SUMMARY: The testable implication of the GHG theory is that surface temperature should be responsive to atmospheric CO2 concentration such that a detrended correlation exists between the logarithm of atmospheric CO2 and surface temperature at the time scale of interest. This test is carried out for five temperature series in eight calendar months for the sample period 1979-2018 using comparative analysis to test observational data against the theoretical series. The comparison does not show evidence of the existence of GHG forcing by atmospheric CO2 in the observational data. 

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FIGURE 1: OBSERVED CLIMATE SENSITIVITY VALUES FOR EACH CALENDAR MONTH

FIGURE 2: CORRELATION BETWEEN TEMPERATURE AND LN(CO2)

FIGURE 3: SPURIOUS CORRELATIONS IN TIME SERIES DATA

FIGURE 4: DETRENDED CORRELATION BETWEEN TEMPERATURE AND LN(CO2)

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1. ABSTRACT: The testable implication of the the theory of GHG forcing of surface temperature by atmospheric CO2 is that surface temperature should be responsive to changes in atmospheric CO2 concentration at the time scale of interest. This test is carried out by comparing four observational temperature series against a theoretical series constructed with GHG forcing of atmospheric CO2. The comparison does not show evidence for the existence of GHG forcing of atmospheric CO2 in the observational data. Conventional climate sensitivity estimations in observational data depend on a spurious correlation and this spuriousness likely explains the great instability of sensitivity and the large range of values found in observational data. When that spuriousness is corrected with detrended correlation analysis, no correlation remains to support the existence of climate sensitivity.
2. DATA AND METHOD: This is a comparative analysis of the responsiveness of surface temperature to changes in atmospheric CO2 concentration in accordance with the GHG theory of atmospheric CO2 concentration described by the climate sensitivity function where surface temperature is a linear function of (LN(CO2)). Three types of temperature data are compared. They are (1) direct observations of global mean temperature with satellite mounted microwave sounding units (UAH, RSS), (2) global mean temperature reconstructions from the instrumental record (HAD, GIS), and (3) a theoretical time series of temperatures expected according to the theory of the GHG effect of atmospheric CO2 concentration where CMIP5 forcings ensure that Temp=f(LN(CO2)). The comparison is used to test the hypothesis that observational data contain the GHG effect of atmospheric CO2 concentration in the form of the climate sensitivity function.
3. The time span for the study is 40 year satellite era from 1979 to 2018 when direct measurements of mean global lower troposphere temperatures are available. The other three temperature time series constrained to this sample period so that a direct comparison can be made. Eight calendar months (January to August) of data are available for the full sample period 1979-2018 at the time of this study. The calendar months are studied separately as their trend behaviors have been shown to differ in a related post at this site [LINK] and also because the existence of a seasonal cycle in GHG forcing has been shown to exist [LINK] .
4. DATA ANALYSIS: FIGURE 1 is a tabulation of the observed climate sensitivity values in each of eight calendar months (January to August) and for each of the five temperature series studied (HAD, GIS, UAH, RSS, RCP8.5). It shows relatively low values of climate sensitivity ranging from 1.5<ECS<2.1 for the direct observations in the satellite data from UAH and RSS; somewhat higher values of 2.1<ECS<2.7 for temperature reconstructions from HAD and GIS; and values of ECS≈3 in theRCP8.5 series derived from a theory that holds that ECS=3±1.5. It is noted that although all observed values are within this large range, the theoretical RCP8.5 sensitivity values are higher than the observational empirical values.
5. FIGURE 2 displays the correlations between temperature and LN(CO2) required to support the reliability of the corresponding regression coefficients that are translated into climate sensitivities. Strong and statistically significant correlations are seen in all five time series. As expected, the highest correlations, all eight of them close to the near perfect  correlation of ρ≈1.0, are found in the theoretical RCP8.5 series that was constructed with the GHG forcing of atmospheric CO2. The lowest correlations, 0.55<ρ<0.75 are found in the directly observed satellite data. Intermediate values of 0.7<ρ<0.9 found in the global temperature reconstructions. All observed correlations are statistically significant and they are generally taken as evidence in support of the validity and reliability of the climate sensitivity parameters implied by the corresponding regression coefficients. The implied climate sensitivities are tabulated in Figure 1.
6. FIGURE 3 is a presentation of the tendency in time series field data to translate long term trends into faux correlations even in the absence of responsiveness at a time scale of interest. The first frame presents an example of such a spurious correlation taken from the Tyler Vigen collection of spurious correlations in time series data [LINK]. The bottom frame is a segment of a lecture on spurious correlations in time series data and their examination with detrended correlation analysis provided on Youtube by Alex Tolley [LINK].  These considerations imply that the correlation seen in the source data (Figure 1) may not be reliable and that they must be decomposed so that the part that derives from long term trends can be removed and only the part that derives from responsiveness at the time scale of interest is interpreted into the theory of causation.
7. FIGURE 4 presents the results of detrended correlation analysis that removes the effect of long term trends such that the correlation that survives into the detrended series more faithfully reflects the responsiveness of temperature to LN(CO2) at an annual time scale. In other words, detrended correlation is independent of the bias imposed by trends  in the correlation between source time series. Therefore they can be interpreted in terms of causal relationships. It is true that “correlation does not imply causation” but it is also true that no causation interpretation of the data can be made without correlation at the causation time scale. That is, detrended correlation at the appropriate time scale is a necessary but not sufficient condition for causation.
8. FIGURE 4: The table of detrended correlations shows that the strong and statistically significant correlations seen in the source time series (Figure 2) do not survive into the detrended series in the observational data (HAD, GIS, UAH, RSS) implying that the source correlations were creations of long term trends and not the responsiveness of temperature to changes in LN(CO2) at an annual time scale. A very different result is seen in the theoretical temperature series constructed with the GHG effect of atmospheric CO2 concentration (RCP8.5). Here, strong and statistically significant correlations survive into the detrended series in all eight calendar months and serve as evidence that temperature is responsive to LN(CO2) at an annual time scale.
9. FIGURE 4: COMPARATIVE ANALYSIS: In this comparative analysis, detrended correlations serve as the measure of the responsiveness of temperature to changes in atmospheric CO2. This measure is used to compare direct observations of global mean temperature and reconstructions of global mean temperature (where the presence of a GHG effect of atmospheric CO2 is being tested), with the theoretical series which serves as the benchmark of what we expect to see in the presence of the GHG effect of atmospheric CO2 concentration. The comparison under identical conditions shows no evidence in any of the eight calendar months studied of the responsiveness of direct temperature observations and the HadCRUT4 temperature reconstructions  (UAH, RSS, HAD) to changes in atmospheric CO2 concentration – a necessary condition for the existence of climate sensitivity and for the theory of anthropogenic global warming. A similar result for the GISTEMP global mean temperature reconstruction is also found but with the caveat that the detrended correlations are higher than in the other three observational data series with statistical significance observed in two of the eight calendar months studied (April and May). This unique feature of the NASA GISTEMP series may have implications for how these reconstructions were constructed and whether an assumption of GHG forcing played a role in their reconstruction.
10. CONCLUSION: The testable implication of the GHG theory is that surface temperature should be responsive to atmospheric CO2 concentration such that a detrended correlation exists between the logarithm of atmospheric CO2 and surface temperature at the time scale of interest. This test is carried out for five temperature series in eight calendar months for the sample period 1979-2018 using comparative analysis to test observational data against the theoretical series. The comparison does not show evidence of the existence of GHG forcing by atmospheric CO2 in the observational data. The assistance and encouragement provided by Mr. Ashley Francis of Salisbury, England in carrying out this work is gratefully acknowledged.

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