2.2. Quasi-Periodic Oscillations in the Observed Global-Mean Near-Surface Temperature Data

Figure 2 shows the SSA analysis of the natural variability. Shown therein are the first four Quasi-periodic Oscillations (QPOs) revealed by SSA. These QPOs are oscillations with variable periods and amplitudes. QPO-1, shown in Figure 2(A), has a period of about 60 years and an amplitude of about 0.1˚C. This is the AMO discovered by us in 1994 [10]. The structure of QPO-1 shown by the NOAA, NASA and JMA data differs from the structure shown by the HadCRU data. We shall return to this below. QPO-2, shown in Figure 2(B), has a period of about 20 years and an amplitude of about 0.05˚C. This oscillation was found by us in our 1994 paper [10] and earlier

by Ghil and Vautard [17]. QPO-3, shown in Figure 2(C), has a period of about 10 years and an amplitude of about 0.05˚C. QPO-4 has an irregular period shorter than 10 years and an irregular amplitude whose average is about 0.02˚C. There are three other QPOs in the data (not shown) with even more irregular interannual periods and smaller amplitudes, as well as a stochastic (random) noise component. We will see in Section 3 that it is possible to represent QPOs 1, 2 and 3 mathematically and thus project them forward in time year by year. Alas, it is not possible to do so with QPO-4 and the higher-order QPOs. Accordingly, we shall represent their effect on the global-mean temperature, and the effect thereon by stochastic noise, by their 90 percent confidence interval.

2.2.1. Influence of Start Date on QPO-1

Figure 2 shows that when the start date of the HadCRU data is changed from 1850 to 1880, the structure of its QPO-1 is very similar to that obtained from the NOAA and NASA data that start in 1880. Accordingly we conclude that a start date of 1880 is too late to correctly characterize the structure of QPO-1. It is of interest therefore to examine when the structure of QPO-1 with start date between 1850 and 1880 agrees with the structure of QPO-1 for the start date of 1850.

Figure 3(A) shows the structure of QPO-1 for the Had-CRU temperature observations for start dates from 1850 to 1880 in 10-year intervals, and Figure 3(B) shows the Coefficient of Determination (R²) of QPO-1 for these start dates with QPO-1 for the actual start date of 1850. It is seen that for R² to exceed 0.9, the start date must be no later than 1860. Accordingly, a record length of about 152 years is required to correctly determine the structure of QPO-1. This record length will be attained for the NOAA and NASA data in 2032, and for the JMA data in 2043. This is 19 and 30 years into the future. As this is a long time to wait to determine the structure of QPO-1 from the NOAA, NASA and JMA data, we suggest that these groups extend their temperature observations backward in time to 1850.

2.2.2. Influence of End Date on the HadCRU QPO-1

It is also of interest to examine how the structure of QPO-1 for the HadCRU temperature observations changed by the addition of the temperatures for 2011 and 2012. This is shown in Figure 3(C). The addition of two more years of data has caused both the period and the amplitude of QPO-1 to become more uniform. This is also seen in Figure 3(D) which presents the year-by-year predictable part of the HadCRU temperature observations given by QPO-1 + QPO-2 + QPO-3. This suggests that the analysis reported here should be repeated at least

every five years.

3. Projection of Temperature Changes from 2012 through 2100

The temperature changes due to internal factors consist of two parts: 1) a predictable part that can be projected year to year; and 2) an unpredictable part that cannot be projected year to year. QPOs 1, 2 and 3 are sufficiently regular that they can be predicted on a year-to-year basis. The higher-order QPOs and the stochastic noise cannot be predicted on a year-to-year basis. We will represent these unpredictable temperature changes by their 90- percent confidence interval—the interval between the 5^th and 95^th percentiles of their cumulative distribution function (CDF).

3.1. Predictable Temperature Changes

We have examined 7 methods for representing the predictable part of the natural variability: 1) the “SSA-vector” and “SSA-recurrent” methods implemented in the RSSA package [18,19] of the statistical software R [20]; 2) four auto-regressive (AR) methods: “AR-Burg”, “AROLS”, “AR-MLE” and “AR-Yule-Walker”; and 3) representation by the sine wave y(t) = C + A sin, with parameters C, A, P and f determined by the Kaleidagraph software package [21], with t_o being the start year of the observed temperature dataset. These methods and their evaluation are described in the Appendix. We found the sine-wave representation to be superior to the other 6 methods.

Figure 4 shows the sine-wave representations of QPOs 1, 2 and 3 from 1850 through 2100 for the HadCRU observed temperature data. The C, A, P and f parameters of these fits, and for the fits for the NOAA, NASA and JMA datasets, are presented in Table 1. Therein it is seen that the coefficient of determination between QPO-1 and its fit for the HadCRU dataset is 0.993. We use this fit for the NOAA, NASA and JMA datasets because, as discussed in Section 2.2.1, their duration is too short to correctly characterize QPO-1.

The coefficient of determination for QPO-2 is above 0.867 for the HadCRU, NOAA and NASA datasets, but is only 0.472 for the JMA dataset. In contrast, the coefficient of determination for QPO-3 is highest (0.899) for the JMA dataset and lowest (0.703) for the HadCRU dataset. Accordingly, it should be kept in mind that the combined temperature change due to QPO-1 + QPO-2 + QPO-3 is not fully predictable, but rather only quasipredictable.

In our Causes 2 paper we used our Simple Climate Model to simulate the temperature change from 1765 through 2010, ΔT_SIM(t), due to solar-irradiance variations, volcanoes and humanity, the latter by the emissions of greenhouse gases and aerosol precursors, and land-use changes. In so doing we determined the climate sensitivity, ΔT_2x—the change in global-mean, equilibrium nearsurface temperature due to a doubling of the pre-industrial CO₂ concentration—that best reproduced the observed temperature departures δT_OBS(t) from the 1961-90 mean temperature for the HadCRU, NOAA, NASA and JMA temperature observations. As shown in Table 2, this yielded ΔT_2x = 1.61˚C, 1.99˚C, 1.45˚C and 2.01˚C, respectively. In so doing we converted ΔT_SIM(t) to the simulated temperature departures from the 1961-1990 average temperature by

(1)

where K for each observed dataset was determined such that the average of δT_SIM(t) − δT_OBS(t) from starting year t_o through 2010 was zero, with t_o = 1850 for the HadCRU observations, 1880 for the NOAA and NASA observations, and 1891 for the JMA observations.

We now combine δT_SIM(t) with the fit temperature changes due to QPOs 1, 2 and 3 to obtain the total predictable temperature changes from t_o through 2100,

(2)

In Section 3.3 we describe the two scenarios of future greenhouse-gas emissions for which we simulated ΔT_SIM(t) from t_o through 2100 and then converted them to the corresponding δT_SIM(t) via Equation (1).

3.2. Unpredictable Temperature Changes

We define the unpredictable temperature change, δT_UP(t), by

(3)

The cumulative distribution function (CDF) for δT_UP(t) is shown in Figure 5 for the four observed temperature datasets. It is seen that the CDFs are very well fit by the CDF for a Normal distribution, shown by the red line in each panel, with a coefficient of determination R² > 0.99. Mathematically this fit is μ + σ norm(x), where μ and σ are the median and standard deviation for the Normal distribution. The characteristics of the fit Normal distributions are presented in Table 2. The median (= mean = mode) for the fit for each of the four datasets is essentially zero ˚C, showing that there is no bias in the unpre-

dictable temperature change, δT_UP(t). The standard deviation for the NOAA, NASA and JMA datasets lies between 0.140˚C and 0.162˚C. This is about 50% larger than the standard deviation for the HadCRU dataset of 0.110˚C. Undoubtedly this is because we have used QPO-1 for the HadCRU data as the QPO-1 for the NOAA, NASA and JMA data, this because, as shown in Section 2.2.1, the record length of these three datasets is too short to accurately characterize QPO-1.

The 5^th and 95^th percentiles of the actual CDFs, ΔT_5th and ΔT_95th are shown in Table 2. In Section 4 we will present δT_SIM,P(t) + ΔT_5th, δT_SIM,P(t) and δT_SIM,P(t) + ΔT_95th for t_o ≤ t ≤ 2100 for two scenarios of future greenhouse-gas emissions. It should be noted here that while we include solar-irradiance variations in δT_SIM,P(t) due to the 11-year sunspot cycle, we do not include any other solar variations, nor do we include any future volcanic eruptions, as their timing, duration and intensity cannot now be predicted.

3.3. Future Emission Scenarios

In FP1 we used our Simple Climate Model [11] to calculate the change in global-average near-surface air temperature from 1765 through year 3000 for two scenarios of the future emissions of greenhouse gases (GHGs), a Reference scenario—the Representative Concentration Pathway 8.5 scenario (RCP-8.5)—and a Mitigation scenario—the Fair Plan scenario. The latter is fair in two ways. First, it uses trade-adjusted emissions, that is, the GHG emissions incurred by country X in producing goods and services imported by country Y are considered as the GHG emissions of country Y, not country X. Second, the intensity of GHG emissions of the developed (so-called Annex B or AB) countries and developing (non-Annex B or nAB) countries were decreased differently from unity in 2015 to zero in 2065 such that: 1) the cumulative GHG emissions for AB and nAB countries were equal; and 2) the maximum increase in global-mean near-surface temperature did not exceed the 2˚C limit adopted by the United Nations Framework Convention on Climate Change “to prevent dangerous anthropogenic interference with the climate system” [8,9]. The GHG intensity decreased from unity in 2015 to zero in 2065 linearly for AB countries and more slowly at first for nAB countries, thereby allowing their continuing economic development. In FP2 we examined the effect of: 1) deferring the start year from 2015 to 2030 in 5-year increments; and 2) increasing the phase-out period from 50 years to 100 years in 10-year increments. We found that it is optimum to begin the phase-out in 2020 and complete it in 2100. Accordingly, below we present δT_SIM,P(t) + ΔT_5th, δT_SIM,P(t) and δT_SIM,P(t) + ΔT_95th for t_o ≤ t ≤ 2100 for the RCP-8.5 and FP2 scenarios of future greenhouse-gas emissions.

4. Temperature Departures for the RCP-8.5 and FP2 Emissions Scenarios

Figure 6 presents the simulated temperature changes from 1765 through year 3000 for the Reference RCP-8.5 and FP2 Mitigation scenarios for the climate sensitivities obtained by optimizing the agreement between the simulated and observed temperature deviations from the 1961- 90 mean, the latter for the HadCRU, NOAA, NASA and JMA datasets. The results for the HadCRU, NOAA and NASA (GISS) datasets are from Figure 1(C) of FP1 and Figure 9 of FP2. The results for JMA were obtained for this paper (FP3). The UNFCCC limit of 2˚C is shown by the green line. The simulated temperature changes from t_o through 2012 include the human-caused contributions due to the emission of greenhouse gases and aerosol precursors, and land-use changes. They also include variations in solar irradiance due to the 11-year sunspot cycle and the Maunder Minimum therein, and volcanoes. The simulated temperature changes from 2012 through year 3000 include the human-caused contributions due to the emission of greenhouse gases and aerosol precursors for each scenario, and variations in solar irradiance due to the 11-year sunspot cycle.

Figure 6 shows that the temperature change for the RCP-8.5 scenario exceeds the UNFCCC limit starting from 2047 (JMA) to 2062 (NASA), and peaks circa 2250 with values from 4.4˚C (NASA) to 6.0˚C (JMA). The temperature change for FP2 peaks in about 2080 and ranges from 1.7˚C (NASA) to 2.3˚C (JMA). The temperature changes for JMA and NOAA slightly exceed the 2˚C UNFCCC limit from 2056 to 2130 for JMA, and

from 2064 to 2118 for NOAA. These small exceedances of the 2˚C UNFCCC limit are not worrisome as they can be avoided, if necessary, by tweaking the FP2 scenario sometime before 2050. Indeed, even if the 2˚C limit were not exceeded, the size of the limit and the greenhousegas-emission intensity of the FP2 scenario relative to the RCP-8.5 scenario should be re-examined on a regular basis following our robust adaptive decision strategy [22-25].

Figure 7 presents the departures of the global-mean near-surface temperature from the 1961-90 mean temperature observed from t_o through 2012 for the HadCRU, NOAA, NASA and JMA datasets (black line), and the corresponding temperature departures simulated from t_o through 2100 for the RCP-8.5 (red line) and FP2 (blue line). The simulated temperature departures have been obtained from the simulated temperature changes via Equation (1), to which we have added the predictable natural variability—QPOs 1, 2 and 3—and the unpredictable natural variability characterized by its 90% confidence interval. The green line in each panel shows the temperature departure that corresponds to the UNFCCC 2˚C limit.

Figure 7 shows that by about 2010 the signal of global warming is unmistakable against the background noise of the predictable (year-to-year) and unpredictable (not yearto-year) natural variability. The further along the RCP- 8.5 trajectory we go, the larger the signal-to-noise ratio becomes. Of course this does not mean that there will never be a future period of time when the temperature does not change or even decreases, as it did during 1998- 2008. But as we cautioned in our year-2000 Causes 1 paper [1], “… it is prudent not to expect year-after-year warming in the near future and, in so doing, diminish concern about global warming should global cooling instead manifest itself again”. Moreover, it is abundantly clear from Figures 6 and 7 that it is now time for the UNFCCC to formulate a plan to transition from the Business-As-Usual RCP-8.5 emission scenario to the FP2 scenario that phases out the human-caused emission of greenhouse gases beginning in 2020 and ending 80 years later in 2100.

5. Discussion

We posit that there are three phases of learning: 1) Ignorance; 2) Uncertainty; and 3) Resolution. In Phase 1 we do not know that we have a problem. In Phase 2 we think there may be a problem, but we are unsure of its scope and magnitude. In Phase 3, we know there is a problem and we know its scope and magnitude within some bounds. In this third Phase we either dismiss the problem as being too small to be of concern among the universe of problems, or we accept that we have a problem and

begin to deal with it.

For the Global Warming Problem, Phase 1—Ignorance —began with the onset of the Industrial Revolution circa 1750. In Phase 1 we were ignorant of the climatic consequences of our burning fossil fuel—coal, oil and natural gas. Phase 1 ended in 1979 with the publication of the first report on Global Warming by the National Research Council of the US Academy of Sciences, the so-called “Charney Report” named after Jule Charney, the chair of the ad hoc study group that wrote the report [22].

Phase 2—Uncertainty—began with the Charney Report in 1979. In Phase 2 we understood that burning fossil fuels would have climatic consequences, but we were uncertain about the scope and magnitude of those conesquences. Phase 2 is now ending, and Phase 3—Resolution—is beginning, because we can now bound the Global Warming Problem. Not only can we project the human contribution thereto—the source of the Problem— we can now project the natural variability and, thereby, bound the Problem into the future.

In so doing we can see that we cannot relegate the Global Warming Problem to the dustbin of problems that are too small to be of our concern. Rather, we can now see that absent any action by humanity to deal with the Global Warming Problem, it will exceed the maximum 2˚C global warming adopted by the United Nations Framework Convention on Climate Change “to prevent dangerous anthropogenic interference with the climate system”, and it will do so sometime between 2034 and 2076. Accordingly, it is time to begin to deal with the Global Warming Problem by adopting our Revised Fair Plan to Safeguard Earth’s Climate, FP2, which entails making the transition from the Fossil-fuel Age to the Post Fossil-fuel Age within this century, by phasing out the emission of greenhouse gases from 2020 to 2100.

6. Conclusions

In this paper we have used Singular Spectrum Analysis to analyze four observational datasets of global-mean near-surface temperature—the HadCRU, NOAA, NASA and JMA datasets. For each dataset we have found a trend and several quasi-periodic oscillations or QPOs. As we have shown previously in Causes 2 [3], the trend for each dataset is caused by human beings, predominantly by their emission of greenhouse gases (GHGs), but also by their emission of aerosol-precursor gases and land-use changes. The longest-period and largest-amplitude QPO is properly revealed by the HadCRU dataset which begins in 1850. QPO-1 is not properly revealed by the NOAA and NASA datasets, which begin in 1880, or by the JMA dataset which begins in 1891. Accordingly we recommend that the starting year of the NOAA, NASA and JMA datasets be extended back at least to 1850.

QPOs 1, 2 and 3 are sufficiently regular that each can be fit by a sine wave that can then be used to predict their future behavior year by year. The higher-order QPOs are not sufficiently regular to be predicted year by year into the future. But they and the remainder of the observed temperature departures can be predicted statistically on a non-year-by-year basis.

We project the human-caused changes in global-mean near-surface temperature across the 21^st century for two scenarios of future emissions of greenhouse gases and aerosol-precursor gases, as well as for the 11-year solar sunspot cycle. One scenario is the Representative Concentration Path 8.5 scenario that we used as the Reference scenario in our FP1 and FP2 papers. The other scenario is the FP2 scenario that phases out the emission of greenhouse gases and aerosol-precursor gases from 2020 through 2100 in a fair manner such that the global-mean temperature change from preindustrial time does not exceed the 2˚C limit adopted by the UN Framework Convention on Climate Change “to prevent dangerous anthropogenic interference with the climate system” [9]. The FP2 scenario crafts the trade-adjusted cumulative emission of GHGs by the developing (so-called nonAnnex B or nAB) countries to be equal to trade-adjusted cumulative emission of GHGs by the developed (annex B or AB) countries. This is achieved by designing the emission-phaseout-intensity for the nAB countries from unity in 2020 to zero in 2100 using a cubic function of time, this in contrast to the linear-in-time emissionphaseout-intensity for the AB countries.

To the human-caused increases in global-mean, nearsurface temperature we add the non-human-caused changes therein by QPOs 1, 2 and 3. We convert this predictable part of the temperature change to temperature departure from the 1961-90 mean temperature as we have done heretofore [1,3,10]. We then subtract the observed temperature departures from the simulated temperature departures and fit the cumulative distribution function for the difference by a Normal distribution. The resulting fit is found to be excellent with a coefficient of determination larger than 0.99. To the predictable part of the projected change in temperature across the 21^st century we add the unpredictable part in terms of its 5^th and 95^th percentiles.

We find that we are now entering the Third Phase of Learning—Resolution—wherein the signal of humancause global warming has, or soon will, swamp the noise of natural variability. Accordingly, it is now time for the UN Framework Convention on Climate Change to formulate an international agreement that phases out the emission of greenhouse gases from 2020 through 2100 following our Revised Fair Plan to Safeguard Earth’s Climate.

REFERENCES

Appendix: Representation of the QPOs

We have examined seven methods of representing QPOs using three different approaches.

The first approach is an SSA-forecasting approach implemented in the RSSA package [18,19] of the statistical software R [20] and yields “the new series which is expected to ‘continue’ the current series” [19] based on a given decomposition. Two distinct algorithms are evaluated, “SSA-vector” and “SSA-recurrent”. SSA-vector “sequentially projects the incomplete embedding vectors (either from the original or reconstructed series) onto the subspace spanned by the selected eigen-triples of the decomposition to derive the missed (ending) values of the such vectors” [19]. SSA-recurrent “continues the set of vectors in the subspace spanning the chosen eigenvectors (…) and then derive the series out of this extended set of vectors” [19].

The second approach is built on the idea that the QPOs are narrowband time series and therefore can be predicted fairly well by fitting a low-order auto-regressive (AR) process to the QPOs [23]. The method has been successfully used by [24] to forecast the SOI time series with considerable skill for 30 to 36 months. However [25] concluded that the instrumental temperature record is not long enough to determine its quasi-periodic components reliably enough to allow for a skillful decadal or longer forecast.

Because our temperature record is more than 20 years longer than that of [25], QPOs 1, 2 and 3 are extracted, fitted with an AR process, and the coefficients thereof are used to predict their future evolution. This approach is implemented in the statistical software R [20], and the “AR-Burg”, “AR-OLS”, “AR-MLE” and “AR-YuleWalker” methods of fitting the AR-model are evaluated.

The order M of the AR process is either automatically determined using the Akaike Information Criterion or fixed.

Finally, since QPO-1, QPO-2 and QPO-3 are quite periodic, each is fitted with a sine function that is used to extrapolate into the future.

The forecast skill for the above methods is assessed for 5, 10 and 20-year periods using the HadCRUT4 record. This record is truncated to: 1) 1850 to 1990; 2) 1850 to 2000 and 3) 1850 to 2005 and SSA is used to determine the corresponding “truncated” QPOs 1 through 3. These truncated QPOs are then forecast into the future and are compared to the QPOs of the 1850-2010 time series. The forecasting skills of the different methodologies are evaluated using: 1) visual inspection, V; 2) root-meansquare errors, R; and 3) the correlation coefficient, ρ. In Table A1 we summarize our skill assessment based on four criteria: “+”, “0”, “−” and “/”. A “+” indicates good V, small R and ρ > 0.8. A “0” means a neutral V, that is, neither good nor bad; an average R compared to the overall distribution of R for this particular skill test, and 0.3 < ρ < 0.8. A “−” signifies poor V, large R and ρ < 0.3. A “/” indicates the failure to fit the corresponding QPO. The Score equals the sum of +’s (each = 1), 0’s (each = 0), −’s (each = −1) &/’s (each = −2).

The best method by far is “Sine”, followed by “SSARecurrent”, “AR-Burg” and “AR-YW”. However, the latter is very dissipative in time and hence not applicable for more than a decade. The “SSA-Recurrent” approach shows considerable skill in forecasting QPO-1 for 5 and 10 years, but tends to underestimate the actual amplitudes of QPO-2 and QPO-3. The latter is also a problem of the “SSA-Vector” approach. The other two methods fail the majority of skill tests; “AR-MLE” is especially unsuccessful in fitting an AR process to QPO-1.