Identification of linear response functions from arbitrary perturbation experiments in the presence of noise – Part I. Method development and toy model demonstration

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.