The Sampling Method for Optimal Precursors of ENSO Events

El Ni\~{n}o-Southern Oscillation (ENSO) is one of the significant climate phenomena, which appears periodically in the tropic Pacific. The intermediate coupled ocean-atmosphere Zebiak-Cane (ZC) model is the first and classical one designed to numerically forecast the ENSO events. Traditionally, the conditional nonlinear optimal perturbation (CNOP) approach has been used to capture optimal precursors in practice. In this paper, based on state-of-the-art statistical machine learning techniques, we investigate the sampling algorithm proposed in [Shi and Sun, 2023] to obtain optimal precursors via the CNOP approach in the ZC model. For the ZC model, or more generally, the numerical models with dimension $\mathrm{O}(10^4-10^5)$, the numerical performance, regardless of the statically spatial patterns and the dynamical nonlinear time evolution behaviors as well as the corresponding quantities and indices, shows the high efficiency of the sampling method by comparison with the traditional adjoint method. The sampling algorithm does not only reduce the gradient (first-order information) to the objective function value (zeroth-order information) but also avoids the use of the adjoint model, which is hard to develop in the coupled ocean-atmosphere models and the parameterization models. In addition, based on the key characteristic that the samples are independently and identically distributed, we can implement the sampling algorithm by parallel computation to shorten the computation time. Meanwhile, we also show in the numerical experiments that the important features of optimal precursors can be still captured even when the number of samples is reduced sharply.