Impact of Optimal Observational Time Window on Parameter Optimization and Climate Prediction : Simulation with a Simple Climate Model

Usually, an optimal time window (OTW) centred at the assimilation time to collect measured data for an assimilation cycle, can greatly improve the CDA analysis skill. Here, with a simple coupled 10 model, we study the impact of optimal OTWs on the quality of parameter optimization and climate prediction. Results show that the optimal OTWs of valid atmosphere or ocean observations exist for the parameter being estimated and incorporating the parameter optimization will do some impact on the optimal OTWs for the state estimation. And using the optimal OTWs can enhance the predictability both of the atmosphere and ocean. 15


Introduction
Because of the imperfect model equations, numeric schemes, and physical parameterizations, as well as the biased model parameters, climate models always drift away from the real world (e.g., Delworth et al., 2006;Collins et al., 2006;Zhang, 2011a,b;Zhang et al., 2012;Wu et al., 2012Wu et al., ,2013b;;Liu et al. 2014a,b;Han et al., 2014;).Parameter optimization, which includes the model parameters into control 20 variables, is a promising way to partly compensate for the bias of the values of the model parameters and improve the climate predictability(e.g.Zhang, 2011a,b;Zhang et al., 2012Zhang et al., ,2013b;;Wu et al., 2012Wu et al., ,2013;;Liu et al. 2014a,b;Han et al., 2014;).
In the words of Han et al. (2013), given the importance of the balance and coherence of different model components (or media) in coupled model initialization, it has been realized that for the purpose of 25 climate estimation and model initialization, data assimilation (including model state estimation and parameter optimization) should be performed within a coupled model framework which can reasonably simulate the interaction of major components of the earth climate system, such as the atmosphere, ocean, land, and sea ice and give the assessment of climate changes (e.g.Chen et al., 1995;Zhang et al., 2007;Randall et al., 2007;Chen, 2010;).And in the coupled climate system, the time scale and 30 characteristic variability in different media are usually different.When the observational data in one or more media are assimilated into a model, information is exchanged among different media and between model states and parameters of the couple system.Such an assimilation procedure can sustain the nature of multiple time-scale interaction during climate estimation (e.g.Zhang et al., 2007;Sugiura et Nonlin.Processes Geophys.Discuss., doi:10.5194/npg-2015Discuss., doi:10.5194/npg- -76, 2016 Manuscript under review for journal Nonlin.Processes Geophys.Published: 21 January 2016 c Author(s) 2016.CC-BY 3.0 License. al., 2008), thus producing coherent and balanced coupled model initialization and parameters that may enhance model predictability (e.g., Zhang, 2011b;Yang et al., 2013).
In each component of a coupled data assimilation system, usually an observational time window (OTW) centred at the assimilation time is used to collect measured data (valid observations) for an assimilation cycle, assuming that all the collected data sample the observation at the assimilation time, and the 5 assimilation scheme assimilates all of these valid observations within the OTW into the coupled model states and parameters sequentially.As the previous study (Zhao et al., 2015, manuscript submitted to J. Climate) has shown that there is an optimal OTW in each coupled component for model state estimation so that the assimilation has maximum observational information but minimum variation inconsistency and the optimal observational time windows analyzed from the characteristic variability 10 time scales of coupled media can significantly improve climate analysis and prediction initialization since it helps recovering some important character variability such as sub-diurnal cycle in the atmosphere and diurnal cycle in the ocean.And the larger the characteristic variability time scale is, the larger the corresponding OTW is.The model parameters are lack of direct observations and prognostic equations, parameter optimization completely relies on the covariance between a parameter and the 15 model state (e.g., Zhang, 2011a,b;Zhang et al., 2012;Wu et al. 2012Wu et al. ,2013;;Han et al., 2014;Liu et al. 2014a,b).Thus the observational time window (OTW) of the model state in each media of the coupled climate system will do some impact on the quality of parameter optimization and climate prediction.
Questions we attempt to answer in this study are: 1) Whether or not exists an optimal OTW of atmosphere or ocean observations for parameter optimization so that the assimilation has maximum 20 observational information but minimum variation inconsistency? 2) What is the impact of optimal OTWs of atmosphere or ocean observations on parameter optimization and climate prediction?
In this study, with a simple coupled model and the DAEPC algorithm (Zhang et al., 2012) which is based on the ensemble adjustment Kalman filter (EAKF, e.g.Anderson, 2001;2003;Zhang and Anderson, 2003;), starting from the characteristic variability time scale of each coupled component and 25 model parameter, we first identify the optimal OTW for each component and parameter optimization.
Then we examine the impact of optimal OTWs on parameter optimization and climate prediction.The simple coupled model consists of chaotic (synoptic) atmosphere (Lorenz 1963) and seasonalinterannual slab upper ocean (Zhang et al., 2012) that couples with decadal deep ocean (Zhang 2011a,b).Although the simple coupled model does not have complex physics as a coupled general 30 circulation model (CGCM), it does characterize the interaction of multiple time-scale media in the climate system (Zhang et al., 2013;Zhao et al., 2015, manuscript

2.1
The model Because of the complex physical processes and huge computation cost involved, it is not convenient to use a CGCM to study the influence of observational time window on the quality of parameter optimization and climate prediction (e.g., Zhang 2011a,b;Zhang et al., 2012;Han et al., 2013Han et al., , 2014;;10 Zhao et al., 2015, manuscript submitted to J. Climate).Instead, here we employ a simple decadal prediction model developed by Zhang (2011a).Same as Zhang (2011a), this simple decadal prediction model is based on the Lorenz's 3-variable chaotic model (Lorenz, 1963) and couples the three Lorenz chaotic atmosphere variables to a slab ocean model (e.g., Zhang 2011a,b;Zhang et al., 2012;Han et al., 2013Han et al., ,2014;;Zhao et al., 2015, manuscript submitted to J. Climate) and a simple pycnocline predictive 15 model (e.g., Gnanadesikan, 1999;Zhang 2011a,b;Han et al., 2013Han et al., ,2014;;Zhao et al., 2015, manuscript submitted to J. Climate).This simple coupled model shares the similar fundamental features with the CGCMs to investigate the problems in this study (e.g., Zhang 2011a;Han et al., 2014;Zhao et al., 2015, manuscript submitted to J. Climate).The governing equations of this simple coupled climate model are as follow: where the five model variables represent the atmosphere ( 1 ,  2 and 3 )and the upper ocean (ω for the upper slab ocean) and the deep ocean (η for the deep ocean pycnocline).A dot above the variable denotes the time tendency.The atmosphere model variables are of high frequency and the standard define the magnitudes of the annual mean and seasonal cycle of the external forcing, which are not sensitive to the coupled model and set as (10,1).The coefficients  1 and  2 in the equations of  2 and ω are chosen as (0.1,1), which realize the coupling between the fast atmosphere and the slow slab ocean, and the  1 represents the slab ocean forcing on the atmosphere and  2 in contrast.In addition,  3

5
and  4 denote the linear forcing of the deep ocean and the nonlinear interaction of the slab and deep ocean.For guaranteeing the dominant role of the interaction between atmosphere and the slab ocean in the slab ocean model, the magnitudes of  3 and  4 are smaller than that of  2 and set as 0.01 in this study.In this simple coupled climate model, the seasonal cycle is defined as 10TUs, and thus a model year (decade) is defined as 10(100)TUs.In the words of Zhang (2011a), the deep ocean pycnocline 10 model state η represent the anomaly of the deep ocean pycnocline depth and its time tendency equation is derived from the two-term balance model of the zonal-time mean pycnocline (Gnanadesikan, 1999).
And in the equation ofη, the parameter Γ is a constant of proportionality and the ratio of Γ and   defines the time scale of η.Because η is a deep ocean variable, its time scale is larger than that of the slab ocean variable ω.Here the time scale of η is defined as ~O(100), 10 times of the time scale of ω, namely Γ is set as 100. 5 and  6 denote the linear forcing of the slab ocean and the nonlinear interaction of the slab and deep ocean.Also for guaranteeing that the linear interaction is stronger than the nonlinear interaction and the nonlinear interaction in the slab ocean model is stronger than that in the deep ocean pycnocline model, the number magnitudes of  5 is larger than that of  6 and the magnitudes of  4 is larger than that of 6 .Here,  5 and  6 are set as (1,0.001).So in this study, the  (9.95,28,8/3,0.1,1,1,10,10,1,10,100,0.01,0.01,1,0.001)(e.g., Zhang 2011a,b;Zhang et al., 2012;Han et al., 2013Han et al., ,2014;;Zhao et al., 2015, manuscript submitted to J. Climate).
Zhang (2011b) illustrated that, this simple coupled climate model with the standard parameters 30 described above can share the common feature that different components of various timescales interact with each other to develop climate signals with the real world climate system.In the words of Han et al. (2014), in this simple coupled model, the transient atmosphere attractor, the slow slab ocean and the even-slower deep ocean interact to produce synoptic decadal timescale signals (see Zhang, 2011a;Han et al., 2014;Zhao et al., 2015, manuscript submitted to J. Climate).
And the EAKF algorithm is a sequential implementation of ensemble Kalman filter (Kalman, 1960;Kalman and Bucy, 1961) under an "adjustment" idea.The assumption of independence of observation 15 error allows the EAKF to sequentially assimilate observations into corresponding model states and parameters (Zhang, 2003;Zhang et al., 2007).On one hand the EAKF algorithm can provide much computation convenience for data assimilation and parameter optimization, on the other hand it can maintains much the non-linearity of background flows as much as possible (e.g., Anderson, 2001;2003;Zhang and Anderson, 2003).

20
Based on the two-steps of EAKF (Anderson, 2001;2003), the first step computes the observational increment (e.g., Zhang et al., 2007;Wu et al., 2012Wu et al., ,2013) ) using where ∆ , denotes the observational increment of the th ensemble member of the th observation  , ;  ̅   is the posterior mean of the th observation; ∆ ́, is updated ensemble spread of the th 25 observation for the ensemble member;  ,  is the th prior ensemble member of the th observation.
Once the observation increment is computed as above, it can be projected onto related model variables and parameters using the following uniform linear regression formula: Where ∆ , represents the observation increment of  , and (  ,   ) defines the error covariance  Zhang et al., 2012;Wu et al., 2012Wu et al., ,2013;;Han et al., 2014;Liu et al., 2014a,b;).The application of Eq.( 4) to the coupled model states when the reliable

35
observations are available implements CDA for state estimation in a straight forward manner (Zhang et al., 2007;Zhang 2011a;) parameter-state covariance and have an enhancive parameter correction with observation information, the application of Eq.( 4) for parameter optimization must be delayed until the coupled model state estimation reaches a quasi-equilibrium, where the errors of model states become mainly contributed from model parameter errors.(e.g., Zhang, 2011a,b;Zhang et al., 2012;Wu et al., 2012Wu et al., ,2013;;Han et al., 2014;Liu et al., 2014a,b;).Once the model parameters are optimized by the Eq.( 4), the updated 5 parameters will further promote the state estimates in the next data assimilation cycle.
In addition, the inflation scheme is essential for the parameter optimization.In this study, the inflation scheme for the DAEPC algorithm follows Zhang et al. (2012), which is formulated as Same as Zhang et al. (2012), β ℓ and β ̃ℓrepresent the prior and the inflated ensemble of the ℓth parameter.
10 σ ℓ, and σ ℓ,0 are the prior spreads of β ℓ at time  and the initial time.α 0 is the constant tuned by a trialand-error procedure.σ ℓ is the sensitivity of the model state with regard to β ℓ (see the similar experiment of model sensitivities on parameter described in Zhang (2011b) and Zhang et al. (2012)).And the β ̅ ℓ represents the ensemble mean.The Eq.( 5) indicates that if the prior spread of β ℓ is less than α 0 σ ℓ ⁄ times the initial spread, it will be enlarged to this amount (e.g., Zhang 2011a.b;Zhang et al., 15 2012;Wu et al., 2012Wu et al., ,2013;;Han et al., 2014;Liu et al., 2014a,b;).

Biased twin experiment framework setup
In this study, a bias twin experiment framework is designed.Same as Zhang (2011a), in the biased twin experiment, while the "truth" model uses the standard parameter values listed in section 2.1, the assimilation model uses biased parameter values that have 10% overestimated error than the 20 corresponding standard values.In this study we assume that the parameter errors are the only source of model errors.The "truth" model produces the true solution of the model states and observations are sampled from the "truth".The model starts from the initial condition (0,1,0,0,0) and integrates forward 10000TUs (1TU= 100∆) for sufficient spin-up.After the spin-up, the model integrates for another 10000TUs for producing the "truth" and observations.The observations are produced through sampling

25
the "true" model state values at an observational frequency and superimposed with a white noise which simulates the observational error.Here, the observational intervals of all the valid observations in this simple coupled model are all assumed to be 1 time step.(Although in the real observation system, the atmosphere observations are available more frequently than the ocean and less frequently than the time step.In this study, we are concerned about the influence of the observational time window on the 30 quality of coupled data assimilation (including the state and parameter estimation) and the optimal observational time window for each component and model parameters.If the observational intervals are set too large, it may damage the characteristic variability.Thus, we hope the observational time interval is small as much as possible.So, for simplicity, the observational intervals are all set to be 1 time step).The standard deviations of the observation errors are 2 for  1 ,  2 ,  3 and 0. assimilation experiments.And it also starts from the initial condition (0,1,0,0,0) and spins up for 10000TUs.A Gaussian white noise with the same standard deviations as the corresponding observational errors (2 for  1 ,  2 and 3 , 0.5 for ω, 0.06 for η) is added on the model states at the end of spin-up to form the ensemble initial condition.In all of the assimilation experiments, same as the previous studies (e.g., Zhang 2011a,b;Han et al., 2013;2014), the assimilation intervals are set to be 5 time steps for  1 ,  2 ,  3 and 20 time steps forω, respectively.The total data assimilation period is 10000TUs, and parameter optimization is started after 3000TUs when state estimation reaches its "quasi-equilibrium" (e.g., Zhang, 2011a,b;2012;Wu et al., 2012Wu et al., ,2013;;Han et al., 2014).And another 2000TUs will be the spin-up of the parameter optimization.All statistics are computed using the results of the last 5000TUs.In this study, the observations including in the observational time windows In order to investigate the impact of the OTWs on climate prediction, we conduct some forecast experiments aiming to the five cases (SPE_Without_OTW, SPE_With_S_P_OTW, SPE_With_S_OTW, SPE_With_P_OTW, SEO_OOTW).Table 1 lists the details of the twin 5 experiment frameworks.

Impact of OTWs on the quality of the parameter optimization and climate prediction
In this section, under the biased twin experiment framework, we will show the influence of OTWs on the quality of the parameter optimization and climate predictability.process, but also it may complex the investigation in this study and be not suitable in reality.Thus in this study we only consider the case that the state estimation and parameter optimization use the same OTWs.
In the biased experiment framework, the coupled models set with the biased values of all the parameters and initialized from the perturbed ensemble initial conditions.The CTL experiment is set

30
The first step we set the OCN-S-P-OTW as 0 (means a single observation and no window).Next, we use L to represent the length of an OTW, meaning that the OTW includes L valid observations at either side of the assimilation time, so the total number of observations within the OTW is 2L+1.Fig. 1 shows the root-mean-square errors (RMSEs) of  1,2,3 ω and η ,where the  1,2,3 is the arithmetical average of the RMSEs of the atmosphere model sates.

35
From Fig. 1, we can learn that the optimal ATM-S-P-OTW and OCN-S-P-OTW for state estimation are But for the parameter being estimated, the optimal OTWs are 0 and 20, respectively.And the RMSE of the parameter being estimated () is reduced about 37% but increase about 28% compared to the 5 experiment of SEO and SPE_Without_OTW, respectively, when using the optimal OTWs for state estimation (2 and 10).Figure 2 shows that the optimal OTWs (ATM-S-OTW and OCN-S-OTW) for state estimation are about 1 and 17, respectively, with the evidence of the lowest RMSEs of the 1,2,3 and ω.And the RMSEs of the 1,2,3 , ω and η are respectively reduced about 20% (42%), 61% (17%) and 15%(2%) compared to the experiment of SPE_Without_OTW (SEO_With_OOTW).Also the optimal OTWs for state estimation are smaller than the corresponding ones in the SEO_With_OOTW experiment (3 and 21, respectively).But the optimal OTWs for the parameter being estimated are about 0 and 6, respectively.And the RMSE of the parameter being estimated () are reduces about 38.4% but increases about 11% compared to the experiment of SEO and SPE_Without_OTW, respectively, when 20 using the optimal OTWs for state estimation (1 and 17).
In the SPE-P-OTW experiment, we only use the OTWs for parameter estimation and assimilate the observations at the assimilation time into the model states.The results are as Fig. 3. Also from Fig. 3, we can learn that the optimal OTWs (ATM-P-OTW and OCN-P-OTW) for state estimation are 0. The results are as same as the SPE_Without_OTW experiment.But for the RMSE of 25 the parameter being estimated, the optimal OTWs are about 0 and 20, respectively.And the RMSE of the parameter are reduced less than 2% compared to the experiment of SPE_Without_OTW experiment when the OTWs are set as 0 and 20, respectively.
The results of above experiments show that the optimal OTWs (ATM-S-P-OTW and OCN-S-P-OTW in the SPE_With_S_P_OTW; ATM-S-OTW and OCN-S-OTW in the SPE_With_S_OTW) for state 30 estimation are smaller than the corresponding ones in the SPE_OOTW experiment.And the RMSE of the model states are reduced greatly when using these optimal OTWs for state estimation.But these optimal OTWs are not optimal for the parameter optimization.And the optimal OTWs (ATM-S-P-OTW and OCN-S-P-OTW in the SPE_With_S_P_OTW; ATM-P-OTW and OCN-P-OTW in the SPE_With_P_OTW) for parameter optimization are about 0 and 20, respectively.

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The ATM-S-OTW and OCN-S-OTW aims to projecting more of the observational information of state variables onto the model state estimation and then do some impact on the parameter estimation with the observations at the assimilation time.And the ATM-P-OTW and OCN-P-OTW aim to projecting more of the observational information of state variables onto the model parameter being estimated and then do some impact on the model states in the next assimilation cycle.So adjusting the ATM-S-OTW and OCN-S-OTW (ATM-P-OTW and OCN-P-OTW) will do some impact on the estimation of the model parameter (states).
Each parameter in the coupled climate model in this study takes a globally uniform value and the characteristic variability time scales can be considered as 0, which are much smaller than those of the model states and cause that the optimal OTWs for state estimation are smaller than the corresponding 5 ones in the SEO_With_OOTW experiment (3/21).And the optimal OTWs of atmosphere observations for parameter optimization are much smaller than those of slab ocean observations, which is owing to characteristic variability time scales of the atmosphere model states are much smaller than that of the slab ocean model state.And when using the optimal OTWs of slab ocean observations for parameter optimization, the RMSE of the parameter being estimated are reduced slightly (less than 5%), which is 10 owing to that the parameter () is not sensitive to the slab ocean observations.

Impact of the OTWs on climate prediction
Compared to the SEO_Without_OOTW experiment, above three experiments improve the quality of state or parameter estimation to some degree, but we are not sure that which case (SEO_OOTW, SPE_Without_OTW, SPE_With_S_P_OTW (OTWs are set as 2 and 10, respectively),

15
SPE_With_S_OTW (OTWs are set as 1 and 17, respectively), SPE_With_P_OTW (OTWs are set as 0 and 20, respectively); the RMSEs of the model states in the SPE_With_S_P_OTW case is smallest and the RMSE of the model parameter being estimated in the SPE_With_P_OTW case is smallest) is of the best skill of prediction.Thus we will conduct some prediction experiments to investigate the impact of the OTWs on the climate prediction.

20
We launch 20 forecasts (each forward up to 50TUs (5000 time steps)) with the initial conditions selected every 50TUs apart during 8000-9000TUs.And in this twin experiment framework, we evaluate forecast skills using the anomaly correlation coefficient (ACC) and root-mean-square-error (RMSE) of forecasts verified with the "truth" (Zhang 2011b;Zhang et al. 2012).And the ACCs and RMS errors of typical "weather" forecasts ( 2 , in 1.5TUs, for instance), SI (ω, in 5-10TUs) and     x , w ,  and parameter k with respect to the lengthen of the ATM-S-OTW, respectively, at the condition that the x , w ,  and parameter k with respect to the lengthen of the OCN-S-OTW, respectively, with the optimal ATM-S-OTW as 1.
submitted to J. Climate).The simple model helps us understand the essence of the problem we want to address here.Using the DAEPC algorithm with the simple coupled model, we first establish a biased twin experiment framework where the degree to which the state and parameter estimation based a certain OTW recovers the truth is an 35 assessment of the influence of the OTW on the quality of parameter optimization and climate prediction.With this biased twin experiment framework, we identify the optimal OTW for the model parameters and examine the impact of optimal OTWs on the quality of parameter optimization and climate prediction.
Nonlin.Processes Geophys.Discuss., doi:10.5194/npg-2015Discuss., doi:10.5194/npg--76, 2016     Manuscript under review for journal Nonlin.Processes Geophys.Published: 21 January 2016 c Author(s) 2016.CC-BY 3.0 License.This paper is organized as follow.Section 2 briefly describes the simple coupled model, the ensemble adjustment Kalman filter for state estimation and parameter optimization and the biased twin experiment framework.Then the influence of OTW on the quality of the parameter optimization and climate prediction are investigated in section 3. Summary and discussions are given in section 4.

30 between
the prior ensemble of the model state or parameter variables and the model estimated observation ensemble.σ  is the standard deviation of the model estimated ensemble of   .The term ∆ , is the contribution of the th observation to the model state or parameter variables for the th ensemble member (e.g., Zhang 2011a.b; . However because of the model parameters are lack of the internal variability and prognostic equation, effective parameter estimation is very difficult before the uncertainty of model states have been sufficiently constrained by observation.And in order to achieve a signal-dominant Nonlin.Processes Geophys.Discuss., doi:10.5194/npg-2015-76,2016 Manuscript under review for journal Nonlin.Processes Geophys.Published: 21 January 2016 c Author(s) 2016.CC-BY 3.0 License.

10 ( 30 (
scheme.) 20 climate model, the characteristic variability time scale at which the flow varies in different media is different.Sustaining the characteristic variability of the different components in the coupled model is the key to improving the quality of the coupled data assimilation.If an OTW is too large, it may damage the characteristic variability of the model, which will adversely impact the state estimation.Thus, the OTW must be smaller than the corresponding characteristic variability time scale 20 of the component.As the previous study(Zhao et al., 2015, manuscript  submitted to J. Climate) has shown that the characteristic variability time scale of the model atmosphere ( 2 ), upper ocean (ω) and deep ocean (η) is about 1TU, 10TUs (1 model year) and 100TUs (1 model decade), respectively, through the power spectrum analysis.And the optimal Atmosphere observation time window (ATM-OTW) and slab ocean observation time window (OCN-OTW) in the CDA_NoMul_OTW_bias 25 experiment are about 3 and 21, respectively, (Here 3/21 represent that there are 3/21 valid observations beside each side of the Central Time Point (right at the assimilation time, namely there are 7/43 valid observations within the ATM-OTW/OCN-ATW) which is much smaller than the corresponding characteristic variability time scale of each component (100/1000).And the larger the characteristic variability time scale is, the larger the corresponding optimal OTW is.30 In this study, each model parameter in the coupled climate model takes a globally uniform value, which do not change with time.Thus, we can think that the characteristic variability time scales of the model parameters in this study are all about 0TU.And the characteristic variability time scales of the model states are much larger than those of the model parameters.Using the observational information of the model states to correct the biased model parameters will do some impact on the optimal 35 observational time windows for the parameter optimization and climate prediction.
Nonlin.Processes Geophys.Discuss., doi:10.5194/npg-2015-76,2016 Manuscript under review for journal Nonlin.Processes Geophys.Published: 21 January 2016 c Author(s) 2016.CC-BY 3.0 License.windows for the model parameters In order to study the impact of the observational time window on the quality of the parameter optimization, we should set two observational time windows for the parameter optimization.Here there are two ways to establish these two observational time windows: the model state and parameter estimation use the same or different observational time windows (ATM-OTW and OCN-OTW).When 5 the model state and parameter estimation use the same observation time windows, there are two observational time windows for model state and parameter optimization (ATM-S-P-OTW and OCN-S-P-OTW).Otherwise, there are two observational time windows for the state estimation and another two observational time windows for the parameter optimization (ATM-S-OTW, OCN-S-OTW and ATM-P-OTW, OCN-P-OTW).But the four OTWs case not only complex the adjusting and computation 10

15 without
the observational constraint and the model ensemble is integrated for 10000TUs, serving as the reference to other assimilation experiments in the biased twin experiment framework.The SEO experiment just assimilate the observations at the assimilation time into the model states without observational time windows and parameter optimization and the SEO_OOTW experiment uses the optimal observational time windows (3 and 21 for ATM-OTW and OCN-OTW,Zhao et al., 2015,   20    manuscript submitted to J. Climate) for state estimation.And in this study we consider that characteristic variability time scales of all the model parameters are same (0 TU).So for simplicity, we can choose one parameter to investigate the impact of the observational time windows on the parameter optimization.In this study we choose the model parameter  (the standard value is 28 and overestimated value is 2.8, namely the RMSE of the parameter is 2.8 if without parameter optimization) 25 to conduct the SPE experiments.As the SEO experiment does, the SPE_Without_OTW just assimilate the observations at the assimilation time into the model state and parameter estimation.In the SPE_With_S_P_OTW, the state estimation and parameter optimization use the same observational time windows.In the SPE_With_S_P_OTW experiment, there are two OTWs, which collect the valid atmosphere and slab ocean observations, called the ATM-S-P-OTW and OCN-S-P-OTW, respectively.

25 decadal
scheme with the coupling covariance between the model states in different media is the important effective methods to improve the accuracy of coupled model state and parameter estimation.So the impact of the optimal observational time window on the multi-variate adjustment scheme using the 10

Figure 2 .
Figure 2. The same as Fig. 1 but only using the observational time windows for state estimation.abcg) represent the RMSEs of the 1,2,3x , w ,  and parameter k with respect to the lengthen of the ATM-S-OTW, respectively, at the condition that the Geophys.Discuss., doi:10.5194/npg-2015-76,2016   Manuscript under review for journal Nonlin.Processes Geophys.Published: 21 January 2016 c Author(s) 2016.CC-BY 3.0 License.OCN-S-OTW is set as 0. And defh) represent the RMSEs of the 1,2,3

5 Figure 3 .
Figure 3.The same as Fig.1 but only using the observational time windows for parameter estimation.abcg) represent the RMSEs of the 1,2,3x , w ,  and parameter k 10 And the parameters   and   in the equation of ω represent the heat capacity and damping coefficient of the upper ocean, respectively.The frequency of ω is much lower than that of the atmosphere model variables, thus the slab ocean model state must have a much slower time scale than atmosphere model 30 variables and the heat capacity should be much larger than the damping rate, namely   ≫   .Here   is set as 10, which represents that the period of the external forcing is similar Nonlin.Processes Geophys.Discuss., doi:10.5194/npg-2015-76,2016 Manuscript under review for journal Nonlin.Processes Geophys.Published: 21 January 2016 c Author(s) 2016.CC-BY 3.0 License.with the time scale of the upper ocean and defines the time scale of the model seasonal cycle.  and