What is the Climate Time Constant? Refining the Estimate I.
In my most recent post,, I noted that the current empirical estimate of the time constant given by Schwartz, Scafetta and in my earlier blog post is roughly 8 ±2 years. However, I also noted that Schwartz admitted uncertainties in the method itself. We all discussed a range of difficulties with the approximate methods. I’m now embarked on incremental improvements on the “lucia” method. (This method is similar to Schwartz’s method as described in his response to comments. However, I do a formal least squares regression to get my time constant.)
As my readers and I are all brainstorming ways to improve the estimate, I’m simply going to report the results form my first step in improving my estimate. So, this is sort of a ’status report’ I am going to present the “back of the envelope” fix for two of the issues. These are:
- The inherent bias associated with estimating the correlation in the residuals and
- Estimating the “true” uncertainties.
I am deferring addressing the other issues as a matter of expediency; the order in which I address issues should not be taken to imply that other issues are of lesser importance. Also, for now, I am using annual average data, and running tests with the coarsest possible resolution; my intention is to thereby identify which issues are most important before refining the method of estimating the time constant.
For those who don’t wish to read further: The main results are after identifying a method to correct for bias, my current estimates for the time constant for the climate are between 15.5 years based on GISS Met data and 8.6 years based on Land/Ocean annual average data. Both methods contain a great deal of uncertainty; so much that it’s not worth estimating formally. The uncertainty should drop dramatically when I ultimately repeat the analysis with monthly data.
For the remaining discussion, I will be assuming people are somewhat familiar with Schwartz97, and with my previous analysis based on his method with my extension to include measurement imprecision in the GMST data! (My previous analysis is discussed here.)
What I did.
Step 1:
I created 10 independent series of synthetic data that consisted of AR(1) noise with a time constant of 12 years, plus white noise, u. The AR(1) noise is thought to represent the “true temperature” of a simple climate system; the white noise is though to represent measurement uncertainty. The ratio of white noise to AR(1) noise was set such that the ratio of the measurement noise to the “true temperature” was 1. This happens to corresponds roughly to the amount of measurement noise admitted by the measurement groups.
Data were created at monthly intervals for a period of 125 years. I then averaged the monthly data to create annual average data. I computed the autocorrelation as a function of lag time for each string.
The autocorrelation as a function of lag time was compared to the known true, underlying autocorrelation for the true process; see figure 1:
In this figure, I multiplied the bias by (-1) and plotted the computed bias as a function of the known autocorrelation. I also computed standard errors for the autocorrelation at each point, calculated a “t” value for 10 data points and created 95% confidence intervals. (These may not be correct as I have not checked to see if the errors are normally distributed.)
I then fit a straight line to the bias as a function of lag time for later use. Obviously, the line doesn’t fit splendidly. However, it does reveal that qualitatively, the bias tends to be negative, and the absolute magnitude increases as the autocorrelation decreases.
Should this method show promise, in later analysis, I will be running a larger number of synthetic data strings and using monthly data. This should reduce the uncertainty in in the calculated bias.
Note that, in principle, this method can also be used to estimate the uncertainty in the estimate of the time constant using 125 years of data. However, what I’ve basically found is that using 125 years of annual average data results in standard errors in the estimated value of the time constant on the order of ± 4 to 5 years. This uncertainty is two or three times larger than one would estimate based on the uncertainty in the in the best fit line to the natural log of correlation as a function of lag time. So, I clearly can’t use that idea as a method to estimate the uncertainty.
In addition, even the ±4-5 years represent the lack of repeatability under the assumption that the time constant really is 12 years, and the noise is the level I used in the synthetic data. I need to think about these uncertainties a bit more to decide what the real uncertainty intervals should be. However, it should also be noted that the standard error should diminish, and possibly dramatically, when I apply the method to monthly data. So, I should ultimately be able to obtain relatively decent estimates of the time scale (contingent on acceptance of the method as valid.)
Step 2:
After estimating the bias in the time autocorrelation as a function of bias, calculated the autocorrelation in the temperature anomalies reported by GISS for both Met and Land Ocean data using a 125 year string of data. I then corrected the computed autocorrelation using the best fit estimate from the linear regression I obtained in the previous analysis. (Yes, that would be the regression on the very uncertain highly scattered data. ) In all cases, this increased the correlation estimated based on the data, raising the value.
I then plotted the natural log of the autocorrelation as a function of lag time, limiting analysis to those cases that had positive values of the autocorrelation. The results for both corrected and uncorrected cases are shown below. I used EXCEL’s LINEST to obtain the slope and intercept, and computed the time constant as the negative of the inverse of the slope. The natural log of the autocorrelation and the data are shown below.
Figure 2: The natural log of the autocorrelation of temperature is illustrated as a function of lag time. Autocorrelations were corrected by estimating the bias associated with a 12 year time constant and the noise value that matches the data. (This was done iteratively; it required 1 iteration.)
Using a bias correct that assumed the true time scale is 12 years results in an experimental estimate of the bias of 15 years when computed using Met data and 8 years when computed using Land Ocean data.
Provisional conclusions
For now, this “extended method of Schwartz” is resulting in roughly 12 year time constants. However, the calculation is crude, uses annual average data, and I’ve only corrected for bias using a coarse test.
I’ll be refining this after we explore other issues like detrending, and the possibility of “spikes” of noise in the actual spectrum of the GMST. (Arthur Smith is looking at this.)
Nevertheless: For now, it appear that correcting for the bias does elevate the time constant noticably, and ought to be done.
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45 Responses to “What is the Climate Time Constant? Refining the Estimate I.”
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pliny May 8th, 2008 at 4:10 pm
Lucia,
One thing has been bothering me since reading your December post. You use regression, while others just form a ratio, and estimate a mean. I think regression is fine, but the regression line should be constrained to pass through the origin.
Here’s why. Your autocorrelation is assumed to be an exponential, and you are estimating the coefficient. Now the autocorrelation is always 1 for zero lag. You should be scanning the range of eligible functions to get the best fit coefficient. Instead, you are allowing ineligible functions. Straight lines not passing through the origin correspond to autocorrelations that are not 1 at zero lag. SS and Tamino don’t do this, so your results won’t correspond to theirs. And I think they are right.
pliny May 8th, 2008 at 10:06 pm
Lucia,
I’ve re-read your December white-noise justification for using general regression, and I now see how it works. So I agree that your regression is OK, noting that it includes an estimate of the measurement, which adds to its variance. It also depends on the measurement error being a white noise process - with successive errors independent.
So for me your analysis is now looking very good. But as you say, the uncertainty of the relaxation time estimate is huge. And we’ve now gone from SS’s original 5 years to 8.6 years for one dataset, 15.5 years for the other. Hopefully the monthly breakdown will help.
steven mosher May 9th, 2008 at 11:04 am
Annan actually mentioned you today and gave you lots of linkies, but he was rather condescending
Raven May 9th, 2008 at 12:08 pm
Do these discussions of time constant and CO2 sensitivity depend on the assumption that natural factors (other than volcanos) are insignificant?
For example, let’s say someone establishes a solar climate link and demonstrates that 50% of the warming over the last 50 years was due this solar effects. Would the time constant change or only estimated CO2 sensitivity derived from the time constant?
lucia May 9th, 2008 at 1:23 pm
Raven–
Actually, the Schwartz method requires volcanos and other factors to matter.
Steve–
I went to Annan’s blog to comment, but I have to get a blogger/google identity. I think I have one… somewhere… but sheesh! (I hate hurdles in front of commenting. That said: his blog, his policy.)
I guess I can respond later here. But.. erhm… no. I’m not going to apologize for saying two of their criticisms are totally wrong. They are totally wrong. I was thinking of not discussing that particular JGR comment “above the fold”, but I guess maybe I should. Oh well….
You know, it’s always interesting to watch referrers when someone links me. My stats says those links sent me 27 readers. (But I think the way WP stats counts, that might mean 9 readers each clicked the three links. Or, maybe they are all Annan checking the links as he writes his post. We’ll see if we get the flood that some other bloggers can send!
Raven May 9th, 2008 at 4:53 pm
Lucia says: Actually, the Schwartz method requires volcanos and other factors to matter.
The issue is how does one take a time constant and use it to estimate the CO2 sensitivity. Schwartz says that a time constant of 8-12 years is within the range of IPCC estimate of 3.0 +/- 1.5. I don’t see how such a claim could be made without making assumptions about the magnitude of the other factors at work. If these assumptions are built into the estimation process then this approach is as flawed as GCMs because it has no way to seperate the effect of CO2 from the effect of any unknown unknowns.
lucia May 9th, 2008 at 6:35 pm
Raven– Schwartz’s 2007 paper discusses three things: 1) time constant, 2) heat capacity and 3) the sensitivity. The estimates for the first two are used to estimate the third.
There have been fewer comments on his estimate for heat capacity.
Raven May 9th, 2008 at 7:36 pm
Lucia,
I could estimate capacitance of a black box electric circuit by applying a known change in voltage to the circuit and measuring the response over time. I could also estimate the voltage change if I have some way to calculate the capacitance. However, neither estimate tells me much about the relative contribution of different unknown voltage sources. That is why I think that using the heat capacity and time constant to estimate CO2 sensitivity is only possible if one assumes that all of the other climate forcings are known.
Am I misunderstaning something fundemental about Schwartz’s method?
EJ May 9th, 2008 at 8:45 pm
Being a civil engineer, and often deferring to a geologist, I am hoping you might do some runs with geological time scales. A thousand years is a hiccup!
I think it not only illustrative, but prudent. There has to be some “time constants ?scales?” in that record.
Maybe I am wrong, but a post for discussion would be cool!
The short term (800 yr) scale findings are warranted, but, IMHO, so are the long term (800K) cycles.
Jorge May 10th, 2008 at 4:32 am
Raven,
May I try to give an electronic analog of what I think the model is.
Imagine a current source in parallel with a resistor and a parallel capacitor. In this version the voltage across the resistor and capacitor represents the surface temperature, the current source represents heat flow /m2, the capacitor represents the heat capacity of the earth /m2 and the resistor is the climate sensitivity in ºC/W/m2
It is fairly easy to show that the equation in electrical terms is the same as the one that Lucia has been describing.
Call the current flowing from the current source I, the current into the capacitor I1 and the current through the resistor I2. Clearly I = I1 + I2 but I1 = C dV/dt and I2 = V/R so I = CdV/dt + V/R. If we divide all terms by C we get the result:
I/C = dV/dt + V/RC or dV/dt = -V/RC +I/C This is the same form as this equation from Lucia - dT/dt = -T/τ + α F
On can clearly see that RC is the tau and 1/C is the alpha. So if RC is calculated and C is assumed it is possible to find R as well. In terms of the real world, this is very simplified but I think we can say that we are assuming a pertubation of the W/m2 arriving at the earth´s surface and we assume that some is captured by the heat capacity and some escapes from the surface by an amount proportional to the rise in surface temperature.
What is not clear to me is how perturbations in the W/m2 really come about and I have some doubts about the climate sensitivity being constant. I have to admit I don´t really understand how the time constant can be derived from simply observing the time series of temperature or, in the electrical case, the voltage. Obviously it can, or Lucia would not be doing it!!! In any event, I don´t think this exercise provides any useful contribution to the CO2 attribution debate. The W/m2 could come from anywhere and it seems likely that things such as altered cloud cover could change the value of R.
I quite like this electrical model as it reveals the exact assumptions and gives us a chance to modify it in ways that maintain a physical meaning. It also has the advantage that we can use something like SPICE to do the sums for us.
pliny May 10th, 2008 at 4:36 am
Raven,
I really like your idea of an electrical analogy. I would put it like this - you have a resistor R and capacitor C in parallel, earthed at one end. To the other end you apply a current source I and measure the voltage V. In the analogy, C=(known)heat capacity, V=temperature, I=(GHG) heat flux, and R is the (unknown) sensitivity. RC is the time constant. I say R is the sensitivity because if you apply a constant I, eventually R=V/I.
We need to infer RC from the dynamic response, and we can’t just apply a pulse. People often try to use something like a volcalo as a “pulse”, but the response is noisy. So the approach here is to suppose that the V that we see is driven by a white noise process I, which acquires autocorrelation because of the R-C smoothing. So RC is inferred from the autocorrelation.
Then everything depends on the assumption of a white noise driver, which seems to me a stretch. That may be why it’s so hard to get a stable answer.
(Jorge, I see we have the same idea. Yours appeared just before I posted. I’m leaving it there because I’ve referred to it on another thread).
lucia May 10th, 2008 at 5:48 am
Pliny/Jorge/Raven–
The electrical engineering analog is precisely correct. And yes, it need to be driven by white noise to result in AR(1) response.
For the electrical engineers, this problem appears in loads of text books on Kalman filtering.
The thing about this particular problem is it reappears everywhere and is in some sense, the simplified models for lots and lots of stuff. Other places you’ll find the model:
1) Brownian motion. In this problem, small particles with mass M are driven by white noise due to molecular colisions. The time constant term arises because for very small particles moving slowly the resistance to motion is linear with velocity “V”. So we get M dV/dt = A V + F . The time constant ends up being A/V. In this problem the “F” is really white.
2) Behavior of thermocouples. Thermocouples are small and essentially have a uniform temperature T. The if we linearize the heat transfer rate, we get C dT/dt = A T + F. If we put the thermocouple in a temperature field of “white noise” we would get the AR(1) response. (One never does this. In reality, one uses the thermocouple to measure the properties of the force F. So, generally, you want the time constant so small that it responds to the properties. The temperature measured by the thermocouple will NOT be AR(1).
So, yes. For the Schwartz method to work, the external forcing “F”, must be white. And yes, if the equation looks like one you recognize, it is!
pliny May 10th, 2008 at 6:08 am
Lucia,
So do you agree that the “whiteness” of F is the only property we are using to determine τ, and so the sensitivity? It seems to me that then there are two difficulties:
1. We don’t really have any evidence that it’s true. Normally you assume noise is white because you don’t have any better assumption. That may do for error analysis etc, but it seems a weak basis for making assertions about the numerical value of sensitivity, and
2. We can’t find a value of τ which exactly corresponds to a “white” driver. In fact, I suspect we don’t really come close (and it should be tested). That so, it seems that a big range of values of τ will give almost optimal “whiteness”. A big error range.
Jorge May 10th, 2008 at 6:18 am
pliny,
No problem.
Thanks for explaining how the RC smoothing gives autocorrelation when driven with white noise. I think that was the part I have been missing from the beginning.
My difficulty with CO2 attribution is that an increase in W/m2 can be inferred from radiation models, but only if the atmospheric profile is assumed to be unaltered apart from the addition of extra CO2. In terms of the model, we have to assume that R is independent of V. In reality R is quite complicated as it depends on convection and moisture content as well as radiation effects. If R were to actually get smaller with increasing V, the resulting change of V with respect to a change of I would also be smaller.
This would be a negative feedback in IPCC terms and shows that the standard assumption of positive feedback actually implies that R increases with the applied V. It would be a nuisance if R were non-linear as it means the time constant and sensitivity would vary with the size of the forcing. It would also add frequency components to the observations that were not present in the forcing, white noise or other.
Now that we seem to have discovered cycles in the temperature observations I think it is even harder to tell a trend caused by CO2 from the upswing of one of these cycles. It reminds me of Gordon Brown, the UK Chancellor, saying he would balance the budget over the economic cycle. After changing the start and end points he still could not achieve it, so he redefined how the budget was calculated.
pliny May 10th, 2008 at 6:40 am
Jorge,
The first thing to say is that I don’t think this model of the atmosphere should be taken too far. I think everyone agrees that there are really many different timescales, and that the earth is not a simple capacitor.
In electrical terms, it’s really a linearised model, dealing with increments of I and V. So the R and C’s are more like the sort of transient impedances that you would get in analysing a transistor (hfe etc). So that is another issue - more CO2 could feed back in, but it is more complicated.
One should also say that the IPCC do investigate a huge range of feedbacks. There’s a whole chapter in the AR3 here.
The various cycles (ENSO, PDO) are one indication that the assumption of a white noise driver is shaky.