The Analog Data Assimilation

Abstract

In light of growing interest in data-driven methods for oceanic, atmospheric, and climate sciences, this work focuses on the field of data assimilation and presents the analog data assimilation (AnDA). The proposed framework produces a reconstruction of the system dynamics in a fully data-driven manner where no explicit knowledge of the dynamical model is required. Instead, a representative catalog of trajectories of the system is assumed to be available. Based on this catalog, the analog data assimilation combines the nonparametric sampling of the dynamics using analog forecasting methods with ensemble-based assimilation techniques. This study explores different analog forecasting strategies and derives both ensemble Kalman and particle filtering versions of the proposed analog data assimilation approach. Numerical experiments are examined for two chaotic dynamical systems: the Lorenz-63 and Lorenz-96 systems. The performance of the analog data assimilation is discussed with respect to classical model-driven assimilation. A Matlab toolbox and Python library of the AnDA are provided to help further research building upon the present findings.

Full text

The Analog Data Assimilation
REDOUANE LGUENSAT AND PIERRE TANDEO
IMT Atlantique, Lab-STICC, UniversitéBretagne Loire, Brest, France
PIERRE AILLIOT
Laboratoire de Mathématiques de Bretagne Atlantique, University of Western Brittany, Brest, France
MANUEL PULIDO
Department of Physics, Universidad Nacional del Nordeste, and CONICET, Corrientes, Argentina
RONAN FABLET
IMT Atlantique, Lab-STICC, UniversitéBretagne Loire, Brest, France
(Manuscript received 23 November 2016, in final form 31 July 2017)
ABSTRACT
In light of growing interest in data-driven methods for oceanic, atmospheric, and climate sciences, this work
focuses on the field of data assimilation and presents the analog data assimilation (AnDA). The proposed
framework produces a reconstruction of the system dynamics in a fully data-driven manner where no explicit
knowledge of the dynamical model is required. Instead, a representative catalog of trajectories of the systemis
assumed to be available. Based on this catalog, the analog data assimilation combines the nonparametric
sampling of the dynamics using analog forecasting methods with ensemble-based assimilation techniques.
This study explores different analog forecasting strategies and derives both ensemble Kalman and particle
filtering versions of the proposed analog data assimilation approach. Numerical experiments are examined for
two chaotic dynamical systems: the Lorenz-63 and Lorenz-96 systems. The performance of the analog data
assimilation is discussed with respect to classical model-driven assimilation. A Matlab toolbox and Python
library of the AnDA are provided to help further research building upon the present findings.
1. Introduction
The reconstruction of the spatiotemporal dynamics of
geophysical systems from noisy and/or partial observa-
tions is a major issue in geosciences. Variational and
stochastic data assimilation schemes are the two main
categories of methods considered to address this issue
[see Evensen (2007) for more details]. A key feature of
these data assimilation schemes is that they rely on re-
peated forward integrations of an explicitly known dy-
namical model. This may greatly limit their application
range as well as their computational efficiency. First,
thorough and time-consuming simulations may be
required to identify explicit representations of the
dynamics, especially regarding finescale effects and
subgrid-scale processes as for instance in regional geo-
physical models (Hong and Dudhia 2012). Such pro-
cesses typically involve highly nonlinear and local
effects (Wilby and Wigley 1997). The resulting numer-
ical models may be computationally intensive and even
prohibitive for assimilation problems, for instance re-
garding the time integration of members with different
initial conditions at each time step. Second, as explained
in Van Leeuwen (2010), ‘‘with ever-increasing resolu-
tion and complexity, the numerical models tend to be
highly nonlinear and also observations become more
complicated and their relation to the models more
nonlinear’’ (p. 1991). In such situations, standard data
assimilation techniques may find difficulties, including
Supplemental information related to this paper is avail-
able at the Journals Online website: https://doi.org/10.1175/
MWR-D-16-0441.s1.
Corresponding author: Redouane Lguensat, redouane.lguensat@
imt-atlantique.fr
OCTOBER 2017 L G U E N S A T E T A L . 4093
DOI: 10.1175/MWR-D-16-0441.1
2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright
Policy (www.ametsoc.org/PUBSReuseLicenses).

nonlinear particle filters which are prone to the ‘‘curse of
dimensionality.’’ Third, difficulties may occur when
geophysical dynamics involve uncertain model param-
eterizations or space–time switching between different
dynamical modes that need to be estimated online (Ruiz
et al. 2013) or offline (Tandeo et al. 2015b). Dealing with
such situations may not be straightforward using classi-
cal model-driven assimilation schemes.
Meanwhile, recent years have witnessed a prolifera-
tion of satellite data, in situ monitoring, as well as
numerical simulations. Large databases of valuable
information have been collected and offer a major
opportunity for oceanic, atmospheric, and climate sci-
ences. As pioneered by Lorenz (1969), the availability of
such datasets advocates for the development of analog
forecasting strategies, which make use of ‘‘similar’’
states of the dynamical system of interest to generate
realistic forecasts. Analog forecasting strategies have
become more and more popular in oceanic and atmo-
spheric sciences (Nagarajan et al. 2015;McDermott and
Wikle 2016), and have benefited from recent advances in
machine learning (Zhao and Giannakis 2014). They
have been applied to a variety of systems and applica-
tion domains, including among others, rainfall now-
casting (Atencia and Zawadzki 2015), air quality
analysis (Delle Monache et al. 2014), wind field down-
scaling (He-Guelton et al. 2015), climate reconstruction
(Schenk and Zorita 2012), and stochastic weather gen-
erators (Yiou 2014).
In this work, we examine the extension of the analog
forecasting paradigm for data assimilation issues.
Given a representative dataset of the dynamics of the
system, this extension that we call analog data assimi-
lation (AnDA) consists of a combination of the implicit
analog forecasting of the dynamics with stochastic fil-
tering schemes, namely, ensemble Kalman and particle
filtering schemes (Evensen and Van Leeuwen 2000).
This idea was first introduced in Tandeo et al. (2015a)
where the relevance of the proposed analog data as-
similation is shown for the reconstruction of complex
dynamics from partial and noisy observations. Tandeo
et al. derived filtering and smoothing algorithms called
the analog ensemble Kalman filter and smoother, which
combine analog forecasting and the ensemble Kalman
filter and smoother. A similar philosophy was followed
independently in Hamilton et al. (2016) where the au-
thors combine ideas from Takens’s embedding theorem
and ensemble Kalman filtering to infer the hidden dy-
namics from noisy observations. Hamilton et al. called
their algorithm the Kalman–Takens filter.
Whereas these two previous works provide proofs of
concept, our study further investigates and evaluates
different analog assimilation strategies and their detailed
implementation. Our contributions are threefold. First, we
present and examine various analog forecasting strategies,
including locally linear ones that were not considered in
previous works, and evaluate their performance for analog
data assimilation. Second, in addition to the ensemble
Kalman algorithms, we propose and examine a novel
implementation of the analog forecasting combined with a
particle filter. Finally, in the online supplemental material,
we provide a unified computational framework, through
both a Matlab Toolbox and a Python Library, to pave the
way for practical use and future research (https://github.
com/ptandeo/AnDA).
The work is organized as follows. In section 2,we
briefly present the general concepts of data assimilation
and introduce the key ideas of analog data assimilation.
Different analog forecasting strategies are introduced in
section 3.Section 4 describes the different components
of the proposed analog data assimilation framework and
the associated algorithms. Numerical experiments for
two classical chaotic dynamical systems are reported in
section 5.Section 6 further discusses our work, high-
lights our key contributions, and proposes possible di-
rections for future work.
2. General context
a. Model-driven data assimilation
Classically, data assimilation is based on the following
discrete state space (Bocquet et al. 2010):
x(t)5M[x(t21), h(t)], (1)
y(t)5H[x(t)] 1«(t), (2)
where time t2f0, ...,Tgrefers to the times in which
observations are available. For the sake of simplicity we
assume observations are at regular time steps.
In (1),Mcharacterizes the dynamical model of the
true state x(t), while h(t) is a random perturbation
added to represent model uncertainty. The observation
equation (2) describes the relationship between the
observation y(t) and x(t). Observation error is consid-
ered through the random noise «(t). Here, for the sake of
simplicity, we consider an additive Gaussian noise «with
covariance Rin (2) and the observation operator H5H
is assumed linear.
Data assimilation aims to reconstruct the state se-
quence fx(t)gfrom a series of observations fy(t)g.Two
types of data assimilation schemes are extensively studied
in the literature: variational and stochastic. Variational
data assimilation proceeds by minimizing a cost function
based on a continuous formulation of (1) and (2) (see
Lorenc et al. 2000), while stochastic data assimilation
4094 MONTHLY WEATHER REVIEW VOLUME 145

schemes rely on the sampling and/or maximization of the
posterior likelihood of the state sequence given the ob-
servation series (see Kalnay 2003). These classical data
assimilation schemes are regarded as ‘‘model driven,’’ in
the sense that they combine observations with forecasts
provided by a numerical model M.
b. Data-driven data assimilation
The proposed assimilation framework relies on a
similar state-space formulation. The key feature is to
substitute the explicit dynamical model Min (1) by a
‘‘data driven’’ dynamical model involving an analog
forecasting operator, denoted by A, namely,
x(t)5A[x(t21), h(t)]. (3)
Henceforth, this state-space model will be referred to
as AnDA. A sequential and stochastic data assimilation
scheme including filtering and smoothing, is used in-
volving different Monte Carlo realizations of the state at
each assimilation time. We sketch the proposed AnDA
methodology for one realization in Fig. 1.
The analog forecasting operator Arequires the ex-
istence of a representative dataset of exemplars of the
considered dynamics. This dataset is referred to as the
catalog and denoted by C. The reference catalog is
formed by pairs of consecutive state vectors, separated
by the same time lag. The second component of each
pair is referred to as the successor of the first component
hereafter. The catalog may be issued from observational
data as well as from numerical simulations. In the last
case, one can have a catalog issued from numerical
simulations (based on physical equations), and wants to
perform data assimilation without running the model
again. This is for instance useful for operational pre-
diction centers that do not have the computational re-
sources to integrate a forecast model, but do have access
to a large database of numerical simulations or analysis
data of a large prediction center. In this respect, we
discuss also the situation where the catalog comprises
noisy versions of the true states (section 5d).
Given a catalog C, the analog forecasting operator A
is stated as an exemplar-based statistical emulator of the
state xfrom time tto time t1dt. For any state x(t),
we emulate the following state at time t1dt based on
its nearest neighbors in catalog C. Given the analog
forecasting operator, we present associated stochastic
assimilation schemes, namely the analog ensemble
Kalman filter/smoother (Tandeo et al. 2015a) and the
analog particle filter.
3. Analog forecasting strategies
a. Analog forecasting operator
Let us consider a kernel function, denoted by g, in the
state space (Schölkopf and Smola 2001). Among the
classical choices for kernels, we consider here a radial
basis function (also referred to as a Gaussian kernel):
g(u,y)5exp(2lku2yk2), (4)
where lis a scale parameter, (u,y) are variables in the
state space X, and k.kis the Euclidean distance or an-
other appropriate distance function. Note that the pro-
posed analog forecasting operator may be applied to
other kernels or subspace reduction methods to effi-
ciently retrieve relevant analog situations. This is dis-
cussed in section 6.
Given the considered kernel, the analog forecasting
operator Ais defined as follows: for a given state x(t),
we denote by ak[x(t)] its kth nearest neighbor (or analog
situation) in the reference catalog of exemplars C, and
by sk[x(t)] the known successor of state ak[x(t)]. Here-
inafter, we refer by Kto the number of nearest neigh-
bors (analogs), and by covwthe weighted covariance. The
normalized kernel weight for every pair fak[x(t)], sk[x(t)]g
is given by
vk[x(t)] 5gfx(t), ak[x(t)]g

K
K51
gfx(t), ak[x(t)]g
. (5)
Several ideas can be explored to define the analog
forecasting operator A. The natural first option consists
in deriving the forecast using the weighted mean of the
Ksuccessors. This approach, that we call here the locally
constant operator, was considered in many analog
forecasting related works (McDermott and Wikle 2016;
FIG. 1. The evolution in time of one particle or member. The
catalog implicitly represents the dynamics of the system from ex-
emplars of historical datasets. The observations are shown by black
asterisks, and their variance is shown by the corresponding
error bar.
OCTOBER 2017 L G U E N S A T E T A L . 4095

Zhao and Giannakis 2014;Hamilton et al. 2016), and is
also known in statistics as Nadaraya–Watson kernel
regression. One can also use as analog forecasting op-
erator the weighted mean of the anomalies between the
Kanalogs and their successors and adding it to the state
to derive the forecast. The operator, referred to as lo-
cally incremental, is seen as more physically sound and
relates more closely to a finite-difference approximation
of the underlying differential equations. Finally, we in-
troduce in this work a new analog forecasting operator
that makes use of local linear regression techniques
based on weighted least squares estimates. This operator
that we call the locally linear operator is known to make
an efficient use of small datasets and to reduce biases
(Cleveland 1979). Note that the locally constant and
locally incremental operators are two special cases of
the locally linear operator.
Figure 2 shows an illustration of the three analog
forecasting operators used in this work. Hereafter, we
denote the forecasted state as xf(t1dt). The three an-
alog forecasting operators are defined as follows for two
sampling schemes: a Gaussian sampling and a multino-
mial one. Hereinafter, dZ() denotes a delta function
centered on Z.
dLocally constant analog operator: for the Gaussian
case, the forecasted state is sampled from a Gaussian
distribution whose mean mLC and covariance SLC are
the weighted mean and the weighted covariance
estimated from the Ksuccessors and their weights:
xf(t1dt);N(mLC,SLC ), (6)
where mLC 5K
k51vk[x(t)]sk[x(t)] and SLC 5
covv(sk[x(t)]k2〚1,K〛
). While in the multinomial case,
the forecasted state is drawn from the multinomial
discrete distribution that samples the successor sk[x(t)]
with a probability of vk:
xf(t1dt);
K
k51
vk[x(t)]dsk[x(t)](). (7)
dLocally incremental analog operator: instead of
considering a weighted mean of the Ksuccessors as
FIG. 2. A simplified illustration of the considered analog forecasting strategies in the case of two analogs (nearest neighbors). Two
situations for the state x(t) are shown: (top) a situation where x(t) lies in the convex hull spanned by catalog exemplars and (bottom)
a situation where x(t) lies farther from its analogs. The second situation is expected to occur more often for high-dimensional space as well
as for states, which are less likely. The latter may model extreme events or outliers.
4096 MONTHLY WEATHER REVIEW VOLUME 145

in the locally constant operator, we consider the value
of the current state plus a weighted mean of the K
increments tk, that is, the differences between ana-
logs and successors tk[x(t)] 5sk[x(t)]2ak[x(t)]. The
Gaussian sampling is given by
xf(t1dt);N(mLI,SLI ), (8)
where mLI 5x(t)1K
k51vk[x(t)]tk[x(t)] 5
K
k51vk[x(t)]fx(t)1tk[x(t)]gand SLI 5
covv(fx(t)1tk[x(t)]gk2〚1,S〛
) and the multinomial
sampling resorts to
xf(t1dt);
K
k51
vk[x(t)]dx(t)1tk[x(t)](). (9)
dLocally linear analog operator: we fit a multivari-
ate linear regression between the Kanalogs of the
current state and their corresponding successors us-
ing weighted least squares estimates (see Cleveland
1979). The regression gives slope a[x(t)] and inter-
cept b[x(t)] parameters, and residuals jk[x(t)] 5
sk[x(t)] 2(a[x(t)]ak[x(t)] 1b[x(t)]). The Gaussian
sampling comes to
xf[(t1dt)] ;N(mLL,SLL), (10)
with mLL 5a[x(t)]x(t)1b[x(t)] and SLL 5
cov(jk[x(t)]k2〚1,K〛
), while the multinomial sampling
is given by
xf(t1dt);
K
k51
vk[x(t)]dmLL1jk[x(t)]() . (11)
The choice of one operator over another depends
mostly on the available computational resource and the
complexity of the application. Locally constant and lo-
cally increment operators are less time and memory
consuming than the locally linear operator, and while
they can be of comparable performance in case of a flat
regression function, the locally linear is expected to
better deal with curvier regression functions at the ex-
pense, however, of the requirement of a larger number
of analogs to fit the regression (Hansen 2000). The lo-
cally linear and the locally incremental are more suitable
for samples near or outside the boundary of the select
analogs (as depicted in Fig. 2), this may be particularly
relevant in geoscience applications where chaos and
extreme events are of high interest.
b. Global and local analogs
The global analog strategy is the direct application of
the introduced analog forecasting strategies to the entire
state vector. We also introduce a local analog fore-
casting operator. For a given state x(t), the analogs
ak[xl(t)] in the reference catalog, and their associated
successors sk[xl(t)] for each component lof the state
x(t) are defined according to a component-wise local
neighborhood, typically fxl2n(t), ...,xl(t), ...,xl1n(t)g
with nbeing the width of the considered component-
wise neighborhood, such that the evaluation of the
kernel function and the computation of the associated
normalized weights vk[xl(t)] only involve this local
neighborhood.
The idea of using local analogs is motivated by the fact
that points tends to scatter far away from each other in
high dimensions, which make the search for skillful an-
alogs nearly impossible for high-dimensional state
space. For instance, Van den Dool (1994) has shown
that finding a relevant analog at synoptic scale over the
Northern Hemisphere for atmospheric data would re-
quire 1030 years of data to match the observational
errors at that time. Conversely, analog forecasting
schemes may only apply to systems or subsystems as-
sociated with low-dimensional embedding. Following
this analysis, the analog forecasting of the global state is
split as a series of local and low-dimensional analog
forecasting operations. Note that such local analogs also
reduce possible spurious correlations.
4. Analog data assimilation
The analog data assimilation is stated as a sequential
and stochastic assimilation scheme, using Monte Carlo
methods. It amounts to estimating the so-called filter-
ing and smoothing posterior likelihoods, respectively,
p[x(t)jy(1), ...,y(t)] the distribution of the current
state knowing past and current observations and
p[x(t)jy(1), ...,y(T)] the distribution of the current
state knowing past, current, and future observations. We
investigate both ensemble Kalman filter/smoother and
particle filter.
a. Analog ensemble Kalman filter and smoother
(AnEnKF/AnEnKS)
Ensemble Kalman filters (EnKF) and smoothers
(EnKS) (Burgers et al. 1998;Evensen 2007) are partic-
ularly popular in geoscience as they provide flexible
assimilation strategies for high-dimensional states. They
rely on the assumption that the filtering and smoothing
posteriors are multivariate Gaussian distributions, such
that the following forward and backward recursions are
derived. The next two paragraphs present the AnEnKF
and AnEnKS equations, which are equivalent to those
of the EnKF and EnKS described in Tandeo et al.
(2015b), except for the update step where we use the
analog forecasting operator.
The forward recursions of the AnEnKF correspond to
the stochastic EnKF algorithm proposed by Burgers
OCTOBER 2017 L G U E N S A T E T A L . 4097

et al. (1998) in which observations are treated as random
variables. The AnEnKF algorithm starts at time t51by
generating the vectors xf
i(1)"i2f1, ...,Ngusing a
multivariate Gaussian random generator with mean
vector xband covariance matrix B. The index iof the
state vector corresponds to the ith realization of the
Monte Carlo procedure (called member or particle).
Then the update step proceeds from t52tot5Tby
applying the analog forecasting operator to each mem-
ber of the ensemble following (3) to generate xf
i(t). The
forecast state is represented by the sample mean xf(t)
and the sample covariance Pf(t). In the analysis step,
following (2),Nsamples of yf
i(t) are generated from a
multivariate Gaussian random generator with mean
Hxf
i(t) and covariance R. The observations are then
used to update the Nmembers of the ensemble as
xa
i(t)5xf
i(t)1Ka(t)[y(t)2yf
i(t)], where Ka(t)5
Pf(t)H0[HP f(t)H01R]21is the Kalman filter gain. The
filtering posterior distribution is then represented by the
sample mean xa(t) and the sample covariance Pa(t).
The analog ensemble Kalman smoother combines the
analog forecasting operator and the classical Kalman
smoother, here, Rauch–Tung–Striebel smoother [see
Cosme et al. (2012) for more details]. Given the forward
recursion, the backward recursion starts from time t5T
with filtered state, "i2f1, ...,Ng, such as xs
i(T)5
xa
i(T) and Ps(T)5Pa(T). Then, we proceed backward
from t5T21tot51. At each time t, we compute
xs
i(t)5xa
i(t)1Ks(t)[xs
i(t11) 2xf
i(t11)], where Ks(t)5
Pa(t)M0[Pf(t11)]21is the Kalman smoother gain. Note
that we empirically estimate Pa(t)M0as the sample co-
variance matrix of the ensemble members as in Pham
(2001) or Tandeo et al. (2015b) in the case of a nonlinear
operator H. The smoothing posterior distribution is
represented by the sample mean xs(t) and the sample
covariance Ps(t).
We note that the following way of extending EnKF
and EnKS to become analog-based algorithms can be
applied in the same way to other flavors of EnKF such as
the square root ensemble Kalman filter (EnSRF). We
chose stochastic ensemble-based Kalman filters and
smoothers as an illustration in this work, even if they are
not the first choice in practice for atmospheric and
oceanic applications because of issues related to per-
turbing observations with noise (Bowler et al. 2013).
Besides, the work of Hoteit et al. (2015), where the au-
thors address this issue, suggests that the stochastic
EnKF is worth a reevaluation for oceanic and atmo-
spheric applications.
b. Analog particle filter (AnPF)
We also implement particle filtering techniques for the
proposed analog data assimilation strategy. Contrary to
the Kalman filters, particle filters do not assume a Gaussian
distribution of the state. The key principle is to estimate
the posteriors of the state from a set of particles (equiva-
lent to members in the terminology used for ensemble
Kalman filters).
Given an analog forecasting operator, we consider an
application of the classical particle filter (Van Leeuwen
2009). From an initialization similar to the EnKF, the
particle filter applies a forward recursion from time t51
to t5Tas follows. At time step t, we first apply the
considered analog forecasting operator Ato forecast
xf
i(t)"i2f1, ...,Ngfrom previous filtered particles
xa
i(t21). Then, following (2), we compute particle
weights pi(t)as
pi(t)}f[y(t)2Hxf
i(t); R], (12)
where f(;R) is a centered multivariate Gaussian dis-
tribution with covariance R. Weights pi(t) are normal-
ized to total one. We then proceed to a systematic
resampling from the multinomial distribution defined by
the particles fxf
i(t)gand their corresponding weights
fpi(t)g. The analyzed state xa(t) is typically computed as
the sample mean
xa(t)51
N
N
i51
pi(t)xf
i(t), (13)
but one may also consider the posterior mode as the
filtered state.
In theory, particle smoothers may also be considered.
Different strategies have been proposed in the past but
they showed numerical instabilities in preliminary experi-
ments with the considered analog forecasting operator.
We do not further detail the considered implementation
but discuss these aspects in section 6.
5. Numerical experiments
To evaluate the relevance and performance of the
proposed analog data assimilation, we consider numerical
experiments on dynamical systems extensively used in the
literature on data assimilation: Lorenz-63 and Lorenz-96
models. The experiments for evaluating the effect of the
size of the catalog, the impact of noisy catalogs, and cat-
alogs with parametric model error are conducted using the
Lorenz-63 model. To evaluate the global and local analog
forecasting operators we use the Lorenz-96 model, an
extended dynamical nonlinear system with 40 variables.
a. Chaotic models
We first consider the chaotic Lorenz-63 system.
From a methodological point of view, it is particularly
4098 MONTHLY WEATHER REVIEW VOLUME 145

interesting because of its nonlinear chaotic behavior and
low dimension. Several works have used this system
(e.g., Miller et al. 1994;Anderson and Anderson 1999;
Pham 2001;Chin et al. 2007;Hoteit et al. 2008 or Van
Leeuwen 2010). The Lorenz-63 model is defined by
dx1(t)
dt 5s[x2(t)2x1(t)],
dx2(t)
dt 5x1(t)[g2x3(t)] 2x2(t),
dx3(t)
dt 5x1(t)x2(t)2bx3(t), (14)
and behaves chaotically for certain sets of parameters,
such as (s510, g528, b58/3). Here, we use the explicit
(4, 5) Runge–Kutta integrating method (cf. Dormand and
Prince (1980)) with time step dt 50:01 (nondimensional
units). As in Van Leeuwen (2010) only the first variable of
the Lorenz-63 system (x1) is observed every 8 integration
time steps (i.e., with dt 50:08). Considering the analogy
between the Lorenz-63 and atmospheric time scales, it is
equivalent to a 6-h time step in the atmosphere.
The Lorenz-96 model is another chaotic model largely
used for evaluating data assimilation techniques in
geophysics (Anderson 2001;Whitaker and Hamill 2002;
Ott et al. 2004;Anderson 2007,2012;Hoteit et al. 2012).
It is defined by
dxj(t)
dt 5[2xj22(t)1xj11(t)]xj21(t)2xj(t)1F, (15)
where j51, ...,nand the boundaries are cyclic [i.e.,
x21(t)5xn21(t), x0(t)5xn(t), and xn11(t)5x1(t)]. The
three right-hand side terms in (15) simulate an advec-
tion, a diffusion, and a forcing term, respectively. As
in Lorenz (1996), we choose n540 and external forcing
of F58 for which the model behaves chaotically.
Equation (15) is solved using the Runge–Kutta fourth-
order scheme with integration time step dt 50:05, cor-
responding to a time step of 6h in the atmosphere.
Observations are taken from half of the state vector
(20 observed components randomly selected) every 4
time steps (i.e., dt 50:20).
b. Experimental details
The considered experimental setting is as follows. To
avoid divergence of the filteringmethods, weuse N5100
members/particles for the Lorenz-63 and N51000
members/particles for the Lorenz-96 for both model-
driven and data-driven strategies. We use the same co-
variance matrix Rwith a noise observation variance set to
2. To avoid any spinup effect, the initial state conditions is
chosen as the ground truth mean and a covariance matrix
Bwith noise variance 0.1. To compare the technique
performances, we use the root-mean-square error (RMSE)
on all the components of the state vector and for all
assimilation times. As training dataset for the catalog
and test dataset for RMSE computation, we use 103and
100 Lorenz times, respectively.
The analog forecasting operator involves two free
parameters, namely, Kthe number of nearest neighbors
and lthe scale parameter of the Gaussian kernel in (4).
Two strategies can be considered for K: either a pre-
defined number of nearest neighbors, or a predefined
threshold on distance dth to select the analogs that are
closer than dth. For the sake of simplicity, we consider
in this work the first alternative and set Kto 50. Besides,
we use for lthe following adaptive rule: l[x(t)] 5
1/md[x(t)], where md[x(t)] is the median distance be-
tween the current state x(t) and its Kanalogs. Note
that a cross-validation procedure could be used to op-
timize the choice of Kand l. All analog forecasting
operators are fitted for forecasting time horizon corre-
sponding to the time step of the numerical simulations
(i.e., dt 50:01 for Lorenz-63 experiments and dt 50:05
for Lorenz-96 experiments). Numerical experiments
(not reported here) show that this parameterization
provides on average the best forecasting performance
with respect to the forecasting time horizon.
c. Experiments with Lorenz-96 model
1) EXPERIMENT 1
The first numerical experiment consisted only in the
application of analog forecasting (without assimilation)
from a catalog. We build a database using Lorenz-96
equations, then we split the samples randomly to 2/3 for
training the analog forecasting operators and 1/3 for test.
Finally, we compare the RMSE w.r.t. ground truth data
as a function of Lorenz-96 forecast time. For local ana-
logs, we consider n52 the width of the considered
component-wise neighborhood. Figure 3 shows the re-
sults of this experiment using the three choices for the
analog forecasting operator A. The locally linear ap-
proach outperforms the two other approaches confirm-
ing that its forecasts are with lower bias compared to the
other approaches. However, it also involves more pa-
rameters, which increases the variance of the forecasts.
This bias-variance trade-off supports the greater gen-
eralization capabilities of the locally linear operator,
when the dynamics can well be approximated locally
by a linear operator.
Figure 3 also compares local and global analog strat-
egies. When using locally constant operator, local ana-
logs are always better than global analogs. Searching for
nearest neighbors on 40-dimensional vectors results
OCTOBER 2017 L G U E N S A T E T A L . 4099

most likely in irrelevant analogs. This affects heavily the
locally constant operator more than the two other op-
erators, since it computes a weighted mean of their as-
sociated successors. The locally constant operator also
limits novelty creation in the dynamics by always drag-
ging the forecast near the mean of the Ksuccessors, and,
according to these experiments, it seems poorly adapted
to complex and highly nonlinear systems. Regarding the
locally incremental and locally linear strategies, local
analogs are more relevant than global ones for pre-
diction in a near future (less than 0.5 in Lorenz-96 time
for locally linear operator and less than 0.25 in Lorenz-
96 time for locally incremental).
2) EXPERIMENT 2
We conducted a second experiment for evaluating the
impact of analog forecasting in data assimilation using
the Lorenz-96 model. We run the AnEnKS with 1000
ensemble members, only 20 variables are observed every
0.20 time steps. Figure 4 shows analog data assimilation
experiments with the locally linear forecasting method us-
ing the Lorenz-96 model. Figures 4a and 4b show the true
state and the observations, respectively. The reconstructed
state with global analogs is shown in Fig. 4c and the one
with local analogs in Fig. 4d. The local analog data assim-
ilation experiment clearly outperforms the global analog
data assimilation experiment.
3) EXPERIMENT 3
A third experiment with the Lorenz-96 system was
conducted. For the local analog strategy, we further
compare the proposed AnDA algorithms, namely,
AnEnKF, AnPF, and the AnEnKS using 1000 ensemble
members/particles, in Table 1. Two main conclusions
can be drawn: (i) EnKF algorithms outperform the
particle filter and (ii) the locally linear analog fore-
casting operator gives the best reconstruction per-
formance. We noticed that the AnPF suffers in the
40-dimensional Lorenz-96 system from sample impov-
erishment and degeneracy. Despite additional experi-
ments with different settings, for instance, w.r.t. the
number of ensemble members, the number of analogs
as well as using jittering (i.e., perturbing the particles
with a small noise), the AnPF still suffered from the
aforementioned issues.
d. Experiments with Lorenz-63 model
1) EXPERIMENT 1
In the proposed AnDA, the size of the catalog is ex-
pected to be a critical parameter. For Lorenz-63 dy-
namics, we conducted different AnDA experiments
varying the size of the catalog S5f101,10
2,10
3,10
4gin
Lorenz-63 times. We consider the same setting as in
Tandeo et al. (2015a) where the locally constant method
with a Gaussian sampling was used for the AnEnKF,
then we compare the three AnDA algorithms using 100
ensemble members/particles. As reported in Fig. 5, the
RMSE decreases when the size of the catalog increases
for all AnDA algorithms. Regarding filtering-only (i.e.,
no smoothing) AnDA algorithms, the AnPF (blue)
outperforms the AnEnKF (green). This is an expected
result since particle filters handle better nonlinear
models and non-Gaussian probability distributions, al-
though at a high cost in terms of computational com-
plexity and execution time. The AnEnKS (red) clearly
gives the lowest RMSE. This supports the additional
benefit of the smoothing step performed by the
AnEnKS. The zoom shown in the right panel of Fig. 5
highlights how the smoothing step corrects the piecewise
effects resulting from the filtering step.
2) EXPERIMENT 2
Modeling uncertainty is a critical source of error in
data assimilation. In this experiment we evaluate
whether AnDA can manage a situation in which the
catalog is composed by multiple numerical simulations,
which may have parametric model error. In (14), pa-
rameters gand bdefine the center of the two attractors
whereas scontrols the shape of the trajectories. In
Fig. 6, we depict trajectories using three sets of param-
eters with different values for s:u15(10, 28, 8/3) (red),
u25(7, 28, 8/3) (blue), and u35(13, 28, 8/3) (green).
We generate three catalogs with Lorenz-63 trajectories
FIG. 3. Results of the analog forecasting performance as a func-
tion of the horizon. Different analog forecasting methods are
plotted: locally constant (green), locally incremental (blue), and
locally linear (red) analog operators with local (straight line) and
global (dashed line) analog strategies. The black dashed line cor-
responds to a persistent prediction over time.
4100 MONTHLY WEATHER REVIEW VOLUME 145

for these three set of parameters, with 103Lorenz time
steps each. Merging these three catalogs into a global
catalog, we apply the proposed AnDA using as obser-
vations the true integration resulting from Lorenz-63
model with u1parameter values. As a by-product of the
analog strategy, we can infer the underlying model pa-
rameterization from the observed partial observations.
The reported experiments (Fig. 6) apply the AnPF
procedure with the locally constant analog method and a
multinomial sampling scheme using 100 particles. Such a
choice was motivated by the desire of keeping track of
the particles and their source catalog, which is harder to
achieve with the other AnDA algorithms, since the
particles would be elements from the catalog and the
AnPF assigns a weight to each particle. This make it
easier to select at each time the particle with the biggest
weight and to know from which catalog it came from.
At every assimilation time step, we determine which
parameterization most ensemble members come from,
and then calculate the proportion of the presence of
each parameterization. As expected, the true parame-
terization (red, parameterization u1)ismorerepre-
sented. The proportions for u1,u2,andu3are around,
TABLE 1. RMSE of the reconstruction of Lorenz-96 state evolution
using different forecasting strategies and data assimilation techniques.
The catalog size corresponds to 103Lorenz-96 times (equivalent to
13 yr) and the number of members/particles is N51000.
Method
Locally
constant
Locally
incremental
Locally
linear
Gaussian
AnEnKF 1.826 1.785 1.403
AnPF 3.174 4.224 4.4616
AnEnKS 1.320 1.287 0.970
Multinomial
AnEnKF 1.814 1.774 1.413
AnPF 2.989 4.412 4.729
AnEnKS 1.313 1.288 1.093
FIG. 4. Lorenz-96 trajectories obtained using analog data assimilation procedures with the locally linear forecasting
strategy, when only 20 variables are observed every 0.20 time steps. (top left) True simulation of the model with 40
variables, (top right) noisy and partial observations, (bottom left) reconstructed state trajectories via the AnEnKS with
global analogs, and (bottom right) reconstructed state trajectories via the AnEnKS with local analogs [taking into
account the 5 (n52) nearest state components]. Only 10 Lorenz-96 cycles are shown for better visibility.
OCTOBER 2017 L G U E N S A T E T A L . 4101

60%, 16%, and 24%, respectively, proving the ability
of the methodology to detect the source of the noisy
and partial observation (here, only coming from u1).
To analyze the results more thoroughly, we calculate
the RMSE of the reconstruction using (i) the three
catalogs as shown before, (ii) only the good catalog,
and (iii) only the two ‘‘bad’’ catalogs. The RMSEs
are (i) 1.287, (ii) 1.207, and (iii) 1.424, respectively.
These results show that having other catalogs with
different parameterization degrade the RMSE but
the filter is still performing well. This experiment
gives insights on the problem of the assimilation of
variables that may switch between different dynami-
cal modes. Analog data assimilation can deal with this
problem in a simpler manner than classical data as-
similation, through the concatenation of the catalogs
issued from different parameterizations into a single
catalog.
FIG. 5. Reconstruction of Lorenz-63 trajectories for different catalog sizes in the analog data assimilation pro-
cedures, when only the first component of the state is observed every 0.08 time steps. (left) RMSE as a function of
the size of the catalog for different analog data assimilation strategies: AnEnKF (green), AnPF (blue), and
AnEnKS (red). For benchmarking purposes, data assimilation results with true Lorenz-63 equations are given in
straight lines. (right) Time series of the first component of the true state (black solid line), associated noisy ob-
servations (black asterisks), mean reconstructed series (solid lines), and 10 analyzed members/particles (dashed
lines) with analog data assimilation strategies, namely AnEnKF (green), AnPF (blue), and AnEnKS (red), using
a catalog of 103Lorenz-63 times (equivalent to 8 yr).
FIG. 6. Identification of Lorenz-63 model parameterizations using a multiparameterization catalog in the analog
data assimilation, when only the first component of the state is observed every 0.08 time step. (left) Examples of
Lorenz-63 trajectories generated with three different parameterizations: u15(10, 28, 8/3) (red), u25(7, 28, 8/3)
(blue), and u35(13, 28, 8/3) (green). (right) Result of the AnPF on the first Lorenz-63 variable using the three
catalogs associated with parameterizations fuig1,2;3for 3 3103Lorenz-63 times (equivalent to 3 38 yr) when only
observations from parameterization u15(10, 28, 8/3) are provided. The figure shows the AnPF particles trajectories
(blue), the AnPF result (red), and the true trajectory (green).
4102 MONTHLY WEATHER REVIEW VOLUME 145

3) EXPERIMENT 3
Whereas previous experiments consider catalogs
produced from noise-free trajectories, here we evaluate
the sensitivity of the AnDA procedures when the cata-
log may involve noisy trajectories of the considered
system. Acquisition systems typically involve such noise
patterns, which may relate for instance to both envi-
ronmental constraints and measurement uncertainties.
We simulate noisy catalogs for Lorenz-63 dynamics as
follows: we artificially degrade the transition between
consecutive states with a Gaussian additive noise. We
performed experiments with different noise variances
c25f0:5, 1, 2gto evaluate the sensitivity of AnDA
procedures with respect to the signal-to-noise ratio. As
illustrated in Fig. 7, the trajectories of these experiments
are extremely noisy. Table 2 reports the RMSE of the
different AnDA algorithms with the locally linear ana-
log forecasting operator and 100 ensemble members/
particles. As expected, the RMSE increases with the variance
of the additive noise. The AnEnKS clearly outperforms
the other AnDA algorithms, which highlights its greater
robustness. Figure 7 further illustrates that the AnEnKS
is able to correctly track the true state of the system,
even for highly degraded catalogs (c252, green curve).
For a high signal-to-noise ratio (i.e., low perturbations)
(c250:5, red curve), reconstructed trajectories are very
close to the ones obtained with a noise-free catalog.
6. Conclusions and perspectives
The present paper demonstrates the potential of data-
driven schemes for data assimilation. We propose and
evaluate efficient yet simple data-driven forecasting
strategies that can be coupled with classical stochastic
filters (viz., the ensemble Kalman filter/smoother and
the particle filter). We set a unified framework that we
call the analog data assimilation (AnDA). The key
features of the AnDA are twofold: (i) it relies on a data-
driven representation of the state dynamics, and (ii) it
does not require online evaluations of dynamical models
based on physical equations. The relevance of the
AnDA is tangible when the dynamical system of interest
demands tremendous and time-consuming physical
modeling efforts and/or uncertainties are difficult to
assess. In cases when large observational or model-
simulated datasets of the considered system are avail-
able, AnDA can both support or compete with classical
data assimilation schemes. As a proof concept, we
demonstrate the relevance of the proposed methodol-
ogy to retrieve the chaotic behavior of the Lorenz-63
and Lorenz-96 models. We performed numerical ex-
periments to evaluate critical aspects of the method,
especially the relevant combinations of analog fore-
casting strategies and of stochastic filters as well as the
exploitation of noisy and noise-free catalogs.
FIG. 7. Results of the reconstruction of Lorenz-63 trajectories from noisy catalogs: (left) examples of noisy
Lorenz-63 trajectories for different noise levels: c2
150:5 (red), c2
251 (blue), and c2
352 (green). (right) Results of
the AnEnKS using noisy catalogs corresponding to 103Lorenz-63 times (equivalent to 8 yr) when only observations
with variance R52 are provided. We also plot the 95% confidence interval computed from the smoothing
covariances.
TABLE 2. RMSE of the reconstruction of Lorenz-63 trajectories
from noisy catalogs: we vary the variance of an additive Gaussian
noise in the creation of the catalogs and apply analog data assim-
ilation procedures with the locally linear operator with a catalog
size of 103Lorenz-63 times, when only the first component of the
state is observed every 0.08 time step with observation noise vari-
ance R52.
Method c2
150:5c2
251c2
352
AnEnKF 1.926 2.136 2.681
AnPF 1.652 1.961 2.313
AnEnKS 1.233 1.561 2.142
OCTOBER 2017 L G U E N S A T E T A L . 4103

All the reported experiments were carried out using
the AnDA Python/Matlab library (https://github.com/
ptandeo/AnDA), which includes the Lorenz-63 and
Lorenz-96 systems. In the spirit of reproducible re-
search, the user can conduct the different experiments
shown in this paper.
Overall, the reported results demonstrate the rele-
vance of the proposed analog data assimilation methods,
even with highly damaged catalogs. They suggest that
AnEnKS combined with locally incremental or locally
linear analog forecasting leads to the best reconstruction
performance, the locally incremental version being the
most robust to noisy settings. Moreover, the flexibility of
the analog data assimilation demonstrates the potential
for the identification of hidden underlying dynamics
from a series of partial observations.
The main pillar of our data-driven approach is the
catalog. As such, analog data assimilation deeply relates
to the quality and representativity of the catalog. In our
experiments, we assumed that we were provided with
large-scale catalogs of complete states of the system of
interest. While catalogs built from numerical simula-
tions fulfill this assumption, observational datasets (e.g.,
satellite remote sensing or in situ data) typically involve
missing data, which may require specific strategies to be
dealt with in the building of the catalogs. In this respect,
local analogs obviously appear much more flexible than
global ones, as partial observations provide relevant
exemplars for the creation of catalogs for local analogs.
The application of analog data assimilation to high-
dimensional systems is another future challenge. As
detailed in Van den Dool (1994), the number of ele-
ments in a catalog shall grow exponentially with the
intrinsic dimension of the state to guarantee the re-
trieval of analogs at a given precision. This makes un-
realistic the direct application of analog strategies to
state space with an intrinsic dimensionality above 10.
As a consequence, global analog forecasting operators
are most likely inappropriate for high-dimensional sys-
tems. By contrast, local analogs provide a means to de-
compose the analog forecasting of the high-dimensional
state into a series of local and low-dimensional analog
forecasting operations. This is regarded as the key ex-
planation for the much better performance reported for
the local analog data assimilation for Lorenz-96 dy-
namics using catalogs of about a million of exemplars
(Fig. 4). For real-world applications to high-dimensional
systems, for instance to ocean and atmosphere dynam-
ics, the combination of such local analog strategies to
multiscale decompositions (Mallat 1989) arise as a
promising research direction as illustrated in Fablet
et al. (2017). Such multiscale decompositions are ex-
pected to enhance the spatial redundancy, with a view to
building the requested catalogs of millions to hundreds
of millions of exemplars (for an intrinsic dimensionality
between 4 and 7, see the appendix) from observation or
simulation datasets over a few decades. Another im-
portant aspect that controls the effective size of the
catalog is the evolution of the system in time. The more
nonlinear the dynamics, the greater the number of re-
quested exemplars in the global catalog to learn the
forecast operator and the spread of the prediction.
We believe that this study opens new research ave-
nues for the analysis, reconstruction, and understanding
of the dynamics of geophysical systems using data-
driven techniques. Such techniques will benefit from
the increasing availability of large-scale historical ob-
servational and/or simulated datasets. Beyond the wide
range of possible applications, future research should
further investigate methodological issues. First of all,
our study demonstrates the relevance of the analog
particle filter, but as mentioned in section 5, the AnPF
suffers from degeneracy and sample impoverishment.
We may point out that complementary experiments with
particle smoother schemes (not shown in this paper)
resulted in numerical instabilities. The derivation of the
analog particle smoother then remains an open ques-
tion. In addition to advanced particle filters as proposed
in Van Leeuwen (2010) and Pitt and Shephard (1999),
one might also benefit from the straightforward appli-
cations of the analog procedure in reverse time, which is
not generally possible for model-driven schemes. A
second direction for future work lies in the design of
the kernel used by the analog forecasting operators.
Whereas we considered a Gaussian kernel, other kernels
have been proposed in the literature; for example, using
Procrustes distance instead of the Euclidean distance
(McDermott and Wikle 2016) or different weighing
strategies (Delle Monache et al. 2011). The explicit
derivation of the mapping associated with a kernel as
considered in Zhao and Giannakis (2014) may also be a
promising alternative to state the analog data assimila-
tion in a kernel-derived lower-dimensional space. The
theoretical characterization of the asymptotic behavior
of analog data assimilation schemes is also an interesting
avenue of research. Similarly to the theoretical analysis
of ensemble Kalman filters and particle filters (Le Gland
et al. 2009), the derivation of convergence conditions,
possibly associated with reconstruction bounds, would
be of key interest to bound the reconstruction perfor-
mance of the proposed analog schemes with respect to
their model-driven counterpart.
Acknowledgments. We thank all the researchers from
various fields who provided careful and constructive
comments on the original paper especially Bertrand
4104 MONTHLY WEATHER REVIEW VOLUME 145

Chapron, Valérie Monbet, and Anne Cuzol. The au-
thors would also like to thank Phi Viet Huynh for his
valuable contribution to both the AnDA Matlab tool-
box and the AnDA Python library. We are also grateful
to the two anonymous reviewers, whose comments helped
to improve the manuscript. We thank Geraint Jones for his
English grammar corrections. This work was supported
by ANR (Agence Nationale de la Recherche, Grant ANR-
13-MONU-0014), Labex Cominlabs (Grant SEACS), the
Brittany council, and a ‘‘Futur et Ruptures’’ postdoctoral
grant from Institut Mines-Télécom.
APPENDIX
Operational Count of the AnDA Applied for
High-Dimensional Applications
This appendix aims at giving an estimate of the oper-
ations involved when applying the AnDA for a realistic
large-scale application. We discuss the computational
cost of the analog forecasting, which is specific to the
AnDA. The latter directly relates to the cost of the
K-nearest neighbor (K-NN) step.
In case of large-scale catalogs, an exhaustive search
strategy is not suitable and the use of space-partitioning
data structures, the most popular ones being K-d trees
(Bentley 1975) and Ball trees (Omohundro 1989), ap-
pears necessary. These structures speed up the K-NN
search, at the expense of an approximate search for
nearest neighbors. Let us denote by Dthe dimension of
the system of interest. Making a choice between K-d
trees or ball trees depends mostly on the dimensionality
of the system. The K-d trees are known to perform well
in dimensions D,20, while ball trees are more suitable
to dimensions higher than 20 but come with a high cost
of space partitioning (Witten et al. 2016). In this appendix
we focus on the use of K-d trees, which are natural can-
didates for local analogs with a small component-wise local
neighborhood nor using a preliminary dimensionality re-
duction algorithm (such as empirical orthogonal func-
tions). A comparison between K-d trees and ball trees is
out of the scope of this work.
Let Ndata be the size of the catalog (the number of
samples from where to look for analogs), and Kthe
number of nearest neighbors to be retrieved. Let us re-
call that nis the size of the local neighborhood used for
the search for local analogs. Van den Dool (1994)
derived a relationship between the local neighborhood
size and the amount of the data needed to find an analog
with a given precision. With the assumption that the
components of the states follow a multivariate Gaussian
distribution and have the same variance sd2, finding
Ksamples that have a distance lower than «for all
the components of the neighborhood with a probability
of 95%, needs the number of data to be on average as
follows:
dGlobal analogs:
Nglobal $Kln(0:05)
ln(1 2aD)’3K
aD, (A1)
dLocal analogs:
Nlocal $Kln(0:05)
ln(1 2a2n11)’3K
a2n11, (A2)
where ais the integral of the standard Gaussian prob-
ability density function from 2«/(
ffiffiffi
2
psd)to 2«/(
ffiffiffi
2
psd).
We present now the operational count for one en-
semble member (or particle) involved in the forecasting,
for both global and local analogs. In each case, we dis-
tinguish the computational cost of the creation of the
K-d trees and the search of Knearest neighbors:
dGlobal analogs:
dCreation of the K-d tree: O[DNglobal log(Nglobal)].
dSearch for Kglobal analogs: O[KDlog(Nglobal )].
dLocal analogs:
dCreation of DK-d trees (for every dimension in D):
O[D(2n11)Nlocal log(Nlocal)].
dSearch for Klocal analogs of component-wise
neighborhood n:O[DK(2n11)log(Nlocal)].
Note that using local analogs requires constructing a
K-d tree for every dimension in D. Construction of the
K-d trees can be done offline (1 ‘‘big’’ K-d tree for the
global strategy and D‘‘small’’ K-d trees for the local
strategy), then the cost of these construction can be
amortized over the high number of queries that needs to
be answered during analog data assimilation. However,
in terms of memory storage, storing a global K-d tree
could be prohibitive, contrarily to small local K-d trees
that can be created, used, then freed for the creation of
the next K-d tree of the next dimension (if there is no
sufficient memory to stock Dsmall local K-d trees).
Keep in mind that we need to have (2n11) Dfor
local analogs to be of relevance.
Let us take an example using the Lorenz-96 model:
D540, n52. Looking for K550 analogs, with an
a50:15 we would need Nglobal ’1035, which is very
prohibitive; however, we would only need Nlocal ’
23106samples using local analogs.
REFERENCES
Anderson, J. L., 2001: An ensemble adjustment Kalman filter for
data assimilation. Mon. Wea. Rev.,129, 2884–2903, doi:10.1175/
1520-0493(2001)129,2884:AEAKFF.2.0.CO;2.
OCTOBER 2017 L G U E N S A T E T A L . 4105

——, 2007: Exploring the need for localization in ensemble data
assimilation using a hierarchical ensemble filter. Physica D,
230, 99–111, doi:10.1016/j.physd.2006.02.011.
——, 2012: Localization and sampling error correction in ensemble
Kalman filter data assimilation. Mon. Wea. Rev.,140, 2359–
2371, doi:10.1175/MWR-D-11-00013.1.
——, and S. L. Anderson, 1999: A Monte Carlo implementation of
the nonlinear filtering problem to produce ensemble assimi-
lations and forecasts. Mon. Wea. Rev.,127, 2741–2758,
doi:10.1175/1520-0493(1999)127,2741:AMCIOT.2.0.CO;2.
Atencia, A., and I. Zawadzki, 2015: A comparison of two tech-
niques for generating nowcasting ensembles. Part II: Analogs
selection and comparison of techniques. Mon. Wea. Rev.,143,
2890–2908, doi:10.1175/MWR-D-14-00342.1.
Bentley, J. L., 1975: Multidimensional binary search trees used for
associative searching. Commun. ACM,18, 509–517, doi:10.1145/
361002.361007.
Bocquet, M., C. A. Pires, and L. Wu, 2010: Beyond Gaussian
statistical modeling in geophysical data assimila-
tion. Mon. Wea. Rev.,138, 2997–3023, doi:10.1175/
2010MWR3164.1.
Bowler, N. E., J. Flowerdew, and S. R. Pring, 2013: Tests of dif-
ferent flavours of EnKF on a simple model. Quart. J. Roy.
Meteor. Soc.,139, 1505–1519, doi:10.1002/qj.2055.
Burgers, G., P. Jan van Leeuwen, and G. Evensen, 1998:
Analysis scheme in the ensemble Kalman filter. Mon. Wea.
Rev.,126, 1719–1724, doi:10.1175/1520-0493(1998)126,1719:
ASITEK.2.0.CO;2.
Chin, T., M. Turmon, J. Jewell, and M. Ghil, 2007: An ensemble-
based smoother with retrospectively updated weights for
highly nonlinear systems. Mon. Wea. Rev.,135, 186–202,
doi:10.1175/MWR3353.1.
Cleveland, W. S., 1979: Robust locally weighted regression and
smoothing scatterplots. J. Amer. Stat. Assoc.,74, 829–836,
doi:10.1080/01621459.1979.10481038.
Cosme, E., J. Verron, P. Brasseur, J. Blum, and D. Auroux, 2012:
Smoothing problems in a Bayesian framework and their linear
Gaussian solutions. Mon. Wea. Rev.,140, 683–695, doi:10.1175/
MWR-D-10-05025.1.
Delle Monache, L., T. Nipen, Y. Liu, G. Roux, and R. Stull, 2011:
Kalman filter and analog schemes to postprocess numerical
weather predictions. Mon. Wea. Rev.,139, 3554–3570, doi:10.1175/
2011MWR3653.1.
——, I. Djalalova, and J. Wilczak, 2014: Analog-based
postprocessing methods for air quality forecasting. Air
Pollution Modeling and Its Application XXIII,D.Steyn
and R. Mathur, Eds., Springer, 237–239, doi:10.1007/
978-3-319-04379-1_38.
Dormand, J. R., and P. J. Prince, 1980: A family of embedded
Runge–Kutta formulae. J. Comput. Appl. Math.,6, 19–26,
doi:10.1016/0771-050X(80)90013-3.
Evensen, G., 2007: Data Assimilation: The Ensemble Kalman Fil-
ter. Springer-Verlag, 280 pp., doi:10.1007/978-3-540-38301-7.
——, and P. J. Van Leeuwen, 2000: An ensemble Kalman smoother
for nonlinear dynamics. Mon. Wea. Rev.,128, 1852–1867,
doi:10.1175/1520-0493(2000)128,1852:AEKSFN.2.0.CO;2.
Fablet, R., P. H. Viet, R. Lguensat, and B. Chapron, 2017: Data-
driven assimilation of irregularly-sampled image time series.
IEEE Int. Conf. on Image Processing (ICIP 2017), Beijing,
China, IEEE, WQ-PB.2.
Hamilton, F., T. Berry, and T. Sauer, 2016: Ensemble Kalman fil-
tering without a model. Phys. Rev. X,6, 011021, doi:10.1103/
PhysRevX.6.011021.
Hansen, B., 2000: Econometrics. Department of Economics, Univer-
sity of Wisconsin, 427 pp., http://www.ssc.wisc.edu/;bhansen/
econometrics/Econometrics.pdf.
He-Guelton, L., R. Fablet, B. Chapron, and J. Tournadre, 2015:
Learning-based emulation of sea surface wind fields from
numerical model outputs and SAR data. IEEE J. Sel. Top.
Appl. Earth Obs. Remote Sens.,8, 4742–4750, doi:10.1109/
JSTARS.2015.2496503.
Hong, S.-Y., and J. Dudhia, 2012: Next-generation numerical
weather prediction: Bridging parameterization, explicit
clouds, and large eddies. Bull. Amer. Meteor. Soc.,93, ES6–
ES9, doi:10.1175/2011BAMS3224.1.
Hoteit, I., D.-T. Pham, G. Triantafyllou, and G. Korres, 2008: A
new approximate solution of the optimal nonlinear filter for
data assimilation in meteorology and oceanography. Mon.
Wea. Rev.,136, 317–334, doi:10.1175/2007MWR1927.1.
——, X. Luo, and D.-T. Pham, 2012: Particle Kalman filtering:
A nonlinear Bayesian framework for ensemble Kalman
filters. Mon. Wea. Rev.,140, 528–542, doi:10.1175/
2011MWR3640.1.
——, D.-T. Pham, M. Gharamti, and X. Luo, 2015: Mitigating obser-
vation perturbation sampling errors in the stochastic EnKF. Mon.
Wea. Rev.,143, 2918–2936, doi:10.1175/MWR-D-14-00088.1.
Kalnay, E., 2003: Atmospheric Modeling, Data Assimilation and
Predictability. Cambridge University Press, 345 pp.
Le Gland, F., V. Monbet, and V.-D. Tran, 2009: Large sample as-
ymptotics for the ensemble Kalman filter. Research Rep. RR-
7014, INRIA, 25 pp., https://hal.inria.fr/inria-00409060/document.
Lorenc, A., and Coauthors, 2000: The Met. Office global three-
dimensional variational data assimilation scheme. Quart. J. Roy.
Meteor. Soc.,126, 2991–3012, doi:10.1002/qj.49712657002.
Lorenz, E. N., 1969: Atmospheric predictability as revealed by
naturally occurring analogues. J. Atmos. Sci.,26, 636–646,
doi:10.1175/1520-0469(1969)26,636:APARBN.2.0.CO;2.
——, 1996: Predictability—A problem partly solved. Proc. Seminar
on Predictability, Reading, United Kingdom, ECMWF, 18 pp.,
https://www.ecmwf.int/sites/default/files/elibrary/1995/10829-
predictability-problem-partly-solved.pdf.
Mallat, S. G., 1989: A theory for multiresolution signal decomposi-
tion: The wavelet representation. IEEE Trans. Pattern Anal.
Mach. Intell.,11, 674–693, doi:10.1109/34.192463.
McDermott, P. L., and C. K. Wikle, 2016: A model-based approach
for analog spatio-temporal dynamic forecasting. Environ-
metrics,27, 70–82, doi:10.1002/env.2374.
Miller, R. N., M. Ghil, and F. Gauthiez, 1994: Advanced data as-
similation in strongly nonlinear dynamical systems. J. Atmos.
Sci.,51, 1037–1056, doi:10.1175/1520-0469(1994)051,1037:
ADAISN.2.0.CO;2.
Nagarajan, B., L. Delle Monache, J. P. Hacker, D. L. Rife,
K. Searight, J. C. Knievel, and T. N. Nipen, 2015: An evalua-
tion of analog-based postprocessing methods across several
variables and forecast models. Wea. Forecasting,30, 1623–
1643, doi:10.1175/WAF-D-14-00081.1.
Omohundro, S. M., 1989: Five balltree construction algorithms.
International Computer Science Institute, Berkeley, CA, 22
pp., http://ftp.icsi.berkeley.edu/ftp/pub/techreports/1989/
tr-89-063.pdf.
Ott, E., and Coauthors, 2004: A local ensemble Kalman filter for
atmospheric data assimilation. Tellus,56A, 415–428, doi:10.3402/
tellusa.v56i5.14462.
Pham, D. T., 2001: Stochastic methods for sequential data assimilati on
in strongly nonlinear systems. Mon. Wea. Rev.,129, 1194–1207,
doi:10.1175/1520-0493(2001)129,1194:SMFSDA.2.0.CO;2.
4106 MONTHLY WEATHER REVIEW VOLUME 145

Pitt, M. K., and N. Shephard, 1999: Filtering via simulation: Aux-
iliary particle filters. J. Amer. Stat. Assoc.,94, 590–599,
doi:10.1080/01621459.1999.10474153.
Ruiz, J. J., M. Pulido, and T. Miyoshi, 2013: Estimating model
parameters with ensemble-based data assimilation: A review.
J. Meteor. Soc. Japan,91, 79–99, doi:10.2151/jmsj.2013-201.
Schenk, F., and E. Zorita, 2012: Reconstruction of high resolution
atmospheric fields for northern Europe using analog-upscaling.
Climate Past,8, 1681–1703, doi:10.5194/cp-8-1681-2012.
Schölkopf, B., and A. J. Smola, 2001: Learning with Kernels:Sup-
port Vector Machines,Regularization,Optimization, and Be-
yond. MIT Press, 648 pp.
Tandeo, P., and Coauthors, 2015a: Combining analog method and
ensemble data assimilation: Application to the Lorenz-63
chaotic system. Machine Learning and Data Mining Ap-
proaches to Climate Science, V. Lakshmanan et al., Eds.,
Springer, 3–12, doi:10.1007/978-3-319-17220-0_1.
——, M. Pulido, and F. Lott, 2015b: Offline parameter estimation
using EnKF and maximum likelihood error covariance esti-
mates: Application to a subgrid-scale orography parametri-
zation. Quart. J. Roy. Meteor. Soc.,141, 383–395, doi:10.1002/
qj.2357.
Van den Dool, H., 1994: Searching for analogues, how long must we
wait? Tellus,46A, 314–324, doi:10.3402/tellusa.v46i3.15481.
Van Leeuwen, P. J., 2009: Particle filtering in geophysical systems.
Mon. Wea. Rev.,137, 4089–4114, doi:10.1175/2009MWR2835.1.
——, 2010: Nonlinear data assimilation in geosciences: An ex-
tremely efficient particle filter. Quart. J. Roy. Meteor. Soc.,
136, 1991–1999, doi:10.1002/qj.699.
Whitaker, J. S., and T. M. Hamill, 2002: Ensemble data assimilation
without perturbed observations. Mon. Wea. Rev.,130, 1913–1924,
doi:10.1175/1520-0493(2002)130,1913:EDAWPO.2.0.CO;2.
Wilby, R. L., and T. Wigley, 1997: Downscaling general circulation
model output: A review of methods and limitations. Prog.
Phys. Geogr.,21, 530–548, doi:10.1177/030913339702100403.
Witten, I. H., E. Frank, M. A. Hall, and C. J. Pal, 2016: Data
Mining: Practical Machine Learning Tools and Techniques.
4th ed. Morgan Kaufmann, 654 pp.
Yiou, P., 2014: AnaWEGE: A weather generator based on ana-
logues of atmospheric circulation. Geosci. Model Dev.,7, 531–
543, doi:10.5194/gmd-7-531-2014.
Zhao, Z., and D. Giannakis, 2014: Analog forecasting with
dynamics-adapted kernels. Nonlinearity,29, 2888–2939,
doi:10.1088/0951-7715/29/9/2888.
OCTOBER 2017 L G U E N S A T E T A L . 4107