To gain understanding of a physical system, it is very helpful to know how it responds to perturbations. Considering a small time-dependent perturbation f:R→R, the resulting time-dependent response R:R→R can from a very general point of view be written as 1R(t)=∫0tχ(t-s)f(s)ds, where the linear response function χ:R→R is a characteristic of the considered system. In fact, under a number of assumptions – among which smoothness and causality are the most important – Eq. () is the first term of a functional expansion of the response R into the perturbation f around the unperturbed state f(⋅)=0, known as Volterra series . In this framing, the key to gaining insight into the system is the linear response function χ: by knowing this function one has at hand not only a powerful tool to predict the response for sufficiently small but otherwise arbitrary perturbations, but also a means to study the internal dynamic modes of the unperturbed system by analyzing the temporal structure of the response function.

Linear response functions have been successfully applied within different contexts in many fields of science and technology. In physics, for example, material constants like the magnetic susceptibility or the dielectric function must be understood as linear response functions that can be obtained by Kubo's theory of linear response via the fluctuation–dissipation theorem from an auto-correlation of the unperturbed system. However, applications of these functions range far beyond physics into fields like neurophysiology and climate . In climate science, in particular, applications of linear response functions in the context of Ruelle's developments in response theory (see below) are a recent topic e.g.,. On the other hand, these functions have already been successfully employed as a heuristic tool to study climate and the carbon cycle for decades e.g.,. Yet another perspective is that from engineering sciences, in which the impulse response – that to a large extent corresponds to the linear response function – and the closely related transfer function (or system function) characterize linear time-invariant (LTI) systems, widely applied in fields such as signal processing and control theory . Regardless of which viewpoint a particular community takes to investigate the linear response of a system, a fundamental step in this investigation is the identification of the appropriate linear response function, the topic of the present study.

From a theoretical point of view, the existence of a linear response is by no means obvious: structurally stable dynamical systems are the exception , so that already small parameter changes typically lead to topological changes in their sets of stable and unstable solutions. Not every such bifurcation must prevent a linear response in macroscopic observables, but the question remains how in view of microscopic structural instability macroscopic linearity can prevail. A key result in this field is Ruelle's rigorous demonstration of the existence of a linear response for the structurally stable class of uniform hyperbolic systems . It is believed that this result transfers to large classes of nonequilibrium systems . An example may be the Lorenz system at standard parameters, for which numerical analysis revealed evidence for a linear response despite non-hyperbolicity . Recent investigations by indicate that the thermodynamic limit must be invoked to reconcile microscopic structural instability with macroscopic differentiability. Results on the existence/absence of a linear response have been particularly obtained for iterative maps , which are known for their notoriously rich bifurcation structure. Well studied is also the linear response of stochastic systems for whose quasistatic response rigorous mathematical results also exist .

In practical applications where the response function must be recovered from data, its identification may be a challenging task. The reason is that the identification problem is generally ill-posed, so that by classical numerical methods one obtains a recovery severely deteriorated by noise (see below). In addition, existent methods to identify these functions from data require one to perform special perturbation experiments. In the present study, we develop a method to identify linear response functions, taking data from any type of perturbation experiment while fully accounting for the ill-posedness of the problem.

The generality of our method allows for derivation of response functions in cases hardly possible before. Examples are problems where performing perturbation experiments is computationally expensive, so that one must use data that were not designed for the purpose of deriving these functions. In the geosciences, this may be the case when one is interested in characterizing by response functions the dynamics of Earth system models – extremely complex systems employed to simulate climate and its coupling to the carbon cycle. In principle, with our method one can derive these functions, taking simulation data from Earth system model intercomparison exercises such as C4MIP – the Coupled Climate-Carbon Cycle Model Intercomparison Project  – that are already available in international databases. In Part 2 of this study we explore this possibility by investigating in an Earth system model the response of the land carbon cycle to atmospheric CO2 perturbations. Because of the relationship between the linear response function and the impulse response and the transfer function in LTI systems, our work can also be seen from the viewpoint of the engineering sciences as a contribution to the corpus of methods to solve system identification problems .

In the field of climate science, the typical method to identify linear response functions is by means of the impulse response function, which is the response to a Dirac delta-type perturbation e.g.,. This method has become so widely known that often the terms linear response function and impulse response function are used interchangeably. Indeed, in the particular case where perturbations are weak, the two concepts coincide. However, this is not true in general: if the impulse strength is large so that nonlinearities become important, the impulse response function differs from the linear response function.

Other studies have proposed to identify linear response functions by making use of other types of perturbations. and used a weak periodic forcing to derive response functions in the Fourier space (also called susceptibilities). , , , , , , and identify the linear response function using step experiments, where the perturbation is a Heaviside-type function. Additional studies have proposed to compute the linear response of the system using the invariant measure of the unperturbed system and by means of shadowing methods .

As noted by , in principle the linear response function of a system can be derived by taking data from an arbitrary type of perturbation experiment. One method would be to apply a Laplace transform to Eq. (), so that χ(t) can in principle be computed by the inverse Laplace transform 2χ(t)=L-1{L{R}/L{f}}, where L{⋅} is the Laplace transform operator. In fact, a first step towards the derivation of χ(t) from the general Eq. () was taken by , although the problem was not systematically discussed.

Deriving χ(t) from perturbation experiment data is not a trivial problem. For the general case where the perturbation is different from a Dirac delta-type function, the problem is ill-posed e.g.,. This basically means that attempts to recover the exact χ(t) yield a solution with large errors due to an amplification of the noise in the data. On the other hand, when f(t) is a Dirac delta-type function with sufficiently small perturbation strength, so that the response can be considered linear, the impulse response gives directly the linear response function, i.e., χ(t)=R(t). However, even in this case noise may hinder the recovery: because the perturbation is only one “pulse” with small perturbation strength, the response may have a too low signal-to-noise ratio because of internal variability , giving once more a recovery with large errors.

To remedy these noise problems, a method intended to “damp” the noise in the response is usually employed. In , a step experiment with large perturbation strength is used to obtain a better signal-to-noise ratio in the response but at the cost of enhancing the effect of nonlinearities. An alternative approach is employed by and , who employ an ensemble of simulation experiments and take the ensemble-averaged response so that the level of noise is reduced. However, especially for complex models such as Earth system models, ensembles of simulations can be computationally extremely expensive, so that such a procedure may not be feasible.

Instead of trying to improve the signal-to-noise ratio of the data by improved experiment design, here we are interested in deriving χ(t) from a single realization of a given experiment by accounting for the ill-posedness of the problem. For this purpose, we employ regularization theory. Although this theory offers a variety of methods to solve ill-posed problems see, e.g.,, currently no general all-purpose method exists. Typically, methods rely on some type of prior information about the problem . Hence, they must be tailored according to the particularities of each application. Here, we develop a method that under certain assumptions solves the ill-posed problem when, in addition to the data from a single arbitrary perturbation experiment, data from an associated unperturbed – or control – experiment are also given to obtain independent information about the noise level (Sect. ). First, we assume that the response function is well approximated by a sum of decaying exponentials, meaning that potential oscillatory contributions to the response function are so small that they can be considered to be part of the noise. The response function is obtained by applying Tikhonov–Phillips regularization. The regularization parameter is chosen via the discrepancy method. An essential ingredient of the discrepancy method is the noise level, which is usually not known a priori. For this reason, we propose a method to estimate the noise level by taking advantage of the information given by a spectral analysis of the perturbation experiment and the control experiment. If the desired response function is known to be monotonic, the noise estimate can be further adjusted. In Sect. , the method is demonstrated to give reliable results under appropriate conditions of noise and nonlinearity. In Sect. , we compare the derived method with two existent methods in the literature to identify the response function in the time domain. Results and technical details are discussed in Sect. . Additional calculations are shifted to the Appendices.

As a preparation for introducing our method in Sect. , in the present section we derive its basic ansatz, which takes into account, in addition to the response formula (Eq. ), also the noise in the data. Depending on the application context, the noise may arise for different reasons, such as errors in the measurements or stochastic components in the system. As will be seen, our basic ansatz is in principle applicable to all those cases. However, to make the connection to modern applications of linear response functions that arise in the context of Ruelle's developments (e.g., climate), here we derive this ansatz starting from considerations of linear response theory . Ruelle considered systems of type 3ddtx=A0(x)+A1(x)f(t), where x(t) is the possibly infinite dimensional state vector and the perturbation f(t) couples to the unperturbed system ddtx=A0(x) via the field A1(x). In the present context Eq. () could, e.g., represent the dynamics of the Earth system perturbed by anthropogenic emissions f(t). Considering an observable Y(x), Ruelle proved that the ensemble average of its deviation from the unperturbed system 〈ΔY〉 can be expanded in the perturbation f(t): 4〈ΔY(x(t))〉=∫0tχ(t-s)f(s)ds+O(f2), where the order symbol O(f2) represents terms that vanish in the limit f(⋅)→0 more quickly than the leading linear term. This expansion describes the response of a system that is noisy as a result of its chaotic evolution: starting from different initial states, one obtains different values for ΔY(x(t)). Compared to Eq. (), in Eq. () the linear response function does not describe the response in observables directly, but only in their ensemble average, i.e., in an average over the initial states of the unperturbed system. For the recovery of linear response functions from numerical experiments, this would mean that one had to perform many experiments starting from different initial states to obtain the appropriate ensemble averages. Using tailored perturbation experiments, it was demonstrated in several studies e.g., that linear response functions can indeed be obtained in this way but at the expense of a large numerical burden from the need to perform many experiments. Instead, the aim here is to obtain the linear response functions from a given experiment and only from a single realization. Since we are dealing with a single realization, Eq. () becomes 5ΔY(t)=∫0tχ(t-s)f(s)ds+η(t)+O(f2), where η(t) is a noise term that must show up as a consequence of dropping the ensemble average. Accordingly, the noise η(t) is introduced here as the difference between the noisy response in a single realization ΔY(t) and the response in the ensemble average 〈ΔY(x(t))〉 (compare Eq. ). In addition, we assume linearity in the perturbation. As a consequence, the present study is based on the ansatz 6ΔY(t)=∫0tχ(t-s)f(s)ds+η(t), where now the response ΔY(t) is divided into a deterministic term ∫0tχ(t-s)f(s)ds and a noise term η(t).

The linearity assumption is on purpose: in the present approach to derive the linear response function (see next section), hereafter called the RFI method (response function identification method), we first use Eq. () to obtain χ(t) and justify the linearity assumption a posteriori by analyzing how robustly the response can be recovered for different perturbation strengths. Dropping the nonlinear terms has the advantage that one can use the corpus of linear methods to derive χ(t) from Eq. (). Note that, in practice, however small the perturbation may be, the nonlinear terms do not vanish. Therefore, the contribution of nonlinearities is in this way distributed between χ(t) and η(t), which will be different from the previous χ(t) and η(t) in Eq. (). How strongly nonlinearities affect the numerical identification of χ(t) depends on the estimation of η(t), which is a crucial part of our RFI method and the main novelty introduced here to deal with the ill-posedness of the problem to identify χ(t).

In addition, although we derived Eq. () starting from considerations of linear response theory, it is clear that this ansatz can also be employed in any other context where it may be assumed that the response formula (Eq. ) applies and that the data are contaminated by additive noise.

In this section we derive the RFI method. As mentioned above, the aim of this method is to obtain the linear response function using data from a single realization of a given perturbation experiment. For this purpose, an essential step is our novel estimation of the noise term η(t), which requires additional data from an unperturbed (control) experiment.

Starting from the ansatz Eq. (), the method is based on the idea that the noise term η(t) can be estimated using information on the internal variability from the control experiment in combination with a spectral analysis of the perturbation experiment. The identification of the linear response function proceeds as follows: first, we choose a functional form for χ(t). Second, Eq. () is discretized for application to the discrete set of time series data, which results in a matrix equation. Then, assuming that the solution obeys the Picard condition (see below), we estimate the high-frequency components of the noise term η(t) in Eq. () via a spectral analysis of the matrix equation applied to the data from the perturbation experiment. Next, assuming that the spectral distribution of noise is similar in the control and perturbation experiments, we also estimate the low-frequency components of η(t). The final estimate of η(t) is then used in a regularization procedure to determine the regularization parameter and thereby find an approximate solution for χ(t). In case χ(t) is known to be monotonic, the approximated solution is further adjusted by checking for monotonicity.

This section is organized as follows. In the first subsection, we introduce the assumption for the functional form of the linear response function. In Sect. , we present the discretized problem. In Sect.  we briefly review some elements of regularization theory employed in our method, in particular Tikhonov–Phillips regularization (Sect. ) and the discrepancy method (Sect. ). In Sect.  we present our noise estimation procedure by which the regularization parameter is determined. Finally, in Sect.  we show how this procedure can be further improved in the presence of a monotonicity constraint.

In application to real data, the presence of noise and nonlinearities may complicate the recovery of linear response functions. Therefore, by using artificial data generated from a toy model, in the present section we analyze the robustness of the RFI method in the presence of such complications. Robustness for real data is studied in Part 2.

As a last test of the quality of the results given by the RFI method in application to the toy model, in this section we compare our method against two existent methods in the literature to identify response functions in the time domain. The comparison is performed for the particular case where the response function is known to be monotonic and also for the more general case where it is not. As a side issue, this section also reveals some insight into the relation between the quality of the recovery of χ(t) as measured by the prediction of responses and the quality of the recovery of χ(t) itself.

In climate science, the most commonly used method is to obtain χ(t) from an impulse response, i.e., the response to a perturbation of Dirac delta type e.g.,. Here we call it the pulse method. Although this method is conceptually straightforward, in some cases it might not yield satisfactory results. Since the perturbation is only one “pulse”, depending on the observable of interest it may give a response with a small SNR. As a consequence, the recovered response function may be severely affected by noise. On the other hand, if the strength of the pulse is made large to obtain a good SNR, the linear regime may be exceeded. In this case, the impulse response does not correspond anymore to the linear response function.

The second method consists of deriving the linear response function from a step response, i.e., the response to a Heaviside-type perturbation e.g.,. Here we call it the step method. Due to the special form of this “step” perturbation, the linear response function can in principle be derived from 42χ(t)=1fstepddtΔYstep(t), where fstep is the step perturbation and ΔYstep is the corresponding response. Unfortunately, such derivation involves numerical differentiation, which is known to be an ill-posed problem . Because the problem is ill-posed, noise is amplified, potentially resulting in large errors in the derived linear response function.

These two methods therefore share two limitations: first, they require a special perturbation experiment; second, because of noise in the data they might yield a response function with large errors. In principle, the second limitation may be overcome by using instead of a single response the ensemble average over multiple responses. However, this comes at the expense of the numerical burden of performing multiple experiments, which is especially large when dealing with complex models such as state-of-the-art Earth system models.

The main advantages of the RFI method lie precisely in overcoming these two limitations: it recovers the response function from any type of perturbation experiment and automatically filters out the noise by regularization.

For the results of this section, we performed ensembles of 200 simulation experiments with the toy model (see Sect. ). Each ensemble member is defined by a realization of the noise η*(t) with a fixed standard deviation (see Eq. ). Each realization was added via Eq. () to three experiments: 1 %, step (2×f0), and pulse (4×f0). Note that because of the issue with the SNR mentioned above, we had to employ for the pulse experiment twice the forcing strength employed for the step experiment. Further, for each ensemble member an additional realization of the noise was generated to serve as a control experiment to compute the noise estimate for the RFI method (step 1 of Fig. ).

We computed the response function by the pulse and step method as follows. For a pulse experiment the forcing is f(t)=aδ(t) with forcing strength a, so that the response is given by 43ΔYpulse(t)=∫0tχ(t-s)aδ(s)ds=aχ(t).

Therefore, for the pulse method we took the response from the pulse experiment and obtained the response function by 44χ(t)=1aΔYpulse(t).

The recovery by the step method was calculated by taking the response from the step experiment and applying Eq. (). The derivative was computed by forward difference.

To obtain comparable results with these two methods, we recovered the response function by the RFI method from the same pulse and step experiments. To compare the quality of the results using also an experiment not decidedly tailored for the identification, we include additionally the recovery from the 1 % experiment.

To obtain a quantitative comparison for the quality of the recovery for each method, we define the recovery error: 45εr:=||χ-χ*||||χ*||, where χ is the recovered response function and χ* is the “true” response function, which is known because we use the toy model. In contrast to the prediction error that measures the quality of the recovery of χ(t) by means of the response (see Eq. ), the recovery error εr measures the quality of the recovery of χ(t) itself. Another reason for introducing the recovery error is to compare its results with results from the prediction error. By doing that, we can gain insight into how much the prediction error can be trusted as an indirect measure of the quality of recovery in real applications, where the “true” response function is not known.

First, we compare the pulse and step methods against the full RFI algorithm, i.e., the RFI algorithm taking monotonicity into account (step 6 in Fig. ). Results are shown in Fig. . In the first row of subfigures, we took for the recovery the ensemble average over the 200 responses for each experiment. For the RFI method, we took the ensemble average over the control experiments as well to estimate the noise (step 1 of Fig. ). As shown in Fig. a1, with this approach all the methods recover the response function almost perfectly. The quality of the recovery is quantified by the recovery error in Fig. b1. The RFI method shows the smallest values for the step and pulse experiments when compared to the step and pulse methods. Overall, the step method clearly shows the largest value. To quantify the quality of the prediction, we plot in Fig. c1 the prediction error (Eq. ). As seen, values are even smaller than for the recovery error. Overall, we see a similar pattern: the step method again stands out, with other methods showing much smaller error values.

In the second row, we compare results by taking only a single response for the recovery. Since the quality of the recovery by the different methods may vary depending on the particular noise realization, we again performed 200 simulations to obtain better statistics but this time deriving the linear response function for each ensemble member separately. Figure a2 shows an example of recovery for one of the ensemble members. As expected, the recoveries by the pulse and step methods largely deviate from the true response function. For the pulse method, the large errors result from the low SNR of the pulse response: even taking twice the forcing strength of the step experiment, the SNR of the pulse response is of order 100 against order 101 for the step and 1 % responses. For the step method, on the other hand, the large errors are not a result of a low SNR but of the noise amplification associated with the ill-posedness of numerical differentiation. In contrast to the recovery by these two methods, because of regularization the recoveries by the RFI method are smoother and visually seem to better fit the true response function. To quantitatively check these results, we plot in Fig. b2 for each method the average and standard deviation over the 200 values of the recovery error (one for each ensemble member). The figure shows that the pulse and step methods indeed display the largest average recovery error, with the pulse method having a much larger spread. Such spread is probably related to the low SNR in the response from the pulse experiment. The results from the 1 % and pulse experiments by the RFI method are better, showing comparable error magnitudes. The smallest average recovery error is obtained from the RFI method using the step experiment. In Fig. c2 we show the average and standard deviation over the 200 values of the prediction error (Eq. ). The smallest average prediction errors are obtained from the RFI method using the 1 % and step experiments. The largest errors are obtained for the pulse method and the RFI method using the pulse experiment. In contrast to the situation for the recovery error, for the prediction error no substantial difference between the two is found. Note also that when comparing recoveries from the same response (i.e., comparing “Pulse” with “RFIpulse” and “Step” with “RFIstep”), the RFI method gives better results than both the pulse and step methods. Another interesting point is that prediction errors for the step method remain approximately unchanged by taking the ensemble mean and a single response (compare “Step” in Fig. c1 and c2). Overall, as in the first row, the prediction error shows for each individual method values smaller than the recovery error. However, now there is a difference between the plots for the recovery and prediction error: although the pulse and step methods show the largest averages, with values of comparable size for the recovery error, for the prediction error the pulse method has the largest average, with a value much larger than the step method.

This difference can be better understood as follows (see , for more details, including the influence of the forcing scenario). Because Eq. () is ill-posed, the convolution operator acts on χ(t) as a “low-pass filter” see, e.g.,. This means that high frequencies in χ(t) are suppressed by convolution and show up damped in the response ΔY(t). Hence, recoveries with large errors only at high frequencies tend to give relatively small prediction errors. Because of the low SNR, the pulse method yields a recovery of χ(t) with large errors at both high and low frequencies. Although the errors at high frequencies are damped in the prediction, errors at low frequencies are not. Hence, the large recovery error results in a large prediction error. On the other hand, because of the good SNR for the step response, the step method gives a relatively good recovery of χ(t) at low frequencies, with large errors concentrated at high frequencies. As a result, the large recovery error results in only a small prediction error. This suppression of high-frequency errors might also explain why the prediction error for the step method remains unchanged when recovering the response function from a single response instead of the ensemble average. By comparing the recovery of χ(t) by the step method in Fig. a1 and a2, one sees that the main difference is indeed at high frequencies (the recovery in Fig. a2 is quite “noisy” but follows the long-term trend). This is because the noise amplification has a larger effect on the recovery from the single response due to its larger noise level. However, since low frequencies are well recovered in both cases, the resulting prediction errors are almost the same.

Overall, the analysis of Fig.  suggests two main conclusions. First, as expected, the prediction error indeed gives an indication of the quality of the recovery, since good recoveries result in good predictions. However, care should be taken when judging the recovery only from the prediction error, because a good prediction does not necessarily imply a good recovery: due to the ill-posedness, Eq. () might damp large high-frequency recovery errors, so that they do not show up in the prediction. Nevertheless, from a good prediction error one can still infer a good recovery at low frequencies, because at these frequencies large recovery errors result in large prediction errors. Since regularization filtering leaves only low-frequency terms in the recovery, the RFI method shows in Fig.  small prediction errors associated with small recovery errors.

Second, by taking only a single response – and not the ensemble average – the full RFI algorithm gives on average smaller recovery and prediction errors than the pulse and step methods when comparing results obtained from the same experiment.

However, the results above cover only the case where the full RFI algorithm is employed. In the following, we also analyze the case where monotonicity is not taken into account. For this purpose, we repeated in full detail the exercise that led to Fig.  but did not apply the additional noise-level adjustment to enforce monotonicity of the response function. Figure a shows the results for the recovery error. Once more, the RFI method gives smaller values than the step and pulse methods when comparing the recovery from the same responses. In addition, now the recovery for the RFI method using the step experiment even improved in comparison to Fig. b2. The reason may be related to the numerical check for monotonicity: depending on the tolerance value that is used to judge whether the recovered response function is monotonic, the additional adjustment might actually overestimate the noise level, leading to slightly worse results.

Yet the improvement brought by the additional noise-level adjustment is clear when looking at the recovery error for the 1 % experiment. Compared to Fig. b2, the average error increases substantially, and the spread is much larger (see inset for the whole value). As explained in Sect. , this deterioration results from cases where the noise in the response is such that the spectral similarity assumption does not hold. Since here the noise estimate resulting from this assumption is not further improved by the monotonicity check, the result is a poor recovery (see Fig. b for an example). However, because the large errors are mostly at high frequencies, even poor recoveries are still sufficiently good for predictions, as shown by the small mean prediction error in Fig. c (see “RFI1%”). Therefore, in contrast to the case where monotonicity is taken into account, here some small prediction errors are associated with large recovery errors.

Nevertheless, we find that, although extreme, such poor recoveries are not frequent. In fact, extreme cases with recovery error εr > 1 account for 6.5 % of the recoveries. This suggests that the large deterioration in the mean and spread of the recovery error in subfigure (a) is not a result of overall poor recoveries but of only few extreme cases. To check this hypothesis, we plot in subfigure (d) the mean and standard deviation, excluding these cases from the calculations. Indeed, the result is much better, showing values comparable to the case where monotonicity is taken into account (compare “RFI1%” in Fig. d and Fig. b2). Overall, this result indicates that at least for models of this type – where in the perturbation experiment the spectral distribution of noise does not change drastically compared to the control experiment – although monotonicity plays a role in avoiding large recovery errors, statistically most recoveries are still relatively good even without this additional improvement.

Existent methods to identify linear response functions from data require tailored perturbation experiments. Here, we developed a method to identify linear response functions from data using only information from an arbitrary perturbation experiment and a control experiment. The RFI method addresses the ill-posedness inherent to the identification problem by applying Tikhonov–Phillips regularization. The regularization parameter is computed by the discrepancy method, which involves the estimation of the noise level. For this purpose, we take advantage of information given by a spectral analysis of the perturbation experiment and by the control experiment. Assuming that the Picard condition holds, we estimate from the perturbation experiment the high-frequency components of the noise. Then, assuming that the spectral distribution of noise is approximately the same for the perturbed and control experiments (spectral similarity assumption), we estimate from the control experiment the low-frequency components of the noise. The obtained noise-level estimate can be further adjusted if the linear response function is known to be monotonic. The robustness of the method in the presence of noise and nonlinearity was demonstrated in Sect. . Additional sensitivity tests showing the robustness of the method under changes in the parameters for the recovery are shown in Appendix .

As discussed in Sect. , the developed method to identify linear response functions is an alternative approach to existent methods in the literature, which require special perturbation experiments and often give results with large errors caused by noise. In contrast, the RFI method accounts in a systematic way for the noise and can be directly applied to data from any type of perturbation experiment once a control experiment is also given. Because it filters out the noise, its results show in the cases analyzed here a higher quality compared to results from previous methods when applied to the same data from a toy model, and because it can identify response functions from any type of perturbation experiment, the method is particularly suitable for application to data from the C4MIP carbon cycle model intercomparison as shown in Part 2 of this study.

The main novelty of the method is the estimation of the noise level (steps 1–3 of Fig. ), which is known to be critical for the application of regularization theory. When solving a problem by regularization, the most crucial step is the computation of the regularization parameter. To compute this parameter in a way that the solution converges to the “true” solution for decreasing noise, methods need to account for the noise level . However, in practical applications the noise level is rarely known. Therefore, methods to obtain good estimates are needed. Our new method to estimate the noise level consists essentially of two steps: first, estimating the high-frequency components from data and then the low-frequency components from the control experiment. While the second step is completely novel, the main idea behind the first step was already brought up in earlier studies e.g., and has recently been further developed by methods to compute the “Picard parameter” , which is different from the imax⁡ that we use in our method. The Picard parameter is computed as the index for which the components |ui•ΔY| start to level off. Typically, from this index onwards the data can be interpreted as noise. For this reason, one might think that from the Picard parameter one can obtain all data components dominated by noise and thereby estimate the noise level. However, this is not generally true: for instance, if the noise has large low-frequency components such as in Fig. a2, then the components |ui•ΔY| level off at an index larger than that at which the data start to be dominated by noise, so that in this case the Picard parameter does not determine all data components dominated by noise. In our RFI method, the interest lies in obtaining not all data components dominated by noise but only enough components to obtain the overall level of the high-frequency noise. For this purpose, we define instead of the “Picard parameter” the more conservative index imax⁡, above which the singular values are zero and by the Picard condition the “true” data components must also be zero. In this way, we unambiguously identify data components that contain only noise (see Eq. ). These components give the high-frequency noise level, so that in the second step the remaining low-frequency noise components can also be estimated from the control experiment (step 3 of Fig. ).

Because our noise-level estimation is not particularly related to the problem of identifying response functions, it can in principle be applied to solve also other types of linear ill-posed problems see, e.g.,. In general, all one needs for the application are the following.

A problem of the type46y=Ax+η,where given the matrix A and the noisy data y one is interested in finding x.

Then, as long as both the Picard condition and the spectral similarity assumption hold, the method gives a reasonable noise estimate – since then, by assumption, the noise estimate is simply a scaling of the noise in the control experiment (see Sect. ) – by which the regularization parameter can be determined.

While the Picard condition is necessary for a solution to be recoverable from an ill-posed problem, the validity of the spectral similarity assumption is less clear. An intuitive explanation for this assumption can be thought as follows. Since here the interest lies in identifying linear response functions, the perturbation to the system must be sufficiently weak so that the response can be considered linear. If the noise in the control experiment depends on the perturbation, a sufficiently weak perturbation will modify its characteristics only slightly. The RFI method accounts partially for this change by adjusting the overall level by which the noise increases. Nevertheless, it assumes that since the characteristics of the noise change only slightly, then the spectral components of the noise in the perturbed experiment can be thought of as having the same relative contributions as those in the control experiment. When in addition the response function is known to be monotonic, the estimate of the noise can be further improved (step 6 of Fig. ), this time by adjusting the relative contribution of the spectral components: since the high-frequency region is known from the spectral analysis of the response, then the components of the noise are adjusted in the low-frequency region; this is done iteratively until the resulting response function becomes monotonic. Such additional adjustment has been demonstrated to give good results in the applications in the present study and subsequent Part 2 for the special case where the response function can be considered monotonic.

Although it is assumed that χ(t) is given by the spectral form (Eq. ), this is not essential for our method. In principle, any functional form can be assumed for χ(t), or even none – in which case one would recover χ(t) pointwise. However, compared to the simpler pointwise recovery of χ(t), assuming Eq. () has some advantages. The most obvious is that in contrast to the pointwise approach, with Eq. () both χ(t) and the spectrum can be recovered together. If χ(t) is recovered pointwise, the spectrum has to be derived in a second step from χ(t), which is also an ill-posed problem . Further, the description (Eq. ) restricts the function space for the recovered χ(t), forcing lim⁡t→∞χ(t)=0 as is expected for most problems of interest, which greatly simplifies the problem compared to the case where χ(t) can assume any form. Our ansatz (Eq. ) also has advantages in comparison with the typical multi-exponential ansatz (Eq. ) assumed in most previous studies (see discussion in Sect. ). When assuming that χ(t) is given by a sum of few exponents, an important problem is how to choose the number of exponents. The typical methods to choose this number rely on “quality-of-fit” criteria, but for ill-posed problems these criteria can be unreliable because in these problems a good fit does not mean that the derived parameters are close to the “true” parameters see, e.g., the famous example fromp. 272. In our approach, as long as the distribution of timescales is appropriately prescribed and the data quality is sufficiently good, numerical results indicate that the solution is approximately independent of the number of exponents (Appendix ). Moreover, compared to the multi-exponential approach, our ansatz (Eq. ) has two additional advantages: the first is that it leads to the linear problem of finding only the spectrum q(τ) – in contrast to the nonlinear problem of finding both the timescales τi and the weights gi from Eq. () – which permits an analytical solution and thereby gives more transparency to the method. The second is that compared to the assumption of only a few timescales, the ansatz of a continuous spectrum of timescales is typically more realistic for real systems, which is, e.g., the case for the carbon cycle study presented in Part 2. One limitation is however that our ansatz (Eq. ) restricts the solution to systems with exponentially relaxing responses and vanishing oscillatory contributions.

In the present paper the robustness of our method has been investigated only for artificial data taken from toy model experiments. In this analysis, we not only knew the “true” response function underlying the data, but also had control over the two complications that may hinder its recovery, namely the level of background noise and nonlinearities. Under these ideal conditions, we could carefully examine the quality of the response functions identified by our RFI method. Nevertheless, such conditions are hardly met in practice. Therefore, the applicability of our method must be investigated as well for real problems. Such an investigation is presented in Part 2 of this study.