-1=e^ipi

Okay, I tried the above regression and got nonsense. I tried simplifying it by removing the flux data (since it is strongly correlated with irradiance), but that didn't help much. Since I'm using the Gauss-Newton method to try to estimate the model, maybe the problem is that my initial guess is too far off (I'm currently first performing a linear regression to get an initial guess).

I’ll try to go over the basic idea of the model. Suppose initially the climate is in equilibrium and that you have a small change in forcing dF at time t = 0, and that overtime the climate will decay exponentially towards equilibrium with a decay time of τ. Then the temperature as a function of time after t = 0 will be T(t) = T(0) + (γdF)(1 – exp(-t/τ)), where γ is a constant that gives the equilibrium change in temperature when multiplied with the change in forcing dF. For a month that occurs after time t = 0, one can calculate the average temperature of this month. If the start of month i is starti and the end of the month is endi, then the average temperature of this month is: Integral(t = starti to endi; T(0) + (γdF)(1 – exp(-t/τ))dt)/(endi - starti) = T(0) + (γdF)(1 + τ/(endi - starti)*(exp(-endi/τ) – exp(-starti/τ))) For simplicity, let’s call μ(starti, endi,dF) the average temperature of month i given the forcing dF at t = 0. Let dμ(starti, endi, endi+1,dF) be the change in average temperature between two consecutive months i and i+1. If starti occurs after t = 0 then dμ will equal μ(endi, endi+1,dF) - μ(starti, endi,dF). If endi+1 occurs before t = 0 then dμ = 0. And if endi occurs at t=0 then dμ = μ(endi, endi+1,dF) - T(0). Note that dμ is proportional to both γ and dF. Therefore, we can write dμ(starti, endi, endi+1,dF) = γ*dF*η(starti, endi, endi+1), where η(starti, endi, endi+1) depends on the month i. To generalize things a bit, let’s suppose that the change in forcing dF, doesn’t happen at t = 0 but at the end of month j. Then the change in temperature between two consecutive months i and i+1 due to a change in forcing that occurred between months j and j+1 is γ*dF*η(starti – endj, endi – endj,endi+1 – endj). To make the notation a bit simpler, let ρ(i,j) = ρ(starti, endi, endi+1,endj) = η(starti – endj, endi – endj,endi+1 – endj). Now let’s make things more realistic. Suppose that there is still only one feedback response to a change in forcing and that this feedback response is exponential with decay time τ. Suppose that for every month the forcing is constant, but between consecutive months, say month j and month j+1, the forcing changes by dFj. Suppose that we know dFj for all integers j and we wish to calculate the change in temperature from month i to month i + 1. Then the change in temperature from month i to month i + 1 will be γ*(dFi*ρ(i,i) + dFi-1*ρ(i,i-1) + dFi-2*ρ(i,i-2) + ...). That is, the change in temperature from month i to month i+1 will depend on the forcing change from month i to month i + 1 as well as all earlier forcing changes. Now let’s make this more realistic. We do not have data going all the way back to infinity. Suppose we have data that only goes back as far as month m. Note that for all temperature changes after month m, since the response decays exponentially towards equilibrium, for all practical purposes we can represent all the forcing changes that occurred before month m as an unknown characteristic forcing dG that occurs at the end of month m-1. This gives that the change in temperature from month i to month i + 1 is γ*(dFi*ρ(i,i) + ... + dFm*ρ(i,m) + dG*ρ(i,m-1)). Now let’s make things even more realistic. Realistically, there is not just a single exponential response with a single decay time, but many exponential response times with different decay times (maybe even a continuum of decay times). Following the suggestion of Van Hateren, it might be reasonable to approximate the true impulse response function with a finite number of exponential response functions. The fastest decay time of a response is approximately half a year, so let τ1 = 0.5 years be the first decay time. In addition, Van Hateren suggested that a factor of 4 between consecutive decay times might be reasonable enough. So let τ2 = 2 years, τ3 = 8 years, τ4 = 32 years and τ5 = 128 years be the next 4 decay times. Finally, since 1959-2012 only covers a time span of 54 years, it probably isn’t a good idea to go higher than a decay time of 128 years since the data won’t be long enough to distinguish between higher decay times, so let’s just stick with the above 5 decay times. Also, this coverage of decay times from 0.5 years to 128 years should be sufficient enough to give a reasonable approximation of the equilibrium climate sensitivity. It is also worth pointing out that the decay time of ocean uptake of additional atmospheric CO2 is on the order of 100 years, which suggests that a 128 year decay time should be enough to estimate the equilibrium climate sensitivity. Let γs be the γ associated with each decay time τs, ρs be the ρ associated with decay time τs and dGs be the characteristic change in forcing at the end of month m-1 for decay time τs. Then the change in temperature from month i to month i + 1 becomes: Sum(s = 1 to 5; γs*(dFi*ρs(i,i) + ... + dFm*ρs(i,m) + dGs*ρs(i,m-1))) Now let’s make this more realistic. In reality, there is not just one factor that can cause a change in radiative forcing for Earth, but many. Furthermore, different types of radiative forcings may influence the Earth by different amounts. For example, an increase in solar forcing has a stronger effect in equatorial regions than polar regions where as an increase in CO2 forcing has a more even effect across the globe. Also, solar irradiance may have a strong negative correlation with cosmic rays. Therefore, one may want to allow for the possibility that different types of forgings have different magnitudes of impact on global temperatures. I’ll assume that there are 4 types of factors that can change radiative forcing over the 1959-2012 time period: greenhouse gases, solar irradiance, cosmic rays and volcanic aerosols. Therefore, I have to introduce 3 unknown constants (call them Solar, Cosmic and Volcano) to allow for the possibility that global temperatures may respond different to these 4 types of forcings. Let dGHGj be the dFj due to greenhouse gases, dSj be the dFj due to changes in solar irradiance, dCj be the dFj due to changes in cosmic rays, and dVj be the dFj due to changes in volcanic aerosols. Then the change in temperature from month i to month i + 1 becomes: Sum(s = 1 to 5; γs*(dGHGi*ρs(i,i) + ... + dGHGm*ρs(i,m) + Solar*(dSi*ρs(i,i) + ... + dSm*ρs(i,m)) + Cosmic*(dCi*ρs(i,i) + ... + dCm*ρs(i,m)) + Volcano*(dVi*ρs(i,i) + ... + dVm*ρs(i,m)) + dGs*ρs(i,m-1))) Now let’s try to account for natural variation to make things more realistic. Changes in radiative forcing are not the only reason why global temperatures may change. Global temperatures may change due to natural variation. To try to account for natural variation, I will use indices such as the variation in the length of day (LOD), atmospheric angular momentum (AAM), Southern Oscillation Index (SOI), Pacific Decadal Oscillation Index (PDO) and North Atlantic Multidecadal Oscillation Index (NAO). If one adds these factors, then the change in temperature from month i to month i+1 becomes: Sum(s = 1 to 5; γs*(dGHGi*ρs(i,i) + ... + dGHGm*ρs(i,m) + Solar*(dSi*ρs(i,i) + ... + dSm*ρs(i,m)) + Cosmic*(dCi*ρs(i,i) + ... + dCm*ρs(i,m)) + Volcano*(dVi*ρs(i,i) + ... + dVm*ρs(i,m)) + dGs*ρs(i,m-1))) + β1*dLODi + β2*dAAMi + β3*dSOIi + β4*dPDOi + β5*dNAOi Where β1, β2, β3, β4, and β5 are unknown constants, dLODi corresponds to the change in the length of day from month i to month i+1, dAAMi corresponds to the change in atmospheric angular momentum from month i to month i+1, dSOIi corresponds to the change in the southern oscillation index from month i to month i+1, dPDOi corresponds to the change in the pacific decadal oscillation index from month i to month i+1, and dNAOi corresponds to the change in the north atlantic multidecadal oscillation index from month i to month i+1. This equation has 18 unknowns (distributed over 30 terms). One can try to estimate the 18 unknowns by turning this equation into a regression equation and performing a non-linear regression.

It's impossible to measure temperatures at all infinity points on the globe. You just need sufficient global coverage (which we have), then you can interpolate to get an estimate. Satellite data helps too.

Update on the data compilation: For total solar irradiance, I take annual total solar irradiance for 1958-2012 (http://lasp.colorado.edu/lisird/tss/historical_tsi.csv?&time%3E=1958-06-29&time%3C=2013-06-29). I then interpolate for estimates of monthly total solar irradiance. In addition, I use monthly solar flux data (http://www.esrl.noaa.gov/psd/data/correlation/solar.data) as well since solar flux might be more strongly correlated with cosmic rays (also, this data set is monthly, so might catch things that the annual data will miss). For annual length of day data, I detrend it by 1.7 ms per century since this is the long run trend (http://en.wikipedia.org/wiki/Tidal_acceleration#Quantitative_description_of_the_Earth.E2.80.93Moon_case). I then lag the data by 6 years for reasons explained earlier. Finally, I interpolate to get monthly data. For AAM, I detrend any seasonal effects. I did the same with the SOI. Now I could use the MEI (Multivariate ENSO Index http://www.esrl.noaa.gov/psd/data/correlation/mei.data ) instead of the SOI. The MEI supposedly better represents ENSO. The problem with MEI is that it uses surface temperature data in its index, so the MEI might be correlated with global warming. Looking at the data set, there does seem to be a clear positive trend. I have detrended the MEI (both for seasonal variation and for long term, although the MEI claims to be seasonally detrended already), but it is probably better to use SOI since SOI does not use temperature data and has no long term trend. I have also seasonally detrended the PDO index. Like the MEI, PDO does use temperature data so there is some concern that it might be correlated with global warming. However, if I look at the trend from similar phases in the pacific decadal oscillation (say from 1954-2013) then there does not seem to be a long term trend in the PDO, so using the seasonally detrended data is probably fine. Lastly, I have seasonally detrended the AMO index (which like the SOI has no long term trend). I also recalculated the adjusted HadCRUT4 temperature data for 1959-2014 (accounting for the linear trend). For the volcanic aerosol data, it doesn’t need to be detrended. Anyway, this means I have sufficient monthly data from January 1959 – September 2012 to perform calculations.

Okay, so I've computed some green house gas forcing time series data from 1954-2013 to use in a future regression. I am only considering CO2, CH4 and N2O. Here is my methodology: For methane, 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 I use from 1959-1984 ice core data from Antarctica and Greenland http://cdiac.ornl.gov/ftp/trends/atm_meth/EthCH498B.txt. There is a slight overlap in the data sets from 1985-1992. I take the difference in the two data sets over 1985-1992 and perform a linear regression. The results of the linear regression suggest that in 1985, the ice core data is ~24.12 ppb greater than the cap grim data. To make the data sets comparable, I lower all the ice core data by 24.12 ppb. Then, I take all of the annual data (monthly methane data is available for 1985-2013, but I want to have the data consistent over both periods and I wish to detrend seasonal effects, so I ignore this) and perform cubic-spline interpolation to get estimates of seasonally detrended monthly atmospheric methane. 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 1958-2004. ftp://daac.ornl.gov/data/global_climate/global_N_cycle/data/global_N_perturbations.txt . Unfortunately, the data prior to 2000 is very inaccurate and has missing years, so I perform a quadratic regression over the snowpack data to get estimates of atmospheric N2O during those years. As I did with methane, I look at the overlapping years (2000-2004) and I perform a linear regression of the difference in the estimates. The linear regression suggests that in 2000, the snowpack data is 0.0165 ppb greater than the global instrumental data. To make the data sets comparable, I therefore decrease the estimates of the snowpack data by 0.0165 ppb. Finally, like with the methane data, I take these annual estimates and perform cubic-spline interpolation to get estimates of monthly seasonally detrended atmospheric N2O. Next, I take the monthly detrended N2O & methane data plus the monthly detrended atmospheric CO2 data (ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_mm_mlo.txt) and I compute the combined estimated change GHG forcing compared to 1750 for these 3 gases using the IPCC’s 2001 formula (http://www.esrl.noaa.gov/gmd/aggi/aggi.html).

I guess I should use CH4 and N2O data as well. Using the IPCC's radiative forcing formulas, I calculate that only about 78% of the change in radiative forcing for greenhouse gases from 1959-2014 was due to CO2 (excluding water vapour of course).

@ TimG - Are you fine with the 1959-2014 time period? Data should be fairly reliable during this time period and less subject to 'adjustments'. I'm trying to get enough monthly data sets to estimate climate sensitivity using a time series approach that combines the Van Hateren, CSALT and my earlier approaches. I think I have most of the data sets: temperature: http://www.cru.uea.ac.uk/cru/data/temperature/HadCRUT4-gl.dat CO2:ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_mm_mlo.txt Volcanic Aerosols: http://data.giss.nasa.gov/modelforce/strataer/tau.line_2012.12.txt Atmospheric Angular Momentum: http://www.esrl.noaa.gov/psd/data/correlation/glaam.data.scaled Southern Oscillation Index: http://www.ncdc.noaa.gov/teleconnections/enso/indicators/soi/data.csv Pacific Decadal Oscillation Index: http://www.ncdc.noaa.gov/teleconnections/pdo/data.csv North Atlantic Multidecadal Oscillation index: http://www.esrl.noaa.gov/psd/data/correlation/nao.data Though I am stuck on a few things. I don't think monthly length of day data going back to 1959 is available (I can find monthly data starting in 1990 at the earliest). Monthly instrumental solar irradiance data starts in 1978, though there are lots of reconstructions and land based measurements. Monthly solar flux data is available though http://www.esrl.noaa.gov/psd/data/correlation/solar.data. I'm also a bit unsure what to do about nitrous oxide and methane as they are also important 'green house gases', I might have to track down some data sets... http://www.epa.gov/climatechange/science/indicators/ghg/ghg-concentrations.html. Also, the volcanic aerosol data I have ends in September 2012.

I was thinking about how I should incorporate the variations in the length of day into the model. The problem with length of day variations is it has a very different mechanism for affecting climate than most of the other factors I am considering, and therefore there is little reason to believe that climate should respond to length of day variation on the time scale as other factors. The length of day fluctuations result from a transfer of angular momentum between the Earth's core+mantle and the Earth's crust (also the tidal effects of the moon are extending the length of the day by 1.4 ms per century, but that is more or less a constant affect). Where as the other major factors that could cause climate change (changes in CO2, changes in solar irradiance, volcanic aerosols, etc) do it by affecting the radiative balance of earth and therefore are expected to have similar time scales of response. So I thought I would try to see if the Earth's temperatures respond to length of day variation, and if so what the time scale of response is (and if the time scale of response is comparable to the time scale of response to changes in other factors). To do this, I first have to detrend temperature data on other factors. I decided to do this using a relatively simple model: dT(t)/dt = (A*dln(CO2(t))/dt + B*dTSI(t)/dt + C*dAOD(t)/dt) + k*(A*dln(CO2(t-1))/dt + B*dTSI(t-1)/dt + C*dAOD(t-1)/dt) + k2*(A*dln(CO2(t-2))/dt + B*dTSI(t-2)/dt + C*dAOD(t-2)/dt) + k3*(A*dln(CO2(t-3))/dt + B*dTSI(t-3)/dt + C*dAOD(t-3)/dt) + ... all the way to infinity where T is the temperature, CO2 is the atmospheric CO2 concentration, TSI is the total solar irradiance, and AOD is the average optical depth at 550 nm (basically is a proxy for volcanic aerosols). Basically, the idea of this model is that there is a simply exponential impulse response in temperature to a change in forcing (with decay time -1/ln(k)). Now I know that an exponential response is unrealistic, as I explained earlier on this page. But I am just trying to perform a simple detrending so that I can see if the response time to Length of Day variances is similar. The above equation can be simplified to: dT(t)/dt = A*dln(CO2(t))/dt + B*dTSI(t)/dt + C*dAOD(t)/dt + k*dT(t-1)/dt This is very similar to the model of post #90, but might be a bit more intuitive. Anyway, I tried to use the data from 1850-2008 to estimate the parameters (though I am mostly interested in the residual). My 95% confidence intervals for B and k are 1.21 +/- 0.40 and 0.56 +/- 0.12 respectively. I can get a rough estimate of equilibrium climate sensitivity from this since ECS should be B/(1-k)*ln(2) in this model. This gives a climate sensitivity of (1.91 +/- 0.82) C. Of course, one of the reasons for such a low ECS is that the assumption of an exponential response gives an unrealistically low decay time (~1.72 years). In any case, after the above regression, the residual is the change in temperature that has been detrended of other factors. Next I detrend the length of day variation data so that it has zero average slope over the 1850-2008 period. I can then perform the regression: dR(t)/dt = D*dLOD(t)/dt + h*dR(t-1)/dt where R is the residual of the first regression, and LOD is the variation in the length of day. I want to test if h is similar to k (because if it is, then I can treat the length of the day similar to how I treat solar irradiance, CO2 or volcanic aerosols). I get a 95% confidence interval for h to be -0.10 +/- 0.15. So not only is h different from k, but h might even be negative. A negative h indicates that one cannot treat climate responses in length of day variation as an exponential response. Some scientists have suggested that the changes in temperature lag changes in LOD by 6-7 years and that changes in temperature move in the opposite direction as changes in the variation in the LOD. Perhaps this is due to: an increase in the LOD means the angular momentum of the Earth's crust reduces, which means that the rotational energy of the Earth's crust reduces. By conservation of energy, perhaps this 'missing' energy is turned into heat, which travels slowly from the bottom of the Earth's crust to the surface. This would take a long time, perhaps even 6-7 years (I don't know though). If this is the case, then perhaps I should simply lag LOD by 6 years. As a final test, I compared how well LOD fits to the residual for 5, 6 and 7 year lags. The 6 gives a better fit than 5 or 7.

I know what you mean. I have first hand experience with corruption in science.

@ Time - with respect to ice cores not having a wide geographic distribution, this is both true and false. It is true if you don't want to go that far back in time (various ice cores from alpine areas exist), but if you want to go back hundreds of thousands of years, then you are limited primarily to antarctica and greenland. These proxies are okay provided that you correct for them using the appropriate amplification factor (the rule of thumb is to divide the Antarctica ice core temperature anomaly data by a factor of 2). Interestingly, I was on a climate site like 2 weaks ago pressing some people on a related question (primarily, the basis of the 'rule of thumb' dividing antarctica ice core data by a factor of two). After enough pressing, I was eventually lead to a paper that reasonably answers my question (Shakun and Carlson (2010)). It seems that the geographic distribution of ice cores for recent times is decent enough to understand get a global representation of temperature and from there get estimates of the amplification factors needed to relate older ice core data to global temperatures. Of course there is also sedimentary core data, in addition to ice core data, which can give representation to equatorial parts of the earth over longer timescales. With respect to tree-ring data. It is not just the thickness of each ring that is used. But other information such as density and the ratios of certain isotopes. So you can get more than 1 dimension of information out of tree-ring data. Also, the tree-ring data is generally regressed on recent instrumental data to make estimates. I agree that there is plenty of reason to be skeptical of tree ring data. But that doesn't mean it can't be used to get somewhat decent estimates of past climate.

My claim wasn't that it was definitely warmer during the medieval warm period. Rather, current temperatures are within the 95% confidence interval of the temperatures during the medieval warm period. So the medieval warm period was arguably warmer. Look, just go to google and type medieval warm period temperature reconstructions or something:

Please provide your the evidence that current temperatures are warmer than the medieval warm period at the 95% confidence level. Though I doubt you can do this because such conclusive evidence does not exist. Your deflection tactics are quite obvious (you are trying to bring up the NW passage and recently declining sea ice volumes as if that somehow demonstrates that current global temperatures are warmer than the medieval warm period.). The hypothesis that current sea ice volumes are less than they were during the medeival warm period has an even lower chance of being true than the hypothesis that current global temperatures are warmer than the medeival warm period since changes in ice volumes are a slow feedback to changes in global temperature. Sigh, why do I bother. The concept of uncertainty in science is probably too difficult for you to grasp. You refuse to even grasp the basics of photosynthesis and cellular respiration. To be fair, there are tree-ring reconstructions, ice-core data and sedimentary-core data.

my claim was that it was arguably warmer. Not that it was warmer. I don't think the uncertainty of the data is small enough to make a definite conclusion about if current temperatures are higher or lower than the medieval warm period (though northern European temperatures were definitely higher).

There is tree ring data. Ice core data. Sedimentary core data.

2014 is statistically tied with like half a dozen other years as the warmest year 'on record'. And 'on record' doesn't even go that far back. It was arguably warmer during the medeival warm period.

I guess I'll have to use the van Hateren method if I want to use time series analysis to get good estimates of climate sensitivity. Though one issue is that recent instrumental data is good for getting the fast response, but not so good for getting the slow response (due to not many years of observation), where as data that covers longer time periods (such as reconstructions for climate data over the past 1000 years, or paleoclimate data) is good for the slow response, but terrible for the fast response. So maybe if you want to get a good estimate of climate sensitivity, you need to use all the data: instrumental, tree ring, ice core, sedimentary core, etc. I might have to take a bit of a break before revisiting this with the van Hateren approach.

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.

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.

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.

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.

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.

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.

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.

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...

Yes. Not only do they spend the 4th most on military, but they control Mecca (which means that if any nation were ever to attack Saudi Arabia, the Saudi's could invoke sura 9 of the quran, and suddenly that invader is at war with hundreds of millions of muslims globally). Plus they have spread their crazy wahabist ideology globally, which means they have many puppet terrorist groups that could do their bidding.