Climate signals are the results of interactions of multiple timescale media such as the atmosphere and ocean in the coupled earth system. Coupled data assimilation (CDA) pursues balanced and coherent climate analysis and prediction initialization by incorporating observations from multiple media into a coupled model. In practice, an observational time window (OTW) is usually used to collect measured data for an assimilation cycle to increase observational samples that are sequentially assimilated with their original error scales. Given different timescales of characteristic variability in different media, what are the optimal OTWs for the coupled media so that climate signals can be most accurately recovered by CDA? With a simple coupled model that simulates typical scale interactions in the climate system and “twin” CDA experiments, we address this issue here. Results show that in each coupled medium, an optimal OTW can provide maximal observational information that best fits the characteristic variability of the medium during the data blending process. Maintaining correct scale interactions, the resulting CDA improves the analysis of climate signals greatly. These simple model results provide a guideline for when the real observations are assimilated into a coupled general circulation model for improving climate analysis and prediction initialization by accurately recovering important characteristic variability such as sub-diurnal in the atmosphere and diurnal in the ocean.

Currently, the interactions between the earth climate system's major components, such as the atmosphere, ocean, land, and sea ice, have been reasonably simulated by coupled climate models, which can also give the evaluation of climate changes (Randall et al., 2007). However, because of the uncertainties and errors in models (e.g., parameterization is only an approximation to sub-grid processes and the dynamical core is imperfect), models always tend to produce different climate features and variability from the real world (e.g., Delworth et al., 2006; Collins et al., 2006; Zhang et al., 2014). Due to the significant importance of preserving the balance and coherence of different model components (or media) during the coupled model initialization, data assimilation for state estimation and prediction initialization should be performed within a coupled climate model framework (e.g., Chen et al., 1995; Zhang et al., 2007; Chen, 2010; Han et al., 2013). The characteristic variability timescales of different media within the coupled frameworks are usually different. When the observed data included in one or more components of the coupled system framework are assimilated, the observational information will be able to be transferred among different media through the coupled dynamics so that all media gain consistent and coherent adjustments. Such an assimilation procedure is called coupled data assimilation (CDA), which can sustain the nature of multiple timescale interactions during climate estimation and prediction initialization (e.g., Zhang et al., 2007; Sugiura et al., 2008; Singleton, 2011), thus producing better climate analysis and prediction initialization and therefore improving the coupled models' predictability (Yang et al., 2013). Zhang et al. (2007) developed the first CDA system in a fully coupled general circulation model, version 2 of the Geophysical Fluid Dynamics Laboratory Coupled Model (GFDL CM2). The National Centres for Environmental Prediction (NCEP) also started using coupled models to generate first-guess forecasts for their Climate Forecast System Reanalysis (CFSR, Saha et al., 2010). Despite the enormous benefits and demand for CDA, it remains both theoretically and technically challenging to implement strong CDA in fully coupled models, including the estimation of the coupled model error covariance matrix and the huge computational costs (e.g., Han et al., 2013; Lu et al., 2015; Liu et al., 2016).

During the coupled data assimilation process, an observational time window (OTW) is usually used to collect measured data in each medium for an assimilation cycle (e.g., Pires et al., 1996; Hunt et al., 2004; Houtekamer and Mitchell, 2005; Laroche et al., 2007) to increase observational samples. As in Hunt et al. (2004), we expand the EnKF to include a time window in which the observations are treated as the exact assimilation times, even though their times are different in the window. That is, we just assume that all the collected data sample the “truth” variation at the assimilation time and will be sequentially assimilated with their original error scales. Thus the OTW is applied in a three-dimensional data assimilation fashion rather than a four-dimensional one. Apparently, while a large OTW provides more observational samples at the assimilation time, the assimilation process blends more data from different times and may distort the variability being retrieved. Given the fact that climate signals are the results of interactions of multiple timescale media, correct variability retrieved for each medium so that correct scale interaction is maintained in CDA is particularly important for climate analysis and prediction initialization. In this study we attempt to answer the following two questions. (1) What is the impact of varying OTWs for each coupled component within the coupled model framework on the quality of CDA? (2) Based on this impact, does an optimal OTW exist so that assimilation fitting has maximum observational information but minimum variability distortion?

With a simple conceptual coupled climate model and a sequential implementation of the ensemble Kalman filter, this study first analyses the characteristic variability timescale of each coupled medium and identifies the corresponding optimal OTW. Then the impact of an optimal OTW on the quality of CDA and its linkage with the corresponding timescale of characteristic variability are investigated. The simple coupled model consists of three typical components, including the synoptic atmosphere (Lorenz, 1963) and the seasonal–interannual slab upper ocean (Zhang et al., 2012) coupling with the decadal deep ocean (Zhang, 2011a, b). Although the simple conceptual coupled model does not share the similar complex physics with a coupled general circulation model (CGCM), it does reasonably simulate the typical interactions between multiple timescale components in the coupled climate system (see Zhang et al., 2013). The simple coupled model helps us understand the essence of the problem by revealing the relationship between the optimal OTWs and corresponding timescales of characteristic variability as well as their impact on CDA. The low-cost nature of the simple model also provides convenience for a large number of CDA experiments with different OTWs in optimal OTW detection. The ensemble Kalman filter (e.g., Evensen, 1994, 2007; Whitaker and Hamill, 2002; Anderson, 2001, 2003) used in this study is the ensemble adjustment Kalman filter (EAKF, e.g., Anderson, 2001, 2003; Zhang and Anderson, 2003). Using the EAKF with the simple coupled model, we first establish a twin experiment framework. Within such a framework, the degree to which the state estimation based on a certain OTW recovers the truth is an assessment of the influence of the OTW on the quality of CDA. In such a way, the optimal OTW of each medium is detected and the impact of optimal OTWs on CDA is evaluated. We also discuss the influence of model bias on an optimal OTW through biased twin experiment setting.

This paper is organized as follows. Section 2 briefly describes the simple conceptual coupled model, the ensemble adjustment Kalman filter, as well as the twin experiment framework including perfect and biased settings. Also with a simplest case, we first show the influence of OTWs on assimilation quality and its linkage with the timescale of characteristic variability in this section. Then Sect. 3 presents results on detection of the optimal OTWs for different media and the impact of optimal OTWs on CDA. The influence of realistic assimilation scenarios on optimal OTWs is discussed in Sect. 4. Finally, a summary and discussions are given in Sect. 5.

Due to the complicated physical processes and huge computational cost
involved, it is inconvenient to use a CGCM to investigate the impact of the
different OTWs on the analysis of climate signals so as to detect each
coupled medium's optimal OTW. Instead, here we employ a simple coupled
“climate” model developed by Zhang (2011a). This simple model is based on
Lorenz's three-variable chaotic model (Lorenz, 1963) that couples with a slab
upper ocean (Zhang et al., 2012) and a simple pycnocline predictive model
(Gnanadesikan, 1999). Although very simple with low computational cost, in
terms of multi-scale interaction inducing low-frequency climate signals, this
model shares a fundamental character with a CGCM, and it is very suitable for
addressing the problem that is concerned here. And for the readers'
convenience, here we simply review some key aspects of this conceptual
coupled model. With all quantities being given in non-dimensional units, the
governing equations are

Following the study of Han et al. (2014), the fourth-order Runge–Kutta
time-differencing scheme is used in this paper to resolve this simple coupled
model, and the time step equals 0.01 TU (1 TU

Zhang (2011b) illustrated that, given the model parameters described above, the constructed simple coupled model can effectively simulate a fundamental feature of the real-world climate system in which different timescales interact with each other to develop climate signals. That is, the synoptic to decadal timescale signals are produced by the interactions between the transient atmosphere attractor, the slow slab ocean, and the even slower deep ocean (see Zhang, 2011a; Han et al., 2014). Again, although the simple coupled model does not have complex physics and cannot consider the issue of impact of localization and imbalance as in a CGCM, it can help us investigate the fundamental issue we want to address here more directly and clearly.

Following Zhang (2011a), during the state estimation, the error statistics evaluated from ensemble model integrations, such as the error covariance between model states, will be used in an ensemble filter to extract observational information to adjust the model states (e.g., Evensen, 1994, 2007; Anderson, 2001; Hamill et al., 2000; Zhang, 2011a, b; Zhang et al., 2012; Han et al., 2014). In this study, a derivative of the Kalman filter (Kalman, 1960; Kalman and Bucy, 1961) called the ensemble adjustment Kalman filter (EAKF, Anderson, 2001, 2003; Zhang and Anderson, 2003; Zhang et al., 2007), which is a sequential implementation of the ensemble Kalman filter under an “adjustment” idea, is used to implement the CDA scheme. The assumption of independence of observational error allows the EAKF to sequentially assimilate observations into corresponding model states (Zhang and Anderson, 2003; Zhang et al., 2007). While the sequential implementation provides much computation convenience for data assimilation, the EAKF maintains as much of the nonlinearity of background flows as possible (Anderson, 2001, 2003; Zhang and Anderson, 2003).

Based on the two-step implementation of the EAKF scheme (Anderson, 2001,
2003), the observational increment at an observation location is first
computed. The observation is denoted as

Although many sophisticated inflation algorithms (e.g., Anderson, 2007, 2009; Li et al., 2009; Miyoshi, 2011) exist for atmosphere data assimilation, the inflation scheme for a coupled model is a new subject due to the multiple-timescale nature of the system. Furthermore, trial-and-error experiments show that the usual form of inflation (e.g., only inflate the atmosphere model states or inflate all the model states equally) will lead to the analysis becoming unstable. Thus, in this paper, for simplicity and computational convenience as well as convenience for comparison, no inflation is used in our assimilation experiments, just as in Han et al. (2014).

The schematic for the assimilation interval, the length of the OTW,
as well as the observational interval in terms of the model integration time
step. Here

In this study, a perfect twin experiment framework and a biased twin
experiment framework are designed, respectively. In both perfect and biased
twin experiments, a “truth” model using the standard parameter values
listed in Sect. 2.1 is used to generate the “true” solution of the model
states and produce the observations sampling the “truth”. Starting from the
initial condition (0, 1, 0, 0, 0), the “truth” model is firstly integrated
forward for 10 000 TUs (i.e., 1000 model years) for sufficient spinup and
then integrated forward for another 10 000 TUs to generate the “truth”
model states. The observations are produced by sampling the “truth”
solution of the model states at an observational interval and superimposing
with a white noise simulating the observational errors. As schematically
shown in Fig. 1, all the observational intervals used in this study are
assumed to be 1 time step (0.01 TU). Although in the real climate system,
the oceanic observations are usually available less frequently than those in
the atmosphere (that is, the oceanic observation interval is larger than that
we set here), for this proof-of-concept study we will set the time interval
of the oceanic observations as small as possible. The standard deviations of
the observational errors are 2 for

We first want to learn some basics from the perfect experiment which
represents an idealized data assimilation regime. In the perfect twin
experiment framework, the assimilation model also uses the standard parameter
values, but starts from different initial conditions using the Gaussian white
noises with the same standard deviation as observational errors (2 for

Then we use the biased experiment setting to simulate the real-world scenario. The biased twin experiment framework is similar to the perfect one except that the assimilation model in the biased twin experiment framework has a systematic discrepancy from the observations. Thus, in the biased twin experiment framework, the parameters included in the assimilation model will have 10 % errors relative to the standard values. The errors in the parameters will be the only model error source.

Figure 1 also illustrates the assimilation update intervals (the
assimilation intervals are 5 time steps for atmosphere, 20 time steps for the
slab ocean in all assimilation experiments, and 100 time steps for the deep
ocean when using the

In order to exhibit the influence of the OTW on the quality of climate
analysis, we show three simple assimilation experiments (the time series of

Time series of the absolute errors of the slab ocean variable
(

From Fig. 2, we can see that a small OCN-OTW (a total of 11 observations in the oceanic OTW) can make a much better ocean analysis than the standard CDA (comparing the red line with the green line). We can understand that this is because an OTW can provide more observational information, thus enhancing the observational constraint so as to improve the accuracy of climate analysis. However, comparing the blue line to the green/red line, it is clear that an overly large OTW degrades the quality of the ocean analysis. The results of these simple assimilation experiments tell us that, if an appropriate OTW is used, we can gain optimal climate analysis. How can we determine such an optimal OTW? Next, starting from analyzing the characteristic variability of each coupled medium, we will discuss the methodology of how to determine an optimal OTW for each medium in a coupled climate system.

The power spectrum (green) of

The key to improving the accuracy of climate analysis in CDA is by accurately recovering the characteristic variability of different media in the coupled system. Thus we can assume that the length of an optimal OTW for each medium will have some relationship with the corresponding characteristic variability timescale. Then, we should first analyze the timescale of characteristic variability in each medium.

Figure 3 presents the power spectrum of

An optimal OTW aims to provide maximal observational information that best
samples the characteristic variability of that medium during the data
blending process. Thus the length of the optimal OTW should be smaller than
the corresponding characteristic variability timescale, which means that the
optimal OTW in the atmosphere must be much smaller than 1 TU (100 time
steps), and in the ocean, the optimal OTW must be much smaller than 50 TUs
(5000 time steps). If we take observations for

Variations of root mean square errors (RMSEs) of

In this section, with the perfect model framework described in Sect. 2.3, we first conduct a series of CDA experiments with different ATM-OTWs and different OCN-OTWs to detect the optimal OTW for each medium. The assimilation scheme is the simple univariate adjustment scheme serving as a proof-of-concept study. To eliminate the dependency of results on initial states, each experiment is repeated 20 times starting from the 20 independent initial conditions described in Sect. 2.2. Then the mean value and the spread of 20 cases of RMSEs are plotted in Fig. 4.

Figure 4a shows that the optimal ATM-OTW is 1; i.e., the optimal ATM-OTW includes only three atmosphere observations, with which the assimilation produces the lowest RMSE of the atmosphere and the smallest spread. (In this study each assimilation experiment will be repeated 20 times starting from 20 different independent initial ensemble conditions. Here the spread just represents the standard deviation of these 20 cases' results. Thus it will be smallest when using the optimal OTW.) In these experiments for detecting the optimal ATM-OTW, the ocean assimilation is kept as the standard setting (i.e., no OTW, 0.2 TU update interval). Then we keep the ATM-OTW as 1 and change the length of OCN-OTW to produce Fig. 4b.

From Fig. 4b, we can see that the optimal OCN-OTW is about 10 (i.e., each
OTW includes a total of 21 observations), with which the lowest

Same as panels

To further understand the relationship between the optimal OTW and
characteristic variability timescale, we also examine the

We also check the variation of the 20-case mean ensemble spread in the space
of OTWs as shown in Fig. 5. The mean and standard deviation of the ensemble
spreads of

The auto-correlation coefficient of

To understand the essence of optimal OTWs, we show the auto-correlation for
each model state and mark the time correlation coefficients at the timescales
of optimal OTWs for

Same as Fig. 4 but using a multi-variate adjustment scheme. In panel

In this section, we first show the impact of the multi-variate adjustment scheme on the optimal OTWs in a perfect model setting. Then we discuss the influence of model bias through a biased model framework. We will also investigate the impact of coupling strength on the optimal OTWs.

While the experiments with the univariate adjustment scheme provide us with a
direct understanding of the influence of the OTWs on CDA, we want to check
whether or not it also applies to the multi-variate adjustment scheme. So we
repeat the experiments described in Sect. 3 but with the multi-variate
adjustment scheme. The results are shown in Fig. 7. Here the multi-variate
adjustment scheme is only limited to the atmospheric observations (i.e., only
the cross-covariances among

The perfect experiment framework provides a direct guideline for the relationship between the optimal OTW and the corresponding characteristic variability timescale. However, in reality, the numerical model has errors and is biased with the observation. It is as necessary to investigate the influence of model bias on optimal OTWs as on the quality of CDA.

Same as Fig. 4 but using the biased model setting. In
panel

With the biased model experiment framework described in Sect. 2.3, we repeat
all the experiments above for detection of the optimal OTWs. The results are
shown in Fig. 8. Compared to the results in the perfect model setting, the
results in the biased model setting have two differences. First, the optimal
ATM-OTW and OCN-OTW are larger than their counterparts in the perfect model
setting, becoming 3 and 20 (that is, the total observations are 7 and 41,
respectively). Second, the RMSE curves in the space of OTWs show more
concavity and sensitive variation. This is more distinguishable in the curve
of

The power spectrum of

Panel

Comparing the results from two experiment frameworks, we can see that regardless of perfect or biased model setting used in the assimilation experiments, the optimal OTW must be associated with the corresponding characteristic variability timescale in the medium. It is clear that while using observations in an OTW increases observational information, an overly large OTW can distort the characteristic variability of coupled media during the information blending process. Therefore choosing an optimal OTW that is much smaller than the medium's characteristic variability timescale is very important. The simple model results suggest that the length of an optimal OTW is about 1–5 % of the medium characteristic timescale, with which characteristic variability of the medium can be retrieved most accurately.

In this study, the OTW validates the observations in a time window to the
analysis time and all the observations included in the OTW are sequentially
assimilated with their original error scales. Another general approach is to
assimilate the average of the observations included in the OTW, but the
observational errors decrease as

Also, among the above assimilation experiments in this study, we have not considered the temporal offset induced by the difference between the time of observations in the OTW and the analysis time. Here we can use the de-correlation coefficients to weight the observations included in the OTW and avoid overweighting them. The comparison of these two assimilation approaches (non-weighted and weighted) has been conducted (the results are not shown). From the comparison we learn that the lengths of the optimal OTWs obtained by these two assimilation schemes are similar, except that the RMSEs in the weighted observation experiment will be lower than that in the non-weighted one when using longer OTWs (when the length of ATM-OTW is greater than 4 and/or that of the OCN-OTW is larger than 50). This is owing to the high correlation between the observation included in the optimal OTWs and model states at the analysis time (exceeding 0.995). Thus the influence of the temporal offset can be ignored and the results obtained by these two schemes will almost be the same when using the shorter OTWs. When we use the longer ones, the correlation will decrease and the influence of the temporal offsets will be obvious in that the results of the weighted observation experiment will be better. For the CDA systems in the CGCMs, owing to the complex physics and dynamics, the influence of the time offsets will be obvious and the weights of the observations will be very necessary. But from this simple model case, we can see that regardless of using the weighted observations, the relationship between the characteristic variability timescales and the optimal OTWs will be robust, and the essence of this study is established.

Changing the coupling strength (controlled by the coupling coefficients

Then we examine the difference in the optimal OTW of

On the one hand, these experiments can further illustrate the idea that a close relationship between the length of the optimal OTW and the corresponding characteristic variability timescale exists. On the other hand, for a realistic CDA system, the coupling physics could be very complicated and affected by many factors. The results of this simple model give the insights that when determining the length of the optimal OTWs for a realistic CDA system, we can only consider such factors that have an obvious influence on the characteristic variability timescales. In this way, the process of determining the optimal OTWs in a realistic CDA system can be greatly simplified and make it possible to apply the method of using the optimal OTWs to the realistic CDA system.

With a simple conceptual climate model and the EAKF method, the impact of OTWs on the quality of CDA has been investigated in this study. This simple conceptual coupled model consists of a synoptic atmosphere (Lorenz, 1963) and seasonal–interannual slab upper ocean (Zhang et al., 2012) coupling with a decadal deep ocean (Zhang, 2011a, b), and reasonably simulates the typical interactions between multiple timescale components in the climate system. Determined from the characteristic variability timescale in each coupled medium, an optimal OTW provides maximal observational information to best fit the characteristic variability of the medium during the data blending process. With correct scale interactions within the coupled system, CDA can recover the climate signals most accurately by incorporating all observations in the optimal OTWs into the coupled model, although in an idealized and simple model circumstance, the conclusion addressing the best fitting characteristic variability in each medium with the optimal OTW is comprehensive and therefore provides a guideline for improving climate analysis and prediction initialization when real observations are assimilated into a CGCM. For example, as learned from the simple model results, we may consider improving the quality of climate analysis and prediction initialization by accurately recovering some important characteristic variability in the atmosphere (sub-diurnal variations, for instance) and ocean (diurnal cycle in the tropical oceans, for instance).

However, the current work can only serve as a proof-of-concept study. Although CDA with the optimal OTWs has shown promising improvement in this simple model, serious challenges still exist for detecting optimal OTWs in the real world with a CGCM for improving climate analysis and prediction. First, the characteristic variability timescales in different media of the real world are complex, and great challenges remain to identify the characteristic variability of the different component models and the real atmosphere and upper and deep ocean, which need to be further studied. Also, in a real ocean model, the upper and deep ocean is inseparable, which bring some troubles in using different OTWs for different parts of the same ocean model. Second, due to model biases, characteristic variability in a CGCM may be different from the real world. The combination of variability of the real world and that of the model may further complicate the problem. Therefore, model bias and its influence on model variability need to be thoroughly analyzed before an optimal OTW is determined. Thirdly, the coupling physics between different coupled components are very complicated and are impacted by many factors for a realistic CDA system. Even though we only consider the factors which will obviously impact the characteristic variability timescales when determining the length of OTWs for different coupled components, it remains a heavy workload. In addition, in this study we assume that all observations in the OTWs have equal weights to contribute to the observational constraint. In the real observation case, the observation far away from the assimilation time should have less contribution to the state estimation at the assimilation time. How to take the time correlation into account in a sequential algorithm needs to be studied before implementing optimal OTWs in the assimilation with CGCM and real observations.

Data can be obtained by contacting the author Xiong Deng (xiongdeng407@hrbeu.edu.cn).

The authors declare that they have no conflict of interest.

This work was supported by National CMOST Key research & development projects 2017YFC1404100 and 2017YFC1404102, the NSFC (nos. 51379049, 41676088, and 41775100), the Fundamental Research Funds for the Central Universities of China (nos. HEUCFX41302, HEUCFD1505, and HEUCF160410), the Young College Academic Backbone of Heilongjiang Province (no. 1254G018), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Heilongjiang Province (no. LC2013C21), and Harbin Engineering University and China Scholar Council (awarded to Xiong Deng for two and a half years' study abroad at UW-Madison – NOAA/GFDL Joint Visiting Program). We thank Liwei Jia, Wei Zhang, Xuefeng Zhang, Wei Li, Lianxin Zhang, and Shuo Yang for their comments and suggestions on the early version of this manuscript. Also, special thanks to three anonymous reviewers for their critical comments that contributed to great improvements in the original manuscript. Edited by: Amit Apte Reviewed by: three anonymous referees