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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 5K

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)1K

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.

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