> **Initial Perturbations for NCEP Ensemble Forecast System**

*Mozheng **Wei**, Zoltan **Toth**. Richard **Wobus**, Yuejian **Zhu**
and Craig H. **Bishop***

NCEP Environmental Modeling Center, 5200 Auth Road, Rm. 207, Camp
Springs, MD 20746, USA

(Mozheng.Wei@noaa.gov)

**Abstract:** The initial perturbations used for the operational global
ensemble prediction system of the National Centers for Environmental
Prediction (NCEP) are generated through the breeding method with a
regional rescaling mechanism. Limitations of the system include the use
of a climatologically fixed estimate of the analysis error variance and
the lack of an orthogonalization in the breeding procedure. In order to
ameliorate these shortcomings We compare the initial perturbations
generated by the Ensemble Transform Kalman Filter(ETKF), Ensemble
Transform (ET) and ET with rescaling methods. The experiments were
carried out in operational environment as NCEP global ensemble forecast
system with information on the actual distribution and error
characteristics of real-time observations. The perturbations generated
by 4 methods are studied in detail and compared with the current NCEP
operational ensemble forecast system.

**1. Introduction**

Ensemble forecasts start from a set of different states that are sampled
from a probability density function which is approximated using a finite
sample of initial perturbations. However, how to best generate these
initial perturbations for an ensemble forecasting system is still a
research issue.

The first generation initial perturbation techniques include: singular
vectors (SVs) used at the European Center for Medium-Range Weather
Forecasts (ECMWF). SVs are used to identify the directions of fastest
forecast error growth for a finite time period (Molteni *et al.* 1996);
bred vectors (BVs) used to sample amplifying analysis errors through
breeding cycles have been used at the National Centers for Environmental
Prediction (NCEP). Generation of BVs are similar to data assimilation
cycles (Toth and Kalnay 1993; 1997). However, both SVs and BVs cannot
accurately represent the true uncertainties in analysis as a good
ensemble forecast system expects. Initial perturbations in both ensemble
systems are not consistent with the data assimilation systems that
generate the analysis fields. A comparison of performance between the
ECMWF and NCEP ensemble forecast systems was described in in Wei and
Toth (2003).

Another major method in the first generation is the perturbed
observation (PO) approach developed at the Meteorological Service of
Canada (MSC) (Houtekamer et al. 1996). The PO approach generates initial
conditions by assimilating randomly perturbed observations using
different models in a number of independent cycles. The initial
perturbations generated by the PO method are more representative of
analysis uncertainties in comparison with SVs and BVs. However, the data
assimilation quality from ensemble based Kalman fitler is still lagging
behind 3-D Var and 4-D Var. A comprehensive summary of current
methodologies and performance of the three ensemble forecast

systems from ECMWF, MSC and NCEP can be found

in Buizza *et al.* (2005)

In this paper, we compare various methods that may be classified as
second generation initial perturbation techniques, namely ETKF, ET, ET
with breeding and pure breeding. Singular vectors with Hessian norm can
also be classified as a second generation method. The ETKF and ET
methods put forward by Bishop *et al*. (2001) and Bishop and Toth (1999)
respectively were initially used for the adaptive observation studies.
Later Wang and Bishop (2003) showed how ETKF could be used to generate
ensemble perturbations in an idealized observation framework. It was
further extended to operational environment with NCEP real-time
observations by Wei *et al.* (2005).

A common feature of the second generation techniques is that the initial
perturbations are more consistent with the data assimilation systems.
This is the case for ETKF based ensemble. However, the quality of data
assimilation from ETKF is still lagging behind NCEP operational 3D-VAR
system. The ensemble cloud has to be adjusted to center around the
analysis field generated by 3D-VAR system. To overcome this drawback, we
adopt the ET technique to generate initial perturbations.

Section 2 provides a brief basic description of the ET formulations for
initial perturbations. Section 3 presents the major results from ET
comparisons with the NCEP operational bred perturbation-based ensemble
system. Discussion and conclusions are given in Section 4.

**2. Basic formulation**

The initial perturbations of the NCEP global ensemble forecast system
are generated by a breeding method. This method is well established,
widely used and well documented. The operational implementation at NCEP
can be found in Toth and Kalnay (1993, 1997). The ETKF formulation
(Bishop *et al*. 2001) is based on the application of a Kalman filter,
with the forecast and analysis covariance matrices being represented by
*k* forecast and *k* analysis perturbations. More details can be found
in Bishop *et al*. (2001), Wang and Bishop (2003) and Wang *et al.*
(2004). Results of ETKF with NCEP real-time observations can be found in
Wei *et al.* (2005). The ET method was formulated in Bishop and Toth
(1999) primarily for target observation studies. In this paper, we adopt
this technique for ensemble forecasting. Let$$

$\begin{array}{r}
Z^{f} = \frac{1}{\sqrt{k - 1}}\lbrack z_{1^{f}},z_{2^{f}},\text{.}\text{.}\text{.}\text{.}\text{.},z_{k^{f}}\rbrack\text{     and } \\
Z^{a} = \frac{1}{\sqrt{k - 1}}\lbrack z_{1^{a}},z_{2^{a}},\text{.}\text{.}\text{.}\text{.}\text{.},z_{k^{a}}\rbrack
\end{array}$ (1)

where the *n* dimensional state vectors
$z_{i^{f}} = x_{i^{f}} - x^{f}\text{   and  }z_{i^{a}} = x_{i^{a}} - x^{a}\text{  }(i = 1,\text{ 2, }\text{.}\text{.}\text{.}\text{.}\text{.}k)$are
$k$ ensemble forecast and analysis perturbations, respectively. In our
experiments, $x^{f}$is the mean of $k$ ensemble forecasts and $x^{a}$ is
the analysis from the independent NCEP operational data assimilation
system. Unless stated otherwise, the lower and upper case bold letters
will indicate vectors and matrices respectively. The $n \times n$
forecast and analysis covariance matrices are formed, respectively, as

$P^{f} = Z^{f}Z^{f^{T}}\text{     and    }P^{a} = Z^{a}Z^{a^{T}}$,

where $T$ indicates the matrix transpose. For a given set of forecast
perturbations $Z^{f}$ at time *t*, the analysis perturbations
$Z^{a} = Z^{f}Τ$. Suppose we have obtained the analysis covariance
matrix from the operational data assimilation system, then
$P^{a} = (Z^{f}T)(Z^{f}T)^{T}$. The ET solution is $Z^{a} = Z^{f}T$,
where $T = \text{CΓ}^{\frac{- 1}{2}}$, $C$ contains column orthonormal
eigenvectors ($c_{i}$) of $Z^{f^{T}}P^{a^{- 1}}Z^{f}$ and $\Gamma$ is a
diagonal matrix containing the associated eigenvalues ($\lambda_{i}$)
that is,
$C = \lbrack c_{1},c_{2},\text{.}\text{.}\text{.}\text{.}\text{.},c_{k}\rbrack$
and$\Gamma = \text{diag}(\lambda_{1},\lambda_{2},\text{.}\text{.}\text{.}\text{.}\text{.}\text{, }\lambda_{k})$.
Although the forecast perturbations are, by definition, centered about
the ensemble mean, i.e. $\sum_{i = 1}^{k}z_{i^{f}} = 0\text{.}0$, the
analysis perturbations produced by the ET defined above are not centered
around the analysis ($\sum_{i = 1}^{k}z_{i^{a}} \neq 0\text{.}0$). A
simple transformation that will preserve $P^{a}$ and center the analysis
perturbations about the analysis is the simplex. Similar to the ETKF
experiments, $C^{T}$ is one of the solutions of this transformation.
Hence, $Z^{a} = Z^{f}\text{TC}^{T}$ will be used as our initial analysis
perturbations for the next cycle forecasts.

The experimental setup in this paper is the same as in Wei *et al.*
2005. Our experiments run from Dec 31, 2002 to Feb. 17, 2003, however,
our study will only focus on the 32-day period from 01/15/2003 to
2/15/2003. There are 10 ensemble members in all the systems. The
observations for ETKF are from the NCEP global data assimilation system
which also produces the analysis field for all experiments. All ensemble
systems are cycled every 6 hours in accordance with the NCEP data
assimilation system in which new observations are assimilated in
consecutive 6-hour time windows centered at 00, 06, 12 and 18 UTC. The
model used is the NCEP operational forecast model (GFS) at T126L28.

**3. Results**

![](media/image1.wmf){width="3.102777777777778in"
height="2.1743055555555557in"}

Fig. 1. Averaged correlation over 10 members between forecast and
analysis perturbations as a function of time for Breeding(thick black),
ETKF (thin) black, ET (red) and ET/rescaling (green). The mean
correlation over all levels, correlations at levels 1000mb, 500mb and
2mb are shown in solid, dotted, dashed and dash-dotted lines
respectively.

First, let's look at the correlations between individual forecast and
analysis perturbations. A high correlation between forecast and analysis
perturbations indicates more time continuity between perturbations at
different cycles. Fig. 1 shows that all four ensembles generate analysis
perturbations that have higher correlations with the associated forecast
perturbations, though ET based ensemble produces highest correlation
values (red). Correlations from ETKF and ET with regional rescaling are
similar, but below that of pure ET and higher than that from breeding.
The reasons for this high correlation in ETKF ensemble is due to the
simplex transformation which was discussed in Wei *et al.* 2005, while
ET including simplex transformation produces analysis perturbations that
have even higher correlations than that in ETKF. The results of ET (red)
and ET with regional rescaling (green) show that the regional rescaling
reduces the correlation between analysis and forecast perturbations.

![](media/image2.wmf){width="2.660416666666667in"
height="3.423611111111111in"}

Fig. 2. Vertically averaged global distribution of energy spread of
analysis perturbations (left) and the ratios of the analysis and
forecast spread (right), for ET (top), ETKF (2^nd^ row), breeding (3^rd^
row) and ET/rescaling (bottom row).

The energy spreads of analysis perturbations are shown in Fig. 2 on the
left panels from top to bottom for ET, ETKF, breeding and ET/rescaling
respectively. The ratios of analysis and forecast spread are shown on
the right panels for the respective ensembles. The results show that
pure ET transformation does not show the spread distribution as the mask
breeding, although it produces the analysis perturbations that have
highest correlations with their forecast and perturbations. The spread
distribution from ET is more similar to that of ETKF ensemble.

Since the transformation does not produce the spread distribution as the
initial variance, we impose the regional rescaling as in breeding after
the ET transformation. The spread distribution and the ratios between
analysis and forecast spreads are shown in bottom of Fig.2. Both
distributions from ET with regional rescaling are similar to those of
breeding. The advantages of this technique are that the analysis
perturbations are more orthogonal, have higher correlations with the
forecast perturbations. Other advantages are explored in the following,
such as growth rates, ability to explain forecast errors.

Shown in Fig.3 is the effective degrees of freedom (EDF) of subspace
spanned by 5 perturbations from all four systems. Overall, the
perturbations from ET span a higher dimensional subspace at nearly all
levels (red). When the regional rescaling is imposed on the ET, the EDF
is compromised slightly (green). Both ETKF and breeding perturbations
have similar EDF at most levels.

![](media/image3.wmf){width="2.751388888888889in"
height="2.3118055555555554in"}

Fig.3. The effective degrees of freedom of subspace spanned by 5
temperature perturbations in 2-d grid point space at each pressure level
(solid: analysis; dotted: forecast) from ETKF (thick), breeding (thin),
ET(red), ET/rescaling ensembles.

Fig. 4 shows the PECA values in the N. Hemisphere for ETKF (thick),
breeding(thin), ET(red) and ET/rescaling(green). PECA (Wei and Toth
2003) describes the patter correlations between the forecast errors and
individual perturbations or (optimally combined perturbations). In
general, ET/rescaling and breeding ensembles have higher PECA than ET
and ETKF, especially for short lead time. For longer lead times, ETKF
and breeding ensembles have similar PECA values.

To estimate how much the forecast variance can be explained by the
ensemble variances, we use the scattered plots shown in Fig. 5. Details
about this kind of plots can be seen in Wei *et al*.2005. In terms of
this measure, ET ensemble perform best, with ET/rescaling performs lower
than ETKF and breeding. The anomaly correlations and RMS errors from the
4 ensemble systems which measure the forecast performances show similar
scores for all the systems (not shown), however for a longer lead times
(after 5 days), ET/rescaling and breeding ensembles have slight
advantage with ETKF lowest.

**4. Discussion and Conclusions**

In this paper, we have carried out experiments with four ensemble
forecast systems based on different techniques for generating initial
perturbations: ET, ET with rescaling, ETKF and breeding. Results are
presented for a 32-day period using the NCEP operational analysis/
forecast system, and focusing on analysis perturbation characteristics.
ET with rescaling looks most promising with best scores in 7 measures.
It generates more orthogonal perturbations with initial analysis
variances consistent with data assimilation. Another advantage is that
ET method is computationally efficient compared with ETKF.

![](media/image4.wmf){width="2.341666666666667in"
height="2.4972222222222222in"}

Fig. 4. The PECA values for ETKF (solid) and breeding(dotted), ET(red)
andET/ rescaling (green) ensembles from 5 perturbations only. Shown in
thick and thin lines are PECA from 5 optimally combined perturbations
and the average PECA from 5 individual perturbations.

![](media/image5.wmf){width="2.8333333333333335in"
height="1.3979166666666667in"}

Fig. 5. Derived ensemble variance and forecast error variances at all
grid points for 500mb temperature, for ET and ETKF (left panel) and
breeding and ET/rescaling (right panel) for globe. The average value
from each of 320 bins is indicated by solid lines. Dotted lines show the
results from 20 bins only. The linear regression line from 320 bins is
displayed by a dashed line.

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