HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-5547-2017Analysis and modelling of a 9.3 kyr palaeoflood record: correlations,
clustering, and cyclesWittAnnetteannette.witt@ds.mpg.deMalamudBruce D.bruce.malamud@kcl.ac.ukhttps://orcid.org/0000-0001-8164-4825MangiliClaraBrauerAchimMax Planck Institute for Dynamics and Self-Organisation,
Göttingen, GermanyDepartment of Geography, King's College London, London, UKGFZ German Research Centre for Geosciences, Potsdam, Germanynow at: Section of Earth and Environmental Sciences, University of
Geneva, Geneva, SwitzerlandAnnette Witt (annette.witt@ds.mpg.de) and Bruce D. Malamud
(bruce.malamud@kcl.ac.uk)14November20172111554755818September201621October20162July201717July2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/5547/2017/hess-21-5547-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/5547/2017/hess-21-5547-2017.pdf
In this paper, we present a unique 9.5 m palaeo-lacustrine record of 771
palaeofloods which occurred over a period of 9.3 kyr in the
Piànico–Sèllere Basin (southern Alps) during an interglacial period
in the Pleistocene (sometime from 780 to 393 ka) and analyse its
correlation, clustering, and cyclicity properties. We first examine
correlations, by applying power-spectral analysis and detrended fluctuation
analysis (DFA) to a time series of the number of floods per decade, and find
weak long-range persistence: a power-spectral exponent
βPS≈ 0.39 and an equivalent power-spectral
exponent from DFA, βDFA≈ 0.25. We then examine
clustering using the one-point probability distribution of the inter-flood
intervals and find that the palaeofloods cluster in time as they are Weibull
distributed with a shape parameter kW= 0.78. We then examine
cyclicity in the time series of number of palaeofloods per year, and find a
period of about 2030 years. Using these characterizations of the correlation,
clustering, and cyclicity in the original palaeoflood time series, we create
a model consisting of the superposition of a fractional Gaussian noise (FGN)
with a 2030-year periodic component and then peaks over threshold (POT)
applied. We use this POTFGN+Period model to create 2 600 000
synthetic realizations of the same length as our original palaeoflood time
series, but with varying intensity of periodicity and persistence, and find
optimized model parameters that are congruent with our original palaeoflood
series. We create long realizations of our optimized palaeoflood model, and
find a high temporal variability of the flood frequency, which can take
values of between 0 and > 30 floods century-1. Finally, we
show the practical utility of our optimized model realizations to calculate
the uncertainty of the forecasted number of floods per century with the
number of floods in the preceding century. A key finding of our paper is that
neither fractional noise behaviour nor cyclicity is sufficient to model
frequency fluctuations of our large and continuous palaeoflood record, but
rather a model based on both fractional noise superimposed with a long-range
periodicity is necessary.
Introduction
Risk estimates of floods are often based on instrumental records that cover a
comparable small time period (e.g. a maximum of 150 years) and might be
influenced by anthropogenic activities. Therefore, palaeoflood sequences
represent a promising source for understanding the long-term hazard dynamics
of the unperturbed climate system (e.g. Kochel and Baker, 1982; Stedinger and
Cohn, 1986; Baker, 1987, 2006; Ely et al., 1993; O'Connor et al., 1994; Knox,
2000; Yang et al., 2000; Greenbaum et al., 2000; Redmond et al., 2002; Benito
et al., 2004; Czymzik et al., 2010; Huang et al., 2013; Swierczynski et al.,
2013; Wirth et al., 2013). One important aspect of these dynamics is to
understand to what extent flood frequencies can be considered constant over
time, and more specifically, whether floods in a given geographic region have
any correlations with themselves and whether the floods are clustered in
time. Here we examine correlations, clustering, and cycles in a 9336-year
record of palaeofloods using varved (annual) sediments from palaeolake
Piànico–Sèllere (northern Italy), and construct a model to take into
account the observed behaviour. In this introduction, we present the idea of
correlations and clustering in time series, and then
summarize the overall organization of this paper.
We first consider correlations in a time series. Consider an
uncorrelated time series (e.g. a white noise), where values are independent
of one another, i.e. it is equally likely at each time step to have values
above or below the median value. To illustrate correlations, take a flood
intensity time series, with flood intensity the number of floods per year; a
“flood” here might be defined in many different ways. If the flood
intensity time series is uncorrelated, when there is a year with a large
intensity (a large number of floods occurring, above the median number of
floods), it is equally probable to have the next year a number of floods that
is above or below the median, i.e. a flood intensity value that is
“large” (more floods) or “small” (zero or few floods). In contrast, for
a flood intensity time series that exhibits positive correlations, adjacent
values will have flood intensities that are on average closer to each other
(in intensity) than for an uncorrelated time series; large values tend to
follow large ones, and small follow small. Temporal correlations are also
referred to as persistence or memory (see Witt and Malamud, 2013, and
references therein). Two examples of positive correlations in time series are
given in Fig. 1a and b, using a tree ring standardized growth index for
Bristlecone pine, White Mountain, California, USA, for the years AD 0–1962
and cosmic ray neutron counts per hour, Beijing, China, 1 January to
11 March 2008. In these two Fig. 1 panels, successive values in both series
are positively correlated with one another, and are examples of persistent
time series.
Example of correlations, clustering, and cyclicity (a and
b following Witt et al., 2010). (a) Tree ring standardized
growth index for Bristlecone pine, White Mountain, California, USA, for the
years AD 0–1962 (Ferguson et al., 1994). (b) Cosmic ray neutron
counts per hour, Beijing, China, 1 January to 11 March 2008 (NGDC, 2008). In
(a) and (b) successive values in each both series are
positively correlated with one another, and are examples of persistent time
series. (c) Maximum daily discharge, Q, for station 05474500 on
the Mississippi River at Keokuk, Iowa, for 116 water years, 1878–1993 with
data from Slack and Landwehr (1992) and USGS (2017) and described in Malamud
et al. (1996). (d) Partial-duration flood series, where floods are
the largest 116 maximum daily discharge from (c) over the 116-year
period, with daily discharges separated by at least 30 days to be considered
a flood. In our partial duration series, 48 of the 116 years have 0 floods,
33 years have 1 flood, 26 years have 2 floods, and 6/2/1 years have
3/4/5 floods. The value of these maximum daily discharges for each flood
event is projected to the x-axis (daily
discharge = 0 m3 s-1). Below this is shown (green dots) all
events along one line, an example of a data series that is strongly clustered
in time. Clustering is due to (i) seasonal effects (cyclicity) and
(ii) longer-term cycles.
Correlations can be both short-range (where only values in a time series
close to each other are correlated) or long-range (where all values in the
time series are correlated with one another). We will find that the unequally
spaced palaeoflood time series used in this paper exhibit both long-range
correlations and long period cyclical behaviour, and will focus on long-range
(vs. short-range) persistent models in this paper. Long-range correlations
have been discussed and documented for many processes in the environmental
and Earth Sciences (see Box et al., 2013; Witt and Malamud, 2013, and
references therein), with examples including river run-off and precipitation
(Hurst, 1951; Mandelbrot and van Ness, 1968; Montanari et al., 1996;
Koscielny–Bunde et al., 2006; Mudelsee, 2007; Khaliq et al., 2009; Ghil et
al., 2011), atmospheric variability (Govindan et al., 2004) and temperatures
over short- to very-long timescales (Pelletier and Turcotte, 1997; Fraedrich
and Blender, 2003). The idea of long-range correlations is commonly used to
characterize observational and experimental measurement data across many
other scientific disciplines: For instance Peng et al. (1993b, 1994) found
long-range persistence in the nucleotide organization of DNA sequences,
Kurths et al. (1995) detected long-range correlations in time series from
radio astronomy, Peng et al. (1993a) and Penzel et al. (2003) analysed
long-term recordings of heart rate variability.
The palaeoflood time series we will examine in this paper is unequally
spaced in time. Some examples given above include unequally spaced time
series, such as Ghil et al. (2011), who studied long-range persistence of the
unequally spaced Nile River time series. The palaeoflood time series we
examine is also integer valued, i.e. is not a continuous one-point
probability distribution. Some long-range persistence research has
investigated integer-valued time series. For example, the example given above
for Peng et al. (1993b, 1994) research on long-range persistence of DNA's
nucleotide organization represented the DNA as sequences with four different
types of symbols. In another two examples, Altmann et al. (2012) studied
correlations in texts as represented by binary sequences and
Schaigorodsky et al. (2014) investigate long-range memory in the opening
moves of chess games.
An alternative to examining the temporal correlations of flood
intensities (here taken as the number of floods per year) is to examine
whether the flood intensities over a given threshold cluster in
time. Clustering is the grouping of values in time more than one would expect
if the process that created them were random. An example of clustering, in
unequally spaced flood magnitudes over a given threshold, is given in
Fig. 1c and d. Here we observe the clustering on an annual cycle.
Correlation and clustering (or lack thereof) have been studied for different
extreme natural events, including earthquakes (e.g. Livina et al., 2005;
Hainzl et al., 2006; Lennartz et al., 2008; Davidsen and Kwiatek, 2013),
volcanic eruptions (e.g. Nathenson, 2001), floods (e.g. Pelletier and
Turcotte, 1997; Milly and Wetherald, 2002), and tropical temperatures (e.g.
Blender et al., 2008).
In this paper, we use a record of 771 detrital layers of the laminated
sediments of palaeolake Piànico–Sèllere (northern Italy) covering a
time span of 9336 years. These 771 layers are interpreted as indictors of
flood events which occurred somewhere during the period of 780 to 393 ka
(thousands of years ago). In Sect. 2 we give a detailed description of the
palaeoflood sequence. In Sect. 3 we explain key definitions and methods
used in this paper. Then in Sect. 4 we analyse the distribution of the
palaeoflood frequencies per year, decade, and century compared to a Poisson
(stationary) model, the one-point probability distribution of the inter-flood
intervals as a potential indicator of clustering, the temporal correlations
among the flood events, and cyclicity. In Sect. 5 we model the record of
palaeoflood timings as elements of a long-range correlated synthetic time
series that exceed a threshold. For this model type, scaling relationships
for the distribution of the interevent intervals and the temporal
correlations are derived in dependence on correlation properties of the time
series and on the applied threshold. In Sect. 5 we also construct a model
that consists of multiple realizations of synthetic time series that captures
the same clustering and correlation properties that we find in our
palaeoflood record. Finally, in Sect. 6 we provide a summary of the paper.
Temporal succession, histograms, and autocorrelation of the detrital
layers of the 9336-year Piànico–Sèllere, Italy, palaeoflood record (data available at Mangili et al., 2017).
The number of observed detrital (flood) layers per year (blue dots, 0≤nyear≤3 floods yr-1), decade (orange dots, 0≤ndecade≤8 floods decade-1), and century (green dots,
0≤ncentury≤31 floods century-1) are presented over
the 9336 years of the record examined. The x-axis represents relative age,
with t=1 year the most recent varve and increasing values indicating further
back in time. The grey bar represents a sediment gap of 65 years (leaving
9271 years of the record examined over the 9336 years). In addition, for each
time resolution is given (upper right) the coefficient of variation
cv=σ/μ, where σ is the standard deviation and
μ is the mean of the given time series. Shown to the right of each
palaeoflood time series is the respective histogram of the number of floods
per year (blue bars), decade (orange bars), and century (green bars).
Data
As discussed in the introduction, this paper focuses on the correlation,
clustering, and cyclicity of palaeofloods. We present here for the first time
and use a very comprehensive flood record at sub-annual resolution, obtained
from the varved sediments of the Piànico–Sèllere Basin, located in
the Borlezza Valley (Province of Bergamo, Italy; Fig. 2a). These data have been lodged online at the World Data Centre PANGEA (Mangili et al., 2017). This palaeolake
sequence was first described in the mid 1800s (e.g. Stoppani, 1857), mainly
for its fossil content. A detailed stratigraphic study was published by
Moscariello et al. (2000).
Palaeolake Piànico–Sèllere is located at the foothills of the
southern Alps in Italy. Its sediments (45∘48′ N,
10∘2′ E, 280–350 m a.s.l.) are visible in outcrops (Fig. 2b).
The sediment formation is more than 48 m thick and extends for about 600 m
laterally. It includes four fine-grained laminated stratigraphic units
(described more fully below). The size of the palaeolake has been
reconstructed to about 3 km in length and 500–800 m in width (Casati,
1968). The palaeocatchment area of Piànico–Sèllere Lake has been
estimated to be less than 13 km2 (Moscariello et al., 2000).
The lacustrine sequence that forms the Piànico Formation (Moscariello et
al., 2000) is a 48 m thick stratigraphic interval that includes four units.
For the palaeoflood data set created here, only the 9.5 m unit called BVC
(Banco Varvato Carbonatico or carbonate varved bed) will be
considered (described more fully below). The age of the sediment is still
debated. Tephrochronological dating of the sequence gives 393 ± 12 ka
(Brauer et al., 2007), which corresponds to the interglacial period at about
400 ka, i.e. Marine Isotope Stage (MIS) 11. This interglacial period is
considered the best analogue to the Holocene because of similar orbital
parameters (e.g. Loutre and Berger, 2003). However, Pinti et al. (2001) dated
a tephra layer in the varved BVC sequence at 779 ± 13 ka, assigning
the sequence to the interglacial at about 780 ka (MIS 19). For the purposes
of the paper here, we will be less concerned with the actual age and more
concerned with the period of overall time that has elapsed, based on
interpreting the BVC alternating layers (rhythmites) as varves.
The BVC 9.5 m unit constitutes an almost continuous succession of about
15 500 rhythmites (alternating layers of dark and light sediment),
0.2–0.8 mm thick (Moscariello et al., 2000). These endogenic calcite
rhythmites have been interpreted to be varves (annual cycles) as they have a
structure very similar to Holocene Alpine lake varves, in which calcite
precipitation takes place in spring and summer (Kelts and Hsü, 1978;
Moscariello et al., 2000). The Piànico–Sèllere varved sequence
formed under interglacial conditions, as testified by the flora remains
included in the sediments (e.g. Amsler, 1900; Maffei, 1924; Rossi, 2003;
Martinetto, 2009) and the oxygen stable isotope values of this interval
(Mangili et al., 2007). The calcite varves consist of two layers (Fig. 2b):
a lightly coloured and ∼ 0.5 mm thick spring/summer layer formed by up
to 96 % of endogenic calcite, and a dark and thin winter layer
constituted of organic remains, diatom frustules, and occasional detrital
grains (Moscariello et al., 2000; Mangili et al., 2005). Using wavelet
analysis, Brauer et al. (2008) found that the varve thickness is partially
modulated by solar activity (88 and 208-year cycles), and probably also by
the thermohaline ocean circulation (512-year cycle). In the BVC stratigraphic
interval, the upper 60 % (9271 out of 15 500 varves) were examined here
in detail using the same methodology and location as described in Mangili et
al. (2005), but extending the vertical profile from 896 varves (Mangili et
al., 2005) to 9271 varves (here). Note that at varve 4019 of 9271, there was
found to be a “gap” of time, consisting of 65 missing varves, bringing the
total sequence to 9336 varves (with 65 varves missing); this gap is described
below in much more detail.
We briefly describe our microfacies sampling methodology (see Mangili et al.,
2005 for further details). Continuous vertical profiles of sediment samples
were collected from two outcrops stratigraphically similar, 150 m apart:
(i) the Main Section, for which data are presented in this paper,
and (ii) the Wall Section, a control section which we used to
evaluate uncertainties in the Main Section results. Both sections had more
than 30 key marker layers so the two outcrops could be correlated with each
other. For each vertical section, the outcrops were cleaned with a sharp
knife until a smooth and vertical surface was obtained. Then a block of
sediment was carved out in situ to enable easy pushing of a special
stainless box (33 cm × 5 cm) with removable side walls onto the
sample. Samples were taken with at least a 5 cm vertical overlap, ensuring
that a marker layer, allowing correlation, was present in both samples. The
samples were then slowly dried at room temperature to avoid shrinkage and
cracking and covered with a transparent epoxy resin, resulting in resin
impregnation of the surface layer (1–2 mm) of the sediments. Samples were
cut into two halves with the fresh surface again carefully dried and
impregnated. From one half, 10 cm thick samples with a 4 cm overlap for
final thin-section (120 mm × 35 mm) preparation were cut out. The
thin sections were analysed with a petrographic microscope. For measurement
of varve and detrital layer thickness, 100× magnification was chosen.
Abbreviations and notation.
AbbreviationDescriptionBVCBanco Varvato Carbonatico (carbonated varved bed)DFADetrended fluctuation analysisFGNFractional Gaussian noiseMISMarine isotope stageMLEMaximum likelihood estimationIEOTInterevent occurrence timePOTPeaks over thresholdPSPower spectrum or power-spectral analysisSymbolDescriptionUnitsαPower-law exponent of the fluctuation function, with βDFA=2α-1.(unitless)βDFAStrength of long-range persistence (based on the power-law exponent α of the fluctuation function)(unitless)βmodelStrength of long-range persistence (model parameter)(unitless)βPSStrength of long-range persistence (based on the power-law exponentof the power-spectral density)(unitless)γPower-law exponent of the autocorrelation function(unitless)Δ,ΔjInterevent occurrence time plotted on the x-axis in real time (Δ) and in natural time (Δj)(years)Δ‾, Δ‾jMean interevent occurrence time Δ‾=Δ‾j(years)θ, θ12Threshold (no subscript: general; subscript 12: separates one and two floods per time unit)(unitless)Λ(t)Time-dependent flood rate(floods yr-1)Λmodel(t)Time-dependent flood rate of model realizations(floods yr-1)Λ‾model(t)Mean of time-dependent flood rate taken over many model realizations(floods yr-1)λRate parameter of the Poisson distribution(unitless)λyear, λdecade, λcenturyFlood rate per year, per decade, per century(floods yr-1,floods decade-1,floods century-1)λWScale parameter of the Weibull distribution(unitless)μMean of the values considered(variable dependent)σ, σ2Standard deviation, variance(variable dependent)τTime lag in real time(years, decades, centuries)τΔTime lag in natural time (of the interevent occurrence time series Δj, j=1, 2, …, N)(unitless)AmodelAmplitude of the periodic component (model parameter)(unitless)ArateAmplitude of the periodic component of the time-dependent flood rate fit to the palaeoflood time series.(unitless)a1, a2, a3Parameters for the modelling the cyclicity of the annual flood series(floods yr-1)C(τ), C(τΔ)Autocorrelation function, depends on the lags τ and τΔ(variable dependent)cvCoefficient of variation (cv=σ/μ)(unitless)FFluctuation function(unitless)jIndex of natural time, 1≤j≤NΔ(unitless)kInteger-valued variable(unitless)kyear, kdecade, kcenturyNumber of occurrences of nyear, ndecade, ncentury with a specific value during a given period of time considered. For example, if there are 80 times that nyear=3 floods yr-1 over 9.3 kyr, then kyear=80 years (at nyear=3 floods yr-1).(years, decades, centuries)kWShape parameter of the Weibull distribution(unitless)lInteger-valued variable(unitless)nIntensity (no. of occurrences) of the random variable X (in Eq. 1)(unitless)nyear, ndecade, ncenturyNumber of detritic layers (floods) per varve (i.e. per year), per 10 varves (i.e. per decade), and per 100 varves (i.e. per century).(floods yr-1,floods decade-1,floods century-1)
Continued.
SymbolDescriptionUnitsNfloodsTotal number of floods (detritic layers) (Nfloods=771 floods)(floods)N, Nyear, Ndecade, NcenturyTotal number of years/decades/centuries where varves are presentin the palaeoflood data set (Nyear=9271 years, Ndecade=925 decades, Ncentury=92 centuries) with N the place holder(years, decades, centuries)NvarvesTotal number of varves (9336 varves)(varves)NΔTotal number of interevent occurrence times (NΔ=739)(unitless)PexpOne-point probability density distribution of theexponential distribution(unitless)PλOne-point probability density distribution of the Poisson distribution(unitless)PWOne-point probability density distribution of the Weibull distribution(unitless)S(f)Power-spectral density depending on the frequency f(variable dependent)st, t=1, …, NRunning sum(variable dependent)t=1, 2, …, NRelative varve index or time index for our palaeoflood time series. Note that the index “skips over” the 65-year gap, so for N=Nyear it goesfrom t=1 to 9271 years.(years)XDiscrete random variable in Eq. (1)(unitless)xyear(t), xdecade(t), xcentury(t)Number of floods per year, decade, century dependingon time t (model output)(floods yr-1,floods decade-1, floods century-1)y(t)Model input depending on time t(unitless)
Approximately 8 % of the varve couplets contain also one or two detrital
layers (Fig. 2b), which we describe in more detail below. The detritus is
mainly constituted of fine, silt-size Triassic dolomite from the catchment
(Mangili et al., 2005). Only in the thickest described layers are found
fine-sand sized particles in minor amounts. No detrital layers contain
gravel. Due to their composition and grain size, these detrital layers can
easily be distinguished from the background endogenic sedimentation of the
lake. These detrital layers are considered to be the result of channelized
streamflow that originated in the hills surrounding the
Piànico–Sèllere Basin and triggered by extreme precipitation events
(Mangili et al., 2005). The site of the outcrops from which the samples have
been taken have been explicitly selected to avoid gravel sediments which
could cause hiatuses through erosion. We interpret the site to be a
low-energy sedimentary environment in a distal position of the inflowing
water. The position of a detrital layer within a varve allows the
identification of the season in which the extreme precipitation event/flood
took place: a “spring” detrital layer settled before the beginning of
endogenic calcite precipitation, a “summer” detrital layer is within the
varve summer layer and a “fall/winter” detrital layer is at the top of the
calcite layer or included in the winter layer (Mangili et al., 2005). Due to
the reduced thickness of the varve winter layer (0.06 mm mean), fall and
winter detrital layers are not distinguished here. Another layer type (called
matrix-supported) has also been observed in the varved sediments (Mangili et
al., 2005); these layers are thought to result from reworking processes
within the lake, are not linked to extreme precipitation events and will,
therefore, not be taken into consideration in this paper.
In our analysis, we will focus on the temporal succession of detrital layers
(flood events), the series of events that is graphically presented in
Fig. 2c, and which consists of 771 single palaeoflood events occurring
during a time interval of T= 9336 years. There are 9271 varves present,
each taken to represent 1 year, in addition to a “gap” of 65 years (varve
index 4019–4083) which is due to a slump at the Main Section. The varve
structure there was not recognizable. To ascertain the temporal period of the
gap, we correlated three marker layers between the Main Section's disturbed
interval with an undisturbed sequence at the Wall Section, where we counted
the number of varves, and were able to conclude that the gap in the Main
Section is 65 years. The flood events have each been given a specific varve
index (relative age) in the time series. In this paper, we will use the
following notation (see also Table 1 for a list of all abbreviations and
notations used in this paper) for the observed data with respect to varves
and detrital (flood) layers: nyear(t), t= 1, …,
Nvarves is the number of detrital layers per varve (flood events
per year) where the index t is the varve index (starting from the top
varve) and represents relative age in years (see above), and
Nvarves= 9336 years is the length of the observational period,
i.e. the total number of varves including the “gap” of 65 years.
The timings of the considered 771 palaeoflood events are transformed into
three event series (Fig. 3), the number of floods (detrital layers) per
year (nyear), per decade (ndecade), and per century
(ncentury), over the 9271 years of palaeoflood data (over a
9336-year record). These three data sets are integer-valued time series which
contain Nyear= 9271, Ndecade= 925, and
Ncentury= 92 data points, and will provide the basis for our
time series analyses in Sect. 4. In these three palaeoflood time series
(Fig. 3) which represent our data, we observe that there are time periods
with little fluctuation of the number of floods per century (i.e. the 24th to
18th centuries with values over the range 10–16 floods century-1).
However, we also see sudden transitions from many to a few floods (as from 24
to 4 floods century-1 from the 39th to 38th centuries) as well as from a
few to many floods (as from 7 to 25 floods century-1 from the 46th to
45th centuries).
The histograms of these data are also given in Fig. 3, on the far right of
each time series. The number of years (kyear) with no floods
(nyear= 0 floods yr-1) is kyear= 8530 years
(92.0 % of the record). Similarly, for nyear= 1, 2, 3
floods yr-1, the number of years (respectively) with those values are
kyear= 712 (7.7 %), 28 (0.3 %), and 1 (0.01 %)
years. For Nyear= 9271 years examined, there are 741 years
which have 1 to 3 floods (detrital layers), a total of 771 floods. We will
revisit these values in Sect. 4.1. The data cover Ndecade= 925 non-overlapping decades (because of the 65-year gap, 8 of the decades
within the 9336 years are disrupted). The number of decades
(kdecade) with no floods (ndecade= 0 floods
decade-1) is kdecade= 453 decades (49.0 % of the
record). Similarly, for ndecade= 1, 2, 3 floods decade-1,
kdecade= 283 (30.6 %), 123 (13.3 %), and 43 (4.6 %)
decades. For 4 ≤ndecade≤ 8 floods decade-1,
kdecade= 23 (2.5 %) decades. No decades have
ndecade > 8 floods decade-1. If we consider the
data on the century resolution, we have Ncentury= 92
non-overlapping centuries, as the 65-year gap falls completely within
1 century. The number of centuries (kcentury) with no floods
(ncentury= 0 floods century-1) is kcentury= 1 century (1 % of the record). For 1 ≤ncentury≤ 5
and 6 ≤ncentury≤ 10 floods century-1,
kcentury= 26 and 35 centuries (28 and 38 % of the record)
respectively. For ncentury > 10 floods
century-1, kcentury then decreases (Fig. 3), with a
maximum value of ncentury= 31 floods
century-1 which is reached once (i.e. kcentury= 1 century).
Data series of detrital layers (floods) and interevent occurrence
times (IEOTs) from the Piànico–Sèllere, Italy, palaeoflood sequence.
(a) In panel (a.1) are shown the number of detrital layers
(floods) per varve, nyear, given as a function of varve index t=1 to 9336 years (blue dots – data available at Mangili et al., 2017) as shown in Fig. 3. In panel (a.2)
a 100-year portion of (A.1) (from t=8090 to 8190 years) is expanded with
an illustration of 4 flood years that have detrital layers (floods) in them,
and the interevent occurrence times Δ between them. These 4 flood
years, each with nyear≥1 flood in them, appear as yellow
symbols in all panels. (b) Interevent occurrence times, Δ,
temporally located (i.e. as a function of the relative age) when they
occurred. (c) Interevent occurrence times, Δj, plotted as
a function of “natural” time, where each Δ is no longer represented
temporally when it occurs, but rather successively one after another, j=1, 2, 3, …, 741; interevent occurrence times of Δj=0 are
not shown. The detrital varve index j includes only those years (varves)
where there are nyear≥1 floods yr-1 (i.e. at least one
detrital layer in that year).
The thicknesses of the detrital layers representing the 771 flood events that
we investigate in this paper range from 0.02 to 23.00 mm (with quartiles of
25 %: 0.07 mm, 50 %: 0.10 mm, and 75 %: 0.18 mm) (see Appendix
Fig. A1). Although detrital layers, which are mainly constituted of
silt-sized dolomite, indicate the occurrence (season and year) of an extreme
precipitation event (flood), the thickness is not taken as representative of
the magnitude (e.g. peak discharge or flow velocity) of the event. This is
because the thickness of a layer can also be influenced by the distance
between the source of the detritus and the studied section and the amount of
detritus available at the time of the extreme precipitation and weaker vs.
stronger rainfall during the extreme precipitation events (Kämpf et al.,
2012, 2015). In addition, over the 9336-year period we investigate, the
source of given rainfall events might result in different thickness
signatures (Swierczynski et al., 2012, 2013). We also note that although we
can identify when a flood is thought to have occurred, this does not preclude
the possibility that other floods occurred but were not identified as a
detrital layer. We do acknowledge research (e.g. Corella et al., 2014;
Schillereff et al., 2014) where the type and thickness of detritus in
Holocene varves are used to make interpretations of the strength of a
hydro-meteorological event; however, due to the age of our sediments this
sort of analysis of event magnitude was not possible here. In this paper, we
therefore focus on just the temporal attributes of the 771 floods themselves
over this 9336-year period in the Pleistocene. See data availability for
access to the palaeoflood database.
In terms of the relative temporal uncertainty of the floods (based on the
varve sequence), a flood in a given year (represented by the detrital layers)
can (very rarely) erode some of the sediment underneath it, forming a hiatus
of missing sediment. This limitation is discussed in detail in Mangili et
al. (2005), who analysed, for 10 % of the outcrop presented in this
paper, the same stratigraphic intervals in both the Main Section and the Wall
Section, again 150 m apart. Out of the 896 varves they examined in both
sections, they found that 3 varves out of 896 were not present (e.g. eroded
away by a flood) in the Wall Section but were present in the Main Section,
and that 4 varves out of 896 were not present in the Main Section but present
in the Wall Section. Overall, this represented a total of 889 out of 896
varves (99.2 %) in common between the two outcrops, which means that if
we extend their results to the other 90 % of the outcrop, some
uncertainty might exist, with up to 1 % of the varves “missing” in
time, which should have a very small impact on the relative timing of floods
discussed in this paper.
The analyses in this paper will concentrate on the fluctuations in flood
frequency over time (i.e. the number of floods per year and per decade),
examining the temporal correlations, clustering, and
cyclicity of the flood frequencies. Before proceeding to the
analyses of these time series, we provide next (Sect. 3) the definitions
and methods used in subsequent sections.
Definitions and methods for the analysis of event series
Both correlations and clustering were introduced in Sect. 1. Here we provide
more in-depth definitions and explanations of clustering and correlation
methods that will be used in examining our palaeoflood time series. In
Sect. 3.1, we introduce interevent occurrence times (IEOTs). Then, in
Sect. 3.2, Poisson processes, a model for non-correlated and non-clustered
time series. Section 3.3 describes the Weibull distribution (in the context
of IEOTs) as an indicator of clustering. Section 3.4 introduces
autocorrelation as a method to quantify short-range and long-range
correlations in a given time series. In Sect. 3.5, we describe
power-spectral analysis and detrended fluctuation analysis as methods for
quantifying long-range correlations. Finally, in Sect. 3.6 we briefly
introduce fractional noises.
Interevent occurrence times
In this paper, we will consider the number of palaeofloods per year, decade,
and century as event series in time. As indicators of clustering and/or
correlations (or lack thereof), we will apply statistical methods to the
event magnitudes (i.e. the number of floods per year, decade, or century). We
will also consider the time intervals between successive events, i.e. the
interevent occurrence times (IEOTs) which we introduce here. We will later use the statistical distribution
of the IEOTs as an indicator of clustering (or lack of) in the original time
series.
We introduce IEOTs in the context of our yearly palaeoflood time series (top
panel of Fig. 3) where, as previously, t is the relative age in years in
our time series and nyear is the number of floods per year.
First, we define a flood year to be any year t with at least one
flood in it, i.e. nyear(t) ≥ 1 flood yr-1. Second,
we define IEOT as the time interval Δ between successive years that
have one or more floods, i.e. more formally, nyear(t) ≥ 1 flood yr-1 and nyear(t+Δ) ≥ 1
flood yr-1, but for all years in-between,
nyear(t+l) = 0 floods yr-1 for l= 1, 2,
…, Δ-1.
In Fig. 4a we illustrate finding the IEOTs: we present the number of
detrital layers per year (Fig. 4a.1, which is the same as the top panel in
Fig. 3) with as inset (Fig. 4a.2) an example of four detrital layers with
IEOTs of Δ= 52, 3, and 26 years. In Fig. 4b we give all the
Δ as a function of relative age t for the entire palaeoflood data
set. The values of Δ range from 1 to 125 years (with quartiles of
25 %: 3 years, 50 %: 7 years, and 75 %: 15 years) and a mean
Δ‾= 12.5 years. We then use a subscript j to indicate
successive IEOTs, Δj, which we plot using “natural” time, in
other words where each Δ is no longer represented temporally when it
occurs, but rather successively one after another, j= 1, 2, 3, …,
NΔ, with the total number of natural time intervals NΔ= 739 (we discard the j value corresponding to the gap). In
Fig. 4c we plot the same Δ from Fig. 4b, but now as a function of
the detrital varve index (natural time), j= 1, 2, 3, …, NΔ.
The use of natural time (i.e. the time between all events is “equal” in
natural time) to represent unequally spaced events in time series has been
mainly used to examine seismic-related time series (e.g. Uyeda et al., 2009;
Rundle et al., 2012), but has found use in a variety of other disciplines
ranging from biology and environmental sciences to cardiology (see Varotsos
et al., 2011, for a review of its use in various disciplines).
Poisson process as a model for an uncorrelated non-clustered time
series
The standard example of an event series that is uncorrelated in time,
non-clustered, and stationary in time is the realization of a Poisson
process. In later analyses (Sect. 4.1) we will compare the statistical
distribution of the magnitudes in our three palaeoflood time series
(Fig. 3) to those of a Poisson process, thus inferring the lack of
correlations and clustering (if a Poisson process) or the presence of
correlations or clustering (if not a Poisson process). If a real-world
process (e.g. flooding) by
which a time series results is assumed to be Poissonian, then many synthetic
realizations of that Poisson process can be created, and their statistical
properties confronted with those of the observed “real-world” time series.
Poisson processes have been found in some cases to model the temporal
occurrence of floods (e.g. Kirby, 1969; Todorovic and Zelenhasic, 1970) and
in other cases to not be an appropriate model (e.g. Mudelsee et al., 2004,
who showed that the winter floods of the Elbe River from 1500 to 2000 cannot
be modelled by a stationary Poisson process).
For a time series to be considered a realization of a Poisson process, the
following must be true (Cox and Lewis, 1978):
the time series elements are non-negative integers;
the time series considered is stochastic (i.e. has no obvious trends
and periodicities and no deterministic components);
the one-point probability distribution of the time series elements
follows a Poisson distribution, that is, a positive integer-valued random
variable X, whose probability densities Pλ are defined as (Cox
and Lewis, 1978)
PλX=n=λnn!exp-λ,n=1,2,3,…
where “!” means factorial, exp is the exponential function, λ is
the rate parameter with λ > 0, and n is the intensity
(number of occurrences) of the “random” variable per given time unit (e.g.
hour, year, decade, century). The rate parameter λ is a physical
quantity and thus has units (time unit)-1, but the Poisson process as
given in Eq. (1) is a mathematical model and Pλ does not have
units.
the times between successive events (interevent occurrence times,
IEOTs) are independent in time (i.e. there is no correlation between one IEOT
and others, so events occur independently of one another);
the one-point probability distribution P of the IEOTs Δ is
exponential with
PexpΔ=λexp-λΔ,Δ≥0
with the same rate parameter λ as in Eq. (1) and the time unit of
the original time series being small enough that almost all time units have
just no or one event (i.e. λ≪1).
The Poisson process as defined in Eq. (1) does not lead to any temporal
clustering. The one-point probability distribution for the Poisson process
for 0<λ≤ 1 had no mode, and for λ> 1 it has a defined mode.
Weibull distribution of IEOTs as an indicator of clustering
If the Δj (the IEOTs), in addition to not being exponentially
distributed, are Weibull distributed, this is taken as an indicator of
clustering of the original time series (Bunde et al., 2005; Witt et al.,
2010). The two-parameter Weibull probability distribution (Weibull, 1951) is
a standard waiting time distribution, i.e. frequently used for modelling the
time intervals between successive events (Cox and Lewis, 1978). The
continuous (vs. discrete, as in Eq. (1) for the Poisson distribution)
two-parameter Weibull probability distribution is given by (Weibull, 1951)
PWΔ=kWλWΔλWkW-1exp-ΔλWkW,Δ≥0
where the two parameters are λW for scale and
kW for shape, and Δ is any real number ≥0. When the
shape parameter kW= 1.0, the two-parameter continuous Weibull
distribution (Eq. 3) turns into
PWΔ=1λWexp-ΔλW,Δ≥0,
that is, the exponential distribution which describes the IEOT distribution
of a Poisson process with a parameter λ=1/λW (see
Eq. 1).
For shape parameters 0.0 < kW < 1.0, the
two-parameter Weibull distribution (Eq. 3) is heavy-tailed (i.e.
asymptotically scale invariant) with a tail parameter of (1-kW),
which means that the probability density for very large values Δ in
Eq. (3) scales with a power law with the tail parameter as power-law
exponent.
Autocorrelation as a method for quantifying temporal (short-range and
long-range) correlations
There are a variety of methods that can be used to explore and quantify
temporal correlations both in observed time series and realizations of a
modelling process. As discussed in Sect. 1, correlations (persistence) have
been studied in many environmental time series. Measures for correlations
quantify the statistical dependence between variables, and in particular,
measures for temporal correlations describe the intensity of this
relation between time series elements with a fixed temporal distance. To
quantify the strength of correlations, we use autocorrelation analysis
applied to the time series of number of floods per year (nyear),
decade (ndecade), and century (ncentury) (Fig. 3).
The autocorrelation function C (e.g. Priestley, 1982) for the number of
floods is defined as
Cτ=1Nσ2n∑j=1N-τnj-n‾nj+τ-n‾
where nj is a time series of the number of floods per year (decade,
century) with j= 1,2, …, N, with N=Nyear,
Ndecade or Ncentury (i.e. the total number of years,
decades, and centuries in the record), n‾ is the mean number of
floods (per year, decade, century) for the entire record, τ is the time
lag (in years, decades, centuries), and σ2(n) is the variance of
the time series. Positive values of the autocorrelation function are
indicative of a process that on average, for a given lag τ, has
positive persistence between values separated by that lag τ. In other
words, if there are a large number of floods in a given year, on average,
τ years later will also be followed by a large number of floods
(compared to the mean number of floods per year overall). The converse is
also true: if there are few floods in a given year, on average, τ years
later will also be followed by only a few floods (compared to the mean number
of floods per year overall). Negative values of the autocorrelation function
are indicative of negative persistence, whereby if there are a large number
of floods in a given year, on average, τ years later will be followed
by a very few number of floods (compared to the mean number of floods per
year overall).
If the correlations are essentially linear and thus can be described by the
autocorrelation function, two types of correlations can be considered:
(i) short-range correlations (Priestley, 1982; Box et al., 2013) and
(ii) long-range correlations (e.g. Taqqu and Samorodnitsky; 1992; Beran,
1994; Malamud and Turcotte, 1999).
Short-range correlations are characterized by a decay of the autocorrelation
function C(τ) (Eq. 5) that is bounded by an exponential decay for
large lags, τ:
Cτ≤γ0exp-γτ,τ>τ0,
where τ0, γ0, and γ are non-negative constants. In
particular, this definition applies for time series with a finite correlation
length (C(τ)= 0 for τ > τ0). Statistical
models for short-range correlated time series include autoregressive (AR) and
moving average (MA) processes (Priestley, 1982).
In contrast to short-range correlated time series, long-range correlated time
series approach a power-law decay of the autocorrelation function C(τ)
(Eq. 5) for large lags τ:
Cτ∼τ-1-β,0.0<β<1.0.
The parameter β (0.0 <β< 1.0) is the
strength of the long-range correlations. The autocorrelation function has two
limiting values: β= 0.0 (which represents short-range persistence
between the time series elements) and β= 1.0 (pink or 1/f noise).
Koutsoyiannis and Montanari (2007) have shown that the statistical
uncertainty of the mean value of (hydrological) time series is increased in
the presence of correlations, and in particular long-range correlations.
Power-spectral analysis and detrended fluctuation analysis (DFA) for
quantifying long-range correlations
In the last section, we used the autocorrelation function as one method to
quantify long-range (and short-range) correlations. Here we describe two more
methods for quantifying long-range correlations: power-spectral analysis and
detrended fluctuation analysis. We focus on long-range (vs. short-range)
correlations as these form the main part of our analyses and modelling in
later sections. These two methods are both more robust than autocorrelation
in quantifying long-range correlations (Witt and Malamud, 2013).
Long-range correlations are reflected by a scaling of the power-spectral
density S (square of the modulus of the Fourier coefficients appropriately
normalized) with frequency f. The power-spectral density S exhibits a
power-law scaling such that (Taqqu and Samorodnitsky, 1992; Beran, 1994)
Sf∼f-β,
where f is the frequency and the relationship holds (for long-range
persistence) over all β (Pelletier and Turcotte, 1999; Witt and
Malamud, 2013). Positive exponents (β > 0.0) in Eq. (8)
represent positive (long-range) persistence and negative ones (β < 0.0) anti-persistence. The specific case of β= 0.0
corresponds to an uncorrelated time series (e.g. a white noise), and a value
of β= 1.0 is known also as a 1/f or pink noise (Mandelbrot and van
Ness, 1968; Bak et al., 1987). Some examples of these long-range persistent
time series are given in the next Sect. 3.6. In this paper, to avoid
confusion with other estimators (e.g. DFA; see next), we will indicate the
measurement of the strength of long-range persistence by power-spectral
analysis using the notation βPS.
Another common method for quantifying long-range correlations is detrended
fluctuation analysis (DFA) (e.g. Peng et al., 1994; Kantelhardt et al.,
2001). Here, the scaling properties of long-range correlated time series are
quantified in terms of the fluctuation function. In this paper the number of
floods per year, decade, or century can be analysed by DFA; we call these
time series here xt, t= 1, …, N. The fluctuation
function is based on the running sums (or profile) of the considered time
series xt, t= 1, …, N:
st=∑i=1txi.
This time series of the running sums st, t= 1, …, N is
then split up into non-overlapping segments of length l. For the kth
segment of the running sums, sk,i, i= 1, …, l, the
fluctuation is determined as the variance of the difference of this segment
and its best-fitting polynomial trend tk,i, i= 1, …, l (with
the polynomial order k, usually between 1 and 4),
F2k,l=1l∑i=1lsk,i-tk,i2.
The fluctuation of the time series is the mean of the fluctuation of the
segments:
F2l=F2k,lk,
where 〈〉k is the mean value taken over all
fluctuations of length k segments. If the underlying time series xt,
t= 1, …, N has long-range correlations, then the fluctuation
function F(l) exposes for long segment lengths l a power-law scaling and
will scale as (Peng et al., 1992)
Fl∼lα.
The strength of long-range correlations, β, is related to the scaling
parameter of the fluctuation function, α, as β= 2α-1.
If polynomials of order k are considered, then the resultant estimate of
the long-range dependence is called DFAk (e.g. DFA1, DFA2). In this paper
we will calculate α using DFA1 to DFA4. We will provide results using
the notation βDFA (to indicate the use of DFA) based on
βDFA= 2α-1.
For Eqs. (7) (autocorrelation analysis), (5) (power-spectral analysis), and
(9) (DFA), in each case we have an inverse power-law decay with increasing
temporal scales (lag, frequency, temporal segment length), which defines a
self-affine time series (the time series is statistically self-similar when
comparing different temporal scales) (Mandelbrot and Van Ness, 1968). If the
power-law exponent held over “all” temporal scales (which it rarely does),
then the correlation length (the largest lag or temporal scale at which there
are still statistical correlations in the time series) would be infinite. One
potential significance of this “infinite” correlation length is that
“all” values are statistically correlated with one another in the time
series. A second significance is that the time series can be explained as a
stochastic fractal (Malamud and Turcotte, 1998). In the case of our
palaeoflood series this means potentially that the flood timings are
organized in time as the points of a Cantor dust (see Ott, 1993, for a
discussion of Cantor sets).
Examples of synthetic fractional Gaussian noises with different
modelled strengths of long-range persistence, 0.0≤β≤1.0. The
presented synthetic data series (unitless in magnitude and time), which have
512 elements each, are normalized to have a mean of 0 and a standard
deviation of 1, and were created by Fourier filtering (see Appendices 1 and 2
in Witt and Malamud, 2013, for further details).
Extensive details about power-spectral analysis, DFA, and software written in
R (R Core Team, 2013) for performing time series analysis using these
techniques are given in the review paper by Witt and Malamud (2013). This
latter study also investigates biases of both techniques which typically
occur when they are applied to time series that have a one-point probability
distribution that is strongly non-Gaussian and/or the time series has very
few values (e.g. just a few thousand data).
Fractional noises as examples of long-range correlated time
series
Fractional noises are standard examples of long-range correlated time series
(Malamud and Turcotte, 1999; Mandelbrot, 1999), which we will use here to
help us model long-range persistent characteristics of our palaeoflood time
series. Examples of six fractional noises are shown in Fig. 5, with the
strength of long-range persistence ranging from β= 0.0 (a white
noise) to β= 1.0 (a pink noise), in 0.2 increments. In the white
noise in Fig. 5, the values are uncorrelated with each other, and as we
increase the strength of long-range persistence, the values, although drawn
from the same underlying probability distribution, become more correlated
with one another (and their clustering increases).
Histograms and autocorrelation of the palaeoflood time series with
the corresponding Poisson model. (a) Histogram (also shown in
Fig. 3) of the number of floods per year (blue bars), decade (orange bars),
and century (green bars), with each compared to the Poisson model with
respective rate parameters λyear=0.083 floods yr-1,
λdecade=0.83 floods decade-1, and λcentury=8.3 floods century-1. Diamonds represent the mean
Poisson model value and the error bars the standard deviation over 100 realizations. (b) Autocorrelation function (ACF)
(Eq. 5) of the number of floods per year (blue squares), decade (orange
squares), and century (green squares). Also shown for the ACF of the number
of floods per decade is the best-fitting power-law model to the ACF (black
dotted line) (Eq. 13). Also shown are the 97.5th percentile, i.e. the upper
bound of the 95 % confidence interval of the ACF (for lags τ>0) of an uncorrelated signal with the same one-point
probability distribution (i.e. realization of a Poisson model).
Results of statistical metrics, clustering, correlations, and
cyclicity of the palaeoflood sequence
In this section, we first statistically analyse the 771 timings of
palaeofloods in terms of their elementary metrics of the number of
palaeofloods per year, decade, and century compared to a Poisson process
(Sect. 4.1). We then examine the probability of IEOTs compared to a Weibull
distribution as a potential indicator of clustering (Sect. 4.2). Next, we
use autocorrelation of the palaeoflood time series to characterize the
temporal correlations (Sect. 4.3) and autocorrelation of the IEOTs as
another potential indicator of clustering (Sect. 4.4). We then characterize
the temporal correlations of the time series using detrended fluctuation
analysis and power-spectral analysis (Sect. 4.5). Finally, we characterize
the cyclicity of the palaeoflood time series using a sinusoidal model
(Sect. 4.6). As the techniques we used are designed for time series with a
continuous one-point probability distribution (e.g. Gaussian, log-normal, or
Levy distribution) vs. the integer values that are found in our palaeoflood
time series, we throughout also show the statistical significance of our
findings.
Frequency-size and autocorrelation analysis of the interevent
occurrence times (IEOTs) and comparison to the Poisson model.
(a) Frequency density of the IEOTs between the flood years given as
a function of IEOTs in years (pink vertical bars, semi-logarithmic scale),
the corresponding best-fit Poisson model (grey diamonds) (Eq. 2) which was
considered in Fig. 6, and the best-fit Weibull distribution (black
diamonds) (Eq. 3), both with 95 % confidence
intervals. (b) Autocorrelation function (Eq. 5) of the IEOTs
between flood years (those years with n≥1 flood). Also given is the
best-fitting power-law model (dashed black line) (Eq. 14) and the upper
limit of the 95 % significance for the autocorrelation function of a
non-correlated time series with the same one-point probability distribution
(horizontal dotted line, pink).
Number of palaeofloods per year, decade, and century, compared to a
Poisson process
We now examine the number of floods per year, decade, and century, as if the
process that created them resulted in a series of floods uncorrelated in
time. Here, we will consider a time series to be the realization of the
underlying process. If we are able to show that our time series has
properties different from a Poisson process (Sect. 3.2), we can infer that
it is correlated or clustered. We will use as a model a Poisson process with
a constant rate parameter to create model time series of flood events per
year, decade, and century (i.e. similar to the palaeoflood time series shown
in Fig. 3). We will see below that the resultant model realizations will be
stationary, uncorrelated in time, and non-clustered, and then show in
subsequent sections that this modelling is inappropriate for our palaeoflood
time series.
We first interpret the number n in Eq. (1) as the number of detrital layers
per varve (nyear, floods yr-1). The rate parameter λ controls the probability of an event: the mean number of events per year
(i.e. per varve) is n‾year=λyear
and the variance of the number of events is σ2nyear=λyear. This equation holds only
mathematically but not physically, i.e. it holds for the numbers but not for
the units. The probability in Eq. (1) is not time dependent, and rather only
depends on (if we take our unit of time to be 1 year) the probability of
nyear= 1 flood yr-1, nyear= 2
floods yr-1, etc., and not the relative temporal spacing of the floods
in time. To model the series of the number of flood events per year in the
9.3 kyr Piànico–Sèllere palaeoflood time series, we use a
(constant) rate parameter of λ= 0.083 floods yr-1 that is the
mean number of detrital layers per varve, i.e. (771 floods)/(9271
years) = 0.083 floods yr-1, where we discard the 65-year gap for the
total number of years considered.
As discussed in Sect. 3.2, another consequence of considering the number of
floods per year as a Poisson process is that the time interval Δ
between two successive events (i.e. the IEOTs between two successive floods)
follows an exponential one-point probability distribution P for very small
time units (Eq. 2) with rate parameter λ. Now, the relative
ordering between single floods is taken into account. For example, if in a
given decade we have three varves, each with one flood, Eq. (2) will be
different if the 3 years with floods are in years 1, 2, and 3 (Δ1=Δ2= 1 year, Δ3≥ 8 years because the fourth flood
will be in a subsequent decade) vs. one flood each in years 1, 5, and 9 of
the decade (i.e. Δ1=Δ2= 4 years, Δ3≥ 2 years). Note that the time interval Δ can have non-integer
values, for instance if two or more events (floods) occur in a single time
unit (within 1 year).
For our palaeoflood time series (Fig. 3), we now consider whether the
histograms (horizontal bars on the right-hand side of Fig. 3) of the
observed number of years, decades, and centuries with a given number of
floods per year, decade, and century follow Poisson distributions. For each
palaeoflood time series, we create a Poisson model which consists of 100
realizations of a Poisson process, each realization with 9271, 927, and 92
(respectively, for year, decade, and century) time series elements and with
rate parameters λ=λyear= 0.083 floods yr-1=λdecade= 0.83 floods decade-1=λcentury= 8.3 floods century-1 (the measurements
values are different as well as the units; the differences cancel out because
of multiplication of measurement values and units). We discard the 65-year
gap for the purposes of this model. Each realization therefore has values
that are uncorrelated in time and follow a Poisson distribution (Eq. 1).
Many common software programs (e.g. Excel, R, Matlab) are able to easily
generate such realizations.
We give the results of our Poisson model for each respective time resolution
(year, decade, century) in Fig. 6a as diamonds (mean of the 100
realizations) ± error bars (standard deviation of the 100 realizations).
We find that for the yearly data, the number of years observed
(kyear) with a given number of floods per year, nyear= 0, 1, 2, 3 floods yr-1, follows closely the number of years given by
the Poisson model. For example, returning to the values given above, the
number of years with nyear= 0, 1, 2, 3 floods yr-1 is
observed to be kyear= 8530, 712, 28, and 1 years
(respectively), and with the Poisson model kyear= 8531 ± 26, 709 ± 25, 30 ± 5, and 1 ± 1 years
(mean ± standard deviation) (respectively). In contrast, for both the decadal and
centennial data, the observed data set contains decades and centuries with
many fewer or greater floods compared to the model data (see Fig. 6a). For
example, the number of decades with ndecade= 0, 1, 2 floods
decade-1 is observed to be kdecade= 453, 283, and 123
decades (respectively), with the Poisson model giving kdecade= 406 ± 15, 338 ± 13, and 140 ± 11 decades
(mean ± standard deviation) (respectively).
We therefore conclude that at the yearly scale the observed data follow a
Poisson process, but at the decadal and centennial scales, the time series
cannot be modelled as a Poisson process. In subsequent sections we will
explore whether clustering, correlations, or both are responsible for this.
Probability of interevent occurrence times (IEOTs) compared to a Weibull
distribution as a potential indicator of clustering
Here, we analyse the one-point probability distributions of the interevent
occurrence times (IEOTs) (Sect. 3.1). As discussed in Sect. 3.2, a
realization of a Poisson process is characterized by “events” that are
uncorrelated with one another and the Δj have an exponential
distribution (Eq. 2). Conversely, it means that if the Δj (the
IEOTs) are non-exponentially distributed, the process is non-Poisson. In
Fig. 7a, we give the frequency density of Δ (the one IEOT that
includes the sediment gap is excluded) of the flood record along with an
exponential distribution which represents the interevent occurrence times of
events in a Poisson process. The distribution of the IEOTs has a higher
number of both short and long time intervals compared to an exponential
distribution. This observation is supported by statistical hypothesis testing
where an Anderson–Darling test (Anderson and Darling, 1952) was adjusted to
integer values of the IEOT distribution (using a method described in
Choulakian et al., 1994; Arnold and Emerson, 2011), and the test rejected the
null hypothesis that this distribution can be explained as the result of a
Poisson process (p-value < 0.002).
We have shown above that our data are not a Poisson process, and now wish to
further characterize the distribution of the IEOT. For the palaeoflood IEOTs,
we use a maximum-likelihood estimation (MLE) and assume a two-parameter
Weibull distribution (Eq. 3 and discussed in Sect. 3.3). We find (see
Fig. 7a) a best-fit shape parameter for the Weibull distribution of
kW= 0.91 ± 0.024, where the error bars represent the 95 % confidence intervals. However, this result is questionable as
the MLE technique is tailored to a Weibull distribution as a
continuous probability distribution, but the interevent occurrence
times Δ have discrete (integer) values. Therefore, we use benchmarks
(500 realizations of a Weibull distributed model for random numbers) for
fitting Weibull distributions to integer-valued data. We find that the MLE
estimator tends to over-fit the shape parameter by 0.12, i.e. we can correct
our estimate of the shape parameter to kW= 0.79 ± 0.024.
Autocorrelation of palaeoflood sequences to quantify
correlations
We now use the autocorrelation function (ACF) discussed in Sect. 3.4 and
apply it to our palaeoflood sequence as a potential indicator of
correlations. In our analyses of the palaeoflood sequence, only pairs of
years, decades, or centuries nj,nj+τ which do
not contain the 65-year gap in the 9336-year sequence of floods (detrital
layers) are considered. The autocorrelation function (Eq. 5) applied to
the time series of the number of floods per year, decade, and century is
given as correlograms in Fig. 6b (squares). This correlogram of the yearly
floods shows significant positive correlations with a decaying trend and no
obvious periodic components for lags of 1 ≤τ≤ 200 years. Due to
the strongly fluctuating behaviour of the annual data's correlogram (squares)
in Fig. 6b it is difficult to determine the functional form of the decay,
i.e. to decide whether the decay is exponential (thin tailed, short-range
persistent) or approaching a power law (heavy tailed, long-range persistent).
Therefore, we apply the autocorrelation function to the time series of the
number of floods per decade (with respect to non-overlapping 10-year time
intervals) and find (Fig. 6b) a power-law behaviour
Cτ∼τ-γACF,τ>0
with a power-law exponent for the decadal time series of
γACF= 0.34 ± 0.04 (fitted for lags 1 ≤τ≤ 20 decade, which is equivalent to 10 ≤τ≤ 200 years)
with uncertainties ±1 standard error of the exponent. Comparing to
Eq. (7), γACF= (1-βACF), giving
βACF= 0.66 ± 0.04. We will return to this power-law
model (Eq. 13) in Sects. 4.5 and 5.2. We conclude that for the floods per
decade time series, when using the autocorrelation function, the data exhibit
long-range correlations over the range 10 ≤τ≤ 200 years.
The time series of the number of floods per century contains only 92 data
points. We therefore have calculated the autocorrelation function C only
for lags τ= 1 century and τ= 2 century as graphically presented
in Fig. 6b. Based on this figure, we see significant positive correlations
and thus confirm the results found for the temporal correlations of the
number of floods per year and decade. Due to the very few lag values,
however, we cannot discuss the functional shape of this autocorrelation
function.
We conclude that the three considered time series nyear,
ndecade, and ncentury have positive autocorrelations
for lags of τ < 200 years. For the number of floods per
decade, ndecade, we find indications of a power-law shape of the
autocorrelation function which indicates long-range persistence of the time
series over lags τ > 200 years. However, we will see later
(Sect. 4.6) for lags τ > 200 years that a power-law
model cannot be conclusively fit, and thus over all lags, we cannot
conclusively use ACF as a robust quantifier of long-range correlations.
Autocorrelation of interevent occurrence times (IEOTs) as a potential
indicator of clustering
Another indicator of clustering is positive temporal correlations of the
IEOTs (Sect. 3.1). Such correlations are particularly caused by a
“lumping” of small (large) IEOTs which results in time intervals with an
increased (decreased) flood rate which correspond to flood clusters (phases
of little flooding activity). In Fig. 7b, we examine correlations (vs.
clustering) of the IEOTs by applying the autocorrelation function (Eq. 5) to
the Δj for the IEOTs. We find significant positive correlations of
the Δj for flood year lags 1 ≤τΔ≤ 20 (no
units). This indicates that for the IEOT, small values of Δj tend
to follow small ones, and large ones tend to follow large ones. Similar to
Eq. (13), the correlogram (see Fig. 7b) of Δj exhibits a
power-law behaviour
CτΔ∼τΔ-γΔ,γΔ>0
with a power-law exponent of γΔ= 0.45 ± 0.14 (fit
for lags of 1 ≤τΔ≤ 20 IEOTs, which is equivalent to
12.5 ≤τ≤ 250 years). Power-law correlations for IEOTs
are reported for a class of theoretical models by Bunde et al. (2003) and
Eichner et al. (2007), who have studied peaks over thresholds of long-range
correlated time series. We will explore this in more detail below.
In summary, we have analysed the one-point probability distribution and the
autocorrelation function of the IEOTs of the palaeofloods with respect to a
time resolution of 1 year (i.e. we have not used time resolutions between
palaeofloods that are sub-annual, such as 3 months). The distribution of the
IEOTs can be well approximated by a Weibull distribution with a shape
parameter of kW=0.78. Therefore, the IEOTs are more likely to be very
short or very long temporal periods compared to a random (uncorrelated)
occurrence of the IEOTs, and therefore the palaeoflood time series is not a
realization of a Poisson process. Furthermore, the IEOTs have positive
temporal correlations. This is particularly caused by a clustering of the
very high and very low IEOTs and thus by a temporal clustering of the floods.
The autocorrelation function seems to follow a power-law behaviour.
Detrended fluctuation analysis (DFA) and power-spectral analysis for
quantifying long-range correlations and as a potential indicator of
cyclicity
We now apply both DFA and power-spectral analysis (both discussed in
Sect. 3.5) to our time series given in Fig. 3 so that we can quantify the
degree of correlations in our data, at yearly, decadal, and centennial
scales. As discussed in Sect. 3.5, both methods are more statistically
robust than autocorrelation in quantifying long-range correlations. Methods
for quantifying long-range correlations have systematic and random errors in
the resultant estimator for time series with very asymmetric probability
distributions and the finite size effects of the time series. The systematic
error is how much the resultant estimator deviates from the underlying
“correct” value, and random errors are the spread of the estimated
quantity. Witt and Malamud (2013), using 17 000 synthetic benchmark series,
studied systematically these two types of errors for power-spectral analysis,
DFA, and semivariogram analysis. ACF gives similar results of robustness to
semivariogram analysis due to the two methodologies being very similar. Witt
and Malamud (2013) found that the systematic and random errors (biases and
standard errors) of DFA and PS analysis are significantly smaller
(particularly for time series with non-Gaussian one-point probability
distribution, and with very few values in the time series) compared to
semi-variogram analysis, and both DFA and PS analysis are appropriate for a
much broader range of long-range persistence strengths compared to
semi-variogram analysis. We therefore only consider power-spectral analysis
and DFA when considering the quantification of long-range correlations.
Detrended fluctuation analysis (DFA) of the 9271-year
Piànico–Sèllere, Italy, palaeoflood record: the fluctuation function
F for the number of floods per decade shown as a function of the segment
length l and for different orders of the detrending (see the legend) on a
double logarithmic scale. Segments containing parts of or the entire gap were
excluded. Also shown are the best-fitting power-law function (Eq. 12) for
DFA3 and the corresponding power-law exponent α and its corresponding
value of βDFA= 2α-1.
We start with our yearly flood record nyear which contains 771
floods and extends over 9336 years. It has a heavily skewed one-point
probability distribution with a coefficient of variation of cv= 3.46. Witt and Malamud (2013) showed that very asymmetric one-point
probability distributions lead to a systematic underestimation of the
persistence strength for both techniques. Due to the palaeoflood time series
elements being integer valued and in a small range (0 ≤nyear≤ 4 floods yr-1), both techniques lead to uncertain results.
Therefore, we will not apply power-spectral analysis and DFA to the annual
palaeoflood data set. But we will quantify long-range persistence of the
decadal and centennial flood series, as the data are less heavily skewed
(cv= 1.30 and 0.77 respectively), and where there is a wider
range of integer values (0 ≤ndecade≤ 8 floods decade-1 and 0 ≤ncentury≤ 31 floods century-1).
In Fig. 8 we show the results of the DFA analysis of ndecade.
The fluctuation functions for different orders of detrending are shown on
logarithmic axes. The almost linear shape indicates power-law behaviour of
the fluctuation function (see Eq. 12) for DFA3 and DFA4. DFA1 (DFA2)
follows this linear shape just for segment lengths up to 70 (100) decades. We
expect a slowly varying trend (a long-term cyclicity > 700
(1000) years) to cause this dependence on the type of detrending and will
analyse it in more detail in Sect. 4.6. The power-law exponent for
the decadal palaeoflood time series was estimated (for DFA3) as
αDFA= 0.62, which corresponds to a strength of long-range
persistence of βDFA= 0.25. This is a weak strength of
long-range persistence. The third data set that counts the number of floods
per century contains just 92 time series elements, which is not enough for a
reliable DFA.
Power-spectral analysis: the power-spectral density (periodogram) is
presented as a function of the frequency (measured per decade) for the number
of floods per decade. Also given is the best-fitting power-law function
(Eq. 8) and the corresponding power-law exponent. Additionally, presented
are the 95th percentiles of the power spectra created for synthetic data
based on peaks over thresholds of fractional noises (see Sect. 5.1) for
different strengths of persistence (see the legend and the description in the
text). Highlighted are the three values of the power-spectral density that
exceed all of these 95th percentiles; the corresponding cycle lengths are
given.
Power-spectral analysis for the decadal palaeoflood series
ndecade is shown in Fig. 9. Although the power-spectral density
S has a lot of scatter, it is clearly oriented along a line in the log–log
axes, and thus exposes power-law (self-affine) behaviour (see Eq. 8). The
persistence strength (i.e. the magnitude of the power-law exponent of the
power-spectral density S) is estimated as βPS= 0.39. This
again indicates a weak strength of long-range persistence. The centennial
palaeoflood series ncentury is not suited for power-spectral
analysis due to the short length of Ncentury= 92 centuries.
We also see the hint of long-term periodic components superimposed onto the
long-range persistence that is identified using power-spectral analysis. In
Fig. 9 are given three coloured lines that represent the 95th
percentiles (95 %), described in detail in a later section
(Sect. 5.1). What we observe is that three values (large blue dots,
outlined with a grey ellipse) are above the three 95 % lines (i.e. they are significant with p < 0.05) at a
cyclicity of 1300 to 2300 years. Again, as for DFA, we will explore this
long-term cyclicity (fluctuation) that appears to be superimposed onto the
long-range persistent signal in the next subsection.
We now consider both the systematic and random errors that can occur when
using DFA and power-spectral analysis by using the techniques given in Witt
and Malamud (2013) and their systematic study using thousands of benchmark
synthetic time series to calibrate the “error” in each method for a given
one-point probability distribution, length of time series, and strength
(uncalibrated) of correlations. We ignore any underlying periodicity as a
contributing factor to any errors. We find for our palaeoflood per decade
time series with 927 values and a coefficient of variation of cv= 1.3 that βPS would lead to a calibrated
βPS∗=0.52 with (0.32, 0.71) the 95 % confidence
interval and that calibrating the value of βDFA would give
βDFA∗=0.37 with (0.13, 0.63) the 95 % confidence
interval.
However, we have already indicated that there might be a long-range
fluctuation (cyclicity) superimposed onto our long-range persistence signal.
Therefore, we are unable to “calibrate” our βPS and βDFA using the techniques of Witt and Malamud (2013) and will
instead proceed with an investigation of the period and amplitude of the
long-term cyclicity (see Sect. 4.6), which will lead us to a model-based
approach (see Sect. 5). In summary, by applying DFA and power-spectral
analysis we find evidence for long-range correlations in our palaeoflood data
set. For the decadal palaeoflood time series and over temporal scales from
yearly to millennial, the estimated strengths of persistence for the
(uncalibrated) values are βPS= 0.39 and βDFA= 0.25.
Cyclicity and fluctuations in the 9.3 kyr palaeoflood record
Several climate cycles on millennial scales are known from the analysis of
long-term palaeoclimate proxy data (e.g. ice cores). They are related to
Heinrich events (Heinrich, 1988; Bond et al., 1992) and the
Dansgaard–Oeschger (Dansgaard et al., 1993; Johnsen et al., 1992; Voelker,
2002; Fisher, 2016) cycle which are caused by ocean ice dynamics and
variations in solar activity (Gray et al., 2010). The corresponding cycle
lengths range between 1000 and 5000 years. These long-term cyclicity
processes might play a role as a background signal in the flood
frequency described in our palaeoflood data set.
In Sect. 4.5 we quantified the long-range persistence of our three
palaeoflood records. We also found that by applying (i) DFA there was an
indication of a long-term cyclicity with a period > 700–1000
years, and (ii) with power-spectral analysis a hint of long-term cyclicity
with a period of 1300–2300 years. We now further explore potential long-term
temporal cycles and fluctuations in our data.
Autocorrelation analysis of the decadal and centennial palaeoflood
records (see Fig. 3) and its Poisson model for lags of up to 7500 years.
Autocorrelation function (ACF) (Eq. 5) of the number of floods per decade
(a, orange squares) and century (b, green triangles). Also
shown are the 97.5th and 2.5th percentiles, i.e. the upper and lower bounds of
the 95 % confidence interval, of the ACF (for lags τ>0) of an uncorrelated signal with the same one-point
probability distribution (i.e. realization of a Poisson model; see
Sect. 3.2). See also Fig. 6b, which is a similar analysis, but only up to
lags τ of 200 years.
Analysis of flood rate fluctuations. (a) Shown is the
time-dependent flood rate Λ (black line) and its 95 % confidence
bands (green bands) as a function of the relative age t=1 to 9336 years
for the palaeoflood data set shown in Fig. 3 which was computed by using a
kernel density estimator. The grey colour indicates results that might be
affected by boundary effects. The estimator is based on a Gaussian kernel
with a width of ±3σ, where σ=250 years (Appendix C),
shown in the inset figure. Also shown are the average flood rate (dashed grey
horizontal line) of λyear=0.83 floods yr-1 and the
best-fit sinusoidal curve (red curved line, Eq. 16). (b) For a
comparison, the number of floods per century ncentury (green
bars) is presented. The grey vertical bar indicates the gap in the data set.
We first return to our autocorrelation analysis results presented in
Sect. 4.3 (Fig. 6b) where we used lags τ up to 200 years and found,
(i) for the decadal and centennial palaeoflood time series, positive
correlations (C > 97.5 % of the Poisson model) for τ < 200 years, and (ii) for the yearly palaeoflood time series,
much weaker correlations (in places below the confidence limits, and with
significant scatter).
We now extend the same autocorrelation analysis up to lags τ of 7500
years for all three time series. For the yearly palaeoflood time series, we
found significant scatter, and so we just show the results for the
autocorrelation analysis applied to the decadal and centennial palaeoflood
time series (Fig. 10). As before, we also show the lower and upper bounds
of the 95 % confidence interval of the ACF (for lags τ > 0) of the Poisson model. We now observe a strong oscillating
signal on both sides of the 95 % confidence interval, for about three
periods, with minima at about 900, 3100, and 5000 years, and maxima at about
2000, 4000, and 5500–6500 years. In other words, we see a strong indicator
of a cycle with a period in the range of 1900–2100 years. This is not
contradictory with our previous observations using DFA (power-spectral
analysis) of cyclicity with a period of > 700–1000 years
(1300–2300 years). In terms of correlations, for both the decadal and
centennial autocorrelation signals, the cycle overwhelms the signal, so no
short-term or long-term correlation behaviour can be concluded from
Fig. 10.
In the hydrology community, semivariograms are also commonly used for
studying temporal correlations in time series (e.g. Chiverton et al.,
2015). The variogram was developed by a French professor of mining and
engineering, Matheron (1963). We therefore apply semivariograms to our
yearly, decadal, and centennial palaeoflood time series for lags of up to
7500 years, with an explanation of semivariograms and results given in
Appendix B. Results are comparable to that found with the autocorrelation
analysis, with a cyclicity at a period of about 1900–2100 years. Due to the
long-term cyclicity, the semivariogram does not approach a limit value for
large lags.
Previously (Sect. 4.1 to 4.3), our modelling considered the flood rate per
year to be constant (λyear= 0.83 floods yr-1) (see
Eq. (1), Fig. 6, and Fig. 7). We will now model the flood rate
λyear as a function of time t (in years) and use the
symbol Λ(t), t= 1, …, 9336 years, to denote the long-term
fluctuations of the flood rate in our palaeoflood record. First, to visualize
the long-term fluctuations of the flood rate, we use a kernel density
estimator (see Appendix C) which computes a weighted mean of the number of
floods per year nyear for a sliding time window. We applied
Gaussian (based on a standard deviation of σ= 250 years) weights,
dealt appropriately with the gap and the values close to the boundaries of
the observational interval, and also computed 95 % confidence intervals
for Λ (for technical details, see Appendix C). The results
(Fig. 11a) show that the time-dependent flood rate Λ varies on
millennial scales. We have high values of Λ > 0.15
floods yr-1 at the beginning of the record, a first minimum with
Λ= 0.065 floods yr-1 at a relative age of
t= 1000 years, a second maximum at t= 2000 years, a second
minimum at t= 3400 years, and an absolute maximum with Λ= 0.16 floods yr-1 at a relative age of t= 4300 years. The
long-term fluctuations of the flood rate Λ(t) continue for larger
relative ages, but get weaker. As the 95 % confidence intervals of the
absolute maximum and some of the minima do not overlap, the fluctuations are
statistically significant. This long-term fluctuation can also be seen in the
time series of ncentury (Fig. 11b), but due to scatter the
millennial-scale cyclicity is not well defined.
We then model the cyclical behaviour of the flood frequency by fitting a
sinusoidal function to the annual palaeoflood time series nyear.
A best-fit sinusoidal function is obtained by applying (with period T)
least-squares regression to time t in years:
nyeart=a1+a2sin(2πt/T)+a3cos(2πt/T)+εt,
with a1 a constant very similar to the average flood rate λyear= 0.083 floods yr-1 given in Eq. (1); the term
“a2sin(2πt/T)+a3cos(2πt/T)” is the best-fit sinusoidal
function with optimized parameters a2 and a3, and ε(t)
is the residual of the statistic model. Using trigonometric functions, we can
rewrite the middle two terms of the right-hand side of Eq. (16) such that
nyeart=a1+Aratesin((2πt/T)+Φ)+εt,
with the amplitude Arate= (a22+a32)0.5 and
the phase Φ= arctan(a3/a2).
We have calculated the best-fitting sinusoidal model for periods
1500 < T < 2500 years (with a step size of ΔT= 10 years) and have found that the model leads to the smallest variance of
the residuals (σ2(ε)= 0.081) when T= 2030 years (see
Fig. D1), where we find just one minimum and a very low curvature
(platykurtic) shape. The model with T= 2030 years best explains the
original palaeoflood time series when considering the time-dependent flood
rate as a sine wave. The corresponding model parameters are (when T= 2030
years) a1= 0.082 floods yr-1 (≈λyear= 0.083 floods yr-1), amplitude Arate= 0.049
floods yr-1, and phase Φ= 39∘. The value for
Arate is almost 60 % of the average palaeoflood rate
(λyear). This model is graphically presented in Fig. 11
as a red line.
When the periodic rate model is compared to the kernel density estimate of
the time-dependent rate Λ, we find (i) a good agreement of the
positions of the maxima and minima, and (ii) constant values of the maxima
and of the minima of the periodic rate model but very different values for
the five maxima of the time-dependent flood rate Λ, and that
(iii) the periodic rate model is far outside the 95 % confidence levels
of the time-dependent rate Λ for the time interval of
5800 < t < 6700 years, which is just 10 %
of the length of the palaeoflood record. We can conclude that the periodic
rate model captures the most important features of the time-dependent rate
Λ.
Schematic of the POTFGN+Period model for peaks over thresholds (POTs) of fractional Gaussian noises (FGN) (see Fig. 13
for POTs of FGN + Period): (a)
shown is a synthetic fractional Gaussian noise (unitless in magnitude) with a
persistence strength of βmodel= 0.5. The synthetic time
series resolution is set at Δt=1 year. See Fig. 5 for an example
of other fractional Gaussian noises. (b) Three thresholds are set
(dashed horizontal lines). All noise values that are below the threshold
θ01 are considered as years with 0 floods yr-1. All noise
values between the thresholds θ01 and θ12 (θ12
and θ23) are considered as years with 1 (2) floods yr-1. And,
all noise values that exceed the threshold θ23 are considered years
with 3 floods yr-1. The thresholds θ01, θ12 and
θ23 are chosen such that the resultant one-point probability
distribution is the same as our flood series. (c) Time series of the
modelled number of floods per year (blue circles) for 1000 years. For
(a), (b), and (c) are also shown the histogram of
the number of values at a given size.
POTFGN+Period model realizations. Shown are
(a) long-range persistent flood time series that are modelled with
the proposed POTFGN+Period model (see Fig. 12) as peaks over
thresholds (POTs) of a fractional Gaussian noise (illustrated in Fig. 5)
with different persistence strengths βmodel=0.0, 0.2, 0.4,
0.6, 1.0 and no superimposed periodic component (Amodel=0.0), and
(b) flood time series xyear with a long-term periodic
cyclicity which are modelled as POTs of a Gaussian white noise
(βmodel= 0.0) superimposed with a sine wave (T=2030
years) with different sine wave amplitudes Amodel=0.0, 0.2,
0.4, 0.6, 1.0.
Summary of clustering, correlation, and cyclicity of the 9.3 kyr
Piànico–Sèllere palaeoflood record
In Sect. 4 we have seen that the decadal number of floods cannot be
explained by a Poisson process due to the palaeoflood time series' one-point
probability distribution and long-range persistence (strength βPS≈ 0.39 using power-spectral analysis and
βDFA≈ 0.25 using DFA). Furthermore, we found our
palaeoflood series to be clustered as the interevent occurrence times are
approximately Weibull distributed (shape parameter kW= 0.78 ± 0.02) and long-range correlated (power-law exponent of the
autocorrelation function γΔ= 0.45 ± 0.14). We also
found evidence for long-term cyclicity (period T≈ 2030 years). In
the next section we will suggest a minimal model that captures these
identified characteristics of clustering, correlation, and cyclicity.
Creating a peaks over threshold (POT) model to capture
correlations and clustering of the observational data
In the previous section, we examined a 9.3 kyr palaeoflood record from
Piànico–Sèllere that occurred sometime during the period 780 to
393 ka, and found long-range persistence indicated by power-law behaviour of
the power-spectral density, the fluctuation function, and the autocorrelation
function as well as temporal clustering and long-period cyclicity of the
flood time series. We now introduce a model using peaks over threshold (POT)
of synthetic time series that consist of a fractional Gaussian noise
(FGN) + period, which we will abbreviate as a POTFGN+Period
model. This model is based on the idea of long-range persistence and
cyclicity that (i) captures the correlation and clustering properties of the
palaeoflood observational data, (ii) captures the long-period cyclicity, and
(iii) depends on a minimum number of parameters. In this section, we will
first present a method by which we can create a general model to incorporate
correlations, clustering, and cyclicity (Sect. 5.1). We next quantify the
correlation, clustering, and cyclicity properties of this model (Sect. 5.2)
and specify the parameters of this general model to reflect the specific
properties of our observed 9.3 kyr palaeoflood data (Sect. 5.3). We then
confront the model with the optimized parameters with the palaeoflood data
(Sect. 5.4) and finally discuss properties and a use of this specified
model by analysing long model realizations (Sect. 5.5).
A POT model utilizing fractional noises and
long-term cyclical behaviour (POTFGN+Period)
To describe our palaeoflood record, we model a series of events along a timeline such that they exhibit correlation and clustering.
As discussed in Sect. 3.6, fractional noises exhibit linear correlations
(i.e. by linear, we mean correlations that can be quantified by
autocorrelation function or power-spectral analysis) that are long-range
persistent (Beran, 1994). POTs of fractional noises result in event series that exhibit long-range persistence. Here the events
are considered to be the values of the considered fractional noise that
exceed a high threshold (i.e. the POTs) (Altmann and Kantz, 2005; Bunde et
al., 2005; Eichner et al., 2007; Olla, 2007; Santhanam and Kantz, 2008;
Moloney and Davidsen, 2009). It has been shown that the interevent occurrence
times of such POT values follow a heavy-tailed or Weibull distribution and
have long-term correlations (Bunde et al., 2005). Similar results have been
found to hold for processes with multifractal and other non-linear
correlations (Bogachev et al., 2007, 2008; Olla, 2007), but not for
non-linear deterministic systems (Schweigler and Davidsen, 2011). We have
taken the idea of using POTs of fractional noises and have modified it in
order to gain long-term periodic fluctuations in the flood frequency and an
annual number of floods ranging from 0 ≤xyear(t)≤ 5
floods yr-1.
The long-period cyclical behaviour of our 9.3 kyr palaeoflood data set was
taken into account by superimposing a sine wave with a period of T= 2030
years onto an input fractional Gaussian noise. The resulting time series is
formally different from the periodic fractional noises introduced by
Montanari et al. (1999). An integer-valued time series of floods per year was
created from the strongly fluctuating input signal by introducing a group of
thresholds. The model POTFGN+Period was implemented using the
following three-part schema.
Step 1. Creation of a superposition of a fractional noise and a periodic signal (FGN+Period) to input to our POT model
Begin with a realization yt, t= 1, …,
9271 years of a fractional Gaussian noise (FGN) with a given long-range
persistence strength (βmodel).
Normalize the FGN so that it has mean μ= 0.0 and standard deviation σ= 1.0.
An example of this normalized FGN is illustrated in Fig. 12a.
Add to the normalized FGN a sine wave with period T= 2030 years and a given
amplitude (which can be varied) Amodel. The value T= 2030 years is
based on the results of Sect. 4.6. The result of the normalized FGN
superimposed with a sine wave is a periodic FGN.
Step 2. POTs to obtain the number of floods per year
Begin with our periodic FGN from Step 1.
Decide the maximum number of floods per year, xmax, the model will
produce. For this paper, we used xmax= 5 floods yr-1, i.e.
the model can produce xyear= 0, 1, 2, 3, 4, and 5
floods yr-1.
Define xmax thresholds where the thresholds θ01, θ12, θ23, …are xmax horizontal lines which
will intersect our periodic FGN. The thresholds θ01, θ12,
θ23, …are chosen such that the resultant one-point
probability distribution of the number of floods per time unit is a Poisson distribution with a rate parameter of
λyear= 0.083 floods yr-1, i.e. the same as in our
palaeoflood series.
Determine for each value of the periodic FGN yt, t= 1,
…, 9271 years the highest of the xmax thresholds which is
exceeded. The ordinal number of this threshold is the modelled number of
floods per year xyear (as illustrated in Fig. 12b). If no
threshold is exceeded, the number of floods per year is set to zero,
xyear(t)= 0 floods yr-1.
The result is an integer-valued time series that models the number of floods
per year (as illustrated in Fig. 12c) that has the same one-point
probability distribution, the same number of data points, and the same
long-term period (T= 2030 years) as our original palaeoflood record. The
model depends on two model parameters (long-range persistence strength
βmodel and amplitude of periodic signal Amodel).
Step 3. Number of floods per decade and century
For computing the modelled number of floods per decade xdecade
the number of floods per year xyear is summed up over 10
consecutive years.
For computing the modelled number of floods per century xcentury the
number of floods per decade xdecade is summed up over 10
consecutive decades.
Analysis of model realizations of the POTFGN+Period
model (peaks over thresholds (POTs) of a (FGN+period)) with different pairs of parameters for persistence strength and
amplitude of the periodic component. Given for all six panels are the two
model parameters of the POTFGN+Period: (i) strength of long-range persistence of the underlying fractional Gaussian noise (FGN)
βmodel=0.00 to 1.00 (step size 0.02) on the x-axis and
(ii) the amplitude of the superimposed periodic component (period T=2030 years), Amodel=0.00 to 1.00 (step size 0.02) on the
y-axis. For each of the 51 × 51 = 2601 pairs of model
parameters (βmodel,Amodel), 1000 realizations were
produced. Each realization is a synthetic floods per year series which is
constructed as described in Sect. 5.1 and illustrated in Fig. 12. In
(a), (b), (c), and (d) are given
respectively the mean of the measured βPS,
βDFA, kW, and Arate (see the legend
for colours and contours) for the 1000 realizations in each of the
51 × 51 cell parameter pairs (βmodel,
Amodel). In (e), we then take measured values
from each of the panels (a) to (d) and show in (e)
their corresponding (βmodel, Amodel) values. The
measured values are chosen from each panel to be the same range (with error
bars) as our Piànico palaeoflood data as follows: (i ) strength of
persistence, 0.370 ≤βPS≤0.410, 5000 values from panel
a, magenta dots; (ii) strength of persistence, 0.23 ≤βDFA≤ 0.27, 5000 values from b), brown dots;
(iii) shape parameter of the Weibull distribution, 0.77 ≤kW≤ 0.81, 5000 values from (c), blue dots; and (iv) sinusoidal
flood rate, 0.0450 ≤Arate≤ 0.0530, 5000 values from
(d), green dots. When a given realization has βPS,
βDFA, kW, and Arate, all of which
satisfy the preceding conditions (i.e. all four colours for a given
realization appear in e), then we give a black dot. In
(f), the 2-D probability density of model parameters
(βmodel, Amodel) is shown whose persistence
strength (βDFA), shape parameter of the Weibull distribution
(kW), and best-fitting sinusoidal flood rate have similar values
to that of the Piànico palaeoflood record (see the legend for the colour
code). Marked is the mode (red bullet) and the area that contain
approximately 50 % of the points (blue dashed lines).
The resultant outputs of our POTFGN+Period model, the
integer-valued synthetic flood time series xyear, depend on two
parameters which are both related to the strongly fluctuating input signal:
the strength of long-range persistence βmodel and the
amplitude of the superimposed sine wave Amodel. In Fig. 13, 12
model realizations are shown. The six floods per year time series presented
in Fig. 13a are constructed with strength of long-range persistence
βmodel= 0.0, 0.2, …, 1.0 but without a long-term
periodic component (Amodel= 0.0). We see visually from
Fig. 13a that higher values of persistence lead to stronger clustering of
the annual number of floods, i.e. years with many floods tend to follow years
with many floods, and years without floods tend to follow years without
floods. The six synthetic floods per year time series presented in Fig. 13b
are constructed with white noise (βmodel= 0) and with an
increasing amplitude of the long-term periodic component Amodel= 0.0, 0.2, …, 1.0. In Fig. 13b we can visually observe that for
high values of Amodel the years with many floods cluster
periodically with a period of T= 2030 years.
We now return to the three 95 % lines in Fig. 9. These were derived as
follows: we simulated 1000 realizations of the model
POTFGN+Period with Amodel= 0.0 and
βmodel= 0.1, 0.3, or 0.5. For a fixed value of
βmodel the power-spectral density (the periodogram) was
computed for all resultant event time series. For each frequency f the 1000
values of S(f) were ordered from smallest to biggest and the 950th of 1000 values formed the three 95 % coloured lines in Fig. 9.
In summary, the time series presented in Fig. 13a and b show that high
values of either of the model parameters lead to strong clustering. In the
next subsection, we will investigate the dependence of the clustering and
correlation properties on the model parameters.
POTFGN+Period model with optimized parameters.
(a) Shown (black time series) is a realization of a fractional
Gaussian noise (FGN) with a long-range persistence strength of
βmodel=0.24 (normalized to a mean = 0.0 and
variance = 1.0) that is superimposed with a periodic component (period:
T= 2030 years; amplitude: A= 0.30). This synthetic time series is
given a time resolution of Δt=1 year and a length Nyear= 9271 years. Peaks over thresholds (POTs) are considered (as in Fig. 12)
for five thresholds (three shown here), θ01, θ12, etc.,
where input (FGN + period) values < θ01 translate to
xyear= 0 flood yr-1, input values between θ01
and θ12 result in xyear=1 flood yr-1, etc. The
resultant model series of the number of floods per year (grey and blue
circles) for 9271 years is constructed to have a Poissonian one-point
distribution and is shown at the bottom of (a). (b) Shown
are 50 more model realizations of the POTFGN+Period model as
number of floods per year (see the legend for the colour code), with maximum
xyear= 4 floods yr-1. (c) Time-dependent flood
rate Λ(t) given as a function of time t (see Fig. 11a, text and
Appendix C) using 1000 model realizations similar to (b). We first
compute a kernel density estimator for each of the 1000 model realizations.
Then we take the mean of these at each time step, giving us a mean Λ(t) (dashed black line). We then compute the 95 % confidence band limits
(dark grey bands), by ordering the 1000 values of Λ(t) for each time
step t and choosing the 25th and 975th values. As the flood rate of the
first and last 500 years might be affected by edge effects, the corresponding
dark grey colour banding is changed to light grey. For a comparison, the
corresponding time-dependent flood rate for the palaeoflood data (solid black
line), as shown in Fig. 11a, is given. (d) Black diamonds
represent the mean number of floods per year, decade, and century based on
1000 realizations of our POTFGN+Period model as given in the
previous panels, with error bars 95 % confidence intervals. For
comparison are shown (i) the histogram (also shown in Fig. 6a) of the
original palaeoflood time series number of floods per year (blue bars),
decade (orange bars), and century (green bars), and (ii) the corresponding
Poisson model (grey diamonds, also shown in Fig. 6a) with respective rate
parameters λyear=0.083 floods yr-1,
λdecade= 0.83 floods decade-1, and
λcentury=8.3 floods century-1.
Quantifying correlation, clustering, and cyclicity properties of the
POTFGN+Period model
In the previous subsection, we have shown that it is possible to use the POTs
of a fractional noise that has been superimposed with a 2030-year period to
create an integer-valued time series with long-range persistence
(correlation) properties and periodicity. In addition, we have been able to
capture the same “number” of values in our synthetic flood time series as
in our palaeoflood record.
Now we want to evaluate the strength of clustering caused by persistence and
the contribution of the long-term periodicity for different model parameters
of our POTFGN+Period model. This is done by creating ensembles of realizations for different pairs of model parameters and by measuring
the strength of long-range persistence, the shape parameter of the Weibull
distribution of the interevent occurrence times, and the periodic rate
fluctuations of these realizations, and by comparing them with the values
measured for the palaeoflood record. The model has two parameters:
(i) βmodel, the long-range persistence strength of
the underlying fractional Gaussian noise, and (ii) Amodel, the
amplitude of the superimposed periodic component with a period T= 2030 years. Both parameters were sampled with a step size of 0.02 from
0.00 to 1.00, resulting in a square grid of 51 × 51 = 2601 pairs
of model parameters. For each pair of model parameters
(βmodel,Amodel), 1000 synthetic flood series
(model realizations) were produced using the peaks-over-thresholds method
described in Sect. 5.1 and illustrated in Fig. 12. The differences
between the realizations for one and the same pair of model parameters are
caused by different noise realizations.
The amplitude of the best-fitting sinusoidal flood rate Arate
assuming a period of T= 2030 years and the shape parameter kW
of the best-fitting Weibull distribution of the interevent occurrence times
were determined for each model realization (xyear). Further, each
resultant palaeoflood yearly series (xyear) was aggregated to
give the number of floods per decade (xdecade), and then (using
power-spectral analysis and detrended fluctuation analysis) its strength of
long-range persistence βPS and βDFA was
computed.
In Fig. 14a to d, for each of the 51 × 51 “cells” of the
parameter space of our POTFGN+Period model, are given the mean
values of measured βPS, βDFA, kW,
and Arate (see the legend for colours and contours) for that
cell's (βmodel, Amodel) 1000 model realizations.
The graphs show that on average an increase in the model parameter
long-range persistence strength of the underlying fractional
Gaussian noise, βmodel, leads to an increased strength in
long-range persistence, βPS and βDFA, of the
synthetic floods per decade series xdecade, and to a decrease in
the shape parameter kW of the Weibull distribution of the
modelled interevent occurrence times (of xyear). Moreover, an
increase in the Amodel parameter of our POTFGN+Period
model (i.e. the amplitude of the superimposed periodic component
with period T= 2030 years) leads to a larger best-fitting sinusoidal
flood rate Arate. However, in both cases the intensity of the
increase depends also on the second model parameter, as otherwise the images
presented in Fig. 14a to d would have horizontally or vertically striped
structures. Furthermore, the mean values of βPS, βDFA, kW, and Arate (Fig. 14a to d) do
not give information about the spread of these measured parameters for a
specific model parameter. This implies that for a specific cell, we do not
have the information about the probability to have a model realization that
is similar to the Piànico palaeoflood data set.
Parameter fitting of the POTFGN+Period model
Here we identify pairs of model parameters (βmodel,
Amodel) that correspond to flood time series whose strength of
long-range persistence (βPS or βDFA),
interevent occurrence time distribution (measured as the shape parameter of
the Weibull distribution kW), and best-fitting sinusoidal flood
rate Arate have values close to those measured for the
palaeoflood record from Piànico. From the 2.6 × 106
synthetic flood series realizations generated in Sect. 5.2, the
realizations' parameters are within the range of our palaeoflood original
series parameters as follows: 162 213 realizations (6.2 % of the total
number of model realizations) with 0.370 ≤βPS≤ 0.410, 157 782 realizations (6.1 %) with 0.23 ≤βDFA≤ 0.27, 132 025 realizations (5.1 %) with
0.76 ≤kW≤ 0.81, and 152 056 realizations (5.8 %)
with 0.0450 ≤Arate≤ 0.0530. In Fig. 14e the
corresponding model parameters of these realizations are presented as clouds
of points and each point stands for a POTFGN+Period model
realization that is similar to the Piànico palaeoflood record with
regards to a single measure of correlation or clustering. These points are
located in specific areas: for instance, model realizations with a strength
of long-range persistence βPS similar to the value of the
palaeoflood record (pink dots) are found for
βmodel < 0.8 and on the right-hand side of the line
connecting the points (βmodel= 0.4, Amodel= 0) and (βmodel= 0.0, Amodel= 0.8).
Realizations with values of Arate close to 0.049, i.e. the value
of the best-fitting sinusoidal flood rate of the palaeoflood record, are
centred around Amodel= 0.32 (green dots) with a spread that
grows with increasing values of βmodel. Next, we will
identify the model parameters βmodel and Amodel
which lead to model time series that best match our palaeoflood record.
Figure 14e also shows that there are just a very few model parameters that
belong to all four clouds and thus are in the desired ranges of
βPS, βDFA, kW, and
Arate. To identify optimum model parameters, we choose
POTFGN+Period model realizations where the following four
parameters intersect: βPS and βDFA
(characterize strength of long-range persistence, pink and brown dots
respectively), kW (characterizes clustering, blue dots), and
Arate (characterizes cyclicity strength, green dots). In other
words, we find where the brown, pink, blue, and green dots intersect. Just
143 realizations (presented as black dots in Fig. 14e) out of the
2.6 × 106 model realizations are part of this intersection.
They are located in a small area of the 2-D parameter space: 0.10 ≤βmodel≤ 0.30 and 0.20 ≤ Amodel≤ 0.38. It should be noted that this area does not intersect with the
x-axis or the y-axis, meaning that both model parameters are necessary
for an appropriate description of the palaeoflood data. Because 143
realizations were not sufficient for effectively investigating (and
visualizing) the 2-D probability distribution of βmodel and
Amodel, for 0.06 ≤βmodel≤ 0.34 and
0.16 ≤Amodel≤ 0.42 (a slightly extended area as to that
just found) we created more realizations (10 000 per cell) to gain a total
of 1518 realizations with the desired long-range persistence, clustering, and
cyclicity properties. The 2-D probability density of these points has been
approximated by a normalized 2-D histogram (Fig. 14f). The grid cell of
βmodel= 0.25 and Amodel= 0.30 (presented as a
red bullet) has the highest probability. This pair of model parameters is
considered to be the best fitting. Approximately 50 % of the 1518 model
realizations that are similar to the palaeoflood data set are found for model
parameters 0.19 < βmodel < 0.29 and
0.26 < Amodel < 0.34; this range is indicated
by the blue dashed lines in Fig. 14f. These ranges give an estimate of the
accuracy of POTFGN+Period model parameter determination.
In summary, the model evaluation provides evidence that both model parameters
of the strongly fluctuating input signal (the strength of long-range
persistence of the fractional noise and the amplitude of the sine wave
superposed onto it) of our POTFGN+Period model are required for
an appropriate modelling of the clustering properties of the palaeoflood
record. Optimum parameters including error bars (βmodel= 0.24 ± 0.05 and
Amodel= 0.30 ± 0.04) have been determined. One
realization of the model with optimum parameters is given in Fig. 15a.
Model confrontation
In the last subsection, we specified the parameters of our
POTFGN+Period model in order to reproduce the observed clustering
and long-term periodic properties. Now we will investigate further model
properties such as the distributions of floods per decade or per century and
confront these properties with the values measured for the palaeoflood record
from Piànico.
For a comparison of the properties of the palaeoflood data and of the
POTFGN+Period model with the specified parameters, we have
created 1000 model realizations (50 of these xyear series are shown
in Fig. 15b). For each of the 1000 model realizations the time-dependent
flood rate Λmodel(t) (Sect. 4.6) was computed which
enabled the calculation of the mean time-dependent flood rate
Λ‾modelt and its 95 %
confidence intervals (Fig. 15c). The mean time-dependent flood rate
Λ‾modelt fluctuates
periodically with a 2030-year period and replicates the long-term changes in
rate fluctuations qualitatively. However, the amplitudes of the rate changes
for some time intervals (t= 0 to 600 years, 3800 to 4800 years, and 5500
to 6800 years) are not correctly captured, i.e. they are outside the 95 %
confidence intervals of Λmodel. This is due to the
stationary character of our palaeoflood model and the non-stationary
character of the long-term rate fluctuations.
Statistics of long-range persistence, clustering, and cyclicity
properties of the specified (βmodel= 0.24,
Amodel= 0.30) palaeoflood model; 1000 model realizations
were created and their long-range correlations (βPS and
βDFA), clustering (kW), and cyclicity
properties (Arate) were quantified. Given is for each measure
the mean value and the 2.5th and 97.5th percentiles of the model realizations,
the value measured for the palaeoflood record, and the percentile of the
value measured with respect to the values of the model.
MeasureModel: mean valuePalaeoflood recordPercentile of(2.5th and 97.5th percentiles)palaeofloodrecordβPS (computed for ndecade)0.26 (0.14, 0.40)0.3997 %βDFA (computed for ndecade)0.21 (0.07, 0.34)0.2574 %kW0.79 (0.74, 0.84)0.7948 %Arate0.050 (0.038, 0.063)0.04948 %
For the ensemble of all model realizations, the histograms of the number of
floods per year, per decade, and per century were computed, such that the
mean number of floods per time unit with 95 % error bars could be
estimated (Fig. 15d). The number of Piànico palaeofloods per year,
decade, and century (as discussed in Sect. 4.1 and seen in Fig. 6a) fit
well within the 95 % confidence intervals of the considered histograms.
Finally, for the ensemble of all POTFGN+Period (with specified
parameters) model realizations, the clustering and correlation properties
were quantified: the persistence strength (βPS, βDFA), the shape parameter of the Weibull distribution of the
interevent occurrence times (kW), and the time-dependent flood
rate (Λmodel(t)) were computed (see Table 2). Table 2
shows that the correlation properties (βPS, βDFA) of the model are on average weaker than those of the
palaeoflood record. Nevertheless, the 95 % confidence intervals from the
model contain the original palaeoflood values for βPS,
βDFA, kW, and Arate. The clustering
properties (kW, Arate) of the data are excellently
captured by the model as the mean values of the corresponding measures
computed for the model realizations are very close to the values measured for
the data.
In summary, we find a good agreement between the POTFGN+Period
model and the palaeoflood data series in particular with respect to the
distribution of floods per time unit and the clustering properties.
Using and analysing realizations from the POTFGN+Period
model
We have shown in Sect. 5.4 above that our POTFGN+Period model
reproduces several properties of the palaeoflood time series, including the
distribution of floods per year, decade, and century as well as long-range
persistence, clustering, and cyclicity properties. We now use realizations
from the POTFGN+Period model with optimized parameters to explore
how the number of floods per time unit (e.g. per decade or century) is
related to the number of floods in the following time unit (e.g. does a
century (decade) with a few floods tend to follow a century (decade) with a
few floods?). To do this we create one long model realization with
109 years (108 decades, 107 centuries). We computed the
probability of floods per century for given numbers of floods in the
preceding century.
Using the optimized POTFGN+Period model for
forecasting the number of floods in 1 century based on the number of floods
in the previous century: based on the palaeoflood model realizations for 109 years are shown (a) the 2-D probability of the
number of floods per century (xcentury(j)) and the number of
floods in the succeeding century (xcentury(j+1)) (background grid
colours; see the legend for the scale). The solid grey diagonal line
represents the equality of the number of floods in both centuries
(xcentury(j+1)=xcentury(j)). (b) The
2-D probability distribution shown in (a) is cut vertically at
xcentury=0, 8, and 16 floods century-1 and the
corresponding probability densities of floods per centuries after a century
with xcentury=0, 8, and 16 floods century-1 are presented
as coloured lines (see the legend). Also given is the probability density of
floods per century of the palaeoflood model.
Figure 16a shows the 2-D probability distribution of the number of floods
per century (ncentury(j)) and the number of floods in the
succeeding century (ncentury(j+1)). This distribution is
concentrated along the diagonal line ncentury(j+1) =ncentury(j) and thus indicates positive correlations in the number
of floods per century time series realizations. Figure 16b presents the
distribution of the number of floods per century if the preceding century
contained 0, 8, or 16 floods century-1. These distributions are unimodal
with a systematic shift in the mode and a dispersion that is increasing with
the numbers of floods in the preceding century. In other words, the
uncertainty of the forecasted number of floods per century increases with the
number of floods in the preceding century.
Our results shown in Fig. 16 imply that low numbers of floods
century-1 follow low ones, and
that high numbers of floods century-1 follow high ones. In other words, a
century with no floods is much more likely to follow a century with no or a
very few floods rather than a century with many floods, and a century with
many (e.g. 20) floods is much more likely to follow a century with many (e.g.
20) floods than a century with no or a very few floods. However, this holds
only in the statistical sense, and successions of centuries with many floods
can certainly follow centuries with very few floods (and vice versa). For
example, in the palaeoflood data set we have found a “jump” from 24 to 4
floods century-1 (between the 39th and 38th centuries). In our model the
probability that a century with 4 floods will follow a century with 24 floods
is 0.02 % on average, but will be much higher for time periods with a
high amplitude of the periodic component of the input signal. We also observe
that in our original palaeoflood time series there is a jump from 7 to 25
floods century-1 between the 46th and 45th centuries. A jump of this
magnitude in our optimized POTFGN+Period model has a probability
of 3.9 % based on our long synthetic realization, and is thus well
captured by the model. Furthermore, we found for the
POTFGN+Period model that medium changes in flood frequencies are
fairly likely, as for instance a transition from 25 to < 10 floods
century-1 has a probability of > 20 %. In a similar
manner, we have also examined the summary statistics for the number of floods
per decade using our long synthetic time series realization. For the
simulated 108 decades, we find between 0 and 13 floods decade-1,
although > 10 floods decade-1 is very unlikely. Similar to
the centennial data, we find indications of correlations that decades with
few (many) floods are followed by decades with few (many) floods, i.e. on
average, low values follow low ones and high values follow high ones.
Summary
We have presented a 9.3 kyr comprehensive flood record at sub-annual
resolution, obtained from the varved interglacial Pleistocene sediments of
the Piànico–Sèllere Basin. A lacustrine sediment unit of 9.5 m
thickness has been considered which consists of an almost continuous
succession of about 15 500 varves that are interpreted as annual cycles.
Approximately 8 % of the varves contain 1–3 detrital layers which are
considered to be the result of channelized streamflow that originated in the
hills surrounding the Piànico–Sèllere Basin and triggered by extreme
precipitation events (Mangili et al., 2005). Our analysis was aimed at
understand the temporal succession of detrital layers.
The analysed palaeoflood record is unique as it comprises the relative
timings of 771 flood events, is continuous (except a gap of 65 years,
which is less than 1 % of the length of the 9336-year time period), and was not
affected by anthropogenic influences. This palaeoflood record provides an
example of the high natural variability of flood frequency over a long period
(e.g. variations of 0–31 floods century-1). Because of the
comprehensive data set, the temporal succession of palaeofloods could be
extensively studied as to its underlying statistics of correlations,
clustering, and cyclicity. We showed that the correlations of the decadal
number of floods over the 9.3 kyr palaeoflood are long range, with a
long-range persistence strength of βPS≈ 0.39
(power-spectral analysis) and βDFA≈ 0.25 (DFA).
Additionally, the flood frequency is modulated by a long-term cyclicity with
a period of T= 2030 years. The palaeofloods are also shown to be
temporarily clustered as the interevent occurrence times are Weibull
distributed (with a shape parameter of kW= 0.78).
We have derived a model (POTFGN+Period) that is based on POTs of
a fractional Gaussian noise (FGN) superimposed by a long period signal. This
model allows us to construct many realizations of an event series with
properties similar to those identified for the original palaeoflood time
series. Should more information (e.g. higher resolution, extending the data
series in time) become available related to our original palaeoflood series,
then the parameters of correlation, clustering, and cyclicity are likely to
be slightly changed, and a modified model can then be easily created.
We used our model to create 2 600 000 synthetic flood series with different
parameters, and then confronted the clustering, correlation, and cyclicity
properties of our original palaeoflood record with the synthetic series to
come up with optimized parameters in our POTFGN+Period model.
Based on the optimized POTFGN+Period model, we found that it is
important when modelling our original palaeoflood time series that the model
needs both long-range correlations and a slowly varying
component (periodicity) to capture the correlation, clustering, and cyclicity
properties of the palaeoflood record. In other words, the observed temporal
flood clusters in our 9.3 kyr time series cannot be explained by either
long-range correlations or slow cyclical changes; rather, both components
need to be present.
As an example of the application of our optimized POTFGN+Period
model we use it to create long simulations with parameters similar to that of
our palaeoflood time series. We show that centuries with no or a few floods
tend to follow each other and that centuries with many floods tend to follow
centuries with many floods. Those centuries that are extremely dry or wet we
mostly attribute to the influence of noise in both the 9.3 kyr palaeoflood
record and the corresponding model that we have created. We also found that
the uncertainty of the forecasted number of floods per century increases with
the number of floods in the preceding century.
Our research in this paper combines the statistical analysis of correlations,
clustering, and cyclicity in a very complete and unique interglacial record
of floods from the Pleistocene, allowing one to create a model to simulate
many realizations with similar parameters to our original series. We believe
that this approach is applicable to other environmental event time series
(e.g. palaeo-hazards); where event magnitude is unknown, the timings are
unequally spaced in time, and where the one-point probability distribution of
the number of events per time unit is strongly asymmetric (i.e.
non-Gaussian). This is true for many palaeo and historical environmental
event series (e.g. earthquakes, wildfires, and volcanic eruptions). We also
believe our approach of creating an optimized model that is congruent with
the correlation, clustering, and cyclicity parameters of the original event
series is one that is generally applicable, and useful for creating many
“realizations”.
The palaeoflood data used here, the number of detrital layers per year, are available at https://doi.org/10.1594/PANGAEA.879779 (Mangili et al., 2017).
Detrital layer thickness figure
Detrital layer thickness: (a) thickness and temporal
location of the detrital layers in the varved sediment; 771 layers are
located within the 9271 varves. The x-axis represents relative age, with
0 years representing the most recent varve and increasing values indicating
further back in time. The grey bar represents a sediment gap of 65 years. The
detrital layer thickness (green vertical bars), shown on a logarithmic scale,
ranges from 0.002 to 23 mm. Some layers are just reported and not measured
(black dots in a horizontal line towards the top of panel a). The
detrital layers are unequally distributed over time. A histogram of the layer
thickness is given in (b).
Semivariogram analysis of palaeoflood time series
In the hydrological sciences, semivariograms are a standard tool for
investigating second-order correlations. The semivariogram s(τ) is a
function of the lag τ as
sτ=12N∑j=1N-τnj+τ-nj2
where N is the length of the time series. The semivariogram is a linear
function of the autocorrelation function C (see Eq. 5). In case of a
stationary time series, the semivariogram s saturates at a certain level
(the sill).
Semivariogram (Eq. B1) of the three palaeoflood time series,
nyear, ndecade and ncentury for lags 0≤τ≤7500 years. Also shown is the 95 % range of the
semivariogram of the best-fitting Poisson model (see Sect. 3.2).
Shown in Fig. B1 are semivariograms of the three palaeoflood time series,
nyear, ndecade, and ncentury, for lags
0 ≤τ≤ 7500 years. Also shown in each panel is the corresponding
95 %
range of the best-fitting Poisson model. The semivariograms of the number of floods per
decade and per century (ndecade and ncentury) are
significantly different from the semivariograms of the Poisson model.
Furthermore, there is no sill; i.e. due to a long-term cyclicity, the
semivariogram does not approach a limit value for large lags.
Computing the time-dependent flood rate with kernel density
estimators
Determining the time-dependent flood rate is necessary for detecting gradual
changes in flood frequency and for identifying long-term cyclicity in flood
occurrence. Here, the time-dependent flood rate and its corresponding
95 % confidence intervals have been determined by kernel density
estimation (Silverman, 1986). The kernel K is a non-negative function which
is symmetric with respect to 0. For our purposes we use a Gaussian kernel
KG with bandwidth σ:
KGτ=12πσ2exp-τ22σ2,
with τ a discrete time step (for this paper, τ= 1 year). The
kernel is truncated such that it is defined only for -3σ≤τ≤3σ. This specific kernel is a discretized and truncated Gaussian
probability density function. The bandwidth parameter σ that is the
standard deviation of the Gaussian distribution controls the width of the
kernel. For this paper, we have applied a bandwidth of σ= 250 years,
and therefore the Gaussian kernel that is created has a width of 1500 years
(range of ±3σ). The idea of kernel density estimation is to
replace each flood in the floods per year function nyear by a
smooth kernel and to consider the superposition of all kernels as a measure
for the number of floods per year. This is implemented by convolving
Eq. (C1) with the time series of the number of floods per year,
nyear. At step one, the time-dependent flood rate Λ(t)
with respect to the specific kernel KG is computed for times t= 1, …, T by
Λt=∑τ=-3σ3σnyeart-τ1t-τKGτ∑τ=-3σ3σ1t-τKGt,
with 1 the temporal range function of our time series, i.e.
1(t)= 1 for values of t in the temporal range of the
time series and 1(t)= 0 for values outside (i.e. t≤ 0, t > T or t is inside in the gap). Thus, the
numerator is the convolution of the kernel density KG and the
number of floods per year nyear and the denominator is the
convolution of the kernel density and the range function of the time series.
Note that the range function is only needed for approximately handling
nyear values that are near the two ends of the time series and
the gap.
After convolution and derivation of the time-dependent flood rate Λ(t), in step 2, we now derive their respective 95 % confidence
intervals. This requires the computation of the effective number of data
points N(t) for each time point t, which describes the number of time
series elements that contribute to the kernel density estimate of the rate,
Λ(t):
Nt=2σ∑τ=-3σ3σ1t-τKGτ.
This effective bandwidth N(t) is smaller than 2σ for time points
t which are close to the beginning and end of the observational interval
(e.g. t= 1 or t=T), and equal to 2σ for time points t far
from these interval ends.
The distribution of the time-dependent rate Λ(t) is related to a
two-parameter Gamma distribution (Johnson et al., 1993). This allows us to
calculate the corresponding 95 % confidence intervals, i.e. the lower
(Λ0.025) and upper (Λ0.975) endpoints of the 95 %
confidence intervals, as
Λ0.025t=2FΓ0.025,0.5N(t)R(t),1/Nt,Λ0.975t=2FΓ0.975,{0.5N(t)Rt}+1,1/Nt,
with FΓ the cumulative distribution function of the two-parameter
Gamma distribution and (i) 0.025 and 0.975 the considered quantile, (ii) 0.5
N(t)R(t) and ({0.5N(t)R(t)}+1) the shape parameter, and (iii) the value 1 the
scale parameter. For comparison, we also calculated confidence intervals of
the time-dependent rates by applying a bootstrap-based method as introduced
by Mudelsee (2014) in his Sect. 6.3.2. The two methods for calculating
confidence intervals resulted in very similar values.
Variance of residuals for fitting a sinusoidal
function
Shown is the variance of the residuals (σ2(ε))
for fitting a sinusoidal function (Eq. 16) to the floods per year series
nyear as a function of the period T of this periodic function.
The function for the variance of the residuals has a minimum at
Tmin=2030 years (shown as a star in the figure).
AB and CM jointly
carried out the sampling and field work for this study. CM carried out varve
and detrital layer microfacies analysis under the project supervision of AB.
AW and BDM did the statistical analyses, model development, and theoretical
framework, with AW carrying out most of the statistical analysis programming.
AW and BDM jointly led the paper for writing and graphical presentation of
the results.
The authors declare that they have no conflict of
interest.
Acknowledgements
We would like to thank the GeoResearch Centre Potsdam (GFZ),
Gabriele Arnold, Michael Koehler, and Dieter Berger for providing high-quality thin sections. We also thank Erwin Zehe
and an anonymous reviewer for their thoughtful comments which improved the
science and readability of the manuscript.
The article processing charges for this open-access
publication were covered by the Max Planck Society. Edited by: Andràs Bàrdossy Reviewed
by: Erwin Zehe and one anonymous referee
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