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Doug Hoffman: Orbital Climate Cycles Reaffirmed, Classical Statistics Denied

Doug Hoffman, The Resilient Earth

Instrument data from the last 160 years indicate a general warming trend during that span of time. However, when this period is examined in the light of palaeoclimate reconstructions, the recent warming appears to be a part of more systematic fluctuations. Specifically, it is an expected warming period following the 200-year “Little Ice Age” cold period.

Moreover, a new study of the natural variability of past climate, as seen from available proxy information, finds a synthesis between the Milankovitch cycles and Hurst–Kolmogorov (HK) stochastic dynamics—a result that shows multi-scale climate fluctuations cannot be described adequately by classical statistics.

During the early 19th century, scientists were trying to come to grips with physical evidence that showed Earth had undergone a great freezing in the recent geological past. Eventually, through efforts of scientists like Louis Agassiz, the theory of ice ages became widely accepted. It became clear that during the Pleistocene (2,588,000–12,000 years before present [BP]) there have been many such glacial periods, interleaved with shorter interglacials. The Holocene era, whose genial warmth we are now enjoying, is but the latest such interglacial period.

Often referred to by laymen as ice ages, glacials have ranged in length from 35–45 thousand years in early the Pleistocene to 90–120 thousand years during the last million years. In the mid 20th century, Milankovitch provided an explanation for this cycle of cold and warm periods based on Earth’s orbit variations, though it took years of careful measurement and recalculation for his theory to be accepted. But it has not always been so.

Indeed, Earth’s climate has varied widely over the Phanerozoic, the past 550 million years or so that complex life has dominated our planet’s ecosystem. The temperature has swung between warm periods when no ice huddled at the poles to frigid times when the sea-levels sank and ice built up on the continents and ocean surface. In a paper appearing in Surveys in Geophysics, Yannis Markonis and Demetris Koutsoyiannis, both from the National Technical University of Athens, set the scene this way:

It is now well known that a succession of glaciation and deglaciation periods has not occurred all the time, but only in large periods defining an ‘icehouse climate’, such as the current (Pliocene-Quaternary) icehouse period that started about 2.5 million years ago, as well as the Ordovician and the Carboniferous icehouse periods, each of which lasted 50–100 million years (Crowell and Frakes 1970). In contrast, the ‘hothouse climates’ are characterized by warmer temperatures, abundance of carbon dioxide (concentrations up to 20–25 times higher than current) and complete disappearance of polar icecaps and continental glaciers.

For those who study Earth’s climate, it is natural to look to times past for a hint as to how the world will change in the future. Trouble is, the Earth system that generates global climate is a moving target, constantly changing. As the ancient Greek philosopher Heraclitus of Ephesus said, “you cannot step twice into the same river.” That is because the water is constantly moving; the banks and river bed also change over time. So it is with the entire planet. Climatologists of all stripe have come to realize that simple statistical tools are not sufficient to capture the variability of nature. In a long and detailed paper, Markonis and Koutsoyiannis have endeavored to apply a form of stochastic dynamics, in conjunction with orbital forcings, to create a better view of Earth’s past climate variability:

We overview studies of the natural variability of past climate, as seen from available proxy information, and its attribution to deterministic or stochastic controls. Furthermore, we characterize this variability over the widest possible range of scales that the available information allows and we try to connect the deterministic Milankovitch cycles with the Hurst-Kolmogorov (HK) stochastic dynamics. To this aim, we analyze two instrumental series of global temperature and eight proxy series with varying lengths from 2 thousand to 500 million years. In our analysis we use a simple tool, the climacogram, which is the logarithmic plot of standard deviation versus time scale, and its slope can be used to identify the presence of HK dynamics. By superimposing the climacograms of the different series we obtain an impressive overview of the variability for time scales spanning almost 9 orders of magnitude—from 1 month to 50 million years. An overall climacogram slope of –0.08 supports the presence of HK dynamics with Hurst coefficient of at least 0.92. The orbital forcing (Milankovitch cycles) is also evident in the combined climacogram at time scales between 10 and 100 thousand years. While orbital forcing favours predictability at the scales it acts, the overview of climate variability at all scales suggests a big picture of irregular change and uncertainty of Earth’s climate.

Acknowledging that orbital forcing can not account for cycles shorter than 10,000 years or longer than 100,000 directly, the authors constructed a paleoclimate record based on a number of proxies, all free of anthropogenic influences. From this they claim to have found evidence for Hurst-Kolmogorov (HK) stochastic dynamics on a truly breathtaking range of timescales.

Harold Edwin Hurst, a British hydrologist working around the same time as Milankovitch’s discovery, observed that “although in random events groups of high or low values do occur, their tendency to occur in natural events is greater. This is the main difference between natural and random events”. In other words, in a natural process events of similar type are more likely to occur in groups. This is similar to the phenomenon of noise bursts discovered by Benoit Mandelbrot (famous for the eponymous Mandelbrot Set).

In 1958, while working at IBM’s Thomas J. Watson Research Center, Mandelbrot examined the problem of line noise. He discovered that, contrary to his intuition that the noise would be random, it occurred in bursts. Furthermore, on shorter and shorter time scales, the distribution of the noise spikes always remained a scaled-down version of the whole—fractal behavior. Mandelbrot’s work eventually showed that the noise was both consistent and erratic, some kind of inescapable natural feature of the system that did not disappear with increased signal strength.

Similarly, Hurst observed event timings that were not random, but instead were dictated by the nature of the systems involved. Based on Hurst’s work a statistical measure, the “Hurst exponent,” was devised as a measure of long term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. The values of the Hurst exponent, or Hurst coefficient, range between 0 and 1 and my be interpreted broadly as follows:

  • A Hurst exponent value H close to 0.5 indicates a random walk (a Brownian time series). In a random walk there is no correlation between any element and a future element and there is a 50% probability that future return values will go either up or down. Series of this type are hard to predict.
  • A Hurst exponent value H between 0 and 0.5 exists for time series with “anti-persistent behavior.” This means that an increase will tend to be followed by a decrease (or a decrease will be followed by an increase). This behaviour is sometimes called “mean reversion” which means future values will have a tendency to return to a longer term mean value. The strength of this mean reversion increases as H approaches 0.
  • A Hurst exponent value H between 0.5 and 1 indicates “persistent behavior”, that is the time series is trending. If there is an increase from time step [t-1] to [t] there will probably be an increase from [t] to [t+1]. The same is true of decreases, where a decrease will tend to follow a decrease. The larger the H value is, the stronger the trend. Series of this type are easier to predict than series falling in the other two categories.

Unknown to Hurst, Andrey Kolmogorov, the eminent Russian mathematician, had already proposed a stochastic process that described this behavior a decade earlier (1940). As is often the case in science, both Kolmogorov’s process and the natural behavior observed by Hurst only became widely known after the works of Mandelbrot and Wallis decades later. In the Surveys in Geophysics paper the authors have coined the term Hurst-Kolmogorov (HK) in order to acknowledge the contribution of the two pioneering researchers.

It is the author’s claim that use of HK stocastic dynamics overcomes some fundamental problems with everyday statistical analysis. “The multi-scale fluctuations cannot be described adequately by classical statistics, as the latter assumes independence (or weak dependence) and underestimates the system’s uncertainty on long time scales, sometimes by two, or even more, orders of magnitude,” they state. “Moreover, traditional stochastic autoregressive (AR) models cannot describe these fluctuations in an adequate way, because the autocorrelation functions of these models decay faster than those of the processes they try to model.”

Those interested in the actual mathematics behind Markonis and Koutsoyiannis’ analysis should download the PDF available on-line. For our purposes it will suffice to jump ahead to their results and the use of climacograms.

A climacogram is a logarithmic plot of standard deviation versus time scale, and its slope can be used to identify the presence of HK dynamics. “By superimposing the climacograms of the different series we obtain an impressive overview of the variability for time scales spanning almost 9 orders of magnitude—from 1 month to 50 million years,” the author’s claim. Example climacograms are shown below.

Example climacograms.

The figure above shows climacograms for (a) a white noise (purely random) process, (b) a purely periodic process with period 100, (c) an AR(1) (Markov) process with ρ = 0.75, and (d) an HK process with H = 0.9. A more meaningful climacogram is the one below, showing the ten temperature observation series and proxies used in the study.

Combined climacogram of the ten temperature observation series and proxies.

The dotted line with slope –0.5 represents the climacogram of a purely random process. The horizontal dashed-dotted line represents the climatic variability at 100 million years, while the vertical dashed-dotted line at 28 months, represents the corresponding scale to the 100-million-year variability if climate is assumed to be random. As can be seen there is a fairly good correlation over a wide number of timescales (note again that the graph is logarithmic). The only wrinkle is the deviation of the EPICA data (the green °s). This is explained by the next diagram.

Theoretical climacograms.

Shown are theoretical climacograms of an HK process with H = 0.92 and two periodic processes with periods 100 and 41 thousand years, all having unit standard deviation at monthly scale, along with the climacogram of the synthesis (weighted sum) of these three components with weights 0.95, 0.30 and 0.15, respectively. The empirical climacogram of a time series simulated from the synthesis process with time step and length equal to those of the EPICA series is also plotted. What this illustrates it that the deviation shown in the previous climacogram can be explained as the result of two of the Earth’s orbital variations, excentricity and obliquity, showing up in the EPICA proxy data.

This would appear to be the “fingerprint” of the Milankovitch Cycles in the proxy record but the case for the longer, 100,000 year, cycle is not as accepted as the 41,000 year cycle. Indeed, the authors state that, “the duration of each of the last four glacial cycles increased from 80 to 130 thousand years, which suggests that major climate shifts were aperiodic.” That caveat aside, Markonis and Koutsoyiannis conclude that there is ample evidence of orbital forcing in the proxy climate record.

What are we to conclude from this paper? I find two salient points: first, that the influence of the Milankovitch Cycles over the glacial-interglacial cycle has been reinforced by this work and, second, the need to look beyond classical, garden variety statistics when analyzing climate data has once again been demonstrated. More practically, the high H factor associated with temperature change means that if you have a single hot year you are apt to have several in a row, and if you have a single cold year more will probably follow, but the cycling between hot and cold continues.

In other words, a string of hot years does not a long-term trend make. It strikes me that a large number of practicing climate scientists do not appreciate the inadequacies of classical statistics. They continue to apply inappropriate measures and wonder later why their predictions do not come true. In this day of cheap computers and widely available free statistical software anyone can run correlations on massive datasets. Sadly, statistics in the hands of a climate alarmist is like a loaded gun in the hands of a child.

Be safe, enjoy the interglacial and stay skeptical.

The Resilient Earth, 6 November 2012