What is the correct value of Climate Sensitivity?

[WestCoastRunner]

Has anyone seen the movie 'A Beautiful Mind'?

[-1=e^ipi]

I'll try to do a rough estimation of the amount of CO2 (in terms of metric tons of carbon) that was in the atmosphere during pre-industrial times.

As I explained in post #163, there is approximately 10329 kg of air above a square meter of Earth at sea level.

The radius of the Earth is 6371 km.

This means that there is approximately 4*pi*(6371000m)^2*10329kg = 5.268 x10^18 kg of air on Earth.

If we assume an atmosphere of 78% N₂, 21% O₂ and 1% Argon, then the molar mass of the atmosphere is 28.96 g/mol.

Thus there are approximately 1.819 x 10^20 mols of gas in the Earth's atmosphere.

If pre-industrial times had 275 ppm of CO2, then this means that there were approximately ~ 5.00 x 10^16 mols of CO2 in the atmosphere during pre-industrial times.

Since the molar mass of carbon is 12 g/mol, this means that the atmosphere had ~ 600 billion tons of carbon in the atmosphere during pre-industrial times.

Dividing this by 275 gives 2.18 billion tons, which is the amount of carbon that needs to be burned in order to increase atmospheric CO2 by 1 ppm.

Though most of the estimates I have done previously gave values higher than 2.18 billion tons. This could suggest that I am underestimating the effect of oceans to absorb CO2 (so it appears like more carbon is needed to raise atmospheric CO2).

[-1=e^ipi]

@ Bonam - I am doing linear approximations as well. Though I am trying to get a rough idea of the time lags of global warming (particularly how long it takes for the oceans to catch up to surface temperatures). That is why your model (not really sure what else to call it, give me a different name if you prefer) is a bit insufficient for me (though a nice comparison). The reason for this is my assumption of constant decay did not appear to hold, which means that I couldn't properly estimate equilibrium climate sensitivity.

As for graphing the model vs actual temperature / CO2, I haven't bothered to do that yet because the models I have presented so far to explain both of them are insufficient.

[Bonam]

Dividing this by 275 gives 2.18 billion tons, which is the amount of carbon that needs to be burned in order to increase atmospheric CO2 by 1 ppm.

Though most of the estimates I have done previously gave values higher than 2.18 billion tons. This could suggest that I am underestimating the effect of oceans to absorb CO2 (so it appears like more carbon is needed to raise atmospheric CO2).

To state this more clearly, 2.18 billion tons is the amount of CO2 that needs to be added to the atmosphere to increase concentration by 1ppm. Not all CO2 that is released through oxidizing carbon ends up permanently residing in the atmosphere, since some is absorbed by the oceans (and other processes).

2.6 billion tons of C to increase CO2 concentration by 1ppm is not unreasonable, all that's saying is that for every 1ppm increase in the atmosphere, an additional 0.42 billion tons C are absorbed by oceans/plants/etc (in the short term; in the long term, I think they'd absorb much more).

Edited February 18, 2015 by Bonam

[On Guard for Thee]

Has anyone seen the movie 'A Beautiful Mind'?

I havent but when I peruse this thread now I certainly see a one track mind. Remind you of Nero at all...

[-1=e^ipi]

Also, if the ocean contains ~ 38000 billion tons of carbon (actually, I'll use a better estimate of 37,400 billion tons http://www.skepticalscience.com/human-co2-smaller-than-natural-emissions.htm) and at pre-industrial times there was ~600 billion tons (and assuming that the change in carbon in the ocean since pre-industrial times is relatively small compared to the total carbon in the oceans, which is reasonable given how much more carbon is in the ocean relative to the atmosphere and also the time difference of only 1.5 centuries), then the ratio of carbon in the atmosphere to carbon in the ocean in equilibrium is about 0.016. Therefore, by conservation of mass, the value of G in my recent regressions should be ~-0.016.

Edit: Sorry, it G should be the inverse of this (-62.5).

Edited February 19, 2015 by -1=e^ipi

[On Guard for Thee]

That wasnt carbon it was salt. Tons and tons.

[Bonam]

If the equilibrium ratio of atmospheric to oceanic CO2 is assumed constant and other CO2 sinks are negligible compared to the oceans, that suggests that you need 138 billion tons of C for a 1ppm atmospheric CO2 increase. Observationally, we see it's instead in the 2-3 billion ton range for short time scales. Therefore, the explanation must be that the timescale for CO2 absorbing into the oceans is long compared to the periods studied (~100 years).

Edited February 18, 2015 by Bonam

[-1=e^ipi]

@ Bonam - Yes I agree about your comment regarding 2.18 billion tons. However, I am trying to take into account the effect of the carbon sink absorbing some of that CO2 in my regression model. So the value of the rate of increase of atmospheric CO2 due to human emissions should be 1 ppm per 2.18 billion tons after taking into account the effect of the oceans (and maybe plant life) absorbing excess CO2.

Edit: Yes with respect to your second comment. I am trying to measure that timescale. So far it appears like the decay time for this is ~70 years.

Edited February 18, 2015 by -1=e^ipi

[Bonam]

My first order estimate for the timescale based on the numbers thrown around in this thread would be a process of thinking like this:

• It takes 2.6 billion tons of released carbon to raise CO2 by 1ppm
• But only 2.18 billion is added to the atmosphere
• Therefore 0.42 billion tons are absorbed by the oceans
• At equilibrium, the oceans would instead have absorbed 138 billion tons, meaning 137.58 billion tons remain to be absorbed
• 137.58/138 = 0.997 of released CO2 that will eventually be absorbed by ocean remaining in atmosphere after 1 year
• The characteristic timescale is the time it takes to go to 1/e = 0.37
• 0.997 ^ 330 = 0.37

Therefore my first order estimate would be that the timescale is more like ~330 years.

Thoughts?

Edited February 18, 2015 by Bonam

[-1=e^ipi]

Maybe, but I think that estimate is very reliant on the 2.6 billion tons value, which has a fair amount of uncertainty. Example: If you used 2.9 billion tons instead, then you would get 191 years instead of 300 years. There is also a fair amount of rounding error, so I don't think you can conclude much more than the order of magnitude is ~ a century.

[Bonam]

That's true... was just illustrating a thought process based on the numbers we have at hand without doing a whole lot of math. I usually like to do simple back of the envelope calculations like that to get an order of magnitude estimate before doing detailed work.

[cybercoma]

This is actually a good discussion guys. The methods are interesting. Thanks.

[-1=e^ipi]

So I tried to perform additional restrictions. I tried fixing D = -0.016 and Ts(1876) = 13.5C (roughly pre-industrial temperatures). One thing I noticed is that with all the additional restrictions I am performing, I can rewrite the regression equation as:

dCO2/dt = D*(CO2(0) - C*Ts(0) + C*T(t) - CO2(t))

+ DH*(C*IT(t) + (G-1)*ICO2(t) + (– C*Ts(0) – G*CO2s(0) + CO2(0))*t)

+ H*(CO2(0) - G*CO2s(0) + (G-1)*CO2(t))

+ E*Human(t)

+ E(D+H)*IHuman(t)

+ EDH*IIHuman(t)

This has 6 explanatory factors for the 3 unknowns. So it is much easier to perform the regression (which means matlab has less rounding error when doing matrix inversions). Performing the Gauss-Newton estimation gives me the following 95% confidence intervals:

E: (0.00025 +/- 0.0005) million tons of carbon per ppm

D: (-0.001 +/- 0.012) per year

H: (-0.007 +/- 0.012) per year

E suggests that ~3.9 billion tons of carbon are needed to increase atmospheric CO2 by 1 ppm. The two decay times are not statistically significant. This isn't a very useful result, so I thought I would try to restrict E = 1/2180. With the new restriction I can rewrite the regression equation as:

dCO2/dt – E*Human(t) = D*(CO2(0) - C*Ts(0) + C*T(t) - CO2(t) + E*IHuman(t))

+ DH*(C*IT(t) + (G-1)*ICO2(t) + (– C*Ts(0) – G*CO2s(0) + CO2(0))*t + E*IIHuman(t))

+ H*(CO2(0) - G*CO2s(0) + (G-1)*CO2(t) + E*IHuman(t))

This has 3 explanatory variables for 2 unknowns, so is even easier to estimate. But if I estimate this I get:

D: (-0.02 +/- 0.02) per year

H: (0.00 +/- 0.02) per year

Again, I can't get statistically significant estimates of the two decay values I am interested in.

Looking at the residual, I think the issue is data quality, especially before 1959 where I am using ice-core data. I tried to vary the restrictions and do other things, but get similar results. Maybe I should try to use only data after 1959 and use monthly data rather than annual data.

[-1=e^ipi]

Okay, so I have tried to compile a 'better' data set to estimate this model with because I think the data prior to 1959 is not of good enough quality.

I can get monthly Mauna Loa CO2 data here: ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_mm_gl.txt

I'll use the seasonally detrended monthly data because I don't want the seasonal effects.

For temperature, I can use the monthly HadCRUT4 data here: http://www.cru.uea.ac.uk/cru/data/temperature/HadCRUT4-gl.dat

The problem is that I don't think this data set is seasonally detrended.

To seasonally detrend the temperature data, I can simply perform the regression:

temperature = β₁*January + β₂*February + ... + β₁₂*December + β₁₃*time + error

I add the time variable because I need to take into account the fact that later months in the year occur later in my data set (since I am interested in January 1959 to December 2008). The month variables are just dummy variables. I can then seasonally detrend the data set if I take each temperature, subtract the dummy variable of that month, and then add the mean of the dummy variables.

For CO2 emission data, I am still stuck with: http://cdiac.ornl.gov/ftp/ndp030/global.1751_2008.ems

I can't seem to find freely available monthly CO2 emission data.

To get estimate monthly CO2 emission data, I have to interpolate. However, the data given above is the emissions for the entire year (as opposed to the rate of emissions at a given point in time). Therefore, I have to turn this emission data (from 1959 to 2008) into 51 points on the integral from the start of 1959 to a later point in time. Once I do this, I can perform a cubic spline interpolation to get the estimates of this integral at the end of each month from January 1959 to December 2008. Then I can simply take the derivative to get approximations of the emissions for each month.

I realize that doing this will remove season affects on CO2 emissions (for example, January probably has higher emissions than May because people in the Northern Hemisphere are heating their homes). However, I want to remove seasonal affects, so this isn't an issue for me.

The other thing is that with monthly data, the months have an unequal number of days. So any regression I do will have to be weighted. I'm not sure how to do a Gauss-Newton non-linear regression with weights, but I can probably figure it out.

Edited February 19, 2015 by -1=e^ipi

[Bonam]

The other thing is that with monthly data, the months have an unequal number of days. So any regression I do will have to be weighted. I'm not sure how to do a Gauss-Newton non-linear regression with weights, but I can probably figure it out.

Given the size of other errors and uncertainties, the difference between 30 and 31 days (or the 28/29 day Februaries) is totally negligible. No point complicating things to account for it. Every month can be approximated as 30.4 days just fine.

[-1=e^ipi]

I just realized an error in post #181. G should not be ~ - 0.016, but the inverse of this, which is -62.5. That means my estimates in #189 are even more meaningless than they already were.

Also, a 62.5:1 ratio of ocean dissolved CO2 to atmospheric CO2 in equilibrium might be an overestimate (since some of the carbon in the oceans in that 37400 billion ton value might be in other forms such as methane ice).

http://www.waterencyclopedia.com/Bi-Ca/Carbon-Dioxide-in-the-Ocean-and-Atmosphere.html

"Over the long term (millennial timescales), the ocean has the potential to take up approximately 85 percent of the anthropogenic CO ₂ that is released to the atmosphere."

So maybe a value of -0.85/0.15 = - 5.67 is more appropriate.

I wonder if it is possible to approximately derive this value using the Henry's constant and knowledge of the amount of water on earth...

Let's see, the volume of water on Earth is ~1.386 x 10^9 km^3

http://water.usgs.gov/edu/gallery/global-water-volume.html

The ratio of dissolved carbon per unit volume in water and dissolved carbon per unit volume in air is approximately 0.8317 at a temperature of 288 K according to wikipedia.

Now the average temperature of the water in the oceans should be below the average surface temperature of 15C, but below the temperature at which water is the most dense (4C). So if I assume that all water is 10C (or 283 K) as a compromise.

According to wiki, I should therefore the ratio by exp(2400*(1/283 - 1/288)) to get 0.9636.

In post #177 I showed that there are 1.819 x 10^20 mols of gas in the Earth's atmosphere.

If this gas follows the ideal gas law (PV = nRT -> V = nRT/P) then I can get the equivalent volume of this air if all the air had a temperature of 15 C and pressure of 101325 Pa (i.e. the conditions at sea level, which is relevant for considering CO2 concentrations in air vs CO2 concentrations in ocean).

This gives 4.30 x 10^9 km^3.

This means that in equilibrium there should be 0.9636*1.386/4.30 = 0.311 as much CO2 in the oceans as the atmosphere in equilibrium, which would suggest in equilibrium, the oceans would absorb 0.311/1.311 = 23.7% of the CO2 emitted. This is very different from the 85% value, or the 98.4% value...

[-1=e^ipi]

Maybe the reason for the discrepancy indicated in my last post is the fact that oceans are less acidic than pure-water. The current pH of the ocean is ~8.1, where as the pH of pure water is 7.0. Perhaps having a more basic aqueous solution allows for more carbonic acid to be absorbed (relatively to atmospheric CO2 concentrations) in equilibrium.

This link on nitric acid suggests that the Henry's constant is inversely proportional to the acidity constant. http://webbook.nist.gov/cgi/cbook.cgi?ID=C7697372&Mask=10

And the pH depends logarithmically on the acidity constant (pH is a log base 10 scale). http://en.wikipedia.org/wiki/Acid_dissociation_constant

This suggests that in equilibrium, sea water would be able to have 10^(8.1 - 7.0) = 12.59 times more dissolved CO2 (plus carbonic acid and other things) for a given atmospheric CO2 concentration. Thus that 0.311 should be replaced with ~ 3.92. So in equilibrium, the oceans would absorb ~ 3.92/4.92 = 79.7% of the CO2 emitted.

This is reasonably close to the 85% value. So I guess I should assume that G ~ -5.67.

Though, again chemistry isn't my strongest point. So maybe I need to consult a chemist.

[-1=e^ipi]

So I retried the regressions of post #189 with G = -5.67 for the 1876-2009 data set. In both cases the decay values were negative, which makes no sense. Here are the 95% confidence intervals for the second regression:

D: (-0.020 +/- 0.001) year^-1
H: (-0.0014 +/- 0.0011) year^-1

I guess I need to use the post 1959 monthly data set after all.

Edited February 20, 2015 by -1=e^ipi

[-1=e^ipi]

I think I figured out what's going on here and why I am getting negative values.

- The time scales of decay for both the temperature sink and the carbon sink are both long relative to the time spawn of the data.

- The effect of the temperature sink and the effect of the carbon sink on changes in CO2 are both strongly correlated.

- The effect of the temperature sink and the effect of the carbon sink are in opposite directions.

Put these 3 together, and it suggests at least two stable 'solutions' when performing the Gauss-Newton estimate (one where both decay values are negative, and another where both decay values are positive).

In fact, if I write (D2, H2) to correspond to the solution where D and H are negative, then the positive solution (D1, H1) should be roughly:

D1 = G*H2*(meanCO2 - CO2s(1876))/C/(meanT - Ts(1876));

H1 = C*D2*(meanT - Ts(1876))/G/(meanCO2 - CO2s(1876));

where meanT is the mean of the temperature across the entire data set, and meanCO2 is the mean of the CO2 concentrations across the entire data set. You can get the above relation if you take the first two taylor approximations of the change in CO2 concentrations over time after assuming that the decay rates are negligibly slow.

This would suggest that

D = (0.029 +/- 0.021) year^-1

H = (0.00097 +/- 0.00017) year^-1

is the other Gauss-Newton solution.

So this would suggest that the timescale of decay for the heat sink is ~34 years and that the timescale of decay for the carbon sink is ~1032 years. Of course since 34 years is less than the 133 years of the dataset, the premise I used to get these values may be sufficiently violated to make these estimates inconclusive.

Edited February 20, 2015 by -1=e^ipi

[-1=e^ipi]

Maybe there is a physical reason why the decay time of the carbon sink is longer than the decay time of the heat sink.

The ocean can be warmed via both convection and conduction/diffusion.

The ocean can absorb CO2 via diffusion. However, since the surface of the ocean is generally warmer than below the surface of the ocean, and warmer water can hold less CO2, convection cannot result in the ocean absorbing CO2. Since diffusion takes longer then convection for oceans, a longer decay time for the carbon sink than the heat sink should be expected.

Edited February 20, 2015 by -1=e^ipi

[-1=e^ipi]

If the time scale of decay of the earth's carbon sink is on the order of 1000 years (so is long compared to the timescale of the data) then this suggests that the model of post #149 is approximately correct (because the carbon sink term becomes a constant + a term proportional to CO2). Unfortunately, when I tried to estimate the model of post #149 last, I had 2 mistakes in my code. I just tried to run the model of post #149 again, the results are okay but the terms are not individually statistically significant. So maybe I should take the model of post #149 and apply some additional restrictions (like I did the past 2 pages) to get a good estimate of the decay time of the Earth's heat sink to equilibrium.

Edited February 20, 2015 by -1=e^ipi

[On Guard for Thee]

Dont you guys ever get bored regurgitating this nonsense!

[-1=e^ipi]

Okay, I tried a few things. One of the things I tried was to use the 1959-2008 monthly & seasonally detrended data set with the older model that assumed that the decay time of the earth's carbon sink to equilibrium is large relative to the 50 year time period of observation. So my model is:

dCO2/dt = E*Human(t) + C*dTs/dt + G*dCO2s/dt

dTs/dt = D*(T(t) - Ts(t))

dCO2s/dt = H*(CO2(t) - CO2s(t))

which gives me the equation:

dCO2/dt = [(D+H)*CO2(0) – GH*CO2s(0) – CD*Ts(0)] + E*Human(t) + E(H+D)*IHuman(t) + EDH*IIHuman(t) +CD*T(t) + CDH*IT(t) + (H(G-1)-D)*CO2(t) + HD(G-1)*ICO2(t) + DH(– C*Ts(0) – G*CO2s(0) + CO2(0))*t;

from which I can try to estimate the unknown parameters using the Gauss-Newton non-linear estimation method.

I also figured out how to perform the weighted Gauss-Newton non-linear estimate, and I corrected for the fact that not all the months have an equal number of days.

When I tried to estimate the above model, most of the coefficients were individually not-statistically significant (since the estimates are all correlated with each other). So like earlier, I have to add restrictions.

I assume that C = 29.55, is the increase in CO2 ppm that is released by the earth's carbon sink when it warms by 1 celcius.

I assume that A = 280 B, since 280 ppm corresponds to roughly the atmospheric CO2 concentrations that the Earth had during recent times where there was comparable solar activity as the late 20th century.

I assume that E = 1/2180, since it takes ~ 2.18 billion tons of carbon to be burned by humans to increase atmospheric CO2 by 1 ppm.

Now even with these assumptions, because I do not know what the temperature of the Earth's heat sink was at in January 1959, the estimates are not statistically significant because the uncertainty on the heat sink characteristic temperature in 1959 is too large. I could try to fix this temperature to a reasonable value (such as a temperature anomaly of -0.2C, which corresponds to ~13.8C), in which case I can simplify the earlier equation to:

dCO2/dt – E*Human = B*(275 – CO2(t))
+ D*(CO2(0) – CO2(t) + C*T(t) –C*Ts(0) + E*IHuman)
+ BD*(275*t – ICO2)

which as 3 explanatory variables and only 2 unknowns, so is much easier to estimate (and doesn't have the issue of multiple regions of convergence for the Gauss-Newton estimation, like the more complicated models do).

A problem is that any result I get from an estimation of the above equation will depend on my choice of the heat sink temperature in 1959. To try to deal with this issue, I first assume that the temperature of the heat sink in January 1959 is ~13.8 C (which is roughly what the temperatures were the few decades prior to 1959). Then I perform the weighted Gauss-Newton estimation of the above equation using data from January 1959 to December 2008. This gives me a value for D, which is the rate of decay of the heat sink towards equilibrium. I can then use this estimate of D, plus the fact that the characteristic temperature of the earth's heat sink in 1850 should be ~13.5 C (as this was roughly pre-industrial temperatures), plus all of the HadCRUT4 temperature data from 1850-1958 to re-estimate the heat sink temperature in 1959. I can then use this new value of the characteristic heat sink temperature in 1959 to perform the Gauss-Newton estimation again, and keep re-estimating the characteristic heat sink temperature in 1959 until convergence is reached.

Using this method, I get that the characteristic heat sink temperature of the Earth in 1959 is ~13.859 C, and I get the following estimates for B and D:

B: (0.0165 +/- 0.0023) year^-1

D: (0.029 +/- 0.051) year^-1

Now assuming that the decay of the earth's carbon sink to equilibrium is large relative to 50 years and that the earth's oceans can store ~85% of carbon emitted by humans in the long run, this value of B suggests that the decay time of the earth's carbon sink is (0.0165/5.667)^-1 = ~344 years. Since 344 > 50, this suggests that the assumption is consistent with the evidence.

The value of D suggests that the decay time of the earth's heat sink is ~34 years, although with a large uncertainty.

These results seem moderately conclusive.

Also, if I use the calculated values to plot the heat sink temperature vs the surface temperature, I get that the temperature difference between the two has been increasing at a rate of 0.007 C per year. In comparison, surface temperatures have been increasing by about 0.013C per year during this period. So the characteristic heat sink has only been warming at about half the rate of surface temperatures.

[-1=e^ipi]

Just to clarify a few things, the reason I have been constantly assuming a constant decay rate towards equilibrium is because it makes the math and calculations relatively easy. But realistically this assumption is false. Realistically there are many different mechanisms that decay at their own rate, and the sum of multiple exponential functions each with different decay rates does not give you an exponential function. I was just hoping that the exponential assumption was good enough to get decent results. When it was clear to me that it wasn't, that's why I started asking what the timescale was of different responses to climate change, because if I could understand these time scales then I might be able to replace the single exponential decay model with a multiple exponential decay model.

However, even things like the ocean heat response do not have a very precise characteristic response time. I guess this is because deeper waters have a longer response time than shallower waters (so the pacific has a longer decay time than the atlantic, the atlantic has a longer decay time than the indian, the indian has a longer decay time than the arctic, the arctic has a longer response time than the mediterranean and so on). From the results I have above, I don't think I'll be able to conclude more than the earth's heat sink response is on the order of decades and the earth's carbon sink response is on the order of millennia.

I tried doing some online searching to maybe find some idea of how to tackle this issue. The main problem is that the functional form of the impulse response function (if the concept of an impulse response function is a valid assumption) to a change in radiative forcing is unknown, so I cannot estimate it. The impulse response function can probably be treated as the sum of exponential response functions, but the decay times of these response functions is unknown, and one might need an infinite number of exponential response functions to represent the true impulse response function. Most likely the density of exponential response functions decreases with the decay time of those exponential functions for the representation of the true impulse response function, but not much else is known a priori. I thought about maybe representing the impulse response function as an exponential response function with a decaying decay rate (where the decay rate has a constant decay rate), but I'm not sure how I could test the validity of the assumption for the functional form. I could look at general circulation model impulse response functions and see if there are any patterns in the decay rate to see if I can get an approximate functional form, but that would depend highly on all the assumptions that go into the general circulation model.

I've found other people making the assumption of constant decay towards equilibrium (example: http://www.gfdl.noaa.gov/blog/isaac-held/2011/03/05/2-linearity-of-the-forced-response/) even James Hansen. But few attempt to relax the assumption of constant decay towards equilibrium.

This recent 2012 paper http://arxiv.org/ftp/arxiv/papers/1111/1111.5177.pdf by Van Hateren is probably one of the best climate science papers that I have read in a while (plus it is free to read). Van Hateren has a really simple and effective approach to tackling this problem. He/she just approximates the impulse response function with 6 exponential response functions (with 0.5, 2, 8, 32, 128 and 512 year decay times respectively). The choice of the 0.5 year decay time is based upon the fact that the fastest response mechanisms have a decay time of this value (he/she references a paper that showed that the response to the Pinotubo effect was 6-7 months, and as I have shown earlier in this thread, the direct response to additional CO2 has a decay time of ~0.5 years). He/she then decides to multiple the decay time by 4 to get new decay times (since you want decay times dense enough to represent the impulse response function, but you should have higher density for faster response times). Going up to a 512 year response time covers should be sufficient to calculate the equilibrium climate sensitivity (though if he goes up to 2048 years, that is probably enough to calculate the earth system sensitivity).

Van Hateren also covers some other things (such as the ocean carbon response to atmospheric CO2 is on the order of 100 years) and obtains estimates of the equilibrium climate sensitivity of ~2.0 C and ~2.5 C. He/she considers recent instrumental data (he/she uses similar data sets that I have used such as HadCRUT), but he/she also considers the response of the climate to solar forcing over the past 1000 years. His/her uncertainties are relatively small and his/her methodologies are pretty sound. It's far more sound than anything I have done in this thread.

Is this the best approach to take? Just treat the climate response function as a sum of exponential response functions with pre-determined decay times.