Fig. 1.
Time series of (a) the fast ‘atmospheric’ variable (x 2) and (b) the slow ‘oceanic’ variable (w) in the spinup run of the simple coupled model with the standard values of parameters described in Section 2.2, starting from (x 1, x 2, x 3, w)=(0,1,0,0). The dotted-black lines are results of a de-coupled version of the model by setting c 1=c 2=0 in eq. (3), by which the ‘atmospheric’ model is degraded to the original Lorenz model (Lorenz, 1963). The dotted-dashed lines are the ensemble standard deviation of the corresponding variable in the coupled model by adding a Gaussian noise on c 2 with a standard deviation of 0.05.
Fig. 2.
Time series of (a) x 1,2,3 (solid, dotted and dotted-dashed) and (b) w observations (*,+and x) at an interval of 0.1 time unit (TU) and the control model integration (lines) with κ=29, after the spinup of 104 TUs described in Section 2.2. The observations are taken from the ‘truth’ integration with the standard values of parameters listed in Section 2.2 where κ=28, superimposed by a white noise with a standard deviation of 2 (0.5) for x 1,2,3 (w).
Fig. 3.
(a) Time series of the κ adjustments (red) and estimated κ values (green) in the first 10 ensemble members of the traditional parameter estimation using only x 1 observations to estimate κ (traditional parameter estimation (TPE) , see the detailed description in Section 3). The dotted-(dashed-)black line marks the truth (erroneously guessed initial) value of the κ being estimated. For the convenience of visualisation, the values of adjustment increments in TPE are imposed on the erroneously guessed initial value. (b) Time series of observational increments of x 1 of member 1 (red) where the observations (asterisks) and the truth (blue-dashed) as well as the model estimate (solid-black) of x 1 in member 1 are plotted as the reference. (c) Time series of the ensemble spread (standard deviation) of x 1 (denoted by σx1) in TPE (red). To detect the time scale of developing a reliable ensemble spread when a model parameter is perturbed in the model system, the σx1 and σ w produced by the free model ensemble integrations starting from a κ ensemble perturbed by a Gaussian noise described in Section 2.3 are also plotted by dotted-black and dotted-blue lines. (d) Time series of the correlation coefficient between κ and x 1 (denoted by ρ(κ,x 1)) (red) evaluated by the prior ensemble of κ and the observation-estimated ensemble of x 1 in TPE (red). The corresponding ρ(κ,w) is also plotted by the blue line to show the different time scale of developing a reliable correlation between the parameter and the model state when κ is perturbed. The transient values of ρ(κ,x 1) between 1 and –1 reflect the transient nature of the estimated atmosphere ensemble between attractor lobes.
Fig. 4.
(a) Time series of the estimated κ values in the first 10 ensemble members of PET0 (green) and PET1 (red). The PET0 [PET1] experiment is the same as traditional parameter estimation (TPE) except that parameter estimation starts at the 2000th-step model ensemble integration without (with) the state estimation using the x 1 observations (see the detailed description in Section 3.2). (b) Time series of the values of x 1 of the first 10 ensemble members in PET0 (dotted-green) and PET1 (dotted-red), and their ensemble means [solid-blue for PET0 , solid-red for PET1] as well as the truth (dotted-black). (c) Time series of three factors of the signal-to-noise ratio (rs2n,1,2,3) (see the detailed description in Section 3.2) of the estimated ensemble of x 1 in PET0 (green) and PET1 (red). (d) Time series of the signal-to-noise ratio () of the estimated ensemble of x 1 (solid) and w (dotted-dashed) in PET0 (green) and PET1 (red) and DAEPC (see the detailed description in Section 4.4).
Fig. 5.
Time series of the root mean square (Rms) of ensemble analysis increments of (a) x 1,2,3 and (b) w, normalised by the ensemble spread and climatological standard deviation (see the context of Section 4.4) produced by the ensemble filtering state estimation using x 1 observations at every 0.1 time unit (TU). The thick dashed line marks an estimated time scale for state estimation to reach a ‘quasi-equilibrium’.
Fig. 6.
Time series of (a) the averaged ensemble standard deviations of x 1,2,3 and (b) the ensemble standard deviation of w when each of five model parameters is perturbed by a Gaussian noise with a standard deviation of 5% of its default value.
Table 1. List of experiments for data assimilation scheme for enhancive parameter correction (DAEPC) proof-of-concept studies
[i] TPE, traditional parameter estimation.
Fig. 7.
Time series of (a) the ensemble mean of the κ values estimated by traditional parameter estimation (TPE) (green) and the data assimilation scheme for enhancive parameter correction (DAEPC) (red) and (b) the errors of the ensemble mean of x 2 and (c) the errors of the ensemble mean of w produced by state estimation only (SEO) (blue), TPE (green) and DAEPC (red) using all x 1,2,3 and w observations at every 0.1 time unit (TU).
Fig. 8.
Variation of x 3 in x 1 space over the period of 110–115 time units (TUs) in the assimilations of data assimilation scheme for enhancive parameter correction (DAEPC) (red) and state estimation only (SEO) (green) when the default value of σ is set to be 11.95 in the assimilation model. The assimilations try to recover the ‘true’ attractor produced by the model integration with σ=9.95 (dotted-black), using the observations that sample the ‘truth’ every 0.1 TU. The blue line is the free model run with σ=11.95 using the same initial conditions as in the assimilations of DAEPC and SEO.
Fig. 9.
Time series of the errors of the ensemble mean of parameters σ, κ, b, c 2 and O d that are simultaneously estimated using observations of all x 1,2,3, and w at every 0.2 time unit (TU).
Fig. 10.
Time series of the errors of the ensemble mean of w in the (a) first and (b) second thousand time units (TUs) produced by data assimilation scheme for enhancive parameter correction (DAEPC) (red) and state estimation only (SEO) (green) using a perfect/biased (dashed/solid) model, with observations at every 0.2 TU. In DAEPC, the whole set of five model parameters (see Table 2) are simultaneously estimated using the observations. The dots denote the initial conditions from which 20 forecasts start in Figs. 11 and 12.
Table 2. Five parameters that are simultaneously estimated using observations and their values in the assimilation experiments
[i] DAEPC, Data assimilation scheme for enhancive parameter correction.
Table 3. The time-mean and Rms errors of the ensemble mean produced by state estimation only (SEO) and the DAEPC scheme using a perfect/biased model ‘dynamical core and physical scheme’ during the period of 1000–2000 time units (TUs)
[i] DAEPC, Data assimilation scheme for enhancive parameter correction.
Fig. 11.
Variations of (a,c) the Rms errors and (b,d) anomaly correlation coefficients (ACCs) of the forecasted ensemble mean of x 1 (panels a, b) and w (panels c, d) with the forecast lead time based on 20 forecast cases initialised from state estimation only (SEO) and DAEPC using perfect/biased ‘dynamical core’ and ‘physical scheme’ shown in Fig. 10.
Fig. 12.
Variations of anomaly correlation coefficients (ACCs) of the forecasted ensemble mean of (a) x 1 and (b) w with the forecast lead time based on 20 forecast cases initialised from state estimation only (SEO)-produced initial conditions with data assimilation scheme for enhancive parameter correction (DAEPC)-corrected parameters applied to the prediction model (dashed) and DAEPC-produced initial conditions with uncorrected parameters applied to the prediction model (dotted-dashed). The prediction model is set with a biased ‘dynamical core and physical scheme’ described in Section 5.2. The original ACCs produced by SEO and DAEPC shown by the dotted and solid lines in Fig. 11 are also marked here as references. The thin dotted black lines mark a 0.6 ACC level in both panels.
