## 1. Introduction

In meteorology and other environmental sciences, an important challenge is to estimate the state of the system as accurately as possible. In meteorology, this state includes pressure, humidity, temperature and wind at different locations and elevations in the atmosphere. Data assimilation (DA) refers to mathematical methods that use both model predictions (also called background information) and partial observations to retrieve the current state vector with its associated error. An accurate estimate of the current state is crucial to get good forecasts, and it is particularly so whenever the system dynamics is chaotic, such as it is the case for the atmosphere.

The performance of a DA system to estimate the state depends on the accuracy of the model predictions, the observations, and their associated error terms. A simple, popular and mathematically justifiable way of modeling these errors is to assume them to be independent and unbiased Gaussian white noise, with covariance matrices

### a. Problem statement

Hereinafter, the unified DA notation proposed in Ide et al. (1997) is used.^{1} Data assimilation algorithms are used to estimate the state of a system, **x**, conditionally on observations, **y**. A classic strategy is to use sequential and ensemble DA frameworks, as illustrated in Fig. 1, and to combine two sources of information: model forecasts (in green) and observations (in blue). The ensemble framework uses different realizations, also called members, to track the state of the system at each assimilation time step.

The forecasts of the state are based on the usually incomplete and approximate knowledge of the system dynamics. The evolution of the state from time *k* − 1 to *k* is given by the model equation

where the model error ** η** implies that the dynamic model operator

*k*, because it may depend on time-dependent external forcing terms. At time

*k*, the forecast state is characterized by the mean of the forecast states,

**x**

^{f}, and its uncertainty matrix, namely

^{f}, which is also called the background error covariance matrix and is noted as

The forecast covariance ^{f} is determined by two processes. The first is the uncertainty propagated from *k* − 1 to *k* by the model _{k} (the green shade within the dashed ellipse in Fig. 1 and denoted by ^{m}). The second process is the model error covariance *k* in Eq. (1). Given that model error is largely unknown and originated by various and diverse sources, the matrix ^{m} and ^{f} (full green ellipse in Fig. 1). In the illustration given here, a large contribution of the forecast covariance ^{f} is due to ^{m} can be too small, as a consequence of the ensemble undersampling of the initial condition error (i.e., the covariance estimated at the previous analysis). In that case, inflating ^{m}.

DA uses a second source of information, the observations **y**, which are assumed to be linked to the true state **x** through the time-dependent operator _{k}. This step in DA algorithms is formalized by the observation equation

where the observation error ** ϵ** describes the discrepancy between what is observed and the truth. In practice, it is important to remove as much as possible the large-scale bias in the observation before DA. Then, it is common to state that the remaining error

**follows a Gaussian and unbiased distribution with a covariance**

*ϵ*DA algorithms combine forecasts with observations, based on the model and observation equations, given in Eqs. (1) and (2), respectively. The corresponding system of equations is a nonlinear state-space model. As illustrated in Fig. 1, this Gaussian DA process produces a posterior Gaussian distribution with mean **x**^{a} and covariance ^{a} (red ellipse). The system given in Eqs. (1) and (2) is representative of a broad range of DA problems, as described in seminal papers such as Ghil and Malanotte-Rizzoli (1991), and still relevant today as referenced by Houtekamer and Zhang (2016) and Carrassi et al. (2018). The assumptions made in Eqs. (1) and (2) about model and observation errors (additive, Gaussian, unbiased, and mutually independent) are strong, yet convenient from the mathematical and computational point of view. Nevertheless, these assumptions are not always realistic in real DA problems. For instance, in operational applications, systematic biases in the model and in the observations are recurring problems. Indeed, biases affect significantly the DA estimations and a specific treatment is required; see Dee (2005) for more details.

From Eqs. (1) and (2), noting that **y** are given, the only parameters that influence the estimation of **x** are the covariance matrices

### b. Illustrative example

In either variational or ensemble-based DA methods, the quality of the reconstructed state (or hidden) vector **x** largely depends on the relative amplitudes between the assumed observation and model errors (Desroziers and Ivanov 2001). In Kalman filter–based methods, the signal-to-noise ratio ||^{f}||/||^{f} depends on

The importance of ^{f}||/||*x* and simple linear dynamics. This simplified setup avoids several issues typical of realistic DA applications: the large dimension of the state, the strong nonlinearities and the chaotic behavior. In this example, the dynamic model in Eq. (1) is a first-order autoregressive model, denoted by AR(1) and defined by

with *η* ~ *Q*_{t}), where the superscript *t* means “true” and *Q*^{t} = 1. Furthermore, observations *y* of the state are contaminated with an independent additive zero-mean and unit-variance Gaussian noise, such that *R*^{t} = 1 in Eq. (2) with *x*) = *x*. The goal is to reconstruct *x* from the noisy observations *y* at each time step. The AR(1) dynamic model defined by Eq. (3) has an autoregressive coefficient close to one, representing a process that evolves slowly over time, and a stochastic noise term *η* with variance *Q*^{t}. Although the knowledge of these two sources of noise is crucial for the estimation problem, identifying them is not an easy task. Given that the dynamic model is linear and the error terms are additive and Gaussian in this simple example, the Kalman smoother provides the best estimation of the state (see section 2 for more details). To evaluate the effect of badly specified *Q* and *R* errors on the reconstructed state with the Kalman smoother, different experiments were conducted with values of {0.1, 1, 10} for the ratio *Q*/*R* (in this toy example, we use *Q*/*R* instead of ||^{f}||/||

Figure 2 shows, as a function of time, the true state (red line) and the smoothing Gaussian distributions represented by the 95% confidence intervals (gray shaded) and their means (black lines). We also report the root-mean-square error (RMSE) of the reconstruction and the so-called coverage probability, or percentage of *x* that falls in the 95% confidence intervals (defined as the mean ± 1.96 the standard deviation in the Gaussian case). In this synthetic experiment, the best RMSE and coverage probability obtained, applying the Kalman smoother with true *Q*^{t} = *R*^{t} = 1, are 0.71% and 95%, respectively. Using a small model error variance *Q* = 0.1*Q*^{t} in Fig. 2a, the filter gives a large weight to the forecasts given by the quasi-persistent autoregressive dynamic model. On the other hand, with a small observation error variance *R* = 0.1*R*^{t} in Fig. 2b, excessive weight is given to the observation and the reconstructed state is close to the noisy measurements. These results show the negative impact of independently badly scaled *Q* and *R* error variances. In the case of overestimated model error variance as in Fig. 2c, the mean reconstructed state vector and thus its RMSE are identical to Fig. 2b. In the same way, overestimated observation error variance like in Fig. 2d gives similar mean reconstruction, as in Fig. 2a. These last two results are due to the fact that in both cases, the ratio *Q*/*R* are equal, respectively, to 10 and 0.1. Now, we consider in Figs. 2e and 2f the case where the *Q*/*R* ratio is equal to 1, but, respectively, using the simultaneous underestimation and overestimation of model and observation errors. In both cases, the mean reconstructed state is equal to that obtained with the true error variances (i.e., RMSE = 0.71). The main difference is the gray confidence interval, which is supposed to contain 95% of the true trajectory: the spread is clearly underestimated in Fig. 2e and overestimated in Fig. 2f, with coverage probability of 36% and 100%, respectively.

Example of a univariate AR(1) process generated using Eq. (3) with *Q*^{t} = 1 (red line), noisy observations as in Eq. (2) with *R*^{t} = 1 (black dots), and reconstructions with a Kalman smoother (black lines and gray 95% confidence interval) with different values of *Q* and *R*, from 0.1 to 10. The optimal values of RMSE and coverage probabilities are 0.71% and 95%, respectively.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0240.1

Example of a univariate AR(1) process generated using Eq. (3) with *Q*^{t} = 1 (red line), noisy observations as in Eq. (2) with *R*^{t} = 1 (black dots), and reconstructions with a Kalman smoother (black lines and gray 95% confidence interval) with different values of *Q* and *R*, from 0.1 to 10. The optimal values of RMSE and coverage probabilities are 0.71% and 95%, respectively.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0240.1

Example of a univariate AR(1) process generated using Eq. (3) with *Q*^{t} = 1 (red line), noisy observations as in Eq. (2) with *R*^{t} = 1 (black dots), and reconstructions with a Kalman smoother (black lines and gray 95% confidence interval) with different values of *Q* and *R*, from 0.1 to 10. The optimal values of RMSE and coverage probabilities are 0.71% and 95%, respectively.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0240.1

We used a simple synthetic example, but for large dimensional and highly nonlinear dynamics, such an underestimation or overestimation of uncertainty may have a strong effect and may cause filters to collapse. The main issue in ensemble-based DA is an underdispersive spread, as in Fig. 2e. In that case, the initial condition spread is too narrow, and model forecasts (starting from these conditions) would be similar and potentially out of the range of the observations. In the case of an overdispersive spread, as in Fig. 2f, the risk is that only a small portion of model forecasts would be accurate enough to produce useful information on the true state of the system. This illustrative example shows how important is the joint tuning of model and observation errors in DA. Since the 1990s, a substantial number of studies have dealt with this topic.

### c. Seminal work in the data assimilation community

In a seminal paper, Dee (1995) proposed an estimation method for parametric versions of **y** − **x**^{f}). Maximization is performed at each assimilation step, with the current innovation computed from the available observations. This technique was later extended to estimate the mean of the innovation, which depends on the biases in the forecast and in the observations (Dee and da Silva 1999). The method was then applied to realistic cases in Dee et al. (1999), making the maximization of innovation likelihood a promising technique for the estimation of errors in operational forecasts.

Following a distinct path, Desroziers and Ivanov (2001) proposed using the observation-minus-analysis diagnostic. It is defined by **y** − **x**^{a}) with **x**^{a} being the analysis (i.e., the output of DA algorithms). The authors proposed an iterative optimization technique to estimate a scaling factor for the background ^{f} and observation

Later, Chapnik et al. (2004) showed that the maximization of the innovation likelihood proposed by Dee (1995) makes the observation-minus-analysis diagnostic of Desroziers and Ivanov (2001) optimal. Moreover, the techniques of Dee (1995) and Desroziers and Ivanov (2001) have been further connected to the generalized cross-validation method previously developed by statisticians (Wahba and Wendelberger 1980).

These initial studies clearly nurtured the discussion of the estimation of observation ^{f} error covariance matrices in the modern DA literature. For demonstration purposes, the algorithms proposed in Dee (1995) and Desroziers and Ivanov (2001) were tested on realistic DA problems, using a shallow-water model on a plane with a simplified Kalman filter, and using the French ARPEGE three-dimensional variational framework, respectively. In both cases, although good performances have been obtained with a small number of iterations, the proposed algorithms have shown some limits, in particular with regard to the simultaneous estimation of the two sources of errors: observation and model (or background). In this context, Todling (2015) pointed out that using only the current innovation is not enough to distinguish the impact of

### d. Methods presented in this review

The main topic of this review is the “joint estimation of

Timeline of the main methods used in geophysical data assimilation for the joint estimation of

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0240.1

Timeline of the main methods used in geophysical data assimilation for the joint estimation of

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0240.1

Timeline of the main methods used in geophysical data assimilation for the joint estimation of

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0240.1

Comparison of several methods to estimate error covariance

On the one hand, moment-based methods assume equality between theoretical and empirical statistical moments. A first approach is to study different type of innovations in the observation space (i.e., working in the space of the observations instead of the space of the state). It has been initiated in DA by Rutherford (1972) and Hollingsworth and Lönnberg (1986). A second approach extracts information from the correlation between lag innovations, namely innovations between consecutive times. On the other hand, likelihood-based methods aim to maximize likelihood functions with statistical algorithms. One option is to use a Bayesian framework, assuming prior distributions for the parameters of

The four methods listed in Fig. 3 will be examined in this paper. Before doing that, it is worth mentioning existing review work that have attempted to summarize the methods in DA context and beyond.

### e. Other review papers

Other review papers on parameter estimation (including ** η** and

**are Gaussians and**

*ϵ*In the statistical community, the recent development of powerful simulation techniques, known as sequential Monte Carlo algorithms or particle filters, has led to an extensive literature on the statistical inference in nonlinear state-space models relying on likelihood-based approaches. A recent and detailed presentation of this literature can be found in Kantas et al. (2015). However, these methods typically require a large number of particles, which make them impractical for geophysical DA applications.

The review presented here focuses on methods proposed in DA, especially the moment- and likelihood-based techniques that are suitable for geophysical systems (i.e., with high dimensionality and strong nonlinearities).

### f. Structure of this review

The paper is organized as follows. Section 2 briefly presents the filtering and smoothing DA algorithms used in this work. The main families of methods used in the literature to jointly estimate error covariance matrices

## 2. Filtering and smoothing algorithms

This review paper focuses on the estimation of

The EnKF and EnKS estimate various state vectors **x**^{f}(*k*), **x**^{a}(*k*), **x**^{s}(*k*) and covariance matrices ^{f}(*k*), ^{a}(*k*), ^{s}(*k*), at each time step 1 ≤ *k* ≤ *K*, where *K* represents the total number of assimilation steps. Kalman-based algorithms assume a Gaussian prior distribution

Then, filtering and smoothing estimates correspond to the Gaussian posterior distributions

of the state conditionally to past/present observations and past/present/future observations, respectively.

The basic idea of the EnKF and EnKS is to use an ensemble *N*_{e} to track Gaussian distributions over time with the empirical mean vector

The EnKF/EnKS equations are divided into three main steps, ∀*i* = 1, …, *N*_{e} and ∀*k* = 1, …, *K*—the forecast step (forward in time):

the analysis step (forward in time):

and the reanalysis step (backward in time):

with ^{f}(*k*) and ^{s}(*k*) being the filter and smoother Kalman gains, respectively. Here, ^{f}(*k*) and *N*_{e} ensemble members.

In some of the methods presented in this review, the ensembles are also used to approximate _{k} and _{k} by linear operators **M**_{k} and **H**_{k} such as

with the dagger indicating the pseudoinverse and

In Eq. (4b), the innovation is denoted as **d** and is tracked by

## 3. Moment-based methods

To constrain the model and observational errors in DA systems, initial efforts were focused on the statistics of relevant variables that could contain information on covariances. The innovation, given in Eq. (4b), corresponds to the difference between the observations and the forecast in the observation space. This variable implicitly takes into account the

Two main approaches have been proposed in the literature to address this issue. They are based on the idea of producing multiple equations involving

### a. Innovation statistics in the observation space

This first approach, based on the Desroziers diagnostic (Desroziers et al. 2005), is historical and now popular in the DA community. It does not exactly fit the topic of this review paper (i.e., estimating the model error ^{f}. However, this forecast error covariance is defined by **M**_{k}. Thus, even if DA systems do not use an explicit model error perturbation controlled by ^{f} has similar effects, compensating for the lack of an explicit model uncertainty.

Desroziers et al. (2005) proposed examining various innovation statistics in the observation space. It is based on a different type of innovation statistics between observations, forecasts, and analysis, with all of them defined in the observation space: namely, ^{f}(*k*) and *k*) are correctly specified, the Desroziers innovation statistics should verify the equalities:

with *E* being the expectation operator. Equation (6a) is given by using Eq. (4b):

and then applying the expectation operator and using the definition of ^{f} and **d**^{o−f} does not depend explicitly on ^{f}. Thus, the combination of Eqs. (6a) and (6b) can be used as a diagnosis of the forecast and observational error covariances in the system. A mismatch between the Desroziers statistics and the actual covariances, namely the left- and right-hand side terms in Eqs. (6a) and (6b), indicates inappropriate estimated covariances ^{f}(*k*) and *k*).

The forecast covariance ^{f} is sometimes badly estimated in ensemble-based assimilation systems. The limitations may be attributed to a number of causes. The limited number of ensemble members produces an over or, most of the time, underestimation of the forecast variance. Another limitation is the inaccuracies in methods used to sample initial condition or model error. The underestimation of the forecast covariance produces negative feedback, and the estimated analysis covariance ^{a} is thus underestimated, which in turn produces a further underestimation of the forecast covariance in the next cycle. This feedback process leads to filter divergence, as was pointed out by Pham et al. (1998), Anderson and Anderson (1999) or Anderson (2007). To avoid this filter divergence, inflating the forecast covariance ^{f} has been proposed. This covariance inflation accounts for both sampling errors and the lack of representation of model errors, like a too-small amplitude for ** η** and

**from Eqs. (1) and (2). In this context, the diagnostics given by the Desroziers innovation statistics have been proposed as a tool to constrain the required covariance inflation in the system.**

*ε*We distinguish three inflation methods: multiplicative, additive and relaxation-to-prior. In the multiplicative case, the forecast error covariance matrix ^{f} is usually multiplied by a scalar coefficient greater than 1 (Anderson and Anderson 1999). Using innovation statistics in the observation space, adaptive procedures to estimate this coefficient have been proposed by Wang and Bishop (2003) and Anderson (2007, 2009) conditionally to the spatial location, Li et al. (2009), Miyoshi (2011), Bocquet (2011), Bocquet and Sakov (2012), Miyoshi et al. (2013), Bocquet et al. (2015), El Gharamti (2018) and Raanes et al. (2019). To prevent excessive inflation or deflation, some authors have proposed assuming a priori distribution for the multiplicative inflation factor. The most usual a priori distributions used by the authors are Gaussian in Anderson (2009), inverse-gamma in El Gharamti (2018) or inverse chi-square in Raanes et al. (2019).

In practice, multiplicative inflation tends to excessively inflate in the data-sparse regions and inflate too little in the densely observed regions. As a result, the spread looks like exaggeration of data density (i.e., too much spread in sparsely observed regions, and vice versa). Additive inflation solves this problem but requires many samples for additive noise; these drawbacks and benefits are discussed in Miyoshi et al. (2010). In the additive inflation case, the diagonal terms of the forecast and analysis empirical covariance matrices is increased (Mitchell and Houtekamer 2000; Corazza et al. 2003; Whitaker et al. 2008; Houtekamer et al. 2009). This regularization also avoids the problems corresponding to the inversion of the covariance matrices.

The last alternative is the relaxation-to-prior method. In application, this technique is more efficient than both additive and multiplicative inflations because it maintains a reasonable spread structure. The idea is to relax the reduction of the spread at analysis. We distinguish the method proposed in Zhang et al. (2004), where the forecast and analysis ensemble perturbations are blended, from the one given in Whitaker and Hamill (2012), which multiplies the analysis ensemble without blending perturbations. This last method is thus a multiplicative inflation, but applied after the analysis, not the forecast. Ying and Zhang (2015) and Kotsuki et al. (2017b) proposed methods to adaptively estimate the relaxation parameters using innovation statistics. Their conclusions are that adaptive procedures for relaxation-to-prior methods are robust to sudden changes in the observing networks and observation error settings.

Closely connected to multiplicative inflation estimation is statistical modeling of the error variance terms proposed by Bishop and Satterfield (2013) and Bishop et al. (2013). From numerical evidence based on the 10-dimensional Lorenz-96 model, the authors assume an inverse-gamma prior distribution for these variances. This distribution allows for an analytic Bayesian update of the variances using the innovations. Building on Bocquet (2011), Bocquet et al. (2015), and Ménétrier and Auligné (2015), this technique was extended in Satterfield et al. (2018) to adaptively tune a mixing ratio between the true and sample variances.

Adaptive covariance inflations are estimation methods directly attached to a traditional filtering method (such as the EnKF used here), with almost negligible overhead computational cost. In practice, the use of this technique does not necessarily imply an additive error term ** η** in Eq. (1). Thus, it is not a direct estimation of

^{f}in order to compensate for model uncertainties and sampling errors in the EnKFs, as explained in Raanes et al. (2019, their section 4 and appendix C). Several DA systems work with an inflation method and use it for its simplicity, low cost, and efficiency. As an example of inflation techniques, the most straightforward inflation estimation is a multiplicative factor

*λ*of the incorrectly scaled

The estimated inflation parameter *k* can be noisy. The use of temporal smoothing of the form *λ*(*k*), denoted as *λ*(*k* + 1) is updated using the previous estimate *λ*(*k*) and the Gaussian distribution with mean *k* and when sufficient observations are available, an estimate of *k*) is possible using Eq. (6b). For instance, Li et al. (2009) proposed estimating each component of a diagonal and averaged *k*). The Miyoshi et al. (2013) implementation is summarized in the appendix as algorithm 1.

The Desroziers diagnostic method has been applied widely to estimate the real observation error covariance matrix **d**^{o−f} and its correlations in space, assuming that horizontal correlations in **d**^{o−f} samples are purely due to ^{f}. Weston et al. (2014) and Campbell et al. (2017) then included the interchannel observation error correlations of satellite radiances in DA and obtained improved results when compared with the case using a diagonal

To conclude, the Desroziers diagnostic is a consistency check and makes it possible to detect if the error covariances ^{f} and

### b. Lag innovation between consecutive times

Another way to estimate error covariances is to use multiple equations involving **d**(*k*) = **d**^{o−f}(*k*) defined in Eq. (4b) and past innovations **d**(*k* − 1), …, **d**(*k* − *l*). Lag innovations were introduced by Mehra (1970) to recover **d**(*k*)}_{k≥1} is completely characterized by the lagged covariance matrix _{l} = Cov[**d**(*k*), **d**(*k* − *l*)], which is independent of *k*. In other words, the information encoded in {**d**(*k*)}_{k≥1} is completely equivalent to the information provided by {_{l}}_{l≥0}. Moreover, for linear systems in a steady state, analytic relations exist between *E*[**d**(*k*)**d**(*k* − *l*)^{T}]. However, these linear relations can be dependent and redundant for different lags *l*. Therefore, as stated in Mehra (1970), only a limited number of

Bélanger (1974) extended these results to the case of time-varying linear stochastic processes, taking **d**(*k*)**d**(*k* − *l*)^{T} as “observations” of *m*^{2} × *m*^{2}, where *m* refers to the dimension of the observation vector. However, this difficulty has been largely overcome by Dee et al. (1985) in which the matrix inversion is reduced to *O*(*m*^{3}) by taking the advantage of the fact that the big matrix comes from some tensor product.

More recent work has focused on high-dimensional and nonlinear systems using the extended or ensemble Kalman filters. Berry and Sauer (2013) proposed a fast and adaptive algorithm inspired by the use of lag innovations proposed by Mehra. Harlim et al. (2014) applied the original Bélanger algorithm empirically to a nonlinear system with sparse observations. Zhen and Harlim (2015) proposed a modified version of Bélanger’s method, by removing the secondary filter and alternatively solving

Here, we briefly describe the algorithm of Berry and Sauer (2013), considering the lag-zero and lag-one innovations. The following equations are satisfied in the linear and Gaussian case, for unbiased forecast and observation when ^{f}(*k*) and *k*) are correctly specified:

Equation (9a) is equivalent to Eq. (6a). Moreover, Eq. (9b) results from the fact that, developing the expression of **d**(*k*) using consecutively Eqs. (2), (1), (4a), and (4d), the innovation can be written as

Hence, the innovation product **d**(*k*)**d**(*k* − 1)^{T} between two consecutive times is given by

and, assuming that the model ** η** and observation

**error noises are white and mutually uncorrelated, then**

*ϵ**E*[

**(**

*η**k*)

**d**(

*k*− 1)

^{T}] = 0 and

*E*[

**(**

*ϵ**k*)

**d**(

*k*− 1)

^{T}] = 0. Last, developing

*E*[

**d**(

*k*)

**d**(

*k*− 1)

^{T}], Eq. (9b) is satisfied.

The algorithm in Berry and Sauer (2013) is summarized in the appendix as algorithm 2. It is based on an adaptive estimation of *k*) and *k*), which satisfies the following relations in the linear and Gaussian case:

In operational applications, when the number of observations is not equal to the number of components in state **x**,

In this adaptive procedure, joint estimations of

with *ρ* being the smoothing parameter. When *ρ* is large (close to 1), weight is given to the current estimates *ρ* is small (close to 0) it gives smoother *ρ* is arbitrary and may depend on the system and how it is observed. For instance, in the case where the number of observations equals the size of the system, Berry and Sauer (2013) uses *ρ* = 5 × 10^{−5} in order to estimate the full matrix

The algorithm in Berry and Sauer (2013) only considers lag-zero and lag-one innovations. By incorporating more lags, Zhen and Harlim (2015) and Harlim (2018) showed that it makes it possible to deal with the case in which some components of **M** and **H** = [1, 0], meaning that only the first component of the system is observed. This is a linear, Gaussian, stationary system, and Mehra’s theory implies that two parameters of

To summarize, methods based on lag innovation between consecutive times have been studied for a long time in the signal processing community. The original methods (Mehra 1970; Bélanger 1974) were analytically established for linear systems with Gaussian noises. Inspired by these foundational ideas, empirical methods have been established for nonlinear systems in DA (Berry and Sauer 2013; Harlim et al. 2014; Zhen and Harlim 2015). Although these methods have not been tested in any operational experiment, the idea of using lagged innovations seems to have significant potential.

## 4. Likelihood-based methods

This section focuses on methods based on the likelihood of the observations, given a set of statistical parameters. The conceptual idea behind what we refer to as likelihood-based methods is to determine the optimal statistical parameters (i.e.,

Early studies in Dee (1995), Blanchet et al. (1997), Mitchell and Houtekamer (2000) and Liang et al. (2012) proposed finding the optimal *k*. Unfortunately, if only the current observations are used, the joint estimation of

The likelihood-based methods are broadly divided into two categories. One approach uses a Bayesian framework. It assumes a priori knowledge about the parameters and estimate jointly the posterior distribution of ^{2} The second one is based on the frequentist viewpoint and attempts a point estimate of the parameters by maximizing a total likelihood function.

### a. Bayesian inference

In the Bayesian framework, the elements of the covariance matrices ** θ**.

The inference in the Bayesian framework aims to determine the posterior density *p*[** θ**|

**y**(1:

*k*)]. Two techniques have appeared, the first based on a state augmentation and the second based on a rigorous Bayesian update of the posterior distribution.

#### 1) State augmentation

In the Bayesian framework, ** θ** is a random variable such that the state is augmented with these parameters by defining

**z**(

*k*) = [

**x**(

*k*),

**]. To define an augmented state-space model, one has to define an evolution equation for the parameters. This leads to a new state-space model of the form of Eqs. (1) and (2) with**

*θ***x**replaced by

**z**. Therefore, the state and the parameters are estimated jointly using the DA algorithms.

State augmentation was first proposed in Schmidt (1966) and is known as the Schmidt–Kalman filter. This technique was mainly used to estimate both the state of the system and additional parameters, including bias, forcing terms and physical parameters. These kinds of parameters are strongly related to the state of the system (Ruiz et al. 2013a). Therefore, they are identifiable and suitable for an augmented state approach. However, Stroud and Bengtsson (2007) and later DelSole and Yang (2010) formally demonstrated that augmentation methods fail for variance parameters like ^{f} is computed using all the ensemble members, each one with a different realization of the random variable ** θ**. Thus,

^{f}and consequently the Kalman gain

^{f}, are mixing the effects of

**. Therefore, after applying Eq. (4d), the update of**

*θ***z**corresponding to the

**parameters is the same for all of the parameters. To capture the impact of a single variance parameter on the prediction covariance and circumvent the limitation of the state augmentation, Scheffler et al. (2019) proposed to use an ensemble of states integrated with the same variance parameter. The choice of an ensemble of states for each variance parameter leads to two nested ensemble Kalman filters. The technique performs successfully under different model error covariance structures but has an important computational cost.**

*θ*Another critical aspect of state augmentation is that one needs to define an evolution model for the augmented state **z**(*k*) = [**x**(*k*), ** θ**(

*k*)]. If persistence is assumed in the parameters such that they are constant in time, this leads to filter degeneracy, since the estimated variance of the error in

*θ*is bound to decrease in time. To prevent or at least mitigate this issue, it was suggested to use an independent inflation factor on the parameters (Ruiz et al. 2013b) or to impose artificial stochastic dynamics for

**, typically a random walk or AR(1) model, as introduced in Eq. (3) and proposed in Liu and West (2001). The tuning of the parameters introduced in these artificial dynamics may be difficult, and this introduces bias into the procedure, which is hard to quantify.**

*θ*#### 2) Bayesian update of the posterior distribution

Instead of the inference of the joint posterior density using a state augmentation strategy, the state **x**(*k*) and parameters ** θ** can be divided into a two-step inference procedure using the following formula:

which is a direct consequence of the conditional density definition. In Eq. (15), *p*[**x**(*k*)|**y**(1: *k*), ** θ**] represents the posterior distribution of the state, given the observations and the parameter

**. It can be computed using a filtering DA algorithm. The second term on the right-hand side of Eq. (15) corresponds to the posterior distribution of the parameters, given the observations up to time**

*θ**k*. The latter can be updated sequentially using the following Bayesian hierarchy:

where *p*[**y**(*k*)|**y**(1:*k* − 1), ** θ**] is the likelihood of the innovations.

Different approximations have been used for *p*[** θ**|

**y**(1:

*k*)] in Eq. (16); these include parametric models based on Gaussian (Stroud et al. 2018), inverse-gamma (Stroud and Bengtsson 2007) or Wishart distributions (Ueno and Nakamura 2016), particle-based approximations (Frei and Künsch 2012; Stroud et al. 2018) and grid-based approximation (Stroud et al. 2018).

The methods proposed in the literature also differ by the approximation used for the likelihood of the innovations. We emphasize that *p*[**y**(*k*)|**y**(1:*k* − 1), ** θ**] needs to be evaluated for different values of

**at each time step, and that this requires applying the filter from the initial time with a single value of**

*θ***, which is computationally impossible for applications in high dimensions. To reduce computational time, it is generally assumed that**

*θ***x**

^{f}and

^{f}are independent of

**, and only observations**

*θ***y**(

*k*−

*l*:

*k*− 1) in a small time window from the current observation are used when computing the likelihood of the innovations [see Ueno and Nakamura (2016) and Stroud et al. (2018) for a more detailed discussion]. A summary of the Bayesian method from Stroud et al. (2018) is given in the appendix as algorithm 3. It was implemented within the EnKF framework and is one of the most recent studies based on the Bayesian approach.

Applications of the Bayesian method in the DA context are now discussed. It has mainly been used to estimate shape and noise parameters of

As pointed out in Stroud and Bengtsson (2007), Bayesian update algorithms work best when the number of unknown parameters in ** θ** is small. This limitation may explain why the joint estimation of parameters controlling both model and observation error covariances is not systematically addressed. For instance, Stroud and Bengtsson (2007) used the EnKF with the Lorenz-96 model for the estimation of a common multiplicative scalar parameter for predefined matrices

^{f}||/||

Widely used in the statistical community, the Bayesian framework is useful incorporating physical knowledge about error covariance matrices and constraining their estimation process. In the DA literature, authors have used a priori distributions for the shape and noise parameters of

### b. Maximization of the total likelihood

The innovation likelihood at time *k*, *p*[**y**(*k*)|**y**(1:*k* − 1), ** θ**] in Eq. (16), can be maximized to find the optimal

**(i.e.,**

*θ*^{f}. When using only time

*k*, it is difficult to disentangle the model and observation error covariances; in application, the aforementioned studies only estimated one of them. Second, the number of observations at each time step is in general limited and, as pointed out by Dee (1995), available observations should exceed “the number of tunable parameters by two or three orders of magnitude.” To overcome these limitations, a reasonable alternative is to use a batch of observations within a time window and to assume

**to be constant in time. The resulting total likelihood expressed sequentially through conditioning is given by**

*θ*Because it is an integration of innovation likelihoods over a long period of time from *k* = 1 to *k* = *K*, Eq. (17) provides more observational information to estimate *p*[**y**(1:*K*)|** θ**] to estimate model error covariance

The likelihood function given in Eq. (17) only depends on the observations **y**. This likelihood can be written in a different way, taking into account both the observations and the hidden state **x**. Indeed, the marginalization of the hidden state to obtain the total likelihood can be produced using the whole trajectory of the state from *k* = 0 to the last time step *K* all at once. It is given by

The maximization of the total likelihood as a function of statistical parameters ** θ** is not possible, since the total likelihood cannot be evaluated directly, nor its gradient with regard to the parameters (Pulido et al. 2018). Shumway and Stoffer (1982) proposed using an iterative procedure based on the expectation–maximization algorithm (hereinafter denoted as EM). They applied it to estimate the parameters of a linear state-space model, with linear dynamics, and a linear observational operator and Gaussian errors. The EM algorithm was introduced by Dempster et al. (1977).

Each iteration of the EM algorithm consists of two steps. In the expectation step (E-step), the posterior density *p*[**x**(0:*K*)|**y**(1:*K*), *θ*_{(n)}] is determined conditioned on the batch of observations **y**(1:*K*) and given the parameters *θ*_{(n)} = [_{(n)}, _{(n)}] from the previous iteration or initial guess. This is obtained through the application of a smoother like the EnKS. Then, the M-step relies on the maximization of an intermediate function, depending on the posterior density obtained in the E-step. The intermediate function is defined by the conditional expectation:

If, as in Eqs. (1) and (2), the observational and model errors are assumed to be additive, unbiased, and Gaussian, the expression for the logarithm of the joint density in Eq. (19) is given by

where **v**^{T}^{−1}**v** and *c* is a constant independent of

The application of the EM algorithm for the estimation of _{(1)} and _{(1)}, an ensemble Kalman smoother is applied with this first guess and the batch of observations **y**(1:*K*) to obtain the posterior density *p*[(**x**(0:*K*)|**y**(1:*K*), *θ*_{(1)}]. Then Eqs. (21a) and (21b) are used to update and obtain _{(2)} and _{(2)}. Next, a new application of the smoother is conducted using the parameters _{(2)} and _{(2)} and the observations **y**(1: *K*), the new resulting states are used in Eqs. (21a) and (21b) to estimate _{(3)} and _{(3)}, and so on. As a diagnostic of convergence or as a stop criterion, the product of innovation likelihood functions given in Eq. (17) is evaluated using a filter. The EM algorithm guarantees that the total likelihood increases in each iteration and that the sequence *θ*_{(n)} converges to a local maximum (Wu 1983). A summary of the EM method (using EnKF and EnKS) from Dreano et al. (2017) is given in the appendix as algorithm 4.

EM is a well-known algorithm used in the statistical community. This procedure is parameter-free and robust, due to the large number of observations used to approximate the likelihood when using a long batch period (Shumway and Stoffer 1982). Although the use of the EM algorithm is still limited in DA, it is becoming more and more popular. Some studies have implemented the EM algorithm for estimating only the observation error matrix

To our knowledge, EM has not been tested yet on operational systems with large observation and state space. In that case, the use of parametric forms for the matrices ** θ** to estimate. For instance, Dreano et al. (2017) and Liu et al. (2017) showed that in the particular cases where covariances are diagonal or of the form

*α*

**in the space of the parametric covariance form can be obtained.**

*θ*## 5. Other methods

In this section, we describe other methods that have been used to estimate

### a. Analysis (or reanalysis) increment approach

This first method is based on previous DA outputs. The key idea here is to use the analysis (or reanalysis) increments to provide a realistic sample of model errors from which statistical moments, such as the covariance matrix **x**^{s} (or analysis **x**^{a}) is the best available representation of the true process **x**. In that case, the following approximation in Eq. (1) is made:

In this approximation, it is implicitly assumed that the estimated state is the truth so that the initial condition at time *k* − 1 is neglected. A similar approximation of the true process by **x**^{a} or **x**^{s} in Eq. (2) can be used to estimate the observation error covariance matrix

Operationally, the analysis (or reanalysis) increment method is applied after a DA filter (or smoother) to estimate the

### b. Covariance matching

The covariance matching method was introduced by Fu et al. (1993). It involves matching sample covariance matrices to their theoretical expectations. Thus, it is a method of moments, similar to the work in Desroziers et al. (2005), except that covariance matching is performed on a set of historical observations and numerical simulations (noted **x**^{sim}), without applying any DA algorithms. It has been extended by Menemenlis and Chechelnitsky (2000) to time-lagged innovations, as first considered in Bélanger (1974).

In the case of a constant and linear observation operator **H**, the basic idea in Fu et al. (1993) is to assume the following system:

with *η*^{sim}. In Eqs. (23b) and (23c), the definitions of ** η** and

**errors remain similar, as in the general Eqs. (1) and (2).**

*ε*Assuming that ** ϵ** is uncorrelated from

**x**and from

*η*^{sim}, then Eqs. (23c) and (23a) yield to the following estimates of

^{sim}(the latter represents the error covariance of the numerical simulations):

where *E* is the expectation operator over time. Then, an estimate of ^{sim} has a unique time-invariant limit.

### c. Forecast sensitivity

In operational meteorology, it is critical to learn the sensitivity of the forecast accuracy to various parameters of a DA system, in particular the error statistics of both the model and the observations. This is why a significant portion of literature considers the tuning problem of

The basic idea is to compute at each assimilation cycle an innovation between forecast and analysis, noted **d**^{f−a}(*k*) = **x**^{f}(*k*) − **x**^{a}(*k*). Then, the forecast sensitivity is given by **d**^{f−a}(*k*)^{T}**d**^{f−a}(*k*) with **d**^{f−a}. The **d**^{f−a}(*k*). The adjoint or the ensemble are thus used to compute the partial derivatives of this forecast sensitivity. w.r.t.

## 6. Conclusions and perspectives

As often considered in data assimilation, this review paper also deals with model and observation errors that are assumed additive and Gaussian with covariance matrices

The discussion starts with the aid of an illustration of the individual and joint impacts of improperly calibrated covariances using a linear toy model. The experiments clearly showed that to achieve reasonable filter accuracy (i.e., in terms of root-mean-squared error), it is crucial to carefully define both

### a. Comparison of existing methods for estimating $\mathsf{Q}$ and $\mathsf{R}$

We mainly focused in this review on four methods for the joint estimation of the error covariances

Most of the methods estimate the model error ^{f}. Moreover, the methods are mainly defined online, meaning that they aim to estimate

Throughout this review paper, as usually stated in DA, it is assumed that model error ** η** and observation error

**, defined in Eqs. (1) and (2), are Gaussian. Consequently, the distribution of the innovations is also Gaussian. The four presented methods use this property to build estimates of**

*ϵ***and**

*η***are non-Gaussian, Desroziers diagnostic and lag-innovation methods are not suitable anymore. However, the EM procedures and Bayesian methods are still relevant, although they must be used with an appropriate filter (e.g., particle filters), not Kalman-based algorithms (i.e., assuming a Gaussian distribution of the state). Recently, the treatment of non-Gaussian error distributions in DA has been explored in Katzfuss et al. (2020), using hierarchical state-space models. This Bayesian framework allows to handle unknown variables that cannot be easily included in the state vector (e.g., parameters of**

*ϵ*The four methods have been applied at different levels of complexity. For instance, Bayesian inference methods (due to their algorithm complexity) and the EM algorithm (due to its computational cost) have so far only been applied to small dynamic models. However, the online version of the EM algorithm is less consuming and opens new perspectives of applications on larger models. On the other hand, methods using innovation statistics in the observation space have already been applied to NWP models.

The four methods summarized in Table 1 show differences in maturity in terms of applications and methodological aspects. This review also shows that there are still remaining challenges and possible improvements for the four methods.

### b. Remaining challenges for each method

The first challenge concerns the improvements of adaptive techniques regarding additional parameters that control the variations of *ρ* in the lag innovations or *ρ*, low

The second challenge concerns considering time-varying error covariance matrices. The adaptive procedures, based on online estimations with temporal smoothing of

A third challenge has to do with the assumption, used by all of the methods described herein, that observation and model errors are mutually independent. Nevertheless, as pointed out in Berry and Sauer (2018), observation and model error are often correlated in real data assimilation problems (e.g., for satellite retrieval of Earth observations that uses model outputs in the inversion process). Methods based on Bayesian inference can, in principle, exploit existing model-to-observation correlations by using a prior joint distribution (i.e., not two individual ones). The explicit taking into account of this correlation can then constrain the optimization procedure. This is not possible in the other approaches described in this review, at least not in their standard known formulations, and the presence of model-observation correlation can deteriorate their accuracy.

A fourth challenge is common to all the methods presented in this review. Iterative versions of the presented algorithms need initial values or distributions for ^{f} in the case of Desroziers). However, as mentioned in Waller et al. (2016) for the Desrorziers diagnostics, there is no guarantee that the algorithms will converge to the optimal solution. Indeed, in such an optimization problem, there are possibly several local and nonoptimal solutions. Suboptimal specifications of

The last remaining challenge concerns the estimation of other statistical parameters of the state-space model given in Eqs. (1) and (2) and associated filters. Indeed, the initial conditions **x**(0) and ** η** and

**. This was initially proposed by Dee and da Silva (1999) and tested in Dee et al. (1999) in the case of maximizing the innovation likelihood, in Dee (2005) in a state augmentation formulation, and was adapted to a Bayesian update formulation in Liu et al. (2017) and in Berry and Harlim (2017). Recently, the joint estimation of bias and covariance error terms, for the treatment of brightness temperatures from the European geostationary satellite, has been successfully applied in Merchant et al. (2020).**

*ϵ*### c. Perspectives for geophysical DA

Beyond the aforementioned potential improvements in the existing techniques, specific research directions need to be taken by the data assimilation community. The main one concerns the realization of a comprehensive numerical evaluation of the different methods for the estimation of

The use of a realistic DA problem, with a high-dimensional state-space and a limited and heterogeneous observational coverage should be addressed in the future. In that realistic case, the observational information per degree of freedom will be significantly lower, and the estimates of

A further challenge for future work is the evaluation of the feasibility of estimating nonadditive, non-Gaussian, and time-correlated noises under the current estimation frameworks. In this way, the need for observational constraints for the stochastic perturbation methods in the NWP community could be considered within the estimation framework discussed in this review.

## Acknowledgments

This work has been carried out as part of the Copernicus Marine Environment Monitoring Service (CMEMS) 3DA project. CMEMS is implemented by Mercator Ocean in the framework of a delegation agreement with the European Union. This work was also partially supported by FOCUS Establishing Supercomputing Center of Excellence. CEREA is a member of Institut Pierre Simon Laplace (IPSL). Author Carrassi has been funded by the project REDDA (250711) of the Norwegian Research Council. Carrassi was also supported by the Natural Environment Research Council (Agreement PR140015 between NERC and the National Centre for Earth Observation). We thank Paul Platzer, a second-year Ph.D. student, who helped to popularize the summary and the introduction, and John C. Wells, Gilles-Olivier Guégan, and Aimée Johansen for their English grammar corrections. We also thank the five anonymous reviewers for their precious comments and ideas to improve this review paper. We are immensely grateful to Prof. David M. Schultz, Chief Editor of *Monthly Weather Review*, for his detailed advice and careful reading of the paper.

## APPENDIX

### Four Main Algorithms to Jointly Estimate $\mathsf{Q}$ and $\mathsf{R}$ in Data Assimilation

Algorithm 1 is an adaptive algorithm for the EnKF as implemented by Miyoshi et al. (2013). The steps of the algorithm are the following:

- initialize inflation factor [for instance *λ*(1) = 1)];

**for** *k* in 1:*K* **do**

** for** *i* in 1:*N*_{e} **do**

- compute forecast

- compute innovation **d**_{i}(*k*) using Eq. (4b);

**end**

- compute empirical covariance

- compute ^{f}(*k*) using Eq. (4c) where *λ*(*k*);

** for** *i* in 1:*N*_{e} **do**

- compute analysis

**end**

- compute mean innovations **d**^{o−f}(*k*) and **d**^{o−a}(*k*) with

- update *k*) from Eq. (6b) using the cross-covariance between

- estimate *λ*(*k*);

- update *λ*(*k* + 1) using temporal smoother;

**end**

Algorithm 2 is an adaptive algorithm for the EnKF by Berry and Sauer (2013). The steps of the algorithm are the following:

- initialize

**for** *k* in 1:*K* **do**

** for** *i* in 1:*N*_{e} **do**

- compute forecast

- compute innovation **d**_{i}(*k*) using Eq. (4b);

**end**

- compute ^{f}(*k*) using Eq. (4c);

** for** *i* in 1:*N*_{e} **do**

- compute analysis

**end**

- apply Eq. (12a) to get **M**_{k} and **H**_{k} given in Eqs. (5a) and (5b);

- estimate

- estimate

- update *k* + 1) and *k* + 1) using temporal smoothers;

**end**

Algorithm 3 is an adaptive algorithm for the EnKF from Stroud et al. (2018). The steps of the algorithm are the following:

- define a priori distributions for ** θ** (shape parameters of

**for** *k* in 1:*K* **do**

** do** *i* in 1:*N*_{e} **do**

** **- draw samples *θ*_{i}(*k*) from *p*[** θ**|

**y**(1:

*k*− 1)];

** **- compute forecast *θ*_{i}(*k*);

** **- compute innovation **d**_{i}(*k*) using Eq. (4b) with *θ*_{i}(*k*);

**end**

** **- compute ^{f}(*k*) using Eq. (4c);

** for** *i* in 1:*N*_{e} **do**

** **- compute analysis

**end**

- approximate Gaussian likelihood of innovations *p*[**y**(*k*)|**y**(1:*k* − 1), ** θ**(

*k*)] using empirical mean

- update *p*[** θ**|

**y**(1:

*k*)] using Eq. (16);

**end**

Algorithm 4 is an EM algorithm for the EnKF/EnKS from Dreano et al. (2017). The steps of the algorithm are the following:

- define ** θ** (shape parameters of

- set *p*[**y**(1:*K*)|*θ*_{(0)}] = +∞;

- initialize *n* = 1, *θ*_{(1)} and ϵ (stop condition);

**while** *p*[**y**(1:*K*)|*θ*_{(n)}] − *p*[**y**(1:*K*)|*θ*_{(n−1)}] > *ϵ***do**

** for** *k* in 1:*K* **do**

** for** *i* in 1:*N*_{e} **do**

** **- compute forecast

** **- compute innovation **d**_{i}(*k*) using Eq. (4b);

**end**

** **- compute ^{f}(*k*) using Eq. (4c);

** for** *i* in 1:*N*_{e} **do**

** **- compute analysis

**end**

**end**

** for** *k* in *K*:1 **do**

** **- compute ^{s}(*k*) using Eq. (4e);

** for** *i* in 1:*N*_{e} **do**

** **- compute reanalysis

**end**

**end**

** **- increment *n* ← *n* + 1;

** **- estimate _{(n)} using Eq. (21a);

** **- estimate _{(n)} using Eq. (21b);

**end**

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