Guest essay by Jeff Patterson
Temperature versus CO2
Greenhouse gas theory predicts a linear relationship between the logarithm of CO2 atmospheric concentration and the resultant temperature anomaly. Figure 1 is a scattergram comparing the Hadcrut4 temperature record to historical CO2 concentrations.
UPDATE: Thanks to an alert commenter, this graph has now been updated with post 2013 data to present:
At first glance Figure 1a appears to confirm the theoretical log-linear relationship. However if Gaussian filtering is applied to the temperature data to remove the unrelated high frequency variability a different picture emerges.
Figure 1b contradicts the assertion of a direct relationship between CO2 and global temperature. Three regions are apparent where temperatures are flat to falling while CO2 concentrations are rising substantially. Also, a near step-change in temperature occurred while CO2 remained nearly constant at about 310 ppm. The recent global warming hiatus is clearly evident in the flattening of the curve above 380 ppm. These regions of anti-correlation were pointed to by Professor Judith Curry in her recent testimony before the Senate Subcommittee on Space, Science and Competitiveness:
If the warming since 1950 was caused by humans, what caused the warming during the period 1910 –1945? The period 1910-1945 comprises over 40% of the warming since 1900, but is associated with only 10% of the carbon dioxide increase since 1900. Clearly, human emissions of greenhouse gases played little role in causing this early warming. The mid-century period of slight cooling from 1945 to 1975 – referred to as the ‘grand hiatus’, also has not been satisfactorily explained.
A much better correlation exists between atmospheric CO2 concentration and the variation in total solar irradiance (TSI). Figure 2 shows the TSI reconstruction due to Krivova .
When the TSI time series is exponentially smoothed and lagged by 37 years, a near-perfect fit is exhibited (Figure 3).
Note that while in general correlation does not imply causation here there is no ambiguity as to cause and effect. Clearly the atmospheric concentration of CO2 cannot affect the sun spot number from which the TSI record is reconstructed.
This apparent relationship between TSI and CO2 concentration can be represented schematically by the system shown in Figure 4. As used here, a system is a black box that transforms some input driving function into some output we can measure. The mathematical equation that describes the input to output transformation is called the system transfer function. The transfer function of the system in Figure 4 is a low-pass filter whose output is delayed by the lag td1 . The driving input u(t) is the demeaned TSI reconstruction shown in Figure 2b. The output v(t) is the time series shown in Figure 3a (blue curve) which closely approximates the measured CO2 concentration (Figure 3a, yellow curve).
In Figure 4, the block labeled 1/s is the Laplacian representation of a pure integration. Along with the dissipation feedback factor a1 it forms what system engineers call a “leaky integrator”. It is mathematically equivalent to the exponential smoothing function often used in time series analysis. The block labeled td1 is the time lag and G is a scaling factor to handle the unit conversion.
In a plausible physical interpretation of the system, the dissipative integrator models the ocean heat content which accumulates variations in TSI; warming when it rises above some equilibrium value and cooling when it falls below. As the ocean warms it becomes less soluble to CO2 resulting in out-gassing of CO2 to the atmosphere.
The fidelity with which this model replicates the observed atmospheric CO2 concentration has significant implications for attributing the source of the rise in CO2 (and by inference the rise in global temperature) observed since 1880. There is no statistically significant signal of an anthropogenic contribution to the residual plotted Figure 3c. Thus the entirety of the observed post-industrial rise in atmospheric CO2 concentration can be directly attributed to the variation in TSI, the only forcing applied to the system whose output accounts for 99.5% ( r2=.995) of the observational record.
How then, does this naturally occurring CO2 impact global temperature? To explore this we will develop a system model which when combined with the CO2 generating system of Figure 4 can replicate the decadal scale global temperature record with impressive accuracy.
Researchers have long noted the relationship between TSI and global mean temperature. We hypothesize that this too is due to the lagged accumulation of oceanic heat content, the delay being perhaps the transit time of the thermohaline circulation. A system model that implements this hypothesis is shown in Figure 5.
As before, the model parameters are the dissipation factor a2 that determines the energy discharge rate; input offset constant Ci representing the equilibrium TSI value; scaling constants G1, G2 which convert their inputs to a contributive DT, and time lag td2. The output offset Co represents the unknown initial system state and is set to center the modeled output on the arbitrarily chosen zero point of the Hadcrut4 temperature anomaly. It has no impact on the residual variance which is assumed zero mean.
The driving function u(t) is again the variation in solar irradiance (Figure 2b). The second input function v(t) is the output of the model of Figure 4 which was shown to closely approximate the logarithmic CO2 concentration. Thus the combined system has a single input u(t) and a single output- the predicted temperature anomaly Ta(t). Once the two systems are combined the CO2 concentration becomes an internal node of the composite system.
Y(t) represents other internal and external contributors to the global temperature anomaly, i.e. the natural variability of the climate system. The goal is to find the system parameter values which minimizes variance of Y(t) on a decadal time scale.
Natural variability is a catch-all phrase encompassing variations in the observed temperature record which cannot be explained and therefore cannot be modeled. It includes components on many different time scales. Some are due to the complex internal dynamics of the climate system and random variations and some to the effects of feedbacks and other forcing agents (clouds, aerosols, water vapor etc.) about which there is great uncertainty.
When creating a system model it is important to avoid the temptation to sweep too much under the rug of natural variation. On the other hand, in order to accurately estimate the system parameters affecting the longer term temperature trends it is helpful to remove as much of the short-term noise-like components as practicable, especially since these unrelated short-term variations are of the same order of magnitude as the effect we are trying to analyze. The removal of these short-term spurious components is referred to as data denoising. Denoising must be carried out with the time scale of interest in mind in order to ensure that significant contributors are not discarded. Many techniques are available for this purpose but most assume the underlying process that produced the observed data exhibits stochastic stationarity, in essence a requirement that the process parameters remain constant over the observation interval. As we show in the next section, the climate system is not even weak sense stationary but rather cyclostationary.
Autocorrelation is a measure of how similar a lagged version of a time series resembles the unlagged data. In a memoryless system, correlation falls abruptly to zero with increasing lag. In systems with memory, the correlation will decrease gradually. Figure 6a shows the autocorrelation function (ACF) of the linearly de-trended unfiltered Hadcrut4 global temperature record. Instead of the correlation gradually decreasing, we see that the correlation cycles up and down in a quasi-periodic fashion. A system that exhibits this characteristic is said to be cyclostationary. Despite the nomenclature, a cyclostationary process is not stationary, even in the weak sense.
With linear detrending, significant correlation is exhibited at two lags, 70 years and 140 years. However the position of the correlation peaks is highly dependent on the order of the detrending polynomial.
Power spectral density (spectrum) is the discrete Fourier transform of the ACF and is plotted in Figure 6b. It shows significant periodicity at 71 and 169 years but again the extracted period will vary depending on the order of the detrending polynomial (linear, parabolic, cubic etc.) and also slightly on the data endpoints selected.
Denoising the Data
From the above it is apparent that we cannot assume a particular trend shape to reliably isolate the “main” decadal scale climatic features we hope to model. Nor can we assume the period of the oscillatory component(s) remains fixed over the entire record. This makes denoising a challenge. However, a technique  has been developed for denoising data which makes no assumptions regarding the stationarity of the time record which combines wavelet analysis with principal component analysis to isolate quasi-periodic components. A single parameter (wavelet order) determines the time scale of the retained data. The implementation used here is the wden function in Matlab™. The denoised data using a level 4 wavelet as described in  is plotted as the yellow curve in Figure 7.
The resulting denoised temperature profile is nearly identical to that derived by other means (Singular Spectrum Analysis, Harmonic Decomposition, Principal Component Analysis, Loess Filtering, Windowed Regression etc.)
Figure 8a compares the autocorrelation of the denoised data (red) to that of the raw data (blue). We see that the denoising process has not materially affected the stochastic properties over the time scales of interest. The narrowness of the central lobe of the residual ACF (Figure 8b) shows that we have not removed any temperature component related to the climate system memory.
The denoised data (Figure 7) shows a long-term trend and a quasi-periodic oscillatory component. Taking the first difference of the denoised data (Figure 9) shows how the trend (i.e. the instantaneous slope) has evolved over time.
There are several interesting things of note in Figure 8. The period is relatively stable while the amplitude of the oscillation is growing slightly. The trend maxed out at .23 ⁰C /decade circa 1994 and has been decreasing since. It currently stands at .036 ⁰C /decade. Note also that the mean slope is non-zero (.05 ⁰C /decade) and the trend itself trends upward with time. This implies the presence of a system integration as otherwise the differentiation would remove the trend of the trend.
A time series trend does not necessarily foretell how things will evolve in the future. The trend estimated from Figure 9 in 1892 would predict cooling at a rate of .6 degrees-per-century while just 35 years later predict 1.5 degrees-per-century of warming. Both projections would have been wildly off base. Nor is there justification in assuming the long-term trend to be some regression on the slope. Without knowledge of the underlying system, one has no basis on which to decide the proper form of the regression. Is the long term trend of the trend linear? Perhaps, but it might just as plausibly be a section of a low frequency sine wave or a complimentary exponential or perhaps it is just integrated noise giving the illusion of a trend. To sort things out we need to approximate the system which produced the data. For this purpose we will use the model shown in Figure 5 above.
As noted, the composite system is comprised of two sub-systems. The first (Figure 4) replicates the atmospheric CO2 whose effect on temperature is assumed linear with scaling factor G1. The parameters of the first system were set to give a best-fit match to the observational CO2 record (see Figure 3).
The remaining parameters were optimized using a three-step process. First the dissipation factor a2 and time delay td2 were optimized to minimize the least-squares error (LSE) of the model output ACF as compared to the ACF of the denoised data (Figure 10, lower left), using a numerical method  guaranteed to find the global minimum. In this step the output and target ACFs are both calculated from the demeaned rather than detrended data. This eliminates the dependence on the regression slope and, since the ACF is independent of the scaling and offset, allows the routine to optimize to these parameters independently. In the second step, the scaling factors G1, G2 are found by minimizing the residual LSE using the parameters found in step one. Finally the input offset Ci is found by solving the boundary condition to eliminate the non-physical temperature discontinuity. The best-fit parameters are shown in Table 1. The results (figure 10) correlate well with observational time series (r = .984).
Figure 10- Modeled results versus observation
|CO2 Lag (years)||td1||37|
|TSI Lag (years)||td2||84|
|Input Offset (W/m2)||C0||-.045|
|Output Offset (K)||C1||.545|
Table 1- Best fit model parameters
The error residual (upper right) remains within the specified data uncertainty (± .1⁰C) over virtually all of the 165 year observation interval. The model output replicates most of the oscillatory component that heretofore has been attributed to the so-called Atlantic Multi-decadal Oscillation (AMO). As shown in the detailed plots of Figure 11, the model output aligns closely in time with all of the major breakpoints in the slope of the observational data, and replicates the decadal scaled trends of the record (the exception being a 10 year period beginning in 1965), including the recent hiatus and the so-called ‘grand hiatus’ of 1945-1975.
Figure 12 plots the scaled, second difference of the denoised data against the model residual. The high degree of correlation infers an internal feedback sensitive to the second derivative of temperature. That such an internal dynamic can be derived from the modeled output provides further evidence of the model’s validity. Further investigation of an enhanced model that includes this dynamic will be undertaken.
Climate Sensitivity to CO2
The transient climate sensitivity to CO2 atmospheric concentration can be obtained from the model by running the simulation with G2 set to zero, giving the contribution to the temperature anomaly from CO2 alone (Figure 13a).
A linear regression on the modeled temperature anomaly (with G2 = 0) versus the logarithmic CO2 concentration (Figure 13b) shows a best fit slope of 1.85 yielding an estimated transient climate sensitivity to doubled CO2 of 1.28 ⁰C. Note however that assuming the model is relevant, the issue of climate sensitivity is moot unless and until an anthropogenic contribution to the CO2 concentration becomes detectable.
These results are in line with the general GHG theory which postulates CO2 as a significant contributor to the post-industrial warming but are in direct contradiction to the notion that human emissions have thus far contributed significantly to the observed concentration. In addition, the derived TCR implies a mechanism that reduces the climate sensitivity to CO2 to a value below the theoretical non-feedback forcing, i.e. the feedback appears to be negative. Other inferences are that the observed cyclostationarity is inherent in the TSI variation and not a climate system dynamic (because a single-pole response cannot produce an oscillatory component) and that at least over the short instrumental time period, the climate system as a whole can be modeled as a linear, time-invariant system, albeit with significant time lag.
In a broader context, these results may contain clues to the underlying climate dynamics that those with expertise in these systems should find valuable if they are willing to set aside preconceived notions as to the underlying cause. This model, like all models, is nothing more than an executable hypothesis and as Professor Feynman points out, all scientific hypotheses start with a guess. The execution of a hypothesis, either by solving the equations in closed form or by running a computer simulation is never to be confused with an experiment. Rather a simulation provides the predicted ramifications of the hypothesis which falsify the hypothesis if the predictions do not match empirical observations.
An estimate of the future TSI is required in order for this model to predict how global temperature will evolve. There are some models of this in development by others and I hope to provide a detailed projection in a future article. In the meantime, due to the inherent system lag, we can get a rough idea over the short term. TSI peaked in the early 80s so we should expect the CO2 concentrations to peak some 37 years later, i.e. in a few years from now. Near the start of the next decade, CO2 forcing will dominate and thus we would expect temperatures to flatten and begin to fall as this forcing decrease. Between now and then we should expect a modest increase. This no doubt will be heralded as proof that AGW is back and that drastic measures are required to stave off the looming catastrophe.
Comment on Model Parametrization
It is important to understand the difference between curve fitting and model parametrization. The output of a model is the convolution of its input and the model’s impulse response which means that the output at any given point in time depends on all prior inputs, each of which is shaped the same way by the model parameter under consideration. This is illustrated in Figure 14. The input u(t) has been decomposed in to individual pulses and the system response to each pulse plotted individually. Each input pulse causes a step response that decays at a rate determined by the dissipation rate, set to .05 on the left and .005 on the right. The output at any point is the sum of each of these curves, shown in the lower panels. The gain factor G simply scales the result and does not affect the correlation with the target function. Thus, unlike polynomial regression, it is not possible to fit an arbitrary output curve given specified forcing function, u(t). In the models of Figures 4 and 5 it is only the dissipation factor (and to a small extent in the early output, the input constant) which determine the functional “shape” of the output. The scaling, offset and delay do not effect correlation and so are not degrees of freedom in the classical sense.
1) Aminghafari, M.; Cheze, N.; Poggi, J-M. (2006), “Multivariate de-noising using wavelets and principal component analysis,” Computational Statistics & Data Analysis, 50, pp. 2381–2398.
2) N.A. Krivova, L.E.A. Vieira, S.K. Solanki (2010).Journal of Geophysical Research: Space Physics, Volume 115, Issue A12, CiteID A12112. DOI:10.1029/2010JA015431
3) Ball, W. T.; Unruh, Y. C.; Krivova, N. A.; Solanki, S.; Wenzler, T.; Mortlock, D. J.; Jaffe, A. H. (2012) Astronomy & Astrophysics, 541, id.A27. DOI:10.1051/0004-6361/201118702
4) K. L. Yeo, N. A. Krivova, S. K. Solanki, and K. H. Glassmeier (2014) Astronomy & Astrophysics, 570, A85, DOI: 10.1051/0004-6361/201423628
5) For a summary of many of the correlations between TSI and climate that have been investigated see The Solar Evidence (http://appinsys.com/globalwarming/gw_part6_solarevidence.htm)
6) STATEMENT TO THE SUBCOMMITTEE ON SPACE, SCIENCEAND COMPETITIVENESS OF THE UNITED STATES SENATE; Hearing on “Data or Dogma? Promoting Open Inquiry in the Debate Over the Magnitude of Human Impact on Climate Change”; Judith A. Curry, Georgia Institute of Technology
8) See wden from MathWorks Matlab™ documentation.
Hadcrut4 global temperature series:
Krivova TSI reconstruction:
Available at http://climexp.knmi.nl/data/ico2_log.dat