1. Introduction
Coupled climate model results are usually biased from those from the real world due to the imperfect numerical implementations and physical parameterization schemes as well as improper parameter values (Zhang et al. 2012). These model biases maintain systematic errors in the traditional climate estimation and produce artifacts in the analyzed variability (e.g. Dee and Da Silva 1998; Dee 2005). The model prediction in a biased model with traditional initialization schemes tends to drift away from observed states and thus has limited forecast skill (Smith et al. 2007).
Regardless of whether the model error arises from dynamical core misfitting, physical scheme approximation, or model parameter errors, the model errors, in principle, could be corrected using data assimilation. Using observations to optimize uncertain parameters in climate models is an important and complex subject. Many efforts have been undertaken to include model parameters into data assimilation control variables (e.g. Daley 1991; Wunsch 1996; Anderson 2001; Bennett 2002; Annan et al. 2005; Aksoy et al. 2006; Evensen 2007; Kondrashov et al. 2008) using four-dimensional variation or ensemble Kalman filter data assimilation methods. Some studies have addressed parameter optimization in a coupled climate model in which the interactions of multiple time scales play important roles in the development and propagation of climate signals. For example, a coupled data assimilation scheme with enhancive parameter correction (DAEPC) is designed to determine how to obtain a signal-dominant state-parameter covariance in order to effectively optimize the coupled model parameters using observations in different system components (Zhang et al., 2012).
Coupled model parameter estimation that uses observational information to tune the coupled model parameters has shown great potential to reduce model biases and improve the quality of climate estimation (i.e., model state estimation) and prediction (i.e., model predictability; Zhang, 2011a,b). Using a simple pycnocline prediction model, Zhang (2011a) investigated the impact of coupled state-parameter optimization on decadal-scale predictions, and showed that when both the coupled model states and parameters are optimized with data assimilation, the coupled model’s predictability is greatly improved. With a DAEPC algorithm in an intermediate coupled model, Wu et al. (2012) introduced a geography-dependent parameter optimization (GPO) scheme to increase the signal-to-noise ratio in the parameter estimation in a perfect model twin experiment framework, and examined the impact of the scheme on climate estimation and prediction (Wu et al. 2013). Recently, the DAEPC algorithm has also been demonstrated successfully in a coupled ocean–atmosphere general circulation model (Liu et al. 2014).
However, all the coupled model studies mentioned above applied the DAEPC algorithm in a perfect model framework. As a result, the impacts of imperfect physical parameterization schemes, which are an important source of model biases, have not been examined. In a coupled model, physical processes are usually approximated by parameterization schemes, which could be biased in their structure due to incomplete understanding and/or representation of the physical processes. In this study, we try to answer this question: To what degree can climate estimation and prediction be improved through physical parameter optimization using observations to minimize the error induced by a biased model structure? As a pilot study, we first construct an intermediate coupled model, which includes physical parameterizations of the heat exchanges among atmosphere, land, and ocean. With the intermediate coupled model with biased physics, a biased assimilation experiment framework is designed to study parameter optimization using a DAEPC algorithm. The biased physics in this study is defined using different outgoing planetary longwave radiation schemes in the assimilation and truth models. Simulated observations sampled from the truth model are assimilated into the assimilation model, and the degree to which the assimilation and prediction, with or without physical parameter optimization, recovers the truth is a measure of the impact of physical parameter optimization on climate estimation and prediction.
Based on such an experimental format, we compare results from five different assimilation experiments: 1) state estimation only (SEO) in which only observations are assimilated into the model state; 2) SEO with perturbed parameters (SEO_PP) in the biased physical optimization scheme; 3) single-valued parameter estimation (SPE) in which each parameter in the biased physical scheme is optimized with a globally uniform value (PP_PO1); 4) SPE in which only the parameters in the biased physical scheme to which the state variables are most sensitive are optimized (PP_PO2); and 5) geography-dependent parameter optimization (i.e., GPO) in which parameters to which the state variables are most sensitive are optimized according to local observational information and model sensitivity thus allowing the parameters to vary geographically.
The paper is organized as follows. The formulation of the intermediate coupled model is detailed in section 2. Section 3 discusses the format of the biased model framework. Section 4 presents the development of model biases due to the use of different outgoing planetary longwave radiation schemes. The impacts of parameter optimization in the biased physical scheme on climate estimation and prediction are examined and discussed in sections 5 and 6, respectively. Finally, a summary and discussion are provided in section 7.
2. An intermediate coupled model
As a first step in addressing the parameter optimization in an assimilation–prediction model that includes a biased physical scheme, we construct an energy-conserving intermediate coupled ocean–atmosphere model. The model is a combination of the intermediate coupled model without physical parameterizations presented by Wu et al. (2012) and the globally resolved energy balance (GREB) model used by Dommenget and Flöter (2011). It should be noted that despite some limitations in the representations of basic processes in the GREB model, such as the absence of an ENSO dynamical mechanism, the GREB model can reasonably simulate aspects of the Arctic winter amplification, the equilibrium land–sea warming contrast, and the interhemispheric warming gradient in the equilibrium 2×CO2 response. In addition, the surface layer humidity equation in the GREB model is removed to not unduly complicate the coupled model constructed for this study.
The intermediate coupled model we construct includes a vorticity advection equation that represents the atmosphere, a 1.5-layer baroclinic ocean with two simple temperature equations for the surface (Liu 1993) and subsurface ocean, and a simple land surface temperature model to complete the bottom boundary condition for the atmosphere over the entire globe. The GREB model is based on a surface energy balance through simple representations of solar and thermal radiation as well as other associated physical processes. The detailed description and the source code of the GREB model can be found online at http://users.monash.edu.au/~dietmard/content/GREB/.
The complete intermediate model includes six prognostic variables to describe the evolution of the three model domains: atmosphere, land, and ocean. The atmospheric variables include the geostrophic atmospheric streamfunction,
a. The atmosphere
In the global barotropic spectral model equation (1a),





















b. The ocean

In the equation of the oceanic streamfunction (2a),




































c. The land







d. Coupling of the model components
The coupling between model components is accomplished mainly through the physical parameterizations of the flux exchanges rather than only by coupling coefficients as in Wu et al. (2012). Dommenget and Flöter (2011) described the parameterizations in detail for solar radiation, thermal radiation, sensible heat flux, latent heat flux, and other terms associated with energy balance. Although the physical processes are greatly simplified relative to a state-of-the-art CGCM, the coupled model is sufficient for the purpose of studying the ensemble coupled parameter optimizations with biased physics.
For simulating a mean climate as close to the true climatology as possible, flux correction terms are imposed on the equations of
e. Model mean states and variability

















Annual mean of (a) atmospheric streamfunction ψ (m2 s−1), (b) atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
The annual-mean absolute errors of the sea surface temperature
The annual-mean absolute error of (a) sea surface temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
To investigate the variability of the state variables, an empirical orthogonal function (EOF) decomposition of the 300-yr time series of the monthly mean anomalies of Ta was conducted. The time coefficients of the first four EOFs were used to perform the power spectrum analysis to show the internal variability of Ta. The characteristic time scales of the first four modes were 100 years, 50 years, 10–25 years, and 6–14 years (Fig. 3), indicating that the model states consist of multiple time scales. Figure 4 shows the spatial distributions for the first four EOF modes of the monthly averaged atmospheric temperature
Power spectrum (solid line) of the (a) first, (b) second, (c) third, and (d) fourth mode of EOF decomposition to the time series of 300-yr anomalies of monthly averaged atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
Spatial distributions of the (a) first, (b) second, (c) third, and (d) fourth mode of EOF decomposition of the time series of 200-yr anomalies of monthly averaged atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
3. Experimental setup
In this section, using the intermediate model described above and an ensemble coupled data assimilation scheme, we designed a biased model framework by setting outgoing planetary longwave radiation schemes different between the assimilation model and the truth model. We assume the different parameterizations of outgoing longwave radiation to be the only source of assimilation model biases aside from errors in the initial model states.
a. Ensemble coupled data assimilation with enhancive parameter correction
A coupled data assimilation scheme with enhancive parameter correction (i.e., DAEPC) (Zhang et al. 2012) is employed to compute the model states and perform parameter optimization. This is a modification to the standard data assimilation with adaptive parameter estimation scheme described by Kulhavy (1993) and Tao (2003). The covariance between a parameter and the model state serves as the critical quantity to project observational information of the model states to the parameter being optimized.
























Unlike the model states, the model parameters in the coupled climate model do not have any dynamically generated internal variability. Therefore, the quality of the covariance between a parameter and the model state [see Eq. (9)] is the only criterion needed to determine if the parameter can be optimized reasonably well. The subsequent parameter updating with observational data can improve the state estimation of the next cycle, and this improved state estimation further enhances the signal-to-noise ratio of the state parameter covariance for the subsequent cycle of parameter correction.
b. Two schemes of outgoing planetary longwave radiation















Default values of parameters in the TW82 parameterization scheme.
The two physical schemes, DF11 and TW82, have completely different implementations of the same physical processes. The TW82 scheme uses a cubic polynomial in
c. Data assimilation experiment with biased physical schemes
The truth model uses the DF11 scheme. Starting from the initial conditions
The assimilation model uses the TW82 scheme. Starting from the same initial conditions used in the truth model run, the assimilation model is spun up for 50 years to produce the biased initial conditions
Starting from the ensemble initial conditions,
List of assimilation experiments and model simulations.
It should be pointed out that the assimilation model differs from the truth model only in the longwave scheme employed [Eq. (13) vs. Eqs. (10)–(12)]. Because of the different parameterization schemes for the longwave radiation, there is no longer a true value for the parameters in the longwave scheme in the assimilation model Eq. (13). The only metrics to evaluate the performance of a parameter estimation scheme are the accuracy of the estimated model states and/or the forecast. This configuration simulates a real world scenario, where the parameters in a climate model have no true values, so that parameter optimization can only be evaluated by examining the accuracy of the estimation and model prediction.
4. The model bias arising from physical schemes
This section examines the model biases induced by different physical schemes, that is, the difference between the annual mean and climatology in the assimilation and truth models. Figs. 5a–d show the spatial distribution of biases in the annual mean of ψ ,
Annual mean of the differences (biases of the assimilation model) in (a) atmospheric streamfunction ψ (m2 s−1), (b) atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
Note that
The bias in ψ (see Fig. 5a) is strongly negative around the globe in the polar and subpolar regions, especially in the NH high latitudes. In the low and middle latitudes, the bias in ψ, while still mostly negative, is weaker and the patterns are more complex. This complexity is partly due to the stronger internal variability of ψ in the tropical and subtropical regions.
Figure 6 shows the climatologic annual cycle of global mean ψ,
Climatology of space averaged (a) atmospheric streamfunction ψ (m2 s−1), (b) atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
5. Parameter optimization with biased model physics
The analyses in the previous section showed that model climate depends heavily on the representation of model physics, even when the flux correction terms work correctly in the coupled model. Differences in outgoing planetary longwave radiation schemes create different model climatology. In this section, we will examine how this difference affects climate estimation and prediction, and how to overcome this problem by optimizing the parameters in TW82 scheme using observations produced by the DF11 scheme. We begin by examining the sensitivities in the assimilation model with respect to the parameters in the TW82 scheme. Then, based upon the model sensitivity, we discuss the impact of parameter optimization on climate estimation.
a. Model sensitivities with respect to physical parameters
It is essential to investigate model sensitivities with respect to parameters prior to parameter optimization. In this study, the sensitivity study is carried out for all 12 parameters in the TW82 scheme (Table 1). The ensemble spread of a model prognostic variable is used to quantitatively evaluate the relative sensitivities. For each parameter, 20 random numbers are drawn from a normally distributed population with a standard deviation of 5% of the default value. This random number is superimposed on the default value of the parameter being perturbed, while other 11 parameters remain fixed at their default values. All 20 ensemble model runs are started from the same initial conditions, S1, and the assimilation model is integrated for 70 years. Sensitivities are calculated using the model output from the last 50 years. This process is repeated for each parameter.
Figures 7a–f show the time-space averaged sensitivities of ψ,
Time–space averaged sensitivity of the (a) atmospheric streamfunction ψ (m2 s−1), (b) oceanic streamfunction
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
The parameter sensitivities of the model states vary significantly with geographic location. For example, Fig. 8a presents the distribution of time-averaged sensitivity of
The spatial distribution of time averaged sensitivity of (a) atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
The relative sensitivity, defined as the sensitivity normalized by the standard deviation, is also an interesting quantity that explains the significance of the sensitivity of a model variable. For example, Fig. 8b shows the distribution of time-averaged relative sensitivity of
b. Impact of parameter optimization on climate estimation
Figures 9a–d show time series of the RMSEs of ψ,
Time series of RMSEs of the (a) atmospheric streamfunction ψ (m2 s−1), (b) atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
The RMSE of
Unlike the RMSEs of sea surface temperature and air temperature, which are reduced significantly during the first 100 days of assimilation, the RMSEs of
The RMSEs of ψ for the SEO, SEO_PP, and PP_PO1 experiments are all reduced by roughly two orders of magnitude relative to that for CTRL (see the small inset panel in Fig. 9a). However, the RMSE of ψ behaves differently than that for temperature in that the RMSE for PP_PO1 is not always smaller than that for the SEO or SEO_PP (blue, red, and black lines in Fig. 8a). There are two reasons for this. First, the RMSE of ψ for SEO has dropped to about 7 × 105 m2 s−1, which is already less than the standard deviation of observational errors of ψ (106 m2 s−1). Therefore, it is difficult to extract an observational signal to further reduce the RMSE of ψ through the adjustment of parameters. Second, physically, all 12 perturbed parameters in the longwave scheme directly impact the temperature field but not the streamfunction, ψ.
Spatial distributions of the RMSEs of
Spatial distributions of the RMSEs of the atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
To further improve the performance of the parameter optimization, we performed the two additional assimilation experiments, PP_PO2 and GPO. Figures 11a–d show the time series of the RMSEs of
Time series of RMSEs of the (a) atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
The benefit of GPO is that it allows the optimized parameter values to vary spatially. This may be justified in that the difference between the assimilation model and truth may depend on the state climatology itself, which varies spatially, so that a spatially uniform parameter may not be the optimal representation of the physics. Compared to PP_PO2, GPO further reduces the RMSEs of
Total RMSEs of
A perfect assimilation experiment without any model bias, SEO_PRFT, is also carried out to serve as a benchmark. SEO_PRFT is the same as SEO except that the assimilation model uses the DF11 scheme and all parameters in the DF11 scheme use the truth values. As can be seen from the solid black lines in Figs. 11a–d, the RMSEs of all four variables,
To investigate the impact of the flux adjustment in the coupled climate model on the parameter optimization within our biased assimilation experimental framework, another four assimilation experiments are conducted. These are the same as SEO_PP, PP_PO1, PP_PO2, and GPO, but without the flux correction terms. Figure 12 shows the time series of the RMSEs of
Time series of RMSEs of the (a) atmospheric temperature
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
The GPO-optimized parameters have distinctive spatial patterns. For example, Fig. 13 shows the spatial distribution of b00. The pattern shows that b00 is not far away from the default value of b00 (243.414) near the equator, but decreases towards high latitudes. In addition, b00 tends to be smaller over land than over the ocean (e.g. the Asia vs the Pacific Ocean and the Africa vs the Indian Ocean in the same latitude). Because of the functional form of the parameterization scheme, this means that b00 has a greater impact on the land than on the ocean, which is consistent with the greater reduction of RMSEs over land than over ocean discussed earlier. In addition, the biased physical scheme can cause larger model errors over the land and at high latitude than over the ocean or at low latitude. This is also consistent with the greater error reduction seen over land in the GPO scheme discussed earlier.
The spatial distribution of the GPO-optimized b00.
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
6. Impact of observation-optimized physical parameters on climate predictions
Although justified to some extent, the parameter optimization, as judged from the analysis, is subject to the criticism of curve fitting. However, from a more practical point of view, the role of parameter optimization should be judged from its effectiveness in climate prediction. In order to evaluate the impact of ensemble coupled data assimilation with biased model physics on model prediction, a 20-member ensemble forecast is performed. This forecast uses 20 initial conditions selected every 2 years apart from the SEO, SEO_PP, PP_PO1, PP_PO2, and GPO analysis fields over the last 40 years. The 20-member ensemble is integrated for 10 years starting from the analyzed states of each of the five assimilation schemes. The global mean of the anomaly correlation coefficient of the forecasted ensemble mean of
We examine the climate prediction skill question first. Figure 14 shows the variation of the ACC (Fig. 14a) and the RMSE (Fig. 14b) as a function of the forecast lead time for the forecasted ensemble means of
Variations of (left) anomaly correlation coefficient (ACC) and (right) RMSE as a function of the forecast lead time for the forecasted ensemble means of the (a),(b) deep ocean
Citation: Journal of Climate 28, 3; 10.1175/JCLI-D-14-00348.1
As with the climate forecast, the parameter optimization also improves the weather forecast skill for the atmosphere. Figs. 14c and 14d show the variations of ACC (Fig. 14c) and RMSE (Fig. 14d) as a function of the forecast lead time for the forecast ensemble means of
7. Summary and discussion
An imperfect dynamical core and empirical physical schemes and improper parameter values are three main sources of model bias. Among these sources of model error, in principle, only the model parameter errors can be treated directly by assimilating observations. Because of our incomplete understanding of atmosphere and ocean physics, we still have a long way to go to adequately improve the estimation and prediction capability of a coupled model through directly improving physical parameterization schemes. In this paper, we studied several methods for using observational information to optimize parameters in an attempt to minimize errors caused by a biased model structure. We designed a biased assimilation experiment framework to study parameter optimization in an energy-conserving intermediate coupled model. Within this framework, the assimilation-prediction model is subject to biased physics from the truth model. The biased physics was implemented by using different outgoing longwave radiation schemes in the assimilation–prediction model and the truth model. A series of assimilation and prediction experiments were performed to investigate the degree to which observational information can be used to optimize the physical parameterization and its associated parameters so as to improve climate estimation and prediction. While stochastic physics, implemented by initially perturbing the parameters of the physical parameterization scheme, can significantly enhance the ensemble spread and improve the representation of the model ensemble, the parameter estimation is better able to reduce model biases induced by the introduction of biased physics. Further, when the biased physics dominates the behavior of the coupled model, the parameter optimization where only the most influential parameters are optimized and allowed to vary geographically leads to better results for climate estimation and prediction. It is also noted that even when there is no direct observational constraint on the state variables of the low-frequency components, such as the deep sea temperature, the quality of the estimation/forecast of the state variables is improved with the DAEPC algorithm (especially the geography-dependent parameter optimization) compared to the traditional state estimation alone (SEO). In addition, compared to the state estimation within a perfect data assimilation framework, the DAEPC algorithm mitigates the model bias to some degree. Because of the substantial difference in the two physical parameterization schemes within the biased data assimilation framework, it is expected to improve the physical parameterizations themselves to mitigate the model bias further together with the DAEPC algorithm.
In spite of the promising results of parameter optimization in the intermediate climate model presented here, much more work is needed to fully understand the impact of model biases in coupled general circulation models (CGCMs) on real world climate estimation and prediction. Imperfect physical schemes are often used in CGCMs. Therefore, methods for optimizing model parameters in multiple biased physical schemes with the DAEPC algorithm need to be further examined to account for the possible compensating or noncompensating effects of the deficiencies in the physical schemes (Yang and Delsole 2009; Zhang 2011b), especially for the real world climate estimation and prediction.
Acknowledgments
The authors would like to thank Drs. X. Yang, G. Vecchi, I. Held, Y.-S. Chang, and A. Wittenberg for their generous discussions. Suggestions from three anonymous reviewers contributed significantly to the final version of this work. This research is sponsored by NSF, 2012CB955200 and partly supported by grants from the National Basic Research Program of China (2013CB430304) and the National Natural Science Foundation of China (under Grants 41030854, 41106005, and 41206178).
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