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What is the correct value of Climate Sensitivity?


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Ok, it's bad science then...

You are in no position to make such an assessment because you refuse to look at the arguments and dismiss them as "conspiracies" because they contradict what your favorite alarmists says. As I said: I know enough about the this topic to separate the good science and bad science. If you claim that all skeptical views are "bad science" then that just means you are blind ideologue no matter how much you wish to believe otherwise. Edited by TimG
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You are in no position to make such an assessment because you refuse to look at the arguments and dismiss them as "conspiracies" because they contradict what your favorite alarmists says.

I have looked at them. I don't need to look at them again. Life is short, and I have moved on to other things.

As I said: I know enough about the this topic to separate the good science and bad science. If you claim that all skeptical views are "bad science" then that just means you are blind ideologue no matter how much you wish to believe otherwise.

Not all skeptical views are blind science, I didn't say that. The science is, by and large, fine. The overall theory of AGW seems correct and you even acknowledge this, so why do we have seemingly intelligent people like JBG throwing sparkdust around, and people like you supporting him ?

If the discussion was around the economics of climate change, I'd likely be on the same side as you so would I be an idiot then ? No, I'd be an intelligent and principled poster of course...

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Again - it's very easy for JBG to wave his hands around, and run around in a circle saying "data can be manipulated" but the danger is that other ridiculous people who vote may be taken in by ridiculous assertions.

No kidding it's easy and dangerous. The denigrating methods and tactics that economic alarmists and extremists have used against climate scientists have been adopted and applied by other's who seek similar attention when espousing ridiculous assertions. Some of which can result in irrational movements against things like vaccinating, gluten, windmills...the list goes on and on.

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I tried to use an a priori justified initial guess to see if it would lead to the gauss-newton estimation converging to something more reasonable:

- If the fast feedback response with decay time 0.5 years of a doubling of CO2 concentrations causes temperatures to rise 1.15 C, then this would suggest that γ1 is 1.15/5.35/ln(2) = 0.3101.
- If the remaining 4 exponential responses result in a climate sensitivity of approximately 3 C (and they are all of equal strength), then this would suggest that γ2 = γ3 = γ4 = γ5 = (3-1.15)/4/5.35/ln(2) = 0.1247.
- The solar irradiance data is given in W/m^2. However, the Earth has an albedo of ~0.35, and the surface area of a sphere divided by the circle of the same radius is 4. Therefore, the coefficient solar in my model should be approximately 1/4*0.65 = 0.1625.

- According to http://data.giss.nasa.gov/modelforce/strataer/, one can convert the aerosol optical depth to a forcing in W/m^2 by multiplying it by a factor of -23. Therefore, the volcano coefficient should be approximately -23.

- I performed a simple exponential decay to equilibrium model and got coefficients for each of the internal variability indices. This coefficients should be roughly the same in the more complicated model.

So I tried the above 'guess' with dG = 0. It still didn't converge to anything realistic either.

- - - - -

A second thing I thought I would try is to integrate my model so that temperature is the dependant variable rather than change in temperature. By integrating, my model becomes:

The difference in temperature between January 1959 and month f + 1 is:

Sum(s = 1 to 5; γs*((Sum i = 1 to f; dGHGi*ρs(i,i) + ... + dGHG1*ρs(i,1))
+ Solar*(Sum i = 1 to f; dSi*
ρs(i,i) + ... + dS1*ρs(i,1))
+ Volcano*(Sum i = 1 to f; dVi*ρs(i,i) + ... + dV1*ρs(i,1))
+ dGs*(Sum i = 1 to f;
ρs(i,0))))
+ β1*(LODf+1 – LOD1) + β2*(AAMf+1 – AAM1) + β3*(SOIf+1 – SOI1) + β4*(PDOf+1 – PDO1) + β5*(NAOf+1 – NAO1))

where month 1 corresponds to January 1959

Estimating this model still yields nonsense. The gauss-newton method stopped converging after only 1 iteration.

- - - - -

So the next thing I thought I would do is try to estimate the above model by using a sequence of linear regressions (alternating between keeping Solar, Volcano and dGs fixed and keeping the γs fixed). This gave slightly better results, but still somewhat meaningless.

The next thing I tried was perform the sequence of linear regressions, but keep going (rather than stopping once the error stops decreasing). What I find is that as the number of iterations approach infinity, the γs approach zero and dG gets large. Obviously zero climate sensitivity is unrealistic. It seems like my attempts at trying to estimate the model are not converging to a reasonable solution. It might not be possible to estimate the model correctly with the 1959-2012 data set and it seems like the dG terms tend to soak up more of the variation.

One option is that I try to use a different data set (say annual data from 1876 to 2011) and hope for the best; though data quality would decrease and some indices like AAM don't go that far back. A second option is to start from a period of time where it is safe for me to assume that dG = 0 (such as say 1750); unfortunately most data sets don't go back far enough to do this. A third option is the initially assume dG = 0, estimate the model for 1959-2012, and then use the estimated parameters to computer dG in 1959 by using annual data prior to 1959 (and repeating until convergence is met).

Edited by -1=e^ipi
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Okay, I constructed a data set for 1876-2012. I removed atmospheric angular momentum from the model because there isn’t good AAM data prior to the 50s. However, AAM is strongly correlated with LOD and SOI, so it should not be a big deal.

Temperature: http://www.cru.uea.ac.uk/cru/data/temperature/HadCRUT4-gl.dat

CO2: ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_annmean_mlo.txt

CH4: I use Cape Grim, Tasmania, Australia Instrumental data from 1985-2013 ftp://ftp.cmdl.noaa.gov/data/trace_gases/ch4/flask/surface/ch4_cgo_surface-flask_1_ccgg_month.txt and 1870-1984 spline smoothed ice core data from Antarctica and Greenland http://cdiac.ornl.gov/ftp/trends/atm_meth/EthCH498B.txt. I interpolate the ice core data for missing years and I decrease the ice core data by 24.12 ppb so that it is continuous with respect to the Cape Grim data.

N2O: For N2O, I use global methane data from 2000-2014 ftp://ftp.cmdl.noaa.gov/hats/n2o/insituGCs/CATS/global/insitu_global_N2O.txt and I use snowpack Antarctica data from 1870-2004. ftp://daac.ornl.gov/data/global_climate/global_N_cycle/data/global_N_perturbations.txt . The data prior to the 90’s is relatively inaccurate and has missing data points, so I perform a cubic fit to the data prior to 1996. The cubic fit is relatively good (R^2 is 0.98) and intersects the data in the 90s around 1993. Therefore, I use the cubic fit for the data prior to 1994. For the 5 overlapping years (2000-2004) between the two data sets, the snowpack data is on average 0.01283 ppb higher than the later data set. Therefore, I reduce the snowpack data by 0.01283 ppb to make the data sets comparable.

TSI: http://lasp.colorado.edu/data/sorce/tsi_data/TSI_TIM_Reconstruction.txt

Volcanic Aerosols: http://data.giss.nasa.gov/modelforce/strataer/tau.line_2012.12.txt

I thought I would try to get estimates of volcanic aerosols for October, November and December for 2012 (so that I have 1 more data point to work with if I want to use annual data). I’ll use the annual volcanic aerosol data from 1850-2011 and I’ll use the volcanic eruptions of VEI > 3 since 1850. I’ll define a variable V which is equal to the number of volcanic eruptions of VEI 4 plus 10 times the number of volcanic eruptions of VEI 5 plus 100 times the number of volcanic eruptions of VEI 6 for that year. Then I’ll regress dAOD(t)/dt = A + B*V(t) – k*AOD(t), where the first term corresponds to aerosol emissions due to volcanic eruptions of VEI less than 4, the second term corresponds to aerosol emissions due to volcanic eruptions of VEI greater than 3, and k is the decay rate of aerosols. A/12 should correspond roughly to emissions of volcanic eruptions of VEI less than 4 for 1 month, and 1 - (1-k)^(1/12) should give the rate of decay of volcanic aerosols between consecutive months.

I get A = 0.0036 and k = 0.4758. Using the fact that there is no volcanic activity of VEI 4 or higher in 2012 and that the optical aerosol thickness in September 2012 is 0.0035, I get that the optical aerosol thickness in October, November and December should be approximately 0.0036, 0.0037 and 0.0038 respectively.

LOD: I get length of day data from http://www.iers.org/IERS/EN/Science/EarthRotation/LODsince1623.html?nn=12932. As before, I lag everything by 6 years and detrend the data by 1.7 ms per century.

SOI: http://www.bom.gov.au/climate/current/soihtm1.shtml

PDO: http://www.ncdc.noaa.gov/teleconnections/pdo/data.csv I am a bit concerned that there might be a long term trend in the PDO due to global warming, but it is hard to tell.

NAO: https://climatedataguide.ucar.edu/climate-data/hurrell-north-atlantic-oscillation-nao-index-station-based

Edited by -1=e^ipi
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I found a mistake in my code which might have led to some of the nonsense results I was getting.

I was using dμ = μ(endi, endi+1,dF) - μ(starti, endi,dF) instead of dμ = μ(endi, endi+1,dF) - T(0) when endi occurs at t=0 (see post #236 for more details).

Edit: retried the regression of post #254. The results are basically the same.

Edited by -1=e^ipi
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So for the annual data from 1876-2012, the regression equation is:

T(Y) - T(1876) =

Sum(s = 1 to 5; γs*((Sum i = 1876 to Y-1; dGHGis(i,i) + ... + dGHG1876s(i,1876))
+ Solar*(Sum i = 1876 to Y-1; dSis(i,i) + ... + dS1876s(i,1876))
+ Volcano*(Sum i = 1876 to Y-1; dVis(i,i) + ... + dV1876s(i,1876))
+ dGs*(Sum i = 1876 to Y-1; ρs(i,1875))))
+ β1*(LODY – LOD1876) + β2*(SOIY – SOI1876) + β3*(PDOY – PDO1876) + β4*(NAOY – NAO1876))

where Y is the year.
ρ(i,j) = 0 if j > i

ρ(i,j) = 1 + τ*(exp(-1/τ) –1) if i = j

ρ(i,j) = τ*exp(-i/τ)*(exp(-1/τ) + exp(1/τ) - 2) if j < i

Where I tried to estimate this using the technique of a sequence of linear regressions with alternating restrictions, it does not converge.

When I try to perform the normal Gauss-Newton technique, it does not converge.

When I modify the Gauss-Newton technique such that the change in the estimate between consecutive iterations is only 1% of the original Gauss-Newton technique, I do get initial convergence. I get that equilibrium climate sensitivity is 2.64 C. Note that in my model, equilibrium climate sensitivity is simply (γ1 + γ2 + γ3 + γ4 + γ5)*5.35*ln(2).

However, if I tell matlab to keep going, I eventually get divergence and head to a case where sensitivity approaches zero and the dG terms get large. So perhaps I need to find a reasonable way to place restrictions on dG.

As for the estimate of the error for climate sensitivity, it is ridiculously large, but I think that is mostly because my method of estimating error completely fails since the model is so non-linear that even the Gauss-Newton method does not converge properly. I think I need to read up more on convergence of non-linear regressions.

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Sorry, I've been writing NO2 instead of N2O the past two pages.

Also, I found a discontinuity in my methane data, which I have corrected for.

I've thought of a way to resolve the issue I've been having with the dG. The problem is that in 1876, it would not be reasonable to assume that global temperatures are in equilibrium since greenhouse gases and solar irradiance were steadily increasing since the 1700s. However, it might be reasonable to assume that global temperatures were in equilibrium in 1700 (at the end of the Maunder Minimum) and that any changes in radiative forcing that occurred before 1700 has negligible impact on changes in global temperatures after 1876. If this is the case then I can rewrite my regression equation as:

T(Y) - T(1876) =

Sum(s = 1 to 5; γs*((Sum i = 1876 to Y-1; dGHGis(i,i) + ... + dGHG1700s(i,1700))
+ Solar*(Sum i = 1876 to Y-1; dSis(i,i) + ... + dS1700s(i,1700))
+ Volcano*(Sum i = 1876 to Y-1; dVis(i,i) + ... + dV1700s(i,1700))))
+ β1*(LODY – LOD1876) + β2*(SOIY – SOI1876) + β3*(PDOY – PDO1876) + β4*(NAOY – NAO1876)) + errorY

Which means that the only thing left to do to estimate the above model is to obtain radiative forcing data from 1700-1875.

Unfortunately, the volcanic aerosol data only goes back to 1850. However, this shouldn't be a big deal because volcanic aerosols quickly decay and volcanic activity there are no long term trends in volcanic activity, so volcanic activity prior to 1850 (even the 1815 Mount Tambora eruption that lead to the year without a summer) should have negligible impact on temperature changes after 1876. So I'll exclude changes in volcanic aerosols before 1850, and my regression equation becomes:

T(Y) - T(1876) =

Sum(s = 1 to 5; γs*((Sum i = 1876 to Y-1; dGHGis(i,i) + ... + dGHG1700s(i,1700))
+ Solar*(Sum i = 1876 to Y-1; dSis(i,i) + ... + dS1700s(i,1700))
+ Volcano*(Sum i = 1876 to Y-1; dVis(i,i) + ... + dV1850s(i,1850))))
+ β1*(LODY – LOD1876) + β2*(SOIY – SOI1876) + β3*(PDOY – PDO1876) + β4*(NAOY – NAO1876)) + errorY

The solar irradiance data I was using (http://lasp.colorado.edu/data/sorce/tsi_data/TSI_TIM_Reconstruction.txt) extends back to 1610, so I can just use the same data.

The law dome CO2 data set that I was using (http://cdiac.ornl.gov/ftp/trends/co2/lawdome.combined.dat) goes back to 948. One problem is that the 20 year smoothed spline data that I was using only goes back to 1832. I can use the 75 year smoothed spline data before then, but then there is a discontinuity in my data set. From 1850-1870, the 75 smoothed spline data is on average 0.362857143 ppm more than the 20 year smoothed spline data. Therefore, I increase the 75 year smoothed spline data by 0.362857143 ppm to make it comparable. I then use the modified 75 year smoothed spline from 1700-1850 and the 20 year smoothed spline from 1851-1958. I then increase the ice-core data by 0.2765 ppm to make them comparable to the Mauna Loa data.

For methane, the 75 year smoothed spline ice core data that I was using (http://cdiac.ornl.gov/ftp/trends/atm_meth/EthCH498B.txt) goes back until 1010, so I can extend my data back to 1700 using the same methodology I used earlier.

For N2O, it is a bit tricky. The snow-pack data (ftp://daac.ornl.gov/data/global_climate/global_N_cycle/data/global_N_perturbations.txt) that I was using starts in 1756 and it is quite noisy prior to 1876. Ice core data for N20 is is available (ftp://ftp.ncdc.noaa.gov/pub/data/paleo/icecore/antarctica/law/law2006.txt), but is still quite noisy and seems to give values lower than the snow-pack data. I take the ice-core data points from 1681 to 1900 and perform a quartic fit. If I compare the fitted values of the ice-pack data with the fitted values of the snow-pack data I find that in 1876, the snow-pack data is approximately 3.787136171 ppb higher than the ice-pack data. Therefore, I increase all of the icepack data by 3.787136171 ppb, use ice-pack data from 1700-1875, and use snow-pack data from 1876-1999. I then subtract 0.01283 ppb from both the snow-pack and ice-pack data to make it comparable to the 2000-2012 instrumental data.

From this I can get estimates of greenhouse gas forcing from these 3 gases from 1700-2012.

Edited by -1=e^ipi
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Okay, I tested the model

T(Y) - T(1876) =

Sum(s = 1 to 5; γs*((Sum i = 1876 to Y-1; dGHGis(i,i) + ... + dGHG1700s(i,1700))
+ Solar*(Sum i = 1876 to Y-1; dSis(i,i) + ... + dS1700s(i,1700))
+ Volcano*(Sum i = 1876 to Y-1; dVis(i,i) + ... + dV1850s(i,1850))))
+ β1*(LODY – LOD1876) + β2*(SOIY – SOI1876) + β3*(PDOY – PDO1876) + β4*(NAOY – NAO1876)) + errorY

I get a climate sensitivity of (1.11 +/- 0.20) C. Seems low... But my estimates of the other coefficients seem reasonable.

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As a robustness check, I decided to remove the 4 internal variability variables and test:

T(Y) - T(1876) =

Sum(s = 1 to 5; γs*((Sum i = 1876 to Y-1; dGHGis(i,i) + ... + dGHG1700s(i,1700))
+ Solar*(Sum i = 1876 to Y-1; dSis(i,i) + ... + dS1700s(i,1700))
+ Volcano*(Sum i = 1876 to Y-1; dVis(i,i) + ... + dV1850s(i,1850))))
+ errorY

I get a climate sensitivity of (1.14 +/- 0.33) C.

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My impulse response function has a weird behavior since 2 of my γ are negative. The impulse response function is peaking after ~3 years.

I should either make the set of γ more dense (such as have them spaced out by a factor of 2 rather than 4) or restrict my γ such that they cannot be negative when performing my estimation.

Edited by -1=e^ipi
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So I tried increasing the number of decay times. I now have 9: 0.5, 1, 2, 4, 8, 16, 32, 64, and 128 year decay times. This lead to a sensitivity of 0 +/- 9 C (there might be too many parameters to estimate).

I tried to impose the restriction that the γ cannot be negative. This lead to a climate sensitivity of 3 C, but with a very large uncertainty (and my current method to evaluate uncertainty is probably invalid now that I've added the restrictions).

I tried the restriction in the simpler model with only 5 decay times. This gives a climate sensitivity of (2.50 +/- 1.67) C. But again, the method to evaluate uncertainty is probably invalid.

Edit: actually, the 2.50 C is invalid.

Edited by -1=e^ipi
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Nevermind, the 2.50 C estimate isn't valid. I checked the residual and it has a clear downward trend. My restricted estimation method could not deal effectively with one of the gammas being zero. In any case, the restriction suggested that the gammas for the decay times of 8 years and 128 years should be zero (or negative).

I retried the model with only 3 decay times. This gave a climate sensitivity estimate of (1.28 +/- 0.03) C (again, I don't think the premises used to obtain the confidence interval are valid). Checking the residual, it seems to be correlated with the Multivariate ENSO Index (0.21 correlation coefficient) and the Atlantic Multidecadal Oscillation Index (0.42 correlation coefficient). So perhaps my current internal variability indices are not sufficient. The problem with using MEI or AMO is that these indices use temperature data and might be correlated with global warming. There are a few people that have tried to detrend this effect though.

Sorry that I haven't been able to get something conclusive yet. Although, the closest results to conclusive that I have are suggesting a lower climate sensitivity.

Edited by -1=e^ipi
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There is a potential issue in the model of post #260. If there is a significant amount of error for T(1876) - LOD(1876) - SOI(1876) - PDO(1876) - NAO(1876), then this will affect all of the results by a constant factor (which the regression will try to correct for with a weird impulse response function). I can modify the model by introducing a constant. The modified model is given below. Note that even though I increase the number of coefficients that need to be estimated by 1, the number of data points available to me also increases by 1, so the uncertainty of my estimates should not increase due to this change.

T(Y) =
Sum(s = 1 to 5; γs*((Sum i = 1876 to Y-1; dGHGis(i,i) + ... + dGHG1700s(i,1700))
+ Solar*(Sum i = 1876 to Y-1; dSis(i,i) + ... + dS1700s(i,1700))
+ Volcano*(Sum i = 1876 to Y-1; dVis(i,i) + ... + dV1850s(i,1850))))
+ β0 + β1*LODY + β2*SOIY + β3*PDOY + β4NAOY + errorY

I also found an error in my code where I was using solar irradiance data before 1876 instead of volcanic aerosol data before 1876. I've also modified my code such that I can calculate climate sensitivity given any set of decay times.

My climate sensitivity estimates are now more robust and more accurate, but the outcome is fairly sensitive to the choice of the decay times. Here are some example results:

Choosing the decay times to be 0.5, 2, 8, 32 years gives a climate sensitivity of (2.24 +/-0.34) C.

Choosing the decay times to be 0.5, 1.5, 4.5, 13.5 and 40.5 years (factors of 3 difference) gives a climate sensitivity of (3.27 +/- 0.97) C.

And since you guys kept saying 42, a decay time of 42 years gives a climate sensitivity of (3.28 +/- 0.17) C.

It seems that a decay time of 128 years is too long to detect however. I would say that these confidence intervals are small enough such that non-linearities do not significantly affect them. Although, these confidence intervals do not account for the specification error in the choice of decay times.

If anyone has a set of decay times they want me to test, just ask me to in this thread. Perhaps I should use some more a priori justified decay times as Bonam has suggested, or change the model to allow for the decay times to vary.

Edit: actually these confidence intervals are off by a factor of 5.35*ln(2).

Edited by -1=e^ipi
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Update:

I've been reading a bit on the literature of the climate impulse response function. Seems like Fortunat Joos is one of the most cited people on this.

This paper http://www.climate.unibe.ch/~joos/papers/hooss01cd.pdf is fairly interesting because it determines impulse response functions for many things (temperature change due to CO2, sea level rise due to CO2, CO2 decay rate towards equilibrium, etc.). On page 7 it gives a two exponential impulse response function for temperature response to CO2. One has a decay time of 12 years, and another with a decay time of 400 years. If one interprets that first decay time to correspond to the equilibrium climate sensitivity and the second decay time to correspond to the earth system sensitivity, then this suggests that earth system sensitivity is approximately 1/0.71 = 1.408 times equilibrium climate sensitivity. This is completely consistent with the claim by Lunt et al. 2012 (which used Pliocene paleoclimate data) that the ESS is approximately 1.4 times the ECS.

If the above paper is correct about a 12 year decay corresponding well to equilibrium climate sensitivity, then if I use a 12 year decay, I should be able to get a good estimate of equilibrium climate sensitivity. Doing this gives me (1.74 +/- 0.18) C, this is pretty low (of course this uncertainty doesn't take into account the large specification error of choosing a 12 year decay time). This webpage (http://unfccc.int/resource/brazil/climate.html) gave a similar 2 decay time model, but with 8.4 year and 410 year decay times.

Another thing I did was I took the impulse response function from the end of this paper (http://www.climate.unibe.ch/~joos/IRF_Intercomparison/Protocol_CO2_impulse_response_modelcomparison_v1.0.pdf) using a freeware program called GetData. I wanted to see what functional forms can represent this impulse response function. The simple exponential decay (1 - exp(-At)) is terrible as expected. However, (1 - exp(-A*t^B ) and (1 - exp(-(A - B*exp(-Ct))*t)) are decent. Maybe I could try using these rather than the Van Hateren approach, although the additional non-linearity will be a pain to deal with. Alternatively, perhaps I should just take one of these GCM impulse response functions as is, and see what climate sensitivity I get from the data under the assumption of this impulse response function.

- - - - -

So most of my recent estimates have been suggesting a climate sensitivity lower than 3 C. To play a climate alarmist advocate, one possible reason for this is that my regression model results in solar and volcano coefficients that may be too high (therefore, it overstates their importance relative to CO2 and gives a low estimate of climate sensitivity). Therefore, it might be interesting to choose lower bounds on what these two coefficients should be, and then try to estimate climate sensitivity from the data using a regression.

For volcanic aerosols, as I explained earlier, the NASA website suggests that one can convert the volcanic aerosol data to forcing in W/m^2 by multiplying by a factor of -23. However, Van Hateren mentioned in his paper that the literature suggests that a forcing of volcanic aerosols only has about 50-60% the effect of an equivalent forcing of greenhouse gases. To play the climate alarmist advocate, I'll choose the lower of these values, which means my volcano coefficient should be no less than -11.5.

Trying to put a lower bound on the solar constant is more tricky. Solar irradiance is correlated with cosmic rays, which may amplify the measured effect of solar irradiance, but since the sun's solar irradiance has a stronger effect in equatorial regions than polar regions, this unequal heating affect should result in a smaller temperature response than if the change in solar irradiance was distributed uniformly (due to the Stefan-Boltzman Law). As I argued earlier, the fact that the ratio of the surface area of a sphere to the area of a circle of equal radius is 4 and the fact that the earth has an albedo of 0.31 (earlier I said 0.35, but 0.31 seems to be a better value http://www.climatedata.info/Forcing/Forcing/albedo.html) suggests that the solar coefficient should be about 0.1725. To play the climate alarmist advocate, I'll pretend cosmic rays don't exist and try to calculate the affect of unequal heating to get a lower bound for the solar coefficient.

The average direct solar irradiance directed upon a square meter of Earth at latitude φ is proportional to:

ϑ*Integral(μ = 0 to 2π; [cos (φ) cos (sin-1 (cos (μ) sin (ψ0 ) ) ) sin (ψ) - sin (φ) cos (μ) sin ( ψ0) ψ ]dμ)

Where ϑ = 1366 W/m^2 is the solar irradiance at 1AU from the sun, ψ0 = 23.4*π/2 is the Earth’s tilt in radians, and ψ = cos-1 (min(0,max(1,tan (φ) tan (sin-1 (cos (μ) sin ( ψ0) ))) ). I’ll spare you guys the derivation (since you guys complain my posts are too long). I’m missing a normalization constant here. In any case, I can normalize this and then make sure the average direct solar irradiance equals ϑ/4/π.

Now consider a simplified representation of the earth, where the temperature distribution depends only on the latitude temperature profile and the rate of heat transfer between two adjacent square meters of the surface is proportional to the temperature difference. Then in equilibrium, one has:

(1-α)*S(φ) + B = G*σ*T4(φ) + k*(d2T(φ)/dφ2 - tan(φ)*dT(φ)/dφ)

Where α = 0.31 is the albedo of earth, S(φ) is the solar irradiance at latitude φ, B = 0.087W/m^2 is the heat flux due to the Earth’s internal energy, G is a factor due to greenhouse gases, σ is the Stefan-Boltzmann constant, k is a constant that determines the rate of heat transfer, and T is the temperature. Again, I’ll spare you guys the derivation.

Now the above equations have no analytical solution. Though it can be approximated numerically with a computer. I need to determine k and G before I can see how the unequal heating of increasing solar irradiance compares to the equal heating of increasing greenhouse gases in terms of equilibrium temperature response.

For an earth with approximately uniform temperature, integrating the second equation gives:

(1-α)*ϑ/4/π + B = G*σ*T4

For a temperature of 288 K, this suggests that G = 0.1925.

If I look at the temperature profile of Earth:
http://www.roperld.com/science/graphics/TempVSLatitude.jpg

At a latitude of 40 degrees, the temperature is approximately 286K, the first derivative of temperature with respect to latitude is approximately -0.7 C per degree, and the second derivative is approximately -0.1 C per degree squared. Putting this into the second equation gives:

(1-α)*S(40*π/180) + B = G*σ*(286 K)4 + k*((-0.1K + 0.7K*tan(40*π/180))*(180/π)^2)

This gives k = -8.78 * 10^-4 W/m2/K.

The other thing I need is an initial temperature profile. I’ll start with a simple sinusoidal temperature profile of 273 + 26*cos(2φ) K. From here, I can write a computer program to start with this initial temperature profile and slowly change the temperature profile based on the imbalance of equation 2 until equilibrium is reached. I can then allow for k and G to vary and impose restrictions that the average temperature is 288K and the equatorial temperature is 300 K to get estimates of k and G.

I get that k = -0.0646 and G = 0.1956.

Now to see the equilibrium temperature response to a change in solar irradiance, I simply increase solar irradiance slightly (say by 0.1 W/m^2) and calculate the change in average temperature once the new equilibrium is reached. To see the equilibrium temperature response to a change in greenhouse gases, I simply decrease G such that the average of G*σ*T4 decreases slightly (say by 0.1 W/m^2) and calculate the change in average temperature once the new equilibrium is reached. I find that the effect of increasing solar irradiance is about 4.41% as effective as increasing greenhouse gas forcing by an equivalent W/m^2.

So Solar = 0.0441 and Aerosol = -11.5 are my lower bounds for these coefficients.

If I use this then the regression becomes:

T(Y) =
Sum(s = 1 to 5; γs*((Sum i = 1876 to Y-1; dFis(i,i) + ... + dF1700s(i,1700))
+ β0 + β1*LODY + β2*SOIY + β3*PDOY + β4NAOY + errorY

Where dF is the sum of the forcing changes (dGHG + 0.0441*dS – 11.5*dA). This is fortunately a linear model, so is very easy to estimate.

This model gives me a climate sensitivity of 0.97 +/- 0.59. The reason why I get this nonsense result is that some of the exponentials are negative. So I have to impose restrictions that they are positive.

Fortunately, I have found a methodology that does this (thanks to stata's website) and allows me to appropriately measure the error. All I have to do is replace each γ with exp(ln(γ)) and then use the ln(γ) as my unknown coefficients. The regression equation becomes:

T(Y) =
Sum(s = 1 to 5; exp(lnγs)*((Sum i = 1876 to Y-1; dFis(i,i) + ... + dF1700s(i,1700))
+ β0 + β1*LODY + β2*SOIY + β3*PDOY + β4NAOY + errorY

Of course, this is now a non-linear regression. But I can estimate this using Gauss-Newton. One issue is that if one of my γ's is supposed to be zero, then ln(γ) will approach minus infinity, which will eventually cause matlab to divide by zero (due to rounding error) and give me nonsense results. Therefore, I have to modify my code slightly to tell matlab not to make any of the lnγ's smaller than say -12 (which will mean that the γ is basically zero). Doing this should not result in an overestimation of my uncertainty (which is good).

Anyway, I chose decay times of 0.5, 1.5, 4.5, 13.5, and 40.5 years (because I was having more computer-rounding issues with longer decay times). Note that my estimates of ln(γ) will all be normal, which means that each of my γ will be log-normal. This means that my estimate of climate sensitivity will be a sum of log-normals that are all correlated with each other (which I'll assume is roughly log-normal as well).

This gives me a 95% confidence interval of [0.69,4.26] C for equilibrium climate sensitivity and a best estimate of 1.71 C. Unlike earlier results, I think that the confidence interval is well approximated here and there isn't much specification error on this model. Interestingly, this excludes the absurdly low climate sensitivity estimates by people like Christopher Monckton and excludes the very high end of the IPCC's confidence interval (which is 4.5C). Also, my residual is very correlated with MEI and AMO, so I might be able to reduce my uncertainty by a fair amount.

One more thing, I forgot to multiply most of my confidence intervals by a factor of 5.35*ln(2) in the last 1-2 pages. Sorry about this mistake.

Edited by -1=e^ipi
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I thought I would try changing my variability indices to see if it would significantly change the estimate.

The problem with using Multivariate ENSO Index (MEI), Pacific Decadal Oscillation Index (PDO) and Atlantic Multidecadal Oscillation Index (AMO) is that they all use temperatures, so there is the issue that global warming may change these indices (reverse causality issue). To try to prevent this, I have take each of these indices are regress it on a constant, on a linear term and on temperature (then I take the residual as my detrended index).

Using the detrended PDO changes my 95% confidence interval to [0.65,4.67]C with a best estimate of 1.74 C.
Replacing the SOI with the detrended MEI changes my 95% confidence interval to [0.70,3.91]C with best estimate of 2.05 C.

Replacing the NAO with the detrended AMO changes my 95% confidence interval to [0.66,4.26]C with best estimate of 2.15 C.

My confidence interval hasn't changed much by modifying these indices. Maybe I need to construct models for how each of these indices vary over time to better detrend them (this could also be used to make temperature projections).

One thing I should mention. For the original post, I didn't take into account the fact that there are other greenhouse gases than just CO2. about 76% of the change in greenhouse gas radiative forcing between 1950 and now is due to CO2. This suggests that my estimate of climate sensitivity from the original post should be 2.25 C rather than 2.95 C. Of course this still has the assumption of constant natural warming (which means that the estimate should be higher than 2.25 C).

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I thought I would try to fit simple models to AMO, PDO and MEI (to get better detrended data, and maybe to use as projections).

So I took AMO, PDO and MEI and looked at the fourier transform.

AMO has strong peaks around ~70 years, 24 years and 10 years.
PDO has strong peaks around 40 years, 27 years, 13 years and 5.7 years.

MEI has ~70 years, 10 years, 7 years, 4 years and 3 years (as well as others).

I tried using the gauss newton algorithm to fit these indices to a constant + a linear factor + temperature + a few sinusoids.

The AMO can be approximated very well by 2-3 sinusoids.
The PDO can be approximated okay by 3 (but it might need more; My computer only has so much computational power, so I haven't done 4 yet).

MEI is very chaotic so is difficult to predict without a very large number of sinusoids.

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You have to shake your head at the intelligence that often gets brought to the discussion on AGW by the naysayers, especially the Republicans.

http://time.com/3725994/inhofe-snowball-climate/

I've had a pack of 35 cm. of global warming in my backyard for more than a month. And it got to -15C last night.
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