**Research article**
23 Dec 2021

**Research article** | 23 Dec 2021

# Inferring the instability of a dynamical system from the skill of data assimilation exercises

Yumeng Chen Alberto Carrassi and Valerio Lucarini

^{1,3},

^{1,2,3},

^{3,4}

**Yumeng Chen et al.**Yumeng Chen Alberto Carrassi and Valerio Lucarini

^{1,3},

^{1,2,3},

^{3,4}

^{1}Department of Meteorology and NCEO, University of Reading, Reading, UK^{2}Department of Physics and Astronomy “Augusto Righi”, University of Bologna, Bologna, Italy^{3}Centre for the Mathematics of Planet Earth, University of Reading, Reading, UK^{4}Department of Mathematics and Statistics, University of Reading, Reading, UK

^{1}Department of Meteorology and NCEO, University of Reading, Reading, UK^{2}Department of Physics and Astronomy “Augusto Righi”, University of Bologna, Bologna, Italy^{3}Centre for the Mathematics of Planet Earth, University of Reading, Reading, UK^{4}Department of Mathematics and Statistics, University of Reading, Reading, UK

**Correspondence**: Yumeng Chen (yumeng.chen@reading.ac.uk)

**Correspondence**: Yumeng Chen (yumeng.chen@reading.ac.uk)

Received: 02 Jul 2021 – Discussion started: 12 Jul 2021 – Revised: 13 Oct 2021 – Accepted: 17 Nov 2021 – Published: 23 Dec 2021

Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov–Sinai entropy and perform numerical experiments on the Vissio–Lucarini 2020 model, a recently proposed generalization of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

We split the Introduction into three parts. The first two are proper introductory discussions providing the context. In part three, we provide the motivations and describe the goals of the present work.

## 1.1 Lyapunov vectors and related measures of chaos in a nutshell

The dynamics of several natural systems, including the atmosphere and the
ocean, are characterized by chaotic conditions which, roughly speaking,
describe the property that a system has sensitivity to initial states. This
means that, even in the presence of a perfect model, small errors in the
initial conditions will grow in size with time, until the forecast becomes de
facto useless (Kalnay, 2002)^{1}. A mathematically sound technique for studying the sensitivity to
initial conditions of a system amounts to studying the properties of its
tangent space. In particular, under fairly general mathematical conditions for
a deterministic *n*-dimensional system whose asymptotic dynamics takes place
in a compact attractor, one can define *n* Lyapunov exponents (LEs)
${\mathit{\lambda}}_{\mathrm{1}},\mathrm{\dots},{\mathit{\lambda}}_{n}$, which are the asymptotic rates of amplification
or decay of infinitesimally small perturbations with respect to a reference
trajectory. Usually, the LEs are ordered according to their value, with
*λ*_{1} being the largest. Unless the system feature symmetries, all the
LEs are distinct, and in the case of continuous time dynamics, one of them
vanishes correspondingly to the direction of the flow and defines the neutral
tangent space. Once ordered from the largest to the smallest, the sum of the
first *k* LEs gives the asymptotic growth rate of a *k*-volume element defined
by *k* displaced infinitesimally nearby the reference trajectory plus the
reference trajectory itself. Additionally, if ${\mathit{\lambda}}_{{n}_{\mathrm{0}}}$ denotes the
smallest non-negative LE, in many practical applications one can estimate the
Kolmogorov–Sinai entropy (or metric entropy) *σ*_{KS}, which
defines the rate of creation of information of the system due to its
instabilities, as ${\mathit{\sigma}}_{\text{KS}}={\sum}_{i=\mathrm{1}}^{{n}_{\mathrm{0}}}{\mathit{\lambda}}_{i}$ (Pesin's identity). Finally,
it is possible to use the spectrum of LEs to define a notion of dimension for
the attractor of a chaotic system. The Kaplan–Yorke conjecture, which
follows from the estimate of the rate of growth of the infinitesimal
*k* volume, indicates that the information dimension of a chaotic attractor is
given by ${D}_{\text{KY}}=p+{\sum}_{i=\mathrm{1}}^{p}{\mathit{\lambda}}_{i}/\left|{\mathit{\lambda}}_{p+\mathrm{1}}\right|$, where *p*
is the largest index such that ${\sum}_{i=\mathrm{1}}^{p}{\mathit{\lambda}}_{i}\ge \mathrm{0}$. In systems
where the phase space contracts (the large class of dissipative systems), one
has *D*_{KY}<*n*. Roughly speaking, larger values of *λ*_{1}, of
*σ*_{KS}, and of *D*_{KY} are associated with conditions of
high instability and low predictability for the flow. This is clearly an
extremely informal presentation of some of the features and properties of the
LE; see Eckmann and Ruelle (1985) for a now classic discussion of these topics.

It is possible to associate each LE with a physical mode. Ruelle (1979) proposed the idea of performing a covariant splitting of the tangent linear space such that the basis vectors are actual trajectories of linear perturbations. The average growth rate of each of the covariant Lyapunov vector (CLVs) equals one of the LE. This idea was first implemented by Trevisan and Pancotti (1998) for studying the properties of the Lorenz 1963 model (Lorenz, 1963). Separate algorithms for the computation of CLVs were proposed in Ginelli et al. (2007) and Wolfe and Samelson (2007); see the recent comprehensive review by Froyland et al. (2013). Note that the CLVs corresponding to the positive (negative) LEs span the unstable (stable) tangent space. Recently, Lyapunov analysis of the tangent space was the subject of a special issue edited by Cencini and Ginelli (2013) and the book by Pikovsky and Politi (2016). Detailed Lyapunov analyses of geophysical flows on models of various levels of complexity have been recently reported (e.g. Schubert and Lucarini, 2015; Vannitsem and Lucarini, 2016; Vannitsem, 2017; De Cruz et al., 2018).

## 1.2 Data assimilation in chaotic systems: the signature and the use of chaos

The properties of the dynamical models have large implications on data assimilation (DA; Asch et al., 2016). Data assimilation refers to the family of theoretical and numerical methods that optimally combine data with a dynamical model with the goal of improving the understanding of the phenomenon under study, enhancing the prediction skill, and quantifying the associated uncertainty. Data assimilation has long been studied and developed in the geosciences. It is an unavoidable piece of the operational numerical weather prediction workflow, but it is nowadays used in a growing range of scientific areas (Carrassi et al., 2018).

Numerical and analytic evidence has emerged recently showing that under certain observational conditions (data types, spatio-temporal distribution, and accuracy), the performance of DA with chaotic dynamics relates directly to the instability properties of the dynamical model where data are assimilated. One can thus in principle use the knowledge of the dynamical features to inform not only the design of the DA that better suits the specific application – e.g. how many model realizations for the Monte Carlo based DA methods, or the length of the assimilation window in variational DA – but also the best possible observational deployment.

A stream of research has shed light on the mechanisms driving the response of
the ensemble-based DA (Evensen, 2009), i.e. its functioning and suitability,
when applied to chaotic systems. A recent comprehensive review can be found in
Carrassi et al. (2022), while we succinctly recall the main findings in the
following. In the deterministic linear and Gaussian case with a Kalman filter
(KF) and smoother (KS), it has been analytically proved that the error
covariance matrices converge in time onto the model's unstable–neutral
subspace, i.e. the span of the backward Lyapunov vectors (BLVs) or of the
covariant Lyapunov vectors (CLVs), associated with the non-negative LEs (Bocquet et al., 2017; Bocquet and Carrassi, 2017). These results have been shown numerically to
hold for the ensemble Kalman filter/smoother in weakly non-linear regimes
(EnKF/EnKS; Evensen, 2009) by Bocquet and Carrassi (2017). In practice, for sufficiently well
observed scenarios, the error of the state estimate is fully confined within
the unstable–neutral subspace. Because this subspace is usually much smaller
than the full system's phase space, the above convergence results imply that
an ensemble size as large as the unstable–neutral subspace dimension, *n*_{0},
suffices to achieve satisfactorily performance, i.e. to track the “truth” and effectively reduce the estimation error along with a substantial
computational saving. The impact of instabilities on non-linear DA, in
particular particle filters (PFs, see e.g. Van Leeuwen et al., 2019), has been
recently elucidated in Carrassi et al. (2022): the number of particles needed to
reach convergence depends on the size of the unstable–neutral subspace rather
than the observation vector size.

The picture above slightly changes in the presence of a degenerate spectrum of LEs, which often arises in systems with multiple scales, associated with the presence of coupling between subsystems with different characteristic dynamical timescales (Vannitsem and Lucarini, 2016; De Cruz et al., 2018). The degeneracy is usually concentrated on the unstable–neutral portion of the LE spectrum. In these cases it is necessary to increase the ensemble size to account for all of the degenerate modes (Tondeur et al., 2020; Carrassi et al., 2022). The necessity for going beyond the number of asymptotic unstable–neutral modes is also connected to the local (in phase space) variability of the degree of instability of the system (Lucarini and Gritsun, 2020). The large heterogeneity of the atmospherics's predictability is due to the presence of substantial variability in the number of unstable dimensions (Lai, 1999) of the unstable periodic orbits (UPOs) populating the attractor and defining the skeletal dynamics of the system (Auerbach et al., 1987). As a result of the fact that the orbit of a chaotic system shadows the UPOs supported on the attractor in some of its regions, certain directions of the stable space experience finite-time error growth due to locally important instabilities, causing the need for a larger ensemble size than the number of the unstable-neutral modes.

In the stochastic scenario, noise is usually injected irrespective of the flow-dependent modes of instabilities. Consequently, with a non-zero probability, error is also injected onto stable directions that would not have been otherwise influential in the long term. The trade-off between the frequency of the noise injection and its amplitude on the one hand, and the dissipation rate of stable modes on the other, determines the amplitude of the long-term error along stable modes (Grudzien et al., 2018a). This mechanism implies the need to include additional members in the ensemble to encompass weakly stable modes that experience instantaneous growth Grudzien et al. (2018b).

The knowledge of the LEs and its associated Lyapunov vectors (LVs) can be used to operate key choices in the implementation of ensemble-based DA schemes aimed at enhancing accuracy with the smallest possible computational cost. This point of view is at the core of DA algorithms that operates a reduction in the dimension of the model (e.g. the assimilation in the unstable subspace, AUS; Palatella et al., 2013), of the data (Maclean and Van Vleck, 2021), or both (Albarakati et al., 2021).

## 1.3 This paper: can data assimilation be used to reconstruct the dynamical properties of the system?

While extremely theoretically appealing and practically useful in
low-to-moderate dimensional problems, the use of the dynamically informed DA
approaches is difficult in high dimensions, where even just computing the
asymptotic spectrum of LEs, let alone the very relevant state-dependent local
LEs (LLEs), is very difficult or just impossible. A major but not exclusive
issue is that LE estimation algorithms require computation of the tangent space
of the dynamical system, a task usually unfeasible for high-dimensional
systems, or impossible when the model equations of are not explicitly
accessible. On the other hand, the existence of a relationship between the DA
and the unstable–neutral subspace suggests reversal of the viewpoint: use DA as a tool for estimating the properties of a given system that would be otherwise
very difficult to compute. As a model-agnostic technique, DA, and in particular ensemble-based methods such as the EnKF, can be applied to any
model without the need of computing the tangent space. This makes the EnKF a
potentially powerful instrument to reveal the stability properties of a
dynamical system. This is the goal of this work. Specifically, we shall
investigate whether we can use DA to infer the spectrum of the LEs and the
Kolmogorov–Sinai entropy (*σ*_{KS}) of the system whereby data
are assimilated.

The paper is structured as follows. In Sect. 2, an upper bound of the root mean squared error of the Kalman filter for the linear dynamics in the asymptotic limit is derived. In Sect. 3 we present the Vissio and Lucarini (2020) (VL20) model and its DA setup. The VL20 model is a recently proposed generalization of the Lorenz '96 (Lorenz, 1996) model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables. Additionally, the presence of a qualitatively distinct set of spatially extended variables allows one to consider non-trivial cases of partial observations for DA exercises. Section 4 presents the main results of the paper by comparing the skill of the performed DA exercises with some fundamental measures of instability of the VL20 model. Finally, in Sect. 5 we discuss our results and present perspectives for future investigations.

We are interested in searching for a further relation between the skill of EnKF-like methods applied to perfect (no model error) chaotic dynamics and the spectrum of LEs. We shall build our derivations on the results mentioned in Sect. 1.2 and reviewed in Carrassi et al. (2022). In this section we set ourselves in a linear and Gaussian context, whereby the Kalman filter (KF) yields the exact solution of the Gaussian estimation problem. Linear results will guide the interpretation of the findings in the general non-linear setting with the EnKF.

At time *t*_{k}, let *x*_{k}∈ℝ^{n} and *y*_{k}∈ℝ^{d} be the state and observation
vector, respectively. The (linear) model dynamics ${\mathbf{M}}_{k}\in {\mathbb{R}}^{n\times n}$ and observation model ${\mathbf{H}}_{k}\in {\mathbb{R}}^{p\times n}$ read

The observation noise, *v*_{k}, is assumed to be a zero-mean Gaussian
white sequence with statistics

with *E*[ ] being the expectation operator, *δ*_{k,l} the
Kronecker's delta function, and **R**_{k} the error covariance matrix of
the observations at time *t*_{k}. For the sake of notation clarity, we assume
that the model dynamics is non-degenerate so that all its Lyapunov exponents
are distinct; we note that the extension to the general degenerate case is
possible.

In general, we can write ${\mathbf{x}}_{\mathbf{k}}={\mathbf{M}}_{\mathbf{k}:\mathbf{l}}{\mathbf{x}}_{\mathbf{l}}$ where *t*_{k}>*t*_{l}, and express *M*_{k:l} using the singular value decomposition (SVD)

where **U**_{k:l} and **V**_{k:l} are non-degenerate orthogonal
matrices and **Λ**_{k:l} the diagonal matrix of singular
values. For ${t}_{l}\to -\mathrm{\infty}$ the left singular vectors,
**U**_{k:l}, converge to the backward Lyapunov vectors (BLVs) at *t*_{k},
and, similarly for *t*_{k}→∞ the right singular vectors,
**V**_{k:l}, converge to the forward Lyapunov vectors (FLVs) at
*t*_{l}. The singular values (SVs) in **Λ**_{k:l} converge to
*n* distinct values of the form $\mathbf{\text{diag}}({\mathbf{\Lambda}}_{k:l}{)}_{i}=\mathrm{exp}({\mathit{\lambda}}_{i}({t}_{k}-{t}_{l}))$, in which *λ*_{i} is the Lyapunov exponents
(LEs) in descending order, ${\mathit{\lambda}}_{\mathrm{1}}>{\mathit{\lambda}}_{\mathrm{2}}>\mathrm{\dots}{\mathit{\lambda}}_{{n}_{\mathrm{0}}}=\mathrm{0}>{\mathit{\lambda}}_{n-\mathrm{1}}>{\mathit{\lambda}}_{n}$. The *n*_{0} non-negative LEs identify the *n*_{0}
unstable–neutral modes. For non-uniformly hyperbolic systems, only one of the exponents vanishes.

Let us define the *information matrix* as follows:

which measures the “observability” of the state at *t*_{k}, with
${\mathbf{\Omega}}_{l}={\mathbf{H}}_{l}^{\mathrm{T}}{\mathbf{R}}_{l}^{-\mathrm{1}}{\mathbf{H}}_{l}$ being the
*precision matrix* of the observations mapped to the model
space. Moreover, let ${\mathbf{U}}_{+,k}^{\mathrm{T}}$ be a matrix whose
columns are the *n*_{0} unstable and neutral BLVs of the dynamics
**M**_{k}. Bocquet and Carrassi (2017) have shown that, if the following three
conditions hold, (i) the unstable–neutral modes are sufficiently
observed, such that

with **I**_{n}∈ℝ^{n} being the identity matrix,
(ii) the neutral modes, ** u**, are subject to the stronger
observation constraint,

which implies that each term of the information matrix should be
positive-definite, and (iii) confining the initial error covariance
matrix to the space of FLVs at time *t*_{0}, then the KF forecast error
covariance matrix, ${\mathbf{P}}_{k}^{\mathrm{f}}$, converges asymptotically to
the sequence

In real applications, the convergence (within numerical accuracy) occurs in long but finite times (Bocquet et al., 2017).

The asymptotic mean squared error of the forecast (MSEF) of the KF solution is given by the trace of Eq. (8),

where, for last equality, we made use of the cyclic property of the matrix trace and the orthogonal relation of the BLVs, ${\mathbf{U}}_{+,k}^{\mathrm{T}}{\mathbf{U}}_{+,k}={\mathbf{I}}_{{n}_{\mathrm{0}}}$.

Equation (9) shows evidence that the asymptotic MSEF
depends on the observation constraint through the information matrix (data
accuracy, encapsulated in **R**, while data type and deployment
are encapsulated in **H**) but also on the unstable–neutral BLVs. Despite
this, it is particularly difficult to use Eq. (9) to derive a direct relation between the MSEF and the spectrum of LEs. This is because
${\mathbf{U}}_{+,k}$ is not invertible in general and because one needs to
make specific (often overly simplified) assumptions on the model dynamics and
observations, i.e. on **M**_{k:l}, **H**_{l} and **R**_{l},
in order to get a treatable expression of the information matrix. Alternatively, rather than through a direct relation, we shall seek informative
bounds for the MSEF in terms of the LEs.

Let us substitute the SVD of **M**_{k:l}, Eq. (4), in the information matrix,

For every *t*_{l}, the individual terms in the summation can be written as

We now define the maximum projection of the precision matrix onto the FLVs as

with $\parallel .\parallel $ being the Euclidean norm, and use it to get an upper bound for the inverse of each term, Eq. (12), in the information matrix summation,

The inequality is based on the Löwner partial ordering of
ℝ^{n×n} (i.e. the partial order defined by the convex cone
of positive semi-definite matrices; see, for example, Bocquet et al., 2017, their
Appendix B). We shall use this partial ordering in the following derivations.

By defining the maximum of *β*_{l} across all $\mathrm{0}\le {t}_{l}\le {t}_{k-\mathrm{1}}$ as

we get the following lower bound for the information matrix:

The bound reflects the effect of assimilating observations (the rhs) compared to the unconstrained free model run (the lhs) – note that ${\mathbf{U}}_{k:l}{\mathbf{\Lambda}}_{k:l}^{\mathrm{2}}{\mathbf{U}}_{k:l}^{\mathrm{T}}={\mathbf{M}}_{k:l}{\mathbf{M}}_{k:l}^{\mathrm{T}}$.

Given that ${\mathbf{U}}_{k:l}{\mathbf{\Lambda}}_{k:l}^{\mathrm{2}}{\mathbf{U}}_{k:l}^{\mathrm{T}}$ is symmetric positive definite, we can invoke the aforementioned partial ordering for this class of matrices and further develop the lower bound of the information matrix as

where ${e}^{\mathrm{2}{\mathit{\lambda}}_{\mathrm{1}}({t}_{k}-{t}_{l})}$ is the largest eigenvalue of ${\mathbf{U}}_{k:l}{\mathbf{\Lambda}}_{k:l}^{\mathrm{2}}{\mathbf{U}}_{k:l}^{\mathrm{T}}$. The lower bound of the information matrix in Eq. (16) then becomes

Under the assumption that the assimilation cycle is uniform in time, e.g. $\mathrm{\Delta}t={t}_{k}-{t}_{k-\mathrm{1}}={t}_{k-\mathrm{1}}-{t}_{k-\mathrm{2}}=\mathrm{\dots}={t}_{\mathrm{1}}-{t}_{\mathrm{0}}$, the summation of the diagonal matrices ${\mathbf{D}}_{l}^{-\mathrm{1}}$ coincides with a geometric series with known sums:

By using the lower bound, Eq. (19), and the orthogonality of the BLVs, ${\mathbf{U}}_{+,k}^{\mathrm{T}}{\mathbf{U}}_{+,k}={\mathbf{I}}_{{n}_{\mathrm{0}}}$, we get a lower bound for the information matrix projected onto the unstable–neutral subspace:

We can thus finally use Eq. (21) in the expression of the MSEF, Eq. (9), and derive the following upper bound:

This upper bound incorporates the key players shaping the relation between the
KF estimation error and the model dynamics. The presence of *β*_{k} and
Δ*t* reflects the observation modulation of the MSEF: the stronger the
data constraint, the smaller *β*_{k} and Δ*t*. The signatures of the
model instabilities are in the term *n*_{0}, the size of the unstable–neutral
subspace, and in *λ*_{1}, the error growth rate along the leading mode of
instability, both related directly to the amplitude of the bound. Under a
Bayesian interpretation, the factor *β*_{k} can be seen as the likelihood of
data and the remaining terms in the bound altogether as the prior
distribution. Note that, if the dynamical model is stable (and independently
of the data), *λ*_{1}=0, ${s}_{i}=\frac{\mathrm{1}}{k}$, and the MSEF goes to zero
asymptotically.

As alluded to at the beginning of the section, a direct expression (e.g. an
equality in place of a bound) relating the model instabilities and the error
can be obtained under strong simplified and somehow unrealistic assumptions on
the form of the model dynamics and of the data, for example, if the linear
dynamics **M**, the observation covariance matrix **R**, and the
observation operator **H** are all scalar matrices. With no need of
these assumptions, and with more generality, the upper bound,
Eq. (26), indicates that the MSEF is determined by a
convolution of model dynamics and observation error.

In the next sections we will perform numerical experiments under controlled scenarios to investigate the conditions for which the bound holds. In particular, we will study the conditions leading to the smallest possible upper bound, such that the output of a converged DA, i.e. its asymptotic MSEF, can be used to infer the LE spectrum of the model dynamics.

## 3.1 The Vissio–Lucarini 2020 model

Our test bed for numerical experiments is the low-order model recently developed by Vissio and Lucarini (2020), hereafter referred to as the VL20. The VL20 model is an extension of the classical Lorenz 96 model (Lorenz, 1996) that includes additional thermodynamic variables. The model is given by the following set of *n* ordinary differential equations (ODEs) (with *n* being an even integer):

where *X* represents the momentum, *θ* is the thermodynamic variable, and the subscript $\mathrm{1}\le i\le n/\mathrm{2}$ is the grid point index. The model is spatially periodic, and the boundary condition is expressed as

In the VL20 model it is possible to introduce a notion of kinetic energy $K={\sum}_{=\mathrm{1}}^{n/\mathrm{2}}{X}_{j}^{\mathrm{2}}/\mathrm{2}$ and potential energy $P={\sum}_{=\frac{n}{\mathrm{2}}}^{n}{\mathit{\theta}}_{j}^{\mathrm{2}}/\mathrm{2}$. Additionally, the model features an energy cycle that allows for the conversion between the kinetic and potential forms and for introducing a notion of efficiency. The parameter *α* modulates the energy transfer between the two forms, while *γ* controls the energy dissipation rate, and *F* and *G* are external forcing defining the energy injection into the system. The model's evolution can be written as the sum of a quasi-symplectic term, which conserves the total energy, and of a gradient term, which describes the impact of forcing and dissipation. In the turbulent regime, the VL20 allows for propagation of signals in the form of wave-like disturbances associated with unstable waves exchanging energy in both potential and kinetic form with the background.
In terms of energetics, the difference between the L96 and the VL20 model mirror the one between a one-layer and a two-layer quasi-geostrophic model because the former features only barotropic processes, while the latter features the coupling between dynamical and thermodynamic processes via baroclinic conversion, which makes its dynamics much more complex (Holton and Hakim, 2013). The VL20 model is thus a very good test bed for research in DA, a further step toward realism from the very successful L96.
Further details on the model as well as an extensive analysis of its dynamical and statistical properties can be found in Vissio and Lucarini (2020).

In all the following experiments, we set *n*=36, implying both model variables *X* and *θ* have 18 components, and consider three model configurations differing in the values of the external forcings: $F=G=\mathrm{10}$, $F=\mathrm{10},G=\mathrm{0}$, and $F=\mathrm{0},G=\mathrm{10}$. Unless otherwise stated, the model runs with the default parameters $\mathit{\alpha}=\mathit{\gamma}=\mathrm{1}$, and it is numerically integrated using the standard fourth-order Runge–Kutta time stepping method with a time step Δ*t*=0.05 time units. A summary of the model instability properties with the chosen configurations is given in Table 1.

## 3.2 Data assimilation setup

Synthetic observations are generated according to Eq. (2) by
sampling a “true” solution of the VL20 model, Eq. (27), and then
adding simulated observational error from the Gaussian distribution
𝒩(**0**,**R**). Observational error is assumed to be
spatially uncorrelated so that the error covariance, **R**, is a
diagonal matrix, and we observe the model components directly, implying that
the observation operator is linear and under the form of a matrix, $\mathbf{H}\in {\mathbb{R}}^{p\times n}$. The observation error variance is set to be
5 % of the variance (i.e. the squared temporal variability) of
the climatology of the corresponding state vector component such that

Linking the observation error to the model variance makes the setup more
realistic, but it ties the error amplitude to the choice of the model
parameters. For example, the model's state vector variance gets very small
when the dissipation is strong, potentially making the **R** matrix
degenerate. Under such circumstances, the corresponding entries in
**R** are set to $\mathrm{5}\times {\mathrm{10}}^{-\mathrm{6}}$.

In line with previous studies (e.g. Carrassi et al., 2022), we work with deterministic EnKFs, whereby it is possible to study the filter performance in relation to the model instabilities without the inclusion of additional noise that is inherent to stochastic versions of the EnKFs (Evensen, 2009). In particular we choose to use the finite-size ensemble Kalman filter (EnKF-N; Bocquet et al., 2015) because it automatically computes the required covariance inflation, thus saving us from running many inflation tuning experiments. The initial conditions for the ensemble are sampled from the Gaussian distribution $\mathcal{N}({\mathit{x}}_{\mathrm{0}}^{t},\mathbf{R})$, with the ${\mathit{x}}_{\mathrm{0}}^{t}$ being the “truth” at *t*_{0}: this choice signifies that the initial condition error is taken to be equal to the observational error.

The performance of DA experiments will be assessed primarily using the root mean square error of the analysis, normalized by the observation variance:

The nRMSEa measures the analysis error independent from observation error,
allowing for a multivariate assessment of the performance. Unless otherwise
stated, observations are taken at every time step, and the experiments last
2000 model time units. With this setting, an experiment comprises 40 000
DA cycles, and when computing time averages of the nRMSEa, we only consider
the last 500 model time units. Finally, and again unless otherwise stated,
we shall adopt *N*=40 ensemble members in the EnKF-N.

Our analysis focuses on the relation between observational design and filter
accuracy and the relation between the model instabilities and the filter
accuracy. By exploiting the novel dynamical–thermodynamical feature of VL20
over its L96 precursor, we will also study the EnKF-N under observational
scenarios that alternatively measure the dynamical variable, **X**, or,
the thermodynamical one, ** θ**.

## 4.1 Data assimilation with the VL20 model: general features

Figure 1 shows the time series of the nRMSEa over the
first 100 time units, for the three main model configurations under
consideration. In all cases, the error drops to below 20 % of the
observational error after approximately 10 time units (corresponding to
200 DA cycles) and then fluctuates with oscillations that only sporadically
lead the error to exceed 0.3. The configuration, $(F,G)=(\mathrm{10},\mathrm{0})$ (red line),
attains the smaller error, while the other two configurations (blue and purple
lines respectively) show comparable error levels slightly larger than
configuration $(F,G)=(\mathrm{10},\mathrm{0})$. Recall that in the configuration of
$(F,G)=(\mathrm{10},\mathrm{0})$, the model is not thermodynamically forced (*G*=0), and is also
slightly stabler than in the other two configurations (cf.
Table 1).

The first connection between the filter performance and the model
instabilities is drawn from Fig. 2 that shows the
nRMSEa as a function of the number of the ensemble members. In line with
previous findings for uncoupled univariate (Bocquet and Carrassi, 2017) and with coupled
models (Tondeur et al., 2020), Fig. 2 shows that, even with a
multivariate model, the error converges to very low values as soon as the
ensemble size exceeds the number of unstable–neutral modes, *n*_{0}, and that it
does not further decrease by adding more members. This behaviour is possible
because error evolution is bounded to be linear or weakly non-linear. This
means that one can in principle induce linearity intentionally in the error
evolution to meet the aforementioned relation between filter accuracy and
ensemble size and use it to infer the number of unstable–neutral modes. In a
DA experiment, a “practical” way to achieve this is by strengthening the
observational constraint (i.e. by increasing the measurements spatial and
temporal density); here we observe the full system's state at every time step.

As mentioned above, the VL20 model represents four main physical mechanisms: (i) conversion between kinetic and potential energy, (ii) the energy injection from external
forcing, (iii) the advection, and (iv) the dissipation. Although these
processes all participate in the evolution of the model, the non-linear interplay
cannot be straightforwardly disentangled. Nevertheless, we shall try to refer
to them when interpreting the outcome of the DA experiments. In particular, in
each experiment we will attempt to identify the prevailing mechanism over the
aforementioned four. We perform three experiments, where we observe the full
system state (i.e. **H**=**I**_{36}), or alternatively
**X** or ** θ** alone (implying in both cases
$\mathbf{H}\in {\mathbb{R}}^{\mathrm{18}\times \mathrm{36}}$). Results are given in
Fig. 3, that displays the time-averaged nRMSEa
(global or for the dynamics or thermodynamics only) over a range of the
coefficient

*α*that modulates the energy transfer rate.

Overall, and as expected, the analysis error is smaller in the observed
variables (cf. the left and middle columns and corresponding colour lines) and
attains the smallest level when **X** and ** θ** are
simultaneously observed (right column). Nevertheless, a few remarkable points
can be raised. First, when the system is fully observed, for large values of

*α*(i.e. for large conversion between available potential energy and kinetic energy) the skills in

**X**and

**become very similar (right column): we conjecture this to be a consequence of the system getting more evenly turbulent with all variables sharing a similar internal variability as energy is exchanged efficiently between the kinetic and potential form. Second, for small values of**

*θ**α*(i.e. small energy conversion), the effect of external forcing becomes dominant and determines the analysis error of

**X**and

**(last column in Fig. 3). For instance, whenever the momentum is externally forced (**

*θ**F*=10), the error in

**X**is systematically smaller than in

**(first and second rows of the last column): DA is more effective in controlling the dynamics than the thermodynamics, even when they are subject to the same observational constraint. The situation is somehow reversed when only the thermodynamical variables are forced ($F=\mathrm{0},G=\mathrm{10}$): the analysis error of the momentum and the thermodynamic variable is undifferentiated. With small values of**

*θ**α*and no forcing for the momentum, the non-linear momentum advection is limited by the small magnitude of the momentum that is not able to activate much the dynamical variables, so that we observe similar analysis error between the thermodynamics variable and the momentum.

Finally, the effect of the energy transfer and advection can be revealed by
looking at the partially observed experiments (left and middle columns). Both
mechanisms involve the momentum, making it more efficacious to observe
**X** than ** θ**, especially in the energy-transfer-dominated regime (large

*α*). However, in an advection-dominated regime (small

*α*), if

**is unobserved,**

*θ***X**has limited capability to constrain the error in

**due to the weak feedback from**

*θ***to**

*θ***X**. On the other hand, observing

**reduces error in**

*θ***X**via the accurate estimate of the advection process of

**(see middle column).**

*θ*Further insight into the role of the driving (unstable) variable and on the interplay between the prevailing physical mechanisms and the analysis error is given by looking at the CLVs (Kuptsov and Parlitz, 2012). In Fig. 4 we show at (normalized time-averaged) absolute amplitude of CLVs components along the state vector: it tells us which variable type/component has the larger influence on each CLV, thus indicating what processes participate more in a specific direction of error growth/decay.

As discussed above, changes in the value of *α* lead to shifting between the advection-dominated regime and an energy mixing one: these two regimes are portrayed
in Fig. 4, by selecting *α*=0.4 and *α*=2.2. Moreover, these two values of *α* correspond roughly to those
giving the largest differences in nRMSEa between the momentum and the
thermodynamic variables (cf. left and middle panels of
Fig. 3). For small energy exchange (*α*=0.4 –
left column in Fig. 4), the model instabilities are
driven by the external forcing, with the driving variable being the one where
energy is injected. This is clearly visible when comparing the amplitudes of
CLVs between **X** and ** θ** in the left panel of
Fig. 4: larger amplitudes of the unstable–neutral CLVs
are found in the forced variables. When the momentum and the thermodynamics
are equally forced (blue lines), the amplitude of the unstable–neutral CLVs
for

**X**and

**is close to each other. The non-linear advection process intensifies the error growth, especially for**

*θ***X**. The non-linear advection and the momentum are of lesser importance if the momentum is not forced ($F=\mathrm{0},G=\mathrm{10}$ – purple lines), while the thermodynamic processes control both the stable and unstable subspace dominantly. The thermodynamic variable on the stable subspace acts as an energy sink to stabilize the dynamical system. The effect of the thermodynamics is shown noticeably by the large relative amplitude of the CLVs of the thermodynamic variable in the stable subspace when the momentum is directly forced (

*F*=10 – blue and red line).

The behavior changes substantially when the energy exchange is the dominant physical mechanism (*α*=2.2 – right column). This causes a stronger
mixing across the model variables so that both **X** and
** θ** play a comparable role in the unstable–neutral
components of the CLVs, leading to similar amplitude of the CLVs for all types
of forcing. Remarkably, the effect of the energy conversion also applies to
the stable components of the CLVs, leading to similar amplitude of the CLVs
between

**X**and

**.**

*θ*The results in Fig. 4 reveal the effect of the prevailing physical mechanisms on determining the driving unstable variables. The figure suggests what variables should in principle be controlled by targeting measurements on the portion of the system's state vector with larger amplitude on the unstable–neutral CLVs.

Along with *α*, the energy in the system is modulated by the dissipation parameter, *γ*: larger values of *γ* imply an efficient removal of energy from the system, thus reducing the system's variability of both the potential and kinetic energy. At dynamical level, the parameter *γ* controls the contraction of the phase space as the sum of all Lyapunov exponents (equal to the average flow divergence) is −*n**γ*. Hence, one expects that larger values of *γ* correspond to weaker instability for the model, as in the case of the classical L96 model (Gallavotti and Lucarini, 2014). Figure 5 is the same as Fig. 3 but for the dissipation, *γ*. Overall, we see that, with large *γ*, the system's internal variability reduces, and we find similar small errors in both **X** and ** θ**. For weaker dissipation, the momentum is better controlled than the thermodynamics.
With partial observations (left and middle columns), the error is much larger than in the corresponding fully observed cases. Similar to Fig. 3, the momentum is generally better reconstructed by the DA than the thermodynamics, although observing the latter appears more efficacious (i.e. it leads to smaller analysis error) than observing the momentum. We think that this is due to the prevailing mechanism being the advection of the thermodynamics given that

*α*=1 in these experiments (cf. also Fig. 3). The amplitudes of the CLVs along the state vector are studied in Fig. 6. We consider the cases

*γ*=0.4 and

*γ*=1.0 for which the difference in the nRMSEa between

**X**and

**is roughly the largest (cf. Fig. 5).**

*θ*With the leading CLVs strongly affected by the external forcing, the amplitude
of the CLVs along the system's components is similar to the pattern of low
energy exchange rate in Fig. 4 where *α*=0.4,
even though here, *α*=1. This confirms that the dynamical regime of our
experiments lies in the regime dominated by advection, and dissipation does
not mix the kinetic and potential energy diffusely as the energy exchange, but
rather it uniformly removes both types of energy without changing the
prevailing physical mechanism. This is also reflected in the consistently low
nRMSEa for the observed variable when varying dissipation rates in the
partially observed experiments (see Fig. 5). The
decreasing analysis error in Fig. 5 corresponds to
the increases of *γ*, which reduces the dimension of the unstable–neutral
subspace with increased relative importance of forced variables in the
unstable–neutral subspace as the fast energy removal reduce the amount of
energy mixing.

The results of Sect. 4.1 confirm the relation between the performance of DA (in terms of analysis error) and the dimension and characteristics of the unstable–neutral subspace. In particular, we conclude that successful DA relies on controlling the error in the unstable–neutral subspace by observing the variable that drives the error growth. The VL20 model enabled the investigation of the relation between the DA and the specific physical mechanisms such as the advection, the energy transfer among dynamics and thermodynamics, and the dissipation. The effect of DA (i.e. its efficacy) is strongly influenced by the form of the coupling between the unobserved and the observed variables that is in turn shaped by the prevailing physical mechanisms.

## 4.2 Inferring the degree of model instability with data assimilation

The derivation in Sect. 2 shows that, in the linear setting, the assimilation error is asymptotically bounded from above by a factor dependent on the observation error, the first LE and the number of unstable–neutral modes of the underlying forecast model. In this section we explore the extent to which this result holds in a non-linear scenario whereby the observational constraint is strong enough such that the error evolution is maintained approximately linear or weakly non-linear. We shall make use of numerical experiments with the VL20 model.

A first insight on the existence of a direct relation between the model instabilities and the skill of the EnKF-N is already provided in Figs. 3 and 5. They display the Kolmogorov–Sinai entropy, *σ*_{KS} (black line), and the first LE, *λ*_{1} (amplified by a factor of 3 – grey line), along with the nRMSEa (discussed in Sect. 4.1). Even just by visual inspection, the figures clearly show the high correlation between the analysis error and both the *σ*_{KS} and *λ*_{1}.

The nature of this relation is further studied in
Fig. 7, which shows scatter plots between the nRMSEa (with black markers) and ${\mathit{\sigma}}_{\text{KS}}/{\mathit{\lambda}}_{\mathrm{1}}$ in a log–log
scale. Data points relative to experiments with forcing values are given in the panels' legends and with varying energy exchange and dissipation rates in the
range $(\mathit{\alpha}\times \mathit{\gamma})\in [\mathrm{0.1},\mathrm{3})\times [\mathrm{0.3},\mathrm{1.8})$. Here, the
EnKF-N assimilates the full state vector at each time step. The analysis error
appears in a linear relationship with either *σ*_{KS} or
*λ*_{1}, as long as $\mathrm{ln}\left(\text{nRMSEa}\right)\ge -\mathrm{4}$. The existence of such an approximate relationship provides the possibility to infer
*σ*_{KS} and/or *λ*_{1} based on the outcome of DA.

The scatter plots also demonstrate the validity of the upper bound (red
markers) of Eq. (26) in Sect. 2. To
compute the bound we set the coefficient related to observation, *β*=1, as
it is compared to analysis errors normalized by observational error. The
nRMSEa is bounded by the theoretical upper bounds for most of the model
configurations considered. The linear relationship of the upper bound can be
explained by its formulation in Eq. (26), where the exponent
${e}^{\mathrm{2}{\mathit{\lambda}}_{\mathrm{1}}\mathrm{\Delta}t}-\mathrm{1}$ can be approximated as 2*λ*_{1}Δ*t* if
2*λ*_{1}Δ*t* is sufficiently small. The spread of upper bounds points
for given *λ*_{1} values (left panel) reflects the various values of *n*_{0} under
similar *λ*_{1} values. The better correspondence (narrower spread of the
scattered points) in the plane nRMSEa with *σ*_{KS} (right
panel) shows the importance of including both the dominant error growth rate,
*λ*_{1}, and the unstable-subspace dimension, *n*_{0}, – both present in
*σ*_{KS} – to better characterize the system's instabilities. The
correspondence between *σ*_{KS} and the theoretical upper bound
could also be a result of the relation between *λ*_{1} and
*σ*_{KS} as in a highly turbulent case, there is a linear relation
between *λ*_{1} and *σ*_{KS} (Gallavotti and Lucarini, 2014).

The linear relation does not hold for numerical experiments when
$\mathrm{ln}\left(\text{nRMSEa}\right)<-\mathrm{4}$ (see the black markers' distribution in the panels'
inset). We explain this behaviour in the following way. The wide clouds of points correspond to all model configurations with very small *σ*_{KS} and *λ*_{1}. In these quasi-stable dynamics, the
error growth in between successive analysis is very little, with occasional
error decay. The observational error, which is random and white in time, will
be often larger than the forecast error and will dominate the analysis error,
thus breaking its direct dependence on the instability-driven forecast
error. In addition, in the weakly unstable model configurations, the most
unstable direction behaves almost like a neutral mode, which also breaks the
assumptions of the theoretical upper bound for unstable models. In fact, we do
not show the weakly unstable results with *σ*_{KS}<1 for both
*σ*_{KS} and *λ*_{1}.

The error bounds in Sect. 2 rely on the assumption of linear error evolution, a condition that we met in our experiments thanks to a strong observational constraint, with (synthetic) measurements covering the full state vector at each time step. These conditions are rarely achievable in practice, so it is relevant to explore how results will change with a lighter observational constraint. There are three direct ways to achieve this by acting on (i) the number/type of measurements, (ii) the measurement error, and/or (iii) the temporal frequency.

The effect of the first is studied in Fig. 8 that is similar to Fig. 7 but for DA experiments whereby only one of each variable in the VL20 is observed.

The impact of partially observing the system causes the emergence of a weakly
quadratic relationship between the analysis error and either
*σ*_{KS} or *λ*_{1}. However, the analysis error is still
uniquely and monotonically related to them, especially for
*σ*_{KS}. A quadratic law requires one additional coefficient to
be determined compared to a linear law; yet the mere existence of such a law
suggests again that one could in principle infer *σ*_{KS} and/or
*λ*_{1} based on the analysis error. With the relaxed observation
constraint, the analysis error can (and indeed do so in several instances)
exceed the theoretical upper bound. However, the general trend of the
numerical experiments still follows the theoretical upper bound.

We study the effect of changing the amplitude of the observational error in Fig. 9. Results reveal that varying the observation error in the range of 5 %–10 % does not break the quasi-linear relationship between the analysis error and *σ*_{KS} or *λ*_{1}.
The nRMSEa is quite insensitive to the observation variance due to the normalization. Nevertheless, the upper bound limit is not violated as in Fig. 7, and the slope of the nRMSEa from the numerical experiment is remarkably similar to the slope of the theoretical bound.

Finally, the impact of varying the observation frequency is explored in
Fig. 10. It is patent that decreasing the
frequency leads to blurring the linear relation between the analysis error and
the *σ*_{KS} or *λ*_{1}. There is a clear deviation from the
trend of the theoretical upper bound and from the uniqueness of the relation
between analysis error and *σ*_{KS}, as soon as the observational
time interval exceeds the error doubling time (that is inverse related to
*λ*_{1}), and DA error evolves beyond the linear regime. However, for
frequent enough observations, a linear relation similar to the upper bound
appears, and, again, one could in principle deduce *σ*_{KS} and/or
*λ*_{1} based on DA.

The larger sensitivity to the observation frequency than to observation noise
(cf. Figs. 9 and 10) is
a direct consequence of the different effects these two factors have in
determining the degree of non-linearity of the error. This is explained, for
the L96 model, in the Appendix of Bocquet and Carrassi (2017) using a dimensional
analysis. The key point is that the observation frequency modulates directly
the magnitude of the non-linear term in the model, namely the
advection. Decreasing the observation noise, while effectively reducing the
analysis error, is not sufficient to keep the error dynamics linear if Δ*t*>0 and the model is chaotic.

It is sometimes of great importance to be able to obtain information on the instability of a system of interest by performing data analysis of suitably defined observables. This is of key importance when one does not have direct access to the evolution equations of the system or when the analysis of its tangent space is too computationally burdensome. As an example, quantitative information on the degree of instability of a chaotic system can be extracted using extreme value theory by studying the statistics of close dynamical recurrences as well as of extremes of so-called physical observables (Lucarini et al., 2014, 2016). The use of such a strategy has shown great potential for the analysis of geophysical fluid dynamical models in a highly turbulent regime (Gálfi et al., 2017) as well for the understanding of the properties of the actual atmosphere (Faranda et al., 2017; Messori et al., 2017).

In this study, we have addressed this problem by taking the angle of DA. The
relation between DA and the instability of the dynamical system where it is
applied has long been studied (see, for example, Miller et al., 1994; Carrassi et al., 2008), and has been used to design
DA techniques in various field of geosciences
(Carrassi et al., 2022; Albarakati et al., 2021). Here, we have reversed this viewpoint
and investigated the possibility of using DA to infer fundamental quantities
of the underlying dynamics, in particular the Lyapunov exponents, *λ*_{i},
or the Kolmogorov–Sinai entropy (*σ*_{KS}). The basic idea is to
look at DA as a control problem and relate our ability to control the system, ceteris paribus, to its underlying instability. We have leveraged a stream of previous works that set the theoretical foundation and that proved
the convergence of the error covariance of the Kalman filters onto the
unstable–neutral subspace of the dynamical system. Based on this, we derived
here an upper bound of the Kalman filter forecast error, i.e. under the
assumptions of a linear model dynamics and a linear observation operator. The
upper bound is very informative as it relates the error's amplitude to all of
the essential descriptors of the model instabilities on the one hand and of
the DA on the other. These are the dimensions of the unstable–neutral subspace,
*n*_{0}, the first Lyapunov exponent, *λ*_{1}, the frequency of the
observation assimilation, Δ*t*, and the observation error, *β*_{k}. By
properly normalizing the bound by the observation error, it can be written as
a function of the model dynamical properties exclusively.

The existence of a relation between *λ*_{1} or *σ*_{KS} and the DA skill, as well as the validity of the bound, has then been investigated in a non-linear scenario using numerical experiments. We have used the EnKF-N (Bocquet et al., 2015) as a prototype of deterministic EnKF (Evensen, 2009) and the new model developed by Vissio and Lucarini (2020). The VL20 is an extension of the widely used Lorenz 96 model that includes a thermodynamic component. While maintaining all of the virtues of a low-dimensional model suitable for investigations on new methods at low computational cost, VL20 is conceptually much richer than the original L96 model. In particular it allows for the exchange of energy between a kinetic and potential form, which, together with forcing and dissipation, provides the fundamental framework for the Lorenz (1955) energy cycle. Additionally, as advection impacts temperature-like variables, one can observe the emergence of more complex dynamical behaviours. By changing the value of its key parameters, and in particular of those determining its forcing and dissipation, the model explores various dynamical regimes, ranging from fixed point, periodic, quasi-periodic, and chaotic behaviour. In terms of DA, the VL20 model has the attractive feature that it includes two qualitatively different set of variables, associated with dynamics and thermodynamics, respectively. Hence, it is possible to explore the problem of having partial observation beyond focusing of the spatial extent of the observations only.

We demonstrate that the skill of the EnKF-N is directly linked to both
*λ*_{1} and *σ*_{KS}. Whenever the error within the EnKF-N
cycles is kept sufficiently linear via a strong observational constraint, the
relation is clearly linear too. By relaxing the observational constraint (by
either reducing the frequency of measurements or by increasing their noise),
deviation from linearity emerges. Nevertheless, the linear relation is very
robust against the level of observational noise (within certain range), while
it turns quadratic once the interval between successive measurements gets too
large and it exceeds the system's doubling time. Similarly, we found out that
the theoretical upper bounds for the errors, derived for a linear system, still
hold, as long as the observational constraint is strong enough, but are then
violated.

The error bound and the linear/quasi-linear relation between the error and
*λ*_{1} or *σ*_{KS} represent two direct ways to infer
*λ*_{1} and *σ*_{KS} by looking at the output of a DA
exercise. First, we can use the bound (Eq. 26) to estimate
*λ*_{1} for a specific dynamical model, based on (normalized) error output
of a DA exercise. This requires the unstable–neutral subspace dimension,
*n*_{0}, that can be obtained, in the case of EnKF-like methods, by looking at
the analysis error convergence for increasing ensemble size; *N*: *n*_{0} will
be equal to ${N}^{*}-\mathrm{1}$, where *N*^{*} is the smallest ensemble size for which the
error reaches its minimum. This procedure will give us an underestimate of
*λ*_{1}. Nevertheless, our results (cf. Fig. 7)
seem to suggest that the amount of the underestimation is small and, notably,
constant across a range of different model configurations (and thus possibly
quantifiable).

Our numerical experiments indicate a second way to estimate *λ*_{1} or
*σ*_{KS} from the skill of DA. The linear/quasi-linear
relationship between normalized DA error and *λ*_{1} or
*σ*_{KS} (cf. Fig. 7) exists for both the
derived upper bound in Eq. (26) and numerical experiments
and is tested under various observation constraints. The existence of the
relationship for the upper bound implies that the relationship may exist for
other dynamical systems as long as the time between analysis Δ*t* is
sufficiently frequent because the upper bound is based on the assumption of a
(quasi-)linear model. To utilize the relationship for a specific dynamical
system, a few DA experiments using different set of parameters of the
dynamical system are required. A linear relation can be obtained by linear
regression from the selected data, by which a relatively accurate *λ*_{1}
or *σ*_{KS} for other parameters of the dynamical system can be
inferred. Unavoidably, for the selected set of parameters, the method requires
the *λ*_{1} or *σ*_{KS} to be known, which possibly can be
obtained by computational methods such as the one proposed by
Wolfe and Samelson (2007). However, the resulting linear relation can lead to a
computationally efficient approach for other sets of parameters of the
dynamical system with an estimate more accurate than the one from the upper
bound in Eq. (26).

The linear relation between error and *λ*_{1} and *σ*_{KS}
will certainly be more complicated with model errors. From Grudzien et al. (2018a) and
Grudzien et al. (2018b), we know that the KF error covariance will no longer be fully
confined within the unstable–neutral subspace but could maintain projections everywhere and thus also on the stable modes. Those
projections would be asymptotically zero in the absence of model noise. While
this remains to be investigated, we argue that the existence of a clear
monotonic relationship between analysis error and *λ*_{1} will still hold
in the presence of model error. The relation to *σ*_{KS} might
also still stand because the correction would come from weakly stable
modes. However, the conjecture needs to be validated by numerical experiments
that are outside of the scope of this paper.

We are currently considering how these results will change when performing DA
for state and parameter estimation. In this context, a relevant recent study
has shown how the minimum number of ensemble members, *N*^{*}, will need to be
increased to include as many members as the number of parameters to be
estimated (Bocquet et al., 2020). By modifying its parameters, the model's
instabilities' properties will change too, potentially inducing a catastrophic
change (a tipping point) in its long-term behaviour. Data assimilation will
then need to infer the best parameter values to track the data signal and keep
the DA solution in the same region of the bifurcation diagram.

The Python script for the plotting and data assimilation experiments is available at https://doi.org/10.5281/zenodo.5788693 (Chen, 2021), which is also dependent on version 1.1.0 of the Python package DAPPER (https://doi.org/10.5281/zenodo.2029295, Raanes and Grudzien, 2018).

All data used in this paper are generated by Python scripts provided in the code availability section.

YC designed and conducted the experiments and prepared the manuscript. AC and VL both provided the original idea and wrote the manuscript. All authors contributed to the development of the work.

At least one of the (co-)authors is a member of the editorial board of *Nonlinear Processes in Geophysics*. The peer-review process was guided by an independent author, and the editors also have no other competing interests to declare.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are thankful to Patrick Raanes (NORCE, NO) for his support with the use of the data assimilation Python platform DAPPER. Yumeng Chen and Alberto Carrassi are thankful for the funding by the UK Natural Environment Research Council. Valerio Lucarini is thankful for the support from the EPSRC and the EU Horizon 2020. We thank two anonymous reviewers for their valuable comments and suggestions for our paper.

This research has been supported by the National Centre for Earth Observation (grant no. NCEO02004), the Horizon 2020 (TiPES (grant no. 820970)), and the Engineering and Physical Sciences Research Council (grant no. EP/T018178/1).

This paper was edited by Takemasa Miyoshi and reviewed by two anonymous referees.

Albarakati, A., Budišić, M., Crocker, R., Glass-Klaiber, J., Iams, S., Maclean, J., Marshall, N., Roberts, C., and Van Vleck, E. S.: Model and data reduction for data assimilation: Particle filters employing projected forecasts and data with application to a shallow water model, Comput. Math. Appl., in press, https://doi.org/10.1016/j.camwa.2021.05.026, 2021. a, b

Asch, M., Bocquet, M., and Nodet, M.: Data assimilation: methods, algorithms, and applications, SIAM, Philadelphia, United States, ISBN: 978-1-61197-453-9, 2016. a

Auerbach, D., Cvitanović, P., Eckmann, J.-P., Gunaratne, G., and Procaccia, I.: Exploring chaotic motion through periodic orbits, Phys. Rev. Lett., 58, 2387–2389, https://doi.org/10.1103/PhysRevLett.58.2387, 1987. a

Bocquet, M. and Carrassi, A.: Four-dimensional ensemble variational data assimilation and the unstable subspace, Tellus A, 69, 1304504, https://doi.org/10.1080/16000870.2017.1304504, 2017. a, b, c, d, e

Bocquet, M., Raanes, P. N., and Hannart, A.: Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation, Nonlin. Processes Geophys., 22, 645–662, https://doi.org/10.5194/npg-22-645-2015, 2015. a, b

Bocquet, M., Gurumoorthy, K. S., Apte, A., Carrassi, A., Grudzien, C., and Jones, C. K. R. T.: Degenerate Kalman Filter Error Covariances and Their Convergence onto the Unstable Subspace, SIAM/ASA Journal on Uncertainty Quantification, 5, 304–333, https://doi.org/10.1137/16M1068712, 2017. a, b, c

Bocquet, M., Farchi, A., and Malartic, Q.: Online learning of both state and dynamics using ensemble Kalman filters, Foundations of Data Science, 3, 305–330, https://doi.org/10.3934/fods.2020015, 2020. a

Carrassi, A., Ghil, M., Trevisan, A., and Uboldi, F.: Data assimilation as a nonlinear dynamical systems problem: Stability and convergence of the prediction-assimilation system, Chaos, 18, 023112, https://doi.org/10.1063/1.2909862, 2008. a

Carrassi, A., Bocquet, M., Bertino, L., and Evensen, G.: Data assimilation in the geosciences: An overview of methods, issues, and perspectives, WIREs Clim. Change, 9, e535, https://doi.org/10.1002/wcc.535, 2018. a

Carrassi, A., Bocquet, M., Demaeyer, J., Grudzien, C., Raanes, P., and Vannitsem, S.: Data Assimilation for Chaotic Dynamics, Springer International Publishing, Cham, 1–42, https://doi.org/10.1007/978-3-030-77722-7_1, 2022. a, b, c, d, e, f

Cencini, M. and Ginelli, F.: Lyapunov analysis: from dynamical systems theory to applications, J. Phys. A-Math. Theor., 46, 250301, https://doi.org/10.1088/1751-8113/46/25/250301, 2013. a

Chen, Y.: yumengch/InferDynPaper: PaperResults, ReleaseVersion, Zenodo [code], https://doi.org/10.5281/zenodo.5788693, 2021. a

De Cruz, L., Schubert, S., Demaeyer, J., Lucarini, V., and Vannitsem, S.: Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models, Nonlin. Processes Geophys., 25, 387–412, https://doi.org/10.5194/npg-25-387-2018, 2018. a, b

Eckmann, J. P. and Ruelle, D.: Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57, 617–656, https://doi.org/10.1103/RevModPhys.57.617, 1985. a

Evensen, G.: Data assimilation: the ensemble Kalman filter, Springer Science and Business Media, Berlin/Heidelberg, Germany, 2009. a, b, c, d

Faranda, D., Messori, G., and Yiou, P.: Dynamical proxies of North Atlantic predictability and extremes, Sci. Rep.-UK, 7, 41278, https://doi.org/10.1038/srep41278, 2017. a

Froyland, G., Hüls, T., Morriss, G. P., and Watson, T. M.: Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study, Physica D, 247, 18–39, https://doi.org/10.1016/j.physd.2012.12.005, 2013. a

Gálfi, V. M., Bódai, T., and Lucarini, V.: Convergence of Extreme Value Statistics in a Two-Layer Quasi-Geostrophic Atmospheric Model, Complexity, 2017, 5340858, https://doi.org/10.1155/2017/5340858, 2017. a

Gallavotti, G. and Lucarini, V.: Equivalence of non-equilibrium ensembles and representation of friction in turbulent flows: the Lorenz 96 model, J. Stat. Phys., 156, 1027–1065, https://doi.org/10.1007/s10955-014-1051-6, 2014. a, b

Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R., and Politi, A.: Characterizing Dynamics with Covariant Lyapunov Vectors, Phys. Rev. Lett., 99, 130601, https://doi.org/10.1103/PhysRevLett.99.130601, 2007. a

Grudzien, C., Carrassi, A., and Bocquet, M.: Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error, SIAM/ASA Journal on Uncertainty Quantification, 6, 1335–1363, https://doi.org/10.1137/17M114073X, 2018a. a, b

Grudzien, C., Carrassi, A., and Bocquet, M.: Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error, Nonlin. Processes Geophys., 25, 633–648, https://doi.org/10.5194/npg-25-633-2018, 2018b. a, b

Holton, J. R. and Hakim, G. J.: An Introduction to Dynamic Meteorology, 4th edn., Academic Press, San Diego, CA, 2013. a

Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, Cambridge, England, https://doi.org/10.1017/CBO9780511802270, 2002. a

Kuptsov, P. V. and Parlitz, U.: Theory and computation of covariant Lyapunov vectors, J. Nonlinear Sci., 22, 727–762, https://doi.org/10.1007/s00332-012-9126-5, 2012. a

Lai, Y.-C.: Unstable dimension variability and complexity in chaotic systems, Phys. Rev. E, 59, R3807–R3810, https://doi.org/10.1103/PhysRevE.59.R3807, 1999. a

Lorenz, E. N.: Available Potential Energy and the Maintenance of the General Circulation, Tellus, 7, 157–167, https://doi.org/10.1111/j.2153-3490.1955.tb01148.x, 1955. a

Lorenz, E. N.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141, 1963. a

Lorenz, E. N.: Predictability: A problem partly solved, in: Proc. Seminar on predictability, Shinfield Park, Reading, 4–8 September 1995, vol. 1, 1996. a, b

Lucarini, V. and Gritsun, A.: A new mathematical framework for atmospheric blocking events, Clim. Dynam., 54, 575–598, 2020. a

Lucarini, V., Faranda, D., Wouters, J., and Kuna, T.: Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems, J. Stat. Phys., 154, 723–750, https://doi.org/10.1007/s10955-013-0914-6, 2014. a

Lucarini, V., Faranda, D., de Freitas, A. C. G. M. M., de Freitas, J. M. M., Holland, M., Kuna, T., Nicol, M., Todd, M., and Vaienti, S.: Extremes and Recurrence in Dynamical Systems, Wiley, New York, 2016. a

Maclean, J. and Van Vleck, E. S.: Particle filters for data assimilation based on reduced-order data models, Q. J. Roy. Meteor. Soc., 147, 1892–1907, 2021. a

Messori, G., Caballero, R., and Faranda, D.: A dynamical systems approach to studying midlatitude weather extremes, Geophys. Res. Lett., 44, 3346–3354, https://doi.org/10.1002/2017GL072879, 2017. a

Miller, R. N., Ghil, M., and Gauthiez, F.: Advanced data assimilation in strongly nonlinear dynamical systems, J. Atmos. Sci., 51, 1037–1056, https://doi.org/10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2, 1994. a

Palatella, L., Carrassi, A., and Trevisan, A.: Lyapunov vectors and assimilation in the unstable subspace: theory and applications, J. Phys. A-Math. Theor., 46, 254020, https://doi.org/10.1088/1751-8113/46/25/254020, 2013. a

Pikovsky, A. and Politi, A.: Lyapunov Exponents: A Tool to Explore Complex Dynamics, Cambridge University Press, Cambridge, England, https://doi.org/10.1017/CBO9781139343473, 2016. a

Raanes, P. and Grudzien, C.: DAPPER v1.1.0, Zenodo [code], https://doi.org/10.5281/zenodo.2029295, 2018. a

Ruelle, D.: Ergodic theory of differentiable dynamical systems, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 50, 27–58, https://doi.org/10.1007/BF02684768, 1979. a

Schubert, S. and Lucarini, V.: Covariant Lyapunov vectors of a quasi-geostrophic baroclinic model: analysis of instabilities and feedbacks, Q. J. Roy. Meteor. Soc., 141, 3040–3055, https://doi.org/10.1002/qj.2588, 2015. a

Tondeur, M., Carrassi, A., Vannitsem, S., and Bocquet, M.: On temporal scale separation in coupled data assimilation with the ensemble kalman filter, J. Stat. Phys., 179, 1161–1185, https://doi.org/10.1007/s10955-020-02525-z, 2020. a, b

Trevisan, A. and Pancotti, F.: Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System, J. Atmos. Sci., 55, 390–398, https://doi.org/10.1175/1520-0469(1998)055<0390:POLVAS>2.0.CO;2, 1998. a

Van Leeuwen, P. J., Künsch, H. R., Nerger, L., Potthast, R., and Reich, S.: Particle filters for high-dimensional geoscience applications: A review, Q. J. Roy. Meteor. Soc., 145, 2335–2365, https://doi.org/10.1002/qj.3551, 2019. a

Vannitsem, S.: Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics, Chaos, 27, 032101, https://doi.org/10.1063/1.4979042, 2017. a

Vannitsem, S. and Lucarini, V.: Statistical and dynamical properties of covariant lyapunov vectors in a coupled atmosphere-ocean model–multiscale effects, geometric degeneracy, and error dynamics, J. Phys. A-Math. Theor., 49, 224001, https://doi.org/10.1088/1751-8113/49/22/224001, 2016. a, b

Vissio, G. and Lucarini, V.: Mechanics and thermodynamics of a new minimal model of the atmosphere, Eur. Phys. J. Plus, 135, 807, https://doi.org/10.1140/epjp/s13360-020-00814-w, 2020. a, b, c, d

Wolfe, C. L. and Samelson, R. M.: An efficient method for recovering Lyapunov vectors from singular vectors, Tellus A, 59, 355–366, https://doi.org/10.1111/j.1600-0870.2007.00234.x, 2007. a, b

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In the words of Ed Lorenz, “Chaos: When the present determines the future, but the approximate present does not approximately determine the future”; see https://tinyurl.com/faf3pnda (last access: 17 December 2021).