ABSTRACT: Correlation analysis shows that the AGW theory of global warming is not supported by the data because projections of the theory in terms of CMIP5 forcings are found to be inconsistent with the observational data in a number of ways.

FIGURE 1: GRAPHICAL COMPARISON OF THE TEMPERATURE SERIES STUDIED

FIGURE 2: OBSERVED WARMING RATES IN EACH TIME SERIES

FIGURE 3: CORRELATIONS AMONG THE TIME SERIES

FIGURE 4: DETRENDED CORRELATION

FIGURE 5: HYPOTHESIS TEST: H0: CORR<=0 AGAINST HA: CORR>0

FIGURE 6: SEASONAL CYCLE OF WARMING RATES

FIGURE 7: SPURIOUS CORRELATIONS IN TIME SERIES DATA

1. The data: The RCP8.5 temperature series is generated by climate models with CMIP5 forcings (discussed in a related post: Correlation of CMIP5 Forcings with Temperature ). Here we compare these temperature projections of climate change theory with the observational record in the satellite era of global temperature measurements. The time span for the study is set to 1979-2018 so that satellite data and surface data may be compared with theory on the same basis and in a common time span. It is acknowledged that satellite data provide the most reliable measures of global mean temperature and also that warming in the observational data and its human cause have been most sustained and “unequivocal” in the post 1979 period when the 30-year cooling that preceded it had ended.
2. Citations: [Santer, Benjamin D., et al. “Comparing tropospheric warming in climate models and satellite data.” Journal of Climate 30.1 (2017): 373-392], [Santer, Benjamin D., et al. “Causes of differences in model and satellite tropospheric warming rates.” Nature Geoscience10.7 (2017): 478], [Santer, Benjamin D., et al. “Tropospheric warming over the past two decades.” Scientific Reports 7.1 (2017): 2336], [Dai, Aiguo, and Christine E. Bloecker. “Impacts of internal variability on temperature and precipitation trends in large ensemble simulations by two climate models.” Climate Dynamics (2017): 1-18], [Coumou, Dim, et al. “Weakened Flow, Persistent Circulation, and Prolonged Weather Extremes in Boreal Summer.” Climate Extremes: Patterns and Mechanisms (2017): 61-73], [Ding, Qinghua, et al. “Influence of high-latitude atmospheric circulation changes on summertime Arctic sea ice.” Nature Climate Change 7.4 (2017): 289], [Sun, Lantao, et al. “Drivers of 2016 record Arctic warmth assessed using climate simulations subjected to Factual and Counterfactual forcing.” Weather and climate extremes 19 (2018): 1-9], [Hughes, Terry P., et al. “Global warming and recurrent mass bleaching of corals.” Nature 543.7645 (2017): 373], [Taylor, Christopher M., et al. “Frequency of extreme Sahelian storms tripled since 1982 in satellite observations.” Nature544.7651 (2017): 475], [Randel, William J., et al. “Troposphere‐Stratosphere Temperature Trends Derived From Satellite Data Compared With Ensemble Simulations From WACCM.” Journal of Geophysical Research: Atmospheres 122.18 (2017): 9651-9667], [Schultz, Natalie M., Peter J. Lawrence, and Xuhui Lee. “Global satellite data highlights the diurnal asymmetry of the surface temperature response to deforestation.” Journal of Geophysical Research: Biogeosciences 122.4 (2017): 903-917]
3. The RCP8.5 “business as usual” emissions based temperature projections are provided in degrees Kelvin by KNMI climate data server at climexp.knmi.nl  .The observed temperature series used for comparison with theory are global mean surface temperature reconstructions HadCRUT4 from the Hadley Centre, and the GISTEMP global series from NASA-GISS; and global mean lower troposphere temperatures derived from satellite data provided by UAH and RSS. Since the observational data are provided only as temperature “anomalies”, they are normalized by adding the average value of the corresponding RCP8.5 series to the anomalies for ease of direct comparison.
4. The calendar months are are not combined into annual means but rather they are studied separately to account for the large seasonal cycle and also in consideration of differences in warming trends among the calendar months reported in a related post at Temperature Trend Profiles & the Seasonal Cycle  . Also important in this comparison is that the RCP data in ºK are not deseasonalized and therefore we must accommodate its seasonal cycle. This is done in this analysis by comparing calendar months and not annual deseasonalized means.
5. Monthly mean temperatures in the five time series studied are displayed graphically in Figure 1 for each of the twelve calendar months. They show a general warming trend across the full span of the data for all twelve calendar months and all five time series studied. Some differences between the theoretical series RCP8.5 and the observational series are apparent in terms of greater random variability and greater differences among calendar months in the data than in the theory. Due to data availability differences as of this writing, the full span of the data is 40 years, 1979-2018 for the months January-July and 39 years 1979-2017 for the months August-December for all five time series.
6. The observed full span OLS warming rates are displayed and compared in tabular and graphical form in Figure 2. The comparison among the five time series shows that the warming trend in the theoretical series RCP8.5 is much higher than the warming trends seen in the data; and with a corresponding standard deviation of trend that is much lower than that seen in the data. The net result of that, seen in the last chart of Figure 2, is that the t-statistics for the statistical significance of the warming rate sets the theoretical series apart from the data with a much higher values of the t-statistic as t>20 compared with observed values in the data of 4<t<14. This comparison shows some evidence that the theory may be an outlier when compared with the four observational data series. This pattern is further explored with correlation analysis.
7. Correlations among the five time series are presented in Figure 3. The table of correlations show very strong positive correlations among the five time series in the range of 0.55<r<0.99. However, the chart appears to indicate that correlations among the data are higher and more reliable than correlations between data and theory. This difference is further explored with detrended correlation analysis.
8. The need for detrended correlation in time series analysis is explained in a related post Spurious Correlations in Climate Science . In brief, correlations in time series data derive from two sources – that imposed by long term trends (spurious) and that due to responsiveness at the time scale of interest (real). These two effects can be separated with detrended correlation analysis as explained in a Youtube video by Alex Tolley included in Figure 7. The top panel of Figure 6 shows that the inclusion of shared trends can create spurious correlations and the bottom panel of Figure 6 shows that detrended correlation analysis is necessary to remove this spurious effect.
9. The results of detrended correlation analysis appear in Figure 4. Here we see a clear distinction between the high correlations among the four data series and the low correlation between the theoretical projection and the four data series. When the annual time scale is imposed and the effect of long term trends is removed we find clear evidence of a divergence between theory and data that was previously hidden by the spurious effect of long term trends.
10. A further evidence of the contrast between correlation analysis (inclusive of trend effects), and detrended correlation analysis (net of trend effects) is seen in Figure 5. Here we see that when the source data are used and the spurious effect of long term trends is included (the first two tables in Figure 5), high correlations are seen among all  variables and all 120 correlations are found to be statistically significant and no distinction is seen between theory and data.  However, when the detrended data are used (bottom two tables in Figure 5), a clear distinction between theory and data emerges for here we find that all 72 detrended correlations among the data are statistically significant but only 9 out of 48 correlations between theory and data are statistically significant.
11. It has been noted in related posts that the warming rate differs among the calendar months [ Temperature Trend Profiles & the Seasonal Cycle ]. A rationale for the seasonal cycle in warming rate is provided in ( Feldman 2015) in terms of a seasonal cycle in atmospheric CO2 concentration. They find that the seasonal swing in GHG forcing can vary from 0.1 to 0.2 watts per square meter. They ascribe this cycle to changing rates at which photosynthesis removes carbon dioxide from the atmosphere noting higher photosynthesis during the boreal spring and summer and lower photosynthesis during the boreal winter. This theoretical seasonal cycle is seen in the RCP8.5 series that was derived from exactly these considerations. However it is not seen in the observational data where in some cases the cycle is exactly reversed.
12. We conclude from the above that  the WMGHG (well mixed greenhouse gas) theory of global warming is not supported by the data because projections of the theory in terms of its CMIP5 forcings are found to be inconsistent with the observational data in a number of ways.

BIBLIOGRAPHY

1. 2010: Moss, Richard H., et al. “The next generation of scenarios for climate change research and assessment.” Nature 463.7282 (2010): 747. Advances in the science and observation of climate change are providing a clearer understanding of the inherent variability of Earth’s climate system and its likely response to human and natural influences. The implications of climate change for the environment and society will depend not only on the response of the Earth system to changes in radiative forcings, but also on how humankind responds through changes in technology, economies, lifestyle and policy. Extensive uncertainties exist in future forcings of and responses to climate change, necessitating the use of scenarios of the future to explore the potential consequences of different response options. To date, such scenarios have not adequately examined crucial possibilities, such as climate change mitigation and adaptation, and have relied on research processes that slowed the exchange of information among physical, biological and social scientists. Here we describe a new process for creating plausible scenarios to investigate some of the most challenging and important questions about climate change confronting the global community.
2. 2011: Meinshausen, Malte, et al. “The RCP greenhouse gas concentrations and their extensions from 1765 to 2300.” Climatic change 109.1-2 (2011): 213. We present the greenhouse gas concentrations for the Representative Concentration Pathways (RCPs) and their extensions beyond 2100, the Extended Concentration Pathways (ECPs). These projections include all major anthropogenic greenhouse gases and are a result of a multi-year effort to produce new scenarios for climate change research. We combine a suite of atmospheric concentration observations and emissions estimates for greenhouse gases (GHGs) through the historical period (1750–2005) with harmonized emissions projected by four different Integrated Assessment Models for 2005–2100. As concentrations are somewhat dependent on the future climate itself (due to climate feedbacks in the carbon and other gas cycles), we emulate median response characteristics of models assessed in the IPCC Fourth Assessment Report using the reduced-complexity carbon cycle climate model MAGICC6. Projected ‘best-estimate’ global-mean surface temperature increases (using inter alia a climate sensitivity of 3°C) range from 1.5°C by 2100 for the lowest of the four RCPs, called both RCP3-PD and RCP2.6, to 4.5°C for the highest one, RCP8.5, relative to pre-industrial levels. Beyond 2100, we present the ECPs that are simple extensions of the RCPs, based on the assumption of either smoothly stabilizing concentrations or constant emissions: For example, the lower RCP2.6 pathway represents a strong mitigation scenario and is extended by assuming constant emissions after 2100 (including net negative CO₂ emissions), leading to CO₂ concentrations returning to 360 ppm by 2300. We also present the GHG concentrations for one supplementary extension, which illustrates the stringent emissions implications of attempting to go back to ECP4.5 concentration levels by 2250 after emissions during the 21^st century followed the higher RCP6 scenario. Corresponding radiative forcing values are presented for the RCP and ECPs
3. 2012: Taylor, Karl E., Ronald J. Stouffer, and Gerald A. Meehl. “An overview of CMIP5 and the experiment design.” Bulletin of the American Meteorological Society 93.4 (2012): 485-498. The fifth phase of the Coupled Model Intercomparison Project (CMIP5) will produce a state-of-the- art multimodel dataset designed to advance our knowledge of climate variability and climate change. Researchers worldwide are analyzing the model output and will produce results likely to underlie the forthcoming Fifth Assessment Report by the Intergovernmental Panel on Climate Change. Unprecedented in scale and attracting interest from all major climate modeling groups, CMIP5 includes “long term” simulations of twentieth-century climate and projections for the twenty-first century and beyond. Conventional atmosphere–ocean global climate models and Earth system models of intermediate complexity are for the first time being joined by more recently developed Earth system models under an experiment design that allows both types of models to be compared to observations on an equal footing. Besides the longterm experiments, CMIP5 calls for an entirely new suite of “near term” simulations focusing on recent decades and the future to year 2035. These “decadal predictions” are initialized based on observations and will be used to explore the predictability of climate and to assess the forecast system’s predictive skill. The CMIP5 experiment design also allows for participation of stand-alone atmospheric models and includes a variety of idealized experiments that will improve understanding of the range of model responses found in the more complex and realistic simulations. An exceptionally comprehensive set of model output is being collected and made freely available to researchers through an integrated but distributed data archive. For researchers unfamiliar with climate models, the limitations of the models and experiment design are described.
4. 2012: Meehl, Gerald A., et al. “Climate system response to external forcings and climate change projections in CCSM4.” Journal of Climate 25.11 (2012): 3661-3683. Results are presented from experiments performed with the Community Climate System Model, version 4 (CCSM4) for the Coupled Model Intercomparison Project phase 5 (CMIP5). These include multiple ensemble members of twentieth-century climate with anthropogenic and natural forcings as well as single-forcing runs, sensitivity experiments with sulfate aerosol forcing, twenty-first-century representative concentration pathway (RCP) mitigation scenarios, and extensions for those scenarios beyond 2100–2300. Equilibrium climate sensitivity of CCSM4 is 3.20°C, and the transient climate response is 1.73°C. Global surface temperatures averaged for the last 20 years of the twenty-first century compared to the 1986–2005 reference period for six-member ensembles from CCSM4 are +0.85°, +1.64°, +2.09°, and +3.53°C for RCP2.6, RCP4.5, RCP6.0, and RCP8.5, respectively. The ocean meridional overturning circulation (MOC) in the Atlantic, which weakens during the twentieth century in the model, nearly recovers to early twentieth-century values in RCP2.6, partially recovers in RCP4.5 and RCP6, and does not recover by 2100 in RCP8.5. Heat wave intensity is projected to increase almost everywhere in CCSM4 in a future warmer climate, with the magnitude of the increase proportional to the forcing. Precipitation intensity is also projected to increase, with dry days increasing in most subtropical areas. For future climate, there is almost no summer sea ice left in the Arctic in the high RCP8.5 scenario by 2100, but in the low RCP2.6 scenario there is substantial sea ice remaining in summer at the end of the century.
5. 2012: Taylor, Karl E., Ronald J. Stouffer, and Gerald A. Meehl. “An overview of CMIP5 and the experiment design.” Bulletin of the American Meteorological Society 93.4 (2012): 485-498. The fifth phase of the Coupled Model Intercomparison Project (CMIP5) will produce a state-of-the- art multimodel dataset designed to advance our knowledge of climate variability and climate change. Researchers worldwide are analyzing the model output and will produce results likely to underlie the forthcoming Fifth Assessment Report by the Intergovernmental Panel on Climate Change. Unprecedented in scale and attracting interest from all major climate modeling groups, CMIP5 includes “long term” simulations of twentieth-century climate and projections for the twenty-first century and beyond. Conventional atmosphere–ocean global climate models and Earth system models of intermediate complexity are for the first time being joined by more recently developed Earth system models under an experiment design that allows both types of models to be compared to observations on an equal footing. Besides the longterm experiments, CMIP5 calls for an entirely new suite of “near term” simulations focusing on recent decades and the future to year 2035. These “decadal predictions” are initialized based on observations and will be used to explore the predictability of climate and to assess the forecast system’s predictive skill. The CMIP5 experiment design also allows for participation of stand-alone atmospheric models and includes a variety of idealized experiments that will improve understanding of the range of model responses found in the more complex and realistic simulations. An exceptionally comprehensive set of model output is being collected and made freely available to researchers through an integrated but distributed data archive. For researchers unfamiliar with climate models, the limitations of the models and experiment design are described.
6. 2013: Zou, Liwei, and Tianjun Zhou. “Near future (2016-40) summer precipitation changes over China as projected by a regional climate model (RCM) under the RCP8. 5 emissions scenario: Comparison between RCM downscaling and the driving GCM.” Advances in Atmospheric Sciences 30.3 (2013): 806-818. Multi-decadal high resolution simulations over the CORDEX East Asia domain were performed with the regional climate model RegCM3 nested within the Flexible Global Ocean-Atmosphere-Land System model, Grid-point Version 2 (FGOALS-g2). Two sets of simulations were conducted at the resolution of 50 km, one for present day (1980–2005) and another for near-future climate (2015–40) under the Representative Concentration Pathways 8.5 (RCP8.5) scenario. Results show that RegCM3 adds value with respect to FGOALS-g2 in simulating the spatial patterns of summer total and extreme precipitation over China for present day climate. The major deficiency is that RegCM3 underestimates both total and extreme precipitation over the Yangtze River valley. The potential changes in total and extreme precipitation over China in summer under the RCP8.5 scenario were analyzed. Both RegCM3 and FGOALS-g2 results show that total and extreme precipitation tend to increase over northeastern China and the Tibetan Plateau, but tend to decrease over southeastern China. In both RegCM3 and FGOALS-g2, the change in extreme precipitation is weaker than that for total precipitation.RegCM3 projects much stronger amplitude of total and extreme precipitation changes and provides more regional-scale features than FGOALS-g2. A large uncertainty is found over the Yangtze River valley, where RegCM3 and FGOALS-g2 project opposite signs in terms of precipitation changes. The projected change of vertically integrated water vapor flux convergence generally follows the changes in total and extreme precipitation in both RegCM3 and FGOALS-g2, while the amplitude of change is stronger in RegCM3. Results suggest that the spatial pattern of projected precipitation changes may be more affected by the changes in water vapor flux convergence, rather than moisture content itself.
7. 2013: Forster, Piers M., et al. “Evaluating adjusted forcing and model spread for historical and future scenarios in the CMIP5 generation of climate models.” Journal of Geophysical Research: Atmospheres 118.3 (2013): 1139-1150. We utilize energy budget diagnostics from the Coupled Model Intercomparison Project phase 5 (CMIP5) to evaluate the models’ climate forcing since preindustrial times employing an established regression technique. The climate forcing evaluated this way, termed the adjusted forcing (AF), includes a rapid adjustment term associated with cloud changes and other tropospheric and land‐surface changes. We estimate a 2010 total anthropogenic and natural AF from CMIP5 models of 1.9 ± 0.9 W m⁻² (5–95% range). The projected AF of the Representative Concentration Pathway simulations are lower than their expected radiative forcing (RF) in 2095 but agree well with efficacy weighted forcings from integrated assessment models. The smaller AF, compared to RF, is likely due to cloud adjustment. Multimodel time series of temperature change and AF from 1850 to 2100 have large intermodel spreads throughout the period. The intermodel spread of temperature change is principally driven by forcing differences in the present day and climate feedback differences in 2095, although forcing differences are still important for model spread at 2095. We find no significant relationship between the equilibrium climate sensitivity (ECS) of a model and its 2003 AF, in contrast to that found in older models where higher ECS models generally had less forcing. Given the large present‐day model spread, there is no indication of any tendency by modelling groups to adjust their aerosol forcing in order to produce observed trends. Instead, some CMIP5 models have a relatively large positive forcing and overestimate the observed temperature change.
8. 2014: Friedlingstein, Pierre, et al. “Uncertainties in CMIP5 climate projections due to carbon cycle feedbacks.” Journal of Climate27.2 (2014): 511-526. In the context of phase 5 of the Coupled Model Intercomparison Project, most climate simulations use prescribed atmospheric CO₂ concentration and therefore do not interactively include the effect of carbon cycle feedbacks. However, the representative concentration pathway 8.5 (RCP8.5) scenario has additionally been run by earth system models with prescribed CO₂ emissions. This paper analyzes the climate projections of 11 earth system models (ESMs) that performed both emission-driven and concentration-driven RCP8.5 simulations. When forced by RCP8.5 CO₂ emissions, models simulate a large spread in atmospheric CO₂; the simulated 2100 concentrations range between 795 and 1145 ppm. Seven out of the 11 ESMs simulate a larger CO₂ (on average by 44 ppm, 985 ± 97 ppm by 2100) and hence higher radiative forcing (by 0.25 W m⁻²) when driven by CO₂ emissions than for the concentration-driven scenarios (941 ppm). However, most of these models already overestimate the present-day CO₂, with the present-day biases reasonably well correlated with future atmospheric concentrations’ departure from the prescribed concentration. The uncertainty in CO₂ projections is mainly attributable to uncertainties in the response of the land carbon cycle. As a result of simulated higher CO₂ concentrations than in the concentration-driven simulations, temperature projections are generally higher when ESMs are driven with CO₂ emissions. Global surface temperature change by 2100 (relative to present day) increased by 3.9° ± 0.9°C for the emission-driven simulations compared to 3.7° ± 0.7°C in the concentration-driven simulations. Although the lower ends are comparable in both sets of simulations, the highest climate projections are significantly warmer in the emission-driven simulations because of stronger carbon cycle feedbacks.
9. 2015: Kay, J. E., et al. “The Community Earth System Model (CESM) large ensemble project: A community resource for studying climate change in the presence of internal climate variability.” Bulletin of the American Meteorological Society 96.8 (2015): 1333-1349. While internal climate variability is known to affect climate projections, its influence is often underappreciated and confused with model error. Why? In general, modeling centers contribute a small number of realizations to international climate model assessments [e.g., phase 5 of the Coupled Model Intercomparison Project (CMIP5)]. As a result, model error and internal climate variability are difficult, and at times impossible, to disentangle. In response, the Community Earth System Model (CESM) community designed the CESM Large Ensemble (CESM-LE) with the explicit goal of enabling assessment of climate change in the presence of internal climate variability. All CESM-LE simulations use a single CMIP5 model (CESM with the Community Atmosphere Model, version 5). The core simulations replay the twenty to twenty-first century (1920–2100) 30 times under historical and representative concentration pathway 8.5 external forcing with small initial condition differences. Two companion 1000+-yr-long preindustrial control simulations (fully coupled, prognostic atmosphere and land only) allow assessment of internal climate variability in the absence of climate change. Comprehensive outputs, including many daily fields, are available as single-variable time series on the Earth System Grid for anyone to use. Early results demonstrate the substantial influence of internal climate variability on twentieth- to twenty-first-century climate trajectories. Global warming hiatus decades occur, similar to those recently observed. Internal climate variability alone can produce projection spread comparable to that in CMIP5. Scientists and stakeholders can use CESM-LE outputs to help interpret the observational record, to understand projection spread and to plan for a range of possible futures influenced by both internal climate variability and forced climate change.
10. 2015: Feldman, Daniel R., et al. “Observational determination of surface radiative forcing by CO 2 from 2000 to 2010.” Nature519.7543 (2015): 339. The climatic impact of CO₂ and other greenhouse gases is usually quantified in terms of radiative forcing¹, calculated as the difference between estimates of the Earth’s radiation field from pre-industrial and present-day concentrations of these gases. Radiative transfer models calculate that the increase in CO₂ since 1750 corresponds to a global annual-mean radiative forcing at the tropopause of 1.82 ± 0.19 W m⁻²(ref. 2). However, despite widespread scientific discussion and modelling of the climate impacts of well-mixed greenhouse gases, there is little direct observational evidence of the radiative impact of increasing atmospheric CO₂. Here we present observationally based evidence of clear-sky CO₂ surface radiative forcing that is directly attributable to the increase, between 2000 and 2010, of 22 parts per million atmospheric CO₂. The time series of this forcing at the two locations—the Southern Great Plains and the North Slope of Alaska—are derived from Atmospheric Emitted Radiance Interferometer spectra³together with ancillary measurements and thoroughly corroborated radiative transfer calculations⁴. The time series both show statistically significant trends of 0.2 W m⁻² per decade (with respective uncertainties of ±0.06 W m⁻² per decade and ±0.07 W m⁻² per decade) and have seasonal ranges of 0.1–0.2 W m⁻². This is approximately ten per cent of the trend in downwelling longwave radiation^5,6,7. These results confirm theoretical predictions of the atmospheric greenhouse effect due to anthropogenic emissions, and provide empirical evidence of how rising CO₂ levels, mediated by temporal variations due to photosynthesis and respiration, are affecting the surface energy balance.