1. Introduction
Turbulent dynamical systems are ubiquitous in many disciplines of contemporary science and engineering (Hinze and Hinze 1959; Townsend 1980; Frisch 1995; Majda and Wang 2006; Vallis 2006; Salmon 1998). They are characterized by both a large dimensional phase space and a large dimensional space of instability with positive Lyapunov exponents. These linear instabilities are mitigated by energy-conserving nonlinear interactions, yielding physical constraints (Majda and Harlim 2013; Sapsis and Majda 2013b; Majda and Harlim 2012; Harlim et al. 2014), which transfer energy to the linear stable modes where it is dissipated resulting in a statistical steady state. Both understanding complex turbulent systems and improving initializations for prediction require filtering for an accurate estimation of full state variables from noisy partial observations. Since the filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model, coping with model errors is of wide contemporary interest (Majda and Harlim 2012; Majda 2012).
Many dynamical models of turbulent systems are summarized as conditional Gaussian systems (Majda and Harlim 2012; Majda 2003; Kalnay 2003; Majda and Gershgorin 2013; Majda et al. 1999). Despite the conditional Gaussianity, such systems nevertheless can be highly nonlinear and able to capture the non-Gaussian features of nature (Berner and Branstator 2007; Neelin et al. 2010). In this paper, we introduce a general conditional Gaussian framework for continuous-time filtering. The conditional Gaussianity means that once the trajectories of the observational variables are given, the dynamics of the unobserved variables conditioned on these highly nonlinear observed trajectories become Gaussian processes. One of the desirable features of such conditional Gaussian filter is that it allows closed analytical formulas for updating the posterior states associated with the unobserved variables (Liptser and Shiryaev 2001) and is thus computationally efficient.
The recently developed nonlinear filter for filtering the stochastic skeleton model for the Madden–Julian oscillation (MJO; Chen and Majda 2016a) belongs to the conditional Gaussian filtering framework, where equatorial waves and moisture were filtered given the observations of the highly intermittent envelope of convective activity. Another application of this exact nonlinear filter involves filtering turbulent flow fields utilizing observations from noisy Lagrangian tracer trajectories (Chen et al. 2014c, 2015), where an information barrier was shown as increasing the number of tracers (Chen et al. 2014c) and a multiscale filtering strategy was studied for the system with coupled slow vortical modes and fast gravity waves (Chen et al. 2015). In addition, a family of low-order physics-constrained nonlinear stochastic models with intermittent instability and unobserved variables, which belong to the conditional Gaussian family, was proposed for predicting the MJO and the monsoon indices (Chen et al. 2014b; Chen and Majda 2015b,a). The effective filtering scheme was adopted for the online initialization of the unobserved variables that facilitates ensemble prediction algorithm. Other applications that fit into the conditional Gaussian framework includes the cheap exactly solvable forecast models in dynamic stochastic superresolution of sparsely observed turbulent systems (Branicki and Majda 2013; Keating et al. 2012), stochastic superparameterization for geophysical turbulence (Majda and Grooms 2014), and blended particle filters for large-dimensional chaotic systems (Majda et al. 2014; Qi and Majda 2015) that captures non-Gaussian features in an adaptively evolving low-dimensional subspace through particles interacting with conditional Gaussian statistics on the remaining phase space.
In this paper, we illustrate three types of applications of the conditional Gaussian filtering framework, where the effect of model error is extensively studied. In addition to the traditional pathwise measures, an information-theoretic framework (Branicki and Majda 2014; Branicki et al. 2013; Majda and Branicki 2012; Majda and Wang 2006) is adopted to assess the lack of information and model error in filtering these turbulent systems.
The first application involves utilizing dyad models (Majda and Lee 2014; Majda 2015) to study the effect of model error due to the ignorance of energy-conserving nonlinear interactions in forecast models in filtering turbulent signals from nature. Such model error exists in many ad hoc quadratic multilevel regression models (Kravtsov et al. 2005; Kondrashov et al. 2005; Wikle and Hooten 2010; Cressie and Wikle 2011) that are utilized as data-driven statistical models for time series of partial observations of nature. However, these models were shown to suffer from a finite-time blow up of statistical solutions (Majda and Yuan 2012; Majda and Harlim 2013). To understand the effect of such model error in filtering, a physics-constrained dyad model (Majda and Lee 2014) is adopted to generate the turbulent signals of nature while a stochastic parameterized model without energy-conserving nonlinearities (Majda and Harlim 2012) is adopted as the imperfect filter. The skill of this stochastic parameterized filter is studied in different dynamical regimes and the lack of information in the filter estimates is compared with that using the perfect filter. Meanwhile, the role of observability (Gajic and Lelic 1996; Majda and Harlim 2012) is explored and its necessity in filtering turbulent systems is emphasized.
The second application of the conditional Gaussian framework is to filter a family of triad models (Majda 2003; Majda et al. 1999, 2001, 2002), which include the noisy Lorenz 63 (L-63) model (Lorenz 1963). The goal here is to explore the effect of model error due to noise inflation and underdispersion in designing filters. The motivation of studying such a kind of model error comes from the fact that many models for turbulence are underdispersed since they have too much dissipation (Palmer 2001) due to inadequate resolution and deterministic parameterization of unresolved features. On the other hand, suitably inflating the noise in imperfect forecast models are widely adopted to reduce the lack of information (Anderson 2001; Kalnay 2003; Majda and Harlim 2012) and also to suppress catastrophic filter divergence (Harlim and Majda 2010a; Tong et al. 2016). Besides filtering a single trajectory, recovering the full probability density function (PDF) associated with the unobserved variables given an ensemble of observational trajectories is also of particular interest. Combining the ensembles of the analytically solvable conditional Gaussian distribution associated with filtering each unobserved single trajectory, an effective conditional Gaussian ensemble mixture approach is proposed to approximate the time-dependent PDF associated with the unobserved variables. In this fashion, an efficient algorithm can be generated for systems with a large number of the unobserved variables, compared with applying a direct Monte Carlo method, which is extremely slow and expensive due to the “curse of dimensionality” (Majda and Harlim 2012; Daum and Huang 2003).
Parameter estimation in turbulent systems is an important issue and this is the third topic within the conditional Gaussian filtering framework of this paper. Regarding the model parameters as augmented state variables, algorithms based on particle or ensemble Kalman filters were designed for parameter estimation (Dee 1995; Smedstad and O’Brien 1991; Van Der Merwe and Wan 2001; Plett 2004; Wenzel et al. 2006; Campillo and Rossi 2009; Harlim et al. 2014; Salamon and Feyen 2009). Although many successful results utilizing these algorithms were obtained, very little mathematical analysis was provided for exploring the convergence rate and understanding the potential limitation of such algorithms. Guidelines for enhancing the efficiency of the algorithms are desirable since a short training period is preferred in many real-world applications. In the conditional Gaussian framework, the closed analytic form of the posterior state estimations facilitates the analysis of both the error and the uncertainty in the estimated parameters for a wide family of models, where detailed mathematical justifications are accessible. Here, focus is on the parameter estimation skill dependence on different factors of the model as well as the observability. In some applications, certain prior information of the parameters is available (Yeh 1986; Iglesias et al. 2014). Yet, none of the existing filtering-based parameter estimation approaches emphasizes exploiting such prior information in improving the algorithms. In this paper, stochastic parameterized equations (Majda and Harlim 2012), involving the prior knowledge of the parameters, are incorporated into the filtering algorithm as the underlying processes of the augmented state variables. This improved algorithm greatly enhances the convergence rate at the cost of only introducing a small model error and it is particularly useful when the system loses practical observability.
The remainder of this paper is as follows. The general framework of the conditional Gaussian nonlinear systems is introduced in section 2. In section 3, an information-theoretic framework for assessing the model error in filtering is proposed. The information measures compensate the insufficiency of the pathwise ones in measuring the lack of information in the filtered solutions. Section 4 deals with dyad models, where focus is on model error in filtering due to the lack of respecting the underlying physical dynamics of the partially observed system. In section 5, a general family of triad model is proposed and the noisy L-63 model is adopted as a test model for understanding the model error in noise inflation and underdispersion. In the same section, the conditional Gaussian ensemble mixture for approximating the PDF associated with unobserved variables is introduced and the model error in filtering the PDF utilizing imperfect models is studied. Section 6 involves parameter estimation, where the skill of estimating both additive and multiplicative parameters is illustrated with detailed mathematical analysis. The comparison of utilizing direct method and stochastic parameterized equations approach is shown for estimating parameters in both linear and nonlinear systems. Summary conclusions are included in section 7.
2. Conditional Gaussian nonlinear systems
3. An information-theoretic framework for assessing the model error
Despite their wide applications in assessing filtering and prediction skill, these pathwise measures fail to assess the lack of information in the filter estimates and the predicted states (Branicki and Majda 2014; Chen and Majda 2015b). As shown in Chen and Majda (2015b), two predicted trajectories with completely different amplitudes can have the same RMS error and anomaly pattern correlation. Undoubtedly, the solution having comparable amplitude as the truth is more skillful than the one with strongly underestimated amplitude, which misses all the extreme events (Majda et al. 2010b; Majda and Harlim 2012; Majda and Branicki 2012) that are important for the turbulent systems. Different from the indistinguishable skill utilizing the pathwise measurements, an information-theoretic framework including the measurement of the lack of information succeeds in discriminating the prediction skill of the two solutions.
In Branicki and Majda (2014), a systematic information-theoretic approach was developed to quantify the statistical accuracy of Kalman filters with model error and the optimality of the imperfect Kalman filters in terms of three information measures was presented. Another application of information theory is illustrated in Branicki and Majda (2015) for improving imperfect predictions via multimodel ensemble forecasts.
Following the general information-theoretic framework in Branicki and Majda (2014) and Chen and Majda (2015b), we consider three information measures in assessing the filtering skill.
- The Shannon entropy
where
- The relative entropy
- The mutual information
where
Each one of the three measures provides different information about the filtering skill. The mutual information 
Because of the importance of measuring the lack of information and quantifying the ability of capturing the extreme events in the filtered solutions, the relative entropy is included in assessing the filtering skill throughout this work. Along with the relative entropy, we nevertheless show the anomaly pattern correlation and the RMS error instead of the mutual information and the Shannon’s entropy since the readers are more familiar with these traditional pathwise measures. Yet, it is important to bear in mind that the mutual information and Shannon’s entropy are the surrogates of the pathwise measures in the information-theoretic framework. Note that there are many other scoring functions that are widely utilized in meteorology in assessing the filtering and ensemble prediction skill (Gneiting and Raftery 2007) and are related to the information measures. Yet, the relative entropy can be explained as a measure of information gain, which cannot be interpreted by the traditional scoring functions, and it is also widely adopted in assessing the model error in atmosphere and ocean science (Majda and Harlim 2012).
In the study of filtering the unobserved single trajectories in sections 4 and 5a, the pathwise filtering skill is assessed. Therefore, 
It is worthwhile remarking that although most of the focus of this paper is on assessing the lack of information in the pathwise filtering solutions, the information-theoretic framework also provides a general framework for quantifying the uncertainty using imperfect models in ensemble prediction. A detailed discussion is included in appendix E.
4. Dyad models
Many turbulent dynamical systems involve dyad and triad interactions (Majda 2015; Majda and Lee 2014; Majda et al. 2009). These nontrivial nonlinear interactions between large-scale mean flow and turbulent fluctuations generate intermittent instability while the total energy from the nonlinear interactions is conserved. In this and the next sections, we study the filtering skill of dyad and triad models, where the effect of different model errors is explored.
In this section, we utilize dyad models to understand the effect of model error due to the ignorance of energy-conserving nonlinear interactions in forecast models in filtering turbulent signals from nature. As discussed in section 1, such model error exists in many ad hoc quadratic multilevel regression models (Kravtsov et al. 2005; Kondrashov et al. 2005; Wikle and Hooten 2010; Cressie and Wikle 2011) for fitting and predicting time series of partial observations of nature, which were shown to suffer from finite-time blow up of statistical solutions and also have pathological behavior of the related invariant measure (Majda and Yuan 2012; Majda and Harlim 2013). Recently, a new class of physics-constrained nonlinear regression models were developed (Majda and Harlim 2013) and the application of these physics-constrained models in ensemble Kalman filtering is shown in Harlim et al. (2014) together with other recent applications to prediction (Chen et al. 2014b; Chen and Majda 2015a,b).
Below, the true signals are generated from the dyad model (8) with different noise levels 
Skill scores for filtering the unresolved process υ using the physics-constrained (perfect) filter in (8) and the stochastic parameterized (imperfect) filter in (9) as a function of 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
The skill scores in the dynamical regime A are shown in Figs. 1a and 1b. The physics-constrained perfect filter (8) has high filtering skill when 
(Dynamical regime A: With practical observability at the attractor). Comparison of the posterior mean estimation of υ across time using the physics-constrained (perfect) filter (8) and the stochastic parameterized (imperfect) filter (9). The time-averaged PDFs of the filtered solutions compared with the truth are also illustrated. (a) The situation with small noise 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
Next, we study the filtering skill in dynamical regime B, at the fixed point of which the system has no observability. Compared with regime A, significant deterioration of the filtering skill is found when 
(Dynamical regime B: Without practical observability at the attractor). As in Fig. 2. (a) The situation with small noise 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
To conclude, the energy-conserving nonlinear feedback plays a significant role in filtering the dyad model (8), especially with large noise 
5. Triad models
Trajectories of the noisy L-63 model (12). (a) 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
The noisy L-63 model (12) equipped with the parameters in (13) is utilized as a test model in this section. Below, we first study filtering the unresolved trajectories given one realization of the noisy observations. Then an efficient conditional Gaussian ensemble mixture approach is designed to approximate the time-dependent PDF associated with the unresolved variables, which requires only a small ensemble of the observational trajectories. In both studies, the effect of model error due to noise inflation and underdispersion is studied. The underdispersion occurs in many models for turbulence since they have too much dissipation (Palmer 2001) due to inadequate resolution and deterministic parameterization of unresolved features while noise inflation is adopted in many imperfect forecast models to reduce the lack of information (Anderson 2001; Kalnay 2003; Majda and Harlim 2012) and suppress the catastrophic filter divergence (Harlim and Majda 2010a; Tong et al. 2016).
a. Model error in filtering the unresolved processes
1) Filtering the deterministic L-63 system utilizing the imperfect forecast model with noise
The detailed derivation of Proposition 1 is shown in appendix B. The results in (16) imply that the error in the posterior mean estimation decays to zero in an exponentially fast rate regardless of the noise level 
Filtering the L-63 model (15) utilizing the imperfect noisy L-63 model (14). (a) Observing y and z and filtering x. The three noise levels 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
The qualitative conclusions are the same in the situation of observing x and filtering y and z (see Fig. 5b). The uncertainty in filtering z is larger than that in filtering y, since y is directly related in the observational process x in (14). The trajectories as a function of time are shown in Figs. 5c and 5d, with nonzero and zero noise 
These results indicate that noise inflation in the imperfect forecast model brings no error regarding the posterior mean estimation and a bounded posterior uncertainty after a short relaxation time provided that the signal is generated from the system with no stochastic noise.
2) Filtering the noisy L-63 system utilizing the imperfect forecast model with no noise in the filtering processes
Filtering the noisy L-63 model (12) utilizing the imperfect forecast L-63 model (17) with no noise in the filtering processes, where x is the observed variable and y and z are the variables for filtering. The noise 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
In Fig. 6d, we show the trajectories with 
3) Filtering the noisy L-63 system utilizing the imperfect L-63 forecast model with different noise amplitudes
Finally, we study the general situation that both the system that generates the true signal in (12) and the imperfect filter (14) contain nonzero noise. Again, we illustrate the situation with observing x and filtering y and z. The other case has the same qualitative results.
Figure 7 shows the filtering skill utilizing the imperfect filter (14), where the model error comes from either the noise 
Filtering the noisy L-63 model (12) utilizing an imperfect forecast model (14) with model error in the noise, where the observational variable is x and the variables for filtering are y and z. (a),(c),(e) Filtering skill as a function of the noise 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
Figure 8 illustrates the posterior mean estimation as a function of time compared with the truth in two underdispersive situations. In the case that the noise 
Comparison of the true signal (blue) and posterior mean estimation (red) in the underdispersion cases, where 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
Therefore, we conclude that underdispersion in the imperfect filter deteriorates the filtering skill while noise inflation within certain range introduces little error.
b. Recovering the time-dependent PDF of the unresolved variables utilizing conditional Gaussian mixture
One important issue in uncertainty quantification for turbulent systems is to recover the time-dependent PDF associated with the unobserved processes. In a typical scenario, the phase space of the unobserved variables is quite large while that of the observed ones remains moderate or small. The classical approaches involve solving the Fokker–Planck equation or adopting the Monte Carlo simulation, both of which are quite expensive with the increase of the dimension, known as the curse of dimensionality (Majda and Harlim 2012; Daum and Huang 2003). For conditional Gaussian systems, the PDF associated with the unobserved processes can be approximated by an efficient conditional Gaussian ensemble mixture with high accuracy, where only a small ensemble of observed trajectories is needed due to its relatively low dimension and is thus computationally affordable. Here, an ensemble of independent observed trajectories is usually obtained from repeated experiments, which is not uncommon in real applications. One typical example involves recovering the PDF of random turbulent flows utilizing Lagrangian tracers (Chen et al. 2014c, 2015), where each observed tracer trajectory evolves independently. Note that the idea here is similar to that of the blended method for filtering high-dimensional turbulent systems (Majda et al. 2014; Qi and Majda 2015; Slivinski et al. 2015; Sapsis and Majda 2013a).
Below, we provide a general framework of utilizing conditional Gaussian mixtures in approximating the time-dependent PDF associated with the unobserved processes. Although the test examples of this approach below are based on the 3D noisy L-63 system, this method can be easily generalized to systems with a large number of unobserved variables. This section deals with the situation with no model error. In section 5c, the skill of recovering the PDF in the appearance of the model error due to noise inflation or underdispersion is explored.
Now we utilize the noisy L-63 model (12) as a test model for the conditional Gaussian ensemble mixture idea in (19). Here we assume x is the observed process, while y and z are the unobserved ones. The tests with different noise 
Figure 9 shows the recovery of the first four central moments (i.e., mean, variance, skewness, and kurtosis) associated with the unobserved variable z with different L. For comparison, we also show the results by adopting the Monte Carlo simulation with a large ensemble number N = 50 000, which is regarded as the truth. Even with 
(a)–(d) Recovering of the mean, variance, skewness, and kurtosis associated with the marginal PDF associated with the unobserved variable z in the noisy L-63 model (12) utilizing the conditional Gaussian ensemble mixture approach (19) with different number of ensembles L in a perfect model setting. As comparison, the recovered statistics utilizing Monte Carlo simulation with N = 50 000 ensemble members are also included. The green dot in (c) indicates the largest skewness in the transition phase, which will be utilized in Fig. 10.
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
In Fig. 10a, we show the model error in (6) utilizing the conditional Gaussian ensemble mixture in recovering the marginal PDF of y at a short-term transition time 
The model error in (6) in recovering the marginal PDF associated with the unobserved variables y and z in the noisy L-63 model (12) utilizing the conditional Gaussian ensemble mixture approach in (19) with different L in a perfect model setting. (a) The model error in the PDF of y as a function of L at a short-term transition phase 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
We have also tested the model error dependence on the ensemble number N utilizing Monte Carlo simulations. To reach a comparable skill with 
c. Recovering the time-dependent PDF of the unresolved variables with model error
Now we study recovering the time-dependent PDF of the unresolved variables in the noisy L-63 model with model error. The true signal associated with the observed variable x is generated from model (12) and the imperfect model with model error in the noise levels in (14) is utilized to recover the unobserved PDFs. Below, the effects of the model error due to both noise inflation and underdispersion are explored.
First, we take 
Recovery of the marginal PDFs associated with the unobserved variables y and z in the presence of model error from noise inflation. The noisy L-63 model (12) with 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
Next, we take 
Recovery of the marginal PDFs associated with the unobserved variables y and z in the presence of model error from underdispersion of noise. The noisy L-63 model (12) with 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
6. Parameter estimation
Given an initial value 
Below we aim at studying the dependence of the error 
a. Estimating one additive parameter in a linear scalar model
It is clear from (26) that decreasing the noise 
Yet, comparing the formulas in (26) and (28), it is obvious that the parameter estimation framework utilizing stochastic parameterized equations (27) leads to an exponential convergence rate for both the reduction of the posterior uncertainty and the error in the posterior mean, which implies a much shorter training phase is needed in the framework in (27). The convergence rate is controlled by the tuning factors in the stochastic parameterized equations. Thus, with a suitable choice of (27b), the convergence rate is greatly improved at the cost of only introducing a small bias in parameter estimation.
b. Estimating one multiplicative parameter in a linear model
In Fig. 13, we show the posterior mean 
Comparison of estimating the multiplicative parameter 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
We first look at the parameter estimation skill utilizing the direct approach (30). Since 
c. Estimating parameters in cubic nonlinear models
We compare the results in (46) of the cubic nonlinear system (40) with those in (35) of the linear system (29). The most significant difference is the role of the noise 
Phase portrait of 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
Below, we study the parameter estimation skill within the framework in (22) utilizing the direct approach in three dynamical regimes as shown in Fig. 14, where regime I 
Parameter estimation of the cubic model (39) in different dynamical regimes. (a) The observational trajectories with 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
The posterior mean 
It is worthwhile remarking that if the noise 
Parameter estimation of the cubic nonlinear model (39) in regime III with 
Citation: Monthly Weather Review 144, 12; 10.1175/MWR-D-15-0437.1
7. Summary, conclusions, and discussion
In this paper, we study filtering the nonlinear turbulent dynamical system (1) through conditional Gaussian statistics. The special structure of the system allows closed the analytic form for the updates of the posterior states (section 2). Information measures (section 3) are adopted for assessing the model error and lack of information in filtering.
The role of energy-conserving nonlinear interactions in filtering the turbulent systems is studied in section 4 based on a dyad model (8). The lack of information in the stochastic parameterized filter (9) is large and the energy-conserving nonlinear feedback is found to be particularly important when the stochastic noise amplitude 
The model error in the stochastic forcing amplitudes is studied in section 5 where the L-63 model (a triad model) is adopted as a test model. Both mathematical analysis (Proposition 1) and numerical experiments (Fig. 7) support that noise inflation leads to little error in filtering the unobserved trajectory while significant model errors are found in the imperfect filter due to underdispersion (Figs. 6–8). An efficient conditional Gaussian ensemble mixture method (19) is proposed in approximating the time-dependent PDF of the unobserved processes, which requires only small ensembles (Fig. 10) and can be generalized to systems with a large number of unobserved variables. This is an alternative to the traditional finite ensemble filter, which is usually quite expensive and sometimes suffers from sampling issues. Again, noise inflation in the imperfect model leads to only a small model error (Fig. 11) while underdispersion results in an obvious gap in estimating the PDF, where a severe underestimation of the variance implies the failure of capturing extreme events (Fig. 12).
The conditional Gaussian framework also allows for systematical study of parameter estimation skill, where the parameters are regarded as the augmented state variables. The convergence rate of the estimated parameters depends largely on the observability. Without practical observability, a slow convergence rate is found utilizing the direct parameter estimation approach (22) (Proposition 2, Proposition 4). On the other hand, a suitable choice of the stochastic parameterized equations for the augmented state variables in (23) leads to an exponentially fast convergence rate at the cost of only introducing a small error (Proposition 3; Fig. 13). In estimating parameters in a cubic nonlinear system, the convergence rate varies in different dynamical regimes utilizing the direct approach (22). The solutions converge to the truth very quickly in a bimodal regime while an extremely slow convergence is found in the nearly Gaussian regime (Fig. 15). Adopting the stochastic parameterized equations (23) again improves the skill of parameter estimation significantly (Fig. 16).
In addition to the three examples extensively discussed in sections 4–6 and those mentioned in section 1, there are many other potential applications of the conditional Gaussian filtering framework in (1)–(3) in atmosphere and ocean sciences. One example is to adopt a simple linear ocean model and filter the ocean flows by fully observing a nonlinear noisy atmosphere, where the ocean–atmosphere coupling is via the noisy temperature flux at the surface with quadratic surface flux exchange. Another example involves recovering the temperature in the full Boussinesq system, where the temperature variable enters linearly into the observed momentum equation with noisy forcing. Furthermore, the conditional Gaussian filtering framework naturally applies to the equations of chemical reaction. In many of these topics, the unobserved dynamics are usually the reduced versions of their fully complex nonlinear physical systems. Yet, this is the typical strategy in dealing with the complicated atmosphere–ocean problems. In fact, the judicious choice of the simplified equations, which offsets part of the model error and the measurements error, has been justified to retain a high skill in many filtering issues (Kalnay 2003; Majda and Harlim 2012).
Finally, we comment on the three underlying assumptions for directly applying the conditional Gaussian filtering framework in (1)–(3): (i) observations are taken continuously in time, (ii) there are no measurement errors beyond the observational uncertainty, and (iii) the state variables collected in 
Developing a systematic framework for optimizing the stochastic parameterized equations will be useful for estimating parameters in more complex systems. The information-theoretic framework described in section 3 is a good candidate for this optimization, which remains as a future work. Other future works involve designing computationally affordable filters based on (3), following the guidelines provided in this work, for more complex turbulent systems. Noticeably, the conditional Gaussian framework in (1)–(3) is also quite useful in studying the ensemble prediction skill and quantifying the uncertainty with model error.
Acknowledgments
This research of A.J.M is partially supported by the Office of Naval Research Grant ONR MURI N00014-12-1-0912. N.C. is supported as a graduate research assistant on this grant. A.J.M also gratefully acknowledges the generous support of the Center for Prototype Climate Modeling of NYU Abu Dhabi Research Institute grant.
APPENDIX A
Observability of Continuous Systems
a. Observability of the dyad model (8)
b. Observability of the L-63 model (12)
On the other hand, if the observed variables are y and z and the filtering variable is x, direct calculations show that the system has no observability when 
APPENDIX B
Detailed Derivation of Proposition 1 in Section 5a of Triad Models
APPENDIX C
Detailed Derivations for Propositions 2–4 in Sections 6a and 6b of Parameter Estimation in Linear Models
a. Detailed derivations of Proposition 2
b. Detailed derivations of Proposition 3
c. Detailed derivations of Proposition 4
APPENDIX D
Detailed Derivations of Proposition 6 in Section 6c of Estimating the Multiplicative Parameter in the Special Cubic System Utilizing the Direct Approach
The derivation of (43) in Proposition 6c follows those in (C20) and (C25). Here, we derive (45).
APPENDIX E
Information-Theoretic Framework in Quantifying the Model Error in Ensemble Prediction
As mentioned in section 3, the information-theoretic framework also provides a general framework in quantifying the uncertainty using imperfect models in ensemble prediction. This appendix includes the details.
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