Studying the response of a climate system to perturbations has practical significance. Standard methods in computing the trajectory-wise deviation caused by perturbations may suffer from the chaotic nature that makes the model error dominate the true response after a short lead time. Statistical response, which computes the return described by the statistics, provides a systematic way of reaching robust outcomes with an appropriate quantification of the uncertainty and extreme events. In this paper, information theory is applied to compute the statistical response and find the most sensitive perturbation direction of different El Ni\~no-Southern Oscillation (ENSO) events to initial value and model parameter perturbations. Depending on the initial phase and the time horizon, different state variables contribute to the most sensitive perturbation direction. While initial perturbations in sea surface temperature (SST) and thermocline depth usually lead to the most significant response of SST at short- and long-range, respectively, initial adjustment of the zonal advection can be crucial to trigger strong statistical responses at medium-range around 5 to 7 months, especially at the transient phases between El Ni\~no and La Ni\~na. It is also shown that the response in the variance triggered by external random forcing perturbations, such as the wind bursts, often dominates the mean response, making the resulting most sensitive direction very different from the trajectory-wise methods. Finally, despite the strong non-Gaussian climatology distributions, using Gaussian approximations in the information theory is efficient and accurate for computing the statistical response, allowing the method to be applied to sophisticated operational systems.

翻译：研究气候系统对扰动的响应具有重要的现实意义。传统计算轨迹偏差的方法常因系统的混沌特性而受限，使得模型误差在较短时间内即掩盖真实响应。统计响应通过计算统计量描述的回归，为获得稳健结果提供了系统化途径，并能对不确定性与极端事件进行合理量化。本文应用信息论计算统计响应，探寻不同厄尔尼诺-南方涛动事件对初始值与模型参数扰动的最敏感方向。研究表明，最敏感扰动方向所涉及的状态变量取决于初始相位与时间尺度：海表温度与温跃层深度的初始扰动通常分别在短期与长期主导海表温度的响应，而纬向平流的初始调整对触发5至7个月中尺度的强统计响应至关重要，这在厄尔尼诺与拉尼娜的过渡阶段尤为显著。研究还发现，由外部随机强迫扰动（如风爆发）引发的方差响应常超越均值响应，导致其最敏感方向与轨迹方法所得结果存在显著差异。最后，尽管气候态分布呈现强非高斯特性，在信息论中采用高斯近似仍能高效精确地计算统计响应，这为该方法应用于复杂业务系统提供了可行性。