The uncertainties in values of coupled model parameters are an important source of model bias that causes model climate drift. The values can be calibrated by a parameter estimation procedure that projects observational information onto model parameters. The signal-to-noise ratio of error covariance between the model state and the parameter being estimated directly determines whether the parameter estimation succeeds or not. With a conceptual climate model that couples the stochastic atmosphere and slow-varying ocean, this study examines the sensitivity of state–parameter covariance on the accuracy of estimated model states in different model components of a coupled system. Due to the interaction of multiple timescales, the fast-varying “atmosphere” with a chaotic nature is the major source of the inaccuracy of estimated state–parameter covariance. Thus, enhancing the estimation accuracy of atmospheric states is very important for the success of coupled model parameter estimation, especially for the parameters in the air–sea interaction processes. The impact of chaotic-to-periodic ratio in state variability on parameter estimation is also discussed. This simple model study provides a guideline when real observations are used to optimize model parameters in a coupled general circulation model for improving climate analysis and predictions.

Nowadays, a coupled atmosphere–ocean general circulation model is widely used as a common tool in climate research and related applications. However, due to the approximation nature of model numeric schemes and physical parameterization, a model always has errors. In particular, one traditionally determines the values of model parameters by experience or a trial procedure which heuristically provides a reasonable estimate but usually is not optimal for the coupled model. Recently, with the aid of information estimation (filtering) theory (e.g., Jazwinski, 1970), research on optimization of coupled model parameters based on instantaneous observational information has grown quickly (e.g., Wu et al., 2013; Liu et al., 2014a, b; Li et al., 2016). Traditional data assimilation that only uses observations to estimate model states (i.e., state estimation) becomes both state estimation (SE) and parameter estimation (also called optimization) (PE) with observations. Such a PE process can be implemented through a variational (adjoint) method (e.g., Stammer, 2005; Liu et al., 2012) or an ensemble Kalman filter (e.g., Zhang et al., 2012) or even a direct Bayesian approach (e.g., Jackson et al., 2004).

In the previous study with a conceptual coupled model, Zhang et al. (2012) pointed out that an important aspect of successful coupled model parameter optimization is that the coupled model states must be sufficiently constrained by observations first. This is because multiple sources of uncertainties exist in a coupled system consisting of different timescale media. If the part of uncertainties in model states, which are not correlated with parameter errors, has not been sufficiently constrained yet, the covariance between the model states and parameters being estimated is noisy (e.g., Dee and Silva, 1998; Dee, 2005; Annan et al., 2005). Without direct observational information, the noise in state–parameter covariance, which is the key quantity to project observed state information onto the parameter, can bring the estimated parameter toward an erroneous value (Zhang, 2011b). This is a general understanding about coupled model parameter estimation. However, since multiple media of the climate system have different timescale variability and different quality of observations so as to have different contributions to the uncertainty of state–parameter covariance, an outstanding question is what the impact of SE accuracy in different media is on coupled model PE. Given the extreme importance of state–parameter covariance for PE, a clear answer for this question must further our understanding on coupled model parameter estimation.

To answer this question, this study uses a simple coupled model to examine the influence of observation-constrained states in each medium on PE for different parameters in different media thoroughly. The model conceptually describes the interactions of three typical timescales of the climate system – chaotic (synoptic) atmosphere, seasonal–interannual upper ocean and decadal deep ocean. A twin experiment framework is used throughout the whole study.

The paper is organized as follows. After the introduction, Sect. 2 gives the methodology, including brief descriptions of the simple coupled model, filtering algorithm and twin experiment framework. Section 3 first presents the results of various PE experiments with different partial SE settings and then analyzes the conditions for successful PE with partial SE. Finally, the summary and discussions are given in Sect. 4.

To clearly address the issue posed in the introduction, this study employs
the simple pycnocline prediction model developed by Zhang (2011a, b). This
conceptual coupled model is based on Lorenz's three-variable chaotic model
(Lorenz, 1963) that is coupled a slab ocean variable (Zhang et al., 2012)
interacting with a pycnocline predictive model (Gnanadesikan, 1999). For the
problem that this concerns, this conceptual coupled model shares the
fundamental features with a coupled general circulation model (CGCM; see
Zhang, 2011a; Han et al., 2013). The model development can be traced in
Zhang (2011a, b) and Zhang et al. (2012) in detail. Here, we only comment on
major points that are relevant to this study. The model includes five
equations:

The filtering method used in this study is the ensemble adjustment Kalman
filter (EAKF; Anderson, 2001). The EAKF algorithm shares all theoretical
derivation of ensemble Kalman filter (EnKF; e.g., Evensen, 1994; Houtekamer
and Mitchell, 1998) that combines an observational probability distribution function
(PDF) with model PDF but under an adjustment idea. After the first version
(Anderson, 2001), the EAKF algorithm had improved its implementation as a
sequential local least squares filter (Anderson, 2003). The EAKF is a member
of ensemble square root filters (Tippett et al., 2003), taking the advantage
of ensemble Kalman filter without perturbing the observation (Whitaker
and Hamill, 2002). While the detailed and exhausted mathematical derivations
can be referred to the aforementioned literature and others (e.g., Zhang and
Anderson, 2003), here we mainly comment on the computational implementation
with a two-step procedure (Anderson, 2003; Zhang et al., 2007) that is
relevant to this study. The first step uses two Gaussian convolutions to
derive the observational increment at the observational location as

The second step regresses the observational increment onto the related model
states or parameters by the model ensemble-evaluated covariance as

Some other relevant aspects of the method are also commented here. Just as
in Zhang and Anderson (2003), based on the trade-off between cost and
assimilation quality, after a series of sensitivity tests on ensemble sizes
of 10 and 20–100, no significant difference in the quality of
standard assimilation is found when the ensemble size is greater than 20.
Thus, a practical ensemble size of 20 is chosen as a basic experiment
setting. We will examine the sensitivity of major conclusions of the
addressed problem in this study to the ensemble size in related places
later. Although the intervals of the atmosphere and ocean observations are
different in the real world, for convenience of comparison, we set a uniform
update interval for SE (in the atmosphere and ocean) and PE as five time steps
as the basic setting in this study (we will also discuss the influence of
update intervals in related places later). The inflation method must be
included in the EAKF PE. Considering that the inflated parameter ensemble
will influence state variables, no inflation is applied to the model
state ensemble. The PE inflation scheme follows Zhang (2011b): when the SD
(spread) of the parameter ensemble is below some limit (40 % of the
initial spread), a factor is applied to inflate the parameter ensemble
spread to this value. During this process, the ensemble structure of
parameter remains unchanged. In addition, to avoid the uncertainty and
complexity of evaluating cross covariance between media that have too
different characteristic timescales (Han et al., 2013), in the SE of this
study, we only allow

List of the successful (S) and failed (F) parameter estimation (PE) cases with partial state estimation (SE) in eight PE experiments (in the parenthesis is the experiment serial number).

Twin experiments are set to test the relation between coupled SE and PE. The
model with the standard parameter values described in Sect. 2.1 is running
10

The first set of PE experiments is done to study the parameters in “air–sea”
interaction. To do that, we use two parameters –

In all PE cases, the initial value of the parameter to be estimated is
deliberately set as biased from the truth (referring to the standard
parameter values described in Sect. 2.1). To maintain the chaotic nature
of the Lorenz equation, parameter values are required to be within a certain
range. This is a constraint for the biased amount of the initial values of a
parameter. Based on some sensitivity studies, the chaotic performance is
more vulnerable to the change of the atmospheric parameter

List of root mean square errors of the state variable and the parameter during the last 100 TUs in eight PE experiments.

Time series of the ensemble mean (solid line) of the
estimated parameter

With the method and experiment settings described in Sect. 2, we test
different PE performances under different SE settings. Generally, with a
full SE (all the atmospheric

In Table 1,

Time series of ensemble means (solid line) of the
estimated parameter

Time series of ensemble means of the estimated parameter

The stability of PE is different among partial SE settings, as shown in Figs. 2 and 3, as the time series of the ensemble mean of the estimated parameters.
Figs. 2b, c and 3b, c show the four successful cases with only atmospheric SE.
Compared to full SE (using observations of

The convergence rate of PE is also obviously different with different SE
settings. The case of

The inflation method is also important in PE (Yang and DelSole, 2009; DelSole and Yang, 2010; Zhang, 2011a, b; Zhang et al., 2012). The partial and full SE cases use the same inflation scheme (Zhang, 2011a, b; Zhang et al., 2012). Shadows in Figs. 1–3 show the range of the parameter ensemble. The zigzag shape of the shadows represents the inflation during PE. In these figures, the width of the shadows shrinks quickly once PE is activated, while some of the mean values move toward the truth slowly (for example, Figs. 2c and 3b). Also from the zigzag shapes, we can see some inflation effects before the parameter converges to the truth. All of these imply that the designed PE is stable and its convergence rate is not very sensitive to the inflation scheme.

Time series of the state variables from the

Time series of the ensemble of parameter

In addition, larger ensemble sizes are used to test the sensitivity of the
conclusion above. The results show that bigger ensemble size has a positive
impact on SE and PE quality but the drawn conclusion from the experiments
above does not change its essence. Also, the ensemble size far exceeds the
problem size in this simple model study. In this regard, further examination
may be necessary in CGCM cases. We also performed the experiments under
different SE update interval settings. Test results show that for the issue
we are addressing, the conclusion is not sensitive to the update interval if
it is within a reasonable range (

In case-3 and case-4 of Table 1, we successfully estimate the oceanic parameter

Wavelet analyses for

The phenomenon above, in estimation of

It is interesting that once the atmospheric states (the Lorenz equation in
this simple model) are constrained by the observations, both the atmospheric
parameter (

In our twin experiment setting, there are three types of model uncertainties:
strong nonlinearity in the atmosphere (chaotic in this case), weaker
nonlinearity in the ocean and biased parameter values. The SE process before
PE aims to control the first and second types of uncertainties by putting
observational constraints on model states. Figure 6 shows the wavelet
analyses for the atmospheric variable

Sampling map of the perturbed parameter anomalies in the
space of model state anomalies for

Without direct observations of parameter values, PE completely relies on the covariance between the parameter and model states for projecting the observational information of states onto the parameter. While the PE projection is carried out by a linear regression equation based on the state–parameter covariance (EnKF/EAKF, for instance), only a linear or quasi-linear relationship between parameters and states in the ensemble is recognized. All failure of PE without direct atmospheric SE could be attributed to the chaotic disturbances in the atmosphere (Lorenz equations in this case) that create difficulties for the system to build up a quasi-linear relationship between the state variable and the parameter.

List of

To investigate the parameter–state relationship in the model background
(prior PE), we conduct a series of parameter perturbation runs corresponding
to eight partial SE experiments (without PE to fix the parameter spread – the
PE process sets the parameter ensemble as an additional system freedom and
makes the relationship of the parameter and model state more complicate). In
that way, the parameter perturbations can be fully transferred to the model
states so that we can study the state–parameter relationship in a straightforward
manner. The results are shown in Figs. 7 and 8, where the horizontal
axis is the ensemble anomaly (vs. ensemble mean) of the state variable and
the vertical axis is the ensemble anomaly of the parameter, and the
background black dots represent the model runs starting from different
initial conditions. Since the parameter ensemble does not change (once
perturbed at the initial time) during the model integration, the lines
constructed by black dots in a perturbation run are parallel to the

The relationship between the states and the parameters can be analyzed
quantitatively. Zhang et al. (2012) defined an ad hoc index to measure the
signal-to-noise ratio (called

The same as Fig. 7 but for the case with SE of

Wavelet analyses for

From the results above, we learned that the PE of

Time series of

Time series of the ensemble of parameter

Time series of the ensemble of the parameter in the

To further test the role of periodic signals in ocean states for oceanic PE,
we conduct oceanic PE on a particular frequency band using the method
described in Appendix C. Some results are shown in Fig. 12 which shows that
using the covariance of

The erroneous values of parameters in a coupled model are a source of model bias that can cause model climate drift. Model bias can be mitigated by PE with observational data. The signal-to-noise ratio in state–parameter covariance plays a centrally important role in the PE process. With a conceptual coupled model, we discuss the issue of how to enhance the signal-to-noise ratio in coupled model PE through further understanding of various aspects of the PE process in a coupled numerical system.

We performed three kinds of comparisons to discuss the issue. The first kind focuses on the PE performance with a two-way coupling model. Results show that atmospheric SE is critically important. The second comparison is carried out by the experiments with the same parameter spread and SE settings as in the first comparison but without the PE process. We use this method to examine the signal-to-noise ratio of state–parameter covariance in different SE settings. Results find that the projection of the observational increment onto the parameter can be easily interrupted under partial SE conditions. In the third kind, we changed the model structure from two-way coupling to one-way coupling, allowing the ocean state to vary forced by the atmosphere without feedback to the chaotic atmosphere. The PE results are better with higher periodic and less chaotic states.

According to all these comparisons, first, we found that due to the interaction of multiple timescales in our conceptual coupled model, the fast-varying component is the major source for producing an inaccurate state–parameter covariance in the system. Enhancing the estimation accuracy of high-frequency states that interact with the parameter is the most important factor to maintain a signal-dominated relationship between the parameter being estimated and model states, and allows for successful coupled model parameter estimation. Second, the chaotic-to-periodic ratio (CPR) of the model state that closely associates with the parameter being estimated determines the required state estimation accuracy. Given limited observational resources, in the future when we work with a realistic model and observing system, the CPR shall be first investigated to increase the opportunities of having successful parameter estimation.

Given the fact that observations are always imperfect, this conceptual
coupled model study tries to provide some general guidelines for CGCM PE
application with the real observing system. However, the results have the
following limitations:

The conceptual coupled model assumes that only the atmosphere is a chaotic uncertainty source. In the real world, this is unnecessarily true (nonlinearity produced by smaller-scale eddies in the ocean could be the part of chaotic uncertainty sources too, for instance).

The atmosphere–ocean interaction is idealized in the conceptual model. In the real world, the air–sea coupling could be complex as it is highly geographically dependent.

The twin experiment assumes that except for the parameters to be estimated, the model “dynamical core” and “physics” are perfect and consistent with the observation. In the real world, the CGCM is biased from the observations.

How the accuracy of state estimation impacts on the coupled model parameter estimation is an interesting and challenging research topic. The spatial and temporal dependence of atmospheric and oceanic circulations could further complicate the issue. For example, the Kuroshio meander in the south of Japan is very different to the Kuroshio meander across the Luzon Strait. The Kuroshio across the Luzon Strait is easily interrupted by the monsoon, but the meander in the south of Japan is a self-sustained dynamic system having multiple equilibria with non-periodic state changes (Taft, 1972; Yu et al., 2013); the uncertainty of the latter comes from the accumulation of the negative vorticities in the ocean. Further, we have already known that the method on a particular frequency can increase the opportunity of success. When such a real problem is addressed through the PE with a CGCM, we may need to make efforts on both adaptive measurements and spectral separation. The PE method shall be improved to perform separately at different timescales. How to speed up the convergent rate in the coupled model PE process is also an important issue. All of these require further research work in order to be clarified.

We use wavelet analysis to exhibit the CPR of different states. Some related wavelet
methods and additional useful examples are available at

A suitable scope of parameter values that maintain the model character is an
important precondition for successful PE. For example, in Eq. (1) when

A chaotic nature naturally lowers predictability of the signal. The
chaotic-to-periodic ratio (CPR) is defined to measure the chaotic degree of
a system within a particular period band as

Previous studies have shown that applying the PE with an averaged covariance in a particular time window can increase the signal-to-noise ratio (Lu et al., 2015, Barth et al., 2015). In our case, it can also effectively increase the CPR of the state variable. Here, we propose an alternative method that has a similar effect to an averaged covariance but is much easier to be implemented. This method applies PE on a particular frequency. The method succeeds in enhancing the CPR by using a designed filter on both the observations and the simulated ensemble results, and it can allow information focusing on a particular frequency more accurately than using the averaging method.

In this study, for the

The authors declare that they have no conflict of interest.

This work is funded by the National Natural Science Foundation of China (41306004), China's National Basic Research Priorities Programmer (2013CB956202) and the National Natural Science Foundation of China (41490640; 41490641). Edited by: A. Turiel Reviewed by: five anonymous referees