Impact of the Time Scale of Model Sensitivity Response on Coupled Model Parameter Estimation

While a coupled climate model reasonably simulates the interaction of major components (atmosphere, ocean, sea ice, land process etc.) of the earth climate system, and gives an assessment of climate changes (Randall et al., 2007), the simulated climate tends to drift away from the real world due to model errors, or model biases (e.g., Collins et al., 2006; Delworth et al., 2006; Smith et al., 2007). There are two types of major sources for model errors (e.g., Zhang et al., 2012). The first type is associated with the imperfect model structure, including an imperfect dynamical core, approximate parameterizations etc., which can be referred to as structural errors. The structural errors can be viewed as "built-in" model errors and are difficult to alleviate through a direct observation-correction process. The other type of model errors is induced by the errors in model parameters. Model parameters are introduced mostly in model physical parameterizations. Physical parameterization is an approximate expression of a certain physical process in the atmosphere and ocean. However, the values for most of the parameters are set empirically and are usually not optimal in the coupled system, despite tuning by a trail-and-error procedure in terms of better climatological fitting. For constraining model bias and quantifying forecast uncertainties, the problem of observation-based parameter estimation in climate modeling has attracted a great deal of attention (e.g., Forest et al., 2000; Andronova and Schlesinger, 2001; Knutti et al., 2002; Gregory et al., 2002).

Derived from data assimilation theory and methodology, parameter estimation (also referred to as parameter optimization in the literature) with observations has become a promising approach to mitigate model bias (e.g., Banks, 1992a, 1992b; Borkar and Mundra, 1999; Aksoy et al., 2006a, 2006b; Zhang et al., 2011), so as to constrain model climate drift with a particular variant for coupled models (Zhang et al., 2012). Customized from state estimation, the estimated parameters are traditionally updated with the frequency of observations available. This may work well in estimating atmospheric parameters because the quickly-varying atmospheric states may instantaneously respond to perturbations of such parameters (Zhang et al., 2012). However, for the parameters in the slow-varying media of a coupled system, such as the ocean, the model may take a long time to transfer the parameter uncertainties into the model states. The covariance used in traditional parameter estimation may be unreliable because, if the update cycle is too short, a signal-dominant covariance may not yet have established. But how does the time scale of the model sensitivity response impact on coupled model parameter estimation? Here, with a simple coupled model, we address this question.

The paper is organized as follows: After this introduction, section 2 presents the methodology, including a description of the model, the "twin" experiment and ensemble filter. The problem of traditional parameter estimation is examined in section 3. The dependence of robust state-parameter covariance on model sensitivity response time scales is also examined in this section. Section 4 presents the results of the impact of the model sensitivity response time scales on parameter estimation. Finally, a summary and discussion are given in section 5.

The impact of the time scales of model sensitivity response on coupled model parameter estimation is a fundamental issue. Here, to address this issue, we employ a simple conceptual coupled "climate" model developed by (Zhang et al., 2011), which is quite simple compared with a coupled general circulation model (CGCM). The simplified model does not change the nature of the problem and it is therefore a good tool to detect the problem and find a potentially deliverable solution for CGCMs. The simple conceptual model takes the following form: $$\left\{\begin{array}{1} \dot{x}_1=-\sigma x_1+\sigma x_2\\ \dot{x}_2=-x_1 x_3+(1+c_1w)kx_1-x_2\\ \dot{x}_3=x_1x_2-bx_3\\ O_{\rm m}\dot{w}\!=\!c_2x_2+c_3\eta+c_4w\eta-O_dw+S_{\rm m}+S_{\rm s}\cos(2\pi t/S_{pd})\\ \Gamma\dot{\eta}=c_5w+c_6w\eta-O_{\rm d}\eta\end{array}\right. \ \ (1)$$ Here, x₁, x₂ and x₃ are the high-frequency variables that represent the atmosphere, while w is a low-frequency variable that stands for the slab ocean, and η represents the slower-varying deep ocean pycnocline. A dot above a model variable denotes time tendency. The definition and standard values of the model parameters are shown in Table 1.

In climate and ocean modeling, we can use parameterization to approximate many physical variables, with one or more parameters playing important roles in the parameterizations. However, usually, a trial-and-error tuning procedure is used to heuristically set the values of such parameters, which could be a reasonable guess for the particular parameterization rather than an optimal guess for the whole coupled model (Zhang et al., 2012). The errors in the values of parameters are an important source that leads to the climate drift of the model away from the real world.

Figure 1.  Timelines for the "truth" model (a) and "biased" model (b). The "truth" model is first integrated for 3× 10⁴ TUs (1 TU = 100 steps) starting from the initiation condition (x₁, x₂, x₃, w, η) = (0, 1, 0, 0, 0) for sufficient spin-up and then integrated for another 7× 10⁴ TUs to generate the "truth" solution of the model states and produce the observations sampled from the "truth". As in the "truth" model, the "biased" model firstly runs for 3× 10⁴ TUs starting from the initiation condition for spin-up. Then, another 2× 10⁴ TUs are extended with the "biased" model to produce 20 independent initial ensemble conditions for each assimilation experiment. The independent initial ensemble conditions are produced by adding a white noise with the same standard deviation as observational errors on the model states apart each 1000 TUs during the second 2× 10⁴-TU biased model integration. Then, starting from these 20 independent initial ensemble conditions, each assimilation experiment using the "biased" model is integrated for 5× 10⁴ TUs, and the parameter estimation is activated after 20 TUs (2000 model integration steps). Finally, the data obtained in the last 3× 10⁴ TUs are used to calculate error statistics for evaluation.

In an ensemble-based filter, the error statistics evaluated from ensemble model integrations, such as the error covariance between model states and model parameters, is used to transform the observational information to optimize the parameters' values (Anderson, 2001; Yang and Delsole, 2009). Here, we choose the ensemble adjustment Kalman filter (EAKF) (Anderson, 2001) to conduct the state and parameter estimation. EAKF is a sequential implementation (Evensen, 1994) of the Kalman filter (Kalman, 1960; Kalman and Bucy, 1961) under an "adjustment" idea. While its sequential implementation is convenient for data assimilation, EAKF maintains the nonlinearity of background flows in the filtering process as much as possible (Anderson, 2001, 2003; Zhang and Anderson, 2003). There are two important steps in the implementation of EAKF parameter estimation. The first step is to calculate the ensemble observational increment, which is identical to state estimation. The second step projects the observational increment onto the relevant parameter. This step is key for us to understand the special perspective of parameter estimation. The default values of parameters can be viewed as the erroneously-set ones and the associated errors can be transferred into the model states through the model integration. Due to the high nonlinearity in a model, the errors of parameters can lead to the model errors. We can then apply the observational increments to the error covariance between the model states and prior parameter ensemble through a local least-squares filtering to perform the parameter estimation (optimization) (Anderson, 2001, 2003). This process can be formulated as \begin{equation} \label{eq2} \Delta\beta_i=\sum_{k=1}^K\frac{c(\beta,\Delta y_k)}{\sigma_k^2}\Delta y_{k,i} .\ \ (2) \end{equation} Here, ∆β_i stands for the adjustment amount for the ith ensemble member; k stands for the observational location or observed variables (in this study, for instance); ∆ y_k,i represents the observation increment of the ith ensemble member; c(β,∆ y_k) defines the error covariance between the prior ensemble of the parameter β and the model-estimated observation ensemble; and σ_k is the standard deviation for the model estimated ensemble. The detailed computational implementation is described in the Appendix.

We use a biased twin experiment framework in this study. On the one hand, a "truth" model is the model described in section 2.1, with standard values for all parameters. The "truth" model is used to generate the "true" solution of the model states and produce the observations sampling the "truth", and its timeline is shown in Fig. 1a. The method for generating "observations" is the same as employed in (Zhang et al., 2012). In order to simulate the feature of the real observing system, the observational intervals are set as 0.05 TU (The "TU" is non-dimensional time unit defined in Lorenz 1963. The physical sense is the time scale by which the "atmosphere" approximately goes through a lobe of the attractor. 1 TU = 100 steps) for x₁, x₂ and x₃, and 0.2 TU for w. The standard deviation of observational errors is 2 for x₁, x₂ and x₃, and 0.5 for w. There is no observation for deep ocean. The obtained "truth" solutions and observations are used in all the assimilation experiments described next. On the other hand, the "biased" model is a model that has one or more biased parameters. The estimated parameter P _error is erroneously guessed with a 50% overestimated error from its standard value ——namely, P _error=(1+0.5)P _true——and then perturbed by the Gaussian random noise centered at P _error with a standard deviation σ _est, which is 10% of the truth (σ _est=10%× P _true). The timeline of the "biased" model is shown in Fig. 1b. After sufficient spin-up, 20 independent initial ensemble conditions for each assimilation experiment are produced. Then, starting from these 20 independent initial ensemble conditions, each assimilation experiment with the biased model setting is integrated for 5× 10⁴ TUs. In this way, we minimize the dependence of the results on initial states. We analyze the mean value of 20 cases and the uncertainty evaluated from these cases.

In order to examine the impact of the time scales of model sensitivity response on coupled parameter estimation, the estimated parameter is updated with different frequencies of the observations, while model states are always updated with the same ones (0.05 TU for the "atmosphere" and 0.2 TU for the ocean) in all parameter estimation experiments (denoted as CDAPE). It should be emphasized that the atmospheric states are adjusted only by the observation of atmosphere (with an interval of 0.05 TU), while the oceanic states are updated only by the oceanic observations (with an interval of 0.2 TU). Here, we are concerned with parameter estimation for the parameters in slow-varying media, so we focus on the performance of oceanic parameters and the estimated parameter is adjusted only using the oceanic observations. According to (Zhang et al., 2012), to allow coupled model states to be constrained by observations sufficiently, the parameter estimation is activated after 20 TUs (2000 steps of model integrations), and the error statistics for evaluation are conducted with the data in the last 3× 10⁴ TUs. In addition, using the same parameters and initial ensemble condition as CDAPE, a free assimilation model control (without observational constraint —— denoted as CTL) and a state estimation only by coupled data assimilation (without parameter estimation——denoted as CDA) are conducted, serving as references for the evaluation of parameter estimation. The state estimation in CDA is same as that in CDAPE.

In order to analyze the results of parameter and model state estimation properly, the sum of the root-mean-square errors (RMSE _sum) for the model states (x₁,x₂,x₃,w,η), and the RMSE of the estimated parameter, are used to evaluate the result in each assimilation experiment (Pan et al., 2011, 2014). The RMSE _sum of model states is calculated as \begin{eqnarray} \label{eq3} {\rm RMSE}_{\rm sum}&=&\left(\dfrac{{\rm RMSE}_{x_1,{\rm PE}}}{{\rm RMSE}_{x_1,{\rm CTL}}}\right) +\left(\dfrac{{\rm RMSE}_{x_2,{\rm PE}}}{{\rm RMSE}_{x_2,{\rm CTL}}}\right)+\nonumber\\ &&\left(\dfrac{{\rm RMSE}_{x_3,{\rm PE}}}{{\rm RMSE}_{x_3,{\rm CTL}}}\right)+\left(\dfrac{{\rm RMSE}_{w,{\rm PE}}}{{\rm RMSE}_{w,{\rm CTL}}}\right)+\nonumber\\ &&\left(\dfrac{{\rm RMSE}_{\eta,{\rm PE}}}{{\rm RMSE}_{\eta,{\rm CTL}}}\right),\ \ (3) \end{eqnarray} where RMSE_{x1, PE} means the RMSE of model state x₁ in the parameter estimation experiment, while RMSE_{x1, CTL} means that in CTL. The RMSE of the model state or parameter is computed from the data obtained in the last 3× 10⁴ TUs of each assimilation experiment by the following equation: \begin{equation} \label{eq4} {\rm RMSE}=\sqrt{\frac{\sum_{i=T_{{\rm start}}}^{T_{\rm end}}(\overline{x_i}-x_{\rm true})^2}{(T_{\rm end}-T_{\rm start})}} , \ \ (4)\end{equation} where x stands for one of the model states (x₁,x₂,x₃,w,η) or the estimated parameter; x _true is the corresponding true value of x; and \(\overline{x_i}\) represents the ensemble mean of that obtained from each assimilation experiment (i is the index of time). T _start denotes the start time to calculate the RMSE and T _end is the end time.

Following (Zhang and Anderson, 2003), an ensemble size of 20 is used in all the assimilation experiments throughout this study.

In this section, we use a simple example to address the potentially unreliability of traditional parameter estimation, especially for slow-varying media of a coupled system. Then, we examine the model sensitivities, focusing on oceanic parameters, and investigate the reliability of state-parameter covariance within a short update interval.

Customized from state estimation in data assimilation, traditionally, a parameter is updated based on the frequency of observations available. To examine the performance of traditional parameter estimation, we first show the results of two experiments —— CDAPE_{Pc2(c2),O obs(w)} and CDAPE_{Pc5(c5),O obs(w)}——in which the oceanic parameters c₂ and c₅ are estimated with the interval of observations (I _obs) of w (see Table 2 for detailed descriptions). Here, I _obs represents 0.2 TU (every 20 model steps) (in real oceanic observations, it is usually daily). The subscript P_c2(c₂) [P_c5(c₅)] represents only parameter c₂ (c₅) being perturbed and estimated in each experiment and, similarly, O _obs(w) represents only the observations of w being used in parameter estimation, in which "obs" means the update interval of the parameter is the same as the observation interval of w being sampled. The examined parameter is perturbed, as mentioned in section 2.3, while other parameters remain as standard values. At the same time, four experiments—— CTL_c2, CDA_c2, CTL_c5, CDA_c5——are conducted, in which the subscript stands for the parameter being perturbed. The ensemble means of the estimated parameter varying with time in the first case of CDAPE_{Pc2(c2),O obs(w)} and CDAPE_{Pc5(c5),O obs(w)} are plotted as the red and blue lines in Fig. 2, while the black line represents the "truth" (both c₂ and c₅ take the value of 1).

From Fig. 2, we can see that, starting from a big error at the initial time, the estimated parameter c₂ (c₅) converges to the "truth" very quickly, but there are some big oscillations in the time series for each case. The biggest deviation for c₅ is about 0.1 (10% of the "truth" value), observed at around 1.17× 10⁴ TUs, while two bigger oscillations for c₂ occur at about 7.4× 10³ TUs and 2.82× 10⁴ TUs, having the biggest deviation of more than 12% from the "truth".

The corresponding time series of model states x₂ and w around the first big oscillation in the estimation experiment CDAPE_{Pc2(c2),O obs(w)} are shown in Fig. 3. Compared to the case of state estimation only by coupled data assimilation (without parameter estimation) CDA_c2, which is quite stable (solid black lines in Fig. 3), the estimated model states x₂ and w in CDAPE_{Pc2(c2),O obs(w)} deviate from the "truth" with a big amplitude (dashed lines in Fig. 3) around the times that the estimated parameter c₂ has computational oscillations. It is clear that the parameter estimation makes that problem. It is worth mentioning that such a deviating phenomenon can be frequently observed in other time periods and other cases starting from different initial conditions. Here, what is shown in Figs. 2 and 3 only serves as an example.

In fact, as the time scale of the "ocean" is relatively longer, the coupled model takes a longer time to respond to the perturbed oceanic parameters (2-10 TUs) (Zhang et al., 2012). Next, we examine the relationship between the time scale of model sensitivity response and parameter estimation frequency in a coupled model.

Figure 2.  Time series of the ensemble means for c₂ (red) and c₅ (blue) in the first case of 20 (described in section 2.3) parameter estimation experiments, CDAPE_{Pc2(c2 ),O obs(w)} and CDAPE_{Pc5(c5),O obs(w)}, with an update interval of 0.2 TU. The "truth" value (1 in this case) of both c₂ and c₅ is marked as the horizontal black line.

Figure 3.  An example of instability caused by insufficiently developed covariance in traditional parameter estimation. Shown is the time series of the errors of ensemble means of (a) x₂ and (b) w between 7300 and 7500 TUs in the first case of 20 experiments, CDAPE_{Pc2(c2),O obs(w)} (dashed line) (traditional CDAPE, with 0.2-TU update interval as observational interval). The errors of state estimation by coupled data assimilation without parameter estimation (CDA) are plotted in black as reference.

As our chief concern is the estimation of parameters in slow-varying media such as the ocean, the first step is to examine the sensitivity of the model to parameters related to the slab ocean (w). There are nine parameters (O _m,c₂,c₃,c₄,O _d, S _m,S _s,c₅,c₆) related to w in the simple model, and the sensitivity of w to each of them is examined in an experiment called SensT_w. To study the sensitivity, each examined parameter is perturbed by adding a white noise with 10% of the standard value as its standard deviation. Starting from the 20-member ensemble initial conditions described in section 2.3, the model ensemble for each parameter is integrated for another 10⁴ TUs. The model sensitivity to each parameter is assessed by examining the 20-case mean temporal evolution of the ensemble spread of model state w. The ensemble spread is calculated as follows at each step: \begin{equation} \label{eq5} w_{\rm std}(i)=\sqrt{\frac{\sum_{j=1}^N(w_{i,j}-\overline{w_i})^2}{N}} ,\ \ (5) \end{equation} where w _std(i) denotes the ensemble standard deviation of model state w at the ith step, w_i,j stands for the state of the ensemble member j, \(\overline{w_i}\) is the ensemble mean of each experiment, and N denotes the ensemble size. The time series of the 20-case mean of the ensemble standard deviation of w over 0-30 TUs for each of O _m, c₂, c₃, c₄, O _d, S _m, S _s, c₅ and c₆ is plotted in Fig. 4.

From Fig. 4, it is apparent that w differs in its sensitivity to different oceanic parameters. Specifically, c₂ is the quickest parameter for the w sensitivity response. It reaches saturation (no longer systematically increasing with time) by about 5-6 TUs. Following c₂ are O _d, O _m and S _m. We refer to these parameters as fast oceanic parameters. In contrast, the slowest parameters for the w sensitivity response are c₅ and c₆. Both show a similar sensitivity response time scale of 10-12 TUs to reach saturation. Such parameters are referred to as slow oceanic parameters.

Figure 4.  Time series of ensemble spread of w when each parameter is perturbed by a Gaussian noise for the cases of c₂ (black), O _m (pink), c₃ (blue), c₄ (yellow), O _d (cyan), S _m (dotted green), S _s (solid green), c₅ (red), and c₆ (dashed green). Shown is the 20-case mean of ensemble spread in 20 experiments with independent initial conditions. The evolution of ensemble spread with model integration times starting from a randomly perturbed parameter is a measure of model sensitivity response with respect to the parameter being examined.

Figure 5.  The variation in RMSE of model states in the space of update intervals from the first case of 20 experiments: (a) CDAPE_{Pc2(c2),OV, win(w)}; (b) CDAPE_{Pc5(c5),OV, win(w)}. The black curve is the result with different update intervals (from 1 TU to 12 TUs, with an increment of 1 TU), and the dashed line is the result of traditional parameter estimation using the observational interval (0.2 TU) to update the parameter being estimated.

The sensitivity results show that, for slow-varying media like ocean, it takes a certain amount of time for the model to respond to the parameter perturbations (or errors). If the time within a parameter estimation cycle is too short for the model to transfer the uncertainty of the parameter to the model state, the signal-to-noise ratio of the state-parameter covariance is low. This will cause the parameter estimation to be unreliable. For example, if the model takes about 5-6 TUs for w to reach a sufficient sensitivity response, that is 30 times longer than the interval of observations of w (0.2 TU). Obviously, the covariance between the estimated parameter and the oceanic model state has not been established sufficiently within such a short observational interval. The parameter estimation with the covariance that has a low signal-to-noise ratio can cause computational oscillations of estimated parameter values, as shown in Figs. 2 and 3. But how does the time scale of model sensitivity response impact on the parameter estimation, and what is a suitable update frequency for these parameters being estimated? We answer these questions in the next section via a series of parameter estimation experiments with different update frequencies.

In this section, we examine one slow oceanic parameter (c₅) and one fast oceanic parameter (c₂) and compare their estimation performance to understand the relationship between the model sensitivity response time scale and the parameter estimation update frequency. We begin by examining the results of a single case, and then use the statistics of 20 cases to prove the robustness of the conclusion.

A series of experiments, CDAPE_{Pc2(c2),OV(w)} and CDAPE_{Pc5(c5),OV(w)}, with different parameter update intervals that are longer than the 0.2 TU observational interval of w is conducted with the first set of ensemble initial conditions we created at 3× 10⁴ TUs, as described in section 2.3. The update intervals vary from 1 TU to 20 TUs, with an increment of 1 TU. The RMSE of the model states calculated by Eq. (5) in the c₂ and c₅ estimation cases is shown in Fig. 5.

Figure 6.  Time series of the ensemble mean of parameters c₂ (blue) and c₅ (red) in the first case of 20 parameter estimation experiments, with update intervals of 2 TUs for c₂ (blue solid) and 5 TUs for c₅ (red solid). The cases using a 0.2-TU update interval for both c₂ and c₅ are also plotted as dotted lines for reference. The black line marks the "truth" value of c₂ and c₅ (both are 1 in this case).

From Fig. 5, we find that the model state errors decrease with the longer update intervals within 5 TUs in both cases, and then increase gradually as the update interval increases. This phenomenon is consistent with the one discovered in (Pan et al., 2014). We observe that, in parameter estimation with a longer update cycle (2 TUs for c₂ and 5 TUs for c₅, for instance), the oscillations in estimation with I _obs (hereafter, I _obs denotes the traditional parameter estimation with the 0.2 TU update interval), as shown in Fig. 2, is eliminated (see Fig. 6), and model variability is recovered more accurately (compare Fig. 7 to Fig. 3). We also notice that, in the c₂ case, the RMSE _sum within 11 TUs of update intervals is less than that of I _obs (Fig. 5a), while in the c₅ case the RMSE _sum is always greater than that of I _obs (Fig. 5b) (shown as a coincident case when the statistics of 20 experiments are discussed later). We understand that the parameter estimation is a result of trade-off between reliable state-parameter covariance and the strength of observational constraint. If the update interval is too short, the state-parameter covariance is unreliable, even though with stronger observational constraint. If the update interval is too long, the observational constraint becomes too weak, so the parameter estimation effect is weak, despite reliable state-parameter covariance. This can explain why the RMSE _sum increases when the update interval is larger than 5 TUs in both cases——a point we discuss in more depth in the next section.

Figure 7.  As in Fig. 3 but for the case using the 2-TU update interval (New CDAPE).

Figure 8.  The variations of 20-case mean RMSE _sum of (a, c) model states and (b, d) parameter c₂ in the space of update intervals in parameter estimation experiments (a, b) CDAPE_{Pc2(c2),OV, win(w)} and (c, d) CDAPE_{Pc2(c2),OV, win(w)}. The experiments in CDAPE_{Pc2(c2),OV, win(w)} use an observational time window of 0.1 TU (10 observations at each update step). In each panel, the solid line represents the ensemble mean of 20 experiments with independent initial conditions, described in section 2.3, while the shading represents the spread of these 20 cases. The blue line and blue shading are the results of parameter estimation with a 0.2-TU update interval, while the red line and the green shading are the results of using different update intervals. Note that, with or without an observational window, the scope of RMSE convexity is quite different, so panels (a, b) and (c, d) use different horizontal axis scales.

Figure 9.  As in Fig. 8 but for the case of parameter estimation of c₅.

Figure 10.  The variations of RMSE _sum of parameter (a) c₂, (b) c₅ and (c) model states in the space of update intervals in multiple parameter estimation experiments CDAPE_{Pc2,c5(c2,c5),OV, win(w)} when c₂ and c₅ are estimated simultaneously with 50% initial bias for both. Panel (d) shows the results of only estimating c₅ when c₂ remains biased ( CDAPE_{Pc2,c5(c5),OV, win(w)}). All other notation is the same as in Fig. 8.

To understand the phenomena in the case studies shown in section 4.1 and eliminate the case-dependence of results, in this section, we discuss the results of the whole 20 experiments with independent initial conditions described in section 2.3. The mean RMSE _sum of 20 experiments for both the model states and the examined parameters (c₂ and c₅) are shown in Figs. 8a and b and Figs. 9a and b. It is clear that, for the c₂ case, the RMSE _sum of both model states and the parameter are less than that of I _obs within 6 TUs of update intervals (Figs. 8a and b), while the RMSE _sum of the c₅ case is less than that of I _obs within 2 TUs of update intervals (Figs. 9a and b). For both cases, when the update interval is greater than a certain magnitude (5 TUs for c₂ and 2 TUs for c₅), the RMSE _sum of both model states and the parameter start to increase and quickly exceed that of I _obs. As we know, besides the state-parameter covariance, the performance of parameter estimation also relies on the strength of observational constraints. In Fig. 6, compared to the I _obs case, the parameter estimation with longer update intervals is more stable but takes much longer to converge due to much weaker observational constraints. Indeed, as the observation is the only information resource to update the model parameter, parameter estimation cannot succeed without sufficient observational constraint. When the update interval increases, the model has a more sensitive response to parameter perturbation, meaning the state-parameter covariance becomes more signal-dominant but the number of observations used to constrain the parameter are obviously less. For example, for parameter estimation with an update interval of 2 TUs, the amount of observational information used to constrain the parameter is only 10% of the amount used in the I _obs case. As the update interval significantly increases, although the state-parameter covariance is more reliable than that of I _obs, the weak observational constraint suppresses the positive impact of reliable covariance, so the parameter estimation diverges.

In the real world, an observational time window is usually used to collect measured data to increase the number of samples of observational information for an assimilation cycle (e.g., Pires et al., 1996; Hunt et al., 2004; Houtekamer and Mitchell, 2005; Laroche et al., 2007). This assumes that all the collected data are the samples of the "truth" variation at the assimilation time (Hamill and Snyder, 2000; Zhang et al., 2011; Gao et al., 2013). In a coupled system, due to different characteristic time scales in different media, how to choose a suitable observational time window in different media so that while sampling information increases the characteristic variability maintains, is an important and interesting research topic, but one that is beyond the scope of this study. Here, we only discuss the impact of observational constraint strength on parameter estimation when the update interval is long. For simplicity, we apply a comparable observational constraint with the I _obs experiment to further understand the impact of model sensitivity response time scales on parameter estimation. Figures 9c and d and Figs. 10c and d are the results of re-running the 20 experiments for c₂ and c₅ estimation but with a 0.1-TU observational time window (each parameter estimation step uses 10 observations), referred to as CDAPE_{Pc2(c2),OV, win(w)} and CDAPE_{Pc5(c5),OV, win(w)} (see Table 2). In this way, for the cases with the 2-TU update interval, the parameter estimation in CDAPE_{Pc2(c2),OV, win(w)} and CDAPE_{Pc5(c5),OV, win(w)} has an equivalent observational constraint strength as I _obs. From Figs. 8c and d and Figs. 9c and d, we can see that the errors of parameter estimation exhibit a wide convex shape when the estimation has a reasonable observational constraint. Of course, when the update interval is longer, the observational constraint becomes weaker and the representation of observations also becomes poorer, so the estimation error starts to increase.

From the analyses above, we conclude that the state-parameter covariance requires some response time to establish a reliable signal (reaching equilibrium) as the model responds to parameter perturbations. Thus, it is necessary to update the parameter with an update interval comparable to the model sensitivity response time scale in parameter estimation.

In order to test the impact of the model sensitivity response time scales on multiple parameter estimation, we conduct a series of experiments—— CDAPE_{Pc2,c5(c2),OV, win(w)}, CDAPE_{Pc2,c5(c5),OV, win(w)} and CDAPE_{Pc2,c5(c2,c5),OV, win(w)} (see Table 2 for detailed descriptions)——with different parameter update intervals that are longer than the 0.2-TU observational interval of w (all experiments with 0.1-TU observational time window). Parameters c₅ and c₂ are perturbed together in all of these experiments. While both are estimated together in CDAPE_{Pc2,c5(c2,c5),OV, win(w)}, only c₅(c₂) is estimated in CDAPE_{Pc2,c5(c5),OV, win(w)} ( CDAPE_{Pc2,c5(c2),OV, win(w)}). The RMSE _sum of model states and parameters from 20-case statistics are shown in Figs. 10a-d. From Figs. 10a-c, we see that the RMSE _sum for the model states and parameters (c₅ and c₂) are reduced quickly as the update intervals increase, meaning that the covariance between model states and the parameter is more stable with longer update intervals in multiple parameter estimation. As in the single parameter estimation cases shown before, when the observational constraint strength becomes too weak and with overly long update intervals, the RMSE shows gradual growth. Clearly, the result of multiple parameter estimation is consistent with that of single parameter estimation. Comparing Fig. 10c to Fig. 10d, we can see that the RMSE is much bigger when both c₅ and c₂ are perturbed but only c₅ is estimated (Fig. 10d), meaning the fast oceanic parameter (c₂) has a big influence on the result of model integration if it is not corrected.

Based on filtering theory, parameter estimation with the observation of model states is a promising approach to mitigate model bias. Customized from traditional state estimation, traditional parameter estimation usually updates the parameter being estimated according to observational frequency. Without direct observations on parameters, the covariance between model states and the estimated parameter plays a critical role in parameter estimation. The sensitivity response time scales of model to parameter perturbations in a coupled system can be different, from hourly to decadal, depending on the nature of the associated physics and characteristic variability of the fluid. With an ensemble filter consisting of a simple coupled model, this study addresses the impact of the time scale of model sensitivity response on coupled model parameter estimation. Meanwhile, the influence of observational constraint strength on parameter estimation is discussed. Results show that it is necessary to update the parameter with an update interval comparable to the model sensitivity response time scale in parameter estimation. These results provide a guideline for when real observations are used to optimize the parameters in a CGCM for improving climate analysis and prediction initialization.

Although the new parameter optimization scheme has shown great promise with a simple model, many challenges remain in applying it to CGCMs. Firstly, it is assumed in this study that the errors of model parameters are the only source of model biases. Actually, the dynamical core and physical schemes themselves are also imperfect and serve as significant sources of model biases in a CGCM. How the new parameter estimation scheme works with multiple model bias sources needs to be examined. Secondly, in order to determine the suitable update interval, the sensitivity of a model state to parameters has to be studied first. So, a thorough examination of the sensitivity of a CGCM with respect to its numerous parameters is an important but challenging issue. When a real-world observing system is combined with a CGCM, all of these issues need to be further investigated and addressed.