perturba-

tions from the ETKF, the Lyapunov exponent, the expo-

nential growth rate of the perturbations in a chaotic system

(Palmer 1993; Magnusson et al. 2008), is calculated as

l51

Dtln

�kDx(t1Dt)kkDx(t)k

�, (10)

where Dt is 6 h and kDx(t)k is the amplitude of the per-

turbations at time step t. Because the 10-day ensemble

forecasts in the KMA EPS are produced in the limited

troposphere (i.e., from the surface to 200 hPa), the

growth rate of the initial ensemble perturbations was

calculated for regions below 200 hPa.

Figure 10 shows the exponential growth rate of the

zonal wind perturbations at 850, 500, and 200 hPa for

a 48-h forecast lead time. Because Eq. (10) assumes that

the perturbation growth is linear, the verification time is

set to 48 h. Overall, the growth rates during DJF are

lower than those during JJA. During both periods, the

growth rates at 850 hPa averaged over the entire globe

had negative values for the short forecast lead time,which

implies that the initial ensemble perturbations decay

during this time (Figs. 10a,d). Because the magnitude of

the 12-h forecast ensemble perturbations is smaller than

that of the initial ensemble perturbations at 850hPa

(Figs. 6a,c), negative growth rates occur at 850hPa. As

the forecast lead time increases, the growth rate over

the entire globe acquires similarly positive values. The

greatest growth rate occurs in the SH, whereas the

smallest rate occurs in the TR at 850 hPa (Figs. 10a,d).

Compared to other regions, the error growth rates in the

TR are small due to the weaker energy source for dy-

namical growth, as described in Whitaker and Hamill

(2012). At 500 and 200 hPa, the initial ensemble per-

turbations exhibited the greatest positive growth rate at

the first time step; subsequently, the growth rates de-

creased as the forecast lead time increased (Figs. 10b,c,e,f).

The perturbations in the NP region had the highest

growth rate for all forecast lead times, whereas those in

the TR had the lowest growth rate (Figs. 10b,c,e,f).

c. The hydrostatic balance of the initial ensembleperturbations

The initial ensemble perturbations from the ETKF,

which are constrained by the forecast error covariance

Pf , generally satisfy geostrophic and hydrostatic bal-

ances. The hydrostatic balance of the initial ensemble

perturbations is necessary because the imbalanced initial

ensemble perturbations grow during the model inte-

gration and degrade the quality of the ensemble fore-

casts. To measure how this balance is maintained with

the initial ensemble perturbations, hydrostatically bal-

anced potential temperature perturbations were calcu-

lated using the initial ensemble perturbations from the

KMA EPS and UM analyses (see Vetra-Carvalho et al.

2012):

u0H 5(«212 1)q0u

[11 («212 1)q]1cp

g

dp0

dzu2[11 («212 1)q] ,

(11)

where « is the ratio of the mass of liquid water to dry air,

q is the specific humidity, and u and q are taken from the

analysis of the KMA UM.

The correlations between the hydrostatically bal-

anced potential temperature perturbations (u0H) and the

potential temperature perturbations (u0) from theETKF

were calculated for each grid and then averaged horizon-

tally over the NP, NH, TR, SH, and SP regions (Fig. 11).

The correlations were greater than 0.9 during JJA and


DJF, with generally similar patterns among whole ver-

tical layers; this finding implies that the initial ensemble

perturbations satisfy the hydrostatic balance. Near the

surface and at altitudes above 6.6 km, the correlations

during JJA and DJF were generally close to 1. The

correlations were between 0.9 and 1 from 0.7 to 6.6 km.

The correlations in the stratosphere are larger than

those in the troposphere due to higher static stability in

the stratosphere than in the troposphere. The correla-

tions during JJA and DJF at the midtroposphere (ap-

proximately 6.6 km) were smaller in the NP region (Fig.

11a) and greater in the TR (Fig. 11c), which is consistent

with the largest and smallest growth rates in the NP region

and the TR, respectively (see Figs. 10b,c,e,f). The corre-

lations in the NH were greater during JJA compared with

DJF (Fig. 11b); in contrast, the correlations in the SH

and the SP region during JJA were smaller than those

during DJF (Figs. 11d,e). The greater correlations during

DJF across the entire globe (Fig. 11f) were caused by the

greater correlations during DJF in the TR, SH, and SP

regions.

Overall, the initial ensemble perturbations maintained

hydrostatic balance during the summer and winter in the

TR and during the summer in each hemisphere, consis-

tent with the smaller perturbation growth rates.

5. Summary and discussion

To accurately estimate the forecast uncertainty in the

ensemble prediction system, initial ensemble perturbations

FIG. 10. The average growth rates of the initial zonal wind perturbations at (a),(d) 850; (b),(e) 500; and

(c),(f) 200 hPa during (left) JJA and (right) DJF. The growth rates are averaged over the NP (gray), NH (purple), TR

(blue), SH (green), and SP (red) regions, as well as the entire globe (black).


should represent the uncertainty of the atmospheric

state at the time of analysis. Due to the limited ensemble

size, the initial perturbations should sample the proba-

bility distribution of the analysis, thereby maintaining

the independence between the ensemble members with

appropriate growth rates. In this paper, the character-

istics of the initial ensemble perturbations based on the

ETKF in the high-resolution operational EPS of the

KMA were investigated over two periods: from 1 June

to 31 August 2011 (JJA) and from 1 December 2011 to

29 February 2012 (DJF).

The characteristics of the transform matrix and the

inflation factor of the initial ensemble perturbations were

investigated. The eigenvalue spectrum and the ensemble

dimension of the transform matrix, which reduces fore-

cast uncertainty with observational information, were

analyzed. The eigenvalue spectra during JJA and DJF

were similar across global and localized domains. The

global eigenvalue spectrum was flatter than the average

of the localized eigenvalue spectra; thus, localization in

the ETKF might bias the filtering of forecast uncertainty

in certain eigenvector directions. Overall, the ensemble

dimension (a measure of how the analysis variances are

distributed over the eigenvector directions) was large

over regions with small errors or small error growth and

with dense observations (e.g., the TR and the summer

hemispheres) during JJA and DJF.

In contrast to the transform matrix, which reduces the

forecast uncertainty to produce the initial ensemble

perturbations, the inflation factor increases the spread of

the initial ensemble perturbations tomatch the ensemble-

mean error. Therefore, the combined effects of these

components are needed to understand the characteristics

of the initial ensemble perturbations from the ETKF; in

this study, the combined effects were investigated using

the ratio of the initial ensemble spread with regard to the

12-h forecast ensemble spread. The vertically averaged

ratiowas large near the TR and generally greater over the

continent compared with over the ocean at the given

latitude during JJA and DJF. This finding partially con-

tradicts those of previous studies (e.g., Wang and Bishop

2003; Wei et al. 2006) concerning the effect of the trans-

form matrix because the inflation factor based on Eq. (7)

had greater values in regions with dense observations. As

Whitaker et al. (2008) discussed, the multiplicative in-

flation factor based on Eq. (7) might produce unrealistic

forecasts or analysis error variance due to the heteroge-

neous observation distributions. The use of other types of

inflation factors [e.g., the additive or the combined ad-

ditive and multiplicative inflation factor discussed by

FIG. 11. The horizontally averaged ensemble-mean correlations between the hydrostatically balanced potential

temperature perturbations and the potential temperature perturbations generated in the ETKF during JJA (solid

line) andDJF (dashed line) over the (a) NP, (b) NH, (c) TR, (d) SH, and (e) SP regions, as well as (f) the entire globe.

The x axis represents the vertical layer.


Whitaker and Hamill (2012)] might address different

aspects of the initial ensemble perturbations.

The ensemble dimension is greater and the ratio is

generally smaller in the NH compared with the SH,

which indicates that the dense observations in the NH

reduce forecast error and that the initial ensembles ap-

proximate the observations. In contrast, in the SH, the

sparse and localized distributions of the observations

differentiate the initial ensembles from the observations

and the reduction of the forecast error variance by the

transform matrix is restricted with regard to magnitude

and direction. To solve this issue, the bias correction of

the forecast error variances estimated via ensemble or

statistical adjustments of the ensemble spread to the

observations (Wang et al. 2007) may be adopted in fu-

ture studies.

The initial ensemble perturbation energy in the bound-

ary layer is located along the sea ice in the NP and SP re-

gions. In the troposphere, the perturbation energy is

greatest nearAntarctica and located at themidlatitude of

the winter hemispheres. The perturbation energy in the

lower and midstratosphere is concentrated zonally in

the TR, and the depth of the large TE layer depends on

the vertical variation of the dominant zonal winds in this

layer. From the upper stratosphere to the model top,

the perturbation energies are broadly spread across the

winter hemispheres where the strong polar night jets

are located.

The growth rates of the ensemble perturbations were

calculated for the troposphere. In the lower tropo-

sphere, the growth rates averaged across the globe were

negative for short forecast lead times during JJA and

DJF; then the growth rates subsequently increased and

converged to positive values. The perturbations showed

the smallest and largest growth rates in the TR and the

SH, respectively. In the mid- and upper troposphere, the

growth rates were greatest at the first forecast time step,

and then decreased as the forecast lead time increased.

The perturbations showed the largest and smallest growth

rates in the NP regions and the TR, respectively, during

JJA and DJF. On average, the perturbations increased

more rapidly during JJA compared with DJF.

The hydrostatic balance of the initial ensemble per-

turbations were investigated by comparing the hydro-

statically balanced potential temperature perturbations

with the potential temperature perturbations generated

from the ETKF. Hydrostatic stability is an instability

criterion associated with the growth of baroclinic per-

turbations (Fleagle 1955); therefore, the hydrostatic bal-

ance of the initial perturbations appears to be consistent

with the growth rate of the initial perturbations. During

JJA and DJF, the initial perturbations were in hydro-

static balance, and these perturbations experienced the

smallest growth rates in the TR. In contrast, the initial

perturbations were in relatively weak hydrostatic balance

in the NP regions where the greatest increase in the

growth rate occurred. In the lower troposphere from

approximately 0.7 to 6.6 km, the initial perturbations

departed from hydrostatic balance during JJA and DJF.

The initial perturbations were closer to hydrostatic bal-

ance states in the SH duringDJF compared with JJA; the

same was true in the NHduring JJA compared withDJF.

These findings imply that hydrostatic balance is main-

tained in the summer in both hemispheres and during JJA

and DJF in the TR, and is associated with the smallest

growth rates of the initial ensemble perturbations.

The observational effect should be reflected appropri-

ately in the ETKF to represent the initial uncertainties by

the initial ensemble perturbations. Because the distribu-

tion and the density of the observations affect the trans-

form matrix and the inflation factor in the ETKF, they

also influence the characteristics and quality of the initial

ensemble perturbations. The adoption of different algo-

rithms for the inflation factor or a bias correction scheme

of forecast error might improve the quality of the initial

ensemble perturbations generated by the ETKF in the

KMA EPS, by relaxing the effect of the observational

distribution smoothly along the latitudinal and longitu-

dinal directions. From a dynamic perspective, the initial

ensemble perturbations of the ETKF in the KMA EPS

are generally in hydrostatic balance and exhibit appro-

priate growth rates, which implies that the initial ensemble

perturbations appropriately describe the uncertainties as-

sociated with atmospheric dynamic features. In the future,

additional studies of the meteorological phenomena in the

upper layer using the initial ensemble perturbations of the

KMA EPS will be conducted.

Acknowledgments. The authors thank the two anony-

mous reviewers for their valuable comments. This study

was supported by the Korea Meteorological Administra-

tion Research and Development Program under Grant

CATER 2013–2030. The authors thank Dr. Young-Youn

Park and Ms. Joohyung Son for their support in the early

stages of this study and theNumericalWeather Prediction

Division of the KoreaMeteorological Administration and

the Met Office for providing the resources necessary for

this study.

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Documents

Characteristics of Initial Perturbations in the Ensemble