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In this paper, IOCAS ICM is chosen to study the SST-OPR of ENSO. The spatial domain of the IOM extends from 31°S to 31°N and from 124°E to 30°E, which covers the tropical Pacific and Atlantic basins. For convenience, only the SSTAs in the region from 31°S to 31°N and from 124°E to 70°W are shown in the following experiments. The region for SSTAs is divided into 134 × 61 grid points for the sea surface. In addition, the CNOP and OPR are solved using the GD method in the ICM and coded using Fortran on the Linux platform.
To seek the SST-OPR of ENSO in the ICM, the non-ENSO years simulated by the ICM are selected as the basic years. Figure 3 shows the temporal evolution of the Niño3.4 index for the three chosen continuous basic years simulated by the ICM. For convenience, the first year is denoted by year(1), the second year by year(2), and so on. The CNOPs for the initial months from July(1) of year(1) to June(3) of year(3) are calculated; these months are shown between the two dashed lines in Fig. 3. Therefore, a total of 24 CNOPs can be obtained with a lead time of 12 months (one year).
Figure 3. Temporal evolution of the Niño3.4 index for the chosen basic years. The months between the black lines are the chosen time period to study the OPRs of ENSO.
Considering the seasonal dependence of the CNOPs in the later experiments, the calendar year is divided into four seasons, from January to March (JFM), April to June (AMJ), and then JAS and OND. In addition, to improve the time efficiency, the MPI parallel strategy is adopted to accelerate the way the CNOPs are solved. All experiments are run on a Tinahe-2 supercomputer system at the National Supercomputer Center in Guangzhou, China, in which each node has two Intel(R) Xeon(R) CPU E5-2692v2 @ 2.20 GHz physical CPUs; each CPU has 12 logical cores, for a total of 24 processors for each node.
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When solving the OPR of ENSO, many parameters are involved. The three crucial parameters are the dimension of the feature space (DoF), the constraint for the initial perturbation δ in Eq. (3) and the value of ω in Eq. (8), which are discussed in this section.
The DoF is determined by the number of features that are decided after performing the SVD analysis. Figure 4 illustrates the change of the eigenvalue and the cumulative eigenvalue ratio with an increasing number of eigenvectors. From Fig. 4, the accumulated eigenvalue ratio increases with increasing eigenvectors, and the ratio reaches more than 95% when the eigenvectors are more than 48. After that, the curve of the ratio tends to become gentler; that is, the ratio changes very slowly. Considering that more features will require more time, it has been found that the adaptive function value and the CNOP pattern obtained do not change much when more features are included. Therefore, the DoF is set to 48 for the subsequent experiments.
The constraint for the initial perturbation δ in Eq. (3) is also a crucial parameter in this study. It is obvious that the larger the δ is, the larger the objective function value. (A larger objective function value means that the SSTA added to the initial state is larger.) However, the SSTA in a real scenario cannot be infinite. Therefore, it is necessary to determine a suitable constraint for the initial perturbation. Here, we adopt the objective function value mentioned in section 2.2 and the average SSTA in the Niño3.4 areas to show the size of the CNOP for different δ values when the initial month is January and the lead time is nine months. Figure 5 shows the change in the objective function value and the SSTA in the Niño3.4 areas (5°S−5°N, 170°E−120°W) as δ changes. From Fig. 5, the objective function value and the SSTA in the Niño3.4 areas increase with increasing δ, and the increasing trend for the SSTA in the Niño3.4 areas decreases when the δ is larger than 10. According to the definition of the National Oceanic and Atmospheric Administration (NOAA) in 2003, an El Niño event can be confirmed to occur if the SSTAs of a three-month running mean in the Niño3.4 index are above 0.5°C for more than three months. Therefore, the initial SST perturbation should be chosen as no more than 0.5°C, and less than 10% of 0.5°C (0.05°C) can be the constraint of the initial SSTA based on experience. From Fig. 5, the average SSTA in the Niño3.4 areas is 0.042 (< 0.05°C) when the δ is set to 10. Therefore, the constraint δ is set to 10 in this study.
In Eq. (8), ω is another crucial parameter when calculating the gradient using the limit in mathematics. In this situation, ω should be a positive real number that should approach 0 but never equal 0. Considering that the evolution
${M_{{\rm{icm}}}}\left({{{S}_0} + {{p}_\omega}} \right) - {M_{{\rm{icm}}}}\left({{{S}_0}} \right)$ in Eq. (8) is 0 when the chosen ω is too small, the following approach in Eq. (10) is adopted to calculate the appropriate ω. As mentioned above, the constraint of the initial perturbation is δ. Suppose that the initial perturbation is a vector that has the same value in all dimensions; then, the average of all dimensions can be calculated using$\sqrt {{\delta ^2}/{\rm{DoF}}} $ , so we obtain the value of ω by reducing the average by 100 times. In Eq. (10), δ is the constraint for the initial perturbations in Eq. (3), and the DoF is the dimension of the feature space mentioned above:The parameters used in the experiments are listed in Table 1. Additionally, other parameters that can be determined according to users’ own requirements are also shown in Table 1, such as the maximum iterations in the SPG2 algorithm and the lead time when solving the CNOP for the ICM.
Parameters Value Meaning DoF 48 Dimension of the feature space δ 10 Constraint of the initial perturbation maxit 50 Maximum iterations in the SPG2 algorithm Lt(months) 9 Lead time when solving the CNOP for the ICM ω $0.01\sqrt {\dfrac{{{\delta ^2}}}{{{\rm{DoF}}}}} $ A parameter in Eq. (10) Table 1. Parameters involved in this study.