Fluctuations of water quality time series in rivers follow superstatistics

•Fluctuations of detrended water quality time series follow q-Gaussian distributions•Superstatistical long timescale is extracted from the data•New type superstatistics observed, which well fitted by mixture χ2-distributions Superstatistics a general method nonequilibrium statistical physics has been applied to variety complex systems, ranging hydrodynamic turbulence traffic delays and air pollution dynamics. Here, we investigate (such as dissolved oxygen concentrations electrical conductivity) measured in rivers provide evidence that they exhibit superstatistical behavior. Our main example recorded River Chess South East England. Specifically, use seasonal detrending empirical mode decomposition separate trends fluctuations for data. With either method, observe heavy-tailed fluctuation distributions, are described log-normal oxygen. Contrarily, find double peaked non-standard conductivity data, model using two combined χ2-distributions. 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Note also captured alpha stable (Shen 2015Shen Zhang Xu Meng Observation alpha-stable noise laser gyroscope data.IEEE Sensors 2015; 16: 1998-2003Crossref (20) so-called κ-distributions κ-distributions, used astrophysical plasmas, typical arising this context, whereas physics, one uses q-Gaussians (Tsallis, 2009Tsallis Introduction Nonextensive Statistical Mechanics: Approaching Complex World. Springer Science & Business Media, 2009Google q related κ κ=1/(q−1). Both approaches equivalent form standard examples generated (more general) approach. common feature real-world consist some long-term trend oscillation short-term fluctuations. Consider connected environment, ambient temperature: This will display strong cycles (Kumar 2009Kumar N. George Kumar Sajish P. Viyol Assessment spatial temporal tropical permanent estuarine system- tapi, west coast India.Appl. Ecol. Environ. 267-276Crossref Day-night add another oscillation, while global warming other influences, deforestation, might induce drift toward higher values. We can decompose full slower (trend) terms fluctuations, methods. particular, consider detrending, i.e., moving averages, via (EMD) (Wu Huang, 2009Wu Huang N.E. Ensemble emprical decomposition: noise-assisted method.Adv. Adaptive Data Anal. 01: 1-41https://doi.org/10.1142/S1793536909000047Crossref (5341) recently shown disentangle signals (Kampers 2020Kampers Wächter Hölling Lind P.G. Queirós S.M. Peinke Disentangling stochastic superposed short localized oscillations.Phys. 384: 126307Crossref (2) Naively, would expect so-extracted distributions. paper, analyze environmental Chess, river located England being actively monitored citizen science project (Heppell Treves, 2020Heppell Treves chess: storymap project.https://tinyurl.com/river-chessDate: 2020Google Key questions include how urban areas local sewage treatment works affect quality. Many quantities determine river. focus particular quantities: concentration Dissolved (or just “oxygen” large parts paper) highly aquatic life, fish, rivers. Meanwhile, (abbreviated “EC” “conductivity”) measures total solutes water. However, it impact humans, e.g., treated effluent fed into For current utilize available ChessWatch (Heppell, Chesswatch:a observatory chess.https://www.qmul.ac.uk/chesswatch/Date: four locations Blackwell Hall (BH) [Red], Little (LC) [Blue], Latimer Park (LP) [Green], Watercress Beds (WB) [Purple]. About twelve months collected within span June 2019 May 2020 evaluated here. LC BH upstream works, LP WB both downstream works. detailed discussion influence EC machine learning be predict understand trajectories found future paper 2021Schäfer, B., C., Rhys, Heppell, C.M. (2021). Spatio-temporal variations conductivity, assessed boosted trees shap, preparation.Google result behave way. structured follows. First, introduce discuss probability density functions (PDFs) EC. Next, subtracted reveal then continue present recap theory given series, specifically adapted our problem Finally, methods extract timescales microscopic parameter β function parameters. Overall, approximately superstatistics, new double-peaked β-distribution site. To obtain initial impression dynamics, visualize Figure 1. Disregarding peaks sites, certain (Figure 1A), winter spring than during summer. On shorter timescale, show obvious stations (Figures 1B 1D). Electrical measure Urban streams tend have mean major comparison their rural counterparts (Conway, 2007Conway T.M. Impervious surface indicator ph specific conductance urbanizing coastal zone New Jersey, USA.J. Manag. 85: 308-316Crossref (75) Rose, 2007Rose effects urbanization hydrochemistry base flow chattahoochee basin (Georgia, USA).J. Hydrol. 341: 42-54Crossref Peters, 2009Peters Effects stream city atlanta, Georgia, USA.Hydrol Process. 23: 2860-2878Crossref (54) arises combination diffuse sources. content health biota, low changes cause harm organisms living chalk (Arroita 2019Arroita Elosegi Jr., R.O. Twenty years metabolism riverine recovery following abatement.Limnol. Oceanogr. 64: S77-S92Crossref Rajwa-Kuligiewicz 2015Rajwa-Kuligiewicz Bialik R.J. Rowiński P.M. lowland over various timescales.J. Hydromech. 63: 353-363Crossref (50) Intriguingly, shows clear deviations Gaussianity, see PDFs 2. sites (red blue) tails. Still, portion variability due cycles, subtract before analysis. Instead its around respective trend. Detrending reduces allows weak stationarity thus allowing forecasting more precision (Contreras-Reyes Idrovo-Aguirre, 2020Contreras-Reyes J.E. Idrovo-Aguirre B.J. Backcasting cross-correlation analysis.Phys. 560: 125109Crossref (16) carry out first need trajectory F(t) (assuming additive model):F(t)=Trend(t)+Fluctuations(t).(Equation 1) achieve separation, employ methods: EMD. Seasonal applies average filtering f deviation between original classified Technically, implement python statsmodels.tsa.seasonal package (statsmodel, 2021statsmodelstatsmodels.tsa.seasonal.https://www.statsmodels.org/stable/generated/statsmodels.tsa.seasonal.seasonal_decompose.htmlDate: 2021Google apply = 6 hr. Alternatively, EMD splits ordered modes slowly changing oscillating modes. Similar Fourier analysis, summing modes, yields As pointed deterministic influences. do following. All hi(t) summed up follows:F(t)=∑i=1Nhi(t),(Equation 2) where N number Since lower numbered represent trend, keep but last m declare remaining i.e.Trend(t)=∑i=1N−mhi(t),(Equation 3) Fluctuations(t)=∑i=N−m+1Nhi(t).(Equation 4) PyEMD (Laszuk, 2017Laszuk Python implementation algorithm.https://github.com/laszukdawid/PyEMDDate: 2017Google chose 2 most cases. procedures demonstrated 3 measurement orange curves, corresponding 6h dropping describes well, preserving settings compromise barely capturing any (green curves) overfitting (essentially reproducing blue data). later study effect results systematically. separated let us now basic idea concept longer complicated indeed aggregation each giving rise simple, non-heavy-tailed distribution. types Scholar)–(Metzler, step T locally Assume know locally, slices, distributed. case, kurtosis snapshot should κGaussian=3. contrast, fully κ. T, test window sizes Δt compute follows:κ¯(Δt)=1tmax−Δt∫0tmax−Δtdt0⟨(u−u¯)4⟩t0,Δt⟨(u−u¯)2⟩t0,Δt2,(Equation 5) tmax length ⟨…⟩t0,Δt expectation slice starting t0. assumed κ¯(T)=κGaussian, windows κ¯(T)=3. After determining split several samples, thereby collection inverse variance β. If themselves χ2-distribution, written follows:f(β)=1Γ(n2)(n2β)n2βn2−1e−nβ2β0,(Equation 6) n degrees freedom β0 β; analytically 2001Beck Dynamical foundations nonextensive mechanics.Phys. 2001; 87: 180601Crossref χ2 obtained integrating (though often, good approximation, approximated q-Gaussian). said χ2, depending what actual is. was originally derived interpreted (Uchiyama kinetic system. But general, series. generic approximation PDFs, follows:p(q,b,μ)=bCq(1+(1−q)(−b(x−μ)2))11−q,(Equation 7) Cq normalization constant, μ shift parameter, shape known entropic index, b scale proportional ⟨β⟩ formed Equation (6). q→1, become 1/2b. specialized book applications q-statistics engineering, (Singh, 2016Singh V.P. Tsallis Entropy Water Engineering. CRC Press, 2016Crossref here may arise process if time-dependent deviation, displays term. While changes, τ gives system relax (local) equilibrium. It defined evaluating decaying autocorrelation c∼e(−t/τ). ensure always equilibrium, assume τ≪T hold. validate supplemental information. laid out, (oxygen conductivity). note leaves 4. could arise, ansatz: Let κ¯ κ¯(T)=κGauss=3, 5. site, investigating concentrations, TLC≈16 investigation Namely, above, (filtering omitted m) likely Hence, dependency 6. Apparently, scales linearly range. Then, increased too (e.g. > 4h oxygen), suddenly increases dramatically. explained follows: moderate tailed. attributed (large frequencies (high filter f), only tailed T) platykurtic behavior, < 3. Based seen here, confident attributing solid cases possible. special case require ≤ not included avoid sites. established, consistency checks: snapshots β-distribution. inspect According approach, Indeed, inspecting plots 7, order 10-100 hr 15 min resolution, contains ~ 100-1000 measurements. damping ratio hypothesis implies (fitted q-Gaussian-like exactly something very interesting: β-distributions alternatively 8), 9). single-peaked, peaks: One close zero values somewhat unusual encountered formalism. They analyzed Remember electric heavily influenced human outflow deeper reason behavior: single-peaked BW influence, site hint natural processes, interaction events flora fauna fluctuations.Figure 9The does χ2-distributionsShow caption(A) fit.(B) single χ2-distribution fitted. lines kernel estimates PDF.View Large Image ViewerDownload Hi-res image Download (PPT) (A) fit. (B) PDF. search suitable description conductivity. extension propose χ2-distributions:f(β)=Wfχ2(β,nχ1,β0)+(1−W)fχ2(β,nχ2,β0),(Equation 8) composed sum χ2-distributions, sharing (originally β) having own degree nχ1 nχ2. weighted weight constant W, ranges 0 excellent fit 9 Supplements further examples. rivers, consistent nonstationary consisting patches whole distinct functions. regardless (seasonal EMD) applied. Using kurtosis, determined lead linear scaling deduced finding robust respect method. Consistent assumptions, quite similar acceleration turbulence. An intriguing (contrary fluctuations) statistics, immediately existing theory. approximate results, still, points additional theoretical β-distributions. extended theory, χ2. Other possibilities extend bivariate (Caamaño-Carrillo 2020Caamaño-Carrillo Contreras-Reyes González-Navarrete Sánchez Bivariate based generalized gamma distribution.Eur. 93: 43Crossref requires illustrating generally homogeneous time. comparable 2013Yalcin Environmental 2013; 392: 5431-5452Crossref (13) tailed, followed Interestingly, tail limited: Regardless location, did 4). Contrary, especially qualitatively behavior 6), diverge doubled-peaked pronounced (upstream). indicate emerge (LP WB) activity. research necessary aspect. imply extreme compared presented means quantify this. compare comparing systematic quantitative way (Kumar, 2011Kumar R.N. assessment variation index sabarmati kharicut canal ahmedabad, Gujarat.Electron. Agric. Food Chem. 2011; 10: 2248-2261Google Moreover, view, desirable expand β-distributions, seem appear naturally context. Tabled 1REAGENT RESOURCESOURCEIDENTIFIERSoftwarePython3https://www.python.orgNumpyhttps://numpy.orgversion 1.20.0Scipyhttps://www.scipy.orgversion 1.6.0Seabornhttps://seaborn.pydata.org/version 0.11.1PyEMDhttps://pypi.org/project/EMD-signal/version 0.2.15 Open table tab Benjamin ( [email protected] ) unique reagents. code reproduce along (both cleaned detrened form) at: https://osf.io/mxcrv/ calculations manuscript were performed libraries referenced above. information body text OSF repository. received funding European Union's Horizon innovation program under Marie Sklodowska-Curie grant agreement No 840825 , Queen Mary University London Centre Public Engagement, Thames install sensors part project. work possible without sensor guardians who maintained throughout monitoring period landowners gave permission B.S. C.B. conceived research, figures, C.M.H. H.R. processed authors contributed writing interpreting results. no competing interests. .pdf (.21 MB) Help pdf files Document S1. Figures S1–S3