Physica D 241 (2012) 671–680
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Physica D
journal homepage: www.elsevier.com/locate/physd
On a nonlinear Kalman filter with simplified divided difference approximation
X. Luo a,∗ , I. Hoteit a , I.M. Moroz b
a
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
b
Mathematical Institute, 24-29 St Giles’, Oxford, OX1 3LB, UK
article
info
Article history:
Received 26 June 2010
Received in revised form
4 December 2011
Accepted 5 December 2011
Available online 16 December 2011
Communicated by B. Sandstede
Keywords:
Data assimilation
Ensemble Kalman filter
Divided difference approximation
abstract
We present a new ensemble-based approach that handles nonlinearity based on a simplified divided
difference approximation through Stirling’s interpolation formula, which is hence called the simplified
divided difference filter (sDDF). The sDDF uses Stirling’s interpolation formula to evaluate the statistics
of the background ensemble during the prediction step, while at the filtering step the sDDF employs the
formulae in an ensemble square root filter (EnSRF) to update the background to the analysis. In this sense,
the sDDF is a hybrid of Stirling’s interpolation formula and the EnSRF method, while the computational
cost of the sDDF is less than that of the EnSRF. Numerical comparison between the sDDF and the EnSRF,
with the ensemble transform Kalman filter (ETKF) as the representative, is conducted. The experiment
results suggest that the sDDF outperforms the ETKF with a relatively large ensemble size, and thus is a
good candidate for data assimilation in systems with moderate dimensions.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
The Kalman filter (KF) is a recursive data assimilation algorithm
that yields optimal estimations in linear systems. However, to extend the KF algorithm to nonlinear systems, some approximations
have to be adopted. One idea is to linearize the nonlinear system
in assimilation so that the KF can be applied to the locally linear
system, which leads to the extended Kalman filter (EKF) [1, ch. 8].
Alternatively, one may approximate the probability density function (pdf), rather than the governing equations, of the underlying
system state by a Gaussian pdf, despite the presence of nonlinearity. Under the Gaussianity assumption, one can use the KF to
update the background mean and covariance to its analysis counterparts, while the estimations of the background statistics are
tackled by the Monte Carlo approximation, which is the rationale
behind the stochastic ensemble Kalman filter (sEnKF) [2–5], the
ensemble square root filter (EnSRF) [6–9], and other similar algorithms [10–13], to name but a few. For convenience, hereafter we
call such nonlinear extensions of the KF the ‘‘nonlinear KFs’’.
In this work we consider one such extension, called the divided
difference filter (DDF) [14,15], which is similar to the EKF in
that, in the presence of nonlinearity, the DDF also conducts local
expansions of the governing equations. However, the expansion in
the DDF is not via Taylor series expansion as in the EKF, but through
Stirling’s interpolation formula [15]. By adopting this formula, one
avoids computing the derivatives of a nonlinear function. Instead,
∗
Corresponding author.
E-mail address: [email protected] (X. Luo).
0167-2789/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.physd.2011.12.003
one uses the divided difference approximations of the derivatives
in computation, and thus circumvents some of the problems in the
EKF.
For applications of the nonlinear KFs to high-dimensional
systems (e.g., models in geophysics), it is often prohibitive to
directly manipulate the full rank covariance matrix of the system
state. Some modifications of the standard nonlinear KFs may
have to be introduced to reduce the computational cost. To this
end, we propose a simplified form of the DDFs in [15], which
can be considered as a hybrid of the EnSRF and the simplified
divided difference approximation, and is thus called the simplified
DDF (sDDF). The idea in the sDDF is to incorporate a simplified
divided difference approximation, derived based on Stirling’s
interpolation formula, to evaluate the background covariance, and
then use an EnSRF algorithm to update the background statistics to
their analysis counterparts. By employing Stirling’s interpolation
formula, symmetric analysis ensembles need to be produced. The
presence of symmetry in the analysis ensembles can reduce the
sampling errors and spurious modes in the evaluation of the
background statistics at the next assimilation cycle, as shown in
the appendix of [16].
This work is organized as follows. In Section 2, we derive the
simplified divided difference approximation based on Stirling’s
interpolation formula. Applying this approximation scheme to the
prediction step of an EnSRF leads to the sDDF, as to be presented
in Section 3. Then in Sections 4 and 5 we conduct experiments to
compare the performance of the sDDF and the ensemble transform
Kalman filter (ETKF) under various ensemble sizes. Finally, we
conclude the work in Section 6.
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X. Luo et al. / Physica D 241 (2012) 671–680
2. Stirling’s interpolation formula and its application to estimating the statistics of nonlinearly transformed random
vectors
This section describes how the divided difference approximation can be derived based on Stirling’s interpolation formula at
the prediction step. To this end, we consider the estimation problem in the following scenario: suppose that one transforms an
m-dimensional random vector x by a nonlinear function F so as
to obtain a transformed m-dimensional random vector η = F (x).
Our objective is to estimate the mean η̄ and covariance Pη of η,
given the known mean x̄ and covariance Px of x. In terms of ensemble filtering, one may quickly identify that this estimation problem
is essentially equivalent to the estimation of the background statistics based on the propagation of an analysis ensemble at the previous assimilation cycle.
To solve the above estimation problem, one can expand F
around x̄ through Stirling’s interpolation polynomials [15]. For
convenience of discussion, here we first define some notations. We
rewrite the random vector x as x = x̄ + δ x, where δ x corresponds
to a random vector with m-dimensional zero mean 0m and m × m
covariance matrix Px . Let Sx be an m × L square root matrix of
Px , such that Px = Sx (Sx )T . Then we can introduce a normalized
L-dimensional random vector δ e, whose mean and covariance are
the L-dimensional zero vector 0L and the L × L identity matrix IL ,
respectively, i.e., E(δ e) = 0L and Cov(δ e) = IL . In addition, we also
define δ ei as the i-th element of δ e. By this definition, one has
E (δ ei ) = 0,
E δ ei δ ej = δij

for i, j = 1, . . . , L,

1
η = F (x̄ + δ x) ≈ F (x̄) + Dδe F (x̄) + Dδ2e F (x̄) ,
(2)
2
where Dδ e and Dδ2e represent the divided difference operators
defined as follows:
Dδ e F (x̄) =
Dδ2e F
(x̄) =
h

δ ei Pi (h/2)Ni (h/2) F (x̄) ,
(3a)
i =1
1

L

h2
(δ ei )2 [Ni (h/2)]2
1
Dδ2e F (x̄) =
4h
+
(6a)
i=1

L
L


δ ei δ ej Ni (h)Nj (h) F (x̄) ,
(6b)
i=1 j=1,j̸=i
where N (h) ≡ [N1 (h), . . . , NL (h)]T . Note that in Eq. (6b), the
summation
L
L


δ ei δ ej Ni (h)Nj (h)F (x̄)
i=1 j=1,j̸=i
contains L(L − 1)/2 terms, which may incur substantial computational cost. For instance, if L = 100, then there is a need to evaluate
about 5000 such terms, which is infeasible in the context of ensemble data assimilation in large-scale systems. To reduce computational cost, we follow [15] and discard these ‘‘cross’’ terms, such
that Eq. (6b) is reduced to
Dδ2e F (x̄) ≈
L
2 
h2 i =1
(δ ei )2 [Pi (h) − I] F (x̄) .
(7)
Since in Eqs. (6) and (7), the parameterized operators become
Ni (h) and Pi (h), we need to generate some symmetric system
states {Xi }2L
i=0 following the definition in Eq. (4), with
X0 = x̄,
Xi = x̄ + h (Sx )i ,
i = 1, 2, . . . , L,
Xi = x̄ − h (Sx )i−L ,
(8)
i = L + 1, L + 2, . . . , 2L.
For convenience, we called such symmetric system states {Xi }2L
i=0
sigma points following [17].
The estimations of the mean and covariance of η are then
carried out based on the transformed sigma points
Zi ≡ F (Xi ) ,
1



h

h
(Sx )i + F x̄ − (Sx )i
2
2
2




h
h
Ni (h/2)F (x̄) = F x̄ + (Sx )i − F x̄ − (Sx )i ,
F
x̄ +
2

η̂ ≈ F (x̄) +
,
(4)
=
2
with the parameter h being the interval length of interpolation, and
(Sx )i the i-th column of the square root matrix Sx of Px .
With some algebra, it can be shown that
[Pi (h/2)]2 F (x̄) =
1
[Pi (h) + I] F (x̄) ;
2
[Ni (h/2)]2 F (x̄) = 2 [Pi (h) − I] F (x̄) ;
[Pi (h/2)Ni (h/2)] F (x̄)
= [Ni (h/2)Pi (h/2)] F (x̄) =
1
2
Ni (h)F (x̄) .
(5)
for i = 0, . . . , 2L
(9)
of {Xi }2L
i=0 . Concretely, to estimate the mean of η , one first
substitutes Eqs. (7) and (6a) into Eq. (2), and then takes the
expectation on both sides of Eq. (2), so that the estimated mean
satisfying
Pi (h/2)F (x̄) =
(δ e)T N (h)F (x̄) ,

L

1
8
(δ ei )2 [Pi (h) − I]
2
2h
i=1
L
L


δ ei δ ej [Pi (h/2)Ni (h/2)]
i=1 j=1,j̸=i


× Pj (h/2)Nj (h/2) F (x̄) .
(3b)
In Eq. (3), Pi (h/2) and Ni (h/2) are parameterized operators
+
Dδ e F (x̄) =
(1)
where δij = 1 if i = j, and δij = 0 otherwise. With this setting,
δ x = Sx δ e corresponds to a random vector with zero mean 0m and
covariance matrix Px .
In [15], the authors consider the second order Stirling’s
interpolation formula

L
1 
Note that in the above equation, I represents the identity operator
such that IF (x̄) = F (x̄).
With Eqs. (4) and (5), Eq. (3) becomes
=
h2 − L
h2
h2 − L
h2
L
1 
h2 i=1
F (x̄) +
Z0 +
E (δ ei )2 [Pi (h) − I] F (x̄)


L
1 
h2 i=1
2L
1 
2h2 i=1
Pi (h)F (x̄)
Zi ,
(10)
while in the deduction of Eq. (10), to derive the first line, the
condition E(δ e) = 0L is applied, such that E(Dδ e F (x̄)) = 0 when
taking the expectation on the right
 hand side of Eq. (2); to derive
the second line, the condition E (δ ei )2 = 1 in Eq. (1) is then
adopted; and to derive the third line, the definitions in Eqs. (4),
(8) and (9) are used.
X. Luo et al. / Physica D 241 (2012) 671–680
Similarly, using Eqs. (2), (7), (10) and (6a), the estimated
covariance is given by
P̂η ≈ E η − η̂

=
L
1 
4h2 i=1
+

T
η − η̂
(Zi − ZL+i ) (Zi − ZL+i )T
L
1 
4h4 i=1
(E(δ ei )4 − 1) (Zi + ZL+i − 2Z0 )
× (Zi + ZL+i − 2Z0 )T ,
(11)
where E(δ ei )4 denotes the fourth order moment of the random
variable δ ei . The derivation of Eq. (11) is omitted for brevity.
Readers are referred to [18, Section 5.3.1.2] for the details. In
particular, if E(δ ei )4 = 1 for all i = 1, . . . , L, then Eq. (11) is
reduced to
P̂η ≈
L
1 
4h2 i=1
(Zi − ZL+i ) (Zi − ZL+i )T .
(12)
Eq. (12) is the simplified form for evaluation of the background
covariance, as to be used in the sDDF in the next section.
The motivation to use this simplified form is based on the
following observation: the m × L matrix
Ŝη =
1
2h
[Z1 − ZL+1 , . . . , ZL − Z2L ]
(13)
is a square root of P̂η , without conducting any matrix factorization.
If one treats the set of sigma points {Xi }2L
i=0 in Eq. (8) as the analysis
ensemble at some assimilation cycle, and Ŝη the square root matrix
of the background covariance at the next cycle, then by applying
the formula in an EnSRF to update Ŝη , one obtains an m × L analysis
square root matrix. In the spirit of Eq. (8), one can generate a new
set of sigma points, with 2L + 1 members again, using the analysis
square root matrix and the analysis mean. Thus by adopting
Eq. (12), there is no extra computational cost needed in order to
produce the same number of sigma points at different assimilation
cycles. This computational advantage is normally not available
in the reduced-rank sigma point Kalman filters. For instance,
in [16,18], the truncated singular value decomposition (TSVD) has
to be conducted at each assimilation cycle in order to prevent
the number 2L + 1 of sigma points growing. The computational
complexity of the TSVD is in the order of m2 L (O (m2 L)), much
lower than O (m3 ) in the full SVD when L ≪ m. However, for high
dimensional systems with large m, the TSVD still incurs substantial
computational cost. In addition, since the simplified form does not
reply on the TSVD to produce sigma points, the ranks of the analysis
covariance matrices does not impose any limitation on the number
of sigma points that one can produce. This is a feature not available
in [16,18].
The simplified form in Eq. (12) introduces certain approximation errors. In the context of ensemble filtering, however, one may
adopt the standard ensemble filtering configuration [19,20], which
equips an ensemble filter with both the covariance inflation [9,21]
and localization [22] techniques to alleviate the effect of approximation errors [23].
673
Eqs. (14a) and (14b) represent the mx -dimensional dynamical
system and my -dimensional observation system, respectively.
Mk,k−1 : Rmx → Rmx is the transition operator, while Hk : Rmx →
Rmy represents the observation operator. We assume that the
model error in the dynamical system Eq. (14a) is characterized by a
white Gaussian random process uk with zero mean and covariance
Qk (denoted by uk ∼ N (uk : 0, Qk )), while the observation error in
Eq. (14b) is described by another white Gaussian process vk with
zero mean and covariance Rk (vk ∼ N (vk : 0, Rk )). In addition, ui
and vj are independent from each other for all indices i and j.
The main idea in the simplified divided difference filter (sDDF)
is as follows: At the prediction step, we apply Eqs. (10) and (13) to
estimate the mean and a square root of the covariance matrix of the
background ensemble, respectively. While at the filtering step, we
adopt the update formulae in one of the EnSRFs [6–9] to update
the background mean and square root matrix to their analysis
counterparts, and to generate the analysis ensemble accordingly.1
In doing this, the computational cost of the sDDF is less than that of
the EnSRF given the same ensemble size, as to be discussed later.
3.1. Prediction step
Without loss of generality, we assume that at time index k − 1,
one has an mx -dimensional analysis mean x̂ak−1 and an associated
mx × n square root matrix Ŝak−1 of the analysis covariance P̂ak−1 such
that P̂ak−1 = Ŝak−1 (Ŝak−1 )T . Based on x̂ak−1 and Ŝak−1 , one can construct
a set of sigma points Xak−1 ≡ {Xak−1,i : i = 0, . . . , 2n} with 2n + 1
members2 in the following manner
Xak−1,0 = x̂ak−1 ,



i
Xak−1,i = x̂ak−1 + hk−1 Ŝak−1
Xak−1,i = x̂ak−1 − hk−1 Ŝak−1

i = 1 , 2 , . . . , n,
i −n
,
(15)
i = n + 1, n + 2, . . . , 2n,

is the ith column of Ŝak−1 , and hk−1 is the length
of interpolation interval (cf. Eq. (8)) with respect to the transition
operator Mk,k−1 . For convenience, we say that the set Xak−1 of sigma
where
Ŝak−1
,
i
points is generated with respect to the triplet (hk−1 , x̂ak−1 , Ŝak−1 ).
If there is no dynamical noise in the dynamical system Eq. (14a),
such that the covariance matrix Qk = 0, one may let
Xbk ≡ {xbk,i : xbk,i = Mk,k−1 (Xak−1,i ), i = 0, . . . , 2n}
be the set of the propagated sigma points, then applying Stirling’s
interpolation formula to Mk,k−1 , with the set Xak−1 of sigma points
in Eq. (15) acting as the interpolation points, one can estimate the
background mean x̂bk and covariance P̂bk in the spirit of Eqs. (10) and
(12). Concretely, the background mean x̂bk is estimated by
x̂bk =
h2k−1 − n
h2k−1
xbk,0 +
1
2n

2h2k−1 i=1
xbk,i ,
(16)
while the background covariance P̂bk by
P̂bk =
1
n


4h2k−1 i=1
xbk,i − xbk,i+n
xbk,i − xbk,i+n

T
.
(17)
3. The simplified divided difference filter
We are interested in the state estimation problem in the
following system:
xk = Mk,k−1 (xk−1 ) + uk ,
(14a)
yk = Hk (xk ) + vk .
(14b)
1 One may construct the hybrid of the simplified divided difference approximation and the stochastic EnKF [2–4] in a similar way, which, however, will not be
investigated in this work.
2 To generate an analysis ensemble with even number of members, one may
simply discard the centre Xak−1,0 of sigma points in Eq. (15) [16,17].
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X. Luo et al. / Physica D 241 (2012) 671–680
If there does exist dynamical noise in the dynamical system
Eq. (14a) such that Qk ̸= 0, then one may construct the propagated
sigma points as
Xbk ≡ {xbk,i : xbk,i = Mk,k−1 (Xak−1,i ) + ξk,i , i = 0, . . . , 2n},
where ξk,i are some realizations of the dynamical noise uk . In
doing this, Eqs. (16) and (17) are again applicable to estimate
the background mean and covariance, respectively. However,
in general the sample covariance of the realizations ξk,i will
underestimate the true covariance of uk . In addition, in Section 2
we have also made some approximations in order to derive
the simplified form for covariance evaluation. For these reasons,
it is rational to provide some compensation for the covariance
estimation form in Eq. (17). Following the idea of covariance
inflation [9,21], we introduce a non-negative covariance inflation
factor δ to P̂bk , such that
P̂bk =
n

(1 + δ)2 
xbk,i − xbk,i+n
4h2k−1
xbk,i − xbk,i+n

T
.
(18)
i=1
As a result, the mx × n matrix
Ŝbk =
1+δ
2hk−1
[xbk,1 − xbk,1+n , . . . , xbk,n − xbk,2n ]
(19)
is a square root matrix of P̂bk , such that Ŝbk (Ŝbk )T = P̂bk .
3.2. Filtering step
To update the background to the analysis, one needs to
first compute the Kalman gain, whose evaluation involves the
observation operator Hk . If Hk is nonlinear, one may apply
Stirling’s interpolation formula once more in evaluation. To this
end, one generates another set of sigma points Xbk ≡ {Xbk,i : i =
to update the background mean x̂bk in Eq. (16) to its analysis
counterpart x̂ak . To update the background covariance P̂bk , we follow
the idea of the ensemble transform Kalman filter (ETKF) [7] to
update the mx × n square root matrix Ŝbk to an mx × n square root
matrix Ŝak of P̂ak . In [7], Ŝak is given by
Ŝak = Ŝbk Tk ,
(26)
where the transform matrix
1
Tk = Ek (Dk + In )− 2
(27)
is an n × n matrix, with In being the n-dimensional identity matrix,
Dk the n × n diagonal matrix whose diagonal elements consisting
1 h
of the eigenvalues of the n × n matrix (Ŝhk )T R−
k Ŝk , and Ek the n × n
1 h
matrix consisting of the corresponding eigenvectors of (Ŝhk )T R−
k Ŝk .
a
Readers are referred to [7] for the deduction of Tk . Given x̂k and
Ŝak , one may generate an analysis ensemble Xak = {Xak,i : i =
0, . . . , 2n} with respect to the triplet (hk , x̂ak , Ŝak ), in the spirit of
Eq. (15), where hk is the length of interpolation interval at the kth
assimilation cycle.
We note that the ensemble size of Xak is equivalent to that of
a
Xk−1 at the previous cycle, and there is no additional computational
cost to achieve this. Therefore, given an analysis ensemble Xak−1
with 2n + 1 members, the computational costs in propagating Xak−1
forward and computing the statistics of the resulting background
ensemble will nearly be the same for both the sDDF and the ETKF.
However, in the sDDF, the dimensions of the square root matrices
Ŝbk and Ŝhk are mx × n and my × n, respectively (cf. Eqs. (19) and (23)),
while in the ETKF, the dimensions of Ŝbk and Ŝhk will be mx ×(2n + 1)
and my × (2n + 1) [7]. As a result, the ETKF requires (slightly)
0, . . . , 2n} with respect to the triplet (dk , x̂bk , Ŝbk ), i.e.,
more computational costs in evaluating the Kalman gain K̂k
1 h
(Eq. (24)) and in conducting SVD on the matrix (Ŝhk )T R−
k Ŝk to obtain
the transform matrix Tk (Eq. (27)).
Xbk,0 = x̂bk ,
4. Experimental setup
 
Xbk,i = x̂bk + dk Ŝbk
Xbk,i
=
where
 i
x̂bk
Ŝbk
− dk
 
Ŝbk
,
i −n
i = 1, 2, . . . , n,
,
(20)
i = n + 1, n + 2, . . . , 2n,
is the ith column of Ŝbk , and dk is the length of
i
Numerical experiments were conducted to compare the
performance of the sDDF and the ETKF. In this section the details of
the experiments setup is first introduced. The experiment results
are presented and discussed in the next section.
interpolation interval with respect to the observation operator Hk .
Let
4.1. The system in assimilation
Ybk ≡ {ybk,i : ybk,i = Hk (Xbk,i ), i = 0, . . . , 2n}
We use the 40-dimensional Lorenz and Emanuel model [24]
(LE98 model hereafter) for illustration, whose governing equations
are given by
(21)
be the set of forecasts of the projections of sigma points Xbk to the
observation space, then applying Eq. (12) with respect to Hk , one
has the estimated innovation covariance matrix P̂in
k , analogous to
Eq. (17), given by
P̂in
k =
n
1 
4d2k i=1
ybk,i − ybk,i+n
ybk,i − ybk,i+n

T
+ Rk .
(22)
In addition, one can construct an my × n square root
Ŝhk =
1
2dk
[ybk,1 − ybk,1+n , . . . , ybk,n − ybk,2n ],
(23)
which we refer to as the projection square root matrix (with respect
to the background ensemble Xbk ), such that the approximate
Kalman gain K̂k is given by
K̂k =
Ŝbk
( )
Ŝhk T

Ŝhk
( ) + Rk
Ŝhk T
−1
.
(24)
With an incoming observation yk , we use the following formula
x̂ak
= x̂bk + K̂k (yk − Hk (x̂bk ))
(25)
dxi
= (xi+1 − xi−2 ) xi−1 − xi + F , i = 1, . . . , 40.
(28)
dt
The quadratic terms simulate advection, the linear term represents
internal dissipation, and F acts as the external forcing term [25]. For
consistency, we define x−1 = x39 , x0 = x40 , and x41 = x1 .
In our experiment we let F = 8. We use the fourth-order
Runge–Kutta method to numerically integrate the system from
time 0 to 30, with a constant integration step of 0.05. The system
state between time 0 and 25 is considered as the transition
trajectory and discarded, while it is the trajectory between time
25.05 and 30 that is used for data assimilation.
We generate the synthetic observations through the following
system
yk = [xk,1 , xk,3 , . . . , xk,39 ]T + vk ,
(29)
where xk,j (j = 1, 3, . . . , 39) denotes the jth element of the state
vector xk at time instant k, vk follows the Gaussian distribution
N (vk : 0, I20 ), with I20 being the 20-dimensional identity matrix.
The observations are made for every 5 integration steps.
X. Luo et al. / Physica D 241 (2012) 671–680
4.2. Measures for filters comparison
For performance comparison between the filters, we calculate
the time-averaged absolute root mean squared error (absolute
rmse for short). Given a set of mx -dimensional state vectors {xk :
xk = (xk,1 , . . . , xk,mx ), k = 0, . . . , T }, with kmax being the
maximum time index, suppose that one has an analysis x̂ak =
(x̂ak,1 , . . . , x̂ak,mx ) of xk , then the time-averaged absolute rmse ê is
defined as
ê =
1
T

T + 1 k=0
ek ,


mx
 1 
√
a
ek ≡ ∥x̂k − xk ∥2 / mx = 
(x̂ak,i − xk,i )2 ,
(30)
mx i=1
where ∥ • ∥2 denotes the Euclidean norm in R

mx 2
mx
such that ∥xk ∥2 =
675
root matrix Ŝa0 . Thus applying Eq. (15), one can generate a (2n + 1)member analysis ensemble at the first cycle. We note that one only
needs to conduct SVD once at the first assimilation cycle. Starting
from the second assimilation cycle, one can apply the formulae in
the sDDF, such that the ensemble sizes of all the background and
analysis ensembles at the subsequent cycles remain to be 2n + 1,
without conducting SVD on the background covariance P̂bk any
more.
The other scenario is n > mx . In this case we do not conduct SVD
on P̂b0 . Instead, we simply select n out of N ensemble members from
the set Xb0 , and follow the method in the EnSRF to compute a square
root matrix Ŝb0 based on these n ensemble members. After updating
Ŝb0 to Ŝa0 , an analysis ensemble with 2n + 1 members can also
be generated in the spirit of Eq. (15). The subsequent procedures
follow those in the sDDF as described in Section 3.
5. Results of the experiments
j=1 xk,j .
A possible problem in directly using ê as the performance
measure is that ê itself may depend on some intrinsic parameters
(e.g., the covariance inflation factor δ ) of the filters, which might
lead to contradictory conclusions at different parameter values. To
avoid this problem, we adopt the following strategy: we relate a
filter’s best possible performance to the minimum absolute rmse
êmin , which is the minimum value of ê that the filter can achieve
within the chosen ranges of the filter’s intrinsic parameters. We say
that A performs better than B if the minimum absolute rmse êAmin
of filter A is less than the minimum absolute rmse êBmin of filter B.
4.3. Initialization of the sDDF
Suppose that at the first data assimilation cycle, one is given
an initial background ensemble, which is not necessarily the
propagation of some symmetric system states from the previous
time as required in the sDDF. For this reason, in the first
assimilation cycle, one is not able to employ the formulae in the
sDDF to compute the background statistics. Instead, one may use
an EnSRF, e.g., the ETKF, to compute the background statistics and
update them to their analysis counterparts. Once the analysis mean
and analysis square root matrix are obtained, one can generate a set
of sigma points accordingly, in the spirit of Eq. (15).
A further issue in the initialization involves the generation
of the ensembles with a specified ensemble size. Suppose that
at the first assimilation cycle, one is given an N-member initial
background ensemble Xb0 = (xb0,1 , xb0,2 , . . . , xb0,N ), while in the
subsequent assimilation cycles, one can at most afford to run 2n +
1-member ensembles, with 2n + 1 ≤ N (the 2n-member case
may follow the idea in footnote 2). To generate such an ensemble
with a specified size, the following idea may be employed. Here we
consider two possible scenarios.
One is that n ≤ mx , the dimension of the system state.
In this case, there exists relatively sparse information from
the background ensemble, hence one may want to use all of
the available information, rather than simply selecting n out
of N members (see the discussion in the n > mx scenario).
To this end, we choose to conduct an SVD on the sample
covariance matrix P̂b0 of Xb0 (with N members) to obtain the first
n leading eigenvalues {σ02,1 , σ02,2 , . . . , σ02,n } and the corresponding
eigenvectors {v0,1 , v0,2 , . . . , v0,n }. We then construct an mx × n
approximate square root matrix
Ŝb0 = [σ0,1 v0,1 , σ0,2 v0,2 , . . . , σ0,n v0,n ]
of P̂b0 . Updating Ŝb0 to its analysis counterpart through the formula
in the ETKF (cf. Eq. (26)), one obtains an mx × n analysis square
5.1. Performance comparison under various ensemble sizes
We first compare the performance of the sDDF and the
ETKF under various ensemble sizes. In the experiments both the
covariance inflation and localization techniques are adopted.
5.1.1. Experiment with 21 ensemble members
In the first experiment we let n = 10 in the sDDF, which
corresponds to an ensemble size p = 2n + 1 = 21 that is
less than the system dimension 40. For comparison, we also let
the ensemble size p = 21 in the ETKF. In the experiment, we
choose to vary the value of the covariance inflation factor δ from
0 to 0.2, with an even increment of 0.05 each time, which is
denoted by δ ∈ {0 : 0.05 : 0.2}. On the other hand, we
follow the setting in [26] to conduct covariance localization, which
brings an additional intrinsic parameter lc , called the length scale
of covariance localization, to the filters. We increase the value of lc
from 50 to 300, with an even increment of 50 each time, which is
denoted by lc ∈ {50 : 50 : 300} accordingly.
In addition, we set the interval lengths of interpolation
(cf. Eqs. (15) and (20)) hk = dk = 0.01 for all k. This setting will
be used in all experiments unless otherwise mentioned. To reduce
statistical fluctuations, we repeat all the experiments for 10 times,
each time with a randomly drawn initial state vector, and some
random perturbations of the initial state as the initial background
ensemble. The size of the initial background ensemble is set equal
to the specified ensemble size p (= 21 in the present experiment).
To initialize the sDDF, we follow the strategy in Section 4.3. The
sDDF in the other experiments is initialized in a similar way.
In Fig. 1 we show the absolute rms errors ê (averaged over
10 experiments) of both the sDDF and the ETKF as functions of δ
and lc . In this small ensemble case, the sDDF under-performs the
ETKF for all values of δ and lc . Moreover, as shown in Table 1, the
minimum error êmin = 1.7146 in the sDDF, and the minimum error
êmin = 1.0648 in the ETKF.
The relatively poorer performance of the sDDF in this case is
largely due to the fact that, given the ensemble size p = 21,
the number of state vectors, when applying Eq. (17) to compute
the background covariances, is only 10, such that the ranks of the
background (hence the analysis) covariance matrices of the sDDF
are at most 9. For distinction, hereafter we use ‘‘sample size’’ to
refer to the number of state vectors in evaluating a background
covariance, and ‘‘ensemble size’’ to refer to the number of state
vectors contained in a background or analysis ensemble. For the
sDDF, the ‘‘sample size’’ is half the ‘‘ensemble size’’ minus 1. In
contrast, for the ETKF, the ‘‘sample size’’ is equal to the ‘‘ensemble
size’’. As a result, the ranks of the covariance matrices in the
676
X. Luo et al. / Physica D 241 (2012) 671–680
(a) sDDF with 21 ensemble members.
(b) ETKF with 21 ensemble members.
Fig. 1. Experiment with 21 ensemble members: averaged absolute rms errors (over 10 experiments) of both the sDDF and the ETKF as functions of the covariance inflation
factor δ and the length scale lc of covariance localization.
Table 1
Minimum absolute rms errors of the sDDF and the ETKF with different ensemble
sizes.
Ensemble size
êmin
11
21
31
41
81
161
321
641
Parameter values
sDDF
ETKF
2.3646
1.7146
1.3438
1.0549
1.0219
0.8537
0.9797
1.0338
1.0457
1.0648
1.0845
1.0995
1.0324
1.1174
1.1252
1.1879
δ ∈ {0 : 0.05 : 0.2},
lc ∈ {50 : 50 : 300}
ETKF are up to = 20 in case that the ensemble members are
independent of each other. In a chaotic system like the LE98 model,
though, the ensemble members normally have certain correlations
with each other, thus the ranks of the covariance matrices are
lower than the maximum possible one. These correlations stem
from the existence of a chaotic attractor in the LE98 model, such
that the trajectories of the ensemble members are attracted into
the basin of attraction, which can be considered as a subspace
embedded in the state space R40 . In [24] it is reported that the
fractal dimension [27] of the LE98 model is approximately 27, less
than the dimension 40 of the state space. This indicates that there
indeed exists such a chaotic attractor, and that there may exist
certain correlations among the ensemble members. Therefore,
even for an ensemble with the number p of members less than the
system dimension, the rank of its sample covariance may be lower
than p − 1.
In Fig. 2 we plot the time series of the ‘‘effective ranks’’ of the
sample covariances computed with various ensemble sizes. Here,
the ‘‘effective rank’’ of an mx × mx sample covariance P̂ is computed
as follows. Let {σj2 , j = 1, 2, . . . , mx } be the eigenvalues of P̂
arranged in a non-ascending order, i.e., σj2 ≥ σl2 if j < l, and
Trace(P̂) =
σj2 be the trace of P̂, then the ‘‘effective rank’’
k
2
k of P̂ is defined as the minimum integer such that
≥
j =1 σ j
c
c
T Trace (P̂), where T ≤ 1 is the cut-off threshold. We choose
T c = 0.99 throughout this work.
mx
j =1
The ‘‘effective rank’’ provides an insight in understanding the
behaviours of the sDDF and the ETKF with various ensemble sizes.
In Fig. 2(a), the ‘‘effective ranks’’ of the sample covariances are
around 7 when the sample size is 11, which roughly depicts the
situation in the sDDF with p = 21. In contrast, the ‘‘effective ranks’’
of the sample covariances are around 12 when the sample size is 21
(cf. Fig. 2(b)), which corresponds to the situation in the ETKF with
p = 21. This shows that the ‘‘effective ranks’’ of the background
covariances of the sDDF are substantially lower than those of the
ETKF, and this difference dominates the benefits of symmetry [16]
in the sDDF. As a result, in this small ensemble scenario, the sDDF
under-performs the ETKF.
5.1.2. Experiment with 41 ensemble members
Next we examine the case n = 20 in the sDDF, which
corresponds to an ensemble size p = 2n + 1 = 41. The same
ensemble size p is used in the ETKF. The covariance inflation factor
δ ∈ {0 : 0.05 : 0.2}, and the length scale lc ∈ {50 : 50 : 300}.
The experiment is also repeated for 10 times to reduce statistical
fluctuations.
In Fig. 3 we show the absolute rms errors ê (averaged over 10
experiments) of both the sDDF and the ETKF as functions of δ and
lc . Compared to the case n = 10 (cf. Fig. 1), the difference between
the sDDF and the ETKF, in terms of ê, is narrowed. In particular,
in the area around the point (δ = 0.18, lc = 200) of Fig. 3(a), the
absolute rms errors ê of the sDDF are lower than the rms errors ê of
the ETKF in Fig. 3(b). In addition, as indicated in Table 1, in the sDDF
the minimum error êmin = 1.0549, and in the ETKF the minimum
error êmin = 1.0995, higher than that in the sDDF now.
Since in this case, the sample size of the sDDF in estimating
the background covariance matrices is 20, Fig. 2(b) provides a
rough depiction of the ‘‘effective ranks’’ in the sDDF. On the other
hand, the sample size of the ETKF in estimating the background
covariance matrices is 41, and Fig. 2(d) indicates that the ‘‘effective
ranks’’ in the ETKF are between 18 and 21. For both filters, their
‘‘effective ranks’’ increase with the sample sizes. Consequently,
their errors in estimating the background covariances are reduced.
This is particularly noticeable for the sDDF. By increasing p from 21
to 41, the absolute rms errors ê in Fig. 3(a) are substantially lower
than those in Fig. 1(a). On the other hand, the performance of the
ETKF when p increases from 21 to 41 seems not improved when
comparing Figs. 3(b) and 1(b). Indeed, the minimum error êmin =
1.0995 when p = 41 and êmin = 1.0648 when p = 21. Similar
results are also found in the experiments below, and in other
works, e.g., [18,28]. This seemingly counter-intuitive phenomenon
will be discussed in Section 5.1.4 with a possible explanation.
X. Luo et al. / Physica D 241 (2012) 671–680
(a) Effective rank with 11 ensemble members.
(b) Effective rank with 21 ensemble members.
(c) Effective rank with 31 ensemble members.
(d) Effective rank with 41 ensemble members.
(e) Effective rank with 81 ensemble members.
(f) Effective rank with 161 ensemble members.
677
Fig. 2. Time series of the effective ranks of the LE98 model with the ensemble sizes of (a) 11, (b) 21, (c) 31, (d) 41, (e) 81, (f) 161, (g) 321, and (h) 641.
5.1.3. Experiment with 161 ensemble members
Now we examine the case n = 80 in the sDDF. The ensemble
size p = 2n + 1 = 161 in both the sDDF and the ETKF. Again the
covariance inflation factor δ ∈ {0 : 0.05 : 0.2}, and the length scale
lc ∈ {50 : 50 : 300}. The experiment is also repeated for 10 times.
In Fig. 4 we show the absolute rms errors ê (averaged over 10
experiments) of both the sDDF and the ETKF as functions of δ and lc .
Comparing Figs. 4(a) and 3(a), it is clear that the performance of
the sDDF is further improved when p increases from 41 to 161. In
this case, the sample size in estimating the background covariances
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X. Luo et al. / Physica D 241 (2012) 671–680
(g) Effective rank with 321 ensemble members.
(h) Effective rank with 641 ensemble members.
Fig. 2. (continued)
(a) sDDF with 41 ensemble members.
(b) ETKF with 41 ensemble members.
Fig. 3. Experiment with 41 ensemble members: averaged absolute rms errors (over 10 experiments) of both the sDDF and the ETKF as functions of the covariance inflation
factor δ and the length scale lc of covariance localization.
(a) sDDF with 161 ensemble members.
(b) ETKF with 161 ensemble members.
Fig. 4. Experiment with 161 ensemble members: averaged absolute rms errors (over 10 experiments) of both the sDDF and the ETKF as functions of the covariance inflation
factor δ and the length scale lc of covariance localization.
X. Luo et al. / Physica D 241 (2012) 671–680
of the sDDF is 80, thus roughly speaking, the ‘‘effective ranks’’ are
between 24 and 27 according to Fig. 2(e). In addition, as reported
in Table 1, the resulting minimum rms errors êmin = 0.8534.
In contrast, the performance of the ETKF does not improve again
when p increases from 41 to 161. In this case, the ‘‘effective ranks’’
in the ETKF are between 29 and 31 according to Fig. 2(f), and in
Table 1 the resulting minimum rms errors êmin = 1.1174, much
higher than that of the sDDF.
5.1.4. Further results
Additional experiments, including the cases n = 5, 15, 40,
160, 320 (corresponding to p = 2n + 1 = 11, 31, 81, 321, 641,
respectively), were also conducted. For brevity, however, the full
details are not presented. Instead, we report their minimum
absolute rms errors in Table 1, together with the choices of δ and
lc in the experiments (while in all the experiments hk = dk = 0.01
for all k in the sDDF).
For the sDDF, the minimum absolute rmse êmin in Table 1
decreases with the ensemble size p until it reaches 161. After
that, êmin arises as p increases further. The behaviour of the ETKF
is slightly different. Its êmin first increases as p grows from 11
to 41, and researches its global minimum at p = 81. After
that, êmin arises as p grows further. Similar phenomena are also
observed in the sDDF and the ETKF when neither the covariance
inflation nor the covariance localization is adopted in the filters
(not presented). This is in contrast with the tuition that, as the
ensemble size increases, the performance of the sDDF or the ETKF
shall be improved. One possible explanation of this phenomenon
is that, according to [29], when assimilating a chaotic system, the
majority of the analysis ensemble members of the ETKF, except
for a few outliers, may spread around a specific location in the
state space after applying the deterministic update formulae in the
ETKF. As a result, when the ensemble size of the ETKF increases, the
filter’s performance is not necessarily improved, since the specific
location may be over-populated by the analysis members. This
makes the majority of the subsequent trajectories of the ensemble
members become too close to a certain location in the state space,
and thus results in a higher estimation error. Since in the sDDF,
deterministic update formulae similar to those in the ETKF are
adopted, the sDDF exhibits similar behaviour to the ETKF.
The experiment results show that, for a relatively small
ensemble size, the ETKF outperforms the sDDF. In contrast, for
a relatively large ensemble size, the sDDF may outperform the
ETKF instead. Our explanation of this phenomenon is the following.
The filters’ performance is influenced by the following two factors:
one is the approximation error of the background covariance, and
the other is the symmetry of the analysis ensembles. On the one
hand, the approximation error of the background covariance is
less when the sample size is larger. On the other, the analytical
result in [16] shows that, in the context of ensemble filtering,
the presence of symmetry in the analysis ensembles can reduce
the errors in estimating the background mean and covariance at
the next assimilation cycle. Since the sample size in estimating a
background covariance of the sDDF is only half that of the ETKF for
a given ensemble size p, the sDDF suffers from more estimation
errors in computing the background covariances. However, this
disadvantage is compensated, to some extent, by the benefits of
symmetry in the analysis ensembles of the sDDF. In the small
ensemble scenario, the disadvantage in estimating the background
covariance dominates the benefits of symmetry, so that the
sDDF under-performs the ETKF. However, as the ensemble size
increases, the effect of sample size deficiency in estimating the
background covariances is mitigated. Indeed, as can be observed in
Fig. 2, for a relatively large ensemble size, say, p = 161, increasing
it to p = 321 only increases the ‘‘effective rank’’ by 2. In contrast,
when increasing p from 11 to 161, one increases the ‘‘effective
rank’’ by about 22. As a result, in the relatively large ensemble
scenario, the benefits of symmetry dominates the effect of sample
size deficiency, so that the sDDF outperforms the ETKF instead.
679
5.2. Sensitivity of the sDDF to the interval lengths hk and dk
To examine the sensitivity of the sDDF to the interval lengths hk
and dk (cf. Eqs. (15) and (20)), in principle one shall vary the values
of hk and dk for each k. However, doing this is computationally
intensive and impractical. Therefore here we still choose to let
hk = h and dk = d for all k in an assimilation run, while h and d are
some constants. In our experiment we let h ∈ {0.01 : 0.05 : 0.25}
and d ∈ {0.01 : 0.05 : 0.25} to examine the performance
of the sDDF with different combinations of h and d. The other
experiment settings are as follows: n = 20 (p = 41), δ = 0.02
and lc = 200. The experiment is also repeated for 10 times to
reduce statistical fluctuations. The experiment result shows that
the absolute rmse ê is insensitive to the values of h and d. Indeed,
the absolute rmse changes very little within the tested ranges of h
and d. The minimum absolute rmse is 1.4262, and the maximum is
1.4331.
6. Conclusions
In this work we introduced an ensemble-based nonlinear
Kalman filter, called the simplified divided difference filter (sDDF),
which can be considered as a hybrid of the ensemble square root
filter (EnSRF) and the simplified divided difference approximation.
With this simplified approximation scheme, the sDDF generates
symmetric analysis ensembles without extra computational cost
in comparison to the EnSRF.
We conducted experiments to compare the performance of
the sDDF and the ensemble transform Kalman filter (ETKF), as
the representative of the EnSRF, with various ensemble sizes. The
results showed that the sDDF under-performed the ETKF in the
small ensemble scenario, while with relatively large ensemble
sizes, the sDDF performed better than the ETKF instead. A possible
explanation is the following: when the ensemble size is relatively
small, the sDDF has larger estimation errors in evaluating the
background covariances than the ETKF, and these errors dominate
the benefits of symmetry in the sDDF such that the sDDF exhibits
larger rms errors. As the ensemble size increases, the effect of
sample size deficiency in the sDDF is mitigated, and the presence
of symmetry in the sDDF makes the sDDF perform better than
the ETKF instead. This may happen, for example, when the
ensemble size is large enough such that the ‘‘effective ranks’’ of
the background covariances are close to, or even higher than, the
fractal dimension (rather than the dimension of the state space) of
a chaotic system. Due to the existence of a chaotic attractor, there
may not be significant accuracy loss in estimating the background
covariances of the sDDF in comparison with the estimates of the
ETKF, so that the symmetry in the sDDF becomes dominant.
Applications of the sDDF may depend on the system in
assimilation, in particular, the dimension of system. For a system
with a moderate dimension (e.g., a network system with a few
hundred or thousand nodes) such that it is affordable to run an
ensemble filter with the ensemble size close to or even larger
than the system dimension, the sDDF may appear to be a better
choice than the ETKF, since in such circumstances, given the same
ensemble size, the computational cost of the sDDF in updating the
background statistics is less than that of the ETKF (cf. Section 3.2).
Moreover, the presence of symmetry in the sDDF also helps to
improve the performance of the filter.
In contrast, for large scale systems like the numerical weather
prediction models, it is impractical for nowadays computers to
run an ensemble filter with the ensemble size close to or even
larger than the system dimension. Therefore it would be more
appropriate to choose the conventional ETKF for data assimilation.
However, with the ever increasing power of parallel computing, we
anticipate that, in the future it may become affordable to do so, and
in such circumstances the sDDF may appear to be a better choice
than the ETKF.
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X. Luo et al. / Physica D 241 (2012) 671–680
Acknowledgements
We would like to thank two anonymous referees for their
constructive suggestions.
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