Guest essay by Jeffery S. Patterson
My last post on this site, examined the hypothesis that the climate is dominated by natural, harmonically related periodicities. As they say in the business, the critics were not kind. Some of the criticisms were due to a misunderstanding of the methodology and others stemmed from an under appreciation for the tentativeness of the conclusions, especially with respect to forecasting. With respect to the sparseness of the stochastic analysis, the critics were well founded. This lack of rigor is why it was submitted as a blog post and not a journal paper. Perhaps it served to spark someone’s interest who can do the uncertainty analysis properly, but I have a day job.
One of the commentators suggested I repeat the exercise using a technique called Singular Spectrum Analysis which I have done in a series of posts starting here. In this post, I turn my attention away from cycles and modeling and towards signal detection. Can we find a signature in the temperature data attributable to anthropogenic effects?
Detecting small signals in noisy data is something I am quite familiar with. I work as a design architect for a major manufacturer of test equipment, half of which is dedicated to the task of finding tiny signals in noisy data. These instruments can measure signals on the order of -100dBm (1 part in 10-13). Detecting the AGW signal should be a piece of cake (tongue now removed from cheek).
Information theory (and a moment’s reflection) tells us that in order to communicate information we must change something. When we speak, we modulate the air pressure around us and others within shouting distance detect the change in pressure and interpret it as sound. When we send a radio signal, we must modulate its amplitude and/or or phase in order for those at the other end to receive any information. The formal way of saying this is that ergodic processes (i.e. a process whose statistics do not change with time) cannot communicate information. Small signal detection in noise then, is all about separating the non-ergodic sheep from the ergodic goats. Singular Spectrum Analysis excels at this task, especially when dealing with short time series.
Singular Spectrum Analysis is really a misnomer, as it operates in the time-domain as opposed to the frequency domain as the term spectrum normally applies. It allows a time-series to be split into component parts (called reconstructions or modes) and sorts them in amplitude order, with the mode contributing most to the original time series first. If we use all of the modes, we get back the original data exactly. Or we can choose to use just some of the modes, rejecting the small, wiggly ones for example to provide the long term trend. As you may have guessed by now, we’re going to use the non-ergodic information-bearing modes, and relegate the ergodic, noisy modes to the dust bin.
SSA normally depends on two parameters, a window length L which can’t be longer than ½ the record length and a mode selection parameter k (k is sometimes a multi-valued vector if the selected modes aren’t sequential but here they are). This can make the analysis somewhat subjective and arbitrary. Here however, we are deconstructing the temperature time-series into only two buckets. Since the non-ergodic components contribute most to the signal characteristics, they will generally be in the first k modes, and the ergodic components will be sequential, starting from mode k+1 and including all remaining L-k-1 modes. Since in this method, L only controls how much energy leaks from one of our buckets to the other, we set to its maximum value to give the finest grain resolution to the division between our two buckets. Thus our analysis depends only on a single parameter k which is set to maximize the signal to noise ratio.
That’s a long time to go without a picture so here are the results of the above applied to the Northern Hemisphere Sea Surface Temperature data.
Figure 1 -SST data (blue) vs. a reconstruction based on the first four eigen modes (L=55, k=1-4)
The blue curve is the data and the red curve is our signal “bucket” reconstructed from the first four SSA modes. Now let’s look in the garbage pail.
Figure 2 – Residual after signal extraction
We see that the residual indeed looks like noise, with no discernible trend or other information. The distribution looks fairly uniform, the slight double-peak due probably to the fact that the early data is noisier than the more recent. Remembering that the residual and the signal sum to the original data, and since there is no discernible AGW signal in the residual, we can state without fear of contradiction that any sign of AGW, if one is to be found, must be found in the reconstruction built from the non-ergodic modes plotted in red in figure 1.
What would an AGW signal look like? The AGW hypothesis is that the exponential rise in CO2 concentrations seen since the start of the last century should give rise to a linear temperature trend impressed on top of the climate’s natural variation. So we are looking for a ramp, or equivalently a step change in the slope of the temperature record. Here’s an idea of what a trendless climate record (with the natural variation and noise similar to the observed SST record) might look like, with (right) and without (left) a 4 °C/century AGW component. The four curves on the right represent four different points in time where the AGW component first becomes detectable: 1950, 1960, 1970 and 1980.
Figure 3 – Simulated de-trended climate record without AGW component (left) and with 4°C/century AWG components
Clearly a linear AGW signal of the magnitude suggested by the IPCC should be easily detectible within the natural variation. Here’s the real de-trended SST data. Which plot above does it most resemble?
Figure 4 -De-trended SST data
Note that for the “natural variation is temporarily masking AGW” meme to hold water, the natural variation during the AGW observation widow would have to be an order of magnitude higher than that which occurred previously. SSA shows that not to be the case. Here is the de-trended signal decomposed into its primary components (note, for reasons too technical to go into here, SSA modes occur in pairs. The two signals plotted below include all four signal modes constituted as pairs)
Figure 5 – Reconstruction of SSA modes 1,2 and 3,4
Note the peak-to-peak variation has remained remarkably constant across the entire data record.
Ok, so if it’s not 4°C/century, what is it? Remember we are looking for a change in slope caused by the AGW component. The plot below shows the slope of our signal reconstruction (which contains the AWG component ,if any), over time.
Figure 6 – Year-to-year difference of reconstructed signal
We see two peaks, one in 1920 well before the effects of AGW are thought to have been detectable and one slightly higher in 1995 or so. Let’s zoom in on the peaks.
Figure 7 – Difference in signal slope potentially attributable to AGW
The difference in slope is 0.00575 °C/year or ~.6 °C/century. No smoothing was done on the first-difference plot above as is normally required, because we have eliminated the noise component which makes this necessary.
Returning to our toy climate model of figure 3, here’s what it looks like with a .6 per century slope (left) with the de-trended real SST data on the right for comparison.
Figure 8 – Simulated de-trended climate record with .6°C/century linear AGW components (see figure 3 above) left, de-trended SST (northern hemisphere) data right
Fitting the SST data on the right to a sine wave-plus-ramp model yields a period of ~65 years with the AGW corner at 1966, about where expected by climatologists. The slope of the AGW fit? .59 °C/century, arrived at complete independent of the SSA analysis above.
As Monk would say, “Here’s what happened”. During the global warming scare of the 1980’s and 1990’s, the quasi-periodic modes comprising the natural temperature variation were both in their phase of maximum slope (See figure 5). This naturally occurring phenomenon was mistaken for a rapid increase in the persistent warming trend and attributed to the greenhouse gas effect. When these modes reached their peaks approximately 10 years ago, their slopes abated, resulting in the so-called “pause” we are currently enjoying. This analysis shows that the real AGW effect is benign and much more likely to be less than 1 °C/century than the 3+ °C/century given as the IPCC’s best guess for the business-as-usual scenario.
- ‘Mind blowing paper’ blames ENSO for Global Warming Hiatus (wattsupwiththat.com)
- Signal Detection: An Important Skill in a Noisy World (perceptualedge.com)
- Where’s the Magic? (EMD and SSA in R) (r-bloggers.com)