Time Constant for Climate: Greater than Schwartz Suggests!
A recent empirical analysis, Schwartz (2007), suggests that the time constant for the earth’s climate may be as low as 5 years. If so, the climate sensitivity of the earth may be much lower than suggested by Climate models. That estimate did not account for the effect of in field measurement uncertainty on the estimate of the time scale. I did a quick and dirty re-analysis of the data and conclude the time scale for the earth’s climate is closer to 8 years.
Briefly, what did I do?
I pretty much assumed the physical model suggested by Schwartz (2007) represented a decent approximation of the earth’s climate. I then revised the analytical method used by Schwartz (2007) to account for uncertainty in individual temperature measurements to determining the time constant for the Global Mean Surface Temperature (GMST). The revised process also permitted me to obtain an in situ estimate of the uncertainty of GISS measurements of GMST.
Using the revised analytical method, time constant pertinent to changes in (GMST) determined from autocorrelation of GMST over 1880-2005 is found to be 8 ± 2 years. This differs from the value of 5 ± 2 years suggested by Schwartz. The in situ uncertainty in GISS measurements of GMST is estimated at 0.1K which is somewhat larger than the 0.05K said to arise due to incomplete station coverage; (see Hansen et al 1988 page 9346.)
Schwartz’ Model & Method
In “Heat Capacity, Time Constant, and Sensitivity of Earth’s Climate System”
Schwartz estimated the time constant of the earth’s climate system by:
- Proposing the simplest possible model of the earth’s climate system: that is, he used a lumped parameterization with only one “lump”. In this model the earth is described as having one temperature, θ which evolves over time as a result of variable radiative forcing, q.
- Developing 1 equation that related the climate’s time constant, τ to the decay rate of an autocorrelation of Rθ with lag time.
- Processing GISS temperature data to estimate the autocorrelation as a function of lag time, and determining the time constant
I’ve pretty much done the same thing, but adjusted the method for processing the data. I’ll now describe the steps with a level of detail I hope makes sense in a blog post.
The Lumped Parameter Model
In a very simplified model of the earth is envisioned as sort of a sphere with some total heat capacity, and one average temperature. The earth gains heat from the sun and loses heat to space. Imagine the energy from the sun’s heat varies, but over long periods of time, there an average value exists and the average does not evolve over time. This quasi-steady state is described as “stationary”. For the purposes of analysis, average values that do not evolve over time simply assumed to exist.
Complex equations describing the evolution for the earth’s surface temperature could be written (see Schwartz 2007, Section 2); these are ultimately linearized near the current equilibrium (or average) value. After some manipulation, this differential equation would be thought to describe the rate of change of the temperature anomoly θ:
dθ/dt = -θ/τ + q’
where:
- t represents time,
- θ(t) is the instantaneous global mean temperature anomaly (GMST),
- τ is the thermal response time of the planet and
- q’ is some forcing function representing variations in heat added to the planet.
The goal of Schwartz 2007, (and my current post) is to estimate τ without every needing to know very much about q’ or even much about the earth. The analysis will be based on examining historical temperature data for the Global Mean Surface Temperature of the earth (i.e. GMST.)
How can we estimate τ using data and equation (1)?
It turns out that equation (1) is the best understood differential equation on the planet. A lot could be said about it, but for the purposes of this discussion, it is sufficient to know that if one assumes q’ shares the spectral characteristics of “white noise”, then a statistical property called “the autocorrelation of θ” is governed by the following equation:
Rθ(t)= e-t/τ
Using the definition of the natural logarithm, this may also be written as:
Ln(Rθ(t))= -t/τ
If one is unfamiliar with the difficulties involved in estimating a time constant, τ, from data collected in the field or lab, one could write this equation as:
τ= -t/Ln(Rθ(t))
The three equations are identical, but expressed in different forms.
- The first form is useful for describing the shape of the autocorrelation function generally.
- The second form shows that if I plot the natural log of Rθ as a function of lag time, should result in a straight line with a slope equal to -1/τ. This is the best for for analytical purposes precisely for this reason.
- I can think of no particular advantage to using the third form. It is, however, the form used by Schwartz (2007) and which appears to have been adopted by Tamino in his post at Real Climate.
Calculation & Plotting of R
To calculate Rθ I downloaded met station temperature data available from GISS. I then entered these into an Excel spreadsheet.
Because the earth has been experiencing a warming trend over the past century, I first de-trended the data to remove the mean warming signal. I did this by fitting an ordinary least squares regression and then subtracting the expected value for a particular year from the data. The method is similar to that described Schwartz applied to obtain figures 5a-c in Schwartz 2007. )De-trending itself can introduce strange “features” into the data, but I will ignore this fact.)
After de-trending, I applied the CORREL() function to obtain an autocorrelation for the temperature data.
As required by its definition, the autocorrelation is equal to Rθ =1 at a lag time of zero and obeys the requirement -1 ≤Rθ≤1.
At short lag times, the autocorrelation decays rapidly with lag time, approaches zero and then drops below zero. It is rather well known that meaningful estimates of the autocorrelation are difficult to obtain near zero. For this reason, and others, I limited any further analysis to lag times that showed no zero crossing for the autocorrelation. (This is, after all, a blog post, and not a journal article. 
)
Then, following equation (2b), I applied the natural log and plotted:
Discussion
Recall that if the simple physical model applies, the graph of Ln(Rθ) as a function of lag time shown directly above should appear as a straight line with an intercept passing through zero.
Examining the graph, we see it sort of resembles a straight line, but the intercept clearly does not pass through zero.
Though I have not revealed the well known reasons the intercept does not pass through zero, I know at least two exist.
So, I plowed ahead and applied a linear regression with these results:
- Based on the slope, I obtain a response time of 7.2 ± 2 years based on the land ocean data and 8± 2 years based on the met station data. (The error bands are standard errors, not 95% confidence intervals. I also treated the errors in slope and intercept as additive. This results in ‘blog quality’ error bands.)
- The linear regression shows a negative value for the intercepts which are estimated at -0.7 ± .3 and -0.6±0.2 based on met data and land ocean data respectively. Both values are found to be statistically significant at a confidence level of 95%.
Why is the intercept non-zero? And more importantly, is that NOT fatal for the model?
First, the non-zero intercept is absolutely not fatal to the simplified model for climate used by Schwartz. Rather, a non-zero intercept is widely known to occur when systems obeying equation 2 are studied experimentally. This is the result of measurement uncertainty of particular two types. These are: the bias in the autocorrelation, Rθ, due to time averaging and the bias in Rθ due to lack of precision in the data.
I’ll discuss the features of averaging bias because, though nearly irrelevant to this particular data set, this bias appear prominently a blog posts criticizing Schwartz (2007), most particularly a guest post at Real Climate written by Tamino. I’ll discuss the lack of precision of bias because this lack of precision is must be understood to estimate τ from the GISS data.
Martin Ringo April 4th, 2008 at 7:57 am
I understand it is hardly de rigueur to comment on posts three months after the fact, but if anyone wants to help here, I’d appreciate.
My question is where does equation come from? Not the provenance, but the meaning. I guess I just don’t understand why the change in temperature should be a constant or a constant plus noise. It would seem to imply an exponential growth, upward or downward, of temperature. What am I missing here?
Thanks.
lucia April 4th, 2008 at 8:42 am
Hi Martin,
I’m going to be cycling back to this. Equation (1) from the IPCC essentially treats the entire climate as one “lump”. The time constant τ is related to the ratio of the thermal heat capacity to the rate at which heat is added to the “lump”.
The expected value of the temperature of the “lump” will vary depending on heat addition rate. So, if “q’” had a step function– suddenly increasing to a new value “Q”, you would expect the temperature to rise to some new steady state level. That would be Qτ, which you can find by solving (1) setting dθ/dt = 0.
The temperature rise would, indeed obey a decaying exploential.
However, under the theory of AGW, the value of the forcing “q’” is expected to rise over time. To leading order, and over short time frames, many of the SRES scenarios assume the forcing is varying somewhat linearly. If the forcing varies over time, it’s possible to solve (1) analytically. For short times, the temperature rises relatively slowly, but after several time multiples of time τ pass, the temperature will be seen to rise linearly as a function of time. I discuss this somewhat here:
http://rankexploits.com/musing.....nsitivity/
The general idea is illustrated in this “cartoon”
I haven’t yet discussed the issue of decomposing the behavior into a “mean” and “random” componenent. But, if you decompose temperature and forcing into mean and random, you can show that when the mean forcing is increasing linearly, you will expect the temperature to decompose into a positive mean trend and a random fluctuation about that.
I need to sort this out better to explain it in a post, because the exact way it works out should have relevance to any sort of montecarlo analysis one might do for statistics.
I hope this helps.
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Francois O April 13th, 2008 at 10:43 am
Quick question: Isn’t detrending really applying a high pass filter to the data? Because the occurence of a trend over a given period might as well be very low frequency noise, not a response to a particular forcing. I’m too lazy to figure out by myself how that affects the autocorrelation. Can you answer that?
It’s not clear what you did that is different from Schwartz. Can you elaborate (or do I have to reread Schwartz, but I told you I’m lazy…)?
lucia April 13th, 2008 at 1:31 pm
Francois: With respect to this data, it’s not clear what detrending means. It’s just done. Since there is no particular reason to expect the underlying trend is linear, that introduced error.
Schwartz plotted… I think log tau/ time, and tried to eyeball the assymtote. In principle, if the number of data are infinite and our two methods should give the same result. The only advantage of examining it my way is: The intercept gives an independent measure of the measurement uncertainty and you can use a fit to get the slope. So, you don’t have to “eyeball”. This is sort of important because with limited amounts of data, Schwartz needs to “eyeball” in the region where noise starts to make it difficult to estimate the autocorreltion (tau). (Both methods assume you can detrend etc. So, with respect to that, both have the same problem.)
Francois O April 14th, 2008 at 5:35 am
About detrending: I faced the same thing with my CO2 model. Allan McRae and others would detrend the CO2 uptake curve because otherwise they didn’t get a good linear correlation with temperatures. But that’s the danger with the “statistical” approach to analyzing data. I decided not to detrend, but rather to try and find the physical model that could explain the discrepancy. You can see I’m a physicist, and not a statistician! I mean, the trend is part of what’s going on. We don’t know a priori what causes it. So it may be related to a linearly increasing forcing, or it could be related to a very long cyclic phenomenon. So, even though the end result is about the same, I’d rather use a high pass filter, rather than detrending, as the physical reasoning behind seems more robust.
lucia April 14th, 2008 at 6:00 am
Francois–
I have the same reservation about the linear detrending– I’m a mechanical engineer. But, for what it’s worth, some statisticians have similar reservations. When possible, physically based models work better.
I fiddled a bit with this equation, and some forcings from NASA, and have created “lumpy”– but I’m trying to find out how “Lumpy” should really be done!
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