1. Introduction



Dozens of papers have been published over the last two decades, demonstrating LRM in temperature records. Many hypothesize that the signal is composed of an LRM noise superposed on a trend driven by external forcing, and hence the methods are designed to eliminate such trends [see a short review and a selection of references in Rypdal et al. (2013)]. The most widely used method is detrended fluctuation analysis (DFA) (Kantelhardt et al. 2001). However, the concept of a slow trend does not always reflect the true nature of deterministic forced variability. Some components of the forcing may be faster than important components of the internal variability, and hence precise separation of internal from forced variability can only be done by using information about the deterministic component of the forcing record. Fortunately, such reconstructions of the forcing records exist and are used as input for historic runs of climate models.
We contend that correct estimation of the LRM properties of the internal climate variability can only be done by analyzing the residual obtained by subtracting the forced deterministic component of the climate signal. We shall show that the climate response function is all we need to predict both the deterministic component of the climate signal and the memory properties of the internal variability.
2. Linear response models




In a quasi-equilibrium state the surface temperature T (temperature of mixed layer) can be almost constant while the total influx into the climate system (the deep ocean) is finite. In such a state dQ/dt in Eq. (1) is nonzero because of an imbalance between the outflux term and influx term on the rhs in this equation.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1

The red curve is the response function R(t) for the two-box model with τtr = 4 yr, τeq = 20 yr, Str = 0.3 K m2 W−1, and Seq = 1.0 K m2 W−1. The blue curve for the same parameters but τeq = 100 yr.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
For a given influx the equilibrium outflux is controlled by the Stefan–Boltzmann radiation law and complex feedback processes that determine the equilibrium climate sensitivity Seq [see, e.g., Eqs. (A5)–(A7) in Rypdal (2012)]. The true value of Seq is subject to considerable controversy because of insufficient knowledge of some of these feedbacks, and because they operate on wildly different time scales (Otto et al. 2013). The estimates of Seq are regularly updated in the literature as the global temperature goes through periods of slower or faster warming (Otto et al. 2013; Aldrin et al. 2012). If the climate system responds on a wide range of time scales, the notion of an equilibrium climate sensitivity may be of little practical interest, since this equilibrium may never be attained in a system that is subject to variability of the forcing. In section 7 we discuss alternatives to this notion.






The simple, linear two-box model contains the essential conceptual elements of our phenomenological response model, since it represents a linear model with more than one relaxation time scale. The state variables are the energy contents of the ocean surface layer and the deep ocean, respectively. The nonlinearities that give rise to the spatiotemporal chaos of the atmosphere and ocean represent unresolved scales that contribute to mean turbulent (anomalous) energy fluxes between the boxes, and to random fluctuations of these fluxes around the mean. The latter represent the stochastic forcing terms. A linear system follows from the ansatz that the mean fluxes are proportional to the temperature difference between the two boxes. The two-box model can trivially be generalized to an n-box model, whose response function may be designed to mimic a power law up to the slowest relaxation scale. The various boxes represent the energy content of different interacting parts of the climate system with different heat capacities and response times. Of course, we do not derive this phenomenological model from first principles, but in this paper we test it against observation data. The model, and the linear approximation, can also be tested against state-of-the-art Earth system models. Work in this direction is in progress. The reality of delayed responses in coupled atmospheric–oceanic GCMs (AOGCMs) has recently been demonstrated by Geoffroy et al. (2013). They fitted the four parameters of the two-box model to runs of 16 different CMIP5 models with step-function forcings, with quite good agreement over 150 yr. For the global response the linearity approximation has been verified in AOGCM simulations (e.g., in Meehl et al. 2004). We also discuss these points further in the concluding section.








3. Dynamic–stochastic models


- Since the observed record in this formulation should be perceived as one realization of a stochastic process produced by the dynamic–stochastic model, the residual difference between this record and the deterministic solution should be perceived as a noise process
given by the stochastic part of the solution to Eq. (14), that is,By using the exponential response model, Eq. (15) produces the Ornstein–Uhlenbeck (OU) stochastic process. On time scales less than τ this process has the same scaling characteristics as a Brownian motion, that is, the PSD has the power-law form P(f) ~ f−2 for f > τ−1. On time scales greater than τ the process has the scaling of a white noise and the PSD is flat for f < τ−1. This means that if we analyze a sample of an OU process whose length is much shorter than τ we cannot distinguish it from a Brownian motion. On the other hand, if we coarse-grain an OU process in time by averaging over successive time windows of length much greater than τ, the resulting discrete-time process is indistinguishable from a white noise. Actually, the PSD has the form of a Lorentzian, P(f) ~ [τ−2 + (2πf)2]−1. For a discrete-time process the direct analog to the Ornstein–Uhlenbeck process is the first-order autoregressive process AR(1). The scaling response model, on the other hand, produces an fGn for −1 < β < 1 and a fractional Brownian motion (fBm) for 1 < β < 3. For these noises and motions the PSD for low frequencies has the power-law form P(f) ~ f−β. In principle, an estimator for the PSD (like the periodogram) applied to the observed residual could be compared to the PSD for the two response models to test the validity of the models against each other. In practice, other estimators in this paper will be used, but the idea is the same. - Formulating the problem as a parametric stochastic model allows systematic estimation of the parameters {F(0), C, σ, τ} for the exponential model, and {F(0), μ, σ, β} for the scale-invariant model. The method is based on maximum-likelihood estimation (MLE), which establishes the most likely parameter set that could produce the observed record from the prescribed forcing. The principles of the MLE employed here are explained in the appendixes.







Frequency-dependent sensitivity S(f) for the exponential response model (blue) and the scale-invariant model (red) for model parameters given in Table 1.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
In principle, the right-hand side of Eq. (17) could be used to estimate S(f) directly from Fourier transforming the temperature and forcing records, and then to compare with Eqs. (18) and (19) to assess the validity of the two response models. The short length of the records, however, makes the Fourier spectra very noisy, and the ratio between them even more so. Additional complications are that the spiky nature of the forcing record to volcanic eruptions and the unknown amplitude of the stochastic forcing component. Hence, we have to resort to the model parameter estimation described above, and to other estimators than the Fourier transform, to settle this issue.
4. Parameter estimation from instrumental records
The temperature datasets analyzed in this section can be downloaded from the Hadley Center Met Office web site. We consider the global mean surface temperature (GMST) as presented by the Hadley Centre Climatic Research Unit, version 3 (HadCRUT3) monthly mean or annual mean temperatures (Brohan et al. 2006). The forcing record is the one developed by Hansen et al. (2005) and used by Hansen et al. (2011), and is shown in Fig. 4a. The forcings decomposed into volcanic, solar, and anthropogenic contributions are shown in Figs. 4b–d, respectively. The forcing records go from 1880 until 2010 with annual resolution, so even if the instrumental temperature record goes further back in time and has monthly resolution, the maximum-likelihood estimation of model parameters only employs the 130-yr records with annual resolution. The analysis of the residual noise signal, however, utilizes the monthly resolution to improve the statistics.

(a) Total forcing for 1880–2010, (b) volcanic forcing, (c) solar forcing, and (d) anthropogenic forcing.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
The results of the MLE method for the exponential and scaling models are given in Table 1. The heat capacity C = 4.2 × 108 J km−2 estimated from the exponential model is very close to that of a 100-m-deep column of seawater, and the time constant 4.3 yr is in the middle of the range (3–5 yr) observed by Held et al. (2010) from instantaneous CO2 experiments with the Geophysical Fluid Dynamics Laboratory (GFDL) Climate Model, version 2.1 (CM2.1). What was also observed in those model runs was an additional slower response that showed that equilibrium was not attained after 100 yr of integration, indicating that the exponential model does not contain the whole story. In Fig. 5a we present the deterministic part of the solutions for both models along with the observed GMST record. Although the solution of the scaling model seems to yield a somewhat better representation of both the multidecadal variability and the response to volcanic eruptions, the difference between the deterministic solutions of the two models is not striking on these time scales. The reason for this can be understood from Fig. 3. It is on time scales longer than a century that the difference will become apparent. For the stochastic part of the response, however, the two models can be tested against data on all observed time scales. Such a test is performed in Fig. 5b, where the residual noise (the observed GMST with the deterministic solution subtracted) has been analyzed by the DFA technique (Kantelhardt et al. 2001). What is plotted here is the DFA(1) fluctuation function of the residual noise versus time scale. For an AR(1) process (stochastic solution of the exponential model) the slope of this curve in a log–log plot is near 1.5 for time scales much less than τ, and near 0.5 for time scales much greater than τ, as shown by the blue dashed curve in the figure. For an fGn the slope of the curve is (β + 1)/2, which has been estimated to yield β ≈ 0.75, as shown by the red dashed curve. The fluctuation functions of the actual observed residuals with reference to the two models are shown as the blue crosses and the red circles in the figure, showing that the residuals are inconsistent with an AR(1) process, but consistent with an fGn process.
The MLE of parameters in the exponential response model and in the scale-free response model. The parameters are estimated from the HadCRUT3 annual temperature record. The parameter


(a) Deterministic part of the solution where blue indicates the exponential response model, red the scale-invariant response model, and black the HadCRUT3 annual temperature record. (b) Blue crosses denote DFA(1) of monthly GMST record with the deterministic solution of the exponential response model subtracted (the residual). Red circles (obscured because they nearly coincide with the blue crosses) denote the same with the deterministic solution of the scaling model subtracted. Blue dashed curve represents ensemble mean of DFA(1) fluctuation function of simulated AR(1) process with estimated parameters from the exponential response model. Shaded blue area represents 2 times std dev of the distribution of DFA(1) over the ensemble. Red dashed curve and shaded area represent the same for an fGn process with estimated parameters from the scaling model.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
In Fig. 6 we demonstrate that the observed record falls within the uncertainty range of the two dynamic–stochastic models. Here we have generated an ensemble of solutions to the two models with the estimated parameters and plotted the 2σ range around the deterministic solutions. The results are shown as the two shaded areas in Figs. 6a and 6b, respectively.

Black curves are the GMST record and shaded areas are the deterministic part of the solution ±2 times std dev of the stochastic part representing the range of solutions to the model for (a) the exponential response model and (b) the scaling response model.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
In Fig. 7 we plot the deterministic scaling response to the total forcing along with the separate responses to the volcanic, solar, and anthropogenic forcing components. During the first half of the twentieth century, solar and anthropogenic forcing contribute equally to the global warming trend. After 1950 there is a significant cooling trend resulting from volcanic aerosols, a weaker warming contribution from solar activity, and a dominating anthropogenic warming.

Deterministic part of forced temperature change for 1880–2010 according to the scale-invariant model from (a) total forcing, (b) volcanic forcing, (c) solar forcing, and (d) anthropogenic forcing. Thin black curves are the GMST record.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
The maximum likelihood estimation employed so far in this section {F(0), C, σ, τ} for the exponential model, and {F(0), μ, σ, β} for the scale-free model, using the model Eq. (10), but neglecting the term Trmn(t). Hence we have used no information about past forcing. If we use reconstructions of forcing throughout the past millennium (Crowley 2000) we can compute F(0) from Eq. (11) and Trmn(t) from Eq. (12) and use the entire Eq. (10), including the term Trmn(t), in the computation of T(t). In Fig. 8a we plot Trmn(t) for the period 1880–2080 computed from the reconstruction and the scaling response model with the parameters {μ, σ, β} given in Table 1. We observe that this remnant rapidly goes from 0 to approximately −0.1 K because of the forcing imbalance created by the variability over the previous millennium. The rapid initial change is an effect of the finite value of the forcing [F = F(0)] around 1880. As t grows, F(t) will fluctuate around the zero equilibrium value and eventually be influenced by the rising trend. This terminates the fast change in Trmn(t), which is followed by a slow decay. In Fig. 8b the black smooth curve is T(t) computed from Eq. (10) and includes Trmn(t). Here, F(0) is computed from Eq. (11) using reconstructed forcing (Crowley 2000) for the period 1000–1880. The red curve is the same as shown in Fig. 7a, but extended to 2080 using a forcing scenario corresponding to a 1% yr−1 increase in atmospheric CO2 concentration. Here F(0) is estimated along with the other model parameters from the instrumental data and Trmn(t) is not part of the statistical model from which the parameter F(0) is estimated. It appears that the two methods yield very similar deterministic solutions for T(t), indicating that the (positive) effect of reestimating F(0) more than compensates the (negative) effect of including Trmn(t). The proximity of the two curves in Fig. 8b shows that the two methods yield very similar results for realistic forcing scenarios for the coming century. For this reason we shall employ the method that does not make use of Trmn(t) in the remainder of this paper.

(a) The remnant from past forcing Trmn from 1880 until 2080. (b) Black curve represents the deterministic solution T(t) including Trmn and with F(0) estimated from forcing data for the period 1000–1880. Red curve represents the deterministic solution estimated by not including Trmn, and with F(0) estimated using forcing data for the period 1880–2010 only. The forcing beyond 2010 used here corresponds to a 1% yr−1 increase in atmospheric CO2 concentration.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
5. Predicting reconstructed records
The DFA fluctuation function plotted in Fig. 5b can demonstrate with statistical confidence that the residual is scaling only up to time scales less than ¼ of the length of the 130-yr record (i.e., for 3–4 decades). Verifying LRM on longer time scales requires longer records. This was done by Rybski et al. (2006) and Rypdal (2012) using detrending techniques like the DFA applied directly to reconstructed temperature records over the last one or two millennia. Here we shall utilize a forcing record for the last millennium (Crowley 2000), shown in Fig. 9, with its decomposition in volcanic, solar, and anthropogenic contributions. Many temperature reconstructions for the Northern Hemisphere exist for this time period [see Rybski et al. (2006) for a selection]. We shall employ our dynamic–stochastic models to the reconstruction by Moberg et al. (2005), which shows a marked temperature difference between the Medieval Warm Period (MWP) and the Little Ice Age (LIA). For the scaling model the parameters estimated from Crowley forcing and Moberg temperature are very close to those estimated from the instrumental records, except for the initial forcing F(0). The initial forcing measures how far the climate system is from equilibrium at the beginning of the record, and this will depend on at what time this beginning is chosen. Considering that the timing of volcanic events and the corresponding temperature responses probably are subject to substantial errors in these reconstructions, this might give rise to errors in the parameter estimates. For this reason we have also estimated F(0) from Crowley forcing and Moberg temperature by retaining the values of the other parameters estimated from the instrumental record and shown in Table 1. The resulting deterministic solutions for the two models are plotted in Fig. 10a, along with the Moberg record. Since only the departures from equilibrium forcing F(0) are estimated from the reconstruction data, these solutions should be considered as “predictions” of the deterministic component of the forced evolution over the last millennium, based on parameters estimated from the modern instrumental records. The exponential model predicts too low temperature in the first half of the record and too strong short-term responses to volcanic eruptions. The scaling model gives a remarkably good reproduction of the large-scale structure of the Moberg record and reasonable short-term volcano responses. The DFA fluctuation functions of the residuals for the two models are plotted in Fig. 10b, and again we observe that the results are consistent with a scaling response over the millennium-long record and inconsistent with the exponential response model.

As in Fig. 4, but for 1000–1978.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1

(a) Deterministic part of the solution with Crowley forcing where blue indicates the exponential response model, red the scaling response model, and black the Moberg annual temperature reconstruction record. (b) Blue crosses denote DFA(1) of Moberg record with the deterministic solution of exponential response model subtracted. Red circles denote the same with the deterministic solution of scaling model subtracted (residual). Blue dashed curve represents ensemble mean of DFA(1) fluctuation function of simulated AR(1) process with estimated parameters from the exponential response model. Shaded blue area represents 2 times std dev of the distribution of DFA(1) over the ensemble. Red dashed curve and shaded area represent the same for an fGn process with estimated parameters from the scaling model.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
Figure 11 shows the scaling response to the total Crowley forcing, along with the responses to the volcanic, solar, and anthropogenic components. The most remarkable feature is that most of the cooling from the MWP to the LIA appears to be caused by volcanic cooling and not by the negative solar forcing associated with the Maunder Minimum, in agreement with recent findings by Schurer et al. (2014). On the other hand, the solar contribution to the warming from the LIA until the mid-twentieth century is comparable to the anthropogenic. After this time the warming is completely dominated by anthropogenic forcing, in agreement with what was shown in Fig. 7.

Deterministic part of forced temperature change for 1000–1978 according to the scaling model with F(0) estimated from the Moberg record and {β, μ, σ} from the instrumental record. Gray curve is the Moberg record and results are shown from (a) total forcing, (b) volcanic forcing, (c) solar forcing, and (d) anthropogenic forcing.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
6. Truncated long-range memory
Does long-range memory imply that we still feel the effect of long past volcanic explosions like the Tambora (1815) eruption? Let us examine this question, which is frequently asked in discussions about LRM in the climate response. On centennial time scales this volcanic eruption represents a delta function forcing
In Fig. 12a we have plotted the deterministic solution corresponding to the instrumental temperature record with a truncated scaling response function with cutoff time tc = 10 yr (green curve), alongside the solution for the untruncated response (red curve). The difference between the two solutions is not remarkable and cannot be used as model selection criterion. In Fig. 12c, however, we have plotted a variogram (log–log plot of the second-order structure function) of the residual obtained by subtracting the deterministic solution from the observed temperature record (dotted line). The result is consistent with an fGn process scaling on time scales up to 102 yr, which was also shown in Fig. 5b [using the DFA(1) fluctuation function]. The dashed line surrounded by the green field is the ensemble mean of variograms of realizations of a simulated stochastic process generated by Eq. (15) with a truncated power-law kernel. The green field indicates 95% confidence for the variogram estimate. It is seen that the scaling properties of the actual residual noise is not captured by the truncated model if we choose tc as small as 10 yr. By increasing tc beyond approximately 30 yr we cannot reject the truncated model based on the instrumental data, but the same study can be made on the data from the millennium reconstructions, and some results are shown in Fig. 12c. Here we observe that the truncated model with tc = 100 yr gives a deterministic solution that gives a considerably poorer fit to the observed record than the full scale-free model, and again the variogram of the noise generated by the truncated model is inconsistent with that obtained from the real residue. We conclude from this that the cutoff time tc in the scale-free response is not less than a century, and hence that predictions made for the twenty-first century based on the scale-free model are supported by the forcing and temperature data available.

(a) Deterministic solution corresponding to the instrumental temperature record with a truncated scaling response function with cut-off time tc = 10 yr (green curve), alongside the solution for the untruncated response (red curve). (b) As in (a), but for the reconstructed record with tc = 100 yr. (c) Variogram of the residual obtained by subtracting the deterministic solution from the observed instrumental temperature record (dots). The green dashed line is the ensemble mean of variograms of realizations of a simulated stochastic process generated by Eq. (15) with a power-law kernel truncated at tc = 10 yr. The green shaded area indicates 95% confidence for the variogram estimate. (d) As in (c), but for the reconstructed record with tc = 100 yr.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
7. Perspectives on climate sensitivity
For predictions of future climate change on century time scales the equilibrium climate sensitivity may not be the most interesting concept. The frequency-dependent climate sensitivity S(f) given by Eq. (17) is more relevant. The transient climate response (TCR), defined as the temperature increase ΔTtr at the time of doubling of CO2 concentration in a scenario where CO2 concentration increases by 1% yr−1 from preindustrial levels, can also readily be computed from the response models. In Fig. 13a this forcing is shown as the dotted curve to the left (the forcing is logarithmic in the CO2 concentration, so the curve is linear). The response curves to this forcing according to the two response models are shown as the blue and red dotted curves to the left in Fig. 13b. At the end of these curves (the time of CO2 doubling after 70 yr) the temperatures represent the respective TCRs. They are both in the lower end of the range presented by Solomon et al. (2007). A more useful definition is to consider the response to the same CO2 increase from the present climate state that is established from the historical forcing since preindustrial times. This response is what is shown as the blue and red full curves in Fig. 13b for the next 70 yr. For the scale-free model the temperature in year 2010 lags behind the forcing because of the memory effects, and the energy flux imbalance dQ/dt established by the historical evolution at this time gives rise to a faster growth in the temperature during the next 70 yr, compared to the CO2 doubling scenario starting in year 1880. The value of ΔTtr (according to the modified definition) is 1.3 K in the exponential response model, but 2.1 K for the scale-invariant model. The latter is very close to the median for the TCR given in Solomon et al. (2007). Another illustration of the memory effect can be seen from the forcing scenario where the forcing is kept constant after 2010 as shown by the dashed line in Fig. 13a. The corresponding responses are given by the blue and red dashed curves in Fig. 13b. The short time constant in the exponential model makes the temperature stabilize in equilibrium after a few years, while in the scale-free model the temperature keeps rising as [2μ1−β/2F(2010)/β](t − 2010)β/2 for t > 2010 yr. As discussed in section 6 this monotonic rise in the temperature will not continue indefinitely, and will stabilize for t − 2010 > tc in a more realistic truncated LRM model. But since our analysis in section 6 indicates that an effective tc is greater than a century, the scaling model is adequate for the time scale scales shown in Fig. 13.

(a) Dotted line is the forcing scenario corresponding to 1% yr−1 increase in CO2 concentration starting in 1880. Solid curve is the historical forcing for 1880–2010 followed by 1% yr−1 increase in CO2 concentration after 2010. Dashed line after 2010 is forcing kept constant at the 2010 level. (b) GMST evolution according to the two response models with parameters given in Table 1 for the three forcing scenarios described for (a). Blue curves indicate the exponential response model and red curves the scaling response model.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1









Hence these crude trend estimates over the last six decades yield results consistent with the existence of an equilibrium climate sensitivity very close to the best estimate of Aldrin et al. (2012). If we suppose, on the other hand, that the linear trend approximation in temperature is not quite correct, the picture may be different. Consider a linearly increasing forcing as in the future 1% CO2 increase scenario shown by the full curve in Fig. 13a, but assume that the temperature evolves according to the scaling response to this forcing shown by the red full curve in Figs. 13b and 14a. By inserting these data into Eq. (23) we obtain the time-varying climate sensitivity shown in Fig. 14b (here the time origin is chosen in year 2010). Using the temperature evolution for the exponential response shown by the blue curves in Figs. 13b and 14a yields the nearly constant climate sensitivity given by the blue curve in Fig. 14b. This demonstrates that the temperature may increase according to the power law ~tβ/2+1 under a linearly increasing forcing and a linearly increasing OHC, provided stronger positive feedback mechanisms take effect on longer time scales and raise the climate sensitivity. In fact, this idea is just a time-domain statement of the concept of a frequency-dependent sensitivity that was formulated in section 3. The scenario of 1% yr−1 increase in CO2 concentration continued 250 years into the future is a very extreme one, and corresponds to a raise in concentration of more than one order of magnitude. However, our results show that within the framework of the scaling model, a scenario where the global temperature increases by more than 10 K while the OHC maintains a positive linear growth rate, is consistent with only a moderate increase in S(t) from 0.5 to 0.8. One important message from these considerations is that a the introduction of a moderately variable time-dependent climate sensitivity will make the scale-invariant LRM response on time scales up to several centuries consistent with energy balance considerations.

(a) Global temperature evolution in response to the 1% yr−1 CO2 increase forcing scenario, starting in year 2010. (b) Evolution of the time-dependent climate sensitivity S(t) in response to this forcing scenario assuming a linear increase in OHC corresponding to a net positive energy flux of 0.25 W m−2. The time origin t = 0 is year 2010. Blue curves indicate the exponential response model and red curves the scaling model.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
8. Discussion and conclusions
We have in this paper considered linear models of global temperature response and maximum-likelihood estimation of model parameters. The parameter estimation is based on observed climate and forcing records and an assumption of additional stochastic forcing. This modeling shows that a scale-invariant response is consistent with the stochastic properties of the noisy components of the data, whereas an exponential model is not. This observation disagrees with Mann (2011), who contends that the purely stochastic solution to the exponential response model [an AR(1) process] is consistent with the global instrumental record. The basic weakness in Mann’s paper is uncritical application of built-in routines for estimation of the memory exponent on time series that do not exhibit scaling properties [such as the AR(1) process]. When the scaling properties are tested, as we do in Fig. 5b, the residuals of the observed and exponentially modeled record are clearly distinguishable. Mann argues that no “exotic physics” is necessary to explain the persistence observed in the global records. Our analysis, however, shows that the exponential response model is insufficient to explain the observed scaling properties. But the physics needed to explain these delayed responses is not particularly exotic, as it may be sufficient to take into account the interactions between the ocean mixed layer and the deep ocean that are observed in AOGCM simulations (Held et al. 2010; Geoffroy et al. 2013).
The scaling model with parameters estimated from the modern instrumental temperature and forcing record successfully predicts the large-scale evolution of the Moberg reconstructed temperature record when the Crowley forcing for the last millennium is used as input. Solutions for the volcanic, solar, and anthropogenic components of the Crowley forcing show that the model ascribes most of the temperature decrease from the MWP to the LIA to the volcanic component, while the rise from the LIA to year 1979 is attributed to both solar and anthropogenic forcing up until about year 1950 and primarily to anthropogenic forcing after this time.
The temperature of the planetary surface (whose most important component is the ocean mixed layer) is driven by radiative forcing and energy exchange with atmosphere and deep ocean, some of which can be modeled as stochastic. Even when the mean energy flux to the surface layer is constant in time the total energy content of the system may vary, and if this variability is large on time scales beyond a century it may have little meaning to operate with the notion of an equilibrium climate sensitivity. Thus, the long-range dependence in the climate response implies that the equilibrium climate sensitivity concept needs to be generalized to encompass a time scale–dependent sensitivity that incorporates the effect of increasingly delayed positive feedbacks. This may have far-reaching implications for our assessment of future global warming under strong anthropogenic forcing sustained over centuries, as illustrated by the difference between the projected warming according to the scaling and exponential response models shown in Fig. 14a.
The great advantage of the response model approach is that it eliminates the influence of correlation structure of the forcing in the temperature signal, and reveals the memory structure of the climate response. It reveals a clean scaling of the residual temperature signal that is maintained at least up to the scales that can be analyzed with reasonable statistics in the millennium-long record, which is a few hundred years.
The importance of the “background” continuum of time scales in climate variability has been stressed by Lovejoy and Schertzer (2013). In a short review of their own work, Lovejoy (2013) shows results based on application of their Haar structure function technique to reanalysis, instrumental, and multiproxy temperature records. For twentieth-century reanalysis local records (75°N, 100°W) they find very weak persistence (β ≈ 0) but a transition to β ≈ 1.8 on longer time scales. For instrumental global records they find a spectral plateau of β ≈ 0.8 on time scales up to a decade but the same transition to β ≈ 1.8 on longer time scales. For the multiproxy NH records they find β ≈ 0.8 and here the transition appears after 5–10 decades. By similar analysis of ice core data they also obtain β ≈ 1.8 on the longer time scales, and argue that this transition constitutes the separation between a “macroweather” regime and a “climate” regime. The analysis presented here does not support that such a transition in the scaling properties of internal variability takes place on decadal time scales in global or hemispheric records. These scaling properties are shown by the DFA fluctuation functions of the residuals in Figs. 5b and 10b, and indicate β ≈ 0.8 scaling throughout the instrumental century-long record and at least up to several centuries scale in the millennium-long multiproxy record, respectively. The transition on multidecadal time scale also fails to show up in the detrended scaling analysis of proxy data in Rybski et al. (2006) and Rypdal et al. (2013). We suggest that the transition in global (NH) multiproxy data reported by Lovejoy (2013) is a consequence of not distinguishing between forced and stochastic response (alternatively, by not properly eliminating “trends” imposed by external forcing). A transition to a more persistent climate regime may perhaps be identified on millennium time scales, but it is an open and interesting question whether the rise of β from a stationary (β < 1) to a nonstationary regime (β > 1) is an actual change in the properties of the climate response or an effect of trends imposed by orbital forcing. However, the transition in scaling of local records from β ≈ 0 to larger β seems to reflect internal dynamics, since local records exhibit (at least over land) very low persistence up to the scales of a few decades that we can reliably estimate from the reanalysis data. Further analysis by application of the response model to local data may help to identify the scale at which this transition in scaling of internal dynamics takes place. Such a transition will naturally take place at the scales where global, purely temporal fluctuations start to dominate over spatiotemporal fluctuations in the local records. For time scales longer than the weather regime, such spatiotemporal fluctuations are associated with the interannual, decadal, and multidecadal modes of the climate system, and hence we would expect that this transition scale exceeds all the characteristic time scales of these modes.
In some recent papers multiple regression models have been constructed to assess the relative influence of volcanic, solar, and anthropogenic forcing components, in addition to the ENSO signal. These have been used to demonstrate that the post-2000 hiatus in global warming disappears after the natural forcing and the ENSO signal have been eliminated, and leaves an essentially linear anthropogenic trend over the last few decades (Foster and Rahmstorf 2011). Such methods have also been used for prediction (Lean and Rind 2009). The problem with this kind of statistical modeling is that the models contain a large number of fitting parameters with a considerable risk of overfitting, and that they lack any physical principle that allows reduction of this number of free parameters. For instance, the weight of different forcing components and the delay time for the response to each of them are left as free fitting parameters, while in the real world these parameters are determined by physics that is to great extent known. Our approach is also statistical, but it is constrained by the physical idea of a linear response function of a particular form. We use weights between force components that are known, and we assume that the response function (and hence the response time) to different forcings is the same. The parameters that are left to be estimated statistically are only those that are not well known from physical modeling. In the LRM response model there are only two parameters that characterize the response (μ and β). In the two-box model there are four parameters (Geoffroy et al. 2013). This means that the risk of overfitting is substantially reduced in the LRM model. Yet a comparative evaluation of these two models against AOGCM experiments should be done.
The deterministic response could be tested against multimodel ensemble means with specified forcings. Such means and the individual runs can be found for instance in frequently asked questions (FAQ) section 10.1, Fig. 1 in Stocker et al. (2013), for the ensembles from phases 3 and 5 of the Coupled Model Intercomparison Project (CMIP3 and CMIP5). The part of this figure that shows the global temperature evolution for those ensembles for natural and natural plus human forcing are shown in Figs. 15a and 15c. The corresponding response model results are shown in Figs. 15b and 15d. There are striking similarities as well as some interesting differences between the multimodel ensembles and the response model ensembles. The overall shapes of the mean response (which in the response model is equivalent to the deterministic response) are very similar, and so are the overall variances in the two ensembles. The response model gives a more accurate description of the observed volcanic responses and the observed hiatus over the last decade, which suggests that the climate models to some extent underestimate the strength of the long-range response. Neither model ensemble means capture quite the observed hiatus, and the reason for this seems to be the importance of the strong El Niño event of 1998, which makes the temperature curve in the following decade appear more flat. Such an event is unpredictable in both model classes, but can appear by chance both in individual climate model runs and in realizations of the stochastic model. Hence, the hiatus appears well within the error bars of both model classes and gives no indication that global warming has come to a halt.

(a) Individual runs and ensemble means of the CMIP3 and CMIP5 ensembles with natural forcing only. The black curves are global instrumental temperature records. (c) As in (a), but with natural and human forcing. (b),(d) As in (a),(c), but for ensembles of realizations of the LRM response model. The black curves represent the HadCrut4 instrumental record. Adapted from Stocker et al. (2013), FAQ 10.1, their Fig. 1.
Citation: Journal of Climate 27, 14; 10.1175/JCLI-D-13-00296.1
The large scatter of the individual climate model runs relative to the ensemble means is described in the stochastic response model as the response to the stochastic forcing. In Figs. 5 and 10 we have analyzed this scatter (the residual) in the observed and reconstructed surface temperature, respectively, and found that they are well described as a persistent fractional Gaussian noise. In a recent paper (Østvand et al. 2014) we have investigated the LRM properties of the global surface temperature in a number of millennium-long climate model simulations. We consistently find persistent LRM scaling in these models, both with historical forcing and in control runs. Hence, the statistics of the internal variability of global temperature as appearing as scatter in model ensembles is well described by the stochastic response model, even though the stochastic forcing term has not been derived from first principles.
This work has received support from the Norwegian Research Council under Contract 229754/E10, from the Tromsø Research Foundation, and from COST action ES1005 (TOSCA). The authors acknowledge useful discussions with Ola Løvsletten.
APPENDIX A
Response Functions and Fractional Gaussian Noise










APPENDIX B
Maximum-Likelihood Estimation




In this paper the parameters of the two models are estimated by maximizing Eqs. (B1) and (B2) numerically.
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This stochastic integral is not convergent, and the definition of a continuous-time fGn should therefore only be seen as a formal expression. How to formulate a mathematically well-defined model is discussed in the appendixes.