---------- Excerpts from Hansen et al 1988 Appendix B ----------
Appendix B: Radiative Forcings
Radiative forcing of the climate system can be specified by the global surface air temperature change ΔT₀ that would be required to maintain energy balance with space if no climate feedbacks occurred... The following formulae approximate the ΔT₀ from the 1D RC model within about 1% for the indicated ranges of composition. The absolute accuracy of these forcings is of the order of 10% because of uncertainties in the absorption coefficients and approximations in the 1D calculations.
CO2
ΔT₀(x) = f(x) - f(x₀);
f(x) = ln(1 + 1.2x + 0.005x² + 1.4*10⁻⁶x³);
x₀ = 315 ppmv, x ≤ 1000 ppmv
...
Specifically, in scenario A CO2 increase as observed by Keeling for the interval 1958-1981 [Keeling et al., 1982] and subsequently with 1.5% yr⁻¹ growth of the annual increment. ...
In scenario B the growth of the annual increment of CO2 is reduced from 1.5% yr⁻¹ today to 1% yr⁻¹ in 1990, 0.5% yr⁻¹ in 2000, and 0 in 2010; thus after 2010 the annual increment in CO2 is constant, 1.9 ppmv yr⁻¹. ...
In scenario C the CO2 growth is the same as in scenarios A and B through 1985; between 1985 and 2000 the annual CO2 increment is fixed at 1.5 ppmv yr⁻¹; after 2000, CO2 ceases to increase, its abundance remaining fixed at 368 ppmv. ...
-------- End excerpts from Hansen et al 1988 Appendix B --------
Unfortunately, Hansen et al did not release their data (and I guess they didn't even bother to version control their code, which apparently has been lost). But there are some hints in their paper.
First of all, they say that they used "CO2 increase as observed by Keeling for the interval 1958-1981 [Keeling et al., 1982]".
When I first prepared the spreadsheet, I was unable to find the Keeling et al., 1982 paper, "Measurements of the concentration of carbon dioxide at Mauna Loa observatory, Hawaii," in Carbon Dioxide Review: 1982. I have since located it:
https://sealevel.info/Keeling1982.pdf
But I found a later Keeling report which has what is probably the same data:
https://sealevel.info/Keeling1986_Table1.png
From that, my best guess for the CO2 concentrations which Hansen et al (1988) used for 1958-1981 were as follows:
1958 315.0 ppmv
1959 316.10
1960 317.02
1961 317.74
1962 318.63
1963 319.13
1964 319.7
1965 320.41
1966 321.09
1967 321.89
1968 322.72
1969 324.21
1970 325.51
1971 326.48
1972 327.60
1973 329.82
1974 330.41
1975 331.00
1976 332.06
1977 333.62
1978 335.19
1979 336.54
1980 338.40
1981 339.46
The two values shown with one decimal digit are my estimates, the rest (two decimal digits) are from Keeling 1986 Table 1. They are similar, but not quite identical, to the currently reported Mauna Loa CO2 data. (I don't know why the data have been slightly revised, since then.)
My estimate of 315.0 for 1958 is because Hansen (1988) says, "x₀ = 315 ppmv".
My estimate for 1964 (the year with several months of data missing) is from the currently reported value, adjusted up by 0.08 ppmv for consistency with surrounding years' deviations in Keeling 1986 from currently reported values.
7/3/2022 UPDATE:
I've found the Keeling 1982 data, and it is different from the Keeling 1986 data above:
Keeling1982.pdf
Keeling1982_Table1.png
Keeling1982_Table1.txt
1958
1959 315.66
1960 316.59
1961 317.30
1962 318.20
1963 318.71
1964
1965 319.99
1966 320.66
1967 321.47
1968 322.30
1969 324.18
1970 325.51
1971 326.47
1972 327.59
1973 329.82
1974 330.40
1975 331.00
1976 332.06
1977 333.63
1978 335.20
1979 336.55
1980 338.36
For 1964, consider:
1982 2022 (ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_annmean_mlo.txt downloaded 6/28/2022)
---- ------ ------
1962 318.20 318.45 -.25 difference
1963 318.71 318.99 -.18
1964 ??? 319.62 interpoate as: -.22 ==> 319.4
1965 319.99 320.04 -.05
1966 320.66 321.37 -.71
So my new best guess is:
1958 315.0 ppmv
1959 315.66
1960 316.59
1961 317.30
1962 318.20
1963 318.71
1964 319.4
1965 319.99
1966 320.66
1967 321.47
1968 322.30
1969 324.18
1970 325.51
1971 326.47
1972 327.59
1973 329.82
1974 330.40
1975 331.00
1976 332.06
1977 333.63
1978 335.20
1979 336.55
1980 338.36
1981 (1981 figure was not included in Keeling 1982)
*** INCOMPLETE -- rest needs to be updated for the above figures:
Guessing the CO2 levels they used after 1988 is a bit tricky. (Too bad the professional standards in climate science are below those of econometrics. https://sealevel.info/FOIA/1075403821.txt#:~:text=McKittrick%20says,codes%20!! ) We're given a 1.5%/year acceleration for the increment, but we aren't told the initial 1981->1982 increment.
But there are clues to be found in the paper.
Our first clue is scenario C. It says that "between 1985 and 2000 the annual CO2 increment is fixed at 1.5 ppmv yr⁻¹; after 2000, CO2 ceases to increase, its abundance remaining fixed at 368 ppmv." That means 1999 was 366.5, 1998 was 365.0, etc., back to 345.5 in 1985.
That leaves three years to fill in, between 339.46 in 1981 and 345.5 in 1985. So, the question is, what initial CO2 increment x from 1981 -> 1982, with +1.5% increases the following three years, gets us to 345.5 in 1945?
That's simple math:
345.5 - 339.46 = 6.04
So:
x × (1 + 1.015 + 1.015² + 1.015³) = 6.04
x = 6.04 / (1 + 1.015 + 1.015^2 + 1.015^3) = 1.47644650737
https://www.google.com/search?q=6.04+%2F+%281+%2B+1.015+%2B+1.015%5E2+%2B+1.015%5E3%29+%3D
That makes the 1984->1985 increment x × 1.015³ = 1.54388818460
Our next clue is this statement from the paper: "In scenario C the CO2 growth is the same as in scenarios A and B through 1985".
That means we can use the values we calculated for scenario C for 1982 thru 1985, also for scenarios A and B. Crucially, it means that in all three scenarios Hansen et al must have assumed that CO2 level was exactly 345.5 ppmv in 1985, and that the 1984->1985 CO2 increment was +1.54388818460 ppmv.
(Of course, by 1988 the actual measured CO2 levels were known for 1982-1987. For instance, according to Keeling 1986, the measured average CO2 level in 1985 was 345.64 ppmv. Although they didn't mention the fact, my guess is that that is probably why Hansen et al chose almost that exact level [345.5 ppmv] for the their modeled 1985 CO2 level.)
Knowing the modeled 1985 CO2 level and the modeled 1984->1985 increment enables us to calculate the rest of the values for scenarios A and B.
For scenario A, the increment increases 1.5% per year, forever. So for 1985->1986 it was 1.015 × 1.54388818460 = 1.56704650737, etc.
For scenario B, the increment increases by 1.5% per year for years 1986 thru 1989 (identical to scenario A), then by 1.0% per year for years 1990 thru 1999, then by 0.5% per year for years 2000 thru 2009, and from 2010 on the increment is fixed at 1.9 ppmv/year. When I entered those formulas into the spreadsheet, I was gratified to see that it calculated a 2008->2009 CO2 increment of almost exactly 1.9 ppmv (1.9026 ppmv).
Dave Burton
February 8, 2021
f(x) = ln(1 + (1.2 * E4) + (5E-3 * (E4^2)) + (1.4E-6 * (E4^2)))
ln(1 + (1.2 * J4) + (5E-3 * (J4^2)) + (1.4E-6 * (J4^2)))
ln(1 + (1.2 * O4) + (5E-3 * (O4^2)) + (1.4E-6 * (O4^2)))
=LN(1 + (1.2 * B4) + (0.005 * (B4^2)) + (0.0000014 * (B4^2)))
=(H4-H$4)*1.25/H$98 =(J4-J$4)*1.25/J$98
=(O4-O$4)*1.25/H$98 =(Q4-Q$4)*1.25/J$98
=(V4-V$4)*1.25/H$98 =(X4-X$4)*1.25/J$98