2.7

ADAPTIVE OBSERVATION STRATEGIES WITH THE LOCAL ENSEMBLE
TRANSFORM KALMAN FILTER
Junjie Liu1*, Eugenia Kalnay1, and Takemasa Miyoshi1 ,University of Maryland, College Park, MD, College Park, MD 20742.

1

1. Introduction
1“Targeted” or “adaptive” observation strategies
to select the optimal location for observations
added to the standard observing system are an
important area of research (e.g., Snyder 1996;
Palmer et al. 1998). The effectiveness of some
adaptive strategies has been tested in field
experiments, such as FASTEX (Snyder 1996; Joly
et al. 1997; Emanuel and Langland 1998),
NORPPEX (Langland et al. 1999a), Winter Storm
Reconaissance Program (Szunyogh et al. 2000)
and Atlantic TOST/TReC. There are two basic
types of adaptive strategies. One is based on theuse of the adjoint model, such as the sensitivity toinitial conditions and singular vectors to identify thesensitive regions in which additional observations
will be taken (Palmer et al, 1998). The other isbased on ensembles such as the ensemble spread
technique (Lorenz and Emanuel 1998; Morss
2002), the Ensemble Transform Kalman Filtering(ETKF) technique proposed by Bishop et al. (2001)
and implemented by Majumdar et al. (2001) in the
FASTEX experiment, the quasi-inverse techniqueof Pu et al. (2001), and the breeding of theanalysis/forecast system of Trevisan and Uboldi(2002). In this paper we explore ensemble-based
adaptive strategies, and propose a more efficientmethod based on Ensemble Kalman Filtering.
The assimilation scheme subjects adaptive
observations to a dynamical process, so that evenwith the same model and standard observations,different assimilation schemes can result indifferent results. Using direct insertion, Lorenz and
Emanuel (1998, referred as LE98 hereafter)
concluded that multiple breeding (an ensemble
based technique) is more efficient thansensitivity-based singular vector technique for the
40-variable Lorenz model. However, using a 1024
member Ensemble Kalman filtering (EnKF),
Hansen and Smith (2000, referred as HS00
1

* Corresponding author address, 2113 CSS, College Park,

MD. Email: jjliu@atmos.umd.edu

hereafter) found that the singular vector approachis competitive with multiple breeding. The Local
Ensemble Kalman Filtering (LEKF), recently
proposed by Ott et al. (2004) does the assimilationon the eigenvector space of background error
covariance. In this way, the analysis increment is
projected onto the unstable space of the
background as in Trevisan and Uboldi (2004,
referred as TU04 hereafter), but more than one
unstable vector is available at the adaptive
observation time. The LEKF belongs to the class ofEnsemble Square Root Filters (Whitaker and
Hamill, 2002, Anderson 2001, Bishop et al, 2002,
Tippett et al, 2003), but unlike other methods thatgain efficiency from assimilating observations one
after the other, LEKF does the assimilation in asmall local patch around each grid point. This
makes it very parallel and allows the simultaneous
assimilation of many observations. Recently, amore efficient version of LEKF, Local Ensemble
Transform Kalman Filtering (LETKF) has been
proposed (Hunt, 2005) that avoids the need for
explicit singular value decomposition at each grid
point. Except for its higher speed, the LETKF yields
the same results as the LEKF. It is being tested on
the NCEP GFS, on the SPEEDY primitive-equation
global model (Molteni, 2003), and on the NASA
finite volume General Circulation Model (fvGCM),
and is under consideration for operational
implementation by Brazil and other countries. In
this paper we consider the application of LETKF toadaptive observations. Section 2 describes themodel and analysis system. Section 3 describes
the formulation of three adaptive strategies, andtheir results are presented in Section 4, together
with a comparison with previously publishedstudies using the same experimental setup.
Section 5 is a summary and conclusions. We follow
the notation of Ide et al. (1997).
2. Descriptions of model, observations andanalysis scheme
As in LE98, HS00 and TU02, the experimentsof this study are based on a simulation in which“truth” is a long model integration started from anarbitrary state, and “observations” are obtained byadding random noise to the true values.
2.1 Forecast model
The 40-variable model first used by Lorenz
and Emanuel is a low-dimensional system whose
error growth shares some common characteristics
with operational NWP systems. The model isgoverned by the following equation:d(1)
x j = ( x j +1 − x j − 2 ) x j −1 − x j + F
dt
The variables ( x j , j=1…J) represent a
meteorological variable on a “latitude circle” withperiodic boundary conditions. As in LE98, J isequal to 40, the time step is 0.05, whichcorresponds to about a 6-hour interval for theatmosphere. F is the external forcing, F=8 isconsidered as the “true” model. A model error isintroduced by setting the forcing equal to 7.6 when
we do forecasts.
2.2 Observations
We follow the experimental set-up of LE98,
observing every “land” grid point (from j= 21 to 40)
each observation time, with an error standard
deviation of σ = F / 40 = 0.2 . A single adaptive
observation point is picked from one of the points
over “ocean” (grid points 1-20). The analysis is the
combination of the 6-hour forecast and bothstandard
observations
andtheadaptiveobservation. The optimality of this additional
observation is evaluated by the analysis error at
the observation time and 10-day forecast from thistime.
2.3 Assimilation scheme
LETKF (Hunt, 2005) is a type of square-root
ensemble Kalman filter. In Ensemble Kalman Filter,an ensemble of K forecasts
xib = Mxia , i = 1,..., K(2)
is carried out in order to estimate the backgrounderror covariance. The ensemble mean is defined
as
b

x =

1 K b
∑ xi .
K i =1The background error covariance is written as

1(3)
X b X bT .
K −1bwhere X is a JｘK matrix whose columns are thebb
K ensemble perturbations x j − x , and J is thePb =dimension of the state vector. The analysis meanstate isabx = x + K(y o − HX b ) ,(4)oy is the observation vector, H is the observation
operator, and K is the Kalman gain matrix thatcan be written as
K = X b [(k − 1)I + (E b ) T R −1 (E b )]−1 (E b ) T R −1bbwhere E = HX . The matrix inversion is

performed in the KxK space of the ensemble
perturbations. The analysis perturbations are
obtained by using Ensemble Transform Kalman
Filter (ETKF) approach (Bishop et al, 2001, Hunt,
2005):

1~
X a X aT = X b P a X bT ,
(5)
K −1
~
P a = [( K − 1)I + E b R −1E bT ] −1 , a KxK matrix,
represents the new analysis covariance in the Kspace of the ensemble forecasts. The analysis
ensemble perturbations are then obtained from:~X a = X b [( K − 1) P a ]1 / 2 .
The LETKF scheme does analysis in the local
patch centered on each grid point. In our
experiments, the local patch size is 9, i.e., it
includes four grid points at each side of every grid
point. The calculation is independent in each local
patch, so the calculation can be done in paralleland because of the small patch size, thecomputations are very fast (Szunyogh et al, 2005).
In order to allow for model errors and nonlinear
error growth we use background error covarianceinflation, but the inflation factor is determined
adaptively (Miyoshi, 2005).
3. Adaptive strategies
The optimality of adaptive observation ismeasured by the RMS error at the analysis time,
which is the reduction of the background error due
to the assimilation of observations. The analysis
error can become “optimal” only if the adaptive
observation is made where the background error islarge. When the true state is not known, thebackground uncertainty can be estimated bymeasures derived from the background error

covariance, such as the ensemble spread or,equivalently, the trace of the background error
covariance. Since it is easy to compute, we only
consider the ensemble-spread method here. If thetruth were known (as would happen only in asimulation), the true background error can becalculated, leading to the (unattainable) true
optimal location. We denote this approach as the“true ensemble error variance”. The analysis error
covariance can be calculated without knowing theactual observation value (Bishop et al, 2001). By
calculating analysis error covariance before theactual analysis is performed, and minimizing theanalysis error variance, one can find the “optimal”
observation location. We refer to this approacha(done locally) as the “local P ” method discussed
below.
3.1 Ensemble spread method
By picking the point with largest ensemble
spread in the ensemble-spread method, the traceof the background error covariance is minimized.
However, the accuracy of adaptive observation is
not explicitly taken into account in this method. If
the type of adaptive observation used happens tohave large errors, the analysis will not be as
improved as with an accurate observation.
Ensemble spread is calculated at each grid
point according to the equation:K

Sj = (K −1)−1 ∑(xi,b j − x j )(xi,b j − x j )Tbb,

j=1…40.

i=1(6)xib, j is the background ensemble at each gridbpoint j, k is the number of ensemble number, x jsummation of the variance at each grid point. 20
adifferent P ’s have to be calculated in order todetermine the adaptive observation point thatmakes the magnitude of the global analysis error
variance smallest. Unlike the ensemble-spreadamethod, the local P
method considers theadaptive observation error variance explicitly,
which allows for adaptive observations of different
types.
3.3 Combined method
Ensemble spread method is simple andalmost cost-free. However, it may not be optimal
due to the possible large observation error
variance in the adaptive observations. Under thiscondition, the overall analysis error variance maya
not decrease substantially. The local P method
allows for observation errors to get a better
analysis, but it computationally much more
expensive. A combined method proposed herecombines the advantages of both methods. We
compute first the ensemble spread, and choose
the 5 grid points with largest ensemble spread fromathe 20 points over ocean. Then, the local P
method is computed only for these 5 grid points.
The grid point that makes the expected trace ofanalysis error covariance smallest is the adaptive
observation point. In this way, the smallest
expected analysis error covariance trace isguaranteed with the adaptive point. Moreover, thisreduces substantially the computation timeacompared with local P method, an importantconsideration if carried out in operations with muchlarger dimension models.
3.4 True ensemble error varianceis the ensemble mean state.
a

3.2 Local P method
Following the idea of Bishop et al. (2001), thealocal P method maximizes the analysis error
variance reduction, which is equal to the difference
between background error variance and analysisaerror variance. We call it local P method because
the analysis error variance is calculated on a local
patch in our implementation.
The analysis error covariance is calculatedafrom equation (5). The variance of P in each
local patch, which is the average of the diagonalavalues of P , is regarded as the variance of theacenter point. The global variance of P is the

The true ensemble error variance method
cannot be computed in operation, but is thestandard that the other methods strive to reach. In
this method, truth is supposed to be known. The
formula of ensemble error variance iskV j = (k − 1) −1 ∑ ( xib, j − x tj )( xib. j − x tj ) T

,

i =1

j=1…40.
(7)
The adaptive point is the point with largest
ensemble variance. The only difference between
thecalculationof
thismethodandensemble-spread method is that the truth is usedinstead of ensemble mean state.

4. Results
4.1 Comparison between different strategies
As in LE98, the Observation-AnalysisForecast (OAF) routine has been carried out
through a 90-day spin-up time and a 5-year testing
period. The analysis RMS error is the average overthe 5-years. The performance of these methods isestimated by 5-year average analysis RMS error.
Fig. 1 shows the analysis RMS error
comparison between different methods. The RMS
errors are very similar over land in every adaptive
strategy (except for random picking not shown
because it has much larger errors, see Table 1),
and they are all below the observation standard
deviation error. Two minimum RMS errors appear
near the land boundaries, and we have verified
that this is due to the systematic error in the forcing.
Without the error in the forcing, the RMS error overland is essentially uniform. Over the ocean, theideal adaptive strategy based on true ensemble
variance, as expected, is the best and only slightly
larger than the observation error with only one
adaptive observation over the whole ocean region.
The performance of the other three operational
possible methods is very similar.

Fig.1. Five-year-average analysis RMS error for different
methods (the straight line is the observation error
standard deviation, the red line is RMS error from true
ensemble variance method, the black line is the RMS
error from ensemble spread method, the green line is thearesult of local P method, the blue line is the resultof combined method ).

The spatial average of the analysis RMS error
is compared in the table 1. The results from all
methods are much better than randomly choosing
the adaptive location. Ensemble-spread method,alocal P method and combined method show
similar performance in this idealized model, but it
may show different results in a more complex
model.
Since the combined method has theadvantages of both ensemble-spread method andalocal P method, we will mainly discuss thismethod in the following. In order to compare withprevious work (LE98, HS00, TU04), the 10-day
forecast RMS error is calculated. Fig. 2a is the
10-day forecast RMS error from true ensemble
variance method. It takes 4 days for the forecast
RMS error to reach 1. Fig. 2b is the RMS error
difference between combined method and true
ensemble variance method. The combined method
is worse initially over ocean, and transported
eastward with time. The magnitude of thedifference is small, initially about 20% of the RMS
error and becoming smaller with time.
Fig. 3 shows the percentage distribution ofadaptive observation points for combined method,
ensemble-spread method and true ensemble
variance method. The percentage distribution has
a similar shape, except that more adaptive
observation points are picked from the “coastal
regions” between data-dense and data-sparse
region for the true ensemble variance method.

ab

Fig.3. Adaptive observation sites distribution for
combined method (black line), true ensemble variancemethod (red line) and ensemble spread method (greenline) (Y-axis is the percentage of each point as adaptive
observation point, X-axis is the grid point.

MethodTime

mean

Operational

analysis

RMS

possibilities

error

Fig. 2 Five-year-average forecast errors. (a) is the result
from true ensemble variance method , (b) is thedifference between the result of combined method andtrue ensemble variance method

True
Ensemblevariance

0.174

Impossible

Backgroundensemble spread

0.210

Possible

Local Pa method

0.211

Possible

Combined method

0.209

PossibleRandom picking

0.480

Possible

Table 1 Summary of average analysis error fordifferent methods
4. 2 Comparison with previous work (LE 98, HS
00, TU 04)
Several articles (LE98, HS00, TU04) havediscussed adaptive strategies with this low-ordersystem and the same land-ocean setup, and usedthe 10-day forecast as one standard to evaluate
the adaptive observation strategies. Although theaverage 10-day forecast from our combined
LETKF method is considerably better than the
results obtained by LE98 and TU04, this difference
cannot be attributed to the adaptive observation
strategy only because LE98 and TU04 used a very

simple assimilation method for the standard
observation (direct insertion). HS00, on the other
hand, performed the data assimilation with an
Ensemble Kalman Filter with a number ofensemble members (1024) much larger than thedimensions of this model (40), so that the accuracyof the HS00 EnKF should be optimal and at least
as good as the accuracy that can be attained with a
15-member LETKF. Therefore, it is reasonable todirectly compare the result from our combined
method with their singular vector approach. Fig. 4
compares the average 10-day forecast errors fromthe HS00 adaptive observations method based on
singular vectors (Fig. 4a) and from the combined
ensemble-based method (Fig. 4b). The initialdifference is small over land, but over ocean the
combined method errors are clearly smaller thanthose of the singular vector approach, and thisadvantage is maintained throughout the 10 days.
Another advantage of the ensemble-based systemis that it does not require the development of linear
tangent and adjoint models.

bFig. 4 (a) 10-day forecast RMS error from Hansen andSmith (2000). Singular vector adaptive observation
strategy is used in this result. (b) 10-day forecast RMS
error of combined method.

5. Discussion and conclusionaSeveral ensemble-based adaptive observation
strategies were tested with a 15-member LETKF
scheme in the Lorenz-40 variable model. Theseamethods include ensemble-spread, local P
method, and a combined method. Our
EnKF-based adaptive observation method isrelated to the breeding of the analysis-forecast
system method developed by TU04 in the sensethat it not only provides the location with largest
forecast error where the adaptive observation
should be located, but also the optimal shape of
the analysis corrections to the forecast which haveto project on the local background error covariance.
Using the same direct insertion data assimilation,
TU04 obtained results better than those of LE98.
Our results cannot be directly compared to theirs
because we used a more advanced data
assimilation method. However, they can be directlycompared with those of HS00 who used a1024-member EnKF for the data assimilation, andsingular vectors to choose the adaptive
observation location. Compared with singular
vector method, the ensemble-based methods
proposed here are clearly better, at least in thisidealized model. The requirement of linear tangent
and adjoint models make the singular vector
method more difficult to implement. Considering
thecomputationalcost
andaccuracy,
ensemble-based adaptive strategies are better.

a

The ensemble-spread method, local P
method and combined method show similar resultsin this model. Ensemble-spread method has a low
computational cost, but it doesn’t explicitly consider
the effect of the adaptive observation error. Local
P a method explicitly considers the observation
error, but is much more expensive. A combined
method has the advantage of both, reducing
observation cost by selecting adaptive observation“candidates” first with the ensemble-spreadamethod, and then using the local P method onthese few “candidates”.
Here, the same standard deviation error has
been used for both the standard observation andadaptive observation, so the ensemble-spreadmethod yield results similar to those of the local
P a method and the combined method eventhough it does not consider the possible effects of
the observation error. However, adaptive
observations usually have different observation
errors with standard observation data set, such as
adaptive observation with GPS dropsonde, themeasured temperature error is only 0.2K, which isaabout 1K for standard observations, the local P
and the combined methods can have an
advantage under this condition. The combinedamethod is better than the local P method for
operational applications because it is lessexpensive.
We plan to compare these three schemes in
the SPEEDY model, using realistic observation
data sets, and considering observation error
differences between standard observation andadaptive observation.
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