This is an important issue for CO2 and the Greenhouse effect.
There are crucial differences between coincidence, correlation and causation these are vitally important to understand in the context of the Anointed's assertion: 'Anthropogenic emissions cause Climate Change', or put in more scientific language used by the IPCC 'The radiative forcing of CO2 increase since pre-industrial times is almost entirely responsible for observed waarming and will, if CO2 levels double (to 554ppm) cause surface temperaaturre to rise by between 2.5⁰C and 4⁰C.
Coincidence is a remarkable concurrence of events or circumstances that have no apparent causal connection with one another. From a statistical perspective, coincidences are inevitable and often less remarkable than they may appear at first glance. The coincidence remarked upon by Hansen and Bolin (see page 'What is Climate Change?') was that in the 70s, the temperature rise from pre-industrial times coincided with a rise in the level of CO2 in the atmosphere.
Correlation is a statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. However, in general, the presence of high correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation). In statistics, the level of correlation is enumerated by use of a correlation coefficient. The most common of these is the Pearson correlation coefficient, which is sensitive to linear relationships between two variables.
The statistical determination of the correlation (Pearson correlation coefficient) between temperature rise and the increase in CO2 is poor.
Coincidence does not imply causation in any circumstances
Very high correlation accompanied by a valid mechanism may well imply causation although, until experimentally deomonstrated or observationally confirmed, causation is not proven.
Considering how these requirements for causation apply in the two cases considered here:
At the fundamental level, the models of the Anointed (a logarithmic Arrhenius type response bolstered significaltly by the 'sensitivity' argument) are based on a coincidence.
The mechanism devised by Anointed modellers around the enhanced Greenhouse effect (as amplified) have so little correlation with reality (observed temperature variations) that the devised mechanism cannot possibly be valid.
This means this coincidence generated theory must be rejected in any proper scientific world - the models are bst described as a mere expression of a fiction from the assumed moral high position of the Climate Change Anointed. In the words of Thomas Sowell, 'they are based on a flawed, utopian vision of how things should be rather than empirical evidence and practical consequences of how things are in the real world'.
A way of thinking about the output of the Cardinal Model is that it results from the statistical practice of variance minimisation and therefore finds where correlation is very high - near perfect.
The empirically derived tanh(CO2) relationship is therefore the relationship where correlation is near perfect.
That enhanced Greenhouse effect (due to increased atmospheric CO2) is the actual operating mechanism is not proven by this perfect correlation. There may well be another explanation: the observed enhancement of the total Greenhouse effect may actually be caused by a different mechanism that also leads to increasing CO2.
However, that the Greenhouse effect is real is beyond contention so there is a high probability that the proposed mechanism is the underlying fundamental mechanism.
I can understand, qualitatively, how the tanh(CO2) relationship comes about: before saturation occurs the increase in Greenhouse effect will be marked as concentration increase. As progressively more CO2 absorption peaks achieve saturation, the only increase in absorption occurs in the broadening/overlap zone. I can even see a way to theoreticall prove a tanh() relationship for Clear Skies conditions. However, the complexities of Real Skies conditions makes any comprehensive theoretical approach impossible.
The Cardinal model gets around this theoretical hurdle by using empirical parameters to match the observed situation to CO2 (and others) concentration.
While not validating vibrational rotational absorption as the only mechanism, it is highly suggestive - certainly very much better than the coincidental observation that temperatures have increase and so has carbon dioxide that underpins the models of the Anointed.
CO2 is evenly distributed throughout the atmosphere with little geographic or altitudinal variation. The Global Monitoring Laboratory (Mauna Loa Laboratory, Hawaii) reports local CO2 and global average CO2. The Cardinal model uses the global average and there is a difference of up to 3ppm in the difference between global and local. However, the annual seasonality follows very similar trends. As noted, the Greenhouse Effect is, to a human observer, instantaneous. Therefore, even allowing for a degree of latency, the monthly temperatures should respond to the monthly variations in CO2.
This and other phenomena are discussed in following pages: Seasonality, Geographic Variability, Temperature Spikes & Longer-term.
However, this and other evidence suggests that there is a major effect of causality in the other direction: increasing temperatures cause increasing CO2.
The conclusions of the Cardinal model suggest such a small increment to the total Greenhouse effect that it may well be the causal direction.