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Nonlin. Processes Geophys., 13, 601–612,
2006www.nonlin-processes-geophys.net/13/601/2006/© Author(s) 2006.
This work is licensedunder a Creative Commons License.
Nonlinear Processesin Geophysics
Synchronicity in predictive modelling: a new view of
dataassimilation
G. S. Duane1, J. J. Tribbia 1, and J. B. Weiss2
1National Center for Atmospheric Research, PO Box 3000, Boulder,
CO 80307, USA2Department of Atmospheric and Oceanic Sciences, UCB
311, University of Colorado, Boulder, CO 80309, USA
Received: 16 March 2006 – Revised: 15 August 2006 – Accepted: 5
September 2006 – Published: 3 November 2006
Abstract. The problem of data assimilation can be viewedas one
of synchronizing two dynamical systems, one repre-senting “truth”
and the other representing “model”, with aunidirectional flow of
information between the two. Syn-chronization of truth and model
defines a general view ofdata assimilation, as machine perception,
that is reminiscentof the Jung-Pauli notion of synchronicity
between matter andmind. The dynamical systems paradigm of the
synchroniza-tion of a pair of loosely coupled chaotic systems is
expectedto be useful because quasi-2D geophysical fluid models
havebeen shown to synchronize when only medium-scale modesare
coupled. The synchronization approach is equivalent tostandard
approaches based on least-squares optimization, in-cluding Kalman
filtering, except in highly non-linear regionsof state space where
observational noise links regimes withqualitatively different
dynamics. The synchronization ap-proach is used to calculate
covariance inflation factors fromparameters describing the
bimodality of a one-dimensionalsystem. The factors agree in overall
magnitude with thoseused in operational practice on an ad hoc
basis. The calcu-lation is robust against the introduction of
stochastic modelerror arising from unresolved scales.
1 Introduction
A computational model of a physical process that providesa
stream of new data to the model as it runs must includea scheme to
combine the new data with the model’s predic-tion of the current
state of the process. The goal of any suchscheme is the optimal
prediction of the future behavior of thephysical process. While the
relevance of the data assimila-tion problem is thus quite broad,
techniques have been in-vestigated most extensively for weather
modeling, because
Correspondence to:G. S. Duane([email protected])
of the high dimensionality of the fluid dynamical state
space,and the frequency of potentially useful new observational
in-put. Existing data assimilation techniques (3DVar, 4DVar,Kalman
Filtering, and Ensemble Kalman Filtering) combineobserved data with
the most recent forecast of the currentstate to form a best
estimate of the true state of the atmo-sphere, each approach making
different assumptions aboutthe nature of the errors in the model
and the observations.
An alternative view of the data assimilation problem issuggested
here. The objective of the process is not to “now-cast” the current
state of reality, but to make the model con-verge to reality in the
future. Recognizing also that a pre-dictive model, especially a
large one, is a semi-autonomousdynamical system in its own right,
influenced but not deter-mined by observational input from a
co-existing reality, it isseen that the guiding principle that is
needed is one of syn-chronism. That is, we seek to introduce a
one-way couplingbetween reality and model, such that the two tend
to be inthe same state, or in states that in some way correspond,
ateach instant of time. The problem of data assimilation
thusreduces to the problem of synchronization of a pair of
dy-namical systems, unidirectionally coupled through a noisychannel
that passes a limited number of “observed” variables.
While the synchronization of loosely coupled regular
os-cillators with limit cycle attractors is ubiquitous in
nature(Strogatz, 2003), synchronization of chaotic oscillators
hasonly been explored recently, in a wave of research spurredby the
seminal work of Pecora and Carroll (1990). Chaossynchronization can
be surprising because it implies that twosystems, each effectively
unpredictable, connected by a sig-nal that can be virtually
indistinguishable from noise, nev-ertheless exhibit a predictable
relationship. Chaos synchro-nization has indeed been used to
predict new kinds of weakteleconnection patterns relating different
sectors of the globalclimate system (Duane, 1997; Duane et al.,
1999; Duane andTribbia, 2004).
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Geosciences Union and the American Geophysical Union.
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602 G. S. Duane et al.: Synchronicity in predictive
modelling
It is now clear that chaos synchronization is surprisinglyeasy
to arrange, in both ODE and PDE systems (Kocarev etal., 1997; Duane
and Tribbia, 2001, 2004). A pair of spatiallyextended chaotic
systems such as two quasi-2D fluid models,if coupled at only a
discrete set of points and intermittentlyin time, can be made to
synchronize completely. The appli-cation of chaos synchronization
to the tracking of one dy-namical system by another was proposed by
So et al. (1994),so the synchronization of the fluid models
suggests a natu-ral extension to meteorological data assimilation
that has notheretofore been recognized.
Since the problem of data assimilation arises in any situ-ation
requiring a computational model of a parallel physicalprocess to
track that process as accurately as possible basedon limited input,
it is suggested here that the broadest viewof data assimilation is
that of machine perception by an arti-ficially intelligent system.
Indeed, the new field of DynamicData Driven Application Systems
(DDDAS) is defined as thereal-time modelling of evolving physical
systems based onselect observations1. Like a data assimilation
system, thehuman mind forms a model of reality that functions well,
de-spite limited sensory input, and one would like to impart suchan
ability to the computational model. In the artificial intelli-gence
view of data assimilation, the additional issue of modelerror can
be approached naturally as a problem of machinelearning, as
discussed in the concluding section.
In this more general context, the role of synchronism
isreminiscent of the psychologist Carl Jung’s notion of
syn-chronicity in his view of the relationship between mind andthe
material world. Jung had noted uncanny coincidences
or“synchronicities” between mental and physical phenomena.In
collaboration with Wolfgang Pauli (Jung and Pauli, 1955),he took
such relationships to reflect a new kind of order con-necting the
two realms. (The new order was taken to explainrelationships
between seemingly unconnected phenomena inthe objective world as
well.) It was important to Jung andPauli that synchronicities
themselves were distinct, isolatedevents, but as described in Sect.
2.1, such phenomena canemerge naturally as a degraded form of chaos
synchroniza-tion.
A principal question that is addressed in this paper iswhether
the synchronization view of data assimilation ismerely an appealing
reformulation of standard treatments,or is different in substance.
The first point to be made isthat all standard data assimilation
approaches, if successful,do achieve synchronization, so that
synchronization defines amore general family of algorithms that
includes the standardones. It remains to determine whether there
are synchroniza-tion schemes that lead to faster convergence than
the stan-dard data assimilation algorithms. It is shown here
analyt-ically that optimal synchronization is equivalent to
Kalmanfiltering when the dynamics change slowly in phase space,so
that the same linear approximation is valid at each point
1http://www.dddas.org
in time for the real dynamical system and its model. Whenthe
dynamics change rapidly, as in the vicinity of a regimetransition,
one must consider the full nonlinear equations andthere are better
synchronization strategies than the one givenby Kalman filtering or
ensemble Kalman filtering. The de-ficiencies of the standard
methods, which are well known insuch situations, are usually
remedied by ad hoc corrections,such as “covariance inflation”
(Anderson, 2001). In the syn-chronization view, such corrections
can be derived from firstprinciples.
This paper takes a broad view of data assimilation by amodel
system, defined as a set of differential equations, thatis coupled
to noisy data obtained from a “true system”, de-fined by the same
set of differential equations, with the pos-sible addition of a
stochastic term to represent model error.We begin by reviewing the
phenomenology of chaos syn-chronization generally in Sect. 2.1, and
an application to geo-physical fluid systems in Sect. 2.2. A brief
review of standarddata assimilation is provided in Sect. 2.3. In
Sect. 3 the syn-chronization approach is compared to standard
approaches.The optimal synchronization problem for a coupled pair
ofstochastic differential equations is framed as a problem
offinding the coupling that gives the tightest synchronizationin a
linear approximation with observational noise. The opti-mal
coupling thus derived can be compared to forms used instandard data
assimilation. The difference becomes large inregions of state-space
where nonlinearities are important. InSect. 4, a comparison of the
two approaches for the full non-linear case is used to estimate
covariance inflation factors thatwould be needed to adjust the
Kalman filter scheme to giveoptimal synchronization, for both
perfect models and modelsincluding stochastic error from unresolved
scales. Section 5concludes by expanding on the view of data
assimilation asmachine perception and discussing automatic model
adapta-tion in the synchronization framework.
2 Background: synchronized chaos and data assimila-tion
2.1 Chaos synchronization
The phenomenon of chaos synchronization was first broughtto
light by Fujisaka and Yamada (1983) and independentlyby Afraimovich
et al. (1986). Extensive research on the syn-chronization of
chaotic systems in the ’90s was spurred bythe work of Pecora and
Carroll (1991), who found that twoLorenz (1963) systems would
synchronize when theX or Yvariable of one was slaved to the
respectiveX or Y variableof the other, despite sensitive dependence
on initial values ofthe other variables. (Synchronization does not
occur if theZvariables are analogously coupled.)
In this paper we consider a weakerdiffusiveform of cou-pling, as
illustrated by the following pair of bidirectionally
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G. S. Duane et al.: Synchronicity in predictive modelling
603
coupled Lorenz systems:
Ẋ = σ(Y − Z)+ α(X1 −X)
Ẏ = ρX − Y −XZ
Ż = −βZ +XY
(1)
Ẋ1 = σ(Y1 − Z1)+ α(X −X1)
Ẏ1 = ρX1 − Y1 −X1Z1
Ż1 = −βZ1 +X1Y1
where α parameterizes the coupling strength. The twoLorenz
systems synchronize rapidly for appropriate valuesof α, and also do
so for unidirectional coupling, defined byremoving the term inα
from the first equation.
For a pair of coupled systems that are not identical, as withan
imperfect model of a physical system, synchronizationmay still
occur, but the correspondence between the statesof the two systems,
that defines thesynchronization manifoldin state space is different
from the identity. In this situation,known asgeneralized
synchronization, we have two differentdynamical systems ẋ=F(x) and
ẏ=G(y), with x, y∈Rn,modified in some manner so as to define two
coupled sys-temsẋ=F̂ (x, y) and ẏ=Ĝ(y, x). The systems are said
tobe generally synchronized iff there is some locally
invertiblefunction8 : Rn→Rn such that||8(x)−y||→0 as t→∞(Rulkov et
al., 1995). Generalized synchronization can beshown to occur even
for very different systems, as with aRossler system coupled to a
Lorenz system, but with a corre-spondence function8 that is nowhere
smooth (Pecora et al.,1997).
It is commonly not the existence, but the stability of
thesynchronization manifoldin state space that distinguishescoupled
systems exhibiting synchronization from those thatdo not (such as
Eq.1 for different values ofα). As thecoupling is weakened, bursts
of desynchronization (a specialcase of on-off intermittency)
interrupt the synchronized be-havior. On-off synchronization, that
can also arise from noisein the communication channel between the
two systems, isa second way that identical synchronization is found
to de-grade (Ashwin et al., 1994). In the data assimilation
appli-cation, it corresponds to “catastrophes” of large model
driftthat can arise from observational noise (Baek et al.,
2004).
2.2 Synchronization between geophysical fluid systems
Pairs of 1D PDE systems of various types, coupled diffu-sively
at discrete points in space and time, were shown tosynchronize by
Kocarev et al. (1997). Synchronization ingeophysical fluid models
was demonstrated by Duane andTribbia (2001), originally with a view
toward predicting andexplaining new families of long-range
teleconnections (Du-ane and Tribbia, 2004).
The uncoupled single-system model, derived from onedescribed by
Vautard et al. (1988), is given by the quasi-
channel A channel Bforcing
a) b)
n=0
c) d)
n=2000
e) f)
FIGURE 1.
Fig. 1. Streamfunction (in units of 1.48×109 m2 s−1)
describingthe forcingψ∗ (a, b), and the evolving flowψ (c–f), in a
parallelchannel model with bidirectional coupling of medium scale
modesfor which |kx |>kx0=3 or |ky |>ky0=2, and|k|≤15, for the
indi-cated numbersn of time steps in a numerical integration.
Parame-ters are as in Duane and Tribbia (2004). An average
streamfunctionfor the two vertical layersi=1,2 is shown.
Synchronization occursby the last time shown(e, f), despite
differing initial conditions.
geostrophic equation for potential vorticityq in a
two-layerreentrant channel on aβ-plane:
Dqi
Dt≡∂qi
∂t+ J (ψi, qi) = Fi +Di (2)
where the layeri=1, 2, ψ is streamfunction, and the Jaco-bianJ
(ψ, ·)= ∂ψ
∂x∂·∂y
−∂ψ∂y
∂·∂x
gives the advective contributionto the Lagrangian derivativeD/Dt
. Equation (2) states thatpotential vorticity is conserved on a
moving parcel, exceptfor forcing Fi and dissipationDi . The
discretized potentialvorticity is
qi = f0 + βy + ∇2ψi + R
−2i (ψ1 − ψ2)(−1)
i (3)
wheref (x, y) is the vorticity due to the Earth’s rotation
ateach point(x, y), f0 is the averagef in the channel,β is
theconstantdf/dy andRi is the Rossby radius of deformationin each
layer. The forcingF is a relaxation term designedto induce a
jet-like flow near the beginning of the channel:Fi=µ0(q
∗
i −qi) for q∗
i corresponding to the choice ofψ∗
shown in Fig.1a. The dissipation termsD, boundary con-ditions,
and other parameter values are given in Duane andTribbia
(2004).
Two models of the form (2), DqA/Dt=FA+DA andDqB/Dt=FB+DB were
coupled diffusively in one direc-tion by modifying one of the
forcing terms:
FBk = µck[q
Ak − q
Bk ] + µ
extk [q
∗
k − qBk ] (4)
where the flow has been decomposed spectrally and the
sub-scriptk on each quantity indicates the wave numberk spec-tral
component. (The layer indexi has been suppressed.)
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Geophys., 13, 601–612, 2006
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604 G. S. Duane et al.: Synchronicity in predictive
modelling
channel A channel B
n=40:
n=280:
FIGURE 2.
T
A
B
A
A
B
A
T
a)
b)
FIGURE 3.
Fig. 2. Flow results as in Fig.1, but with the inter-channel
cou-pling restricted to the 10-local-bred-vector subspace as in Eq.
(5).Synchronization is apparent by the time step shown.
The two sets of coefficientsµck
andµextk
were chosen to cou-ple the two channels in some medium range of
wavenumbersand to force each channel only with the low
wavenumbercomponents of the background flow.
It was found that the two channels rapidly synchronize ifonly
the medium scale modes are coupled (Fig.1), startingfrom initial
flow patterns that are arbitarily set equal to theforcing in one
channel, and to a rather different pattern in theother channel.
(Results are shown for bidirectional couplingdefined by adding an
equation forFA
kanalogous to Eq.4.
The synchronization behavior for coupling in just one di-rection
is very similar.) With unidirectional coupling, thesynchronization
effects data assimilation from theA channelinto theB channel.
One question about data assimilation that may be ad-dressed in
the synchronization context concerns the defi-nition and minimum
number of variables that must be as-similated to give adequate
predictive skill. It has been ar-gued, for instance, that
atmospheric dynamics is locally low-dimensional, and that a small
number of locally selected bredvectors spans the effective state
space in each local region(Patil et al., 2001). Bred vectors are
commonly used to spec-ify likely directions of forecast error (Toth
and Kalnay, 1993,1997).
If Patil et al. (2001)’s argument about low “BV-dimension”is
correct, then it should only be necessary to couple twochannels in
the subspace defined by the properly chosen bredvectors, in each
local region, to synchronize the two chan-nels. That is, it should
be possible to replace Eq. (4) by
FB(x, y) = µBV∑i
[(qA − qB) · bi]kbi(x, y)+ FBext(x, y) (5)
for (x, y)∈rk, where thebi for i=1...10 are an orthonormalset of
vectors formed from ten bred vectorsBi by Gram-Schmidt
orthogonalization in each local region separately.That is, the bred
vectorsBi are first computed globally forthe channel as a whole, as
in Toth and Kalnay (1997). Then,the channel is divided
rectangularly into a 20×16 patch-work of local regionsrk k=1...320.
The set of vectorsbi isformed fromBi by Gram-Schmidt
orthogonalization of thesetBi(x, y):(x, y)∈rk for eachrk and
concatenating the re-
sulting vectors over all regions. The dot product in bracketsin
Eq. (5) is computed separately for each local region:
[v · w]k ≡∑
(x,y)∈rk
v(x, y)w(x, y) (6)
so that 320×10=3200 independent coefficients[(qA −qB)·bi]k are
computed at each instant of time. Thus[bi ·bj ]k=δij for all k. The
overall coupling strength is givenby µBV , and the external forcing
by the jet is defined byFBextk ≡µ
extk [q
∗
k−qBk ] as before.
It is found that two channels coupled in a truncated bredvector
basis according to Eq. (5) do synchronize, as illus-trated in
Fig.2, and do not synchronize with fewer indepen-dent regions or if
fewer bred vectors are used to define thecoupling subspace. The
total number of independent coef-ficients, however, is comparable
to or larger than the totalnumber of Fourier components in the
mid-range of scalesthat was seen to be effective for
synchronization in the cou-pling scheme (4).
The lesson is that the synchronization phenomenon doesnot appear
to be very sensitive to the detailed choice of cou-pling subspace.
A similar conclusion was reached by Yang etal. (2004) for
synchronizing Lorenz systems. Those authorsobtained only small
improvement by using bred vectors orsingular vectors instead of
single-variable coupling. In thepresent case of spatially extended
models, it seems that anybasis that captures the essential physical
phenomena in eachlocal region, phenomena that can be described in
terms of amiddle range of scales, is adequate for synchronization
andhence for data assimilation.
2.3 Data assimilation
Standard data assimilation, unlike synchronization, estimatesthe
current statexT ∈Rn of one system, “truth”, from thestate of a
model systemxB∈Rn, combined with noisy obser-vations of truth. The
best estimate of truth is the “analysis”xA, which is the state that
minimizes error as compared toall possible linear combinations of
observations and model.That is
xA ≡ xB + 3(xobs− xB) (7)
minimizes the analysis error for a stochasticdistribution given
byxobs=xT+ξ whereξ is observationalnoise, for properly chosenn×n
gain matrix3. The standardmethods to be considered in this paper
correspond to specificforms for the generally time-dependent
matrix3.
The simple method known as 3dVar uses a time-independent3 that
is based on the time-averaged statisticalproperties of the
observational noise and the resulting fore-cast error. Let the
matrix
R ≡< ξξT >=< (xobs− xT )(xobs− xT )T > (8)
be the observation error covariance, and the matrix
B ≡< (xB − xT )(xB − xT )T > (9)
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G. S. Duane et al.: Synchronicity in predictive modelling
605
be the “background” error covariance, describing the devia-tion
of the model state from the true state. If both covariancematrices
are assumed to be constant in time, then the optimallinear
combination of background and observations is:
xA = R(R + B)−1xB + B(R + B)−1xobs (10)
The formula (10), which simply states that observations
areweighted more heavily when background error is greater
andconversely, defines the 3dVar method in practical data
as-similation, based on empirical estimates ofR and B. The4dVar
method, which will not be considered here, general-izes Eq. (10) to
estimate a short history of true states from acorresponding short
history of observations.
The Kalman filtering method, that is popular for a vari-ety of
tracking problems, uses the dynamics of the model toupdate the
background error covarianceB sequentially. Theanalysis at each
assimilation cyclei is:
xiA = R(R + Bi)−1xiB + B
i(R + Bi)−1xiobs (11)
where the backgroundxiB is formed from the previous
anal-ysisxi−1A simply by running the modelM:R
n→Rn
xiB = Mi−1→i(xi−1A ) (12)
as is done in 3dVar. But now the background error is
updatedaccording to
Bi = M i−1→iAi−1MTi−1→i + Q (13)
where A is the analysis error covarianceA≡, given conveniently
byA−1=B−1+R−1. The matrixM is thetangent linear modelgiven by
Mab ≡∂Mb
∂xa
∣∣∣∣x=xA
(14)
The update formula (13) gives the minimum analysis error=T rA at
each cycle. The termQ is the covari-ance of the error in the model
itself, as discussed in Sect. 4.
3 Comparison of synchronization with standard meth-ods of data
assimilation
3.1 Optimal coupling for synchronization of stochastic
dif-ferential equations
To compare synchronization to standard data assimilation,we
inquire as to the coupling that is optimal for synchroniza-tion, so
that this coupling can be compared to the gain matrixused in the
standard 3dVar and Kalman filtering schemes.The general form of
coupling of truth to model that we con-sider in this section is
given by a system of stochastic differ-ential equations:
ẋT = f (xT )
ẋB = f (xB)+ C(xT − xB + ξ) (15)
where true statexT ∈Rn and the model statexB∈Rn evolveaccording
to the same dynamics, given byf , and where thenoise ξ in the
coupling (observation) channel is the onlysource of stochasticity.
The form (15) is meant to includedynamicsf described by partial
differential equations, as inthe last section. The system is
assumed to reach an equi-librium probability distribution, centered
on the synchro-nization manifoldxB=xT . The goal is to choose a
time-dependent matrixC so as to minimize the spread of the
dis-tribution.
Note that if C is a projection matrix, or a multiple ofthe
identity, then Eq. (15) effects a form of nudging. Butfor
arbitraryC, the scheme is much more general. Indeed,continuous-time
generalizations of 3DVar and Kalman filter-ing can be put in the
form (15).
Let us assume that the dynamics vary slowly in state space,so
that the JacobianF≡Df , at a given instant, is the same forthe two
systems
Df (xB) = Df (xT ) (16)
where terms ofO(xB−xT ) are ignored. Then the differencebetween
the two Eqs. (15), in a linearized approximation, is
ė = Fe − Ce + Cξ (17)
wheree≡xB−xT is the synchronization error.The stochastic
differential equation (17) implies a de-
terministic partial differential equation, the
Fokker-Planckequation, for the probability distributionρ(e):
∂ρ
∂t+ ∇e · [ρ(F − C)e] =
1
2δ∇e · (CRCT∇eρ) (18)
whereR= is the observation error covariance matrix,and δ is a
time-scale characteristic of the noise, analogousto the discrete
time between molecular kicks in a Brownianmotion process that is
represented as a continuous process inEinstein’s well known
treatment. Equation (18) states thatthe local change inρ is given
by the divergence of a proba-bility currentρ(F−C)e except for
random “kicks” due to thestochastic term.
The PDF can be taken to have the Gaussian formρ=N exp(−eTKe),
where the matrixK is the inverse spread,andN is a normalization
factor, chosen so that
∫ρdne=1.
For background error covarianceB, K=(2B)−1. In the
one-dimensional case,n=1, whereC andK are scalars, substitu-tion of
the Gaussian form in Eq. (18), for the stationary casewhere∂ρ/∂t=0
yields:
2B(C − F) = δRC2 (19)
SolvingdB/dC=0, it is readily seen thatB is minimized (Kis
maximized) whenC=2F=(1/δ)B/R.
In the multidimensional case,n>1, the relation (19)
gen-eralizes to thefluctuation-dissipation relation
B(C − F)T + (C − F)B = δCRCT (20)
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606 G. S. Duane et al.: Synchronicity in predictive
modelling
channel A channel B
n=40:
n=280:
FIGURE 2.
T
A
B
A
A
B
A
T
a)
b)
FIGURE 3.
Fig. 3. An analysis cycle, with trajectories shown for the true
state“T”, the model evolving from the initial analysis “A” to the
nextforecast, or background “B”, and an alternative model run
(dot-ted line) starting from an inferior “analysis” that is further
fromthe initial truth. In the case(a) whereDf (xT )=Df (xA), then
aworse analyis will always produce a worse forecast, but in the
gen-eral case(b) whereDf (xT ) 6=Df (xA), nonlinearities may allow
aworse analysis to evolve to a better forecast (the trajectories do
notactually cross).
that can be obtained directly from the stochastic
differentialequation (17) by a standard proof that is reproduced in
Ap-pendix A.B can then be minimized element-wise. Differen-tiating
the matrix equation (20) with respect to the elementsof C, we
find
dB(C − F)T + B(dC)T + (dC)B + (C − F)dB
= δ[(dC)RCT + CR(dC)T ] (21)
where the matrixdC represents a set of arbitrary incrementsin
the elements ofC, and the matrixdB represents the re-sulting
increments in the elements ofB. SettingdB=0, wehave
[B − δCR](dC)T + (dC)[B − δRCT ] = 0 (22)
Since the matricesB and R are each symmetric, the twoterms in
Eq. (22) are transposes of one another. It is eas-ily shown that
the vanishing of their sum, for arbitrarydC,implies the vanishing
of the factors in brackets in Eq. (22).ThereforeC=(1/δ)BR−1, as in
the 1D case.
3.2 Optimal synchronization vs. least-squares data
assimi-lation
Turning now to the standard methods, so that a comparisoncan be
made, it is recalled that the analysisxA after eachcycle is given
by:
xA = R(R + B)−1xB + B(R + B)−1xobs= xB + B(R + B)−1(xobs− xB)
(23)
In 3dVar, the background error covariance matrixB is fixed;in
Kalman filtering it is updated after each cycle using thelinearized
dynamics. The background for the next cycle iscomputed from the
previous analysis by integrating the dy-namical equations:
xn+1B = xnA + τf (x
nA) (24)
where τ is the time interval between successive analyses.Thus
the forecasts satisfy a difference equation:
xn+1B = xnB + B(R + B)
−1(xnobs− xnB)+ τf (x
nA) (25)
We model the discrete process as a continuous process inwhich
analysis and forecast are the same:
ẋB = f (xB) + 1/τB(B + R)−1(xT − xB + ξ)
+ O[(B(B + R)−1)2] (26)
using the white noiseξ to represent the difference
betweenobservationxobs and truthxT . The continuous approxima-tion
is valid so long asf varies slowly on the time-scaleτ .
It is seen that when background error is small compared
toobservation error, the higher order termsO[(B(B+R)−1)2]can be
neglected and the optimal couplingC=1/δBR−1 isjust the form that
appears in the continuous data assimila-tion equation (26), for δ=τ
. Thus under the linear assump-tion thatDf (xB)=Df (xT ), the
synchronization approach isequivalent to 3dVar in the case of
constant background error,and to Kalman filtering if background
error is dynamicallyupdated over time. The equivalence can also be
shown for anexact description of the discrete analysis cycle, by
comparingit to a coupled pair of synchronized maps. See Appendix
B.
The equivalence between synchronization and standardmethods in
the linear case actually follows easily from acomparison of the
optimization principles that define the twoapproaches. In the
standard approaches, the form (23) min-imizes the expected value
of(xA−xT )2, as compared to allother linear combinations ofxobs
andxB . But if double lin-earization (16) holds then the
minimization ofimplies the minimization of at any futuretime.
To see this, first consider the background error after a pe-riod
of time τ , just before the next analysis, as in Fig.3a.If double
linearization (16) holds, then this projected back-ground error is
related to the initial analysis error by
e(t) = T
[exp
∫ tto
F(t ′)dt ′]
e(to) ≡ Me(to) (27)
where the notationT before the expression in brackets de-notes
time-ordering. The expectations are related by
B==MMT=MAM T (28)
If we consider a more general “analysis”x3A formed from ageneral
linear combination of forecast and observations
x3A ≡ xB + 3(xobs− xB) (29)
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G. S. Duane et al.: Synchronicity in predictive modelling
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and compute a general “analysis error covariance”A3≡
accordingly, we seek tominimize the traceT rA3=. But it is
readilyshown that a solution3 of d3T rA3 = 0 is also a solutionof
d3T rM(t)A3MT (t)=0 for 0
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608 G. S. Duane et al.: Synchronicity in predictive
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by X1, Y1, Z1, based on “observations” of theX, Y,Z sys-tem. It
is seen that the Kalman filter approach (Fig.4b)gives bursts of
desynchronization just after the transitions,as expected, unlike
the coupling (31) (Fig. 4a), although theKalman filter performance
is better on the average. (Theplots in Fig.4 are averages over a
large number of realiza-tions of the stochastic process. Each
realization is basedon exactly the same master system trajectory.
The desyn-chronization phenomenon occurs when the
Lorenz/reversedLorenz transition takes place at certain points on
the mastersystem attractor, but not when it takes place at other
points.)
There are several lessons to be learned from this extremeand
somewhat artificial example. First, the optimal synchro-nization
scheme, with time-varying coupling, would reduceboth the average
error and the bursting phenomenon. Thatwould be the ideal way to do
data assimilation. But sec-ond, the Kalman filtering approach is
almost always betterthan any of the coupling algorithms described
in the synchro-nization literature. Specifically, the bursts
(corresponding to“catastrophes” in the data assimilation
literature) are short-lived even in this extreme example, lasting
for only one ortwo time steps.
In the geophysical realm, effective non-autonomy is com-mon, as
with phenomena influenced by the diurnal or annualcycles for
example. More generally, the highly nonlinear re-gions of phase
space where the assumption (16) fails, andthe optimality of the
Kalman filter is expected to break downcorrespond to regime
transitions. It is in such regions, thattypically occupy small
volumes of phase space, where thesynchronization approach is
expected to improve on standardmethods to a small degree.
4 Synchronization vs. data assimilation for stronglynonlinear
dynamics
In a region of state space where nonlinearities are strong
andEq. (16) fails, the prognostic equation for error (17) must
beextended to incorporate nonlinearities. Additionally, modelerror
due to processes on small scales that escape the digi-tal
representation should be considered. While errors in theparameters
or the equations for the explicit degrees of free-dom require
deterministic corrections, the unresolved scales,assumed
dynamically independent, can only be representedstochastically. The
physical system is governed by:
ẋT = f (xT )− ξM (32)
in place of Eq. (15a), whereξM is model error, with
covari-anceQ≡. The error equation (17) becomes
ė = (F − C)e +Ge2 +He3 + Cξ + ξM (33)
where we have included terms up to cubic order ine, withH
-
G. S. Duane et al.: Synchronicity in predictive modelling
609
a)
-3 -2 -1 1 2 3x
-1.5
-1
-0.5
0.5
1
1.5
f
-d1 d2
b) 1.25 1.5 1.75 2.25 2.5 2.75 3C
0.16
0.17
0.18
B
c) -1 -0.5 0.5 1 e
0.2
0.4
0.6
0.8
PDF
d)1.25 1.5 1.75 2.25 2.5 2.75 3 C
0.021
0.022
0.023
0.024
B
FIGURE 5.
Fig. 5. For a true systeṁx=f (x), with f defined by (38)
describ-ing a two-well potential aboutx=xo = 0, (a), background
errorB is plotted against coupling strengthC for bimodality
parametersd1=1.06, d2=1.26, observation errorR=1 and
inter-observationtime τ=0.1 (b). Although the distribution of true
states is bimodal,the error distributionρ(e) is unimodal(c). B vs.
C for a shorterinter-observation timeτ=0.01 (d) gives a covariance
inflation fac-tor near unity (see text).
Matching the gross structure of the two-well potential tothe
gross structure of the Lorenz ’63 or Lorenz ’84 systems,one can
find values ofG andH , that will give the correctdistances between
the central fixed point and the two out-lying fixed points,
respectively. For the Lorenz ’84 system,these distances ared1=1.06
andd2=1.26 respectively. Theprognostic equation for error (33) has
fixed points in the de-sired positions iffG=.15 andH=−.75, in a
re-scaled timecoordinate for whichF=1.
For the perfect model case,Q=0, the functionB(C) isthen found
numerically to have the form plotted in Fig.5b,with a distinct
optimum at B=0.15, C=1.5, where the inter-observation timeτ=0.1 (6
h in dimensional units) is chosento be an order of magnitude
smaller than the dynamical timescaleF−1 and we assume a large
observation errorR=1, ofthe same order as the typical displacement
ofx about the as-sumed unstable fixed point. Note that although the
true stateof the system is distributed bimodally, the distribution
(35)of synchronization errorρ(e) shown in Fig.5c is
unimodal,because of smearing by observational noise.
The coupling that gives optimal synchronization can againbe
compared with the coupling used in standard data assim-ilation, as
for the linear case. In particular, one can askwhether the
“covariance inflation” scheme that is used as anad hoc adjustment
in Kalman filtering (Anderson 2001) canreproduce theC values found
to be optimal for synchroniza-tion. The formC=τ−1B(B+R)−1 is
replaced by the ad-justed form
C =1
τ
FBFB + R
(39)
Table 1. Covariance inflation factor vs. bimodality
parametersd1, d2, for 50% model error in the resolved tendency.
d1.75 1. 1.25 1.5 1.75 2.
.75 1.26 1.26 1.28 1.30 1.32 1.341. 1.26 1.23 1.23 1.25 1.27
1.29
d2 1.25 1.28 1.23 1.22 1.23 1.24 1.251.5 1.30 1.25 1.23 1.22
1.23 1.24
1.75 1.32 1.27 1.24 1.23 1.23 1.232. 1.34 1.29 1.25 1.24 1.23
1.23
whereF is the covariance inflation factor. For the
exampledepicted in Fig.5b, the optimal valueC=1.5 would be
gen-erated by an inflation factorF=1.2.
The optimization problem was solved numerically with re-sults as
plotted in Table1 for a range of values of the bi-modality
parametersd1 andd2, giving dynamical parametersG=1/d2−1/d1
andH=−1/(d1d2). Results are displayedfor the case where the
amplitude of model error in Eq. (32)is about 50% of the resolved
tendencyẋT , with the resultingmodel error covarianceQ=0.04
approximately one-fourth ofthe background error covarianceB. The
covariance inflationfactors are remarkably constant over a wide
range of param-eters and agree with typical values used in
operational prac-tice.
For the range of bimodality parameters considered, thelarge
stochastic model error makes little difference in theestimated
covariance inflation factors, typically changingF only by about
±0.001. Indeed, for the linear case(d1=d2=∞), the optimal coupling
is still 1/δBR−1, as for aperfect model. The stochastic model error
results in an extraconstant termδQ on the RHS of the
Fluctuation-DissipationRelation (20) that does not affect the
derivatives in the sub-sequent optimization procedure. (However,Q
does enter theprognostic equation (13) for B in the linear case,
just as inKalman filtering.)
As the inter-observation timeτ becomes smaller,B de-creases at
the minimum point, and the form (39) implies adecreasing covariance
inflation factor. For the caseτ=0.01shown in Fig.5d, the required
factor is near unity. (Precisely,F=1.02.) That is, the coupling
required for continuous-timeKalman filtering approaches the optimal
coupling for syn-chronization. That the advantage conferred by the
synchro-nization approach obtains only for sizable
inter-observationtimes suggests some commonality with non-linear
general-izations of Kalman filtering (e.g. Miller and Ghil,
1994).The difference between optimal synchronization and nonlin-ear
Kalman filtering, while likely to be small, merits
furtherinvestigation, since the two approaches are defined by
dif-ferent goals. Similarly, the use of ensembles to
representnon-Gaussian PDF’s in an empirical way (Anderson,
2003)
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610 G. S. Duane et al.: Synchronicity in predictive
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would seem to address some of the same issues treated in
thissection, and a comparison is warranted.
In the synchronization view, the values typically used forthe
covariance inflation factor in the presence of model non-linearity
and stochastic model error are readily explained.Sampling error, as
it affects estimates of background covari-anceB, is not treated
here, but might be expected to enter ina similar way and not to
substantially alter the conclusions ofthe optimal coupling
analysis.
5 Concluding remarks: data assimilation as machineperception
That ad hoc covariance inflation factors used in
operationalpractice can be explained naturally in the
synchronizationview of data assimilation suggests a deep relevance
for thatviewpoint. Nothing in the foregoing discussion of
synchro-nization and data assimilation is limited to
meteorologicalprocesses. In any situation in which a computational
predic-tive model of a physical process receives a stream of new
dataas it is running, the synchronization of the physical
processand the model is the true goal of any assimilation scheme.As
suggested in the introduction, the philosophical basis forthe
proposed use of synchronization appears to be the idea
ofsynchronicity as espoused by Jung and Pauli, originally in
apsychological context.
While a weather-prediction model is not usually viewedas
artificially intelligent software, it forms an internal
rep-resentation of the external world as complex as that of
anyrobot, albeit without the motor component. One could envi-sion
augmenting the statistical data assimilation algorithmswith rules
to discriminate between good and bad observa-tions, and with rules
to transform the observed data in com-plex ways based on the
current model state. Neurobiologicalparadigms may guide the design
of such artificially intelli-gent predictive systems. It is known
that interneuronal syn-chronization across wide distances in the
brain plays a rolein the grouping of percepts that is a
prerequisite to higherprocessing, and may even underlie
consciousness (Strogatz,2003; Von Der Malsburg and Schneider,
1986). These find-ings lend credibility to the suggestion that a
synchronizationprinciple is also fundamental to the relationship
between thebrain and the external world, and that synchronization
shouldbe a cornerstone in the design of a neuromorphic,
artificiallyintelligent predictive model.
Machine learning might also be realized in the synchro-nization
context, so as to correct for deterministic modelerror in the
resolved degrees of freedom. By allowingmodel parameters to vary
slowly, generalized synchroniza-tion would be transformed to more
nearly identical synchro-nization. Indeed, parameter adaptation
laws can be added toa synchronously coupled pair of systems so as
to synchronizethe parameters as well as the states. Parlitz (1996)
showed forexample that two unidirectionally coupled Lorenz
systems
with different parameters:
Ẋ = σ(Y −X)
Ẏ = ρX − Y −XZ
Ż = −βZ +XY
(40)
Ẋ1 = σ(Y −X1)
Ẏ1 = ρ1X1 − νY1 −X1Z1 + µ
Ż1 = −βZ1 +X1Y1
could be augmented with parameter adaptation rules:
ρ̇1 = (Y − Y1)X1
ν̇ = (Y1 − Y )Y1 (41)
µ̇ = Y − Y1
so that the Lorenz systems would synchronize, and addition-ally
ρ1→ρ, ν→1, andµ→0. Note that parameters ceaseto adapt when the
systems are perfectly synchronized withY−Y1=0. Generalizations to
PDEs would allow model pa-rameters to automatically adapt. In
complex cases, a stochas-tic component (in the parameters) might be
necessary to al-low parameters to jump among multiple basins of
attraction,most of which are sub-optimal. The stochastic
approachcould perhaps be extended to a genetic algorithm that
wouldmake random qualitative changes in the model as well,
untilsynchronization is achieved.
The main competing approach to the tracking of realityby a
predictive model is Kalman filtering, or generaliza-tions thereof
that use Bayesian reasoning to estimate the cur-rent state. Further
development of the optimal synchroniza-tion approaches to provide
more refined modifications of theKalman filter in select regions of
state space will be of inter-est in any situation where a Kalman
filter is used to track ahighly nonlinear process.
Conversely, unidirectional synchronization can always beviewed
as a data assimilation problem, by regarding the slavesystem as a
“model” of the master. The synchronizationproperties of a
bidirectionally coupled system can often beinferred from the study
of a corresponding unidirectionalconfiguration. The optimally
modified Kalman filter that isneeded for data assimilation will
therefore also be useful foroptimizing the synchronization of
dynamical systems gener-ally.
Appendix A
Derivation of the fluctuation-dissipation relation
forsynchronously coupled differential equations with noisein the
coupling channel
Consider the stochastic differential equation for
synchroniza-tion error (17), rewritten as:
de
dt= (F − C)e + Cξ (A1)
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wheree is the synchronization error vector,F is a matrix
rep-resenting the linearized dynamics,C is the coupling matrix,andξ
is a time-dependent vector of white noise process sat-isfying=0
and
< ξ(t)ξT (t ′) >= δRδ(t − t ′) (A2)
whereR is the observation error covariance matrix andδ isthe
time over which the physical noise decorrelates. The so-lution to
Eq. (A1) is:
e(t) = e(F−C)te(0)+∫ t
0dt ′e(F−C)(t−t
′)Cξ(t ′) (A3)
Thus the mean synchronization error is
< e(t) >= e(F−C)te(0) (A4)
and the synchronization error variance is
< [e(t)− e(F−C)te(0)][e(t)− e(F−C)te(0)]T >=∫ t0dt ′
∫ t0dt ′′e(F−C)(t−t
′)CCT e(F−C)T (t−t ′′)
(A5)
or
< e(t)eT (t) > = e(F−C)te(0)eT (0)e(F−C)T t
+
∫ t0dt ′e(F−C)(t−t
′)δCRCT e(F−C)T (t−t ′)
(A6)
If C is chosen so that the system synchronizes, in the absenceof
noise, ast→∞, then the first term on the right hand sidedof Eq.
(A6) vanishes in this limit. The system with noiseapproaches a
stationary state with=0 and
< eeT >≡ B =∫
∞
0dte(F−C)tδCRCT e(F−C)
T t (A7)
Differentiating the integrand in Eq. (A7) with respect tot
andand using Eq. (A7) to simplify the resulting expression,
wefind
(F − C)B + B(F − C)T
=
∫∞
0dtd
dt[e(F−C)tδCRCT e(F−C)
T t]
= [e(F−C)tδCRCT e(F−C)T t
]
∣∣∣∞0
(A8)
The last expression in brackets vanishes at the upper limit
forthe case of stable synchronization, so we have
(F − C)B + B(F − C)T = −δCRCT (A9)
which is the fluctuation-dissipation relation (20).
Appendix B
Optimal coupling for synchronization of discrete-time maps
In Sects. 3.1 and 3.2, standard data assimilation was com-pared
to optimal synchronization of differential equations byconsidering
the continuous-time limit of the discrete analy-sis cycle. Instead,
one can leave the analysis cycle intact andcompare it to a
discrete-time version of optimal synchroniza-tion, i.e. to
optimally synchronized maps.
We begin by deriving a fluctuation-dissipation relation(FDR) for
stochastic difference equations. Consider thestochastic difference
equation with additive noise,
x(n+1) = F x(n)+ ξ(n) < ξ(n)ξ(m)T >= R δn,m, (B1)
wherex, ξ∈Rn, F, R aren×n matrices,F is assumed to bestable,
andξ is Gaussian white noise. One can prove by in-duction that the
solution to this equation, with initial condi-tion x(p), is
x(n+ 1) = Fn+1−px(p)+n∑
m=p
Fm−p ξ(n+ p −m) (B2)
We first wish to find the equilibrium covariance matrix0=. If
the initial condition is in the infinite past thenthe equilibrium
covariance is the covariance at any finite it-eration and it is
convenient to choose iteration one. SinceFis stable the initial
condition is forgotten and we obtain:
x(1) =∞∑m=0
Fmξ(−m), (B3)
and then
0 =
∞∑m=0
FmRFmT
. (B4)
One can then show that0 satisfies the matrix FDR
F 0 FT − 0 + R = 0. (B5)
Now consider a model that takes the analysis at stepn toa new
background at stepn+ 1, given by a linear matrixM .That is,xB(n +
1)=MxA(n). Also, xT (n + 1)=MxT
(n).SincexA(n)=xB(n)+B(B+R)−1(xobs(n)−xB(n)), wherexobs=xT+ξ , we
derive a difference equation fore≡xB−xT :
e(n+ 1)=M(I−B(B+R)−1)e(n)+MB(B+R)−1ξ . (B6)
For synchronously coupled maps, on the other hand, wehave
e(n+ 1) = (M − C)e(n+ 1)+ Cξ , (B7)
and with the FDR as derived above:
(M − C)B(M − C)T − B + CRCT = 0 (B8)
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612 G. S. Duane et al.: Synchronicity in predictive
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Differentiating the matrix equation (B8) with respect to
theelements ofC, as in the continuous-time analysis, we find
0 = (M − C)dB(M − C)T + (−dC)B(M − C)T
+ (M − C)B(−dC)T − dB + dCRCT + CRdCT . (B9)
We seek a matrixC for which dB=0 for arbitrarydC, andthus
(−dC)[B(M − C)T − RCT ]
+[(M − C)B − CR](−dC)T = 0 (B10)
for arbitrarydC. The two terms are transposes of one an-other,
and it is easily shown, as in the continuous-time case,that the
quantities in brackets must vanish. This gives theoptimal
matrix
C = MB(B + R)−1 (B11)
which upon substitution in Eq. (B7) reproduces the standarddata
assimilation form (B6), confirming the equivalence.
Acknowledgements.The authors thank S.-C. Yang for
providingsoftware used as a basis for the alternating Lorenz system
experi-ment. The National Center for Atmospheric Research is
sponsoredby the National Science Foundation. This work was
supportedunder NSF Grant 0327929.
Edited by: M. ThielReviewed by: two referees
References
Afraimovich, V. S., Verichev, N. N., and Rabinovich, M.
I.:Stochastic synchronization of oscillation in dissipative
systems,Radiophys. Quantum Electron., 29, 795–803, 1986.
Anderson, J. L.: An ensemble adjustment Kalman filter for
dataassimilation, Mon. Wea. Rev., 129, 2884–2903, 2001.
Anderson, J. L.: A local least-squares framework for ensemble
fil-tering, Mon. Wea. Rev., 131, 634–642, 2003.
Corazza, M., Kalnay, E., Patil, D. J., Yang, S.-C., Morss, R.,
Cai,M., Szunyogh, I., Hunt, B. R., and Yorke, J. A.: Use of the
breed-ing technique to estimate the structure of the analysis
“errors ofthe day”, Nonlin. Processes Geophys., 10, 233–243,
2003,http://www.nonlin-processes-geophys.net/10/233/2003/.
Ashwin, P., Buescu, J., and Stewart, I.: Bubbling of attractors
andsynchronisation of chaotic oscillators, Phys. Lett. A, 193,
126–139, 1994.
Baek, S. J., Hunt, B. R., Szunyogh, I., Zimin, A., and Ott, E.:
Local-ized error bursts in estimating the state of spatiotemporal
chaos,Chaos, 14, 1042–1049, 2004.
Corazza, M., Kalnay, E., Patil, D. J., Yang, S.-C., Morss, R.,
Cai,M., Szunyogh, I., Hunt, B. R., and Yorke, J. A.: Use of the
breed-ing technique to estimate the structure of the analysis
“errors ofthe day”, Nonlin. Processes Geophys., 10, 233–243,
2003,http://www.nonlin-processes-geophys.net/10/233/2003/.
Duane, G. S.: Synchronized chaos in extended systems and
meteo-rological teleconnections, Phys. Rev. E, 56, 6475–6493,
1997.
Duane, G. S.: Synchronized chaos in climate dynamics, in:
Proc.7th Experimental Chaos Conference, edited by: In, V.,
Ko-carev, L., Carroll, T. L., et al., AIP Conference Proceedings
676,Melville, New York, 115–126, 2003.
Duane, G. S. and Tribbia, J. J.: Synchronized chaos in
geophysicalfluid dynamics, Phys. Rev. Lett., 86, 4298–4301,
2001.
Duane, G. S. and Tribbia, J. J.: Weak Atlantic-Pacific
telecon-nections as synchronized chaos, J. Atmos. Sci., 61,
2149–2168,2004.
Evensen, G.: Sequential data assimilation with a non-linear
quasi-geostrophic model using Monte Carlo methods to forecast
errorstatistics, J. Geophys. Res., 99, 10 143–10 162, 1994.
Fujisaka, H. and Yamada, T.: Stability theory of synchronized
mo-tion in coupled-oscillator systems, Prog. Theor. Phys., 69,
32–47,1983.
Jung, C. G. and Pauli, W.: The interpretation of nature and the
psy-che, Pantheon, New York, 1955.
Kocarev, L., Tasev, Z., and Parlitz, U.: Synchronizing
spatiotem-poral chaos of partial differential equations, Phys. Rev.
Lett., 79,51–54, 1997.
Lorenz, E. N.: Deterministic nonperiodic flows, J. Atmos. Sci.,
20,130–141, 1963.
Lorenz, E. N.: Irregularity – a fundamental property of the
atmo-sphere, Tellus A, 36, 98–110, 1984.
Miller, R. N. and Ghil, M.: Advanced data assimilation in
stronglynonlinear dynamical systems, J. Atmos. Sci., 51,
1037–1056,1994.
Parlitz, U.: Estimating model parameters from time series by
au-tosynchronization, Phys. Rev. Lett., 76, 1232–1235, 1996.
Patil, D. J., Hunt, B. R., Kalnay, E., Yorke, J. A., and Ott,
E.: Locallow dimensionality of atmospheric dynamics, Phys. Rev.
Lett.,86, 5878–5881, 2001.
Pecora, L. M. and Carroll, T. L.: Synchronization in chaotic
sys-tems, Phys. Rev. Lett., 64, 821–824, 1990.
Pecora, L. M., Carroll, T. L., Johnson, G. A., Mar, D. J., and
Heagy,J. F.: Fundamentals of synchronization in chaotic systems,
con-cepts, and applications, Chaos, 7, 520–543, 1997.
Rulkov, N. F., Sushchik, M. M., and Tsimring, L. S.:
Generalizedsynchronization of chaos in directionally coupled
chaotic sys-tems, Phys. Rev. E, 51, 980–994, 1995.
So, P., Ott, E., and Dayawansa, W. P.: Observing chaos –
deducingand tracking the state of a chaotic system from limited
observa-tion, Phys. Rev. E, 49, 2650–2660, 1994.
Strogatz, S. H.: Sync: The Emerging Science of Spontaneous
Or-der, Theia, New York, 338 pp., 2003.
Toth, Z. and Kalnay, E.: Ensemble forecasting at NMC – the
gener-ation of perturbations, Bull. Amer. Meteor. Soc., 74,
2317–2330,1993.
Toth, Z. and Kalnay, E.: Ensemble forecasting at NCEP and
thebreeding method, Mon. Wea. Rev., 125, 3297–3319, 1997.
Vautard, R., Legras, B., and Déqúe, M.: On the source of
midlat-itude low-frequency variability. Part I: A statistical
approach topersistence, J. Atmos. Sci., 45, 2811–2843, 1988.
Von Der Malsburg, C. and Schneider, W.: A neural
cocktail-partyprocessor, Biol. Cybernetics, 54, 29–40, 1986.
Yang, S.-C., Baker, D., Cordes, K., Huff, M., Nagpal, G.,
Okereke,E., Villafañe, J., and Duane, G. S.: Data assimilation as
synchro-nization of truth and model: Experiments with the
three-variableLorenz system, J. Atmos. Sci., 63, 2340–2354,
2006.
Nonlin. Processes Geophys., 13, 601–612, 2006
www.nonlin-processes-geophys.net/13/601/2006/
http://www.nonlin-processes-geophys.net/10/233/2003/http://www.nonlin-processes-geophys.net/10/233/2003/