INTRODUCTION
At an elapsed time of t years since 1750 (the start of the industrial revolution with the onset of the use of fossil fuels in vehicles and power plants), atmospheric CO_{2} concentrations, C(t) , increased along with increases in temperatures. Atmospheric measurements of C(t) were not available until 1958 at the Mauna Loa, HI, observatory of the National Oceanic and Atmospheric Administration (NOAA), which has provided the longest record of atmospheric measurements of the total CO_{2} initiated by Charles Keeling in 1958 at the Mauna Loa observatory (Keeling 1960). Based on our knowledge, the anthropogenic fossil component, C _{F} (t) , and non-fossil component, C _{NF} (t) , in C(t) have never been estimated by NOAA at its observatories or at any other observatory from atmospheric measurements of CO_{2} . Despite the lack of knowledge of the components of C(t) , claims have been made in the scientific literature (^{CSIRO 2014} ; ^{Rubino et al. 2013} , ^{2019} ) that all or most of the increase in C(t) since 1800 has been due to the anthropogenic fossil component, C _{F} (t).

Other atmospheric measurements of C(t) began in 2003 at the NOAA observatory in Niwot Ridge, including measurements of the three isotopes of carbon: ^{12} C, ^{13} C, and ^{14} C. Carbon-14 is a radioactive isotope of carbon having a half-life of 5,730 y. Carbon-14 atoms are produced in the atmosphere by interactions of cosmic rays, and they have reached an essentially constant steady state activity, i.e., disintegration rate, in the total world environment (^{Eisenbud and Gesell 1997} ). The age of fossil fuels is much longer than the 5,730 y half-life of the ^{14} C radioactive isotope; consequently, fossil fuels are devoid of the ^{14} C isotope. When the anthropogenic fossil component of CO_{2} is released to the atmosphere, the specific activity of ^{14} C_{,} S(t) in C(t) , decreases. The units of S(t) used in this paper are disintegrations per minute per gram of carbon abbreviated as dpm (gC)^{−1} , the common units used in ^{14} C dating. The ratio R _{S13} of the (^{13} C/^{12} C) atoms and the ratio R _{S14} of the (^{14} C/^{12} C) atoms at the Niwot Ridge observatory are used to calculate two statistics designated respectively in this paper as d13C and D14C , both of which are said to decrease when the anthropogenic fossil component, C _{F} (t) , increases in the atmosphere. As discussed later in Table 1 , values of the annual mean specific activity, S(t) , are calculated in this paper from annual mean values of the D14C statistic.

Both the d13C and D14C statistics represent 1,000 times the relative deviations of their respective (^{13} C/^{12} C) and (^{14} C /^{12} C) atom ratios from those of a 1950 standard (^{Karlen et al. 1964} ) when expressed in per mil, given by the symbol ‰. This magnification increases their underlying relative deviations and slopes in plots by a factor 1,000. While such amplification techniques often are useful for displaying very small changes in quantities of interest, the interpretation of such magnified changes must be attended with some care. In the cases of concern here, the resultant steep slopes in plots likely have led persons throughout the world to conclude that the anthropogenic component has dominated the increase of CO_{2} and caused global warming. We believe that both statistics have been misused to validate the anthropogenic fossil component, C _{F} (t) , as the major cause of the increase of C(t) .

Global carbon cycle and its effect on CO_{2} quantities
The global carbon cycle for CO_{2} is described by the Energy Information Administration (EIA 2020). Natural, two-way exchanges of CO_{2} occur between the atmosphere and its two exchange reservoirs, the oceans and terrestrial biosphere. Two-way exchanges with the atmosphere also occur from changes in land use. The ocean is the largest reservoir of CO_{2} , and it contains 50 times that for the atmosphere and 19 times that for the terrestrial biosphere (^{Water Encyclopedia 2005} ). All of the two way exchanges are considered in this paper to be comprised of both the non-fossil component and the anthropogenic fossil component. Annual changes, DC _{NF} (t) in C _{NF} (t), in the atmosphere relative to the 1750 initial value, C(0) , can be positive or negative depending on the net flow of CO_{2} between the atmosphere and its exchange reservoirs as well as on land use changes. A one-way pathway of anthropogenic fossil CO_{2} into the atmosphere from fossil fuel combustion and industrial fuel processes since 1750 is represented by annual emissions, DE(t) , of anthropogenic fossil CO_{2} to the atmosphere, which have been increasing each year since 1750. These emissions over time t result in increasing annual mean anthropogenic fossil concentrations, C _{F} (t) , that result in specific activities, S(t) , of ^{14} C in C(t) that are increasingly lower than the initial value, S(0) . This dilution of S(0) in C(0) in 1750 by the presence of C _{F} (t) in C(t) corresponds to what is described as the Suess effect (^{Suess et al. 1967} ).

Use of symbols, equations, and significant digital content
In this paper, it is important for the reader to be aware of the following use of symbols, equations, and supplemental digital content (SDC), https://links.lww.com/HP/A210 . All symbols for quantities in equations and the text are italicized. Symbols for expected values of a CO_{2} quantity calculated from an analytical equation are shown in brackets, < >; for example, expected specific activity, <S(t)> , and expected anthropogenic fossil component, <C _{F} (t)> , which is calculated from the value of <S(t)> . Other CO_{2} quantities are considered as random variables either estimated directly from measurements or calculated from other measured CO_{2} quantities. The standard use of sequential numbers in the text for equations is not employed. Equations used to calculate a quantity in a numbered column of a table have that number, but equations listed below a table are not numbered. Equations and their numbers are listed in the figures. Definitions for all symbols used in this paper are listed in the Appendix. Table 2 (https://links.lww.com/HP/A210 ) contains 8 pages of significant digital content of the values of CO_{2} quantities in 1750 through 2018 (^{Skrable et al. 2021} ). Table 2a contains a partial listing of the data in Table 2. All of the values for CO_{2} quantities in Table 2 are plotted in the figures.

Glacial–interglacial cycles
A NOAA link^{3} gives a description and (Fig. 1 ) of glacial-interglacial cycles from 350,000 y ago to the present. Present in the figure is defined as 1950 AD, the convention used in radiocarbon dating because H-bomb testing in the 1950s distorted radiocarbon ages in many materials (personal communication, Bauer 2021^{2} ).

Fig. 1: Glacial-interglacial cycles. Solar radiation varies smoothly through time (top, orange line) with a strong cyclicity of ~23,000 years, as seen in this time series of July incoming solar radiation at 65°N (Berger and Loutre 1991). In contrast, glacial–interglacial cycles last ~100,000 years (middle, black line) and consist of stepwise cooling events followed by rapid warmings, as seen in this time series inferred from hydrogen isotopes in the Dome Fuji ice core from Antarctica (Kawamura et al. 2007). Atmospheric CO_{2} measured from bubbles in Dome Fuji ice (bottom, blue line) shows the same pattern as the temperature time series (Kawamura et al. 2007). Yellow columns indicate interglacial periods.

The cause of glacial-interglacial cycles in the figure is stated to be due to variations in Earth's orbit through time, which changes the amount of solar radiation received by Earth. The figure includes three plots from the top to the bottom of the figure: (1) July insolation at 65° N (Wm^{−2} ), (2) temperature change from present (°C), and (3) atmospheric CO_{2} (ppm). The caption in the figure describes its content. It also is stated in the article: “The most recent glacial period occurred between about 120,000 and 11,500 years ago. Since then, Earth has been in an interglacial period called the Holocene.”

During the last long glacial period, the oceans absorbed a large amount of CO_{2} from the atmosphere. It appears in the figure that Earth is still in the Holocene interglacial period that started 11,500 y ago. Its peak temperature change over the 11,500 years, thus far in 1950, appears to be significantly less than those over the three previous interglacial periods. Its peak CO_{2} appears less than 300 ppm and less than the peak value in the previous interglacial period. Thus, the increase in CO_{2} that Earth has been experiencing since 1800 appears to have started more than 5,000 years ago.

Increase in ^{14} CO_{2} from high altitude nuclear bomb tests
A Wikipedia link^{4} for ^{14} C describes the increase in the concentration of ^{14} CO_{2} in the atmosphere that resulted from high altitude nuclear bomb tests, circa 1955–1963. Based on the figure in the Wikipedia link, ^{14} CO_{2} from the atmospheric bomb tests during this period would be significant in 1955 to about 2005. For the purpose of estimating the anthropogenic fossil and non-fossil components of CO_{2} , measurements of ^{14} C specific activities of atmospheric CO_{2} during this period should be corrected for the contribution from bomb tests. Outside of this period, no correction would be required. The Wikipedia link gives the world inventory of ^{14} C:

Global inventory: ~8500 PBq (about 50 tonnes):
Atmosphere: 140 PBq (840 kg);
From nuclear testing (till 1990): 220 PBq (1.3 tonnes); and
Terrestrial materials: the balance.
Thus, the portion of ^{14} CO_{2} from nuclear testing is about 2.6% of the global inventory, which now is mostly in the ocean reservoir. Some part of this 2.6% is expected to be released to the atmosphere, opposing the dilution effect from burned fossil fuels, although it is expected to be small in comparison to the dilution effect of fossil fuel CO_{2} .

A NOAA link^{5} about the bomb spike provides a plot of the D14C statistic, and the following statement is made: “By the 1980s, most of the “bomb” ^{14} C had been absorbed into the oceans and land biota, leaving slightly elevated levels in the atmosphere. Yet atmospheric ^{14} C levels continue to decrease–now because of fossil fuel CO_{2} emissions.”

In this paper, we calculate annual mean specific activities, S(t) , from annual mean values of the D14C statistic in 2004 through 2012, which are calculated from data obtained from NOAA (^{Lehman and Miller 2019} ). These S(t) values along with our chosen value, S(0) , in 1750 then are used in Table 1 to estimate expected specific activities, <S(t)> , in 1750 through 2018 from an approximation fitting function. Therefore, the <S(t)> values in this paper are not expected to be influenced significantly by ^{14} CO_{2} from nuclear bomb tests.

Table 1 -
Niwot Ridge, Colorado annual mean specific activity,

S(t) , and expected specific activity,

<S(t)> , of

^{14} C in

C(t) in 1750 through 2018.

^{a}

(1)
(2)
Time, t ,
^{14}
C
^{13}
C
S(t)
<S(t)>
Since
Decay
Fraction.
Specific
Specific
1750
D14C
Factor
d13C
Factor
R
_{
S13
}
R
_{
S14
}
Activity
Activity
Year
(years)
(‰)
F
_{
d
}
(‰)
F
_{
13
}
(13C/12C)
(14C/12C)
[dpm (gC)
^{−1}
]
[dpm (gC)
^{−1}
]
1750*
0
−
−
−6.50
0.9631
0.00000
−
16.33
16.33
1775
25
−
−
−
−
−
−
−
16.32
1800
50
−
−
−
−
−
−
−
16.31
1850
100
−
−
−
−
−
−
−
16.28
1900
150
−
−
−
−
−
−
−
16.25
1950
200
−
−
−
−
−
−
−
16.10
1975
225
−
−
−
−
−
−
−
15.80
2000
250
−
−
−
−
−
−
−
15.06
2004
254
64.61
0.9935
–8.348
0.9667
0.01114
1.304 × 10^{−12}
14.87
14.87
2005
255
58.21
0.9934
–8.245
0.9665
0.01114
1.296 × 10^{−12}
14.78
14.81
2006
256
55.57
0.9932
–8.311
0.9666
0.01114
1.293 × 10^{−12}
14.74
14.76
2007
257
50.91
0.9931
–8.294
0.9666
0.01114
1.287 × 10^{−12}
14.68
14.71
2008
258
46.93
0.9930
–8.321
0.9666
0.01114
1.283 × 10^{−12}
14.63
14.65
2009
259
43.22
0.9929
–8.324
0.9667
0.01114
1.278 × 10^{−12}
14.58
14.59
2010
260
38.88
0.9928
–8.369
0.9667
0.01114
1.273 × 10^{−12}
14.52
14.53
2011
261
33.77
0.9926
–8.415
0.9668
0.01114
1.267 × 10^{−12}
14.45
14.47
2012
262
29.99
0.9925
–8.425
0.9668
0.01114
1.262 × 10^{−12}
14.39
14.40
2018**
268
4.734
0.9918
–8.573
0.9671
0.01114
1.232 × 10^{−12}
14.05
13.96
Relative % Change =
−92.67
−0.17
2.70
0.05
−0.02
−5.51
−5.51
−6.07

^{a} Constants and equations:

R _{std13} ≡ (^{13} C/^{12} C) ratio = 0.011237 in 1950. R _{std14} ≡ (^{14} C/^{12} C) ratio = 1.176 × 10^{−12} in 1950.

k = 1.1404 × 10^{13} dpm (gC)^{−1} per unit (^{14} C/^{12} C) ratio.

F _{d} = e^{λ(200 − t)} , for t years after 1750 and λ = ln2 (5,730 y)^{−1} . F _{13} = [0.975(1 + 0.001 d13C )^{−1} ]^{2} .

R _{S13} = (1 + 0.001 d13C )R _{std13} . R _{S14} = (1 + 0.001 D14C ) R _{std14} (F _{d} F _{13} )^{−1} .

S(t) ≡ annual mean specific activity = k R _{S14} in units of dpm (gC)^{−1} .

S _{std14} ≡ standard specific activity = k R _{std14} = 13.41 dpm (gC)^{−1} in 1950.

<S(t)> ≡ expected specific activity of C(t) calculated from Equation (2) for the approximation fitting function, dpm (gC)^{−1} = 16.33-(10^{−22.82} ) t ^{9.542} –0.00049 t .

MATERIALS AND METHODS
The calculation of the annual mean CO_{2} components, C _{F} (t) and C _{NF} (t) , requires starting with estimates of the values present in 1750 (t = 0). The time t is calculated in years from 1750 to the year of the annual mean values of the CO_{2} components; for example, t = 1 y in 1751 and 200 y in 1950. The annual mean total concentrations, C(t) , along with their ^{14} C specific activities, S(t) , present in the atmosphere at time t years after 1750, are evaluated in this paper as follows. In each year, the values of C(t) and S(t) are compared to the initial value, C(0) , and its specific activity, S(0) , of ^{14} C present in 1750. The value of C(0) is assumed to be comprised of an unspecified natural background fossil component and natural non-fossil CO_{2} . The value of C(0) also is treated in our analysis as the initial, natural, non-anthropogenic fossil component, C _{NF} (0) . The comparisons yield estimates of the annual mean values of the anthropogenic fossil (F) component, C _{F} (t) , and the non-fossil (NF) component, C _{NF} (t) , present in C(t) each year. The two components are calculated from simple equations that account for the change, DC _{NF} (t) , each year in the non-fossil component, which results from the redistribution of all CO_{2} among its reservoirs, including the atmosphere, oceans, and terrestrial biosphere as well as from changes in land use. The sum of C(0) and DC _{NF} (t) equals the non-fossil component, C _{NF} (t) , present in the atmosphere at time t years after 1750. Although the initial value, C(0) , in 1750 has a background fossil component, such as emissions from volcanos, it is small in comparison to the anthropogenic fossil component (^{Gerlach 2011} ), and this natural background fossil component is not considered directly in the derivation of the equation for C _{F} (t) . Because the anthropogenic fossil component, C _{F} (t) , is assumed to be zero in 1750, the change, DC _{F} (t) , in the anthropogenic fossil component in the atmosphere in any year since 1750 is identical to the concentration, C _{F} (t) , present in that year. The specific activity, S(t) of C(t) , resulting from the presence of C _{F} (t) at time t has the same value calculated from the dilution of the initial specific activity, S(0) in C(0) , in 1750 by the presence of C _{F} (t) in C(t) at time t . Changes, DC _{F} (t) and DC _{NF} (t) are independent processes that automatically incorporate any redistribution of CO_{2} in its reservoirs and changes in land use. Thus, C(t) can be expressed: C(t) = DC _{F} (t) + DC _{NF} (t) + C(0) . The value of DC _{F} (t) varies each year only with the annual mean specific activity, S(t) in C(t) , in units of dpm (gC)^{−1} . The value of DC _{NF} (t) that occurs each year depends only on the change in the ^{14} C activity per unit volume of the atmosphere in units, for example, of dpm m^{−3} .

Equations used to calculate the values of the d13C and D14C statistics reported by NOAA
Annual mean values of the d13C and D14C statistics are expressed in the common unit of per mil represented by the symbol ‰. Values of both statistics are used in plots by NOAA and others to depict time trends in the anthropogenic fossil component. The NOAA d13C data (^{White et al. 2015} ) are used to calculate the annual mean d13C values listed in Table 1 along with applicable equations and constants listed below the table and discussed here. The NOAA data of D14C (^{Lehman and Miller 2019} ) are used to calculate the annual mean D14C values listed in Table 1 , which are used to estimate the annual mean S(t) values in C(t) in 2004 through 2012. Equations for both statistics are given at the NOAA website at a link^{6} for “Determining Delta Values” and at a link^{7} for “The Data: The Story Told from CO_{2} Samples” (NOAA).

Values of the d13C statistic are obtained by an equation containing the (^{13} C/^{12} C) atom ratios, R _{S13} in C(t) , for a sample, and the ratio, R _{std13} , for a 1950 standard:

$d13C=\left(\frac{{\left(\frac{13C}{12C}\right)}_{\mathit{\text{sample}}}}{{\left(\frac{13C}{12C}\right)}_{s\mathit{\text{tandard}}}}-1\right)1,000\u2030=\left(\frac{{R}_{S13}}{{R}_{\mathit{std}13}}-1\right)1,000\u2030=\left(\frac{{R}_{S13}-{R}_{\mathit{std}13}}{{R}_{\mathit{std}13}}\right)1,000\u2030.$

As indicated by the last equality, the reported value of the d13C statistic is actually a relative delta value multiplied by 1,000‰ and not simply the measured annual mean ratio, R _{S13} , of (^{13} C/^{12} C) atoms in C(t) as sometimes inferred incorrectly in the literature (^{CSIRO 2014} ). It also should be noted for the quantities shown in the second equality that the d13C statistic is a linear function of R _{S13} whose intercept of −1 and slope of (R _{std13} )^{−1} are multiplied by 1,000‰.

The annual mean ratios, R _{S13} , of (^{13} C/^{12} C) atoms in C(t) listed in Table 1 are calculated by the equation for d13C above rearranged: R _{S13} = (1 + 0.001 d13C ) R _{std13} , which is listed below Table 1 . All values in 2004 and afterward have the same value of 0.01114, which is slightly smaller than that of the 1950 standard ratio, R _{std13} , of 0.011237. Thus, it appears that changes in the d13C values from 2004 to 2018 reflect very small changes in the ratios, R _{S13} , of (^{13} C/^{12} C) atoms in C(t) . The slope of a plot of R _{S13} versus d13C values is 0.001 R _{std13} , which equals (0.001)(0.011237) or 0.000011237 per ‰ of d13C , which explains why the ratio, R _{S13} , is insensitive to even large changes in the d13C statistic.

Much of the discussion above for d13C and R _{S13} applies to D14C and R _{S14} . The factor F _{d} in Table 1 is added to the equation for d13C to account for the decay of ^{14} C atoms from the year of the annual mean value for C(t) and its D14C statistic to the year 1950 for the standard. The factor F _{13} in Table 1 is added to account for the fractionation of ^{13} C atoms relative to that of the 1950 standard, which has an assigned d13C value of −25‰ (^{Miller et al. 2013} ). Equations for these two factors are listed below Table 1 . The equation revised from that for d13C yields the annual mean value of the D14C statistic calculated from data provided by NOAA:

$D14C=\left(\frac{{R}_{S14}{F}_{d}{F}_{13}}{{R}_{\mathit{std}14}}-1\right)\mathrm{1,000}\u2030=\left(\frac{S\left(t\right){F}_{d}{F}_{13}}{{S}_{\mathit{std}14}}-1\right)1,000\u2030=\left(\frac{{R}_{S14}{F}_{d}{F}_{13}-{R}_{\mathit{std}14}}{{R}_{\mathit{std}14}}\right)1,000\u2030$

The latter equality shows that the D14C statistic also is a relative delta value multiplied by 1,000‰. The second equality is expressed in terms of the specific activities S(t) and S _{std14} calculated from the respective products of their ratios, R _{S14} and R _{std14} , and the conversion constant k listed below Table 1 . The factors F _{d} and F _{13} in Table 1 change very little from 2004 to 2012, and the first expression for D14C is approximated by a general linear equation of D14C versus R _{S14} multiplied by 1,000‰: y ≈ (A x + B )1,000‰, where y is D14C ; A is [F _{d} F _{13} (R _{std14} )^{−1} ], which approximates the slope of the linear equation before multiplying by 1,000‰; x is R _{S14} ; and B is −1 and the y intercept of the underlying linear equation. Thus, a plot of values of the D14C statistic versus R _{S14} , when expressed in per mil (‰), has the slope A and the y intercept, −1, both multiplied by a factor of 1,000 in this underlying linear equation. The values of R _{S14} and S(t) for anthropogenic fossil CO_{2} are zero; therefore, its D14C value is −1,000‰. Annual mean D14C values for all applicable air sampling events in each year for the Niwot Ridge site are used in this paper to calculate the annual mean ratios, R _{S14} , of (^{14} C/^{12} C) atoms. Annual mean ratios, R _{S14} , of (^{14} C/^{12} C) atoms in C(t) listed in Table 1 are calculated by rearrangement of the equation for D14C : R _{S14} = (1 + 0.001 D14C ) R _{std14} (F _{d} F _{13} )^{−1} , which is listed below Table 1 . It is noted that R _{S14} is the annual mean value in C(t) and in D14C in the year of the measurement and not the year 1950 for the standard ratio, R _{std14} . Thus, the measured value of R _{S14} is independent of all three factors to the right of the bracketed expression for R _{S14} , as well as the value for the d13C statistic in the equation for the F _{13} factor listed below Table 1 . Values of R _{S14} and R _{std14} listed in Table 1 are multiplied by the constant factor k listed below Table 1 to obtain their listed respective specific activities S(t) and S _{std14} . The value for the factor k is 1.1404 × 10^{13} dpm (gC)^{−1} per unit (^{14} C/^{12} C) atom ratio. It is calculated from the reported half-life of 5,730 y for ^{14} C, a constant percent ^{12} C abundance of 98.897, and a constant atomic weight of carbon of 12.0111 g mole^{−1} . The values of the latter two quantities in 2004 through 2012 vary slightly in their sixth significant digit because of a very slight decrease in the R _{S13} ratio. These very small changes have no practical significance on the results in this paper, which apply to an essentially constant value of the factor k from 1750 through 2020.

Process used to obtain expected specific activities, <S(t)> , in 1750 through 2018
NOAA has provided limited D14C data for the Niwot Ridge, CO, observatory beginning in 2004 and ending in 2012; these data have been used in this paper to calculate nine annual mean input specific activities, S(t) , in Table 1 . Despite the fact that the D14C program has continued, we have been unable to obtain from NOAA the data for 2013 through 2020. To make up for the limited D14C data, a process is used to estimate expected specific activities, <S(t)> , from an approximation fitting function applicable to the years 1750 through 2018. The process used to estimate the parameter values for the approximation fitting function is described as follows.

The solid trend line in Fig. 2 is obtained from the approximation fitting function estimated from 10 input-specific activities, S(t) , including our chosen value, S(0) , of 16.33 dpm (gC)^{−1} in 1750 and the 9 input S(t) values in 2004 through 2012. The equations listed at the bottom of Table 1 are used in sequence in the table to calculate the 9 input-specific activities, S(t) . The approximation fitting function is used to estimate the annual mean expected specific activities, <S(t) >, in 1750 through 2018 for the Niwot Ridge site. Naturally, the lack of data points between 1750 and 2004 makes the exact shape of the curve uncertain in this interval. Parameter values are chosen so that the <S(t) > values yield a fit that does not deviate significantly from the points for the input S(t) values, most importantly the value of 14.87 dpm (gC)^{−1} for S(t) in 2004 and our chosen S(0) value of 16.33 dpm (gC)^{−1} in 1750. Parameter values, however, have been chosen so that the 10 points for the S(t) values and the solid line for the expected <S(t) > values in the figure provide a realistic representation of the initial small decreases and later more significant decreases in the specific activities expected after the start of the industrial revolution in 1750 and the increased burning of fossil fuels each year. The S(t) value of 14.05 dpm (gC)^{−1} for 2018, based on extrapolated values of D14C and d13C , is 0.09 dpm (gC)^{−1} more than the expected <S(t) > value of 13.96 dpm (gC)^{−1} in 2018.

Fig. 2: Niwot Ridge, Colorado annual mean specific activity, S(t) , and expected specific activity,<S(t) >, of ^{14} C in C(t) in 1750 through 2018.

Methodology used to calculate C _{F} (t) and C _{NF} (t)
The methodology used to calculate C _{F} (t) and C _{NF} (t) relies on two accepted facts: (1) the initial total mole fraction C(0) of (280 ± 10) ppm before 1750 has been essentially constant for several thousand years (^{Prentice et al. 2018} ) and (2) the production rate of ^{14} C atoms in the atmosphere has been essentially constant for at least 15,000 years (^{Eisenbud 1997} ). Therefore, the steady-state activity of ^{14} C per unit volume of the atmosphere also would have been constant except for the redistribution of CO_{2} in the atmosphere in each year with its exchange reservoirs. The product C(t) S(t) is proportional to the activity per unit volume of the atmosphere, which varies each year depending on whether there is a net input or output, DC _{NF} (t) , of non-anthropogenic fossil CO_{2} in the atmosphere. The change in the product, DC _{NF} (t) S(t) , each year is independent of the value of C _{F} (t) in the atmosphere because it contains no activity of ^{14} C. Also, except for the dilution of S(0) by the anthropogenic fossil component, C(t) , present in the atmosphere each year, the ^{14} C specific activity, S(t) , would have remained constant at our chosen initial value, S(0) , of 16.33 dpm (gC)^{−1} in 1750. This dilution effect is noted as the Suess effect (^{Suess 1967} )^{8} .

The value for the initial specific activity, S(0) , of 16.33 dpm (gC)^{−1} is based upon the following information in the literature. For the average cosmogenic production rate of ^{14} C, we have adopted a value of 2.15 atoms cm^{−2} s^{−1} , based upon the average of the range limits of 1.8 and 2.5 atoms cm^{−2} s^{−1} (^{Mak et al. 1999} ) cited by ^{Taylor et al. (2014)} . Additionally, we have used the value of 7.9 gC cm^{−2} for the areal density of carbon at Earth’s surface, which considers contributions from the atmosphere, terrestrial biosphere, terrestrial humus, marine biosphere, and inorganic ocean carbon (^{Craig 1957} ). The quotient of the ^{14} C atom production rate of 2.15 atoms cm^{−2} s^{ − 1} by the areal density of 7.9 gC cm^{−3} yields our estimate of the steady state value for the specific activity S(0) of 0.2722 dps (gC)^{−1} or 16.33 dpm (gC)^{−1} . The product, C(0)S(0)K , of respectively 280 ppm, 16.33 dpm (gC)^{−1} , and the constant K of 4.996 × 10^{−4} in units of gC m^{−3} (ppm)^{−1} for NTP conditions, yields 2.284 dpm m^{−3} for S(0) in 1750. Our chosen value of 16.33 dpm (gC)^{−1} for S(0) is larger than the value of 13.41 dpm (gC)^{−1} for the 1950 standard and somewhat larger than values reported in the literature for ^{14} C dating.

Derivation of equations for the components of CO_{2}
In the derivation of equations for all components of CO_{2} and their changes since 1750, the annual mean total CO_{2} concentrations, C(t) , along with their ^{14} C specific activities, S(t) , are compared to the initial value, C(0) , and its specific activity, S(0) , present in 1750. Two logic pathways can be used to derive equations for DC _{NF} (t) , C _{NF} (t) , and C _{F} (t) :

Pathway 1: The statement for the change, K DC _{NF} (t) S(t) , of the ^{14} C activity per unit volume of the atmosphere at time t is: K DC _{NF} (t) S(t) = K C(t) S(t) - K C(0) S(0) , which yields:

DC _{NF} (t) = [C(t) S(t) - C(0) S(0)] S(t)^{-1} →

C_{NF} (t) = C(0) + DC_{NF} (t) →

C_{F} (t) = C(t) - C_{NF} (t).

Pathway 2: The equation for DC _{NF} (t) above gives an expression for the specific activity, S(t) :

S(t) = C(0) S(0) [ C(t) − DC_{NF} (t)]^{−1} ,

which reduces to the fundamental equation representing the specific activity, S(t) in C(t) , due to the dilution of S(0) in C(0) by the presence of C _{F} (t) in C(t) :

S(t) = C(0) S(0) [ C(t) − DC_{NF} (t)]^{−1} = C(0) S(0) [ (C_{F} (t) + C_{NF} (t)) − (C_{NF} (t) − C(0))]^{−1} ; thus,

S(t) = C(0) S(0) [C(0) + C_{F} (t)]^{−1} .

This fundamental equation for S(t) , based on the Suess effect, yields alternative equations for C _{F} (t) , C _{NF} (t) , and DC _{NF} (t) :

${\mathrm{C}}_{\mathrm{F}}\phantom{\rule{0.12em}{0ex}}\left(\mathit{t}\right)=\left[\right(\mathrm{S}\left(0\right)\mathrm{S}{\left(\mathit{t}\right)}^{-1}-1]\mathrm{C}\left(0\right)\mathbf{\to}$

${\mathrm{C}}_{\mathrm{NF}}\left(\mathit{t}\right)=\mathrm{C}\left(\mathit{t}\right)-{\mathrm{C}}_{\mathrm{F}}\left(\mathit{t}\right)\mathbf{}\mathbf{\to}$

${\mathrm{DC}}_{\mathrm{NF}}\left(\mathit{t}\right)={\mathrm{C}}_{\mathrm{NF}}\left(\mathit{t}\right)-\mathrm{C}\left(0\right).$

A limited number of D14C values in 1750 through 2018 are available from NOAA for calculating the specific activities, S(t) . Values for expected specific activities, <S(t) > in 1750 through 2018, are estimated in Table 1 from eqn 2, the approximation fitting function described previously. When <S(t) > is substituted for S(t) for pathway 2 above, the equations are:

$<{C}_{\mathit{F}}\phantom{\rule{0.13em}{0ex}}\left(t\right)>=\left[S\left(\mathit{0}\right)<S\left(t\right)>{\phantom{\rule{0.25em}{0ex}}}^{-1}-1\right]C\left(\mathit{0}\right)\mathbf{\to}\phantom{\rule{0.25em}{0ex}}$

${\mathit{C}}_{\mathit{NF}}\left(t\right)=C\left(t\right)-<{C}_{F}\left(t\right)>\mathbf{\to}\phantom{\rule{0.25em}{0ex}}$

${\mathit{DC}}_{\mathit{NF}}\left(t\right)={C}_{\mathit{NF}}\left(t\right)-C\left(\mathit{0}\right),$

where the value for the anthropogenic fossil component now is its expected value, <C _{F} (t) >. These equations are used in Table 2 to calculate <C _{F} (t) >, C _{NF} (t) , and DC _{NF} (t) , and they are displayed in Fig. 3 and Fig. 4 . The equation for <C _{F} (t) > is independent of DC _{NF} (t) and C(t) , and <C _{F} (t) > only varies over time t with the expected specific activity, <S(t) >, in C(t) . It is noted that the equation for <C _{F} (t) > yields smaller values if smaller values of either C(0) or S(0) are selected.

Fig. 3: DC(t) , <S(t) >, <C _{F} (t) >, DC _{NF} (t) , and DE(t) vs. time t .

Fig. 4: C(t) , <S(t) >, <C _{F} (t) >, C _{NF} (t) and DE(t) vs. time t .

RESULTS
The number of digits shown for quantities in cells of the tables often are less than those present; therefore, the reader will not be able to verify all listed quantities. The approximation fitting function listed below Table 1 is used to calculate the expected specific activities, <S(t) > in C(t) , listed in Table 2 and in Table 2a . Values of <S(t) > and C(t) are used in Table 2 and Table 2a in calculating the expected anthropogenic fossil component, <C _{F} (t) >, which then is used along with the total CO_{2} concentration, C(t) , to calculate the non-fossil component, C _{NF} (t) , and other CO_{2} quantities listed in the tables and plotted in Fig. 3 through Fig. 5 .

Table 2a -
Partial world atmospheric CO

_{2} , its

^{14} C specific activity, anthropogenic fossil and nonfossil components, and emissions (1750-2018).

^{a}

Time t
since
(2)
(3)
(4)
(5)
(6)
(7)
1750
C(t)
<S(t)>
<C
_{
F
}
(t)>
C
_{
NF
}
(t)
DC
_{
NF
}
(t)
DC(t)
DE(t)
Year
(years)
(ppm)
(dpm (gC)^{-1} )
(ppm)
(ppm)
(ppm)
(ppm)
(bill. met. tons)
1750
0
276.44
16.33
0.00
276.44
0.00
0.00
0.00
1751
1
276.40
16.33
0.01
276.39
−0.05
−0.04
0.01
1800
50
281.86
16.31
0.42
281.44
5.00
5.42
0.03
1850
100
287.06
16.28
0.84
286.22
9.78
10.62
0.20
1900
150
296.03
16.25
1.40
294.63
18.19
19.59
1.96
1950
200
311.54
16.10
4.03
307.51
31.07
35.10
5.83
2000
250
367.43
15.06
23.39
344.04
67.60
90.99
24.53
2018
268
405.40
13.96
46.84
358.56
82.12
128.96
36.22

^{a} Note: The sum of DE(t) values each year through 2018 is 1.590 × 10^{18} g. In 2018, <C _{F} (t)> of 46.84 ppm present in the atmosphere corresponds to a total mass estimated as 3.664 × 10^{17} g or about 23% of the total emissions. Thus in 2018, 77% of the total emissions is in the atmosphere's exchange reservoirs (see text).

Fig. 5: <S(t) > and < C _{F} (t) > versus DE(t) .

DISCUSSION
Discussion of results in Table 1 and Fig. 2
The primary purpose of Table 1 and Fig. 2 is the estimation of the expected specific activities, <S(t) >, listed in Table 2 (https://links.lww.com/HP/A210 ) and Table 2a, but other unrelated results are provided and discussed here because of their relevance to the primary results and conclusions. The content of Table 1 and Fig. 2 has been discussed in Results and other sections above. Relative % changes in all CO_{2} quantities in 2004 to those in 2018 are listed below Table 2a .

The D14C value of 4.734‰ in Table 1 for 2018 is calculated from a linear fit to the values in 2004 through 2012: D14C = −4.2148 t + 1134.3; R ^{2} = 0.9969.

The d13C value of −8.573‰ in Table 1 for 2018 is an extrapolated value calculated from a linear fit to the values in 2004 through 2014:

$d\mathit{1}\mathit{\text{3C}}=-0.026\phantom{\rule{0.25em}{0ex}}t-1.6052;{R}^{\mathit{2}}=\phantom{\rule{0.25em}{0ex}}0.9648.$

The ratio of the slope in the equation for D14C to that in the equation for d13C is 162, which represents the relative sensitivity of the D14C statistic to that of the d13C statistic for detecting increases in the anthropogenic fossil component from one year to the next.

Relative % changes listed below Table 1 are discussed as follows.

The D14 statistic has the largest relative change of −92.67%. This value reflects the excellent sensitivity of D14C for detecting changes from one year to the next, including those of the anthropogenic fossil component, <C _{F} (t) >, which is reflected by the −5.51% change in the values of R _{S14} and S(t) and by the −6.07% change in the <S(t) > value calculated from the approximation fitting function.
The discrepancy of 0.56% between the % changes for the S(t) and <S(t) > values is due to the discrepancy of 0.09 dpm (gC)^{−1} between the S(t) and <S(t) > values in 2018. This discrepancy of 0.09 dpm (gC)^{−1} is 0.64% of the <S(t) > value of 13.96 dpm (gC)^{−1} .
The largest difference between the S(t) and < S(t) > values in 2004 to 2012 is 0.03 dpm (gC)^{−1} .
The value of 16.33 dpm (gC)^{−1} for our chosen value of S(0) is the same as the expected value <S(0) >.
The d13C statistic has a relative change from 2004 to 2018 of 2.70%, which is much more than the −0.02% change in R _{S13} .
All listed R _{S13} values in 2004 through 2018 are 0.01114. This value is not much different from the value of 0.01116 in 1750 calculated from the d13C value of −6.50‰ (^{Rubino et al. 2013} ).
Discussion of results in Table 2, Table 2a, and Fig. 3 through Fig. 5
Table 2 (https://links.lww.com/HP/A210 ) contains a listing of all the same quantities as those in Table 2a . Each table contains a listing of seven annual mean CO_{2} quantities in 1750 through 2018. The symbol C(t) and the numbers (2) through (7) heading other columns are used as labels for the plots of all CO_{2} quantities in Figs. 3 , 4 , and 5 . The plot for C(t) is labeled as such. The numbers are placed before the symbols in legends, equations, and at the end of a line for a plot in each figure. Equations listed in the figures are used to calculate the quantities listed in columns (2) through (6) of each table. It is important to recognize in the following discussion that the value for <C _{F} (t) > is identical to its annual change, D <C _{F} (t)> since 1750 and that the sum of D <C _{F} (t)> , which is not specifically listed, and the annual change, DC _{NF} (t) , in the non-fossil component equals the annual change, DC(t) , in the total CO_{2} concentration, C(t) .

Results in Table 2 are plotted in the figures versus the time t since 1750, and they are discussed along with plots in Figs. 3 as follows. In 1750 (t = 0), the initial value of 276.44 ppm listed for both C(0) and C _{NF} (0) in Table 2a has been estimated from the Table 2 values of C(t) in 1751 and in 1752 of, respectively, 276.40 ppm and 276.36 ppm. Except for these values and the initial S(0) value of 16.33 dpm (gC)^{−1} , the initial values of other quantities are assumed to be zero. In 1751, a value of 0.01 ppm is listed for C _{F} (t) , and slightly negative values of −0.05 ppm and of −0.04 ppm are listed respectively for DC _{NF} (t) and DC(t) in Table 2 and Table 2a , both of which do not become positive until 1765 and thereafter. These negative values may be associated with the Little Ice Age^{9} when temperatures and CO_{2} concentrations coincidentally had their lowest values in about 1750. Values for the anthropogenic fossil component, <C _{F} (t) >, in Table 2a and in plot 3 of Fig. 3 slowly increase from 0.01 ppm in 1751 to a value of 4.03 ppm in 1950. This <C _{F} (t) > value of 4.03 ppm is 13.0% of the DC _{NF} (t) value of 31.07 ppm and 11.5% of the DC(t) value of 35.10 ppm, which equals the sum of <C _{F} (t) > and DC _{NF} (t) . After 1950, values of <C _{F} (t) >, DC _{NF} (t) , and DC(t ) in Fig. 3 begin to increase rapidly. The increase in <C _{F} (t) > coincides with the rapid increase after 1950 in the values of the annual emissions, DE(t) , of anthropogenic fossil CO_{2} in Table 2 and Table 2a as well as in plot (7) in Fig. 3 . Coincidentally, for the scale used on the primary vertical axis, the values for plot (3) of <C _{F} (t) > and the plot (7) of DE(t) overlap until about 1900. The values in plot (5) of DC _{NF} (t) and in plot (6) of DC(t) overlap each other early on, and then their plots slowly separate until 1950. After 1950, plots of both quantities increase more rapidly and their separation becomes greater each year. This separation is the value of <C _{F} (t) >. In Table 2a in 2018, <C _{F} (t) > is 46.84 ppm and DC _{NF} (t) is 82.12 ppm. Their sum of 128.96 ppm equals the value of DC(t) .

The following discussion applies to quantities plotted in Fig. 4 . The total CO_{2} concentration, C(t) , equals the sum of the value in plot (4) for C _{NF} (t) and in plot (3) for <C _{F} (t) >. Values in plot (2) are for the expected specific activity of ^{14} C, <S(t) >. Equations used for calculating <S(t) >, <C _{F} (t) >, and C _{NF} (t) are listed in the figure. Two plots are provided for the annual CO_{2} emissions reported by the Environmental Information Agency (EIA 2020) listed in Table 2 (Table SDC 1, https://links.lww.com/HP/A210 ) and designated here as DE(t) . Plot 7 for DE(t) uses the scale of the primary vertical axis on the left in Fig. 4 . The unnumbered plot designated as “DE(t) plot with scale of EIA on secondary axis” uses the scale on the right. This same scale is used for the EIA plot of “CO_{2} emissions” in the figure provided in the EIA reference. Consequently, the plot of DE(t) values using the scale of the secondary vertical axis appear here to be 450(40)^{−1} or 11.5 times the values in plot 7. Also, the slopes of the DE(t) plot of EIA at each time t are 11.5 times those in plot (7). Absent plot (4) for C _{NF} (t) and plot (3) for <C _{F} (t) >, one might falsely conclude that the increase in the total CO_{2} concentration, C(t) , in its plot after 1950 (t = 200 y) is due entirely to the increase in the anthropogenic fossil component, C _{F} (t) .

The following discussion applies to quantities plotted in Fig. 5 . Displayed in the figure are plot (2) of <S(t) > and plot (3) of <C _{F} (t) > versus DE(t) ; equations for their second order polynomial fits including R ^{2} values; and points for their annual mean values in 1750 through 2018. The y intercept in plot 2 of S(t) is 16.293 dpm (gC)^{−1} , which is slightly less than the <S(0) > value of 16.33 dpm (gC)^{−1} listed in Table 2a . The y intercept in plot 3 of <C(t) > is zero. The sensitivities are the slopes of the plots of each quantity, and they represent their changes per unit of DE(t) of 1 billion metric tons. These are estimated by the equations displayed for the rates of change of <S(t) > and <C _{F} (t) > with respect to DE(t) . Both the magnitude and slope of the plot for <S(t) > decrease very little, while those for the plot of <C _{F} (t) > increase much more over the entire range of DE(t) from 0 in 1750 to 36.22 billion metric tons in 2018.

Anthropogenic fossil CO_{2} in 2018 relative to the total of its annual emissions
The Table 2a value of the atmospheric concentration, <C _{F} (t) >, of anthropogenic fossil derived CO_{2} in 2018 is 46.84 ppm. It is assumed that <C _{F} (t) > is representative of its molar concentration throughout the entire atmosphere and that the total mass of the atmosphere is 5.148 × 10^{18} kg (^{Trenberth and Smith 2005} ) with an effective molecular weight of 28.96 g mole^{−1} . The number of moles, N _{F} , of anthropogenic fossil CO_{2} present in the atmosphere then is calculated:

${N}_{F}=\left(46.84\times {10}^{-6}\right)\left(5.14\text{8 x}{10}^{21}\mathrm{g}\right){\left(28.96\phantom{\rule{0.25em}{0ex}}\mathrm{g}-{\text{mole}}^{-1}\right)}^{-1}=8.326\times {10}^{15}\phantom{\rule{0.25em}{0ex}}\text{moles}.$

Based on a molecular weight of 44.01 g mole^{−1} for CO, the total mass of anthropogenic fossil CO_{2} present in the atmosphere in 2018 is calculated as 3.664 × 10^{17} g. The Table 2 value of 1,589.86 billion metric tons of anthropogenic fossil-derived CO_{2} emitted into the atmosphere in 1751 through 2018 (^{EIA 2020a} and ^{2020b} ) represents 1.590 × 10^{18} g. The inference is that the quantity of anthropogenic fossil CO_{2} in the atmosphere in 2018 represents about 23% of the total amount of anthropogenic fossil-derived CO_{2} that had been released to the atmosphere since 1750.

Therefore, 77% of the total anthropogenic fossil emissions of CO_{2} then would be present in the atmosphere’s exchange reservoirs in 2018. These results differ significantly from those reported by others: “Using these data, Keeling was able to compare the amount of CO_{2} accumulating in the atmosphere against estimates of the amount of CO_{2} being released by burning fossil fuels. The atmospheric fraction appeared to be approximately 55%, meaning that roughly half of all CO_{2} released by coal, oil, and natural gas was remaining in the atmosphere, thus causing the Keeling Curve’s annual rise” (^{Harris 2010} ).

Values of *S(t) and *D14C statistics required to support claims of the dominance of C _{F} (t)
Larger reductions in values for the annual mean <S(t) > and D14C statistics are required to support claims that the increase, DC(t) , in 2018 (t = 268 y) above C(0) in 1750 (t = 0) has been dominated by or is equal to the anthropogenic fossil component, C _{F} (t) , in 2018. Quantities of CO_{2} in 1750 and 2018 are as follows. Table 2a includes C(0) of 276.44 ppm; S(0) of 16.33 dpm (gC)^{−1} ; C(t) of 405.40 ppm; DC(t) of 128.96 ppm; <S(t) > of 13.96 dpm (gC)^{−1} ; and <C _{F} (t) > of 46.84 ppm, which is a fraction F of 0.3632 of DC(t) . Table 1 includes in 2018: D14C of 4.734‰; F _{d} of 0.9918; F _{13} of 0.9671; and S _{std14} of 13.41 dpm (gC)^{−1} . The equation for the specific activity based on the Suess effect is expressed: S(t) = S(0) [1 + CF(t)C(0) ^{−1} ]^{−1} . Equations for calculating *S(t) and *D14C values required for C _{F} (t) in 2018 to equal a given fraction, F , of DC(t) are expressed:

$*S\left(t\right)=\frac{S\left(0\right)}{1+\left(\frac{\mathit{FDC}\left(t\right)}{C\left(0\right)}\right)},\mathit{\text{whose value gives}}:*D14C=\left(\frac{*S\left(t\right){F}_{d}{F}_{13}}{{S}_{\mathit{std}14}}-1\right)1,000\text{\u2030}$

To achieve an F value of 0.5, *S(t) would have to be 13.24 dpm (gC)^{−1} instead of 13.96 dpm (gC)^{−1} , and *D14C would have to be −52.89‰ instead of 4.734‰. Also C _{F} (t) , DC _{NF} (t) , and C _{NF} (t) would have values, respectively, of 64.48 ppm, 64.48 ppm, and 340.92 ppm. To achieve an F value of 1, *S(t) would have to be 11.14 dpm (gC)^{−1} instead of 13.96 dpm (gC)^{−1} , and *D14C would have to be −203.5‰ instead of 4.734‰. Also, C _{F} (t) would have the value of DC(t) of 128.96 ppm, DC _{NF} (t) the value of 0, and C _{NF} (t) the value of C(0) of 276.44 ppm. These results, especially those for D14C , negate the claims that all or most of the increase of CO_{2} since 1750 has been due to the burning of fossil fuel.

CONCLUSION
Results in this paper and citations in the scientific literature support the following 10 conclusions.

The scientific literature does not appear to provide estimates of either the annual mean values of the anthropogenic fossil component, C _{F} (t) , or of the non-fossil component, C _{NF} (t) , present in the total atmospheric CO_{2} concentration, C(t) , nor their respective changes from values in 1750.
The annual mean values of all CO_{2} quantities provided in this paper automatically account for the redistribution of CO_{2} among its reservoirs, including all of its isotopic forms. Results depend on chosen values in 1750 of 276.44 ppm for C(0) and 16.33 dpm (gC)^{−1} for S(0) , both of which may be somewhat overestimated as indicated in the text. Based on the simple equations used to calculate all CO_{2} quantities, smaller values for these chosen quantities would yield smaller values of the anthropogenic fossil component, C _{F} (t) , and larger values of the non-fossil component, C _{NF} (t) .
In 1950, the <C _{F} (t) > value of 4.03 ppm in Table 2a is 1.29 % of C(t) and 11.48% of the increase, DC(t), of 35.10 ppm since 1750. After 1950, values of the two components of C(t) begin to increase rapidly, and this increase continues through 2018. This rapid increase, however, is not triggered by the greenhouse effect and global warming associated with either the 1950 value of 4.03 ppm for C _{F} (t) or the relatively small increase in the annual change, DC _{NF} (t) , of 31.07 ppm in the non-fossil component, which is 88.5% of the DC(t) value of 35.10 ppm. This DC _{NF} (t) value of 31.07 ppm in 1950 results from the annual redistribution of CO_{2} among its reservoirs, primarily a net release of CO_{2} from the oceans due to increases in temperatures from solar insolation in 1950 and afterwards.
In 2018, the <C _{F} (t) > value of 46.84 ppm is 11.55% of the C(t) value of 405.40 ppm, 36.32% of the DC(t) value of 128.96 ppm, and 57.04% of the DC _{NF} (t) value of 82.12 ppm. These results negate claims that the increase, DC(t) , in C(t) , since 1750 has been dominated by the increase of the anthropogenic fossil component, C _{F} (t) .
In 2018, the total content of anthropogenic fossil CO_{2} in the atmosphere is estimated as 3.664 × 10^{17} g, which is 23% of the total emissions of 1.590 × 10^{18} g since 1750. Thus, in 2018, 77% of the total emissions is estimated to be present in the atmosphere’s exchange reservoirs.
Claims of the dominance of the anthropogenic component, C _{F} (t) , in the increase of the CO_{2} concentration, C(t) , first began in 1960 with: “Keeling Curve: Increase in CO_{2} from burning fossil fuel” (^{Rubino 2013} ). Despite the lack of knowledge of the two components of C(t) , these claims have continued in the scientific literature.
Calculated much larger reductions in values for the annual mean D14C statistic and other reductions in the <S(t) > statistic in Table 1 are required to support the claims that the increase of the total concentration, C(t) above C(0) in 1750 has been dominated by or is equal to the increase in the anthropogenic fossil component, C _{F} (t) . These results negate the claims of the dominance of C _{F} (t) in the increase of C(t) .
Claims of the dominance of the anthropogenic fossil component have involved (a) the misuse of the d13C and D14C statistics to validate these claims when they are expressed in the common unit of per mil (‰), which causes their slopes in plots to be magnified by a factor of 1,000 above what they otherwise would be; (b) the plot of decreasing values of the d13C statistic along with the plot of increasing values of the total CO_{2} concentration, C(t) , in the same figure on different vertical axes to infer or claim that the increase in C(t) is due mostly or entirely to the anthropogenic fossil component, C _{F} (t) , even though large reductions of the d13C statistic reflect no significant changes in its (^{13} C/^{12} C) atom ratios and bear no significant relationship with increases in the anthropogenic fossil component, C _{F} (t) ; and (c) EIA plot of the CO_{2} concentration, C(t) , in the same figure with the plot of the annual emissions, DE(t) , of anthropogenic fossil-derived CO_{2} on different vertical scales, which provides the inference that the anthropogenic fossil component, C _{F} (t) , is responsible for the increase in C(t) when a plot of the non-fossil component, C _{NF} (t) , and of the anthropogenic fossil component, C _{F} (t) , are not included in the EIA figure.
An article on Glacial-Interglacial Cycles (NOAA) suggests that recent increases in CO_{2} and temperatures are due primarily to cyclic changes of solar radiation associated with Earth’s orbit about the sun. The annual change, DC _{NF} (t) , in the non-fossil component has positive increasing values in Table 2 (https://links.lww.com/HP/A210 ) after 1764. It will eventually become negative in the next glacial period when average temperatures decrease again as they have done over all of the previous glacial-interglacial cycles.
The assumption that the increase in CO_{2} since 1800 is dominated by or equal to the increase in the anthropogenic component is not settled science. Unsupported conclusions of the dominance of the anthropogenic fossil component of CO_{2} and concerns of its effect on climate change and global warming have severe potential societal implications that press the need for very costly remedial actions that may be misdirected, presently unnecessary, and ineffective in curbing global warming.
APPENDIX
Definitions of symbols, constants, and standards for CO_{2} quantities are listed alphabetically as follows at time t years after 1750 along with initial values in 1750 and values for 1950 standards.

C(0) ≡ initial total concentration of atmospheric CO_{2} in 1750 = 276.44 ppm.

C(t) ≡ total concentration of atmospheric CO_{2} at time t, ppm.

C _{F} (0) ≡ initial anthropogenic fossil component present in C(0) = 0.

C _{F} (t) ≡ anthropogenic fossil component present in C(t) at time t , ppm.

<C _{F} (t) > ≡ expected anthropogenic fossil component present in C(t) at time t , ppm.

C _{NF} (0) ≡ initial non-anthropogenic fossil component present in 1750 = C(0) = 276.44 ppm.

C _{NF} (t) ≡ non-anthropogenic fossil component present in C(t) at time t , ppm.

DC(t) ≡ increase in C(t) above C(0) , ppm.

DC _{F} (t) ≡ increase in C _{F} (t) above C _{F} (0) = C _{F} (t) , ppm.

<DC _{F} (t) > ≡ expected increase in C _{F} (t) above C _{F} (0) = <C _{F} (t) >, ppm.

DC _{NF} (t) ≡ increase in C _{NF} (t) above C _{NF} (0) , ppm.

DE(0) ≡ initial annual emission of anthropogenic fossil component, C _{F} (t) , in 1750 = 0.

DE(t) ≡ annual emission of anthropogenic fossil component, C _{F} (t) , billions of metric tons.

d13C ≡ 1,000‰ times the deviation of the annual mean ratio, R _{S13} , of (^{13} C/^{12} C) atoms in C(t) from that of a 1950 standard, R _{std13} , relative to R _{std13} , ‰ = −6.5‰ in 1750. The d13C value of the 1950 standard = −25‰.

D14C ≡ 1,000‰ times the deviation of the annual mean ratio, R _{S14} , of (^{14} C/^{12} C) atoms in C(t) multiplied by factors, F _{d} and F _{13} , from that of a 1950 standard, R _{std14} , relative to R _{std14} , ‰.

*D14C ≡ required value of D14C for anthropogenic fossil component, C _{F} (t) , to equal a given fraction F of the increase, DC(t) , of C(t) , from 1750 (t = 0) to 2018 (t = 268 y).

F _{d} ≡ factor in equation for D14C to account for the decay of ^{14} C atoms in the year of the annual mean value for D14C relative to that of the 1950 standard.

F _{13} ≡ factor in equation for D14C to account for the fractionation of ^{13} C atoms in the year of the annual mean value for D14C relative to that of the 1950 standard.

k ≡ constant of 1.404 × 10^{13} dpm (gC^{−1} ) per unit (^{14} C/^{12} C) ratio.

K ≡ constant of 4.996 × 10^{[mimus]1} gCm^{−3} /ppm for converting concentration of CO in ppm to gC m^{−4} under NTP conditions.

R _{std13} ≡ (^{13} C/^{12} C) atom ratio of standard = 0.011237 in 1950.

R _{std14} ≡ (^{14} C/^{12} C) atom ratio of standard =1.176 × 10^{−12} in 1950.

S(t) ≡ annual mean specific activity of C(t) , dpm (gC^{−1} ).

*S(t) ≡ required value of S(t) for anthropogenic fossil component, C _{F} (t) , to equal a given fraction F of the increase, DC(t) , from 1750 (t = 0) to 2018 (t = 268 y).

<S(t) > ≡ expected specific activity of C(t) calculated from eqn (2) for the approximation fitting function, dpm (gC)^{−1} .

t ≡ time in years from 1750 to the year of an annual mean value of a CO_{2} quantity, for example, t equals 1 year in 1751 and 200 years in 1950, the year for the 1950 standard.