# 1. Background

Both natural and anthropogenic drivers cause changes in the Earth’s climate. Crowley [1] states that prior to the Industrial Revolution (pre -1850) 41 to 64% of decadal scale temperature variations were due to changes in solar radiation and volcanism.

Budyko [2] studied “The Effect of Solar Radiation Variations on the Climate of the Earth.” He concludes that secular variations of the mean temperature of the Earth can be explained by variations in short-wave radiation arriving at the surface. He found that comparatively small variations in atmospheric transparency could be sufficient for the development of quaternary glaciations.

Robock [3] wrote an extensive review article on “Volcanic Eruptions and Climate.” In a few days after an eruption, the amplitude of the diurnal cycle of surface air temperature is reduced under the cloud. On a much larger time scale, volcanic effects played a major role in climate change during the Little Ice Age.

After 1850, anthropogenic drivers of the climate (namely, CO_{2},
CH_{4} and N_{2}O) became steadily more important than natural climate
drivers. Crowley’s model indicates that only about 25% of the 20thcentury
temperature increase can be attributed to natural variability.

Working Group 1 [4]: The Physical Science Basis - of the IPCC
Fifth Assessment Report states that the warming of the climate system
is unprecedented since the 1950s. The linear trend line through the
HadCRUT4 data in figure 1 indicates a temperature increase of 0.8°C
from 1850 through 2014. The largest contribution to total radiation
forcing at the surface during the 20^{th} Century is caused by increased
atmospheric concentrations of CO_{2}.

In general, climate scientists look at global warming as a time series
of increasing temperature anomalies along with a separate time series
of increasing CO_{2}. Can changes in CO_{2} alone explain temperature
anomalies, both of which are treated as singular functions in time?
This process seems to be a fault in their analysis procedures.

To illustrate this point, figure 2 is a time series of CO_{2} increases since
(1850 – 2014) and Figure 1 shows the global temperature anomalies of
the Earth for the same period. Both graphs are singular functions in
time. It is evident from the graphs that CO_{2} increases seem to mirror
the variations in temperature. However, if the scale of Figure 2 was
changed, a good comparison would be lacking. Certainly both graphs
indicate increasing magnitudes in time, and since CO_{2} is a GHG, there
is a physical coupling. The logic appears to be sound.

# 2. Cross Correlation Formulation

Cross correlation is a standard method of estimating the degree
to which two series are correlated. Consider two series X(i) and Y(i)
where I = 0, 1, 2… N-1. The cross correlation coefficient **ρ** is defined as

**mX** and **mY** are means of their corresponding series and **σ _{X}** and

**σ**are their respective standard deviations. The denominator in Eq. (1) serves to normalize the correlation coefficients such that (

_{Y}**-1 <= ρ(X,Y)<= +1**), the bounds indicating maximum correlation and 0 indicating no correlation. Also Eq. (1) assumes the time delay is zero. The term

**R**is defined herein as the square of the correlation coefficient (

^{2}**ρ**) and is a measure of how much the variance of

^{2}**Y(i)**is explained by the variance of

**X(i)**.

Calculations utilized the CORREL function of MS Excel spread sheets. The CORREL function is identical to Equation (1).

# 3. Correlation of Temperature Anomalies with Increases of CO_{2}
Concentration

How much does an increases in CO_{2} concentrations explain the
record of global temperature warming? To determine this, data in
Figure 1 are cross correlated with the HadCRUT4 data of figure 2 over
the same period (1850 to 2014)

Calculations for global data indicate a correlation of **ρ = 0.691** and **R ^{2}
= 48%.**, for the Northern Hemisphere

**ρ = 0.683**and R

^{2}= 47% and for the Southern Hemisphere,

**ρ = 0.662**and

**R**.

^{2}= 44%
**R ^{2}** values indicate that less than 50% of the variance of global
temperatures is explained by the variance of CO

_{2}concentrations. These results are important in understanding the contribution of CO

_{2}to the global warming picture.

# 4. Concluding Remarks

Both anthropogenic and natural drivers of the Earth’s climate affect
the temperature anomalies of the HadCRUT4 data set. Thus, the total
variance (100%) may be explained by these two factors. In round
numbers, 50% of the variance is explained by CO_{2} increases, and
perhaps an additional 10% may be due to other greenhouse gases. The
40% of the variance that remains may be attributed to natural climate
drivers such as volcanic activity, solar variations, El Nino cycles,
coming out of the Little Ice Age, pollution particulates, etc. The 40%
value is in keeping with Crowley’s estimate of 25% for the 20^{th} century.

This partitioning of variances is illustrated in figure 3.

There is a lack of robust correlation between the increase in CO_{2}
emissions and global temperature rise. If global warming is caused
primarily by manmade increases in CO_{2}, why is there not a 0.90
correlation? In conclusion, evidence of warming is not evidence of
what causes the warming.

# Competing Interests

The author declare that there is no competing interests regarding the publication of this article.

# Acknowledgments

Thanks go to my daughter Mary Elizabeth who helped greatly in editing this manuscript.

# Abbreviations

GISS: Goddard Institute for Space Studies

MS: Microsoft Corporation

NASA: National Aeronautics and Space Administration

UK: United Kingdom

US: United States