Hydrology and Change - ITIA

IUGG XXV General Assembly Earth on the Edge: Science for a Sustainable Planet Melbourne, Australia, 27 June - 8 July 2011 Plenary lecture

Hydrology and Change Demetris Koutsoyiannis Department of Water Resources and Environmental Engineering School of Civil Engineering National Technical University of Athens, Greece ([email protected], www.itia.ntua.gr/dk/) Presentation available online: itia.ntua.gr/1135/

I would like to thank the President and the officers of IUGG for the invitation to present this lecture, which is a great honour for me. I also thank all of you for attending it. Hydrology is my field and its interplay with change is the subject of my talk.

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Hydrology is the science of the water on Earth: its occurrence, circulation, distribution, physical and chemical properties, and interaction with the environment and the biosphere

Nile River

Aswan Dam Lake Nasser Images from earthobservatory.nasa.gov/IOTD/view.php?id=2416 earthobservatory.nasa.gov/images/imagerecords/46000/46209/earth_pacific_lrg.jpg earthobservatory.nasa.gov/images/imagerecords/1000/1234/PIA02647_lrg.jpg earthobservatory.nasa.gov/images/imagerecords/5000/5988/ISS010-E-14618_lrg.jpg

D. Koutsoyiannis, Hydrology and Change

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As illustrated in this slide, hydrology is the science of water on Earth. Its domain spans several temporal and spatial scales. Of course, large hydrological systems, like the Nile River are of huge importance. The image of the Nile also highlights the connection of water with life: the blue colour of the Nile is not seen, but it is seen the green of the life it gives rise to. I will refer to the Nile later. The spot on the Aswan dam on the Nile wants to emphasize the strong link of hydrology with engineering, and through this, with the improvement of diverse aspects of our live and civilization. The next spot on the Lake Nasser, upstream of the Aswan dam, indicates that the intervention of humans on Nature can really be beautiful.

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It looks like, recently, our scientific community has been amazed that things change…


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I hope you can agree that the title of my talk “Hydrology and Change” harmonizes with a large body of literature, books, conferences, scientific papers and news stories, all of which scream about change. Changing planet, changing world, changing ocean, changing climate... Has our scientific community and our society only recently understood that things change?

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Raphael's “School of Athens” (1509–1510; Apostolic Palace, Vatican City; en.wikipedia.org/wiki/School_of_Athens) 4 D. Koutsoyiannis, Hydrology and Change

In reality, change has been studied very early, at the birth time of science and philosophy. All of these Greek philosophers you see in Raphael's “School of Athens” had something to say about change. I will only refer to two of them, the central figure of the painting, Aristotle, and this lonely figure in front, Heraclitus.

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Heraclitus (ca. 540-480 BC)

Πάντα ῥεῖ Everything flows

Heraclitus (figured by Michelangelo) in Raphael's School of Athens; en.wikipedia.org/wiki/Heraclitus

Alternative versions Τὰ ὄντα ἰέναι τε πάντα καὶ μένειν οὐδέν [from Plato's Cratylus, 401d] All things move and nothing remains still Πάντα χωρεῖ καὶ οὐδὲν μένει [ibid, 402,a] Everything changes and nothing remains still

Δὶς ἐς τὸν αὐτὸν ποταμὸν οὐκ ἂν ἐμβαίης [from Plato's Cratylus, 402a] You cannot step twice into the same river D. Koutsoyiannis, Hydrology and Change

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Heraclitus summarized the dominance of change in a few famous aphorisms. Panta rhei--Everything flows. And: You cannot step twice into the same river (the second time you step into, it is no longer the same river). Notice the simple hydrological notions he uses to describe change: Flow and River.

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Aristotle (384-322 BC) in Meteorologica Change

ὅτι οὔτε ὁ Τάναϊς οὔτε ὁ Νεῖλος ἀεὶ ἔρρει, ἀλλ' ἦν ποτε ξηρὸς ὁ τόπος ὅθεν ῥέουσιν· τὸ γὰρ ἔργον ἔχει αὐτῶν πέρας, ὁ δὲ χρόνος οὐκ ἔχει. ... ἀλλὰ μὴν εἴπερ καὶ οἱ ποταμοὶ γίγνονται καὶ φθείρονται καὶ μὴ ἀεὶ οἱ αὐτοὶ τόποι τῆς γῆς ἔνυδροι, καὶ τὴν θάλατταν ἀνάγκη μεταβάλλειν ὁμοίως. τῆς δὲ θαλάττης τὰ μὲν ἀπολειπούσης τὰ δ' ἐπιούσης ἀεὶ φανερὸν ὅτι τῆς πάσης γῆς οὐκ ἀεὶ τὰ αὐτὰ τὰ μέν ἐστιν θάλαττα τὰ δ' ἤπειρος, ἀλλὰ μεταβάλλει τῷ χρόνῳ πάντα [I.14, 353a 16]

Νeither the Tanais [River Don in Russia] nor the Nile have always been flowing, but the region in which they flow now was once dry: for their life has a bound, but time has not… But if rivers are formed and disappear and the same places were not always covered by water, the sea must change correspondingly. And if the sea is receding in one place and advancing in another it is clear that the same parts of the whole earth are not always either sea or land, but that all changes in course of time Conservation of mass within the hydrological cycle

ὥστε οὐδέποτε ξηρανεῖται· πάλιν γὰρ ἐκεῖνο φθήσεται καταβὰν εἰς τὴν αὐτὴν τὸ προανελθόν [II.3, 356b 26]

Thus, [the sea] will never dry up; for [the water] that has gone up beforehand will return to it

Aristotle in Raphael's “School of Athens”; en.wikipedia.org/wiki/ Aristotle

κἂν μὴ κατ' ἐνιαυτὸν ἀποδιδῷ καὶ καθ' ἑκάστην ὁμοίως χώραν, ἀλλ' ἔν γέ τισιν τεταγμένοις χρόνοις ἀποδίδωσι πᾶν τὸ ληφθέν [II.2, 355a 26]

Even if the same amount does not come back every year or in a given place, yet in a certain period all quantity that has been abstracted is returned D. Koutsoyiannis, Hydrology and Change

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I found it amazing that Aristotle had understood the scale and extent of change much better than some contemporary scholars do: Neither the Tanais, nor the Nile have always been flowing, but the region in which they flow now was once dry. Rivers are formed and disappear. The same parts of the whole earth are not always either sea or land. All changes in course of time. He also understood and neatly expressed the conservation of mass within the hydrological cycle.

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Frequency (per million)

Reinventing change... or worrying about it? 4

Worries for change have receded after 2000...

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Except one...

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What happened at the beginning of the 20th century? Did the world start to change? Did people understand that things change? Or did people start to dislike change and worry about it?

And what happened in the 1970-1980s?

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Data and visualization by Google labs; ngrams.googlelabs.com (360 billion words in 3.3 million books published after 1800; see also Mitchel et al., 2011) D. Koutsoyiannis, Hydrology and Change

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Let us come back to the present and try to investigate our perception of change in quantified terms. To this aim, I found the information provided recently by the Google Labs very useful. This information is the frequency per year of each word or phrase in some millions of books. Here you see that at about 1900 people started to speak about a “changing world” and a little later about “environmental change”. “Demographic change”, “climate change” and “global change” “start” at 1970s and 1980s. In my view, the intense use of these expressions reflects worries for change. So, all worries have receded after 2000, except one: climate change.

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Frequency (per billion)

An unprecedented disbelief that change is real? 4

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Why is it necessary to assure people that “climate change is real” while no one needs to be assured that “weather change is real”?

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Data and visualization by Google labs; ngrams.googlelabs.com D. Koutsoyiannis, Hydrology and Change

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I have made a lot of searches of this type using Google labs but I do not have the time to discuss them in detail. Anyhow, this is the funniest: The statements “climate change is happening”, “... is occurring”. “... is real” contain the same truth as “weather change is real”. You can speculate why the former has been in wide use, while the latter has not.

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Is our focus opposite to importance? 4

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Who can dispute the unprecedented change in the demographic and energy conditions in the last half century? Weren’t these the 20th century determinants and don’t they represent the main uncertainty for the future?

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What does the inflationary use of the term “climate change” mean, given that climate has been in perpetual change? Is “climate change” a scientific term or a trendy popular slogan of political origin?

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Data and visualization by Google labs; ngrams.googlelabs.com D. Koutsoyiannis, Hydrology and Change

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You can also think and speculate whether or not the more frequent use of a phrase indicates higher importance or the opposite.

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Frequency (per billion)

Is change only negative?

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zero appearances of “favorable”, “positive” and “beneficial”

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You can further think whether, according to conventional wisdom, change can only be dramatic and catastrophic.

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Are we becoming more and more susceptible to scare and pessimism? 4

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Even removing “climate” or “climate change” from our phrases, we can see that our society may like scare and pessimism, as testified by the more and more frequent use of words like “apocalyptic” and “catastrophic”.

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Change is tightly linked to uncertainty 0.8

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In any case, I think that there is no doubt that the world is changing and, also, that the future is uncertain. Change is tightly linked to uncertainty.

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Predictability of change

Change

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e.g. acceleration of a falling body

e.g. daily and annual cycles

e.g. consecutive outcomes of dice

Simple systems – Short time horizons Important but trivial

Structured random e.g. climatic fluctuations

Complex systems – Long time horizons Most interesting D. Koutsoyiannis, Hydrology and Change

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This hierarchical chart wants to say that in simple systems the change is regular. The regular change can be periodic or aperiodic. Whatever it is, using equations of dynamical systems, regular change is predictable at short time horizons. But this type of change is rather trivial. More interesting are the more complex systems at long time horizons, where change is unpredictable in deterministic terms, or random. Pure randomness, like in classical statistics, where different variables are identically distributed and independent, is sometimes a useful model, but in most cases it is inadequate. A structured randomness should be assumed instead. As we will see in the next slides, the structured randomness is enhanced randomness, expressing enhanced unpredictability of enhanced multi-scale change.

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What is randomness?

Common dichotomous view: Natural process are composed of two different, usually additive, parts or components—deterministic (signal) and random (noise) Randomness is cancelled out at large scales and does not produce change; only an exceptional forcing can produce a long-term change My view (explained in Koutsoyiannis, 2010): Randomness is none other than unpredictability Randomness and determinism coexist and are not separable Deciding which of the two dominates is simply a matter of specifying the time horizon and scale of the prediction At long time horizons (where length depends on the system) all is random Heraclitus (Fragment 52) Αἰών παῖς ἐστι παίζων πεσσεύων· παιδός ἡ βασιληίη Time is a child playing, throwing dice; the ruling power is a child's Vs. Einstein (in a letter to Max Born in 1926) I am convinced that He does not throw dice

This die is from 580 BC (photo from the Kerameikos Ancient Cemetery Museum, Athens) D. Koutsoyiannis, Hydrology and Change

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But what is randomness? According to the common dichotomous view, natural process are composed of two different, usually additive, parts or components—deterministic (signal) and random (noise). Such randomness is cancelled out at large scales and does not produce change. In this view, only an exceptional forcing can produce a longterm change. My view (which I explain in a recent paper in HESS) is different from this. Randomness is none other than unpredictability. Randomness and determinism coexist and are not separable. Deciding which of the two dominates is simply a matter of specifying the time horizon and scale of the prediction. At long time horizons all is random. This view has been initially proposed by Heraclitus, who said that: Time is a child playing, throwing dice; I illustrate his aphorism with a picture of a die from his epoch (580 BC). As you can see, in this case there is not much change in the dice themselves for several millennia.

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Contemplating the change in rivers: From mixing and turbulence to floods and droughts

Flood in the Arachthos River, Epirus, Greece, under the medieval Bridge of Arta, in December 2005

The bridge is famous from the legend of its building, transcribed in a magnificent poem (by an anonymous poet); see en.wikisource.org/wiki/Bridge_of_Arta; el.wikisource.org/wiki/Το_γιοφύρι_της_Άρτας D. Koutsoyiannis, Hydrology and Change

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Let us now return to rivers and flows. The photo is from a flood in a famous medieval bridge close to my homeland. Several phenomena can help us contemplate the change: From mixing and turbulence to floods and droughts.

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Turbulence: macroscopic motion at millisecond scale Laboratory measurements of nearly isotropic turbulence in Corrsin Wind Tunnel (section length 10 m; cross-section 1.22 m by 0.91 m) at a highReynolds-number (Kang et al., 2003) Measurements by X-wire probes; Sampling rate of 40 kHz, here aggregated at 0.833 kHz—each point is the average of 48 original values

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(attributed to Werner Heisenberg or, in different versions, to Albert Einstein or to Horace Lamb)

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When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first

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Data downloaded from www.me.jhu.edu/meneveau/datasets/Activegrid/M20/H1/m20h1-01.zip D. Koutsoyiannis, Hydrology and Change

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Let us start with turbulence. Turbulence is a very complex phenomenon; according to this saying attributed to Heisenberg or to Einstein, even God may not have answers about it. My view is quite different. Turbulence is God’s dice game and He wants us to be aware that He plays His dice game without having to wait too much: we can see it even in time scales of milliseconds. We can see it visually, just looking at any flow, but nowadays we can also take quantified measurements at very fine time scales. In the graph each measurement represents the average velocity every 1.2 milliseconds from an experiment performed by a group of very kind scientists cited on the slide who made the data available on line at the address seen in the slide. Each value is in fact the average of 48 original data values. Averages at time scales of 0.1 and 1 second are also plotted.

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Pure random processes, assuming independence in time (white noise), have been effective in modelling microscopic motion (e.g. in statistical thermodynamics) Macroscopic random motion is more complex In pure randomness, change vanishes at large scales In turbulence, change occurs at all scales

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Turbulence vs. pure randomness

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To understand the change encapsulated in turbulence we can compare our graph of turbulent velocity measurements with pure random noise with the same statistical characteristics at the lowest time scale. The differences are substantial. At scales 0.1 and 1 second the pure randomness does not produce any change. In turbulence, change is evident even at these scales.

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A river viewed at different time scales—from seconds to millions of years

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 bed will change (erosion-deposition of sediments) Next century: The climate and the river basin characteristics (e.g. vegetation, land use) will change Next millennia: All could be very different (e.g. the area could be glacialized) Next millions of years: The river may have disappeared None of these changes will be a surprise Rather, it would be a surprise if things remained static These changes are not predictable Most of these changes can be mathematically modelled in a stochastic framework admitting stationarity! D. Koutsoyiannis, Hydrology and Change

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So, contemplating the flow in a river, and the river itself, we may imagine several changes. Next second: the hydraulic characteristics, like water level and velocity, will change due to turbulence. Next day: the river discharge will change—even dramatically, in the case of a flood. Next year: The river bed will change because of erosion and deposition of sediments. Next century: The climate and the river basin characteristics, such as vegetation and land use, will change. Next millennia: All could be very different; for example the area could be glacialized. Next millions of years: The river may have disappeared. None of these changes will be a surprise. Rather, it would be a surprise if things remained static. Nonetheless, these changes are not predictable. And amazingly, as we will see, most of these changes can be mathematically modelled in a stochastic framework admitting stationarity! Here we should distinguish the term “change” from “nonstationarity” which has been recently been used in statements like “a nonstationary world”. The world is neither stationary nor nonstationary. The world is real and unique. It is our models that can be stationary or nonstationary as the latter terms need the notion of an ensemble (e.g. of simulations) to apply. 18

The Roda Nilometer and over-centennial change

The Roda Nilometer as it stands today; water entered and filled the Nilometer chamber up to river level through three tunnels In the centre of the chamber stands a marble octagonal column with a Corinthian crown; the column is graded and divided into 19 cubits (a cubit is slightly more than half a meter) and could measure floods up to about 9.2 m A maximum level below the 16th mark could portend drought and famine; a level above the 19th mark meant catastrophic flood (Credit: Aris Georgakakos) D. Koutsoyiannis, Hydrology and Change

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Are these changes just imagination or speculation? Certainly not. The information abounds on the earth’s crust and that is why Aristotle was able to understand the extent of change. In addition, we have records of quantified change. The longest instrumental record is that of the water level of the Nile River. It was taken at the Roda Nilometer near Cairo and extends for more than six centuries. As you see, the measurements were taken at a robust structure, much more elaborate than today’s measuring devices.

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A real-world process

Minimum water level (m)

The Nilometer record vs. a pure random processes 8

Nile River annual minimum water level (663 values)

Annual 30-year average

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Minimum roulette wheel outcome

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Each value is the minimum of m = 36 roulette wheel outcomes. The value of m was chosen so that the standard deviation be equal to the Nilometer series

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Nilometer data: Beran (1994) and lib.stat.cmu.edu/S/beran (here converted into meters)

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The upper graph shows the annual minimum water level of the Nile for 663 years: from 622 to 1284 AD. May I repeat that it is not a proxy, but an instrumental record. Apart from the annual values, 30-year averages are also plotted. These we commonly call climatic values. Due to the large extent of the Nile basin, these reflect the climate evolution of a very large area in the tropics and subtropics. Notice that around 780 AD the climatic value was 1.5 meters, while at 1110 AD it was 4 meters. 2.5 times higher. In the lower panel we can see a simulated series from a roulette wheel which has equal variance as the Nilometer. Despite equal “annual” variability, the roulette wheel produces a static “climate”.

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Hurst’s (1951) seminal paper The motivation of Hurst was the design of the High Aswan Dam on the Nile River However the paper was theoretical and explored numerous data sets of diverse fields Hurst observed that: Although in random events groups of high or low values do occur, their tendency to occur in natural events is greater. This is the main difference between natural and random events

Obstacles in the dissemination and adoption of Hurst’s finding: Its direct connection with reservoir storage Its tight association with the Nile The use of a complicated statistic (the rescaled range) D. Koutsoyiannis, Hydrology and Change

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The first who noticed this behaviour, and in particular the difference of pure random processes and natural processes, was the British hydrological engineer Hurst who worked in Egypt. His motivation was the design of the High Aswan Dam on the Nile River. This is the first page of his seminal paper. He gave the emphasis on the clustering or grouping of similar events in natural processes.

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Kolmogorov (1940)

Kolmogorov studied the stochastic process that describes the behaviour to be discovered a decade later in geophysics by Hurst The proof of the existence of this process is important, because several researches, ignorant of Kolmogorov’s work, regarded Hurst’s finding as inconsistent with stochastics and as numerical artefact

Kolmogorov’s work did not become widely known The process was named by Kolmogorov “Wiener’s Spiral” (Wienersche Spiralen) and later “Self-similar process”, or “fractional Brownian motion” (Mandelbrot and van Ness, 1968); here it is called the Hurst-Kolmogorov (HK) process D. Koutsoyiannis, Hydrology and Change

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Strikingly, earlier the Soviet mathematician and physicist Kolmogorov, a great mind of the 20th century who gave the probability theory its modern foundation and also studied turbulence, had proposed a mathematical process that has the properties to be discovered 10 years later by Hurst in natural processes. Here we call this the HurstKolmogorov (or HK) process, although it is more widely known with other names.

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The climacogram: A simple statistical tool to quantify the change across time scales

Take the Nilometer time series, x1, x2, ..., x663, and calculate the sample estimate of standard deviation σ(1), where the superscript (1) indicates time scale (1 year) Form a time series at time scale 2 (years): x(2)1 := (x1 + x2)/2, x(2)2 := (x3 + x4)/2, ..., x(2)331 := (x661 + x662)/2 and calculate the sample estimate of standard deviation σ(2) Form a time series at time scale 3 (years): x(3)1 := (x1 + x2 + x3)/3,, ..., x(3)221 := (x661 + x662 + x663)/3 and calculate the sample estimate of standard deviation σ(3) Repeat the same procedure up to scale 66 (1/10 of the record length) and calculate σ(66) The climacogram is a logarithmic plot of standard deviation σ(k) vs. scale k If the time series xi represented a pure random process, the climacogram would be a straight line with slope –0.5 (the proof is very easy) In real world processes, the slope is different from –0.5, designated as H – 1, where H is the so-called Hurst coefficient (0 < H < 1) The scaling law σ(k) = σ(1) / k1 – H defines the Hurst-Kolmogorov (HK) process High values of H (> 0.5) indicate enhanced change at large scales, else known as longterm persistence, or strong clustering (grouping) of similar values D. Koutsoyiannis, Hydrology and Change

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The mathematics to describe the HK process are actually very simple as shown in this slide and the calculations can be made in a common spreadsheet.

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Minimum water level (m)

The climacogram of the Nilometer time series

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The classical statistical estimator of standard deviation was used, which however is biased for HK processes

Bias

The Hurst-Kolmogorov process seems consistent with reality 0 The Hurst coefficient is H = 0.89 600 (Similar H values are estimated from the simultaneous record of maximum water levels and from the modern, 131-year, flow record of the Nile flows at Aswan) Essentially, the Hurst-Kolmogorov behaviour, depicted in the climacogram, manifests that longterm changes are much more frequent and intense than commonly perceived and, simultaneously, that the future states are much more uncertain and unpredictable on long time horizons than implied by pure randomness

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Let us see this applied to the Nilometer time series. The -0.5 slope corresponds to white noise (a pure random process). The reality departs substantially from this and is consistent with the HK behaviour with H = 0.89. Essentially, the Hurst-Kolmogorov behaviour manifests that longterm changes are much more frequent and intense than commonly perceived and, simultaneously, that the future states are much more uncertain and unpredictable on long time horizons than implied by pure randomness. Unfortunately, in literature the high value of H is typically interpreted as long-term or infinite memory. This is a bad and incorrect interpretation and the first who pointed this out was the late Vit Klemes back in 1974 (later he became the president of IAHS). So, a high value of H indicates enhanced multi-scale change and has nothing to do with memory.

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The simple-scaling HK process is appropriate for scales > 50 ms, but not for scales smaller than that For small scales, a smoothing effect reduces variability (in comparison to that of the HK process) A Hurst-Kolmogorov process with Smoothing (HKS) is consistent with turbulence measurements at the entire range of scales The HKS process, in addition to the Hurst coefficient, involves a smoothing parameter (α), and can be defined by [σ(k)]2 = [σ(1)]2 / (α2 – 2H + k2 – 2H)

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The climacogram of the turbulent velocity time series

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Data Pure random (H = 0.5) Standard HK process (H = 0.67) HKS procees (HK with smoothing) HKS adapted for bias

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The fitted curve (H = 2/3) is suggestive of the Kolmogorov’s “5/3″ Law: for high frequencies it yields a power spectrum with slope –5/3; this turns to 1 – 2H = –1/3 for low frequencies

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Coming back to the turbulent velocity time series, we observe that its climacogram shows a structure more complex than the simple HK process. Again, there is significant departure from the pure random process. For all time scales, the slope is milder than -0.5 and asymptotically suggests a Hurst coefficient of 2/3.

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Are reconstructions of past hydroclimatic behaviours consistent with the perception of enhanced change?

Lake Victoria is the largest tropical lake in the world (68 800 km2) and is the headwater of the White Nile The contemporary record of water level (covering a period of more than a century) indicates huge changes Reconstructions of water level for past millennia from sediment cores (Stager et al., 2011) suggest that the lake was even dried for several centuries

From Sutcliffe and Petersen (2007)

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These two examples, the Nilometer and the turbulence, gave us an idea of enhanced change and enhanced uncertainty over long time scales. Here there is another example from hydrology. Lake Victoria is the largest tropical lake in the world and is also associated with the Nile. The contemporary record of water level (from a recent paper in Hydrological Sciences Journal) covers a period of more than a century and indicates huge changes. Reconstructions of water level for past millennia from sediment cores— from a brand new study cited in the slide—suggest that the lake was even dried for several centuries at about 14 and 16 thousand years before present.

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Co-evolution of climate with tectonics and life on Earth over the last half billion years 12.0 Permian-Pangea Hothouse

Veizer dO18 Isotopes Long Polyfit Smooth

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Devonian Warm Period

8.0 Eocene Maximum

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Palaeoclimate temperature estimates based on δ18O; adapted from Veizer et al. (2000) D. Koutsoyiannis, Hydrology and Change

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Of course, all these examples reflect changes of the climate on Earth. This graph depicts perhaps the longest palaeoclimate temperature reconstruction and is based on the 18O isotope. It goes back to more than half a billion years. The change has been prominent. Today we are living in an interglacial period of an ice age. We can observe that in the past there were other ice ages, where glacial and integlacial periods alternated, as well as ages with higher temperatures free of ice. The graph also shows some landmarks providing links of the co-evolution of climate with tectonics (Pangaea) and life on Earth.

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6 4 2 0 -2 -4 -6 -8 -10 -12

EPICA

0.5 0 -0.5 -1 -1.5 -2 1870

1890

1910

1930 Year

1950

1970

1990

2010

800

Moberg & Lohle

700

600

500 400 Thousand Years BP

300

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0 -1.5

δT

1

Huybers

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-1

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-0.5

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0

-1

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-1.5 200

400

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Taylor Dome -37

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500

0 0

δ18 Ο

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δ18Ο

1850

1

δ18 Ο

1

δT

CRU & NSSTC

T (oC)

Temperature change on Earth based on From Markonis and Koutsoyiannis (2011) observations and proxies

2

-39

3

-41

4

-43

Zachos 5

-45 10

9

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Thousand Years BP

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δΤ

Veizer δ18 Ο

-30 -32 -34 -36 -38 -40 -42 -44 -46 -48 -50

GRIP

30 25 Million Years BP

10 5 0 -5 -10 500

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300 250 Million Years BP

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0


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These eight graphs depict 10 data series related to the evolution of the climate, specifically temperature, on Earth. The first graph is made from instrumental records over the last 160 years and satellite observations over the last 30 years. The last panel is that of the previous slide. The red squares provide the links of the time period of each proxy with the one before.

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A combined climacogram of all 10 temperature observation sets and proxies NSSTC CRU Lohle Moberg Taylor GRIP EPICA Huybers Zachos Veizer

orbital forcing Wh it

(anti-peristence)

i se , e No e slo p = -0 .5

Standard deviation (arbitrary units)

1

Enhanced change Slop e

Common perception: Pure random change

= -0 .08

This slope supports an HK behaviour with H > 0.92 The HK behaviour extends over all scales

The actual climatic variability at the scale of 100 million years equals that of 3 years of a pure random climate! 0.1 10-1 1.E-01

0

10 1.E+00

1

10 1.E+01

2

10 1.E+02

3

10 1.E+03

From Markonis and Koutsoyiannis (2011)

4

10 1.E+04

5

10 1.E+05

6

10 1.E+06

7

10 1.E+07 Scale (years)

The orbital forcing is evident for scales 4 108 between 10 1.E+08 and 105 years


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Now, if we superimpose, the climacograms of the 10 time series, after appropriate translation of the logarithm of standard deviation, but without modifying the slopes, we get this impressive overview for time scales spanning almost 9 orders of magnitude—from 1 month to nearly 100 million years. First we observe that the real-climate climacogram departs spectacularly from that of pure randomness. Second, the real climate is approximately consistent with an HK model with a Hurst coefficient greater than 0.92. Third, there is a clear deviation of the simple scaling HK law for scales between 10 and 100 thousand years, exactly as expected due to the Milancovich cycles acting at these time scales. Overall, the actual climatic variability at the scale of 100 million years equals that of about 2.5-3 years of a pure random climate! This dramatic effect should make us understand the inappropriateness of classical statistical thinking for climate. It should also help us understand what enhanced change and enhanced unpredictability are. My PhD student Yiannis Markonis and I are preparing a publication of this with further explanations (hopefully it will be published somewhere).

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Clustering in time in other geophysical processes: Earthquakes

Megaquakes Cluster 1

Megaquakes Cluster 2 Low seismicity Cluster 1 The graph, adapted from Kerr (2011), shows megaquake clusters, as well as a low seismicity cluster


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But the characteristics of the HK behaviour are not limited to hydrology and climate. This graph, adapted from a recent article in the Science Magazine, but with my own interpretation, shows how megaearthquakes and periods of low seismicity cluster in time. That is, the HK behaviour is universal...

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Cosmological evolution: from uniformity to clustering Uniformity: Maximum uncertainty at small scales

Time

200 million years after the Bing Bang

7 billion years later

Void

Black hole and galaxy cluster

300 million light years

Clustering: Enhanced uncertainty at large scales Images are snapshots from videos depicting cosmological simulations, timemachine.gigapan.org/wiki/Early_Universe; Di Matteo et al. (2008)


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... and since it is universal, we should find it in the big picture of the universe itself. These snapshots taken from videos depicting cosmological simulation indicate how the universe evolved from the initial uniform soup to clustered structures, where the clusters are either voids or stars, galaxies and black holes.

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Even the use of neutral words, like “and”, “you”, “we”, etc., is subject to spectacular change through the years Data from Google labs; ngrams.googlelabs.com

Frequency (per thousand)

Clustering and change in human related processes

29 28 27 26 25 24 23 22 3.5

and you

we

boy

girl

one

two

3 2.5 2 1.5 1 0.5 0.15

0.1 0.05 0 1800

1850

1900

1950

Year


2000

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Moreover, the HK behaviour is not a characteristic of the physical universe only, but also of mental universes constructed by humans. Here I return to the Google Labs to examine frequencies of simple and unsophisticated words. I chose very neutral and ingenuous words like “and”, “you”, “we”, “one”, “two”, “boy”, “girl”. One might expect that the frequency of using these words should be static, not subject to change. But the reality is impressively different as shown in these graphs. The most neutral and less changing seems to be the world “two”, so let us examine it further...

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1.2 Frequency (per thousand)

The climacogram of the frequency of a very neutral word: “two”

1.15 1.1 1.05 1 0.95

"two", actual average 99% prediction limits (classical statistics)

0.9 0.85 1800

The pure random approach is a spectacular failure The climacogram is almost flat This indicates an HKS process with H close to 1 Even a Markov (AR1) model with high autocorrelation can match this climacogram The two models (HKS and Markov) are indistinguishable in this case (the available record is too short)

1850

1900

1950

2000 Year

0.1 Standard deviation of frequency (per thousand)

Data Pure random (H = 0.5) HKS procees adapted for bias (H = 0.99) Markov process, adapted for bias (ρ = 0.89)

0.01 1

10

100 Scale (years)


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A detailed plot will again reveal change different from pure randomness. The climacogram will indicate that the change is consistent with the HK behaviour with a Hurst coefficient very close to 1.

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Seeking an explanation of long-term change: Entropy

Definition of entropy of a random variable z (adapted from Papoulis, 1991) ∞

Φ[z] := E[–ln[ f(z)/l(z)]] = –∫-∞ f(z) ln [f(z)/l(z)] dz

[dimensionless]

∞

where f(z) the probability density function, with ∫-∞ f(z) dz = 1, and l(z) a Lebesgue density (numerically equal to 1 with dimensions same as in f(z)) Definition of entropy production for the stochastic process z(t) in continuous time t (from Koutsoyiannis, 2011) Φ΄[z(t)] := dΦ[z(t)] / dt [units T-1] Definition of entropy production in logarithmic time (EPLT) φ[z(t)] := dΦ[z(t)] / d(lnt) ≡ Φ΄[z(t)] t [dimensionless] Note 1: Starting from a stationary stochastic process x(t), the cumulative t (nonstationary) process z(t) is defined as z(t) := ∫0 x(τ) dτ; consequently, the discrete time process xiΔ := z(iΔ) – z((i – 1)Δ) represents stationary intervals (for time step Δ in discrete time i) of the cumulative process z(t) Note 2: For any specified t and any two processes z1(t) and z2(t), an inequality relationship between entropy productions, such as Φ΄[z1(t)] < Φ΄[z2(t)] holds also true for EPLTs, e.g. φ[z1(t)] < φ[z2(t)] D. Koutsoyiannis, Hydrology and Change

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Is there a physical explanation of this universal behaviour? In my opinion, yes, and it relies on entropy. Entropy is none other than a measure of uncertainty and its maximization means maximization of uncertainty. When time is involved, the quantity that we should extremize is entropy production, a derivative of entropy.

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Extremizing entropy production (EPLT) The solutions depicted are generic, valid for any Gaussian process, independent of μ and σ, and depended on ρ only (the example is for ρ = 0.543)—see Koutsoyiannis (2011)

2

Markov, unconditional Markov, conditional HK, unconditional+conditional

The conditional EPLT corresponds to the case where the past has been observed

φ (t ) 1.5

As t → 0, the EPLT is maximized by a Markov process

As t → ∞, the EPLT is maximized by an HK process

1

0.5

As t → 0, the EPLT is minimized by an HK process

0 0.001

0.01

The HK process has constant EPLT = H, where H is the Hurst coefficient—the exponent of the power law: H = ½ + ½ ln(1 + ρ)/ln 2

0.1

1

As t → ∞, the EPLT is minimized by a Markov process

10

100

1000 t


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As this graph depicts, the HK process appears to extremize entropy production and the Hurst coefficient has a clear physical meaning: It is the entropy production in logarithmic time. The details can be found in my recent paper in Physica A.

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Concluding remarks

The world exists only in change Change occurs at all time scales Change is hardly predictable in deterministic terms Humans are part of the changing Nature—but change is hardly controllable by humans (fortunately) Hurst-Kolmogorov dynamics is the key to perceive multi-scale change and model the implied uncertainty and risk Hydrology has greatly contributed in discovering and modelling change— however, lately, following other geophysical disciplines, it has been affected by 19th-century myths of static or clockwork systems, deterministic predictability (cf. climate models) and elimination of uncertainty A new change of perspective is thus needed in which change and uncertainty are essential parts

Both classical physics and quantum physics are indeterministic Karl Popper (in his book “Quantum Theory and the Schism in Physics”)

The future is not contained in the present or the past W. W. Bartley III (in Editor’s Foreward to the same book) D. Koutsoyiannis, Hydrology and Change

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General references

Beran, J., Statistics for Long-memory Processes, Chapman and Hall, New York, 1994 Di Matteo, T., J. Colberg, V. Springe, L. Hernquist and D. Sijacki, Direct cosmological simulations of the growth of black holes and galaxies, The Astrophysical Journal, 676 (1), 33-53, 2008 Hurst, H. E., Long term storage capacities of reservoirs, Trans. Am. Soc. Civil Engrs., 116, 776–808, 1951 Kang, H. S., S. Chester and C. Meneveau, Decaying turbulence in an active-grid-generated flow and comparisons with large-eddy simulation, J. Fluid Mech., 480, 129-160, 2003 Kerr, R. A., More megaquakes on the way? That depends on your statistics, Science, 332 (6028), 411, DOI: 10.1126/science.332.6028.411, 2011 Kolmogorov, A. N., Wienersche Spiralen und einige andere interessante Kurven in Hilbertschen Raum, Dokl. Akad. Nauk URSS, 26, 115–118, 1940 Koutsoyiannis, D., A random walk on water, Hydrology and Earth System Sciences, 14, 585–601, 2010 Koutsoyiannis, D., Hurst-Kolmogorov dynamics as a result of extremal entropy production, Physica A: Statistical Mechanics and its Applications, 390 (8), 1424– 1432, 2011 Mandelbrot, B. B., and J. W. van Ness, Fractional Brownian Motion, fractional noises and applications, SIAM Review, 10, 422-437, 1968 Markonis, I., and D. Koutsoyiannis, Combined Hurst-Kolmogorov and Milankovitch dynamics in Earth’s climate, 2011 (in preparation). Michel J.-B., Y. K. Shen, A. P. Aiden, A. Veres, M. K. Gray, The Google Books Team, J. P. Pickett, D. Hoiberg, D. Clancy, P. Norvig, J. Orwant, S. Pinker, M. A. Nowak and E. L. Aiden, Quantitative analysis of culture using millions of digitized books, Science, 331 (6014), 176-182, 2011 Papoulis, A., Probability, Random Variables, and Stochastic Processes, 3rd ed., McGraw-Hill, New York, 1991 Stager, J. C., D. B. Ryves, B. M. Chase and F. S. R. Pausata, Catastrophic Drought in the Afro-Asian Monsoon Region During Heinrich Event, Science, DOI: 10.1126/science.1198322, 2011 Sutcliffe, J. V., and G. Petersen, Lake Victoria: derivation of a corrected natural water level series, Hydrological Sciences Journal, 52 (6), 1316-1321, 2007

Paleoclimate data references

[GRIP] Dansgaard, W., S. J. Jonhsen, H. B. Clauson, D. Dahl-Jensen, N. S. Gundestrup, C. U. Hammer, C. S. Hvidberg, J. P. Steffensen, A. E. Sveinbjornsdottir, J. Jouzel, and G. Bond, Evidence for general instability in past climate from a 250-kyr ice-core record, Nature, 364, 218-220, 1993 [EPICA] Jouzel, J., et al. EPICA Dome C Ice Core 800KYr Deuterium Data and Temperature Estimates, IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series # 2007-091, NOAA/NCDC Paleoclimatology Program, Boulder CO, USA, 2007 [Huybers] Huybers, P., Glacial variability over the last two million years: an extended depth-derived age model, continuous obliquity pacing, and the Pleistocene progression, Quaternary Science Reviews, 26(1-2), 37-55, 2007 [Lohle] Lohle, C., A 2000-Year Global Temperature reconstruction based on non-treering proxies, Energy & Environment, 18(7-8), 1049-1058, 2007 [Moberg] Moberg A., D. Sonechkin, K. Holmgren, N. Datsenko and W. Karlén, Highly variable Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data, Nature, 433 (7026), 613 – 617, 2005 [Taylor Dome] Steig, E. J., D. L. Morse, E. D. Waddington, M. Stuiver, P. M. Grootes, P. A. Mayewski, M. S. Twickler and S. I. Whitlow, Wisconsinan and holocene climate history from an ice core at Taylor Dome, Western Ross Embayment, Antarctica, Geografiska Annaler: Series A, Physical Geography, 82, 213–235, 2000 [Veizer] Veizer, J., D. Ala, K. Azmy, P. Bruckschen, D. Buhl, F. Bruhn, G. A. F. Carden, A. Diener, S. Ebneth, Y. Godderis, T. Jasper, C. Korte, F. Pawellek, O. Podlaha and H. Strauss, 87Sr/86Sr, d13C and d18O evolution of Phanerozoic seawater, Chemical Geology, 161, 59-88, 2000 [Zachos] Zachos, J., M. Pagani, L. Sloan, E. Thomas and K. Billups, Trends, rhythms, and aberrations in global climate 65 Ma to present, Science, 292(5517), 686693, 2001


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