Update 28 April 2014: Climate Dialogue summaries now online
The summary of the second Climate Dialogue discussion on long-term persistence is now online (see below). We have made two versions: a short and an extended version. We apologize for the delay in publishing the summary.

Both versions can also be downloaded as pdf documents:
Summary of the Climate Dialogue on Long-term persistence
Extended summary of the Climate Dialogue on Long-term persistence
[End update]

In this second Climate Dialogue we focus on long-term persistence (LTP) and the consequences it has for trend significance.

We slightly changed the procedure compared to the first Climate Dialogue (which was about Arctic sea ice). This time we asked the invited experts to write a first reaction on the guest blogs of the others, describing their agreement and disagreement with it. We publish the guest blogs and these first reactions at the same time.

We apologise again for the long delay. As we explained in our first evaluation it turned out to be extremely difficult to find the right people (representing a range of views) for the dialogues we had in mind.

Climate Dialogue editorial staff
Rob van Dorland, KNMI
Marcel Crok, science writer
Bart Verheggen

Introduction long-term persistence

How persistent is the climate and what is its implication for the significance of trends?

The Earth is warmer now than it was 150 years ago. This fact itself is uncontroversial. It’s not trivial though how to interpret this warming. The attribution of this warming to anthropogenic causes relies heavily on an accurate characterization of the natural behavior of the system. Here we will discuss how statistical assumptions influence the interpretation of measured global warming.

Most experts agree that three types of processes (internal variability, natural and anthropogenic forcings) play a role in changing the Earth’s climate over the past 150 years. It is the relative magnitude of each that is in dispute. The IPCC AR4 report stated that “it is extremely unlikely (<5%) that recent global warming is due to internal variability alone, and very unlikely (< 10 %) that it is due to known natural causes alone.” This conclusion is based on detection and attribution studies of different climate variables and different ‘fingerprints’ which include not only observations but also physical insights in the climate processes.

Detection and attribution
The IPCC definitions of detection and attribution are:

“Detection of climate change is the process of demonstrating that climate has changed in some defined statistical sense, without providing a reason for that change.”

“Attribution of causes of climate change is the process of establishing the most likely causes for the detected change with some defined level of confidence.”

The phrase ‘change in some defined statistical sense’ in the definition for detection turns out to be the starting point for our discussion. Because what is the ‘right’ statistical model (assumption) to conclude whether a change is significant or not?

According to AR4, “An identified change is ‘detected’ in observations if its likelihood of occurrence by chance due to internal variability alone is determined to be small.” The magnitude of internal variability can be estimated in different ways, e.g. by control runs of global climate models, by characterising the statistical behaviour of the time series, by inspection of the spatial and temporal fingerprints in observations, and by comparing models and observations (e.g. via their respective power spectra, cf. fig 9.7 in AR4).

Long-term persistence
Critics argue though that most if not all changes in the climatological time series are an expression of long-term persistence (LTP). Long-term persistence means there is a long memory in the system, although unlike a random walk it remains bounded in the very long run. There are stochastic /unforced fluctuations on all time scales. More technically, the autocorrelation function goes to zero algebraically (very slowly). These critics argue that by taking LTP into account trend significance is reduced by orders of magnitude compared to statistical models that assume short-term persistence (AR1), as was applied e.g. in the illustrative trend estimates in table 3.2 of AR4. [1,2]

This has consequences for attribution as well, since long term persistence is often assumed to be a sign of unforced (internal) variability (e.g. Cohn and Lins, 2005; Rybski et al, 2006). In reaction to Cohn and Lins (2005), Rybski et al. (2006)[3] concluded that even when LTP is taken into account at least part of the recent warming cannot be solely related to natural factors and that the recent clustering of warm years is very unusual (see also Zorita (2008)[4]). Bunde and Lennartz also looked extensively at the consequences of LTP for trend significance and concluded that globally averaged temperature data on land do show a significant trend, but the sea surface temperatures don’t. [5,6]

These different conclusions translate directly into the question of how important the statistical model used is for determining the significance of the observed trends.

Climate Dialogue
Although the IPCC definition for detection seems to be clear, the phrase ‘change in some defined statistical sense’ leaves a lot of wiggle room. For the sake of a focussed discussion we define here the detection of climate change as showing that some of this change is outside the bounds of internal climate variability. The focus of this discussion is how to best apply statistical methods and physical understanding to address this question of whether the observed changes are outside the bounds of internal variability. Discussions about the physical mechanisms governing the internal variability are also welcome.

Specific questions

1. What exactly is long-term persistence (LTP), and why is it relevant for the detection of climate change?

2. Is “detection” purely a matter of statistics? And how does the statistical model relate to our knowledge of internal variability?

3. What is the ‘right’ statistical model to analyse whether there is a detected change or not? What are your assumptions when using that model?

4. How long should a time series be in order to make a meaningful inference about LTP or other statistical models? How can one be sure that one model is better than the other?

5. Based on your statistical model of preference do you conclude that there is a significant warming trend?

6. Based on your statistical model of preference what is the probability that 11 of the warmest years in a 162 year long time series (HadCrut4) all lie in the last 12 years?

7. If you reject detection of climate change based on your preferred statistical model, would you have a suggestion as to the mechanism(s) that have generated the observed warming?

[1] Cohn,. T. A., and H. F. Lins (2005), Nature's style: Naturally trendy,. Geophys. Res. Lett., 32, L23402, doi:10.1029/2005GL024476
[2] Koutsoyiannis, D., Montanari, A., Statistical analysis of hydroclimatic time series: Uncertainty and insights, Water Resour. Res., Vol. 43, W05429, doi:10.1029/2006WR005592, 2007
[3] Rybski, D., A. Bunde, S. Havlin, and H. von Storch (2006), Long-term persistence in climate and the detection problem, Geophys. Res. Lett., 33, L06718, doi:10.1029/2005GL025591
[4] Zorita, E., T. F. Stocker, and H. von Storch (2008), How unusual is the recent series of warm years?, Geophys. Res. Lett., 35, L24706, doi:10.1029/2008GL036228
[5] Lennartz, S., and A. Bunde. "Trend evaluation in records with long‐term memory: Application to global warming." Geophysical Research Letters 36.16 (2009)
[6] Lennartz, Sabine, and Armin Bunde. "Distribution of natural trends in long-term correlated records: A scaling approach." Physical Review E 84.2 (2011): 021129

Guest blog Rasmus Benestad

The debate about trends gets lost in statistics...

Rasmus E. Benestad, senior researcher, Norwegian Meteorological Institute

Figure 1. The recorded changes in the global mean temperature over time (red). The grey curve shows a calculation of the temperature based on greenhouse gases, ozone, and changes in the sun.

From time to time, the question pops up whether the global warming recorded by a network of thermometers around the globe is a result of natural causes, or if the warming is forced by changes in the atmospheric concentrations of greenhouse gases (GHGs).

In 2005, there was a scientific paper [1] suggesting that statistical models describing random long-term persistence (LTP) could produce similar trends as seen in the global mean temperature. I wrote a comment then on this paper on RealClimate.org with the title Naturally trendy?.

More recently, another paper [2] followed somewhat similar ideas, although for the Arctic temperature rather than the global mean. A discussion ensued after my posting of ‘What is signal and what is noise?’ on RealClimate.org. A meeting is planned in Tromsø, Norway in the beginning of May to discuss our differences - much in the spirit of Climate Dialogue.

Weather statistics
It is easy to make a statement about the Earth’s climate, but what is the story behind the different views? And what is really the problem?

If we start from scratch, we first need to have a clear idea about what we mean by climate and climate change. Often, climate is defined as the typical weather, described by the weather statistics: what range of temperature and rainfall we can expect, and how frequently. Experts usually call this kind of statistics ‘frequency distribution’ or ‘probability density functions’ (pdfs).

A climate change happens when the weather statistics are shifted: weather that was typical in the past is no longer typical. It involves a sustained change rather than short-lived variations. New weather patterns emerge during a climate change.

At the same time, there is little doubt that some primary features of our climate involve short-lived natural ‘spontaneous’ fluctuations, caused by the climate itself. The natural fluctuations are distinct to a forced climate change in terms of their duration [3].

A diagnostic for climate change involves statistical analyses to assess whether the present range and frequency of temperature and precipitation are different from those of the past.

Natural variations
However, the presence of slower natural variations in the climate makes it difficult to make a correct diagnosis.

Long-term persistence (LTP) describes how slow physical processes change over time, where the gradual nature is due to some kind of ‘memory’. This memory may involve some kind of inertia, or the time it takes for physical processes to run their course. Changes over large space take longer time than local changes.

The diffusion of heat and transport of mass and energy are subject to finite flow speeds, and the time it takes for heat to leak into the surroundings is well-understood. Often the rate decays at an exponential rate (with a negative exponent).

There may be more complex behaviour when different physical processes intervene and affect each other, such as the oceans and the atmosphere [4]. The oceans are sluggish and the density and heat capacity of water are much higher than that for the air. Hence the ocean acts like a flywheel, and once it gets moving, it will go on for a while.

There are some known examples of LTP processes, such as the ice ages, changes in the ocean circulation, and the El Niño Southern Oscillation.

We also know that the Earth’s atmosphere is non-linear (‘chaotic’) and may settle into different states, known as ‘weather regimes’ [5-7]. Such behaviour may also produce LTP. Changes in the oceans through the overturning of water masses can result in different weather regimes.

Laws of physics
It is also possible to show that LTP takes place when many processes are combined, which individually do not have LTP. For example, the river flow may exhibit some LTP characteristics, resulting from a collection of rainfall over several watersheds.

We should not forget that long-term changes in the forcing from GHGs also result in LTP behaviour [3].

The climate involves more than just observations and statistics, and like everything else in the universe, it must obey the laws of physics. From this angle, climate change is an imbalance in terms of energy and heat.

It is fairly straight-forward to measure the heat stored in the oceans, as warmer water expands. The global sea level provides an indicator for Earth’s accumulation of heat over time [3].

Hence, the diagnosis (“detection”) of a climate change is not purely a matter of statistics. The laws of physics sets fundamental constraints which lets us narrow down to a small number of ‘suspects’.

Energy and mass budgets are central, but also the hydrological cycle is entangled with the temperatures and the oceans [8]. Modern measurements provide a comprehensive picture: we see changes in the circulation and rainfall patterns.

Classical mistakes
There are two classical mistakes made in the debate about climate change and LTP: (a) looking only at a single aspect (one single time series) isolated from the rest of Earth’s climate and (b) confusing signal for noise.

The term ‘signal’ can have different meanings depending on the question, but here it refers to manmade climate change. ‘Noise’ usually means everything else, and LTP is ‘noise in slow motion’.

There are always physical processes driving both LTP and spontaneous changes on Earth (known as the weather), and these must be subject to study if we want to separate noise from signal.

If an upward trend in the temperature were to be caused by internal ocean overturning, then this would imply a movement of heat from the oceans to the air and land surface. Energy must be conserved. When the heat content increases in both the oceans [9] and the air, and the ice is melting, then it is evident that the trend cannot be explained in terms of LTP.

The other mistake is neglecting the question: What is signal and what is noise? Some researchers have adapted statistical trend models to mimic the evolution seen in measured data [1,2]. These models must be adapted to the data by adjusting number of parameters so that they can reproduce the LTP behaviour.

In science, we often talk about a range of different types of models, and they come in all sorts of sizes and shapes. A climate model calculates the temperature, air flow, rainfall, and ocean currents over the whole globe, based on our knowledge about the physics. Statistical trend models, on the other hand, take past measurements for which they try to find statistical curves that follow the data.

Statistical models are sometimes fed random numbers in order to produce a result that looks like noise [10]. It is concluded that the measured data are a result of a noisy process if these models produce results that look the same. In other words, these models are used to establish a benchmark for assessing trends.

Statistical trend models may, however, produce misguided answers if proper care is not taken. For example, those adapted to data containing both signal and noise cannot be used to infer whether the observed trends are unusual or not. The LTP associated with GHGs can be quite substantial, and forced trends in measured temperatures will fool the statistical models which assume the LPT is due to noise.

The misapplication of statistical trend models can easily be demonstrated by subjecting them to certain tests. The statistical trend models describing LTP make use of information embedded in the data, revealed by their respective autocorrelation function (ACF).

Figure 2 below shows a comparison between two ACFs for temperature for the Arctic (area mean above 60N), taken from two different climate model simulations. One simulation represents the past (black) driven with historical changes in GHGs. The other (grey) describes a hypothetical world where the GHGs were constant, representing a ‘stable’ climate in terms of external forcings.

It is clear that the ACFs differ, and the statistical models used to assess trends and LTP rely on the shape of the ACF.

Figure 2. The upper graph shows the annual mean temperature for the Arctic simulated by a climate model. The black curve shows the year-to-year variations for a run where the model was given the observed GHG measurements. The grey curve shows a similar run, but where the GHGs do not change. The bottom panel shows the ACF, and the black curve indicates that the effect of GHGs on temperature is to increase the LTP. For the assessment of trends, the statistical models should be trained on the grey curve, for which we know there is no forced trend and where all the variations are due to changes in the oceans.

Circular reasoning
It is the way models are used that really matters, rather than the specific model itself. All models are based upon a set of assumptions, and if these are violated, then the models tend to give misleading answers. Statistical LTP-noise models used for the detection of trends involve circular reasoning if adapted to measured data. Because this data embed both signal and noise.

For real-world data, we do not know what degree of the variations are LTP-noise and what are signal.

We can nevertheless specify the degree of forcing in climate model simulations, and then use these results to test the statistical models. Even if the climate models are not completely perfect, they serve as a test bench [11].

The important assumptions are therefore that the statistical trend models, against which the data are benchmarked, really provide a reliable description of the noise.

We need more than a century-long time series to make a meaningful inference about LTP if natural variations have time scales of 70-90 years. Most thermometer records do not go much longer back in time than a century, although there are some exceptions.

It is possible to remedy the lack of thermometer records to some extent by supplementing the information with evidence based on e.g. tree rings, sediment samples, and ice cores3,12,13. Still, such evidence tends to be limited to temperature, whereas climate change involves a whole range of elements.

For all intents and purposes, however, it is important to account for both natural and forced climate change on these time scales. Most people would worry more about the combined effect of these components, as natural variations may be just as devastating as forced. For most people, the distinction between trend and noise is an academic question. For politicians, it's a question about cutting GHG emissions.

Regression
Another side to the story is that the magnitude of the unforced LTP variations may give us an idea about the climate's sensitivity to changing conditions. Often, such cycles are caused by delayed but reinforcing processes. Conditions which amplify an initial response are inherent to the atmosphere, and may act both on forced response (GHGs) as well as internal variations. Damping mechanisms will also tend to strangle oscillations, which is well known from many different physical systems.

The combination of statistical information and physics knowledge lead to only one plausible explanation for the observed global warming, global mean sea level rise, melting of ice, and accumulation of ocean heat. The explanation is the increased concentrations of GHGs.

We can also use another statistical technique for diagnosing a cause [11] which is also used in medical sciences and known as ‘regression analysis’. Figure 1 shows the results of a multiple regression analyses with inputs representing expected physical connections to Earth’s climate. In this case, the regression analysis used GHGs, ozone and changes in the sun’s intensity as inputs, and the results followed the HadCRUT4 data closely.

The probability that this fit is accidental is practically zero if we assume that that the temperature variations from year-to-year are independent of each other. LTP and the oceans inertia will imply that the degrees of freedom is lower than the number of data points, making it somewhat more likely to happen even by chance.

Furthermore, taking paleoclimatic information into account, there is no evidence that there have been similar temperature excursion in the past ~1000 years [12-14]. If the present warming was a result of natural fluctuations, it would imply a high climate sensitivity, and similar variations in the past. Moreover, it would suggest that any known forcing, such as GHGs, would be amplified accordingly. The climate sensitivity may be a common denominator for natural fluctuations and forced trends (Figure 3).

Figure 3. Comparison between trend estimates and the amplitude of 10-year low-passed internal variability in state-of-the-art global climate models.

There may be some irony here: The warming 'hiatus' during last decade is due to LPT-noise [15,16]. However, when the undulations from such natural processes mask the GHG-driven trend, it may in fact suggest a high climate sensitivity – because such natural variations would not be so pronounced without a strong amplification from feedback mechanisms. Figure 3 shows that such natural variations in the climate models are more pronounced for the models with stronger transient climate response (TCR, a rough indicator for climate sensitivity).

For complete probability assessment, we need to take into account both the statistics and the physics-based information, such as the fact that GHGs absorb infrared light and thus affect the vertical energy flow through the atmosphere.

Summary
In summary, we do not really know what the LTP in the real world would be like without GHG forcings, and we don’t know the real degrees of freedom in the measured data record. The lack of such information limits our ability to make a statistics-based probability estimate. On the other hand, we know from past reconstructions and physical reasoning that present warming is highly unusual.

Biosketch
Rasmus Benestad is a physicist by training. Benestad has a D.Phil in physics from Atmospheric, Oceanic & Planetary Physics at Oxford University in the United Kingdom. He has affiliations with the Norwegian Meteorological Institute.
Recent work involves a good deal of statistics (empirical-statistical downscaling, trend analysis, model validation, extremes and record values), but he has also had some experience with electronics, cloud micro-physics, ocean dynamics/air-sea processes and seasonal forecasting.
In addition, Benestad wrote the book ‘Solar Activity and Earth’s Climate’ (2002), published by Praxis-Springer, and together with two colleagues the text book ‘Empirical-Statistical Downscaling’ (2008; World Scientific Publishers). He has also written a number of R-packages for climate analysis posted on http://cran.r-project.org.
Benestad was a member of the council of the European Meteorological Society for the period (2004-2006), representing the Nordic countries and the Norwegian Meteorology Society, and he has served as a member of the CORDEX Task Force on Regional Climate Downscaling.
He is a regular contributor to the well-known climate blog RealClimate.org.

References

1. Cohn, T. A. & Lins, H. F. Nature’s style: Naturally trendy. Geophys. Res. Lett. 32, n/a–n/a (2005).

2. Franzke, C. On the statistical significance of surface air temperature trends in the Eurasian Arctic region. Geophys. Res. Lett. 39, n/a–n/a (2012).

3. Climate Change: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. (Cambridge University Press, 2007).

4. Anderson, D. L. T. & McCreary, J. P. Slowly Propagating Disturbances in a Coupled Ocean-Atmosphere Model. J. Atmospheric Sci. 42, 615–629 (1985).

5. Gleick, J. Chaos. (Cardinal, 1987).

6. Lorenz, E. Deterministic nonperiodic flow. J. Atmospheric Sci. 20, 130–141 (1963).

7. Shukla, J. Predicatability in the Mids of Chaos: A scientific Basis for climate forecasting. Science 282, 728–733 (1998).

8. Durack, P. J., Wijffels, S. E. & Matear, R. J. Ocean Salinities REveal Strong Global Water Cycle Intensification During 1950 to 2000. Science 336, 455–458 (2012).

9. Balmaseda, M. A., Trenberth, K. E. & Källén, E. Distinctive climate signals in reanalysis of global ocean heat content. Geophys. Res. Lett. n/a–n/a (2013). doi:10.1002/grl.50382

10. Paeth, H. & Hense, A. Sensitivity of climate change signals deduced from multi-model Monte Carlo experiments. Clim. Res. 22, 189–204 (2002).

11. Benestad, R. E. & Schmidt, G. A. Solar Trends and Global Warming. JGR 114, D14101 (2009).

12. Mann, M. E., Bradley, R. S. & Hughes, M. K. Global-scale temperature patterns and climate forcing over the past six centuries. Nature 392, 779–787 (1998).

13. Moberg, A., Sonechkin, D. M., Holmgren, K., Datsenko, N. M. & Karlén, W. Highly variable Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data. Nature 433, 613–617 (2005).

14. Marcott, S. A., Shakun, J. D., Clark, P. U. & Mix, A. C. A Reconstruction of Regional and Global Temperature for the Past 11,300 Years. Science 339, 1198–1201 (2013).

15. Easterling, D. R. & Wehner, M. F. Is the climate warming or cooling? Geophys Res Lett 36, L08706 (2009).

16. Foster, G. & Rahmstorf, S. Global temperature evolution 1979–2010. Environ. Res. Lett. 6, 044022 (2011).

Guest blog Demetris Koutsoyiannis

LTP: Looking Trendy—Persistently

Demetris Koutsoyiannis, National Technical University of Athens, Greece

Stochastics and its importance in studying climate

Probability, statistics and stochastic processes, lately described by the collective term stochastics, provide essential concepts and tools to deal with uncertainty useful for all scientific disciplines. However, there is at least one scientific discipline whose very domain relies on stochastics: Climatology. To refer to a popular definition of this domain by IPCC[1] (also quoted in Wikipedia):

Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. The classical period for averaging these variables is 30 years, as defined by the World Meteorological Organization.

“Average”, “statistical description”, “mean”, “variability”, are all statistical terms. Several questions related to climate also involve probability, as exemplified in question 6 of the Introduction of this Climate Dialogue theme:[2]

Based on your statistical model of preference what is the probability that 11 of the warmest years in a 162 year long time series (HadCrut4) all lie in the last 12 years?

Interestingly, similar probabilistic and statistical notions are implied in a recent President Obama statement:[3]

Yes, it’s true that no single event makes a trend. But the fact is, the 12 hottest years on record have all come in the last 15.

The latter statement highlights how important statistical questions are for policy matters and presumes some public perception of probability and statistics, which determines how the message is received.

I have no doubt that the average human being has some understanding of probability and statistics, not only thanks to education, but because life is uncertain and each of us needs to develop understanding of uncertainty and skills to cope with it. However, common experience and perception are mostly related to too simple uncertainties, like in coin tosses, dice throws and roulette wheels. Also education is mainly based on classical statistics in which:

· Consecutive events are independent to each other: the outcome of an event does not affect that of the next one.

· As a result, time averages tend to stabilize relatively fast: their variability, expressed by the probabilistic concept of variance, is inversely proportional to the length of the averaging period.

Adhering to classical statistics, when dealing with climate and other complex geophysical processes, is not just a problem of laymen. There are numerous research publications adopting, tacitly or explicitly, the independence assumption for systems in which it is totally inappropriate. Even the very definition of climate quoted above, particularly the phrase “The classical period is 30 years” historically reflects a perception of a constant climate[4][5] and a hope that 30 years would be enough for a climatic quantity to get stabilized to a constant value—and this is roughly supported by classical statistics. In this perception a constant climate would be the norm and a deviation from the norm would be something caused by an extraordinary agent. The same static-climate conviction is evident in the “weather vs. climate” dichotomy (e.g. the “climate is what you expect, weather is what you get”; (see critical discussions in Refs. [6], [7], [5]).

Figure 1. Probability that a 12-year period contains the specified number of warmest years (n) or more in a 162-year long period, as calculated assuming a random climate and a Hurst-Kolmogorov (HK) climate with Hurst parameter H = 0.92 (see text below for explanation of the latter).

Now let us pretend that, as in classical statistics, climate was synthesized by averaging random events without dependence and try to study on this basis the above question (slightly modified for reasons that will be explained later). So, what is the probability that, in a 162-year long time series, at least n (where n = 1, 2, …12) of the warmest years all lie in a 12-year long period? The reply is depicted in Figure 1 labelled “Random climate”. The first seven points are calculated by Monte-Carlo simulation. For n = 7 years this probability is 0.00005 (1/20 000). The Monte Carlo simulation would require too much time to find the probability that all 12 warmest years are consecutive (n = 12), because this probability is really an astonishingly small number; but I was able to find it analytically and plotted it on the graph. I also approximated with analytical calculations the probability that at least 11 warmest years are clustered within 12 years. From the graph we can conclude that it is quite unlikely that more than 8-9 warmest years would cluster, even throughout the entire Earth’s life (4.5 billion years separated in segments of 162 years). To have 11 warmest events clustering in a 12-year period we would need, on the average, one hundred thousand times the age of the Universe.

Is this overwhelming evidence that something extraordinary has occurred during our lives, or that the independence assumption leads to blatantly irrational results?

No one would believe that the weather this hour does not depend on that an hour ago. It is natural to assume that there is time dependence in weather. Therefore, we must study weather not on the basis of classical statistics, but we should rather use the notion of a stochastic process. Now, if we average the process to another scale, daily, monthly, annual, decadal, centennial, etc. we get other stochastic processes, not qualitatively different from the hourly one. Of course, as the scale of averaging increases the variability decreases—but not as much as implied by classical statistics. Naturally the dependence makes clustering of similar events more likely.

The first who studied clustering in natural processes was Harold Edwin Hurst, a British hydrologist who worked in the Nile for more than 60 years. In 1951 he published a seminal paper[8] in which he stated:

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.

Herodotus said that the Egyptian land is "a gift of the Nile". Nile gave also hydrology and climatology invaluable gifts: one of them is the longest record of instrumental observations in history. Its water levels were measured in so-called Nilometers and archived for many centuries. In the 1920s Omar Toussoun, Prince of Alexandria, published a book[9] containing, among other things, annual minimum and maximum water levels of the Nile at the Roda Nilometer from AD 622 to 1921. Figure 2 depicts the time series of annual minimum levels up to 1470 (849 values; unfortunately, after 1470 there are substantial gaps). Climatic, i.e. 30-year average, values are also plotted. One may say that these values are not climatic in strict sense. But they are strongly linked to the variability of the climate of a large area, from Mediterranean to the tropics. And they are instrumental.

The clustering of similar events, more formally described as Long-Term Persistence (LTP) is obvious. For example, around AD 780 we have a group of low values producing a low climatic value, and around 1110 and 1440 we have groups of large values. Such grouping would not appear in a climate that would be the synthesis of independent random events. The latter would be more flat as illustrated by the synthetic example of Figure 3.

Another way of viewing the long-term variability of the Nile in Figure 2 is by using the notion of trends, irregularly changing from positive to negative and from mild to steep. The long instrumental Nile series may help those who prefer the view of variability in terms of trends to recognize “Nature's style [as] Naturally trendy” to invoke the title of a celebrated recent paper.[10]

Figure 2. Nile River annual minimum water level at Roda Nilometer (from Ref. 9, here converted into water depths assuming a datum for the river bottom at 8.80 m), along with 30-year averages (centred). A few missing values at years 1285, 1297, 1303, 1310, 1319, 1363 and 1434 are filled in using a simple method from Ref. [11]. The estimated statistics are mean = 2.74 m, standard deviation = 1.00 m, Hurst parameter = 0.87.

Figure 3. A synthetic time series from an independent (white noise) process with same statistics as those of the Nilometer series shown in the caption of Figure 2.

Seeking a proper stochastic model for climate

Variability over different time scales, trends, clustering and persistence are all closely linked to each other. The former is a more rigorous concept and is mathematically described by the variance (or the standard deviation) of the averaged process as a function of the averaging time scale, aka climacogram. [6],[12] The variability over scale (the climacogram), is also one-to-one related (by transformation) to the stochastic concepts of dependence in time (the autocorrelation function) and the spectral properties (the power spectrum) of the process of interest. [6][12]

In white noise, i.e., the process characterized by complete independence in time, the variability is infinite at the instantaneous time scale (in technical terms its autocorrelation in continuous time is a Dirac Delta function). No variability is added at any finite time scale. Clearly, this is a mathematical construct which cannot occur in nature (the adjective “white”, suggestive of the white light as a mixture of frequencies, is misleading; the spectral density of white noise is flat, while that of the white light is not).

A seemingly realistic stochastic process, which has been widely used for climate, is the Markov process, whose discrete time version is more widely known as the AR(1) process. The characteristic properties of this process are two:

· Its past has no influence on the future whenever the present is known (in other words, only the latest known value matters for the future).[13]

· It assumes a single characteristic time scale in which change or variability is created (but in contrast to the white noise, this time scale is non-zero and technically is expressed by the denominator of the exponent in an exponential function that constitutes its autocorrelation function). As a result, when the time scale of interest is fairly larger than this characteristic scale, the process behaves like white noise.

It is difficult to explain why this model has become dominant in climatology. Even these two theoretical properties should have hampered its popularity. How could the future be affected just by the latest value and not by the entire past? Could any geophysical process, including climate, be determined by just one mechanism acting on a single time scale?

The flow in a river (not necessarily the Nile) may help us understand better the multiplicity of mechanisms producing change and the multiplicity of the relevant time scales (see also Ref. [14]):

· Next second: the hydraulic characteristics (water level, velocity) will change due to turbulence.

· Next day: the river discharge will change (even dramatically, in case of a flood).

· Next year: The river cross-section will change (erosion-deposition of sediments).

· Next century: The river basin characteristics (e.g. vegetation, land use) will change.

· Next millennium: All could be very different (e.g. the area could be glacialized).

· Next millions of years: The river may have disappeared (owing to tectonic processes).

Of course none of these changes will be a surprise; rather, it would be a surprise if things remained static. Despite being anticipated, all these changes are not predictable.

Does a plurality of mechanisms acting on different scales require a complex stochastic model? Not necessarily. A decade before Hurst detected LTP in natural processes, Andrey Kolmogorov,[15] devised a mathematical model which describes this behaviour using one parameter only, i.e. no more than in the Markov model. We call this model the Hurst-Kolmogorov (HK) model (aka fGn—for fractional Gaussian noise, simple scaling process etc.), while its parameter has been known as the Hurst parameter and is typically denoted by H. In this model, change is produced at all scales and thus it never becomes similar to white noise, whatever the time scale of averaging is.

Specifically, the variance will never become inversely proportional to time scale; it will decrease at a lower rate, inversely proportional to the power (2 – 2H) of the time scale (nb. 0.5 ≤ H < 1, where the value H = 0.5 corresponds to white noise). A characteristic property of the HK process is that its autocorrelation function is independent of time scale. In other words if there is some correlation in the average temperature between a year and the next one (and in fact there is), the same correlation will be between a decade and the next one, a century and the next one, and so on to infinity. Why? Because there will always be another natural mechanism acting on a bigger scale, which will create change, and thus positive correlation at all lower scales (the relationship of change with autocorrelation is better explained in Ref. 6). The HK behaviour seems to be consistent with the principle of extremal entropy production.[16]

The Nilometer record described above is consistent with the HK model with H = 0.87. Are there other records of geophysical processes consistent with the HK behaviour? A recent overview paper[17] cites numerous studies where this behaviour has been verified. It also examines several instrumental and proxy climate data series related to temperature and, by superimposing the climacograms of the different series, it obtains an overview of the variability for time scales spanning almost nine orders of magnitude—from 1 month to 50 million years. The overall climacogram supports the presence of HK dynamics in climate with H 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.

Statistical assessment of current climate evolution

Re-examining the statistical problem of 11 warmest years in 12 within 162 year period, now within an HK framework with H = 0.92, we will find spectacularly different results from those of the random climate, as shown in Figure 1. We may see, for example, that what, according to the classical statistical perception, would require the entire age of the Earth to occur once (i.e. clustering of 8-9 events) is a regular event for an HK climate, with probability on the order of 1-10%.

This dramatic difference can help us understand why the choice of a proper stochastic model is relevant for the detection of changes in climate. It may also help us realize how easy it is to fool ourselves, given that our perception of probability may heavily rely on classical statistics.

Figure 4 gives a close-up of the results, excluding the very low probabilities and also generalizing the “12-year period” to “N-year period” so that it can host, in addition to the Climate Dialogue statistical question 6, the results for the “Obama version” thereof as quoted above. In addition, Figure 4 is based on a slightly higher value of the Hurst coefficient, H = 0.94, as estimated by the Least Squares based on Standard Deviation method[18] for the HadCrut4 record. Both versions result in about the same answer: the probability of having 11 warmest years in 12, or 12 warmest years in 15, is 0.1%.

Figure 4. Probability that a N-year period, where N = 12 or 15, contains the specified number, n, of warmest years or more in a 162-year long period, calculated in the same manner as in Figure 1 with H = 0.94.

If we used the IPCC AR4 terminology[19] we would say that either of these events is exceptionally unlikely to have a natural cause. Interestingly, the present results do not contradict those of a recent study of Zorita, Stocker and von Storch,[20] who examined a similar question and concluded that:

Under two statistical null-hypotheses, autoregressive and long-memory, this probability turns to be very low: for the global records lower than p = 0.001…

I note, though, that there are differences in the methodology followed here and that in Zorita et al.; for example, the analysis here did not consider whether the N-year period (where the n warmest years are clustered) is located in the end of the examined observation period or anywhere else in it (the reason will be explained below).

One may note that the above results, as well as those by Zorita et al., are affected by uncertainty, associated with the parameter estimation but also with the data set itself. The data are altered all the time as a result of continuous adaptations and adjustments. Even the ranks of the different years are changing: for example in the CRU data examined by Koutsoyiannis and Montanari (2007)[21], 1998 was rank 1 (the warmest of all) and 2005 was rank 2, while now the ranking of these two years was inverted. But most importantly, the analysis is affected by the Mexican Hat Fallacy (MHF), if I am allowed to use this name to describe a neat example of statistical misuse offered by von Storch,[22] in which the conclusion is drawn that:

The Mexican Hat is not of natural origin but man-made.

Von Storch [22] aptly describes the fallacy in these words:

The fundamental error is that the null hypothesis is not independent of the data which are used to conduct the test. We know a-priori that the Mexican Hat is a rare event, therefore the impossibility of finding such a combination of stones cannot be used as evidence against its natural origin. The same trick can of course be used to “prove” that any rare event is “non-natural”, be it a heat wave or a particularly violent storm - the probability of observing a rare event is small.

I believe that by rephrasing “11 of the warmest years … all lie in the last 12 years” into “11 of the warmest years … all lie in a 12-year long period ” reduces the MHF effect, but I do not think it eliminates it. That is why I prefer other statistical methods of detecting changes[23], such as the tests proposed by Hamed[24] and by Cohn and Lins [10]. The former relies on a test statistic based on the ranks of all data, rather than a few of them, while the second considers also the magnitude of the actual change, not that of the change in the ranks.

Another test statistic was proposed by Rybski et al.,[25] and was modified to include the uncertainty in the estimation of standard deviation by Koutsoyiannis and Montanari [21], who also applied it for the CRU temperature data up to 2005. Note that, to make the test simple, the uncertainty in the estimation of H was not considered even in the latter version (thus it could rather be called a pseudo-test). Here I updated the application of this test and I present the results in Figure 5.

Figure 5 Updated Fig. 2 in Koutsoyiannis and Montanari [21] testing lagged climatic differences based on the HadCrut4 data set (1850-2012; see explanation in text).

The method has the advantages that it uses the entire series (not a few values), it considers the actual climatic values (not their ranks) and it avoids specifying a mathematical form of trend (e.g. linear). Furthermore, it is simple: First we calculate the climatic value of each year as the average of the present and the previous 29 years. This is plotted as a pink continuous line in Figure 5, where we can see, among other things, that the latest climatic value is 0.31°C (at 2012, being the average of HadCrut4 data values for 1983-2012), while the earliest one was –0.30°C (at 1879, being the average of 1850-79). Thus, during the last 134 years the climate has warmed by 0.61°C. Note that no subjective smoothing is made here (in contrast to the graphs by CRU), and thus the climatic series has length 134 years (but with only 5 non-overlapping values), while the annual series has length 163.

Our (pseudo)test relies on climatic differences for different time lags (not just that of the latest and earliest values). For example, assuming a lag of 30 years (equal to the period for which we defined a climatic value), the climate difference between 2012 and 1982 is 0.31°C – (–0.05°C) = 0.36°C, where the value - 0.05°C is the average of years 1953-82. The value 0.36°C is plotted as a green triangle in Figure 5 at year 2012. Likewise, we find climatic differences for years 2011, 2010, …, 1909, all for lag 30. Plotting all these we get the series of green triangles shown in Figure 5. We repeat the same procedure for time lags that are multiples of 30 years, namely 60 years (red points), 90 years (blue points) and 120 years (purple points).

Finally, we calculate, in a way described in Ref. 21, the critical values of the test statistic, which is none other than the lagged climate difference. The critical values are different for each lag and are plotted as flat lines with the same colour as the corresponding points. Technically, the (pseudo)test was made as two-sided for significance level 1% but only the high critical values are plotted in the graph. Practically, as long as the points lie below the corresponding flat lines, nothing significant has happened. This is observed for the entire length of the lag-30 and lag-60 differences. A few of the last points of the lag-90 series exceed the critical value; this corresponds to the combination of high temperatures in the 2000s and low temperatures in the 1910s. But then all points of the lag-120 series lie again below the critical value, indicating no significant change.

Concluding remarks

Assuming that the data set we used is representative and does not contain substantial errors, the only result that we can present as fact is that in the last 134 years the climate has warmed by 0.6°C (nb., this is a difference of climatic—30-year average—values while other, often higher, values that appear in the literature refer to trends based on annual values). Whether this change is statistically significant or not depends on assumptions. If we assume a 90-year lag and 1% significance, it perhaps is; again I cannot be certain as the pseudo-test did not consider the uncertainty in H. Note, the 1% significance corresponds to ±2.58 standard deviations away from the mean; if we made it ±3 everything would become insignificant.

Irrespective of statistical significance, paleoclimate and instrumental data provide evidence that the natural climatic variability, the natural potential for change, is high and concerns all time scales. The mechanisms producing change are many and, in practice, it is more important to quantify their combined effects, rather than try to describe and model each one separately.

From a practical point of view, it could be dangerous to pretend that we are able to provide credible quantitative description of the mechanisms, their causes and effects, and their combined consequences: We know that the mechanisms and their interactions are nonlinear, as well as that the climate model hind casts are poor.[26],[27] Indeed, it has been demonstrated that, particularly for runoff that is mostly relevant for water availability and flood risk, deterministically projected future traces can be too flat in comparison to changes that can be expected (and stochastically generated) admitting stationarity and natural variability characterized by HK dynamics[28].

Biosketch
Demetris Koutsoyiannis received his diploma in Civil Engineering from the National Technical University of Athens (NTUA) in 1978 and his doctorate from NTUA in 1988. He is professor of Hydrology and Analysis of Hydrosystems at the Department of Water Resources and Environmental Engineering of NTUA (and former Head of the Department). He is also Co-Editor of Hydrological Sciences Journal and member of the editorial board of Hydrology and Earth System Sciences (and formerly of Journal of Hydrology and Water Resources Research). He teaches undergraduate and postgraduate courses in hydrometeorology, hydrology, hydraulics, hydraulic works, water resource systems, water resource management, and stochastic modelling. He is an experienced researcher in the areas of hydrological modelling, hydrological and climatic stochastics, analysis of hydrosystems, water resources engineering and management, hydro-informatics, and ancient hydraulic technologies. His record includes about 650 scientific and technological contributions, spanning from research articles to engineering studies, among which 96 publications in peer reviewed journals. He received the Henry Darcy Medal 2009 by the European Geosciences Union for his outstanding contributions to the study of hydro-meteorological variability and to water resources management.

References



[1] IPCC (2007), Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor and H.L. Miller (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 996 pp (Annex 1, Glossary: http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-annexes.pdf).

[2] Climate Dialogue Editorial Staff (2013), How persistent is the climate system and what is its implication for the significance of observed trends and for internal variability? (This blog post).

[3] Obama’s 2013 State of the Union Address, The New York Times, http://www.nytimes.com/2013/02/13/us/politics/obamas-2013-state-of-the-union-address.html?pagewanted=3&_r=0

[4] Lamb, H. H. (1972), Climate: Past, Present, and Future, Vol. 1, Fundamentals and Climate Now, Methuen and Co.

[5] Lovejoy, S., and D. Schertzer (2013), The climate is not what you expect, Bull. Amer. Meteor. Soc., doi: 10.1175/BAMS-D-12-00094.

[6] Koutsoyiannis, D. (2011), Hurst-Kolmogorov dynamics and uncertainty, Journal of the American Water Resources Association, 47 (3), 481–495.

[7] Koutsoyiannis, D. (2010), A random walk on water, Hydrology and Earth System Sciences, 14, 585–601.

[8] Hurst, H.E. (1951), Long term storage capacities of reservoirs, Trans. Am. Soc. Civil Engrs., 116, 776–808.

[9] Toussoun, O. (1925). Mémoire sur l’histoire du Nil, in Mémoires a l’Institut d’Egypte, vol. 18, pp. 366-404.

[10] Cohn, T. A., and H. F. Lins (2005), Nature's style: Naturally trendy, Geophys. Res. Lett., 32, L23402, doi: 10.1029/2005GL024476.

[11] Koutsoyiannis, D., and A. Langousis (2011), Precipitation, Treatise on Water Science, edited by P. Wilderer and S. Uhlenbrook, 2, 27–78, Academic Press, Oxford ( p. 57).

[12] Koutsoyiannis, D. (2013), Encolpion of stochastics: Fundamentals of stochastic processes, 30 pages, National Technical University of Athens, Athens, http://itia.ntua.gr/1317/, accessed 2013-04-17.

[13] Papoulis, A. (1991), Probability, Random Variables and Stochastic Processes, 3rd edn., McGraw-Hill, New York (p. 635).

[14] Koutsoyiannis, D. (2013), Hydrology and Change, Hydrological Sciences Journal (accepted with minor revisions; currently available in the form of an IUGG Plenary lecture, Melbourne 2011, http://itia.ntua.gr/1135/, accessed 2013-04-17).

[15] Kolmogorov, A. N. (1940). Wiener spirals and some other interesting curves in a Hilbert space, Dokl. Akad. Nauk SSSR, 26, 115-118. English translation in: Tikhomirov, V.M. (ed.) 1991. Selected Works of A. N. Kolmogorov: Mathematics and mechanics, Vol. 1, Springer, 324-326.

[16] Koutsoyiannis, D. (2011), Hurst-Kolmogorov dynamics as a result of extremal entropy production, Physica A: Statistical Mechanics and its Applications, 390 (8), 1424–1432.

[17] Markonis, Y., and D. Koutsoyiannis (2013), Climatic variability over time scales spanning nine orders of magnitude: Connecting Milankovitch cycles with Hurst–Kolmogorov dynamics, Surveys in Geophysics, 34 (2), 181–207.

[18] Tyralis, H., and D. Koutsoyiannis (2011), Simultaneous estimation of the parameters of the Hurst-Kolmogorov stochastic process, Stochastic Environmental Research & Risk Assessment, 25 (1), 21–33.

[19] As Ref. 1, p. 23.

[20] Zorita, E., T. F. Stocker, and H. von Storch (2008), How unusual is the recent series of warm years?, Geophys. Res. Lett., 35, L24706, doi: 10.1029/2008GL036228.

[21] Koutsoyiannis, D., and A. Montanari (2007), Statistical analysis of hydroclimatic time series: Uncertainty and insights, Water Resources Research, 43 (5), W05429, doi: 10.1029/2006WR005592.

[22] von Storch, H. (1999), Misuses of statistical analysis in climate research, in Analysis of Climate Variability, Applications of Statistical Techniques, Proceedings of an Autumn School organized by the Commission of the European Community, Edited by H. von Storch and A. Navarra, 2nd updated and extended edition, http://www.hvonstorch.de/klima/books/SNBOOK/springer.pdf, accessed 2013-04.

[23] Koutsoyiannis, D. (2003), Climate change, the Hurst phenomenon, and hydrological statistics, Hydrological Sciences Journal, 48 (1), 3–24.

[24] Hamed, K. H. (2008), Trend detection in hydrologic data: The Mann-Kendall trend test under the scaling hypothesis, Journal of Hydrology, 349(3-4), 350-363.

[25] Rybski, D., A. Bunde, S. Havlin and H. von Storch (2006), Long-term persistence in climate and the detection problem, Geophys. Res. Lett., 33, L06718, doi: 10.1029/2005GL025591.

[26] Anagnostopoulos, G. G., D. Koutsoyiannis, A. Christofides, A. Efstratiadis and N. Mamassis (2010), A comparison of local and aggregated climate model outputs with observed data, Hydrological Sciences Journal, 55 (7), 1094–1110.

[27] Koutsoyiannis, D., A. Christofides, A. Efstratiadis, G. G. Anagnostopoulos, and N. Mamassis (2011), Scientific dialogue on climate: is it giving black eyes or opening closed eyes? Reply to “A black eye for the Hydrological Sciences Journal” by D. Huard, Hydrological Sciences Journal, 56 (7), 1334–1339.

[28] Koutsoyiannis, D., A. Efstratiadis, and K. Georgakakos (2007), Uncertainty assessment of future hydroclimatic predictions: A comparison of probabilistic and scenario-based approaches, Journal of Hydrometeorology, 8 (3), 261–281.

Guest blog Armin Bunde

How to estimate the significance of global warming when taking explicitly into account the long-term persistence in temperature?

Armin Bunde, Institut für Theoretische Physik, Universität Giessen, Germany

1. Long-term Persistence in climate and its detection

Long-term persistence (LTP), also called long-term correlations or long-term memory, plays an important role to characterize records in physiology (e.g. heartbeats), computer science (e.g. internet traffic) and in financial markets (volatility). The first hint that LTP is important in climate has been given in the classic papers by Hurst more than 50 years ago when studying the historic levels of the Nile River.

We can distinguish between uncorrelated records (''white noise''), short-term persistent records (STP) and long-term persistent records. In white noise all data points x1, x2, ..., xN are independent of each other. In STP records, each data point xi depends on a short subset of previous points xi-1, xi-2,.. xi-m, i.e., the memory has a finite range m. In LTP records, in contrast, xi depends on all previous points. The simplest model for STP is the ''AR1 process'' where x i is proportional to the foregoing data point xi-1, plus a white noise component ηi-1,

Despite the evidence that temperature anomalies cannot be characterized by the AR1 process, most climate scientists have used the AR1 model when trying to describe temperature fluctuations and estimating the significance of a trend. This usually leads to a considerable overestimation of the external trend and its significance.

There are several methods to quantify the memory in a given sequence. (For a recent review see [1] and references therein). The first one is the autocorrelation function C(s) where s = 1,2,3,… is the lag time between 2 data points. For white noise, there is no memory and C(s) = 0. For the AR1 process, C(s) decays exponentially, C(s) ~ exp{- s/S} where S =1/|ln a| is the ''persistence length''. For infinitely long stationary LTP data C(s) decays algebraically,

where ϒ is called correlation exponent.

The first figure shows parts of an uncorrelated (left) and a synthetic long-term persistent (right) record, with ϒ = 0.4. The full line is the moving average over 30 data points. For the uncorrelated data, the moving average is close to zero, while for the LTP data, the moving average can have large deviations from the mean, forming some kind of mountain-valley structure that looks as if it contained some external deterministic trend. The figure shows that it is not a straightforward task to separate the natural fluctuations from an external trend, and this makes the detection of external trends in LTP records a difficult task. I will return to this later.

Figure 1.

One can show analytically [2], that in LTP records with a finite length N, the algebraic dependence of C(s) on s can be seen only for very small time lags s, satisfying the inequality (s/N)ϒ << 1. Already for ϒ = 1/2 and records of length 600 (which corresponds to 50 years of monthly data), this condition can only be met for very small time lag times s, roughly s < 6. For larger time lags, C(s) decays faster than algebraically. This is an artifact of the method called ''Finite Size''-Effect. If one is not aware of this effect, one may be led to the wrong conclusion that there exists no long-term memory in sequences of a finite length.

A similar mistake may happen, when one uses the second traditional method for detecting LTP, the power spectrum (spectral density) S(f). The discrete frequency f is equivalent to an inverse lag time, f=1/s, and a multiple of 1/N. For white noise, S(f) is constant. For STP data, S(f) is constant for f well below m/N (since the data are uncorrelated at time lags s above m), and then decreases monotonously.

For LTP records, S(f) decreases by a power law,

so one may detect LTP also by considering the power spectrum. However, due to the discreteness of f, the algebraic decay cannot be clearly observed at frequencies below 50/N, which again may lead to the wrong conclusion that there is no long-term memory.

In addition to the remarkable finite size resp. discreteness effects, both methods lead an over-estimation of the LTP in the presence of external deterministic trends.

In recent years, several methods (see, e.g.,[1,3]) have been developed where long-term correlations in the presence of deterministic polynomial trends can be detected. These methods include the detrended fluctuation analysis (DFA2) and Haar-wavelet analysis (WT2), where linear trends are eliminated systematically. DFA2 is quite accurate in the time window 8 ≤ s < N/4 while WT2 is accurate for 1 ≤ s < N/50. In both methods, one determines a fluctuation function F(s) which measures the fluctuations of the record in time windows of length s around a trend line. For LTP records with correlation exponent ϒ, F(s) increases as

where α is usually called Hurst exponent. By combining DFA2 and WT2 one can obtain a consistent picture on time scales between s=1 and s=N/4. For a meaningful analysis, the records should consist of more than N = 500 data points. I like to emphasize again that in the case an external deterministic trend cannot be excluded, the evaluation of the LTP and the determination of the Hurst exponent must be done with trend-eliminating methods, e.g., DFA2 and WT2, as described above.

The second figure summarizes the results of our earlier analysis (for references, see [1]) for a large number of atmospheric and sea surface temperatures as well as precipitation and river run-offs. Each histogram shows how many stations have Hurst exponents around 0.5, 0.55, 0.6, 0.65 and so on.

Figure 2.

For the daily precipitation records and the continental atmospheric temperatures the distribution of Hurst exponents is quite narrow. For daily precipitation, the exponent is close to 0.5, indicating the absence of persistence (see also [3]), while for the daily continental temperature records, the exponent is close to 0.65, indicating a “universal” persistence law. Both laws can be used very efficiently as test bed for climate models and paleo reconstructions (for references, see [1] and [3]).

There are also more intuitive measures of LTP, and one of them is the distribution of the persistence lengths l in a record (see, e. g. [3]). In temperature data, l describes the lengths of warm resp cold periods where the temperature anomalies (deviation of the daily or monthly temperature from their seasonal mean) are above resp. below zero. It is easy to show that the distribution P(l) of the persistence length decays exponentially for uncorrelated data, i.e., ln P(l) ~ - l. For LTP data, P(l) decays via a stretched exponential, ln P(l) ~ -lϒ where ϒ is the correlation exponent (see [1]). Accordingly, in LTP records large persistence lengths are more frequent, which is intuitively clear.

2. Detection of external trends in LTP data

For detection and estimation of external trends (“detection problem”) one needs a statistical model. For monthly (and annual) temperature records the best statistical model is long-term persistence, as we have seen in the foregoing section. The main features of a long-term persistent record of length N are determined by the Hurst exponent α. Synthetic LTP records characterized by these two parameters can be easily generated by a Fourier-transformation (see, e.g., [1]) with the help of random number generators.

When using LTP as statistical model we assume that there are no additional short term correlations, generated by ''Großwetterlagen'' (blocking situations). Since the persistence length of these short term correlations is below 14 days, they are not present in monthly data sets.

For the detection problem, one then needs to know the probability W(x) that in a long-term correlated record of length L and Hurst exponent α, the relative trend exceeds x. For temperature data, the relative trend is the ratio between the temperature change (determined by a simple regression analysis) and the standard deviation σ around the trend line. For Gaussian LTP data, an analytical expression for W(x), for given α and N, has been derived in [4], which is easy to implement and can serve also as a very good approximation for Non-Gaussian data. In order to decide if a measured relative trend xmmay be natural or not, one has to determine the exceedance probability at xm. If W(xm) is below 0.025, the trend usually is called significant (within the 95 percent confidence interval), if it is below 0.005, the trend is called highly significant (within the 99 percent confidence interval).

From the condition W(y) = 0.025 one may derive error bars y (within the 95 percent confidence interval) for the expected external trend,

If xm is slightly below y, then the minimum value of the external trend is negative and thus the trend is not significant. But the maximum value of the external trend can be large, and thus an external trend cannot be excluded, even though the trend is not significant. Accordingly, if a trend is not significant since W(xm) is above 0.025, this does not mean that one can exclude the possibility of an external deterministic trend. It only means that one is not forced to assume an external trend in order to describe the variability of the record properly. For example, if we observe a small insignificant positive trend, then this trend may either arise from the superposition of a strong positive natural fluctuation (as in Fig.1b) and a small negative external trend or from a strong negative fluctuation (as in Fig. 1b, but downwards) and a large positive external trend.

These conclusions are independent of the used model and hold also for the STP model. In previous significance analyses, climate scientists usually used the STP model, where the model parameter a has been determined from measuring C(1) of the data, see Sect. 1. The significance of a trend (see below) is clearly underestimated by this model.

3. Detection of climate change within the LTP model

Using our terminology of “significant” and “highly significant” we have obtained a mixed picture of the significance of temperature records, partly reviewed in [1].

(i) The global sea surface temperature increased, in the past 100y, by about 0.6 degree, which is not significant. The reason for this is the large persistence of the oceans, reflected by a large Hurst exponent.

(ii) The global land air temperature, in the past 100y, increased by about 0.8 degrees. We find this increase even highly significant. The reason for this is the comparatively low persistence of the land air temperature, which makes large natural increases unlikely.

(iii) Local temperatures: In local temperature records it is more difficult to detect external trends due to their large variability. We have studied a large number of local stations around the globe. For stations at high elevation like Sonnblick in Austria or in Siberia, we found highly significant trends. For about half of the other stations, we could not find a significant trend. However, when averaging the records in a certain area, this picture changed. Due to the averaging, the fluctuations around the trend line decrease and the temperature increases become more significant.

Our estimations are basically in line with earlier, less rigorous trend estimations in LTP data by Rybski et al [5] and in line with the conclusions of Zorita et al when estimating the probability that 11 of the warmest years in a 162 year long record all lie in the last 12 years.

My conclusion is that the AR1 process falsely used by climate scientists to describe temperature variability leads to a strong overestimation of the significance of external trends. When using the proper LTP model the significance is considerably lower. But also the LTP model does not reject the hypothesis of anthropogenic climate change.

Biosketch

Armin Bunde is professor of theoretical physics in Giessen. After receiving his PhD in theoretical solid state physics at Giessen University, he spent several years as a Post Doc in Antwerp, Saarbrücken, and Konstanz. He received a prestigious Heisenberg Fellowship in 1984 and spent three years at Boston University and Bar Ilan University in Israel, where he worked with H.Eugene Stanley (Boston) and Shlomo Havlin (Israel). In 1985 he received the Carl-Wagner Award. Between 1987 and 1993 he was professor of theoretical physics at Hamburg University, since 1993 he is back in Giessen.

In the last 20 years, his main research areas are disordered materials, percolation theory and applications in materials science, as well as fractals and time series analysis in different disciplines, among them geo science, where he is mainly interested in long-term persistence, extreme events and climate networks. In geo science, he has cooperated intensively with Hans-Joachim Schellnhuber, Donald Turcotte, Hans von Storch, and Shlomo Havlin. He has published more than 300 papers and his Hirsch Index (google) is well above 50.

References

[1] A. Bunde and S. Lennartz, Acta Geophysica 60, 562 (2012) and references therein

[2] S. Lennartz and A. Bunde, Phys. Rev. E (2009)

[3] A. Bunde, U. Büntgen, J. Ludescher, J. Luterbacher, and H. von Storch, Nature Climate Change, 3, 174 (2013), see also: online supplementary information

[4] S. Lennartz and A. Bunde, Phys. Rev. E 84, 021129 (2011)

[5] D. Rybski, A. Bunde, S. Havlin, and H. von Storch, Geophys. Res. Lett. 35, L06718 (2006)

[6] E. Zorita, T.F. Stocker, and H. von Storch, Geophys. Res. Lett. 35, L24706 (2008)

Summary of the Climate Dialogue on long-term persistence

Summary of the Climate Dialogue on Long-Term Persistence

Authors: Marcel Crok, Bart Strengers (PBL), Bart Verheggen (PBL), Rob van Dorland (KNMI)

This summary is based on the contributions of Rasmus Benestad, Armin Bunde and Demetris Koutsoyiannis who participated in this Climate Dialogue that took place in May 2013.


Long term persistence and trend significance
“Is global average temperature increase statistically significant?” To answer this question one needs to make assumptions on the statistical nature of the temperature time series and to choose what statistical model is most appropriate.

If the temperature of this year is not related to that of last year or next year we can use statistics to determine whether the increase in global temperature is significant or not. In such an “uncorrelated climate”, i.e. if the temperature of this year is fully independent of other years, the average value becomes zero (or a fixed value) quickly and deviations from the mean last only shortly. However, if there is (strong) temporal dependence the moving average can have large deviations from the mean. This is called long-term dependence or long-term persistence (LTP).

The three participants agree that LTP exists in the climate (Table 1), but they disagree about the exact definition and the physical processes that lie behind it. Benestad and Bunde describe LTP in terms of “long memory”. Koutsoyiannis holds the opinion that LTP is mainly the result of the irregular and unpredictable changes that take place in the climate (Table 2). Both Bunde and Koutsoyiannis are in favour of a formal (mathematical) definition of LTP, which states that on longer time scales climate variability decreases—but not as much as implied by classical statistics.

Benestad said that ice ages and the El Niño Southern Oscillation (ENSO) are examples of LTP processes. Bunde and Koutsoyiannis disagreed (Table 3).

Table 1

Benestad

Bunde

Koutsoyiannis

Does LTP exist in the climate?

Yes

Yes

Yes

Table 2

What is long-term persistence (LTP)?

Benestad

LTP describes how slow physical processes change over time, where the gradual nature is due to some kind of ‘memory’.

Bunde

LTP is a process with long memory; the value of a parameter (e.g. temperature) today depends on all previous points.

Koutsoyiannis

It is unfortunate that LTP has been interpreted as memory; it is the change, mostly irregular and unpredictable in deterministic terms, that produces the LTP, while the autocorrelation results from change.

Table 3

Benestad

Bunde

Koutsoyiannis

Quasi-oscillatory phenomena like ENSO can be described as LTP.

Yes

No

No

Is LTP relevant for the detection of climate change?

There was confusion about the exact meaning of the IPCC definition about detection. The definition reads: “Detection of change is defined as the process of demonstrating that climate or a system affected by climate has changed in some defined statistical sense without providing a reason for that change. An identified change is detected in observations if its likelihood of occurrence by chance due to internal variability alone is determined to be small.”

Bunde and Koutsoyiannis both think detection is mainly a matter of statistics while Benestad thinks it also involves a physical interpretation of distinguishing unforced internal variability from forced changes.

Table 4

Benestad

Bunde

Koutsoyiannis

Is detection purely a matter of statistics?

No, laws of physics sets fundamental constraints

Yes

Not purely but primarily yes

LTP versus AR(1)
Bunde and Koutsoyiannis argue that LTP is the proper model to describe temperature variability, that climate scientists in general use a Short Term Persistence (STP) model like AR(1) which leads to a strong overestimation of the significance of trends (Table 5). Koutsoyiannis showed that the clustering of warm years, for example, is orders of magnitude more likely to happen if you use an LTP model. Benestad agrees that the AR(1) model may not necessarily be the best model. He argues that in general statistical models and LTP in particular, used for the detection of trends involve circular reasoning when applied to what is called the instrumental period, because in this period the data embed both “signal” and “noise”. LTP or STP or whatever statistical model are meant to describe “the noise” only in his opinion (Table 6). Koutsoyiannis in response gave a few examples why in his opinion the danger of circular reasoning is not justified in this case (see Extended Summary).

Table 5

Benestad

Bunde

Koutsoyiannis

Is LTP relevant/important for the statistical significance of a trend?

Yes (though physics still needed)

Yes, very much

Yes, very much

Table 6

What is the relevance of LTP for the detection of climate change?

Benestad

Statistical LTP-noise models used for the detection of trends involve circular reasoning if adapted to measured data. State of the art detection and attribution is needed.

Bunde

For detection and estimation of external trends (“detection problem”) one needs a statistical model and LTP is the best model to do this.

Koutsoyiannis

LTP is the only relevant statistical model for the detection of changes in climate.

Table 7

Benestad

Bunde

Koutsoyiannis

Is the AR(1) model a valid model to describe the variability in time series of global average temperature?

No, if physics based information is neglected

No

No

Does the AR(1) model leads to overestimation of the significance of trends?

Yes, if you don’t also take into account the physics-based information

Yes

Yes

LTP and chaos
There was disagreement about the relation between LTP and chaos (Table 8). According to Benestad chaos theory implicates the memory of the initial conditions is lost after a finite time interval. Benestad interprets “the system loses memory” as “LTP is not a useful concept”. Koutsoyiannis considers memory as a bad interpretation of LTP: it is the change which produces the LTP and thus LTP is fully consistent with the chaotic behaviour of climate.

Table 8

Benestad

Bunde

Koutsoyiannis

Is the climate chaotic?

Yes

Yes

Yes

Does chaos mean memory is lost and does this apply for climatic timescales as well?

Yes

Chaos is not a useful concept for describing the variability of climate records on longer time scales

No; LTP is not memory

Does chaos exclude the existence of LTP?

Yes, at both weather and climatic time scales

No

No; on the contrary, chaos can produce LTP

Does chaos contribute to the existence of LTP?

No, but chaos may give an impression of LTP

Yes

Yes, LTP does involve chaos

Signal and noise
There was disagreement about concepts like signal and noise. According to Benestad the term “signal” refers to manmade climate change. “Noise” usually means everything else, and LTP is ‘noise in slow motion’ (Table 9). Koutsoyiannis argued that the “signal” vs. “noise” dichotomy is subjective and that everything we see in the climate is signal. To isolate one factor and call its effect “signal” may be misleading in view of the nonlinear chaotic behavior of the system. Bunde does assume there is an external deterministic trend from the greenhouse gases but he calls the remaining part of the total climate signal natural “fluctuations” and not noise (Table 9). All three seem to agree that one cannot use LTP to make a distinction between forced and unforced changes in the climate (Table 10).

Table 9

Signal versus noise

Benestad

The signal is manmade climate change; the rest is noise and LTP is noise in slow motion.

Bunde

My working hypothesis: there is a deterministic external trend; the rest are natural fluctuations which are best described by LTP.

Koutsoyiannis

Excepting observation errors, everything we see in climate is signal.

Table 10

Benestad

Bunde

Koutsoyiannis

Is the signal versus noise dichotomy meaningful?

Yes

Yes

No*

Can LTP distinguish between forced and unforced components of the observed change?

No

No

*

Can LTP distinguish between natural fluctuations (including natural forcings) and trends?

No

Yes

*

* Koutsoyiannis thinks that even the formulation of these questions, which imply that that the description of a complex process can be made by partitioning it into additive components and trying to know the signatures of each one component, indicates a linear view for a system that is intrinsically nonlinear.

Forced versus unforced
According to Bunde Natural Forcing plays an important role for the LTP and is omnipresent in climate. Koutsoyiannis agreed that (changing) forcing can introduce LTP and that forcing is omnipresent, but LTP can also emerge from the internal dynamics alone.

Table 11

Benestad

Bunde

Koutsoyiannis

Does forcing introduce LTP?

Yes

Yes

Yes

Is forcing omnipresent in the real world climate?

Yes

Yes

Yes

What according to you is the main mechanism behind LTP?

Forcings

Natural Forcing plays an important role for the LTP and is omnipresent in climate

I believe it is the internal dynamics that determines whether or not LTP would emerge

Is the warming significant?
The three participants gave different answers on the key question of this Climate Dialogue, namely of the warming in the past 150 years is significant or not. They used different methods to answer the question. Benestad is most confident that both the changes in land and sea temperatures are significant. Bunde concludes that due to a strong Long Term Persistence the increase in sea temperatures are not significant but the land and global temperatures are. Koutsoyiannis concludes that for most time lags the warming is not significant. In some cases it maybe is.

Table 12

BenestadI

BundeII

KoutsoyiannisIII

Is the rise in global average temperature during the past 150 years statistically significant?

Yes

YesIV

NoV

Is the rise in global average sea surface temperature during the past 150 years statistically significant?

Yes

No

No

Is the rise in global average land surface temperature during the past 150 years statistically significant?

Yes

Yes

No

I Benestad’s conclusions are based on the difference between GCM simulations with and without anthropogenic forcing (Box 10.1 or Figs 10.1 & 10.7 in AR5)
II Based on the Detrended Fluctuation Analysis (DFA) and/or the wavelet technique (WT).
III Based on the climacogram and different time lags (30, 60, 90 and 120 years).
IV This change is 99% significant according to Bunde.
V For a 90 year time lag and a 1% significance level it maybe is significant (see guest blog).

Is there a large contribution of greenhouse gases to the warming?
Bunde is more convinced of a substantial role for greenhouse gases on the climate than Koutsoyiannis although he admits he cannot rule out that the warming on land is (partly) due to urban heating. Bunde said he may not fully agree with Koutsoyiannis: “We cannot show in our analysis of instrumental temperature data that GHG are responsible for the anomalously strong temperature increase that we see and that we find is significant, but it is my working hypothesis.” Koutsoyiannis believes the influence of greenhouse gases is relatively weak, “so weak that we cannot conclude with certainty about quantification of causative relationships between GHG and temperature changes”. Benestad on the other hand said the increased concentrations of GHGs is the only plausible explanation for the observed global warming, global mean sea level rise, melting of ice, and the accumulation of ocean heat.

Table 13

Benestad

Bunde

Koutsoyiannis

Is the warming mainly of anthropogenic origin?

The combination of statistical information and physics knowledge lead to only one plausible explanation: GHGs

Yes, it is my working hypothesis

No, I think the effect of CO2 is small

Extended summary of the Climate Dialogue on long-term persistence

Extended Summary of the Climate Dialogue on Long Term Persistence

Author: Marcel Crok
With contributions from Bart Verheggen, Rob van Dorland (KNMI) en Bart Strengers (PBL)


Introduction

This summary is based on the contributions of the three invited scientists who participated in the dialogue entitled “Long-term persistence and trend significance”. We want to thank Rasmus Benestad, Armin Bunde and Demetris Koutsoyiannis for their participation.

The summary is not meant to be a consensus statement. It’s just a summary of the discussion and should give a good overview of how these three scientists view the topic at this moment. This summary was written by Marcel Crok and then reviewed and adjusted by the other editors of Climate Dialogue and the advisory board members. In some cases the editors disagreed about the text. In the summary we make clear when this is the case.

The summary was then reviewed by the three invited participants. They do not necessarily endorse the full text or our selection of the dialogue. We did ask them to check the claims in all the tables though in order to make these consistent with their views.

Long term persistence and trend significance
In science one often asks whether a change in some parameter, variable or process is statistically significant. So we could ask: is the increase in global average temperature statistically significant? Whether an observed trend is significant or not is related to the chance of occurrence and thus on the underlying variability, noise and errors, as well as the temporal stochastic structure thereof.


Figure 2.14 from the IPCC AR5 WGI report. Global annual average land-surface air temperature (LSAT) anomalies relative to a 1961–1990 climatology from the latest versions of four different data sets (Berkeley, CRUTEM, GHCN and GISS).

The temperature time series in the figure above shows variation on annual and multi-decadal scales. However, how do we know if this trend is part of the natural variability of the climate system or whether it is due to some forced changes and whether is significant or not? Is the increase very unlikely or quite normal in terms of natural variability?

This is a statistical problem and thus the way to look at it is making a statistical analysis of the time series to determine the amplitude of natural variability. For instance to answer the question for the year-to-year variability we would need to know for every single time step (year) what the chance is to go up or down and how strong these excursions can be. The difficulty is that we don’t have a data set for the “undisturbed” climate, i.e. the climate without anthropogenic influences, which could be used as a reference period and to assess the significance of the recent warming trend. It is noted, though, that there are a lot of proxy data sets, which can be used to infer the stochastic structure of natural climatic variability. These data sets do not describe the climate precisely, but certainly can give information on its stochastic structure and also are free of anthropogenic influences.

With such issues we enter the arena of the Climate Dialogue about long-term persistence (LTP).

Definition of long-term persistence
Both Bunde and Koutsoyiannis showed figures in their guest blogs to explain the difference between independent or uncorrelated data and long-term correlated data. Below is the graph that Bunde showed:


Figure 1 of Bunde’s guest blog showing the difference between uncorrelated data (left) and data with long-term persistence (right). The coefficient γ (gamma) is a measure of persistence.

As he explained: “For the uncorrelated data, the moving average is close to zero, while for the LTP data, the moving average can have large deviations from the mean, forming some kind of mountain-valley structure that looks as if it contained some external deterministic trend. The figure shows that it is not a straightforward task to separate the natural fluctuations from an external trend, and this makes the detection of external trends in LTP records a difficult task.”

Koutsoyiannis said it as follows: “No one would believe that the weather this hour does not depend on that an hour ago. It is natural to assume that there is time dependence in weather. (…) Now, if we average the process to another scale, daily, monthly, annual, decadal, centennial, etc. we get other stochastic processes, not qualitatively different from the hourly one. Of course, as the scale of averaging increases the variability decreases—but not as much as implied by classical statistics.”

Benestad gave the following description of LTP: “Long-term persistence (LTP) describes how slow physical processes change over time, where the gradual nature is due to some kind of ‘memory’. This memory may involve some kind of inertia, or the time it takes for physical processes to run their course. Changes over large space take longer time than local changes.”

So they all accept that LTP ‘exists’ in the climate or is part of climate. There were disagreements though, even about the concept of LTP. Bunde and Koutsoyiannis are both in favour of a formal (mathematical) definition of LTP, which describes what Koutsoyiannis said above, that on longer time scales variability decreases—but not as much as implied by more “classical statistics” like AR(1)[1].

Benestad found this proposition “somewhat artificial” when dealing with temperature time series. He said a great deal of variance is usually removed before the data is analysed, like seasonal variations and the diurnal cycle. “Most of the variance is tied up to these well-known cycles, forced by regional changes in incoming sunlight. Furthermore, ENSO has a time scale that is ~3-8 years, and is associated with most of the variance after the seasonal and diurnal scales are neglected.” Elsewhere Benestad said: “There are some known examples of LTP processes, such as the ice ages, changes in the ocean circulation, and the El Niño Southern Oscillation.” Bunde disagreed that ENSO is an LTP process: “Rasmus [Benestad] will recognize that ENSO is not an example of LTP, in the same way as other quasi-oscillatory phenomena cannot be described as LTP.”

So there is confusion/disagreement about what LTP really “is”. The reason could be that for Bunde and Koutsoyiannis LTP is a statistical property of climatic time series and according to Bunde, as such, it is not an “abstract issue”.

A key issue seemed to be whether it is possible to talk about LTP in terms of physical processes. Koutsoyiannis thinks the system is just too complex to talk about simple physical causes for observed changes and he does not accept the dichotomy physics vs. statistics as in complex physical systems a statistical description is the most pertinent and the most powerful.

According to Koutsoyiannis it is unfortunate that LTP has been commonly described in the literature in association with autocorrelation and as a result of memory mechanisms. For him it is “the change, mostly irregular and unpredictable in deterministic terms, that produces the LTP”. For Benestad LTP is a manifestation of memory in the climate system.

Summary
To answer the question “is the increase in global average temperature statistically significant?” one needs to make assumptions about the statistical nature of the time series and one needs to choose what statistical model is the most appropriate.

If the temperature of this year is not related to that of last year or next year we can use classical statistics to determine whether the increase in global temperature is significant or not. In such an “uncorrelated climate” the average value becomes zero (or a fixed value) quickly and deviations from the mean last only shortly. If there is (strong) temporal dependence though the moving average can have large deviations from the mean. Bunde and Koutsoyiannis claim the climate displays such long-term dependence or long-term persistence (LTP).

The three participants agree that LTP exists in the climate (Table 1). They disagree about the exact definition though and about the physical processes that lie behind it. Benestad and Bunde describe LTP in terms of “long memory”. Koutsoyiannis says that in his opinion LTP is mainly the result of the irregular and unpredictable changes that take place in the climate (Table 2). Bunde and Koutsoyiannis are both in favour of a formal (mathematical) definition of LTP, which states that on longer time scales climate variability decreases—but not as much as implied by “classical statistics” such as AR(1).

Benestad said that ice ages and the El Niño Southern Oscillation are examples of LTP processes. Bunde and Koutsoyiannis disagreed and said that quasi-oscillatory phenomena cannot be described as LTP (Table 3).

Table 1

Benestad

Bunde

Koutsoyiannis

Does LTP exist in the climate?

Yes

Yes

Yes

Table 2

What is long-term persistence (LTP)?

Benestad

LTP describes how slow physical processes change over time, where the gradual nature is due to some kind of ‘memory’.

Bunde

LTP is a process with long memory; the value of a parameter (e.g. temperature) today depends on all previous points.

Koutsoyiannis

It is unfortunate that LTP has been interpreted as memory; it is the change, mostly irregular and unpredictable in deterministic terms, that produces the LTP, while the autocorrelation results from change.

Table 3

Benestad

Bunde

Koutsoyiannis

Quasi-oscillatory phenomena like ENSO can be described as LTP.

Yes

No

No


Is LTP relevant for the detection of climate change?

The full IPCC definitions of detection and attribution in AR5 are (our emphasis)[2]:

“Detection of change is defined as the process of demonstrating that climate or a system affected by climate has changed in some defined statistical sense without providing a reason for that change. An identified change is detected in observations if its likelihood of occurrence by chance due to internal variability alone is determined to be small.”

Attribution is defined as “the process of evaluating the relative contributions of multiple causal factors to a change or event with an assignment of statistical confidence”. As this wording implies, attribution is more complex than detection, combining statistical analysis with physical understanding.

The definition of detection has been differently interpreted by the members of the Editorial Staff:

Interpretation 1 (Rob van Dorland, Bart Verheggen): the second part clarifies the first part that you need (some defined) statistical model to distinguish between forced and unforced (internal variability) change. In the first part it is stated that this is done without knowing the causeof the forced change, i.e. whether it is anthropogenic or natural (sun, volcanoes etc).

Interpretation 2 (Marcel Crok): The first part of the definition suggests that you only need statistics to do detection. The second part suggests you need more than statistics (physical models), unless a statistical method would be able to distinguish between forced changes and internal variability. So the definition is self-contradictory.

Bunde and Koutsoyiannis both think detection is mainly a matter of statistics while Benestad thinks it also involves a physical interpretation of distinguishing unforced internal variability from forced changes.

Bunde wrote: “For detection and estimation of external trends (“detection problem”) one needs a statistical model.” Koutsoyiannis preferred the word “primarily” instead of “purely”: “I would say it is primarily a statistical problem, but I would not use the advert “purely”. Besides, as we wrote in Koutsoyiannis/Montanari (2007)[3], even the very presence of LTP should not be discussed using merely statistical arguments.”

Benestad wrote: “Hence, the diagnosis (“detection”) of a climate change is not purely a matter of statistics. The laws of physics set fundamental constraints which let us narrow down to a small number of ‘suspects’. For complete probability assessment, we need to take into account both the statistics and the physics-based information, such as the fact that GHGs absorb infrared light and thus affect the vertical energy flow through the atmosphere.”

Bart Verheggen wrote the following analysis of this part of the discussion: “This discussion showed that the participants used a slightly different operational definition of detection. Benestad followed the first interpretation of the IPCC definition, i.e. testing the significance of observed changes relative to what is expected from only unforced internal variability. Bunde and Koutsoyannis take detection to mean testing the significance of observed change w.r.t. some reference period without anthropogenic forcings (but with natural forcings). The latter definition in effect sets a higher bar for detection than the former (as the observed trend has to exceed not just unforced internal variability, but also the effect of natural forcings). These differences are probably rooted in different perceptions of what internal variability is (and whether or not it is different in principle from natural forcings).”

Summary
There was confusion about the exact meaning of the IPCC definition about detection. The definition reads: “Detection of change is defined as the process of demonstrating that climate or a system affected by climate has changed in some defined statistical sense without providing a reason for that change. An identified change is detected in observations if its likelihood of occurrence by chance due to internal variability alone is determined to be small.”

Bunde and Koutsoyiannis both think detection is mainly a matter of statistics and that it is very relevant for the detection of climate change. Benestad on the other hand thinks detection is not mainly a statistical issue and that it also involves a physical interpretation of distinguishing unforced internal variability from forced changes.

Table 4

Benestad

Bunde

Koutsoyiannis

Is detection purely a matter of statistics?

No, laws of physics sets fundamental constraints

Yes

Not purely but primarily yes


LTP versus AR(1)

Bunde’s main conclusion in his guest blog was: “My conclusion is that the AR1 process falsely used by climate scientists to describe temperature variability leads to a strong overestimation of the significance of external trends. When using the proper LTP model the significance is considerably lower.” The AR(1) process refers to the simplest model for short-term persistence (STP). So Bunde is saying several things here: 1) LTP is the proper model to describe temperature variability; 2) climate scientists in general use a STP model like AR(1) and 3) this leads to a strong overestimation of the significance of trends.

In a comment Bunde added that “This crucial mistake appeared also in the IPCC report [AR4] since the authors were (…) not aware of the LTP of the climate. They assumed STP [in table 3.2] and thus got the trend estimations wrong by overestimating the significance.”

Koutsoyiannis agrees with Bunde’s conclusions. In figure 1 of his guest blog he showed that the clustering of warm years, for example, is orders of magnitude more likely to happen if you use an LTP model. “We may see, for example, that what, according to the classical statistical perception, would require the entire age of the Earth to occur once (i.e. clustering of 8-9 events) is a regular event for an HK [Hurst-Kolmogorov] climate[4], with probability on the order of 1-10%.” He added that “this dramatic difference can help us understand why the choice of a proper stochastic model is relevant for the detection of changes in climate.” In a comment he also said that “a Markov [AR(1)] process […] finally produces a static climate […]. The truth is, however, that climate on Earth has never been static.”

Benestad agrees that “the AR1 model may not necessarily be the best model”, but adds that “it is difficult to know exactly what the noise looks like in the presence of a forced signal.” Elsewhere he wrote: “The important assumptions are therefore that the statistical trend models, against which the data are benchmarked, really provide a reliable description of the noise.”

In the discussion of this summary Benestad disagreed with Bunde’s claim that the AR(1)-process is falsely used by climate scientists and the IPCC. According to Benestad, Table 3.2 in AR4 as mentioned by Bunde is not seriously arguing that internal variability is AR(1), but merely uses this method as a crude estimate of the trend significance for that particular plot. Benestad: “The relevant question is whether the trend is anthropogenic or due to LTP (or signal versus noise) and to answer this question, you must look at chapter 9 in AR4 on detection and attribution and in particular figure 9.5 on the comparison between global mean surface temperature anomalies from observations and model simulations, and not at Table 3.2. In chapter 9 there are zero hits on ‘AR(1)’.”

He warns for the danger of circular reasoning when using statistical models. “It is the way models are used that really matters, rather than the specific model itself. All models are based upon a set of assumptions, and if these are violated, then the models tend to give misleading answers. Statistical LTP-noise models used for the detection of trends involve circular reasoning if adapted to measured data. Because this data embed both signal and noise.”

This is a key argument of Benestad. He claims statistical models are useless when applied to what is called the instrumental period, because in this period the data embed both “signal” and “noise” and LTP or STP or whatever statistical model are meant to describe “the noise” only in his opinion. Benestad therefore favours other methods: “State-of-the-art detection and attribution work do not necessarily rely on the AR1 concept, but use results from climate models and error-covariance matrices based on the model results to evaluate trends, rather than simple AR(1) methods.”

Koutsoyiannis in response gave a few examples why in his opinion the danger of circular reasoning is not justified in this case. In his first example he divided the global average time series in two parts: “The HadCrut4 data set is 163 year long. So, let us exclude the last 63 years and try to estimate H [the Hurst coefficient][5] based on the 100-year long period 1850-1949. The Hurst coefficient estimate becomes 0.93 instead of 0.94 for the entire period.” So he disagrees that the global average temperature time series cannot be used because the record is ‘contaminated’ by anthropogenic forcing. He also referred to analyses of proxies made by Koutsoyiannis and Montanari (2007), who estimated high values of the Hurst coefficient (H between 0.86-0.93) for the period 1400-1855 and by Markonis and Koutsoyiannis (2013)[6], who showed that a combination of proxies supports the presence of LTP with H > 0.92 for time scales up to 50 million years.

However, Benestad is in favour of separating forced and unforced climate change (as the definition of detection implies), and part of the 1850-1949 temperature changes are due to (natural) forcing. This implies that it is difficult to draw conclusions like Koutsoyiannis did. There was no further discussion on this issue.

Summary
Bunde and Koutsoyiannis argue that LTP is the proper model to describe temperature variability, that climate scientists (and the IPCC) in general use an STP model like AR(1) and that this leads to a strong overestimation of the significance of trends (Table 5 and 7). Koutsoyiannis showed that the clustering of warm years, for example, is orders of magnitude more likely to happen if you use an LTP model. Benestad agrees that the AR(1) model may not necessarily be the best model, but in general statistical models are useless in his opinion when applied to what is called the instrumental period, because in this period the data embed both “signal” and “noise” and LTP or STP or whatever statistical model are meant to describe “the noise” only in his opinion: “Statistical LTP-noise models used for the detection of trends involve circular reasoning if adapted to measured data because this data embed both signal and noise.” (Table 6). Koutsoyiannis in response gave a few examples why in his opinion the danger of circular reasoning is not justified in this case.


Table 5

Benestad

Bunde

Koutsoyiannis

Is LTP relevant/important for the statistical significance of a trend?

Yes (though physics still needed)

Yes, very much

Yes, very much

Table 6

What is the relevance of LTP for the detection of climate change?

Benestad

Statistical LTP-noise models used for the detection of trends involve circular reasoning if adapted to measured data. State of the art detection and attribution is needed.

Bunde

For detection and estimation of external trends (“detection problem”) one needs a statistical model and LTP is the best model to do this.

Koutsoyiannis

LTP is the only relevant statistical model for the detection of changes in climate.

Table 7

Benestad

Bunde

Koutsoyiannis

Is the AR(1) model a valid model to describe the variability in time series of global average temperature?

No, if physics based information is neglected

No

No

Does the AR(1) model leads to overestimation of the significance of trends?

Yes, if you don’t also take into account the physics-based information

Yes

Yes

LTP and chaos

There was disagreement about the relation between LTP and chaos. Obviously, the participants agree that the climate system possesses a chaotic component, but they differ in the extent and time scales of this component. For Benestad though the implication is that LTP cannot be a valid concept for the climate on longer terms. For example in his first reaction to Bunde’s guest blog Benestad wrote: “I presently think one major weakness in your reasoning is [when you say that] ‘in LTP records, in contrast, xi depends on all previous points.’ This cannot be true if the weather evolution is chaotic, where the weather system loses the memory of the initial state after some bifurcation point.”

In another comment Benestad wrote: “The so-called ‘butterfly effect’, an aspect of ‘chaos’ theory, is well-established with meteorology, which means there is a fundamental limit to the predictability of future weather due to the fact that the system loses the memory of the initial state after a certain time period. (…) For geophysical processes, chaos plays a role and may give an impression of LTP, and still the memory of the initial conditions is lost after a finite time interval.”

Koutsoyiannis referred to some of his papers and said that yes, “these publications show that LTP does involve chaos.” In another comment that dealt with untangling the different causes of climate change he said: “in chaotic systems described by nonlinear equations, the notion of a cause may lose its meaning as even the slightest perturbation may lead, after some time, to a totally different system trajectory (cf. the butterfly effect).” So Koutsoyiannis and Benestad largely agree about how chaotic systems behave. The main difference though is that Benestad interprets “the system loses memory” as “LTP is not a useful concept” and here Koutsoyiannis and Bunde disagree with him. In particular, Koutsoyiannis considers memory as a bad interpretation of LTP: it is the change which produces the LTP and thus LTP is fully consistent with the chaotic behaviour of climate.

Summary
There was disagreement about the relation between LTP and chaos (Table 8). According to Benestad chaos theory implicates the memory of the initial conditions is lost after a finite time interval. Benestad interprets “the system loses memory” as “LTP is not a useful concept”. Koutsoyiannis considers memory as a bad interpretation of LTP: it is the change which produces the LTP and thus LTP is fully consistent with the chaotic behaviour of climate.

Table 8

Benestad

Bunde

Koutsoyiannis

Is the climate chaotic?

Yes

Yes

Yes

Does chaos mean memory is lost and does this apply for climatic timescales as well?

Yes

chaos is not a useful concept for describing the variability of climate records on longer time scales

No; LTP is not memory

Does chaos exclude the existence of LTP?

Yes, at both weather and climatic time scales

No

No; on the contrary, chaos can produce LTP

Does chaos contribute to the existence of LTP?

No, but chaos may give an impression of LTP.

Yes

Yes, LTP does involve chaos


Signal and noise

There was disagreement about concepts like signal and noise. In his guest blog Benestad wrote: “The term ‘signal’ can have different meanings depending on the question, but here it refers to manmade climate change. ‘Noise’ usually means everything else, and LTP is ‘noise in slow motion’.”

Koutsoyiannis disagreed with this distinction: “I would never agree with your term “noise” to describe the natural change. Nature’s song cannot be called “noise”. Most importantly, your “signal” vs. “noise” dichotomy is something subjective, relying on incapable deterministic (climate) models and on, often misused or abused, statistics.”

In another comment Koutsoyiannis elaborated on this point: “The climate evolution is consistent with physical laws and is influenced by numerous factors, whether these are internal to what we call climate system or external forcings. To isolate one of them and call its effect “signal” may be misleading in view of the nonlinear chaotic behaviour of the system.”

Bunde seems to take a position in between Benestad and Koutsoyiannis. He does assume – as a working hypothesis - that there is an external deterministic trend from the greenhouse gases but he calls the remaining part of the total climate signal natural “fluctuations” and not noise. Bunde: “we have to note that we distinguish between natural fluctuations and trends. When looking at a LTP curve, we cannot say a priori what is trend and what is LTP. (...) The LTP is natural, the trend is external and deterministic.”

The distinction in signal and noise is another way of stating what detection aims to do: distinguishing whether the (forced) changes are significantly outside of the bounds of the unforced or internal variability. All three appear to agree that purely based on LTP, this distinction can’t be made.

Summary
There was disagreement about concepts like signal and noise. According to Benestad the term ‘signal’ refers to manmade climate change. ‘Noise’ usually means everything else, and LTP is ‘noise in slow motion’ (Table 9). Koutsoyiannis argued that the “signal” vs. “noise” dichotomy is subjective and that everything we see in the climate is signal. To isolate one factor and call its effect “signal” may be misleading in view of the nonlinear chaotic behaviour of the system. Bunde does assume there is an external deterministic trend from the greenhouse gases but he calls the remaining part of the total climate signal natural “fluctuations” and not noise (Table 9). All three seem to agree that one cannot use LTP to make a distinction between forced and unforced changes in the climate (Table 10).

Table 9

Signal versus noise

Benestad

The signal is manmade climate change; the rest is noise and LTP is noise in slow motion.

Bunde

My working hypothesis: there is a deterministic external trend; the rest are natural fluctuations which are best described by LTP.

Koutsoyiannis

Excepting observation errors, everything we see in climate is signal.

Table 10

Benestad

Bunde

Koutsoyiannis

Is the signal versus noise dichotomy meaningful?

Yes

Yes

No*

Can LTP distinguish between forced and unforced components of the observed change?

No

No

*

Can LTP distinguish between natural fluctuations (including natural forcings) and trends?

No

Yes

*

* Koutsoyiannis thinks that even the formulation of these questions, which imply that that the description of a complex process can be made by partitioning it into additive components and trying to know the signatures of each one component indicates a linear view for a system that is intrinsically nonlinear.


Forced versus unforced

In our introduction we introduced the three climate influences that climate scientists distinguish:

“Most experts agree that three types of processes (internal variability, natural and anthropogenic forcings) play a role in changing the Earth’s climate over the past 150 years. It is the relative magnitude of each that is in dispute. The IPCC AR4 report stated that “it is extremely unlikely (<5%) that recent global warming is due to internal variability alone, and very unlikely (< 10 %) that it is due to known natural causes alone.” This conclusion is based on detection and attribution studies of different climate variables and different ‘fingerprints’ which include not only observations but also physical insights in the climate processes.”

There was a lot of discussion about the physical mechanisms behind LTP. Bart Verheggen of the Climate Dialogue team asked a series of questions about this: Can we agree that forcing introduces LTP? Can we agree that forcing is omnipresent for the real world climate? Is LTP mainly internal variability or the result of a combination of internal variability and natural forcings?

Bunde replied that “Natural Forcing plays an important role for the LTP and is omnipresent in climate (so yes and yes to first two questions).” Koutsoyiannis also agreed that “(changing) forcing can introduce LTP and that it [forcing] is omnipresent. But LTP can also emerge from the internal dynamics alone as the above examples show. Actually, I believe it is the internal dynamics that determine whether or not LTP would emerge.”

Verheggen concluded: “All three invited participants agree that radiative forcing can introduce LTP and that it is omnipresent. It follows that the presence of LTP cannot be used to distinguish forced from unforced changes in global average temperature. The omnipresence of both unforced and forced changes means that it’s very difficult (if not impossible) to know the LTP signature of each. Therefore, LTP by itself doesn’t seem to provide insight into the causal relationships of change. It is however relevant for trend significance, but fraught with challenges since the unforced LTP signature is not known.”

Summary
According to Bunde natural forcing plays an important role for LTP and is omnipresent in climate. Koutsoyiannis agreed that (changing) forcing can introduce LTP and that forcing is omnipresent, but LTP can also emerge from the internal dynamics alone.

Table 11

Benestad

Bunde

Koutsoyiannis

Does forcing introduce LTP?

Yes

Yes

Yes

Is forcing omnipresent in the real world climate?

Yes

Yes

Yes

What according to you is the main mechanism behind LTP?

Forcings

Natural Forcing plays an important role for the LTP and is omnipresent in climate.

I believe it is the internal dynamics that determines whether or not LTP would emerge.

Is the warming significant?

This brings us to one of the key questions in this climate dialogue: do you conclude there is a significant warming trend? The participants used different models and methods to answer this question and understanding their different views requests a detailed understanding of these methods which is outside the scope of this dialogue. So here we just mention the differences and focus on the results.

Benestad preferred to use a regression analysis of the global average temperature against known climate forcings as these may be considered as additional information with respect to any statistical model. The results are shown in his figure 1:


Benestad’s Figure 1. The recorded changes in the global mean surface temperature over time (red). The grey curve shows a model calculation of this temperature based on greenhouse gases (GHGs), ozone (O3), and changes in the sun (S0).

Benestad: “The probability that this fit [in the regression analysis] is accidental is practically zero if we assume that that the temperature variations from year-to-year are independent of each other. LTP and the oceans inertia will imply that the degrees of freedom is lower than the number of data points, making it somewhat more likely to happen even by chance.” Benestad says it is very likely that the main physical causes of the change are clear and that greenhouse gases are the main contributors to the warming since the midst of the 20th century (as also illustrated by figure 9.15 in AR4 or figure 10.7 in AR5).

Koutsoyiannis asked Benestad whether his model shown in his Figure 1 is free of circular reasoning, which means that at least he has split the data into two periods for modelling and validation. Benestad left the question unanswered and there was no further discussion on this issue.

Bunde and Koutsoyiannis use different statistical methods. Bunde explained that “nowadays, there is a large number of methods available that is able to detect the natural fluctuations in the presence of simple monotonous trends. Two of them are the detrended fluctuation analysis (DFA) and the wavelet technique (WT).”

Based on these methods Bunde reached the following conclusions:
“(i) The global sea surface temperature increased, in the past 100y, by about 0.6 degree, which is not significant. The reason for this is the large persistence of the oceans, reflected by a large Hurst exponent.
(ii) The global land air temperature, in the past 100 years, increased by about 0.8 degrees. We find this increase even highly significant. The reason for this is the comparatively low persistence of the land air temperature, which makes large natural increases unlikely.”

Koutsoyiannis used a different method to identify and quantify the LTP which he calls a climacogram. Koutsoyiannis is the most ‘skeptical’ of the three participants when it comes to the significance of trends: “Assuming that the data set we used is representative and does not contain substantial errors, the only result that we can present as fact is that in the last 134 years the climate has warmed by 0.6°C (this is a difference of climatic—30-year average—values while other, often higher, values that appear in the literature refer to trends based on annual values). Whether this change is statistically significant or not depends on assumptions. If we assume a 90-year lag and 1% significance, it perhaps is.”


Koutsoyiannis’ Figure 5: testing lagged climatic differences based on the HadCrut4 data set (1850-2012). Differences are not statistically significant according to Koutsoyiannis, except maybe for the 90 year lag.

When asked specifically if his results mean that “detection” has not yet taken place, Koutsoyiannis replied: “Yes, I believe it has not taken place. Whether it comes close: It is likely.”

Koutsoyiannis argues that the current temperature signal is not outside of the bounds of what could be expected from natural forced and unforced changes, thereby using a stricter rating (1%) than the standard definition of “detection” (5%). He bases his statement on a higher Hurst coefficient than Bunde does which partly explains why Koutsoyiannis and Bunde don’t reach exactly the same conclusions.

Summary
The three participants gave different answers on the key question of this Climate Dialogue, namely of the warming in the past 150 years is significant or not. They used different methods to answer the question. Benestad is most confident that both the changes in land and sea temperatures are significant. Bunde concludes that due to a high Hurst parameter the sea temperatures are not significant but the land and global temperatures are. Koutsoyiannis concludes that for most time lags the warming is not significant. In some cases it maybe is.

Table 12

BenestadI

BundeII

KoutsoyiannisIII

Is the rise in global average temperature during the past 150 years statistically significant?

Yes

YesIV

NoV

Is the rise in global average sea surface temperature during the past 150 years statistically significant?

Yes

No

No

Is the rise in global average land surface temperature during the past 150 years statistically significant?

Yes

Yes

No

I Benestad’s conclusions are based on the difference between GCM simulations with and without anthropogenic forcing (Box 10.1 or Figs 10.1 & 10.7 in AR5)
II Based on the detrended fluctuation analysis (DFA) and/or the wavelet technique (WT).
III Based on the climacogram and different time lags (30, 60, 90 and 120 years).
IV This change is 99% significant according to Bunde.
V For a 90 year time lag and a 1% significance level it maybe is significant (see Koutsoyiannis’ guest blog).

Is there a large contribution of greenhouse gases to the warming?

While Bunde and Koutsoyiannis share similar views about the importance of LTP for detection, there are some differences as well which are reflected in the table above. Koutsoyiannis for example does not agree with Bunde’s conclusion that the increase in the global land air temperature in the past 100 years is significant.

Bunde and Koutsoyiannis seem to disagree about the level of LTP (i.e. the value of the Hurst coefficient) in land surface temperature records. Koutsoyiannis believes that on climatic time scales, sea surface temperatures (SSTs) and land surface temperatures (LSTs) should be highly correlated: “I believe if you accept that the sea surface temperature has strong LTP, then logically the land temperature will have too, so I cannot agree that the latter has “comparatively low persistence”. (…) I believe climates on sea and land are not independent to each other—particularly on the long term.” Bunde though thinks that the persistence in the SSTs is higher than that in LSTs.

Bunde is more convinced of a substantial role for greenhouse gases on the climate than Koutsoyiannis although he admits he cannot rule out that the warming is (partly) due to urban heating. “First of all, from our trend significance calculations we can see, without any doubt, that there is an external temperature trend which cannot be explained by the natural fluctuations of the temperature anomalies. We cannot distinguish between Urban Warming and GHG here, but there are places on the globe where we do not expect urban warming but we still see evidence for an external trend, so we may conclude that it is GHG.”

In another comment Bunde wrote: “as a consequence of the LTP in the temperature data, the error bars are very large, considerably larger than for short-term persistent records. But nevertheless, except for the global sea surface temperature, we have obtained strong evidence from this analysis that the present warming has an anthropogenic origin.” And in another comment: “Regarding GHG [greenhouse gases] I may not fully agree with Demetris [Koutsoyiannis]: We cannot show in our analysis of instrumental temperature data that GHG are responsible for the anomalously strong temperature increase that we see and that we find is significant, but it is my working hypothesis.”

When we asked Koutsoyiannis whether he believes the influence of greenhouse gases is small he answered: “Yes, I believe it is relatively weak, so weak that we cannot conclude with certainty about quantification of causative relationships between GHG and temperature changes. In a perpetually varying climate system, GHG and temperature are not connected by a linear, one-way and one-to-one, relationship. I believe climate models and the thinking behind them have resulted in oversimplifying views and misleading results. As far as climate models are not able to reproduce a climate that (a) is chaotic and (b) exhibits LTP, we should avoid basing conclusions on them.”

Benestad on the other hand wrote: “The combination of statistical information and physics knowledge lead to only one plausible explanation for the observed global warming, global mean sea level rise, melting of ice, and accumulation of ocean heat. The explanation is the increased concentrations of GHGs.”

Summary
Bunde is more convinced of a substantial role for greenhouse gases on the climate than Koutsoyiannis although he admits he cannot rule out that the warming on land is (partly) due to urban heating. Bunde said he may not fully agree with Koutsoyiannis: “We cannot show in our analysis of instrumental temperature data that GHG are responsible for the anomalously strong temperature increase that we see and that we find is significant, but it is my working hypothesis.” Koutsoyiannis believes the influence of greenhouse gases is relatively weak, “so weak that we cannot conclude with certainty about quantification of causative relationships between GHG and temperature changes”. Benestad on the other hand said the increased concentrations of GHGs is the only plausible explanation for the observed global warming, global mean sea level rise, melting of ice, and accumulation of ocean heat.

Table 13

Benestad

Bunde

Koutsoyiannis

Is the warming mainly of anthropogenic origin?

The combination of statistical information and physics knowledge lead to only one plausible explanation: GHGs

Yes, it is my working hypothesis

No, I think the effect of CO2 is small



[1] In statistics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it describes certain time-varying processes in nature, economics, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values. It is a special case of the more general ARMA model of time series. More details, see Wikipedia.

[2] Section 10.2.1 in AR5.

[3] Koutsoyiannis, D., and A. Montanari (2007), Statistical analysis of hydroclimatic time series: Uncertainty and insights, Water Resources Research, 43 (5), W05429, doi: 10.1029/2006WR005592.

[4] Hurst-Kolmogorov is a term that Koutsoyiannis has introduced and which is synonymous to LTP. In his guest blog he explains where it comes from: “A decade before Hurst detected LTP in natural processes, Andrey Kolmogorov, devised a mathematical model which describes this behaviour using one parameter only, i.e. no more than in the Markov [AR(1)] model. We call this model the Hurst-Kolmogorov (HK) model.”

[5] The Hurst coefficient H is a measure for long-term persistence. H is a number between 0.5 and 1. The closer to 1, the more persistent a system is.

[6] Markonis, Y., and D. Koutsoyiannis, Climatic variability over time scales spanning nine orders of magnitude: Connecting Milankovitch cycles with Hurst–Kolmogorov dynamics, Surveys in Geophysics, 34 (2), 181–207, 2013.