Posted on April 5th, 2011 in Isaac Held's Blog
Upper panel: Interdecadal component of annual mean temperature changes relative to 1890–1909. Lower panel: Area-mean (22.5°S to 67.5°N) temperature change (black) and its interdecadal component (red). Based on the methodology in Schneider and Held, 2001 and HadCRUT3v temperatures. More info about the figure.
Perhaps the first thing one notices when exposed to discussions of climate change is how much emphasis is placed on a single time series, the globally averaged surface temperature. This is more the case in popular and semi-popular discussions than in the scientific literature itself, but even in the latter it still plays a significant role. Why such an emphasis on the global mean?
Two of the most common explanations involve 1) the connection between the global mean surface temperature and the energy balance of the Earth, and 2) the reduction in noise that results from global averaging. I’ll consider each of these rationales in turn.
The energy balance of any sub-portion of the atmosphere-ocean system is complicated by the need to consider energy fluxes between this selected portion and the rest of the system. It is only for the global mean that the balance simplifies to one involving only radiative fluxes escaping to space, providing a basic starting point for a lot of considerations. But is there a tight relationship between the global mean surface temperature and the global mean energy budget?
I have already indicated in a previous post (#5) that this coupling is not very tight in many climate models. In these models, the pattern of temperature change in response to an increase in CO2 evolves in time, becoming more polar amplified as equilibrium is approached. And, as a consequence of these changes in spatial pattern, the relationship between the global mean temperature and global mean top-of-atmosphere (TOA) flux changes as well. Among other things, the dynamics governing the vertical structure of the atmosphere is very different in low and high latitudes, and one needs to know how the vertical structure responds to estimate how radiative fluxes respond. There are also plenty of reasons why cloud feedbacks might have a different flavor in high and low latitudes, and might be controlled more by changes in temperature gradients than in local temperature. The potential for some decoupling of global mean surface temperature and global mean TOA flux clearly seems to be there.
[One sometimes sees the claim that it is the 4th power in the Stefan-Boltzmann law that primarily decouples the global mean temperature from the global mean TOA flux. But this is a very weak effect, at most 5% according to my estimate; the effects of differing vertical atmospheric structures in high and low latitudes, and the effects of clouds, are potentially much larger.)
There is a tendency, especially when discussing “observational constraints” on climate sensitivity, to ignore this issue — assuming, say, that interannual variability is characterized by the same proportionality between global mean temperature and TOA fluxes as is the trend forced by the well-mixed greenhouse gases. This is not to say that the internanual constant of proportionality is irrelevant to constraining climate sensitivity. One can imagine, if interannual variability is characterized by one spatial pattern, and the response to CO2 by another pattern, that one might be able to compensate for this difference in pattern when trying to use this information to constrain the magnitude of the response to CO2.
Let’s turn now to the noise reduction rationale.
There is plenty of variability in the climate system due to underlying chaotic dynamics, in the absence of changing external forcing agents. To the extent that a substantial part of this internal variability is on smaller scales than the forced signal, spatial averaging will reduce the noise as compared to the signal. But is global averaging the optimal way to reduce noise?
Suppose one has a time series at each point on the Earth’s surface. There are a lot of different linear combinations of these individual time series that one could conceivably construct; the global mean is just one possibility. Some of these linear combinations will have the property of reducing the noise more than others. One can turn this around and ask which linear combination reduces the noise most effectively.
Tapio Schneider and I examined this question in a paper in 2001. One has to first define what one means by “minimizing noise”. In our case, we define a “signal” by time- filtering the local temperature data to retain variations on time scales of 10 or 15 years and longer and then define the “noise” to be what is left over. We are not saying that this signal is forced by exterrnal agents; it is presumably some combination of forced responses and free low-frequency variations. But the forced response due to slowly varying external agents is presumably captured within this signal. We then maximize the ratio of the variance in the “signal” to the variance of the “noise”. This is an example of discriminant analysis, in which you group the data and look for those patterns that best discriminate between the data in different groups. (Roughly speaking, the different decades are different groups for our analysis, although we do not actually use non-overlapping decadal groups.) The result is a ranked set of patterns and a time series associated with each pattern. The most dominant pattern, the one that reduces the noise most effectively, turns out to be quite different from uniform spatial weighting. The animation at the top of the blog shows the evolution of annual mean temperatures filtered to retain the 4 most discriminating patterns (this is the number of patterns with a ratio of signal to noise greater than one.) Tapio has comparable animations for the individual seasonal means.
A more popular approach to multivariate analysis of the surface temperature record, complimentary to discriminant analysis, is “fingerprinting”. Here models provide one or more patterns (starting with the pattern forced by the well-mixed greenhouse gases) and, using multiple regression, we test the hypothesis that these patterns are discernible in the observed record. These approaches are complimentary because discriminant analysis does not start with a given pattern and test the hypothesis that it is present in the data; it is just a way of describing the data. A purely descriptive analysis can only take you so far, but for some purposes it is advantageous to let the data tell you what the dominant patterns are, rather than having models suggest how to project out interesting degrees of freedom.
In any case, you can do better than take a global mean if you want to reduce the noise level in the data.
The information content in the global mean depends on how many distinct patterns are present. Let’s assume that one has already isolated from the full time series what one might call “climate change”, either through a discriminant analysis or some other algorithm. If the evolution of the signal is dominated by one perturbation pattern , and if we normalize the pattern so that it has an integral over the sphere of one, we can just think of the perturbation to the global mean as equal to , the amplitude of the pattern. If 2 (or more) things are going on that contribute to observed climate changes, you are obviously going to need 2 (or more) pieces of data to describe the observations, and the value of the global mean is more limited.
If the response to CO2 , or the sum of the well-mixed greenhouse gases, is linear, the spatial response of surface temperature could still be a function of the frequency of the forcing changes. If one assumes in addition that this frequency dependence is weak, as in the “intermediate” regime discussed in earlier posts, then one can expect evolution of the forced response that is approximately self-similar, with a fixed spatial structure, in which case the global mean is a perfectly fine measure of the amplitude of the forced response.
It is easy to come up with examples of how an exclusive emphasis on global mean temperature can be confusing. Suppose two different treatments of data-sparse regions such as the Arctic or the Southern Oceans yield different estimates of the global mean evolution but give the same results over data rich regions. And suppose, for the sake of this simple example only, that the actual climate change is self-similar, and is, in fact, entirely the response to increasing well-mixed greenhouse gases. One is tempted to conclude that the method that gives the larger global mean warming suggests a larger “climate sensitivity”. But both would be providing the same estimate of the response to greenhouse gases in data-rich regions.
There are other interesting model-independent multivariate approaches to describing the instrumental temperature record besides the discriminant analysis referred to above. Typically one needs to choose something to maximize or minimize. For example, in this paper, Tim Del Sole maximizes the integral time scale, , where is the autocorrelation function of the time series associated with a particular pattern. I encourage readers to think about other alternatives.
In response to a question below, here (thanks to Tapio) is the spatial weight vector that is multiplied by the data to generate the canonical variate time series for the first (the most dominant) term in the discriminant decomposition of the annual mean data set shown at the top of the post. The lower panel shows the filtered and unfiltered canonical variate time series. The shading is the band within which 90% of the values should lie in the absence of interdecadal variability, as estimated by bootstrapping.
The low weights over land are interesting. Although this is only the first term in the expansion, it suggests that climate variations over land can be estimated by using a discriminant analysis of ocean data alone and then regressing the resulting canonical variates with the land data. This is consistent with dynamical model studies, such as Campo and Sardeshmukh, 2009, of the extent to which land variations are slaved to the ocean on these time scales.
[The views expressed on this blog are in no sense official positions of the Geophysical Fluid Dynamics Laboratory, the National Oceanic and Atmospheric Administration, or the Department of Commerce.]