## 1. Introduction

Because of its potential role in the reduction of model bias and the improvement of climate predictability, parameter estimation for a coupled climate model is emerging as an important area of research. Traditional coupled data assimilation uses observations to adjust state variables only [called state estimation only (SEO)]. Therefore, the generated climate states usually exhibit a systematic error (Dee and Da Silva 1998; Dee 2005). The systematic error in state estimation could lead model predictions to drift toward imperfect model climate (Smith et al. 2007).

Based on the data assimilation theory (e.g., Jazwinski 1970), parameter estimation (e.g., Banks 1992a,b; Anderson 2001; Hansen and Penland 2007) can be realized by the state vector augmentation technique that adds model parameters into control variables of data assimilation. Many efforts have been made to advance parameter estimation. Early studies focused on the four-dimensional variational method (e.g., Navon 1997; Zhu and Navon 1999). Their results showed that initial conditions dominate short-term forecasts while longer time-scale signals rely more on the positive impact of optimized parameters. Based on ensemble Kalman filter (EnKF; Evensen 2006), Annan and Hargreaves (2004) applied parameter estimation in a highly nonlinear model. Annan et al. (2005) estimated the parameters of an intermediate complexity climate model using EnKF. Aksoy et al. (2006a) investigated the performance of EnKF through parameter estimation for the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5) and found that with high-level model sensitivities and covariances between parameters and model states, parameters being estimated converged to true value sufficiently. Similarly, Aksoy et al. (2006b) applied the method to a two-dimensional sea-breeze model and confirmed that simultaneous multiple-parameter estimation can reduce the model state errors effectively. Kondrashov et al. (2008) carried out parameter estimation with an intermediate coupled model using extended Kalman filter and found that estimating both state variables and parameters produced much better results than SEO. Tong and Xue (2008a,b) employed ensemble square root Kalman filter (Whitaker and Hamill 2002) to implement simultaneous state and parameter estimation in single-moment ice microphysics schemes using radar observations. Their results demonstrated that the ensemble-based parameter estimation can correct model errors in microphysical parameterization. Recently, a coupled data assimilation scheme with enhancive parameter correction (DAEPC) is designed to address how to obtain a signal-dominant state-parameter covariance so as to effectively optimize coupled model parameters using observations in different system components (Zhang et al. 2012). The DAEPC has been applied to a simple pycnocline prediction model to improve model decadal predictions (Zhang 2011a,b).

In this study, based on the DAEPC method implemented in the ensemble adjustment Kalman filter (EAKF; Anderson 2001), the impact of geographically varying optimized parameter values on climate estimation and prediction is investigated using an intermediate coupled model. The coupled model consists of a barotropic atmosphere, a 1.5-layer baroclinic ocean, and a simple land model. The assimilation model is biased by erroneously setting the values of all model parameters. By comparing the results of single-value parameter estimation (SPE) and geographic-dependent parameter optimization (GPO) with single- and multiple-parameter cases, we show the superiority of GPO in both climate estimation and prediction.

After briefly describing the intermediate coupled model, section 2 also gives a brief description for the enhancive parameter correction scheme and experimental setup that will be used throughout the paper. Section 3 investigates the geographic dependence of model parameter sensitivities. Section 4 presents the geographic-dependent parameter optimization scheme, and section 5 discusses the impact of GPO on “climate prediction.” A summary and a general discussion are given in section 6.

## 2. Methodology

### a. An intermediate coupled model

To clearly illustrate the impact of the geographic dependence of model sensitivities on parameter optimization and avoid the complexity of a coupled general circulation model (CGCM), an intermediate atmosphere–ocean–land coupled model is first developed here. The coupling scheme follows the work of Liu (1993) where a linear ocean–atmosphere coupled model is designed to study the interannual-scale feedbacks of the atmosphere and ocean in extratropics. In this study, a vorticity advection equation is used to represent the atmosphere to account the nonlinearity of the atmosphere. The atmosphere is coupled with a 1.5-layer “baroclinic” ocean including a slab mixed layer and simulated upwelling through an oceanic streamfunction equation. To provide a complete bottom boundary condition for the atmosphere, a simple land model in which the evolution of land surface temperature is driven by atmosphere–land fluxes is added to the coupled system. All three model components adopt 64 × 54 Gaussian grid and are forwarded by a leapfrog time stepping with a half-hour integration step size. An Asselin–Robert time filter (Robert 1969; Asselin 1972) is introduced to damp spurious computational modes in the leapfrog time integration.

#### 1) The atmosphere

*β*=

*df*/

*dy*,

*f*denotes Coriolis parameter,

*y*represents the northward meridional distance from the equator,

*ψ*represents the geostrophic atmosphere streamfunction,

*μ*is a scale factor that converts streamfunction to temperature,

*λ*is the flux coefficient from the ocean (land) to the atmosphere, and

*T*and

_{o}*T*denote sea surface temperature (SST) and land surface temperature (LST), respectively.

_{l}The terms at the right-hand side simulate the fluxes from ocean and land, serving as the forcing of the atmosphere. A 21 rhomboidal truncation is applied to transform the grid values of *ψ* to spectral coefficients.

#### 2) The ocean

*ϕ*is the oceanic streamfunction;

*h*

_{0}being the reduced gravity and mean thermocline depth;

*γ*denotes momentum coupling coefficient between the atmosphere and ocean;

*K*is the horizontal diffusive coefficient of

_{q}*ϕ*;

*K*and

_{T}*A*are the damping coefficient and horizontal diffusive coefficient, respectively, of

_{T}*T*;

_{o}*κ*is the ratio of upwelling and damping. Because Eq. (2) is more appropriate for extratropical ocean (Liu 1993), in order to enhance the high-frequent signal in tropic,

*K*is set to be constant from 25°N to the North Pole and from 25°S to the South Pole, and reduces linearly toward the equator to 90% of the extratropical value. The linearly varying damping coefficient

_{T}*K*acts as a part of the “dynamic core” that does not engage in parameter estimation. The quantity

_{T}*C*is the flux coefficient from the atmosphere to the ocean. The term

_{o}*s*(

*τ, t*) is the solar forcing that introduces the seasonal cycle:where

*s*

_{0}represents the annual-mean solar forcing with zonal distribution,

*τ*denotes latitude, and

*t*is the days at the current time step. The period of solar forcing is set to 360 days, which defines the model calendar year.

#### 3) The land

*m*represents the ratio of heat capacity between the land and the ocean mixed layer;

*K*and

_{L}*A*are damping and diffusive coefficients of

_{L}*T*, respectively; and

_{l}*C*denotes the flux coefficient from the atmosphere to the land.

_{l}It should be noted that because of our barotropic representation of the atmosphere, it is more appropriate to comprehend the coupling between the atmosphere and the ocean and land in this system as a mathematical way rather than a physical parameterization. However, with the geographic distributions of synoptic and climate prognostic variables, this model is sufficient in its mathematical complexity for our purpose to explore the impact of the geographic dependence of model sensitivities on parameter optimization.

#### 4) Parameter classification

Default values of all parameters are listed in Table 1. The last 14 parameters are empirically determined by trial-and-error tuning. Note that the solar forcing *s*(*τ*, *t*) will not alter once it is determined using the default value of *K _{T}*, so it also acts as a part of the dynamic core. We define a vector

**= (**

*β**λ*,

*μ*,

*h*

_{0},

*γ*,

*K*,

_{q}*κ*,

*K*,

_{T}*A*,

_{T}*C*,

_{o}*m*,

*K*,

_{L}*A*,

_{L}*C*,

_{l}*η*) as a collection of parameters, where

*η*is the Asselin–Robert time filtering coefficient. The

*β*_{t}denotes the standard values of

**in the “truth” model that is used to produce “observations” (i.e., the samples of the truth model states).**

*β*Default values of parameters.

#### 5) Features and variability

We show the annual mean of model states and the variability of *T _{o}* in this section. Starting from initial conditions

**Z**

_{0}= (

*ψ*

^{0},

*ϕ*

^{0},

*ψ*

^{0},

*ϕ*

^{0},

*β*_{t}. Leaving the first 50 years as spinup period, what are showing next is based on the data over the last 300 years. Figure 1 shows the annual mean of

*ψ*(Fig. 1a),

*ϕ*(Fig. 1b),

*T*(Fig. 1c), and

_{o}*T*(Fig. 1d). We can see the western boundary currents, gyre systems, and the Antarctic Circumpolar Current clearly. High-frequency signals of

_{l}*T*in the tropics are also described by the model. The time mean wave train pattern of the atmospheric streamfunction mainly exists in low and midlatitudes. Because of the topography effect, the wave pattern in North Hemisphere is more complicated than that in South Hemisphere. For

_{o}*T*, because of the weak coupling with other components and the simple form of control equation, it basically takes a zonal distribution in the long time mean. Note that the low temperature in tropical lands can be attributed to the linear damping of

_{l}*K*in the solar forcing.

_{T}To investigate the variability of *T _{o}*, we also apply empirical orthogonal function (EOF) decomposition to the time series of the 3000-yr anomalies (monthly mean). Explained variances of the first six modes are 39%, 15%, 7%, 5%, 5%, and 4%, respectively. Time coefficients of the first six modes are used to perform the power spectrum analysis to show the internal variability of

*T*. The characteristic time scales of the first six modes are 400, 50, 30, 10–20, 5–20, and 5–15 years, respectively (Fig. 2), showing the nature of the model ocean variability with multiple time scales (interannual to multicentennial).

_{o}### b. Brief description of a coupled DAEPC

*y*

_{k}_{,i}represents the observational increment of the

*k*th observation

*y*for the

_{k}*i*th ensemble member; Δ

*β*

_{k}_{,i}indicates the contribution of the

*k*th observation to the parameter

*β*for the

*i*th ensemble member; cov(

*β*,

*y*) denotes the error covariance between the prior ensemble of parameter and the model-estimated ensemble of

_{k}*y*; and

_{k}*σ*is the standard deviation of the model-estimated ensemble of

_{k}*y*.

_{k}DAEPC (Zhang et al. 2012) is a modification of the standard data assimilation with adaptive parameter estimation (e.g., Kulhavy 1993; Borkar and Mundra 1999; Tao 2003). Since the successfulness of parameter estimation entirely depends on the accuracy of the state-parameter covariance (Zhang et al. 2012), and that model parameters do not have any dynamically supported internal variability, the accuracy of the ensemble-evaluated covariance is determined by the accuracy of the model ensemble simulating the intrinsic uncertainty of the states for which the observations try to sample. In DAEPC, parameter estimation is activated after state estimation reaches a quasi-equilibrium (QE) where the uncertainty of model states is sufficiently constrained by observations so that the state-parameter covariance is signal dominant. A norm of model state adjustments is used to determine whether the state estimation has reached a QE state (Zhang et al. 2012). Then parameters are adjusted using Eq. (5). The updated parameters are applied to the next data assimilation cycle, which further refines the state estimation.

*σ*and

_{l,t}*σ*

_{l,}_{0}denote the prior spreads of

*t*and the initial time;

*α*

_{0}is a constant tuned by a trial-and-error procedure;

*σ*is the sensitivity of the model state with regard to

_{l}*α*

_{0}/

*σ*times the initial spread, it will be enlarged to this amount.

_{l}### c. Design of “twin” experiment

Starting from **Z**_{0} described in section 2a(5), the truth model is run for 101 years to generate time series of truth with the first 50 years as the spinup period. Observations of model states are generated through adding a Gaussian white noise that simulates observational errors to the relevant true states of the remaining 51 years at specific observational frequencies. The standard deviations of observational errors are 10^{6} m^{2} s^{−1} for *ψ*, 100 m^{2} s^{−1} for *ϕ*, and 1 K for *T _{o}* and

*T*, respectively; while corresponding sampling frequencies are 6 h (for

_{l}*ψ*), 1 day (for

*ϕ*,

*T*, and

_{o}*T*). In this study, the observation locations of

_{l}*ψ*are global randomly and uniformly distributed with the same density of the model grids, while the observation locations of

*ϕ*,

*T*, and

_{o}*T*are simply placed at 5° × 5° global grid points that start from 85°S, 0° at the bottom-left corner to 85°N, 355°E at the top-right corner. Following previous studies (Zhang and Anderson 2003; Zhang et al. 2004; Zhang 2011a,b; Zhang et al. 2012), the ensemble size is set as 20 throughout this study.

_{l}To roughly simulate the real-world scenario in which both the assimilation model and the assimilation initial condition are biased relative to observations, in our assimilation model, all parameters are set with the values of 10% greater than their true values (*β*_{b} = 1.1*β*_{t}). Starting from **Z**_{0}, the biased model is also spun up for 50 years to generate the biased initial model states **Z**_{1} = (*ψ*^{1}, *ϕ*^{1}, *ψ* are produced by superimposing a Gaussian white noise with the standard deviation of 10^{6} m^{2} s^{−1} on *ψ*^{1}, while *ϕ*, *T _{o}*, and

*T*are not perturbed. In addition, initial standard deviations of four most sensitive parameters (see section 3) to be optimized are set to 1% of relevant biased values, while the other 10 biased parameters are not perturbed. We denote the ensemble initial conditions of the coupled model as

_{l}**Π**.

Starting from **Π**, a 51-yr model ensemble control run without any observational constraint (denoted as CTL) and an SEO experiment are first performed. Then all the parameter estimation experiments start after SEO has performed one year where the state estimation has reached its QE (Zhang et al. 2012).

Leaving another 3 years as the parameter estimation spinup, all evaluation for assimilation schemes next are based on the last 47-yr results. Table 2 lists observation-adjusted model variables and observation-optimized model parameters in the assimilation. Here, *ψ ^{o}*,

*ϕ*,

^{o}*ψ*,

*ϕ*,

*T*, and

_{o}*T*denote model states to be estimated; and

_{l}*μ*,

*γ*,

*K*, and

_{T}*K*are parameters to be optimized.

_{L}Observation-adjusted model variables and observation-optimized model parameters in the assimilation.

Given that various time scales exist among component models, the multivariate adjustment scheme is only performed within the ocean component (see Table 2). Because of the leapfrog time stepping, a two-time level adjustment (Zhang et al. 2004) is applied for state estimation. Additionally, in order to remove spurious correlations caused by long distance, the distance factor (Hamill et al. 2001; Zhang et al. 2007) is introduced into the filtering (Hamill et al. 2001; Zhang et al. 2007). For *ψ* and *T _{l}*, the impact radius of observations is set to 500 km; while for

*ϕ*and

*T*, it is set to 1000 km × cos[min(

_{o}*τ*, 60)], where

*τ*denotes the latitude of model grids.

## 3. Geographic dependence of model sensitivities with respect to parameters

In this section we first investigate the model sensitivities with respect to parameters. Here, the sensitivity study is conducted for all 14 parameters with the assimilation model.

The ensemble spread of a model prognostic variable when a perturbation is added on a parameter is used to evaluate the relevant sensitivities quantitatively.

For the *l*th parameter (say *β _{l}*), draw 20 Gaussian random numbers with the standard deviation being 5% of the default value of

*β*to produce perturbations, while the other 13 parameters remain unperturbed. Starting from

_{l}**Z**

_{1}, the assimilation model is forwarded up to 11 years. Model states are perturbed with the ensemble of

*β*. Because of the relatively short time scale of the atmosphere, the sensitivities of

_{l}*ψ*with respect to

*β*are computed using time series of 0.5–1 year. For the ocean and land, sensitivities are calculated with the results of last 10 years. This process is looped for each parameter.

_{l}Figure 3 shows the time–space-averaged sensitivities of *ψ* (Fig. 3a), *ϕ* (Fig. 3b), *T _{o}* (Fig. 3c), and

*T*(Fig. 3d) with respect to 14 parameters. Here

_{l}*K*and

_{T}*K*are the most sensitive parameters because they determine the time scales of

_{L}*T*and

_{o}*T*so that a tiny perturbation can cause a dramatic drift of

_{l}*T*and

_{o}*T*. As a return, the different

_{l}*T*and

_{o}*T*further change the fluxes from the ocean (land) to the atmosphere by which the new curls of wind stress change the ocean currents. Because of the weak coupling with other components and the linear nature of the control equation of

_{l}*T*,

_{l}*K*dominates the sensitivity of

_{L}*T*. According to the weak sensitivities of

_{l}*T*with respect to

_{o}*κ*and

*C*, as well as the small magnitude of the advective term, the nonlinearity is also weak in the

_{o}*T*equation. Therefore,

_{o}*K*also plays a main role in the sensitivity of

_{T}*T*. For

_{o}*ψ*and

*ϕ*, the second sensitive parameters are

*μ*and

*γ*.

Figure 4 presents the geographic-dependent distribution of time-averaged sensitivities of *ψ* with respect to *μ* (Fig. 4a), *ϕ* with respect to *γ* (Fig. 4b), *T _{o}* with respect to

*K*(Fig. 4c), and

_{T}*T*with respect to

_{l}*K*(Fig. 4d). The most sensitive areas of the atmospheric streamfunction are, in order, the Antarctic Circumpolar Current system, the Antarctic continent, and the high latitudes of the Northern Hemisphere. The sensitivities of the oceanic streamfunction mainly focus on western boundary current systems and subtropical gyres. By contrast, the Antarctic Circumpolar Current is the most sensitive region of

_{L}*T*. Subtropical circulations in both hemispheres are the most insensitive areas of

_{o}*T*. The most insensitive region of

_{o}*T*is the Antarctic continent and not much different sensitivity is found for other continents.

_{l}## 4. Geographic-dependent parameter optimization

In this section, based on information of geographic-dependent model sensitivities, we extend the DAEPC (Zhang et al. 2012) to implement geographic-dependent parameter optimization (GPO) that pursues a significant signal enhancement in estimated parameter values. Before parameter optimization starts, it should be ensured that model state reaches the quasi-equilibrium (Zhang et al. 2012). Through computing the norm of model state adjustments, the state estimation spinup period can be roughly determined as one year. We first examine a simple case that only estimates a single parameter using observations.

### a. Single parameter GPO

From section 3, we choose *K _{T}*, the most sensitive parameter, to perform the parameter optimization. Observations of

*T*are used to optimize

_{o}*K*(Table 2) while the sensitivity of

_{T}*T*with respect to

_{o}*K*serves as the sensitivity of

_{T}*K*. We first introduce the traditional one parameter SPE briefly.

_{T}#### 1) Traditional one parameter SPE

SPE assumes that *K _{T}* has no geographic distribution. Starting from

**Π**, SEO is performed during the first year to reach the QE of model states. For each analysis step in the later 50 years,

*ψ*,

*ϕ*,

*T*, and

_{o}*T*are adjusted by corresponding observations first (Table 2). Then all available observations of

_{l}*T*are used to adjust the ensemble of

_{o}*K*sequentially using Eq. (5). Last, the updated ensemble of

_{T}*K*engages in the model integration until the next analysis step. The inflation scheme [Eq. (6)] is introduced into parameter estimation so as to avoid losing ensemble spread. Here, the sensitivity of

_{T}*K*,

_{T}*σ*, is also a single value that is simply the time–space-averaged sensitivity of

_{l}*T*with respect to

_{o}*K*(Fig. 3c). Through several trial-and-error tests,

_{T}*α*

_{0}takes the value 1.0. Additionally, upper and lower bounds (here are ±50% of the default value) are set to constrain the adjustment so that ensemble of

*K*cannot drift far away from the biased value.

_{T}Solid and dashed lines in Fig. 5 show the time series of RMSEs of *T _{o}* for SEO and SPE, respectively. There is an obvious spinup phase for parameter estimation. During this period, the RMSE of SPE is even larger than SEO. With 3-yr parameter estimation, the RMSE of

*T*reaches a stable state. Since all observations of

_{o}*T*are used to estimate the single value of

_{o}*K*, adjustments of observations that locate at sensitive and insensitive areas may be counteracted. In consequence, SPE is a parameter estimation scheme in the global-averaged sense that does not consider the geographic dependence of sensitivity sufficiently. Figure 6c shows the spatial distribution of RMSEs of

_{T}*T*for SPE. Compared with SEO (Fig. 6b), SPE reduces errors in most areas. No pronounced amelioration, however, is found in most sensitive and insensitive areas (check with Fig. 4c).

_{o}#### 2) One parameter GPO

In this subsection, we introduce the main idea of GPO. It is known that uncertainties of parameters can be transferred to model states through model integrating. Therefore, the model sensitivity is geographic dependent, which may impact the signal in parameter estimation. To estimate *K _{T}* better, GPO introduces the geographic dependence of sensitivity into parameter estimation. Through localizing the parameter sensitivity

*σ*and allowing optimized parameter values to vary geographically, signals in covariance can be assimilated sufficiently and optimal parameters can be obtained under local least squares frame. In addition, the introduced inflation scheme [Eq. (6)] is also geographic dependent. On one hand, for a small

_{l}*σ*(insensitive area), the inflation level is large, which enhances the signal-to-noise ratio in the parameter–observation covariance. On the other hand, for a large

_{l}*σ*(sensitive place), the high sensitivity can maintain a high signal-to-noise ratio and the small inflation factor can also prevent the model blowing up due to the excess inflation of the parameter ensemble. Therefore, the inflation scheme further enhances the signal in parameter optimization.

_{l}The following steps outline the main procedures of GPO for the intermediate coupled model and the idealized observing system:

- Step 1: Starting from
**Π**, SEO is applied to the first year to reach the quasi-equilibrium of model state. - Step 2: For each analysis step in the later 50 years:
- Step 2.1: Compute observational increments for observations of
*ψ*,*ϕ*,*T*, and_{o}*T*._{l} - Step 2.2: Perform the state estimation according to Table 2.
- Step 2.3: Compute prior spreads of
*K*for model grids that fall into the specific influence scope of observation, then inflate the ensemble of_{T}*K*and update it using the observational increment of_{T}*T*._{o} - Loop Steps 2.1–2.3 until all observations have been processed. Then for each model latitude do the following:
- Step 2.4: For each ensemble member of
*K*, compute its zonal average, which is assigned to all_{T}*T*model grids at current latitude._{o} - Step 2.5: For each ensemble member of
*K*, compute the mean of adjoining three latitudes, then the mean value is assigned to all_{T}*T*model grids at the current latitude._{o}

- Step 2.1: Compute observational increments for observations of
- Step 3: Integrate the model with the updated ensemble of
*K*until the next analysis step._{T}

Note that *α*_{0} in the inflation scheme is set to 1. The geographic-dependent sensitivity of *T _{o}* with respect to

*K*(see Fig. 4c) serves as the

_{T}*σ*. During the optimization of

_{l}*K*, in order to use as many as possible observations and considering the zonal trait of the sensitivity of

_{T}*T*with respect to

_{o}*K*(see Fig. 4c), grids that locate between 70°S and 70°N are influenced by observations whose latitudes are less than 500 km away, while other grids are adjusted by the observations whose latitudes are less than 3000 km away. The goal of step 2.4 is to eliminate the zonal gradient of

_{T}*K*,

_{T}*K*ensemble is employed. Similarly, step 2.5 is used to smooth the meridional gradient of

_{T}*K*ensemble. Apparently, the gradient smooth scheme depends on the geographic dependence of model sensitivities. In the real world, both model sensitivities and the representation of observations are geographic dependent, so the related gradient smooth scheme and parameter optimize method will be modified correspondingly.

_{T}Table 3 lists the root-mean-square errors (RMSEs) of all model components produced by different data assimilation schemes we compare in this study. From first four rows in Table 3, both SPE and GPO further reduce errors of *T _{o}* greatly from SEO by 68% and 85%, respectively, while SEO dramatically reduces the error of

*T*from CTL (by 96%). Note that because of the computational modes induced by the inconsistent boundary conditions between the SPE-produced SSTs (

_{o}*T*) and the SEO-produced LSTs (

_{o}*T*) over the coastal regions, the RMSEs of

_{l}*ψ*and

*ϕ*for SPE and GPO are a little larger than SEO. This means that it is very important for all components to obtain a consistent observational constraint in a coupled system. For

*T*, because all three schemes have the same

_{l}*K*and the seasonal cycles that depend on the truth of

_{L}*K*, and the fact that

_{T}*K*and

_{L}*K*are the two most sensitive parameters, the RMSE of

_{T}*T*for GPO is the same as that of SEO and SPE. Additionally, the weak and indirect coupling with

_{l}*T*is also a reason.

_{o}Total RMSEs of atmospheric streamfunction (*ψ*), oceanic streamfunction (*ϕ*), SST (*T _{o}*), and LST (

*T*) for all experiments.

_{l}Figure 6d shows the spatial distribution of the RMSE of *T _{o}* for GPO. Compared with CTL (Fig. 6a) and SEO (Fig. 6b), significant improvements are made in global areas. In addition, GPO significantly mitigates the error in most sensitive and insensitive areas for SPE (Fig. 6c). The dotted line in Fig. 5 shows the time series of RMSEs of

*T*for GPO, which indicates the smallest error of

_{o}*T*among all schemes after 3-yr parameter optimization spinup period.

_{o}To present the advantage of GPO in detail, Fig. 7 shows the ensemble mean time series (the 10th year) of *T _{o}* at an insensitive point (48.72°N, 168.75°E; top panel) and a sensitive point (58.63°S, 202.5°E; bottom panel) (see Fig. 4c). Dashed-dot, solid, dashed, and dotted lines represent truth, SEO, SPE, and GPO, respectively. Because of the global average meaning, SPE has no significant improvement of

*T*for these two grids compared with SEO. For SEO, compared with truth, the error of the sensitive point (bottom panel) is larger than that of the insensitive point (top panel) because of the high sensitivity. However, for GPO, as the explanation at the beginning of section 4a(2), time series of

_{o}*T*for GPO significantly approach the truth. Evident undulations exist in SEO, indicating the remarkable model bias (here mainly stems from the uncertainty of

_{o}*K*). For SPE and GPO, however, the undulation is almost invisible, which confirms that the model bias has been reduced sufficiently.

_{T}### b. Multiple parameter

In a CGCM, usually there is an array of vital and sensitive parameters. In this section we examine the general case (i.e., multiple-parameter GPO). Here, the four most sensitive parameters identified by the sensitivity study, *K _{T}*,

*K*,

_{L}*μ*,

*γ*, are chosen to perform simultaneously state estimation and multiple-parameter GPO. Multiple-parameter SPE is also conducted as an important reference. According to Table 2, observations of

*ψ*,

*ϕ*,

*T*, and

_{o}*T*are used to optimize

_{l}*μ*,

*γ*,

*K*, and

_{T}*K*, respectively. Because model states have different sensitivities with respect to various parameters, the single variable adjustment is employed for simplicity. Again we first introduce the traditional multiple-parameter SPE.

_{L}#### 1) Traditional multiple-parameter SPE

Similar to the one parameter case, SPE here assumes that *μ*, *γ*, *K _{T}*, and

*K*have no geographic distributions. Starting from

_{L}**Π**, with 1-yr SEO, for each analysis step in the later 50 yr, state estimation is first performed the same as SEO. Then all observations of

*ψ*,

*ϕ*,

*T*, and

_{o}*T*are used to, respectively, adjust ensembles of

_{l}*μ*,

*γ*,

*K*, and

_{T}*K*. The updated parameter ensembles engage in the next data assimilation cycle and act as prior parameter ensembles. The inflation scheme [Eq. (6)] is also applied. Time–space-averaged sensitivities of

_{L}*ψ*,

*ϕ*,

*T*, and

_{o}*T*with respect to

_{l}*μ*,

*γ*,

*K*, and

_{T}*K*, respectively (Fig. 3), are assigned to

_{L}*σ*values of these four parameters. With trial-and-error tests, the related

_{l}*α*

_{0}values are set to 10

^{6}, 500, 1, and 0.2.

Blue and red lines in Fig. 8 show the time series of RMSEs of *ψ* (Fig. 8a), *ϕ* (Fig. 8b), *T _{o}* (Fig. 8c), and

*T*(Fig. 8d) for SEO and SPE, respectively. Significant ameliorations are made for all model states. Improvement levels of

_{l}*T*and

_{o}*T*are the most significant, which results from the high sensitivities of

_{l}*K*and

_{T}*K*. From total statistics in Table 3, RMSEs of

_{L}*ψ*,

*ϕ*,

*T*, and

_{o}*T*are reduced by 12%, 10%, 61%, and 24%, respectively, from SEO to multiple-parameter SPE.

_{l}First and second columns in Fig. 9 display the spatial distributions of RMSEs of *ψ* (Figs. 8a,b), *ϕ* (Figs. 8d,e), *T _{o}* (Figs. 8g,h), and

*T*(Figs. 8j,k) for SEO (column 1) and SPE (column 2). For

_{l}*T*, according to Figs. 8j,k, SPE reduces errors in almost all continents except the south Antarctic continent. According to the spatial distribution of sensitivity of

_{l}*T*with respect to

_{l}*K*(see Fig. 4d), the Antarctic continent is the only insensitive area. Because SPE uses the global-averaged sensitivity and assumes parameters to be estimated have single values, it prefers to mitigate errors of

_{L}*T*in other places, which causes the

_{l}*T*in Antarctic continent worse than SEO. For

_{l}*ψ*, significant improvements are found in the Antarctic Circumpolar Current, South America, and Africa continents. However, in the Eurasian plate, North American plate, and Antarctic continent, SPE is even worse than SEO. As the error analysis of

*T*, high RMSEs of

_{l}*ψ*in the Antarctic continent attribute to the worse fluxes from land. In the Eurasian plate and North American plate, large errors result from insensitivities of

*ψ*with respect to

*μ*there. From Fig. 4a, it is easy to see that insensitive areas are consistent with the areas where SPE behaves badly (except the Antarctic continent). Because most places have high sensitivities, SPE inclines to improve

*ψ*in sensitive areas, which results in the bad performance in insensitive areas. The moderate sensitivity in the South America continent and the corrected land–atmosphere fluxes there result in the improvement of

*ψ*. For the Africa continent, the mitigation of RMSE of

*ψ*may result from the improvement of the land–atmosphere fluxes. For

*ϕ*, obvious improvements are found in tropical Indian Ocean and tropical Atlantic Ocean. Because of the weak sensitivity of

*γ*, the improvement of

*ϕ*is less than that of

*T*and

_{o}*T*. Results of

_{l}*T*are similar to that of one parameter SPE. From Table 3, the RMSE of

_{o}*T*for multiple-parameter SPE is a little larger than one parameter SPE, which may stem from the combination effect in multiple-parameter estimation.

_{o}#### 2) Multiple-parameter GPO

Here GPO allows optimized parameter values of *μ*, *γ*, *K _{T}*,

*K*to vary geographically. Starting from

_{L}**Π**, with 1-yr SEO, for each analysis step in the later 50 yr, state estimation is first implemented the same as SEO. Then observations of model states are used to sequentially optimize parameter ensembles. Here, the optimizing schemes of

*γ*and

*K*are the same as that of

_{T}*K*in one parameter GPO case. Because of the high observing density and frequency of

_{T}*ψ*in space and time, the impact radius of observation during optimizing

*μ*is the same as state estimation. Furthermore, the zonal and meridional gradient smoothing are not involved. For the optimization of

*K*, grids whose latitudes are higher than 70° are impacted by the observations whose latitudes are less than 3000 km away, while other grids are influenced by the observations whose latitudes are less than 1000 km away. The gradient smoothing is implemented for the optimization of

_{L}*K*. Additionally, the geographic-dependent distributions of the time-averaged sensitivities of

_{L}*ψ*,

*ϕ*,

*T*, and

_{o}*T*with regard to

_{l}*μ*,

*γ*,

*K*, and

_{T}*K*(see Fig. 4), respectively, serve as the relevant spatial

_{L}*σ*values. The

_{l}*α*

_{0}values of

*μ*,

*γ*,

*K*, and

_{T}*K*are set to 1.5 × 10

_{L}^{6}, 1000, 1, and 1.5, respectively.

The green line in Fig. 8 shows the time series of RMSEs of *ψ* (Fig. 8a), *ϕ* (Fig. 8b), *T _{o}* (Fig. 8c), and

*T*(Fig. 8d) for GPO. Relative to SEO (blue line) and SPE (red line), all model states are improved significantly. From the last two rows in Table 3, GPO reduces RMSEs of

_{l}*ψ*,

*ϕ*,

*T*, and

_{o}*T*by 41%, 23%, 62%, and 59%, respectively, from multiple-parameter SPE.

_{l}The last column in Fig. 9 shows the spatial distributions of RMSEs of *ψ* (Fig. 9c), *ϕ* (Fig. 9f), *T _{o}* (Fig. 9i), and

*T*(Fig. 9l) for GPO. For

_{l}*T*, GPO reduces RMSEs of SPE in all lands. However, Antarctic continent is still worse than SEO (Fig. 9j), which may be caused by the following two reasons. One lies in the fact that

_{l}*T*is so insensitive with respect to

_{l}*K*in this place (Fig. 4d) that it needs a very large

_{L}*α*

_{0}to enhance the signal-to-noise ratio of the error covariance. However, too large

*α*

_{0}may blow up

*K*in other places. Therefore, it is difficult to balance the insensitivity of Antarctic continent and the sensitivities of other places only through coordinating

_{L}*α*

_{0}. Even so, GPO improves the

*T*in the Antarctic continent for SPE. The other is that there are too few observations to constrain the ensemble of

_{l}*K*in that place. Improvements of

_{L}*T*in other places further correct the fluxes from the land to the atmosphere. For

_{l}*ψ*, relative to SPE (Fig. 9b), significant improvements are made in the Eurasian plate and North American plate, which contributes to the localization of

*μ*and the improvement of fluxes from the land to the atmosphere. There is also a pronounced amelioration in Antarctic continent, which is consistent with the results of

*T*. Compared with SEO (Fig. 9a), GPO reduces RMSEs of

_{l}*ψ*notably for almost all places except the southernmost of Antarctic continent, which is again in line with the result of

*T*. For

_{l}*ϕ*, the error level is reduced relative to both SPE and SEO. Because of the improvements of both

*γ*and

*ψ*, forcings from the atmosphere are also corrected, which further reduces the error of

*ϕ*. Results of

*T*are nearly the same as one parameter GPO.

_{o}## 5. Impact of GPO on “climate” prediction

To evaluate the impact of GPO on the model prediction, 20 forecast initial conditions are selected every 2 years apart from analysis fields of 5–43 years for SEO, multiple-parameter SPE, and multiple-parameter GPO. Then 20 forecast cases are forwarded up to 10 years for these three assimilation schemes. Note that in order to eliminate the discrepancy of parameter structures in the prediction model and the truth model, which produces observations, the global mean value of each GPO-generated parameter is applied to the prediction model. Here, the global anomaly correlation coefficient (ACC) of the forecasted ensemble mean is used to evaluate the global pattern correlation verified with the truth, and an ad hoc value of 0.6 ACC is employed to evaluate the valid time scale of forecast (Hollingsworth et al. 1980); the global RMSE of the forecasted ensemble mean is used to evaluate the global absolute error relative to the truth. However, in a CGCM with instrumental data, the error of the innovation of forecasts to observations may be a more appropriate quantity to evaluate forecast skills of different data assimilation schemes (see e.g., Fukumori et al. 1999).

### a. Weather forecast skill

With the observation-estimated single-value parameters, SPE extends the valid atmosphere forecast of SEO from 8 to 10 days (see Fig. 10a). By comparison, with better initial conditions (see the initial RMSE of *ψ* for GPO in Fig. 10b) and global means of GPO-produced parameters, GPO shows a smaller RMSE and a higher ACC, which lead to a 13-day valid weather forecast time scale. Note that GPO mixes with SPE after a lead time of 30 days, which is roughly the time scale of the predictability of the atmospheric signals in this intermediate coupled model.

### b. Climate prediction skill

Because *K _{T}* dominates the

*T*equation, biased

_{o}*K*in SEO makes the RMSE of

_{T}*T*(see solid line in Fig. 10d) increase rapidly and dramatically. With the observation-estimated single-value

_{o}*K*and the global mean of observation-optimized geographic

_{T}*K*, SPE and GPO mitigate the RMSE of

_{T}*T*significantly (see dashed and dotted lines in Fig. 10d) while GPO shows the least error of

_{o}*T*. The erroneously set

_{o}*K*causes a low initial ACC (0.36) (see the solid line in Fig. 10c) of the SEO-generated initial condition. With the observation-estimated single-valued

_{T}*K*, the initial ACC is improved (0.48). but still not higher than 0.6. However, with the geographic-dependent parameter optimization, the initial ACC is greatly enhanced (0.65). GPO maintains a higher ACC than SPE for the first 3 years of lead time. It seems that with the global mean of a GPO-generated geographic-dependent parameter, the model has a superior performance compared to the case using a SPE-produced parameter value. Note that when the method is applied to a CGCM for climate estimation and prediction with instrumental data, due to no truth parameters existing, we shall directly apply the GPO-generated parameters into the prediction model.

_{T}## 6. Summary and discussion

An intermediate atmosphere–ocean–land coupled model is developed to investigate the impact of geographic dependence of model sensitivities on parameter optimization. The coupled model consists of a barotropic atmosphere, a 1.5-layer baroclinic ocean with a slab mixed layer, and a simple land model. A biased twin-experiment framework is designed by setting a 10% overestimate error for all model parameters in the assimilation model, while the “truth” model with standard parameter values produces “observations.” Then, observations are assimilated into the assimilation model to implement single-value parameter estimation and geographic-dependent parameter optimization. Experiments show that the new geographic-dependent parameter optimization substantially reduces the model bias and improves the quality of climate estimation, while the single-value parameter estimation cannot effectively extract observational information according to geographically variant sensitivities. With better initial conditions and signal-enhanced parameter values, the geographic-dependent parameter optimization significantly improves the model predictability.

Although many improvements have been shown with the intermediate coupled model and the idealized observing system, several problems remain before applying GPO to CGCMs. First, in a CGCM, besides model bias induced by uncertainty of parameter, physical processes also introduce complicated model biases. How the complex model biases impact on GPO needs to be investigated further. Second, the real observation systems are geographic dependent, which directly impacts the signals in parameter optimization. Using sparse observations in a nonsensitive region will degrade the signal-to-noise ratio of parameter optimization. Third, the constant *α*_{0} in the inflation scheme is determined through trial-and-error tests. In a CGCM, we may need to implement an adaptive inflation correction algorithm (Anderson 2007). Last, as we adopt the state vector augmentation technique to implement the parameter estimation, the computational cost can be almost neglected for SPE. For GPO, the computational cost is equivalent to adding more control variables in state estimation. When the number of parameters being optimized is significantly large, it could be an issue that is worthy of concern as we apply GPO to a CGCM.

## Acknowledgments

The authors thank Drs. Xinyao Rong and Shu Wu for their thorough examination and comments during the development of this intermediate coupled model. The authors thank two anonymous reviewers for their thorough examination and comments that were very useful for improving the manuscript. This research is sponsored by NSF Grant 2012CB955201 and NSFC Grant 41030854.

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