Abstract. Existent methods to identify linear response functions from data require tailored perturbation experiments, e.g. impulse or step experiments. And if the system is noisy, these experiments need to be repeated several times to obtain a good statistics. In contrast, for the method developed here, data from only a single perturbation experiment at arbitrary perturbation is sufficient if in addition data from an unperturbed (control) experiment is available. To identify the linear response function for this ill-posed problem we invoke regularization theory. The main novelty of our method lies in the determination of the level of background noise needed for a proper estimation of the regularization parameter: This is achieved by comparing the frequency spectrum of the perturbation experiment with that of the additional control experiment. The resulting noise level estimate can be further improved for linear response functions known to be monotonic. The robustness of our method and its advantages are investigated by means of a toy model. We discuss in detail the dependence of the identified response function on the quality of the data (signal-to-noise ratio) and on possible nonlinear contributions to the response. The method development presented here prepares in particular for the identification of carbon-cycle response functions in Part II of this study. But the core of our method, namely our new approach to obtain the noise level for a proper estimation of the regularization parameter, may find applications in solving also other types of linear ill-posed problems.

中文翻译：

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在有噪声的情况下通过任意扰动实验识别线性响应函数-第一部分。方法开发和玩具模型演示

摘要。从数据中识别线性​​响应函数的现有方法需要量身定制的扰动实验，例如脉冲或阶跃实验。如果系统嘈杂，则需要将这些实验重复几次，以获得良好的统计数据。相反，该方法在这里开发，数据只从一个单一的扰动实验任意如果另外还有来自不受干扰（对照）实验的数据，则干扰就足够了。为了确定这个不适定问题的线性响应函数，我们调用正则化理论。我们方法的主要新颖之处在于确定适当估计正则化参数所需的背景噪声的水平：这是通过将摄动实验的频谱与其他控制实验的频谱进行比较来实现的。对于已知是单调的线性响应函数，可以进一步改善所得的噪声水平估计。我们的方法的鲁棒性及其优势通过玩具模型进行了研究。我们详细讨论了已识别的响应函数对数据质量（信噪比）以及对响应的可能非线性贡献的依赖性。本文介绍的方法开发尤其为鉴定本研究第二部分中的碳循环响应功能做准备。但是，我们方法的核心，即为合理估计正则化参数而获取噪声水平的新方法，可能会在解决其他类型的线性不适定问题中找到应用。