1. Introduction
The reconstruction of the spatiotemporal dynamics of geophysical systems from noisy and/or partial observations 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 repeated forward integrations of an explicitly known dynamical 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 geophysical models (Hong and Dudhia 2012). Such processes typically involve highly nonlinear and local effects (Wilby and Wigley 1997). The resulting numerical models may be computationally intensive and even prohibitive for assimilation problems, for instance regarding the time integration of members with different initial conditions at each time step. Second, as explained in Van Leeuwen (2010), “with ever-increasing resolution 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 nonlinear particle filters which are prone to the “curse of dimensionality.” Third, difficulties may occur when geophysical dynamics involve uncertain model parameterizations 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 classical model-driven assimilation schemes.
Meanwhile, recent years have witnessed a proliferation 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 sciences. 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 atmospheric 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 application domains, including among others, rainfall nowcasting (Atencia and Zawadzki 2015), air quality analysis (Delle Monache et al. 2014), wind field downscaling (He-Guelton et al. 2015), climate reconstruction (Schenk and Zorita 2012), and stochastic weather generators (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 assimilation (AnDA) consists of a combination of the implicit analog forecasting of the dynamics with stochastic filtering 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 assimilation 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 authors combine ideas from Takens’s embedding theorem and ensemble Kalman filtering to infer the hidden dynamics 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, highlights our key contributions, and proposes possible directions for future work.
2. General context
a. Model-driven data assimilation
In (1), 
Data assimilation aims to reconstruct the state sequence 
b. Data-driven data assimilation
Henceforth, this state-space model will be referred to as AnDA. A sequential and stochastic data assimilation scheme including filtering and smoothing, is used involving different Monte Carlo realizations of the state at each assimilation time. We sketch the proposed AnDA methodology for one realization in Fig. 1.
The evolution in time of one particle or member. The catalog implicitly represents the dynamics of the system from exemplars of historical datasets. The observations are shown by black asterisks, and their variance is shown by the corresponding error bar.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0441.1
The analog forecasting operator 
Given a catalog 
3. Analog forecasting strategies
a. Analog forecasting operator
Several ideas can be explored to define the analog forecasting operator 
Figure 2 shows an illustration of the three analog forecasting operators used in this work. Hereafter, we denote the forecasted state as 
- Locally constant analog operator: for the Gaussian case, the forecasted state is sampled from a Gaussian distribution whose mean
where
- Locally incremental analog operator: instead of considering a weighted mean of the K successors as in the locally constant operator, we consider the value of the current state plus a weighted mean of the K increments
where
- Locally linear analog operator: we fit a multivariate linear regression between the K analogs of the current state and their corresponding successors using weighted least squares estimates (see Cleveland 1979). The regression gives slope
with
A simplified illustration of the considered analog forecasting strategies in the case of two analogs (nearest neighbors). Two situations for the state 
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0441.1
The choice of one operator over another depends mostly on the available computational resource and the complexity of the application. Locally constant and locally 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 expense, however, of the requirement of a larger number of analogs to fit the regression (Hansen 2000). The locally 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 forecasting operator. For a given state 
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 analogs 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 require 
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 filtering and smoothing posterior likelihoods, respectively, 
a. Analog ensemble Kalman filter and smoother (AnEnKF/AnEnKS)
Ensemble Kalman filters (EnKF) and smoothers (EnKS) (Burgers et al. 1998; Evensen 2007) are particularly 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 et al. (1998) in which observations are treated as random variables. The AnEnKF algorithm starts at time 
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 
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 perturbing observations with noise (Bowler et al. 2013). Besides, the work of Hoteit et al. (2015), where the authors address this issue, suggests that the stochastic EnKF is worth a reevaluation for oceanic and atmospheric 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 (equivalent to members in the terminology used for ensemble Kalman filters).
In theory, particle smoothers may also be considered. Different strategies have been proposed in the past but they showed numerical instabilities in preliminary experiments 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 catalogs 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
b. Experimental details
The considered experimental setting is as follows. To avoid divergence of the filtering methods, we use 
The analog forecasting operator involves two free parameters, namely, K the number of nearest neighbors and λ the scale parameter of the Gaussian kernel in (4). Two strategies can be considered for K: either a predefined number of nearest neighbors, or a predefined threshold on distance 
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 analogs, we consider 
Results of the analog forecasting performance as a function 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 corresponds to a persistent prediction over time.
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0441.1
Figure 3 also compares local and global analog strategies. When using locally constant operator, local analogs are always better than global analogs. Searching for nearest neighbors on 40-dimensional vectors results most likely in irrelevant analogs. This affects heavily the locally constant operator more than the two other operators, since it computes a weighted mean of their associated successors. The locally constant operator also limits novelty creation in the dynamics by always dragging the forecast near the mean of the K successors, 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 prediction 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 using 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 assimilation experiment clearly outperforms the global analog data assimilation experiment.
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 
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0441.1
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 forecasting operator gives the best reconstruction performance. We noticed that the AnPF suffers in the 40-dimensional Lorenz-96 system from sample impoverishment and degeneracy. Despite additional experiments 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.
RMSE of the reconstruction of Lorenz-96 state evolution using different forecasting strategies and data assimilation techniques. The catalog size corresponds to 
d. Experiments with Lorenz-63 model
1) Experiment 1
In the proposed AnDA, the size of the catalog is expected to be a critical parameter. For Lorenz-63 dynamics, we conducted different AnDA experiments varying the size of the catalog 
Reconstruction of Lorenz-63 trajectories for different catalog sizes in the analog data assimilation procedures, 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 observations (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 
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0441.1
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), parameters γ and β define the center of the two attractors whereas σ controls the shape of the trajectories. In Fig. 6, we depict trajectories using three sets of parameters with different values for σ: 
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: 
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0441.1
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 parameterization (red, parameterization 
3) Experiment 3
Whereas previous experiments consider catalogs produced from noise-free trajectories, here we evaluate the sensitivity of the AnDA procedures when the catalog may involve noisy trajectories of the considered system. Acquisition systems typically involve such noise patterns, which may relate for instance to both environmental 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 
Results of the reconstruction of Lorenz-63 trajectories from noisy catalogs: (left) examples of noisy Lorenz-63 trajectories for different noise levels: 
Citation: Monthly Weather Review 145, 10; 10.1175/MWR-D-16-0441.1
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 assimilation procedures with the locally linear operator with a catalog size of 
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 available, AnDA can both support or compete with classical data assimilation schemes. As a proof concept, we demonstrate the relevance of the proposed methodology to retrieve the chaotic behavior of the Lorenz-63 and Lorenz-96 models. We performed numerical experiments to evaluate critical aspects of the method, especially the relevant combinations of analog forecasting strategies and of stochastic filters as well as the exploitation of noisy and noise-free catalogs.
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 research, the user can conduct the different experiments shown in this paper.
Overall, the reported results demonstrate the relevance 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 simulations 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 elements in a catalog shall grow exponentially with the intrinsic dimension of the state to guarantee the retrieval of analogs at a given precision. This makes unrealistic 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 systems. By contrast, local analogs provide a means to decompose 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 explanation for the much better performance reported for the local analog data assimilation for Lorenz-96 dynamics using catalogs of about a million of exemplars (Fig. 4). For real-world applications to high-dimensional systems, for instance to ocean and atmosphere dynamics, 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 expected 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 important 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 requested exemplars in the global catalog to learn the forecast operator and the spread of the prediction.
We believe that this study opens new research avenues 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 observational 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 question. In addition to advanced particle filters as proposed in Van Leeuwen (2010) and Pitt and Shephard (1999), one might also benefit from the straightforward applications 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 assimilation 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 performance 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 Chapron, Valérie Monbet, and Anne Cuzol. The authors would also like to thank Phi Viet Huynh for his valuable contribution to both the AnDA Matlab toolbox 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 operations 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), appears necessary. These structures speed up the K-NN search, at the expense of an approximate search for nearest neighbors. Let us denote by D the 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 
Let 
- Global analogs:
- Local analogs:
We present now the operational count for one ensemble member (or particle) involved in the forecasting, for both global and local analogs. In each case, we distinguish the computational cost of the creation of the K-d trees and the search of K nearest neighbors:
Global analogs:
Creation of the K-d tree:
Search for K global analogs:
Local analogs:
Creation of D K-d trees (for every dimension in D):
Search for K local analogs of component-wise neighborhood ν:
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 D small local K-d trees). Keep in mind that we need to have 
Let us take an example using the Lorenz-96 model: 
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