20 Years of Nonlinear Dynamics in Geosciences Rhodes, Greece 11-16 June 2006 Lessons from the long flow records of the Nile: Determinism vs. indeterminism and maximum entropy Demetris Koutsoyiannis Department of Water Resources, School of Civil Engineering, National Technical University of Athens, Greece Aris Georgakakos Georgia Water Resources Institute, School of Civil and Environmental Engineering, Georgia Institute of Technology, USA Data set 1: 160 (30) 140 Contemporary x x, Annual 120 record of 100 monthly flows at 80 60 Annual Aswan 40 30-year average Data period 1870- 20 40 2001 (131 years) (30) 35 September 3 x x, Units: km 30 25 20 15 10 5 6 General observations: (30) x , , 5 February Aperiodic fluctuations x on large time scales 4 Behaviours in 3 different months may 2 be like or unlike 1 0 1870 1890 1910 1930 1950 1970 1990 2010 Year D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 2 Data sets 2 and 3: records of annual maximum and minimum water levels at the Roda Nilometer Data period 640-1452 (813 years): the longest instrumental data set available worldwide – high importance in understanding and modelling hydroclimatic behaviours 10 (30) x , , 8 Units: m Minimum x 6 4 2 0 15 (30) x , , 13 x Maximum 11 9 7 Annual 5 30-year average General observation: 3 Aperiodic fluctuations 600 700 800 900 1000 1100 1200 1300 1400 1500 on large time scales Year AD D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 3 Part 1 Seeking determinism 40 Fifty years of monthly flow Analysis of the x i 30 monthly flows 20 10 0 0 100 200 300 400 500 600 i 50 γ j 600 terms of 30 autocovariance γj vs. 10 lag j -10 -30 0 100 200 300 400 500 600 j 100000 Power spectrum From data 1/12 10000 2/12 From autocovariance 3/12 1000 4/12 General observations: 100 5/12 6/12 10 1. All three plots s (ω k ) highlight the annual 1 cycle 0.1 2. No apparent over- 0.01 0.001 0.01 0.1 1 annual cycle ω k D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 5 Fifty4 years of monthly flow cyclically standardized Analysis of the x i ' 3 2 standardized 1 0 monthly flows -1 -2 -3 0 100 200 300 400 500 600 i 1 General observations: γ j 0.8 600 terms of 1. The annual cycle 0.6 autocovariance γ vs. lag j cannot be ‘removed’ 0.4 j by standardizing the 0.2 0 monthly time series -0.2 2. ‘Linear’ logics 0 100 200 300 400 500 600 j implying additivity of 1000 Power spectrum the type 1/12 100 ‘periodic + aperiodic’ 10 2/12 or ‘signal + noise’ s (ω k ) may be inappropriate 1 0.1 for natural processes From data 0.01 3. No apparent over- From autocovariance annual cycle 0.001 0.0001 0.001 0.01 0.1 ω k 1 D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 6 Seeking over-annual cycles Several studies have claimed 4 .2 2 .2 7 1 2 1 9 6 4 to have detected several 2 5 6 over-annual periods from Study 1 the Nilometer time series Study 2 Here are two recent 5 1 3 2 1 7 6 examples 2 5 6 1 10 100 1000 Period (years) Authors’ opinion: Such claims may be suspicious for several reasons 1. The plethora of periods, which are numbers asymmetric to each other, may rather indicate a stochastic behaviour (all frequencies are significant) 2. Some studies test the significance of detected periods against white noise; apparently, a white noise hypothesis is totally inconsistent with the Nile behaviour 3. Some studies may have undervalued the estimation uncertainty in stochastic processes with high autocorrelation D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 7 10000 Seeking over- annual cycles: 1000 s (ω k ) Aswan and 100 Nilometer annual 10 Entire record First half Second half Aswan, annual series 1 0.01 0.1 1 ω A simple methodology: 100 k A ‘real’ peak in the 10 power spectrum (manifesting s (ω k ) 1 determinism rather than 0.1 a random effect) will appear also at the same 0.01 Nilometer, minimum frequency if we split the 0.001 sample in two halves or 100 in three tertiaries 10 s (ω k ) 1 Conclusion: 0.1 No over-annual cycle 0.01 appears in any of the Nilometer, maximum three time series 0.001 0.001 0.01 0.1 1 ω k D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 8 Algorithmic details in seeking over-annual cycles 1. We split the sample into two halves; if the sample size n is not even we omit one value so that the half-sample size ν = n/2 is integral 2. We estimate the periodograms of the full time series and its two halves at frequencies ωk = k/ν for integral k such that 1/ν ≤ ωk ≤ 1/2 3. For each ωk we calculate a likelihood measure of a ‘real’ peak η(ωk) as the number of peaks in the three periodograms divided by 3 (a real peak should have η(ωk) = 1) 4. We repeat (analogously) steps 1-3 splitting the sample into three tertiaries (here ν = n/3) 5. We plot η(ωk) vs. ωk for the two cases (halves, tertiaries) and locate frequencies ω in whose neighborhood η(ω) = 1 in both cases 6. For these frequencies, we inspect in the periodogram of the full series whether or not the peak is higher than neighbouring peaks; if yes we can say that the identified ω manifest determinism rather than a random effect 1 η (ω k ) 0.75 Power spectrum peaks 0.5 plot, Nilometer, maximum 0.25 0 0 0.1 0.2 0.3 0.4 0.5 ω k D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 9 Seeking a chaotic attractor Natural monthly flows of Standardized monthly Nile, Time lag 1 flows of Nile, Time lag 6 ) 8 ) 8 1 2 3 4 m m 1 2 3 4 , , 7 ε ε 7 5 6 7 8 ( ( 5 6 7 8 2 2 d d 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 ε ε Notation: ε = scale length; m = embedding dimensions (from 1 to 8 as indicated in the legends); d2(ε, m) = local slope of correlation sums Conclusion: No low-dimensional determinism D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 10 Part 2 Stochastic description Multi-scale setting of stochastic analysis A process at continuous time t X(t) i k (k) 1 ⌠ The averaged process at a scale k X := ⌡ X(t) dt i k (i – 1) k Properties of the process at an arbitrary observation scale k = 1 (e.g. annual) Standard deviation σ ≡ σ(1) (1) Autocorrelation function (for lag j) ρj ≡ ρj Power spectrum (for frequency ω) s(ω) ≡ s(1)(ω) Can be derived analytically from Properties of the process at any other scale those at scale k = 1 – depend on ρj Specific properties at any scale of a simple scaling stochastic process (SSS - fractional Gaussian noise) with Hurst exponent H Standard deviation σ(k) = kH – 1 σ (0.5 < H <1) (k) 2H – 2 Autocorrelation function (for lag j) ρj = ρj ≈ H (2 H – 1) |j| Power spectrum (for frequency ω) s(k)(ω) ≈ 4 (1 – H) σ2 k 2H – 2 (2 ω)1 – 2 H D. Koutsoyiannis and A. Georgakakos, Lessons from the long flow records of the Nile 12 Manifestation of aperiodic fluctuations into the statistics of the time series 3 3 Month µ (km ) σ (km ) Cs Ck τ3 τ4 Η ρFGN1 ρ1 ρ2 ρ12 Aug 19.03 4.38 -0.06 -0.28 -0.01 0.11 0.74 0.40 0.58 0.06 0.17 Sep 21.55 5.07 -0.21 -0.18 -0.03 0.11 0.81 0.53 0.74 0.24 0.25 Oct 14.58 4.27 0.17 -0.24 0.01 0.11 0.91 0.77 0.85 0.61 0.50 Nov 8.16 2.31 0.64 -0.14 0.14 0.09 0.85 0.62 0.86 0.72 0.33 Dec 5.57 1.40 1.21 1.91 0.23 0.20 0.90 0.74 0.90 0.75 0.46 Jan 4.33 1.03 0.74 1.05 0.14 0.19 0.87 0.68 0.90 0.75 0.44 Feb 3.17 0.91 0.64 0.53 0.11 0.13 0.82 0.56 0.92 0.76 0.40 Mar 2.76 0.87 0.79 1.25 0.10 0.14 0.83 0.59 0.90 0.75 0.41 Apr 2.52 1.05 0.43 -0.87 0.12 0.01 0.93 0.82 0.80 0.62 0.73 May 2.33 1.01 0.46 -1.11 0.14 -0.02 0.94 0.83 0.94 0.73 0.79 Jun 2.24 0.82 0.82 0.69 0.15 0.07 0.88 0.69 0.78 0.69 0.46 Jul 5.26 1.76 0.74 0.58 0.13 0.13 0.83 0.59 0.57 0.35 0.39 Average 0.53 0.27 0.10 0.10 0.86 0.65 0.81 0.59 0.44 Annual 91.51 18.38 0.48 0.23 0.11 0.13 0.83 0.58 0.32 0.32 Series µ (m) σ (m) Cs Ck τ3 τ4 Η ρFGN1 ρFGN2 ρ1 ρ2 Minima 2.70 0.96 0.70 3.20 0.05 0.16 0.84 0.60 0.46 0.46 0.38 Maxima 9.27 0.75 -0.13 6.24 -0.01 0.22 0.84 0.59 0.45 0.43 0.36 High autocorrelation (ρ) – High Hurst coefficient (H > 0.5) D.