• Aksoy, A., , F. Zhang, , and J. W. Nielsen-Gammon, 2006a: Ensemble-based simultaneous state and parameter estimation with MM5. Geophys. Res. Lett., 33, L12801, doi:10.1029/2006GL026186.

    • Search Google Scholar
    • Export Citation
  • Aksoy, A., , F. Zhang, , and J. W. Nielsen-Gammon, 2006b: Ensemble-based simultaneous state and parameter estimation in a two-dimensional sea-breeze model. Mon. Wea. Rev., 134, 29512970.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903.

  • Anderson, J. L., 2003: A local least squares framework for ensemble filtering. Mon. Wea. Rev., 131, 634642.

  • Anderson, J. L., 2007: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus, 59A, 210224.

  • Annan, J. D., , and J. C. Hargreaves, 2004: Efficient parameter estimation for a highly chaotic system. Tellus, 56A, 520526.

  • Annan, J. D., , J. C. Hargreaves, , N. R. Edwards, , and R. Marsh, 2005: Parameter estimation in an intermediate complexity Earth System Model using an ensemble Kalman filter. Ocean Modell., 8, 135154.

    • Search Google Scholar
    • Export Citation
  • Asselin, R., 1972: Frequency filter for time integrations. Mon. Wea. Rev., 100, 487490.

  • Banks, H. T., 1992a: Control and Estimation in Distributed Parameter Systems. Vol. 11, Frontiers in Applied Mathematics, SIAM, 227 pp.

  • Banks, H. T., 1992b: Computational issues in parameter estimation and feedback control problems for partial differential equation systems. Physica D, 60, 226238.

    • Search Google Scholar
    • Export Citation
  • Borkar, V. S., , and S. M. Mundra, 1999: Bayesian parameter estimation and adaptive control of Markov processes with time-averaged cost. Appl. Math., 25 (4), 339358.

    • Search Google Scholar
    • Export Citation
  • Dee, D. P., 2005: Bias and data assimilation. Quart. J. Roy. Meteor. Soc., 131, 33233343.

  • Dee, D. P., , and A. M. Da Silva, 1998: Data assimilation in the presence of forecast bias. Quart. J. Roy. Meteor. Soc., 124, 269295.

  • Evensen, G., 2006: Data Assimilation: The Ensemble Kalman Filter. Springer, 280 pp.

  • Fukumori, I., , R. Raghunath, , L. Fu, , and Y. Chao, 1999: Assimilation of TOPEX/Poseidon data into a global ocean circulation model: How good are the results? J. Geophys. Res., 104 (C11), 25 64725 665.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , J. S. Whitaker, , and C. Snyder, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 27762790.

    • Search Google Scholar
    • Export Citation
  • Hansen, J., , and C. Penland, 2007: On stochastic parameter estimation using data assimilation. Physica D, 230, 8898.

  • Hollingsworth, A., , K. Arpe, , M. Tiedtke, , M. Capaldo, , and H. Savijarvi, 1980: The performance of a medium range forecast model in winter—Impact of physical parameterizations. Mon. Wea. Rev., 108, 17361773.

    • Search Google Scholar
    • Export Citation
  • Jazwinski, A. H., 1970: Stochastic Processes and Filtering Theory. Academic Press, 376 pp.

  • Kalman, R., 1960: A new approach to linear filtering and prediction problems. Trans. ASME, J. Basic Eng., 82D, 3545.

  • Kalman, R., , and R. Bucy, 1961: New results in linear filtering and prediction theory. Trans. ASME, J. Basic Eng., 83D, 95109.

  • Kondrashov, D., , C. Sun, , and M. Ghil, 2008: Data assimilation for a coupled ocean–atmosphere model. Part II: Parameter estimation. Mon. Wea. Rev., 136, 50625076.

    • Search Google Scholar
    • Export Citation
  • Kulhavy, R., 1993: Implementation of Bayesian parameter estimation in adaptive control and signal processing. J. Roy. Stat. Soc., 42D (4), 471482.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., 1993: Interannual positive feedbacks in a simple extratropical air–sea coupling system. J. Atmos. Sci., 50, 30223028.

  • Navon, I. M., 1997: Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography. Dyn. Atmos. Oceans, 27, 5579.

    • Search Google Scholar
    • Export Citation
  • Philander, G., , T. Yamagata, , and R. C. Pacanowski, 1984: Unstable air-sea interaction in the tropics. J. Atmos. Sci., 41, 604613.

  • Robert, A., 1969: The integration of a spectral model of the atmosphere by the implicit method. Proc. WMO/IUGG Symp. on NWP, Tokyo, Japan, Japan Meteorological Society, 19–24.

  • Smith, D. M., , S. Cusack, , A. W. Colman, , C. K. Folland, , G. R. Harris, , and J. M. Murphy, 2007: Improved surface temperature prediction for the coming decade from a global climate model. Science, 317, 796799.

    • Search Google Scholar
    • Export Citation
  • Tao, G., 2003: Adaptive Control Design and Analysis. John Wiley & Sons, Inc., 640 pp.

  • Tong, M., , and M. Xue, 2008a: Simultaneous estimation of microphysical parameters and atmospheric state with simulated radar data and ensemble square root Kalman filter. Part I: Sensitivity analysis and parameter identifiability. Mon. Wea. Rev., 136, 16301648.

    • Search Google Scholar
    • Export Citation
  • Tong, M., , and M. Xue, 2008b: Simultaneous estimation of microphysical parameters and atmospheric state with simulated radar data and ensemble square root Kalman filter. Part II: Parameter estimation experiments. Mon. Wea. Rev., 136, 16491668.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924.

  • Zhang, S., 2011a: Impact of observation-optimized model parameters on decadal predictions: Simulation with a simple pycnocline prediction model. Geophys. Res. Lett., 38, L02702, doi:10.1029/2010GL046133.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., 2011b: A study of impacts of coupled model initial shocks and state–parameter optimization on climate predictions using a simple pycnocline prediction model. J. Climate, 24, 62106226.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , and J. L. Anderson, 2003: Impact of spatially and temporally varying estimates of error covariance on assimilation in a simple atmospheric model. Tellus, 55A, 126147.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , J. L. Anderson, , A. Rosati, , M. J. Harrison, , S. P. Khare, , and A. Wittenberg, 2004: Multiple time level adjustment for data assimilation. Tellus, 56A, 215.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , M. J. Harrison, , A. Rosati, , and A. T. Wittenberg, 2007: System design and evaluation of coupled ensemble data assimilation for global oceanic climate studies. Mon. Wea. Rev., 135, 35413564.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , Z. Liu, , A. Rosati, , and T. Delworth, 2012: A study of enhancive parameter correction with coupled data assimilation for climate estimation and prediction using a simple coupled model. Tellus, 64A, 10963, doi:10.3402/tellusa.v64i0.10963.

    • Search Google Scholar
    • Export Citation
  • Zhu, Y., , and I. M. Navon, 1999: Impact of parameter estimation on the performance of the FSU Global Spectral Model using its full physics adjoint. Mon. Wea. Rev., 127, 14971517.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Annual mean of (a) atmospheric streamfunction (m2 s−1), (b) oceanic streamfunction (m2 s−1), (c) SST (K), and (d) LST (K).

  • View in gallery

    Power spectrums of the first six modes of monthly averaged SST anomalies. Dashed lines represent 95% confidence upper limits.

  • View in gallery

    Time–space-averaged sensitivities of (a) the atmospheric streamfunction, (b) the oceanic streamfunction, (c) SST, and (d) LST with respect to all parameters. The sensitivity is defined by the ensemble spread of a model state variable induced by the perturbation of parameter.

  • View in gallery

    The geographic-dependent distributions of time-averaged sensitivities of (a) the atmospheric streamfunction with respect to μ, (b) the oceanic streamfunction with respect to γ, (c) SST with respect to KT, and (d) LST with respect to KL.

  • View in gallery

    Time series of RMSEs of SST for SEO (solid line), traditional SPE (dashed line) that uses observations of SST to estimate KT, and GPO (dotted line) that uses the same observations to optimize the geographic-dependent KT. The small panel is the zoomed out version of the large one which includes the first year’s results. Note that parameter estimation (optimization) is activated after 1-yr SEO. Here, results of free run (CTL) are not shown because of the dramatic error.

  • View in gallery

    Spatial distributions of RMSEs (K) of SST for (a) CTL; (b) SEO; (c) SPE, which uses observations of SST to estimate KT; and (d) GPO, which uses the same observations to optimize the geographic-dependent KT. Note that (c) and (d) use the same shade scale.

  • View in gallery

    Two examples of ensemble mean time series (the 10th year) of SST at (top) an insensitive point (48.72°N, 168.75°E) and (bottom) a sensitive point (58.63°S, 202.50°E). SPE uses observations of SST to estimate KT, and GPO uses the same observations to optimize the geographic-dependent KT.

  • View in gallery

    Time series of RMSEs of (a) the atmospheric streamfunction, (b) the oceanic streamfunction, (c) SST, and (d) LST for SEO (blue line); SPE (red line), which uses observations of the atmospheric streamfunction, the oceanic streamfunction, SST, and LST to estimate μ, γ, KT, and KL, respectively; and GPO (green line), which uses the same observations to optimize the geographic-dependent μ, γ, KT, and KL. Note that parameter estimation (optimization) is activated after 1-yr SEO.

  • View in gallery

    Spatial distributions of RMSEs of the atmospheric streamfunction (m2 s−1), the oceanic streamfunction (m2 s−1), SST (K), and LST (K) for (from left to right) SEO; SPE, which uses observations of the atmospheric streamfunction, the oceanic streamfunction, SST, and LST to estimate μ, γ, KT, and KL, respectively; and GPO, which uses the same observations to optimize the geographic-dependent μ, γ, KT, and KL. Note that each variable uses the same color bar for the three schemes.

  • View in gallery

    Variations of (a),(c) ACC and (b),(d) RMSE with the forecast lead time of the forecasted ensemble means of the (a),(b) atmospheric streamfunction and (c),(d) SST for SEO (solid line), SPE (dashed line), and GPO (dotted line), which applies the global means of observation-optimized parameters into prediction model. The dashed line in (a) represents the critical ACC value (0.6) of valid forecast time scale. The small panel is the zoomed out version of (d), which shows the complete evolutions of RMSE of To for these three schemes.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 34 34 2
PDF Downloads 18 18 1

Impact of Geographic-Dependent Parameter Optimization on Climate Estimation and Prediction: Simulation with an Intermediate Coupled Model

View More View Less
  • 1 NOAA/GFDL–Wisconsin Joint Visiting Program, Princeton, New Jersey, and South China Sea Institute of Oceanology, Chinese Academy of Science, Guangzhou, and Key Laboratory of Marine Environmental Information Technology, State Oceanic Administration, National Marine Data and Information Service, Tianjin, China
  • | 2 NOAA/GFDL, Princeton University, Princeton, New Jersey
  • | 3 Center for Climate Research and Department of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, Madison, Wisconsin, and Laboratory of Ocean–Atmosphere Studies, Peking University, Beijing, China
  • | 4 NOAA/GFDL, Princeton University, Princeton, New Jersey
  • | 5 Center for Climate Research and Department of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, Madison, Wisconsin
© Get Permissions
Full access

Abstract

Because of the geographic dependence of model sensitivities and observing systems, allowing optimized parameter values to vary geographically may significantly enhance the signal in parameter estimation. Using an intermediate atmosphere–ocean–land coupled model, the impact of geographic dependence of model sensitivities on parameter optimization is explored within a twin-experiment framework. The coupled model consists of a 1-layer global barotropic atmosphere model, a 1.5-layer baroclinic ocean including a slab mixed layer with simulated upwelling by a streamfunction equation, and a simple land model. The assimilation model is biased by erroneously setting the values of all model parameters. The four most sensitive parameters identified by sensitivity studies are used to perform traditional single-value parameter estimation and new geographic-dependent parameter optimization. Results show that the new parameter optimization significantly improves the quality of state estimates compared to the traditional scheme, with reductions of root-mean-square errors as 41%, 23%, 62%, and 59% for the atmospheric streamfunction, the oceanic streamfunction, sea surface temperature, and land surface temperature, respectively. Consistently, the new parameter optimization greatly improves the model predictability as a result of the improvement of initial conditions and the enhancement of observational signals in optimized parameters. These results suggest that the proposed geographic-dependent parameter optimization scheme may provide a new perspective when a coupled general circulation model is used for climate estimation and prediction.

Corresponding author address: Xinrong Wu, NOAA/Geophysical Fluid Dynamics Laboratory, 201 Forrestal Road, Princeton, NJ 08542. E-mail: xinrong.wu@noaa.gov

Abstract

Because of the geographic dependence of model sensitivities and observing systems, allowing optimized parameter values to vary geographically may significantly enhance the signal in parameter estimation. Using an intermediate atmosphere–ocean–land coupled model, the impact of geographic dependence of model sensitivities on parameter optimization is explored within a twin-experiment framework. The coupled model consists of a 1-layer global barotropic atmosphere model, a 1.5-layer baroclinic ocean including a slab mixed layer with simulated upwelling by a streamfunction equation, and a simple land model. The assimilation model is biased by erroneously setting the values of all model parameters. The four most sensitive parameters identified by sensitivity studies are used to perform traditional single-value parameter estimation and new geographic-dependent parameter optimization. Results show that the new parameter optimization significantly improves the quality of state estimates compared to the traditional scheme, with reductions of root-mean-square errors as 41%, 23%, 62%, and 59% for the atmospheric streamfunction, the oceanic streamfunction, sea surface temperature, and land surface temperature, respectively. Consistently, the new parameter optimization greatly improves the model predictability as a result of the improvement of initial conditions and the enhancement of observational signals in optimized parameters. These results suggest that the proposed geographic-dependent parameter optimization scheme may provide a new perspective when a coupled general circulation model is used for climate estimation and prediction.

Corresponding author address: Xinrong Wu, NOAA/Geophysical Fluid Dynamics Laboratory, 201 Forrestal Road, Princeton, NJ 08542. E-mail: xinrong.wu@noaa.gov

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

The atmosphere is a global barotropic spectral model based on the equation of potential vorticity conservation:
e1
where , 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 To and Tl denote sea surface temperature (SST) and land surface temperature (LST), respectively.

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

The ocean consists of a 1.5-layer baroclinic ocean with a slab mixed layer (Liu 1993) and the simulated upwelling by a streamfunction equation as
e2
where ϕ is the oceanic streamfunction; , is the oceanic deformation radius, with and h0 being the reduced gravity and mean thermocline depth; γ denotes momentum coupling coefficient between the atmosphere and ocean; Kq is the horizontal diffusive coefficient of ϕ; KT and AT are the damping coefficient and horizontal diffusive coefficient, respectively, of To; (Philander et al. 1984), representing the strength of upwelling (downwelling), and 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, KT 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 KT acts as a part of the “dynamic core” that does not engage in parameter estimation. The quantity Co is the flux coefficient from the atmosphere to the ocean. The term s(τ, t) is the solar forcing that introduces the seasonal cycle:
e3
where s0 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

The evolution of LST is simulated simply by a linear equation:
e4
where m represents the ratio of heat capacity between the land and the ocean mixed layer; KL and AL are damping and diffusive coefficients of Tl, respectively; and Cl denotes the flux coefficient from the atmosphere to the land.

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 KT, so it also acts as a part of the dynamic core. We define a vector β = (λ, μ, h0, γ, Kq, κ, KT, AT, Co, m, KL, AL, Cl, η) as a collection of parameters, where η is the Asselin–Robert time filtering coefficient. The βt denotes the standard values of β in the “truth” model that is used to produce “observations” (i.e., the samples of the truth model states).

Table 1.

Default values of parameters.

Table 1.

5) Features and variability

We show the annual mean of model states and the variability of To in this section. Starting from initial conditions Z0 = (ψ0, ϕ0, , ), where ψ0, ϕ0, , and , zonal mean values of corresponding climatological fields, the coupled model is run for 3050 years with βt. 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), To (Fig. 1c), and Tl (Fig. 1d). We can see the western boundary currents, gyre systems, and the Antarctic Circumpolar Current clearly. High-frequency signals of To 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 Tl, 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 KT in the solar forcing.

Fig. 1.
Fig. 1.

Annual mean of (a) atmospheric streamfunction (m2 s−1), (b) oceanic streamfunction (m2 s−1), (c) SST (K), and (d) LST (K).

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

To investigate the variability of To, 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 To. 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).

Fig. 2.
Fig. 2.

Power spectrums of the first six modes of monthly averaged SST anomalies. Dashed lines represent 95% confidence upper limits.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

b. Brief description of a coupled DAEPC

The details of DAEPC can be found in Zhang et al. (2012), and here we only comment on a few aspects that are related to this study. Following Zhang et al. (2012) and Zhang (2011a,b), the EAKF (Anderson 2001) is used to perform the simultaneously state estimation and parameter optimization in this study. EAKF is a sequential implementation of Kalman filter (Kalman 1960; Kalman and Bucy 1961) under an “adjustment” idea. While the sequential implementation provides much computational convenience for data assimilation, the EAKF maintains the nonlinearity of background flows as much as possible (Anderson 2001, 2003; Zhang and Anderson 2003). Based on the two-step EAKF (Anderson 2003; Zhang and Anderson 2003; Zhang et al. 2007), parameter estimation is a process similar to multivariate adjustment in state estimation for a nonobservable variable. The first step that computes the observational increment is identical to state estimation [see Eqs. (2)–(5) in Zhang et al. (2007)]. The second step that projects the observational increment onto relevant parameters can be formulated as
e5
Here Δyk,i represents the observational increment of the kth observation yk for the ith ensemble member; Δβk,i indicates the contribution of the kth observation to the parameter β for the ith ensemble member; cov(β, yk) denotes the error covariance between the prior ensemble of parameter and the model-estimated ensemble of yk; and σk is the standard deviation of the model-estimated ensemble of yk.

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.

The inflation scheme of DAEPC based on model sensitivities of parameters (Zhang et al. 2012; Zhang 2011a,b) is a particular aspect of DAEPC that is extremely important for the application of this study. It is formulated as
e6
where and represent the prior and the inflated ensemble of the parameter ; σl,t and σl,0 denote the prior spreads of at time t and the initial time; α0 is a constant tuned by a trial-and-error procedure; σl is the sensitivity of the model state with regard to ; and the overbar represents the ensemble mean. Equation (6) means that if the prior spread of is less than α0/σl times the initial spread, it will be enlarged to this amount.

c. Design of “twin” experiment

Starting from Z0 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 106 m2 s−1 for ψ, 100 m2 s−1 for ϕ, and 1 K for To and Tl, respectively; while corresponding sampling frequencies are 6 h (for ψ), 1 day (for ϕ, To, and Tl). In this study, the observation locations of ψ are global randomly and uniformly distributed with the same density of the model grids, while the observation locations of ϕ, To, and Tl 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.

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 (βb = 1.1βt). Starting from Z0, the biased model is also spun up for 50 years to generate the biased initial model states Z1 = (ψ1, ϕ1, , ). Then the ensemble initial conditions of ψ are produced by superimposing a Gaussian white noise with the standard deviation of 106 m2 s−1 on ψ1, while ϕ, To, and Tl 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 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, ψo, ϕo, , and represent observations; ψ, ϕ, To, and Tl denote model states to be estimated; and μ, γ, KT, and KL are parameters to be optimized.

Table 2.

Observation-adjusted model variables and observation-optimized model parameters in the assimilation.

Table 2.

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 Tl, the impact radius of observations is set to 500 km; while for ϕ and To, it is set to 1000 km × cos[min(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 lth parameter (say βl), draw 20 Gaussian random numbers with the standard deviation being 5% of the default value of βl to produce perturbations, while the other 13 parameters remain unperturbed. Starting from Z1, the assimilation model is forwarded up to 11 years. Model states are perturbed with the ensemble of βl. Because of the relatively short time scale of the atmosphere, the sensitivities of ψ with respect to βl 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.

Figure 3 shows the time–space-averaged sensitivities of ψ (Fig. 3a), ϕ (Fig. 3b), To (Fig. 3c), and Tl (Fig. 3d) with respect to 14 parameters. Here KT and KL are the most sensitive parameters because they determine the time scales of To and Tl so that a tiny perturbation can cause a dramatic drift of To and Tl. As a return, the different To and Tl 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 Tl, KL dominates the sensitivity of Tl. According to the weak sensitivities of To with respect to κ and Co, as well as the small magnitude of the advective term, the nonlinearity is also weak in the To equation. Therefore, KT also plays a main role in the sensitivity of To. For ψ and ϕ, the second sensitive parameters are μ and γ.

Fig. 3.
Fig. 3.

Time–space-averaged sensitivities of (a) the atmospheric streamfunction, (b) the oceanic streamfunction, (c) SST, and (d) LST with respect to all parameters. The sensitivity is defined by the ensemble spread of a model state variable induced by the perturbation of parameter.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

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), To with respect to KT (Fig. 4c), and Tl with respect to KL (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 To. Subtropical circulations in both hemispheres are the most insensitive areas of To. The most insensitive region of Tl is the Antarctic continent and not much different sensitivity is found for other continents.

Fig. 4.
Fig. 4.

The geographic-dependent distributions of time-averaged sensitivities of (a) the atmospheric streamfunction with respect to μ, (b) the oceanic streamfunction with respect to γ, (c) SST with respect to KT, and (d) LST with respect to KL.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

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 KT, the most sensitive parameter, to perform the parameter optimization. Observations of To are used to optimize KT (Table 2) while the sensitivity of To with respect to KT serves as the sensitivity of KT. We first introduce the traditional one parameter SPE briefly.

1) Traditional one parameter SPE

SPE assumes that KT 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, To, and Tl are adjusted by corresponding observations first (Table 2). Then all available observations of To are used to adjust the ensemble of KT sequentially using Eq. (5). Last, the updated ensemble of KT 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 KT, σl, is also a single value that is simply the time–space-averaged sensitivity of To with respect to KT (Fig. 3c). Through several trial-and-error tests, α0 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 KT cannot drift far away from the biased value.

Solid and dashed lines in Fig. 5 show the time series of RMSEs of To 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 To reaches a stable state. Since all observations of To are used to estimate the single value of KT, 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 To 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).

Fig. 5.
Fig. 5.

Time series of RMSEs of SST for SEO (solid line), traditional SPE (dashed line) that uses observations of SST to estimate KT, and GPO (dotted line) that uses the same observations to optimize the geographic-dependent KT. The small panel is the zoomed out version of the large one which includes the first year’s results. Note that parameter estimation (optimization) is activated after 1-yr SEO. Here, results of free run (CTL) are not shown because of the dramatic error.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

Fig. 6.
Fig. 6.

Spatial distributions of RMSEs (K) of SST for (a) CTL; (b) SEO; (c) SPE, which uses observations of SST to estimate KT; and (d) GPO, which uses the same observations to optimize the geographic-dependent KT. Note that (c) and (d) use the same shade scale.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

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 KT better, GPO introduces the geographic dependence of sensitivity into parameter estimation. Through localizing the parameter sensitivity σl 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 σl (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 σl (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.

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, To, and Tl.
    • Step 2.2: Perform the state estimation according to Table 2.
    • Step 2.3: Compute prior spreads of KT for model grids that fall into the specific influence scope of observation, then inflate the ensemble of KT and update it using the observational increment of To.
    • 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 KT, compute its zonal average, which is assigned to all To model grids at current latitude.
    • Step 2.5: For each ensemble member of KT, compute the mean of adjoining three latitudes, then the mean value is assigned to all To model grids at the current latitude.
  • Step 3: Integrate the model with the updated ensemble of KT until the next analysis step.

Note that α0 in the inflation scheme is set to 1. The geographic-dependent sensitivity of To with respect to KT (see Fig. 4c) serves as the σl. During the optimization of KT, in order to use as many as possible observations and considering the zonal trait of the sensitivity of To with respect to KT (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 KT, . Figure 4c indicates that the zonal gradient of sensitivity can be neglected relative to the meridional gradient. Here, a simply zonal average of the KT ensemble is employed. Similarly, step 2.5 is used to smooth the meridional gradient of KT 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.

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 To greatly from SEO by 68% and 85%, respectively, while SEO dramatically reduces the error of To from CTL (by 96%). Note that because of the computational modes induced by the inconsistent boundary conditions between the SPE-produced SSTs (To) and the SEO-produced LSTs (Tl) over the coastal regions, the RMSEs of 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 Tl, because all three schemes have the same KL and the seasonal cycles that depend on the truth of KT, and the fact that KL and KT are the two most sensitive parameters, the RMSE of Tl for GPO is the same as that of SEO and SPE. Additionally, the weak and indirect coupling with To is also a reason.

Table 3.

Total RMSEs of atmospheric streamfunction (ψ), oceanic streamfunction (ϕ), SST (To), and LST (Tl) for all experiments.

Table 3.

Figure 6d shows the spatial distribution of the RMSE of To 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 To for GPO, which indicates the smallest error of To among all schemes after 3-yr parameter optimization spinup period.

To present the advantage of GPO in detail, Fig. 7 shows the ensemble mean time series (the 10th year) of To 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 To 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 To for GPO significantly approach the truth. Evident undulations exist in SEO, indicating the remarkable model bias (here mainly stems from the uncertainty of KT). For SPE and GPO, however, the undulation is almost invisible, which confirms that the model bias has been reduced sufficiently.

Fig. 7.
Fig. 7.

Two examples of ensemble mean time series (the 10th year) of SST at (top) an insensitive point (48.72°N, 168.75°E) and (bottom) a sensitive point (58.63°S, 202.50°E). SPE uses observations of SST to estimate KT, and GPO uses the same observations to optimize the geographic-dependent KT.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

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, KT, KL, 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, To, and Tl are used to optimize μ, γ, KT, and KL, 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.

1) Traditional multiple-parameter SPE

Similar to the one parameter case, SPE here assumes that μ, γ, KT, and KL have no geographic distributions. Starting from 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, To, and Tl are used to, respectively, adjust ensembles of μ, γ, KT, and KL. 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 ψ, ϕ, To, and Tl with respect to μ, γ, KT, and KL, respectively (Fig. 3), are assigned to σl values of these four parameters. With trial-and-error tests, the related α0 values are set to 106, 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), To (Fig. 8c), and Tl (Fig. 8d) for SEO and SPE, respectively. Significant ameliorations are made for all model states. Improvement levels of To and Tl are the most significant, which results from the high sensitivities of KT and KL. From total statistics in Table 3, RMSEs of ψ, ϕ, To, and Tl are reduced by 12%, 10%, 61%, and 24%, respectively, from SEO to multiple-parameter SPE.

Fig. 8.
Fig. 8.

Time series of RMSEs of (a) the atmospheric streamfunction, (b) the oceanic streamfunction, (c) SST, and (d) LST for SEO (blue line); SPE (red line), which uses observations of the atmospheric streamfunction, the oceanic streamfunction, SST, and LST to estimate μ, γ, KT, and KL, respectively; and GPO (green line), which uses the same observations to optimize the geographic-dependent μ, γ, KT, and KL. Note that parameter estimation (optimization) is activated after 1-yr SEO.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

First and second columns in Fig. 9 display the spatial distributions of RMSEs of ψ (Figs. 8a,b), ϕ (Figs. 8d,e), To (Figs. 8g,h), and Tl (Figs. 8j,k) for SEO (column 1) and SPE (column 2). For Tl, 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 Tl with respect to KL (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 Tl in other places, which causes the Tl in Antarctic continent worse than SEO. For 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 Tl, high RMSEs of 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 To and Tl. Results of To are similar to that of one parameter SPE. From Table 3, the RMSE of To for multiple-parameter SPE is a little larger than one parameter SPE, which may stem from the combination effect in multiple-parameter estimation.

Fig. 9.
Fig. 9.

Spatial distributions of RMSEs of the atmospheric streamfunction (m2 s−1), the oceanic streamfunction (m2 s−1), SST (K), and LST (K) for (from left to right) SEO; SPE, which uses observations of the atmospheric streamfunction, the oceanic streamfunction, SST, and LST to estimate μ, γ, KT, and KL, respectively; and GPO, which uses the same observations to optimize the geographic-dependent μ, γ, KT, and KL. Note that each variable uses the same color bar for the three schemes.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

2) Multiple-parameter GPO

Here GPO allows optimized parameter values of μ, γ, KT, KL to vary geographically. Starting from 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 KT are the same as that of KT in one parameter GPO case. Because of the high observing density and frequency of 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 KL, 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 KL. Additionally, the geographic-dependent distributions of the time-averaged sensitivities of ψ, ϕ, To, and Tl with regard to μ, γ, KT, and KL (see Fig. 4), respectively, serve as the relevant spatial σl values. The α0 values of μ, γ, KT, and KL are set to 1.5 × 106, 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), To (Fig. 8c), and Tl (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 ψ, ϕ, To, and Tl by 41%, 23%, 62%, and 59%, respectively, from multiple-parameter SPE.

The last column in Fig. 9 shows the spatial distributions of RMSEs of ψ (Fig. 9c), ϕ (Fig. 9f), To (Fig. 9i), and Tl (Fig. 9l) for GPO. For Tl, 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 Tl is so insensitive with respect to KL in this place (Fig. 4d) that it needs a very large α0 to enhance the signal-to-noise ratio of the error covariance. However, too large α0 may blow up KL in other places. Therefore, it is difficult to balance the insensitivity of Antarctic continent and the sensitivities of other places only through coordinating α0. Even so, GPO improves the Tl in the Antarctic continent for SPE. The other is that there are too few observations to constrain the ensemble of KL in that place. Improvements of Tl in other places further correct the fluxes from the land to the atmosphere. For 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 Tl. Compared with SEO (Fig. 9a), GPO reduces RMSEs of ψ notably for almost all places except the southernmost of Antarctic continent, which is again in line with the result of Tl. For 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 To are nearly the same as one parameter GPO.

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.

Fig. 10.
Fig. 10.

Variations of (a),(c) ACC and (b),(d) RMSE with the forecast lead time of the forecasted ensemble means of the (a),(b) atmospheric streamfunction and (c),(d) SST for SEO (solid line), SPE (dashed line), and GPO (dotted line), which applies the global means of observation-optimized parameters into prediction model. The dashed line in (a) represents the critical ACC value (0.6) of valid forecast time scale. The small panel is the zoomed out version of (d), which shows the complete evolutions of RMSE of To for these three schemes.

Citation: Monthly Weather Review 140, 12; 10.1175/MWR-D-11-00298.1

b. Climate prediction skill

Because KT dominates the To equation, biased KT in SEO makes the RMSE of To (see solid line in Fig. 10d) increase rapidly and dramatically. With the observation-estimated single-value KT and the global mean of observation-optimized geographic KT, SPE and GPO mitigate the RMSE of To significantly (see dashed and dotted lines in Fig. 10d) while GPO shows the least error of To. The erroneously set KT 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 KT, 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.

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 α0 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.

REFERENCES

  • Aksoy, A., , F. Zhang, , and J. W. Nielsen-Gammon, 2006a: Ensemble-based simultaneous state and parameter estimation with MM5. Geophys. Res. Lett., 33, L12801, doi:10.1029/2006GL026186.

    • Search Google Scholar
    • Export Citation
  • Aksoy, A., , F. Zhang, , and J. W. Nielsen-Gammon, 2006b: Ensemble-based simultaneous state and parameter estimation in a two-dimensional sea-breeze model. Mon. Wea. Rev., 134, 29512970.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903.

  • Anderson, J. L., 2003: A local least squares framework for ensemble filtering. Mon. Wea. Rev., 131, 634642.

  • Anderson, J. L., 2007: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus, 59A, 210224.

  • Annan, J. D., , and J. C. Hargreaves, 2004: Efficient parameter estimation for a highly chaotic system. Tellus, 56A, 520526.

  • Annan, J. D., , J. C. Hargreaves, , N. R. Edwards, , and R. Marsh, 2005: Parameter estimation in an intermediate complexity Earth System Model using an ensemble Kalman filter. Ocean Modell., 8, 135154.

    • Search Google Scholar
    • Export Citation
  • Asselin, R., 1972: Frequency filter for time integrations. Mon. Wea. Rev., 100, 487490.

  • Banks, H. T., 1992a: Control and Estimation in Distributed Parameter Systems. Vol. 11, Frontiers in Applied Mathematics, SIAM, 227 pp.

  • Banks, H. T., 1992b: Computational issues in parameter estimation and feedback control problems for partial differential equation systems. Physica D, 60, 226238.

    • Search Google Scholar
    • Export Citation
  • Borkar, V. S., , and S. M. Mundra, 1999: Bayesian parameter estimation and adaptive control of Markov processes with time-averaged cost. Appl. Math., 25 (4), 339358.

    • Search Google Scholar
    • Export Citation
  • Dee, D. P., 2005: Bias and data assimilation. Quart. J. Roy. Meteor. Soc., 131, 33233343.

  • Dee, D. P., , and A. M. Da Silva, 1998: Data assimilation in the presence of forecast bias. Quart. J. Roy. Meteor. Soc., 124, 269295.

  • Evensen, G., 2006: Data Assimilation: The Ensemble Kalman Filter. Springer, 280 pp.

  • Fukumori, I., , R. Raghunath, , L. Fu, , and Y. Chao, 1999: Assimilation of TOPEX/Poseidon data into a global ocean circulation model: How good are the results? J. Geophys. Res., 104 (C11), 25 64725 665.

    • Search Google Scholar
    • Export Citation
  • Hamill, T. M., , J. S. Whitaker, , and C. Snyder, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 27762790.

    • Search Google Scholar
    • Export Citation
  • Hansen, J., , and C. Penland, 2007: On stochastic parameter estimation using data assimilation. Physica D, 230, 8898.

  • Hollingsworth, A., , K. Arpe, , M. Tiedtke, , M. Capaldo, , and H. Savijarvi, 1980: The performance of a medium range forecast model in winter—Impact of physical parameterizations. Mon. Wea. Rev., 108, 17361773.

    • Search Google Scholar
    • Export Citation
  • Jazwinski, A. H., 1970: Stochastic Processes and Filtering Theory. Academic Press, 376 pp.

  • Kalman, R., 1960: A new approach to linear filtering and prediction problems. Trans. ASME, J. Basic Eng., 82D, 3545.

  • Kalman, R., , and R. Bucy, 1961: New results in linear filtering and prediction theory. Trans. ASME, J. Basic Eng., 83D, 95109.

  • Kondrashov, D., , C. Sun, , and M. Ghil, 2008: Data assimilation for a coupled ocean–atmosphere model. Part II: Parameter estimation. Mon. Wea. Rev., 136, 50625076.

    • Search Google Scholar
    • Export Citation
  • Kulhavy, R., 1993: Implementation of Bayesian parameter estimation in adaptive control and signal processing. J. Roy. Stat. Soc., 42D (4), 471482.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., 1993: Interannual positive feedbacks in a simple extratropical air–sea coupling system. J. Atmos. Sci., 50, 30223028.

  • Navon, I. M., 1997: Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography. Dyn. Atmos. Oceans, 27, 5579.

    • Search Google Scholar
    • Export Citation
  • Philander, G., , T. Yamagata, , and R. C. Pacanowski, 1984: Unstable air-sea interaction in the tropics. J. Atmos. Sci., 41, 604613.

  • Robert, A., 1969: The integration of a spectral model of the atmosphere by the implicit method. Proc. WMO/IUGG Symp. on NWP, Tokyo, Japan, Japan Meteorological Society, 19–24.

  • Smith, D. M., , S. Cusack, , A. W. Colman, , C. K. Folland, , G. R. Harris, , and J. M. Murphy, 2007: Improved surface temperature prediction for the coming decade from a global climate model. Science, 317, 796799.

    • Search Google Scholar
    • Export Citation
  • Tao, G., 2003: Adaptive Control Design and Analysis. John Wiley & Sons, Inc., 640 pp.

  • Tong, M., , and M. Xue, 2008a: Simultaneous estimation of microphysical parameters and atmospheric state with simulated radar data and ensemble square root Kalman filter. Part I: Sensitivity analysis and parameter identifiability. Mon. Wea. Rev., 136, 16301648.

    • Search Google Scholar
    • Export Citation
  • Tong, M., , and M. Xue, 2008b: Simultaneous estimation of microphysical parameters and atmospheric state with simulated radar data and ensemble square root Kalman filter. Part II: Parameter estimation experiments. Mon. Wea. Rev., 136, 16491668.

    • Search Google Scholar
    • Export Citation
  • Whitaker, J. S., , and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 19131924.

  • Zhang, S., 2011a: Impact of observation-optimized model parameters on decadal predictions: Simulation with a simple pycnocline prediction model. Geophys. Res. Lett., 38, L02702, doi:10.1029/2010GL046133.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., 2011b: A study of impacts of coupled model initial shocks and state–parameter optimization on climate predictions using a simple pycnocline prediction model. J. Climate, 24, 62106226.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , and J. L. Anderson, 2003: Impact of spatially and temporally varying estimates of error covariance on assimilation in a simple atmospheric model. Tellus, 55A, 126147.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , J. L. Anderson, , A. Rosati, , M. J. Harrison, , S. P. Khare, , and A. Wittenberg, 2004: Multiple time level adjustment for data assimilation. Tellus, 56A, 215.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , M. J. Harrison, , A. Rosati, , and A. T. Wittenberg, 2007: System design and evaluation of coupled ensemble data assimilation for global oceanic climate studies. Mon. Wea. Rev., 135, 35413564.

    • Search Google Scholar
    • Export Citation
  • Zhang, S., , Z. Liu, , A. Rosati, , and T. Delworth, 2012: A study of enhancive parameter correction with coupled data assimilation for climate estimation and prediction using a simple coupled model. Tellus, 64A, 10963, doi:10.3402/tellusa.v64i0.10963.

    • Search Google Scholar
    • Export Citation
  • Zhu, Y., , and I. M. Navon, 1999: Impact of parameter estimation on the performance of the FSU Global Spectral Model using its full physics adjoint. Mon. Wea. Rev., 127, 14971517.

    • Search Google Scholar
    • Export Citation
Save