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What is the correct value of Climate Sensitivity?


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Okay, so I went through with the non-linear regression of the change in CO2 on a constant, CO2, Human emissions, Volcanic emissions, the integral of Human emissions, the integral of Volcanic emissions, time, and the integral of atmospheric CO2. I used the data from 1876 to 2009.

Basically I did the regression

dCO2/dt = (A - D*CO2(0)) + (B + D)*CO2(t) + E*CO2_emissions + F*Volcanic_Aerosols + E*D*Integral(0 to t; CO2_emissions*dt) + F*D*Integral(0 to t; Volcanic_Aerosols*dt) - (AD)*t - (BD)*Integral(0 to t;CO2(t)*dt))

The 95% confidence intervals for the coefficients A, B, D, E, and F are: (262233 +/- 10191481), (-874 +/- 361), (2.98 +/- 0.0000018)x10^-4, (0.141 +/- 0.000025) and (1.95 +/- 0.000011)x10^-3. Though I will point out that I got the error message 'Matrix is close to singular or badly scaled' when doing the Gauss-Newton method.

Only the constant is not statistically significant, which isn't that relevant. E and F are highly significant, which suggests that both volcanism and human emissions are significant at explaining changes in CO2 concentrations over time.

Most relevant is the estimate of D, which if you invert it suggests that the characteristic temperature of Earth's heat sink has a decay time of 3355 years.

Here we go again. Have you figured out that human breath does not add to the CO2 level.

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Basically I did the regression

dCO2/dt = (A - D*CO2(0)) + (B + D)*CO2(t) + E*CO2_emissions + F*Volcanic_Aerosols + E*D*Integral(0 to t; CO2_emissions*dt) + F*D*Integral(0 to t; Volcanic_Aerosols*dt) - (AD)*t - (BD)*Integral(0 to t;CO2(t)*dt))

The 95% confidence intervals for the coefficients A, B, D, E, and F are: (262233 +/- 10191481), (-874 +/- 361), (2.98 +/- 0.0000018)x10^-4, (0.141 +/- 0.000025) and (1.95 +/- 0.000011)x10^-3. Though I will point out that I got the error message 'Matrix is close to singular or badly scaled' when doing the Gauss-Newton method.

Only the constant is not statistically significant, which isn't that relevant. E and F are highly significant, which suggests that both volcanism and human emissions are significant at explaining changes in CO2 concentrations over time.

If E is your constant for human CO2 emissions and F is your constant for volcanic emissions, and your values are:

E = 0.141

F = 0.00195

does that not suggest that volcanic emissions are a ~1% effect compared to human CO2 emissions, which matches fairly well with the volcanic CO2 emission rates known in the literature to be about 1% of human CO2 emission rates?

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-1, what's the goal of your calculations regarding volcanic aerosols?

A few things. Obviously if you are trying to explain changes in temperature over time you want to use the aerosol data directly. But if I am trying to explain CO2 emissions over time I need a proxy for volcanic CO2 emissions. Since aerosols decay much more quickly that CO2, I have to perform the sharpening to get a decent proxy of volcanic CO2 emissions. This is done so that the unexplained variability in my data is lower so that I can get good estimates of the parameters that I am interested in. There are a few things the regression explaining changes in CO2 over time can help answer. For one, it can help answer 'where the heat is going?' and help make predictions about how temperatures will change in the future. Also, it can help answer how much additional CO2 will be released (via ocean warming and permafrost melting) due to an increase in temperatures.

But as far as I know, average annual CO2 emissions from volcanoes are already fairly well known, at about 200 million tons / year (in comparison to about 35 billion tons from human emissions).

Yeah, that 200 million tons / year value is fairly well known. However, it is an average and there is quite a bit of uncertainty on that value. Also, I need volcanic emissions for each year since 1876 to do my time series analysis. All I need is something that is proportional to volcanic emissions, since the regression scales it for me to explain the data. Also, using CO2 data directly can cause issues of reverse causality which is something you usually want to avoid when doing regressions.

I do this for human emissions as well despite the fact that Human CO2 emissions are far more well known and it is fairly well known how many metric tonnes of CO2 correspond to how many ppm of CO2. I am basically assuming that I do not know what these values are because I don't want to make assumptions that may skew my results and cause me to underestimate my uncertainty.

That's actually one of the problems of GCMs and some of the more pure physics approaches to estimating these interactions. Often they underestimate their uncertainty because they don't take into account the uncertainty on all the scaling parameters that they obtain from the scientific literature.

Also, can you explain what you mean regarding calculating a characteristic heat sink temperature for Earth? Quickly googling around, I can't find much about this term. Articles that talk about a heat sink for Earth are mostly referring to the ocean heat sinks, where heat from the surface is subducted to the deep oceans. If this is the heat sink you are referring to, it's temperature is also known.

It's not a term that is used. I was just using known words to describe a concept that I did not have a word for. Basically, you can think of the oceans + glaciers + permafrost as a 'heat sink'. They don't necessarily warm as fast as the surface. Changes in forcing (due to changes in solar irradiance, CO2, etc.) cause the surface to warm, which then warms the heat sink. If you think of this heat sink as having a single temperature, call it the characteristic temperature, then the rate of heat transfer from the surface to the heat sink is going to be proportional to the temperature difference. Just think of the characteristic temperature as a useful concept for explaining and predicting heat transfer.

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All we really know is a majority think that CO2 is a "moderate" concern. And even if they did support your views on policy their opinion is largely irrelevant since they are not qualified to comment on the science of energy production.

What we know is that a vast number think that producing CO2 could lead to serious consequences therefore there is good reason to see why billions of human beings have concluded that reducing CO2 is the proper course of action. We're as qualified to conclude that as we're wise enough to recognize the limitations of our knowledge while still recognizing the implications what the expert opinion and consensus is saying.

Edited by eyeball
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does that not suggest that volcanic emissions are a ~1% effect compared to human CO2 emissions, which matches fairly well with the volcanic CO2 emission rates known in the literature to be about 1% of human CO2 emission rates?

Not necessarily. Again the units I used for the volcanic emission data is somewhat arbitrary so that has to be taken into account. Though I can check this question for you. Just give me a minute.

Edited by -1=e^ipi
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It's not a term that is used. I was just using known words to describe a concept that I did not have a word for. Basically, you can think of the oceans + glaciers + permafrost as a 'heat sink'. They don't necessarily warm as fast as the surface. Changes in forcing (due to changes in solar irradiance, CO2, etc.) cause the surface to warm, which then warms the heat sink. If you think of this heat sink as having a single temperature, call it the characteristic temperature, then the rate of heat transfer from the surface to the heat sink is going to be proportional to the temperature difference. Just think of the characteristic temperature as a useful concept for explaining and predicting heat transfer.

If the heat sink is glaciers + permafrost + deep ocean, then it's temperature is likely very close to 0 degrees C, and likely to remain static at 0 degrees C even as surface/atmospheric temperatures rise by a few C.

Edited by Bonam
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What we know is that a vast number think that producing CO2 could lead to serious consequences therefore there is good reason to see why billions of human beings have concluded that reducing CO2 is the proper course of action. We're as qualified to conclude that as we're wise enough to recognize the limitations of our knowledge and listen to what the expert opinion and consensus is saying.

Expert opinion in fields related to energy production say that reducing CO2 is not a remotely viable option at this time (in the case of nuclear the viability problem comes from politics rather than economics). The only "solution" that economic experts propose is a carbon price at levels that may slow the growth of emissions slightly but will likely never result in reductions. There are no qualified experts calling for policies that will actually reduce CO2 emissions. Only activists speaking about fields where they have no expertise. Edited by TimG
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@ Bonam:

I just realized I mixed up two numbers when I made my last post. The magnitudes of E and F should be 10.8 and -15601623 respectively.

Actually, looking at my code again, I forgot a squared. Which means my error estimates are all off. I'll correct this, just give me a minute.

Here we go again. Have you figured out that human breath does not add to the CO2 level.

No. And nether do the vast majority of scientists. The consensus on cellular respiration is pretty well established.

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@ Bonam:

I just realized I mixed up two numbers when I made my last post. The magnitudes of E and F should be 10.8 and -15601623 respectively.

Actually, looking at my code again, I forgot a squared. Which means my error estimates are all off. I'll correct this, just give me a minute.

No. And nether do the vast majority of scientists. The consensus on cellular respiration is pretty well established.

So you do understand the closed carbon cycle.

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Expert opinion in fields related to energy production say that reducing CO2 is not a remotely viable option at this time (in the case of nuclear the viability problem comes from politics rather than economics). The only "solution" that economic experts propose is a carbon price at levels that may slow the growth of emissions slightly but will likely never result in reductions.

Expert opinion in fields related to ecological sustainability say that by not having reduced CO2 we've already started making ourselves economically unviable. I heard and felt the result of this opinion from government fisheries and economic scientists years ago, Conservative government scientists too I might add. They included their concerns about climate change in their advice to fisheries policy makers that impacted thousands of people on the coast.

There are no qualified experts calling for policies that will actually reduce CO2 emissions. Only activists speaking about fields where they have no expertise.

I haven't seen anything at all that would suggest you're qualified to make that assessment.

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Okay, after fixing the mistake in the code the 95% confidence intervals for the coefficients A, B, D, E, and F are: (262233 +/- 3588895), (-829 +/- 12326), (3.16 +/- 153.66)x10^-4, (46.22 +/- 1.37) and (-15601554 +/- 0.1). So Human emissions and Volcanic emissions are significant. Not sure why the volcanic parameter is negative, maybe cause of the 6 month lag I used. Also, My method of error estimation is generally an overestimation for non-linear estimations. I could do f-tests to determine more accurate values.

I'll have to go through my code when I have more time to make sure everything is done properly. I'm skeptical of my results.

Edit: These results are nonsense. See post #149.

@ Bonam - if I multiply the mean of volcanic emissions (in the arbitrary units) by the estimate of F I get -1 x 10^-5. Where as if I multiply human emissions in 2008 by the estimate of E I get 4 x 10^-5. So that's only a factor of 4 difference.

Edited by -1=e^ipi
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I'm skeptical of my results.

Yes those results look fishy. The large errors on the first three constants are indicative of the solution not having converged properly. I've done a lot of matlab curve fitting and regression analysis and you pretty much can't trust the result at all until the errors on all the constants are less than a few %. Graph your results to see how well they fit your data points and you'll likely see a pretty terrible fit (with those kinds of error bars on the constants, I'd expect the fit is likely orders of magnitude off for most points).

Edited by Bonam
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I haven't seen anything at all that would suggest you're qualified to make that assessment.

That is just because I don't say things that you agree with. If I parroted the garbage that you want to believe you would be more than willing to accept my assessment of experts.

You also miss the point of this thread. You obsess about experts but you really only listen to experts that you agree with and you don't really care if the so called experts is qualified to make the claims as long as they confirm your beliefs. This bigotry on your part shows that your appeal to experts is just a ruse to hide what is really a ideological belief system. Other people are not like you, they look at evidence for themselves and don't simply take the word of experts even if the opinion of qualified experts within their field of expertise is a good place to start.

Edited by TimG
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I'll do the unrestricted regression for comparison (there are definitely no mistakes with that).

dCO2/dt = H + I*CO2(t) + J*CO2_emissions + K*Volcanic_Aerosols + L*Integral(0 to t; CO2_emissions*dt) + M*Integral(0 to t; Volcanic_Aerosols*dt) + N*t + O*Integral(0 to t;CO2(t)*dt)) + error.

The 95% confidence intervals of H, I, J, K, L, M, N and O are (4.28 +/- 3.64), (-1.39 +/- 1.34)x10^-2, (7.49 +/- 12.00)x10^-4, (9.21 +/- 5.11)x10^-1, (-1.52 +/- 6.15)x10^-4, (-2.39 +/- 3.14), (-4.74 +/- 3.69)x10^-2, and (1.18 +/- 21.32)x10^-4.

Basically nothing is statistically significant for the unrestricted model. The issue might be that I don't have enough data to estimate my parameters. Maybe I should use monthly data or use a longer time period.

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I have to go to sleep because I have work tomorrow. But I might be way overthinking things. Maybe I should just regress change in CO2 on a constant, CO2, Human emissions and Volcano emissions and then just look at the residual.

Edited by -1=e^ipi
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I'll do the unrestricted regression for comparison (there are definitely no mistakes with that).

dCO2/dt = H + I*CO2(t) + J*CO2_emissions + K*Volcanic_Aerosols + L*Integral(0 to t; CO2_emissions*dt) + M*Integral(0 to t; Volcanic_Aerosols*dt) + N*t + O*Integral(0 to t;CO2(t)*dt)) + error.

The 95% confidence intervals of H, I, J, K, L, M, N and O are (4.28 +/- 3.64), (-1.39 +/- 1.34)x10^-2, (7.49 +/- 12.00)x10^-4, (9.21 +/- 5.11)x10^-1, (-1.52 +/- 6.15)x10^-4, (-2.39 +/- 3.14), (-4.74 +/- 3.69)x10^-2, and (1.18 +/- 21.32)x10^-4.

Basically nothing is statistically significant for the unrestricted model. The issue might be that I don't have enough data to estimate my parameters. Maybe I should use monthly data or use a longer time period.

If you can't get a good fit to the data with a certain model, then it's possible the model is simply wrong and can't get a good fit no matter what. Like trying to fit a parabola with a straight line.

I have to go to sleep because I have work tomorrow. But I might be way overthinking things. Maybe I should just regress change in CO2 on a constant, CO2, Human emissions and Volcano emissions and then just look at the residual.

Starting with a simpler model might yield an easier fit.... but throwing out the physics/reality of the situation can also lead you into trouble. But in any case, I'm not sure of why you need to make the distinction between human emissions and volcanic emissions when trying to determine rates of change of CO2 concentration. CO2 is CO2 whatever its source.

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That is just because I don't say things that you agree with. If I parroted the garbage that you want to believe you would be more than willing to accept my assessment of experts. You also miss the point of this thread. You obsess about experts but you really only listen to experts that you agree with and you don't really care if the so called experts is qualified to make the claims as long as they confirm your beliefs.

It's not a matter of belief, it's a matter of evidence - of vast numbers of scientists in long-term agreement on a matter that implies a need to do something, starting 10 years or more ago. Of course I care about qualifications, that's why I'm impressed by sources that consistently include virtually every advanced national academy of science, association of universities, and discipline of science on the planet.

This bigotry on your part shows that your appeal to experts is just a ruse to hide what is really a ideological belief system. Other people are not like you, they look at evidence for themselves and don't simply take the word of experts even if the opinion of qualified experts within their field of expertise is a good place to start.

Only a very small number of people would ignore such a sheer weight of evidence, usually the sort that still cling to theories of globe spanning conspiracies of eco-feminists, terrorists and fraudulent research grant applicants.

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Only a very small number of people would ignore such a sheer weight of evidence, usually the sort that still cling to theories of globe spanning conspiracies of eco-feminists, terrorists and fraudulent research grant applicants.

You have absolutely no idea what the "weight of the evidence" says which makes pretty ridiculous for you to lecture others about ignoring it. All you do is parrot the BS you have been fed by Greenpeace and other ideologues and rent seekers.

There are plenty of people who actually do care "what the weight of the evidence" says and what it does not say and many of those are skeptics. The posts by -1 are his efforts to work through one part of the evidence and make his own decisions.

I realize that your desire to live in a little bubble of fantasy means you cannot accept that rational people can look at the evidence and conclude that your pet policies are a waste of time and money. Your appeal to the authority of experts who don't actually say the things you claim is laughable.

Edited by TimG
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By evidence I meant the weight of consensus. If I see the IPCC, and all the academies and university's and science disciplines etc etc cross over to your side of the scale I'll conclude new information must have come to light that's changed everything. I'll try to digest and understand that new information but if I can't I'll respect the limits of my ability to do so and probably defer to the weight of informed opinion on the matter as I suspect billions of other lay people will too.

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You have absolutely no idea what the "weight of the evidence" says which makes pretty ridiculous for you to lecture others about ignoring it. All you do is parrot the BS you have been fed by Greenpeace and other ideologues and rent seekers.

There are plenty of people who actually do care "what the weight of the evidence" says and what it does not say and many of those are skeptics. The posts by -1 are his efforts to work through one part of the evidence and make his own decisions.

I realize that your desire to live in a little bubble of fantasy means you cannot accept that rational people can look at the evidence and conclude that your pet policies are a waste of time and money. Your appeal to the authority of experts who don't actually say the things you claim is laughable.

Of course he does, you would have to be a naysayer with your head totally buried in the sand to be able to ignore the overwhelming weight of evidence. Or maybe busy playing Xs and Os. :)

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I'm not sure of why you need to make the distinction between human emissions and volcanic emissions when trying to determine rates of change of CO2 concentration. CO2 is CO2 whatever its source.

Because I am trying to account for non-ocean/permafrost emissions to get ocean/permafrost emissions to be able to estimate some useful parameters. The reason I am interested in these parameters is because my earlier regression that had change in temperature as the dependent variable and yielded a climate sensitivity of ~2 had a very low time scale of decay. So I need to figure out a good way to relax the assumption of constant decay rate towards equilibrium for temperature in order to get a better estimate of climate sensitivity. The regression could also help explain other things, such as temperature changes due to changes in the Length of the Day generally lag by ~6 years.

Anyway, I've tried some simpler regressions.

I regressed a constant, CO2, human emissions, and volcanic emissions on changes in CO2. The residual seems to behave very differently prior to 1958 compared to after it. This probably is a result of the fact that I use Mauna Loa CO2 data after 1959, where as before then I have to use ice-core data. The ice core data is less accurate, and was smoothed by 5 years, which means it is unlikely to show any spikes in CO2 output that might be due to volcanic activity.

Using only the 1959-2009 period, volcanism isn't statistically significant and has a negative effect.

From removing volcanism from the equation and looking at the residual, I see no correlation between the residual and volcanic activity. Perhaps I need to use monthly data instead, but I don't have monthly CO2 output data (though I could interpolate and then add dummies to count for seasonal effects).

One possibility is that the volcanic aerosols decrease temperature, which causes a reduction in CO2 output which hides the CO2 output that is due to volcanism. But when I add temperature to the regression, the residual doesn't look much different.

Another possibility is that the CO2 emission data is 'corrupted' and already includes volcanic emissions. So I looked at the residual of both change in CO2 and change in Human emission output after removing the exponential trend. But neither of the residuals resemble volcanic activity.

I tried changing things such as using volcanic aerosol data directly, but I seem to get the same result. As far as I can tell, volcanism cannot significantly explain the variation in changes in atmospheric CO2 over time. Or at least this is true for large volcanic eruptions such as Krakatoa and Pinotubo.

Maybe the vast majority of volcanic CO2 comes from flood volcanoes or from undersea rifts and this CO2 output is relatively constant since minor eruptions are far more frequent. From my understanding of volcanism, this could very well be the case since the volcanoes that create the massive and infrequent erruptions are rarer and they tend to release a higher stratospheric aerosol to CO2 ratio (thus the majority of the volcanic CO2 may not show up in the aerosol data). If that is the case, then perhaps I should just treat volcanic CO2 emissions as a constant and remove the volcanic aerosol proxy from the regression because it doesn't seem to explain anything.

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If I see the IPCC, and all the academies and university's and science disciplines etc

The trouble is you don't have a clue what the IPCC et. al. actually says. Your opinion is based entirely on what Greenpeace and other ideologues tell about what the IPCC says which is quite far from the truth. If you actually looked at the evidence instead bragging about how you are ignorant and proud you might be surprised what it actually says.
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I feel so dumb right now.

I made a very basic mistake back in post #94.

dCO2/dt - (dCO2/dt)_human - (dCO2/dt)_volcano - A - B*CO2 = C*D*(T - Ts). (1)

dCO2/dt - (dCO2/dt)_human - (dCO2/dt)_volcano - A - B*CO2 = C*dTs/dt. (2)

CO2(t) - CO2(0) - Integral(0 to t; (dCO2/dt)_human*dt) - Integral(0 to t;(dCO2/dt)_volcano*dt) - At - B*Integral(0 to t;CO2(t)*dt) = C*(Ts(t) - Ts(0)). (3)

Substituting (3) into (1) gives:

dCO2/dt - (dCO2/dt)_human - (dCO2/dt)_volcano - A - B*CO2(t) = D*(CO2(t) - CO2(0) - Integral(0 to t; (dCO2/dt)_human*dt) - Integral(0 to t;(dCO2/dt)_volcano*dt) - At - B*Integral(0 to t;CO2(t)*dt))

I mixed up T(t) - Ts(t) with Ts(t) - Ts(0).

Rather than substituting, I should add (1) with D*(3). Then I get:

dCO2/dt - (dCO2/dt)_human - (dCO2/dt)_volcano - A - B*CO2(t) + D*CO2(t) - D*CO2(0) - D*Integral(0 to t; (dCO2/dt)_human*dt) - D*Integral(0 to t;(dCO2/dt)_volcano*dt) - ADt - BD*Integral(0 to t;CO2(t)*dt)= C*D*(T(t) - Ts(0))

If I replace change in atmospheric CO2 due to human emissions with E times human emissions and I remove the volcanic emissions by sticking it into the constant term (see my last post that suggests that volcanic CO2 eruptions are roughly constant with time) then I get:

dCO2/dt - E*human_emissions(t) - A - B*CO2(t) + D*CO2(t) - D*CO2(0) - ED*Integral(0 to t; human_emissions(t)*dt) - ADt - BD*Integral(0 to t;CO2(t)*dt)= C*D*(T(t) - Ts(0))

Isolating for dCO2/dt gives:

dCO2/dt = (A + D*CO2(0) - CD*Ts(0)) + (B - D)*CO2(t) + E*human_emissions(t) + ED*Integral(0 to t; human_emissions(t)*dt) + ADt + BD*Integral(0 to t;CO2(t)*dt) + CD*T(t)

There are 7 dependant variables, but only 6 unknown parameters (A, B, C, D, E and Ts(0)).

So I can can estimate this by performing a regression with 1 restriction. For example:

dCO2/dt = β0 + β1*CO2(t) + β2*Human_emissions(t) + β3*Integral(0 to t; Human_emissions(t)*dt) + β4*t + β5*Integral(0 to t;CO2(t)*dt)) + β6*T(t) + error

where the restriction is β1 = β5β2/β3β3/β2. Of course this restriction is non-linear. So I'll have to derive things and write the code to do the Gauss-Newton estimation.

Also, with this regression, I can get an estimate of C, which means that I can calculate the characteristic heat sink temperature. So forget what I said earlier about not being able to calculate it.

Edit: this error was also why I was getting nonsense earlier.

Edited by -1=e^ipi
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Okay, so I ran the unrestricted model: dCO2/dt = β0 + β1*CO2(t) + β2*Human_emissions(t) + β3*Integral(0 to t; Human_emissions(t)*dt) + β4*t + β5*Integral(0 to t;CO2(t)*dt)) + β6*T(t) + error

Over the 1876-2009 period, nothing is very statistically significant. This maybe because the ice core CO2 data acts differently from the Mauna Loa CO2 data. If I do the same thing but over the 1959-2009 period (so only Mauna Loa data) things improve but the β1 and β5 are not very significant, which makes my estimates in the restricted model not significant.

I think this suggests that either there is no excess CO2 decay effect (which wouldn't make sense because then equilibrium wouldn't be reached) or that the CO2 decay effect is too small and the time period is too short a period to measure this CO2 decay effect (more likely, since my understanding is this would be on the order of hundreds or thousands of years).

In that case, perhaps I should drop the CO2 decay effect from the model on the basis that 50-100 years is too short a period for it to be significant. I could always measure it using ice core data that covers hundreds or thousands of years.

Edited by -1=e^ipi
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