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To seek stochastic analogies in key processes related to the hydrological cycle, an extended collection of several billions of data values from hundred thousands of worldwide stations is used in this work. The examined processes are the near-surface hourly temperature, dew point, relative humidity, sea level pressure, and atmospheric wind speed, as well as the hourly/daily streamflow and precipitation. Through the use of robust stochastic metrics such as the K-moments and a secondorder climacogram (i.e., variance of the averaged process vs. scale), it is found that several stochastic similarities exist in both the marginal structure, in terms of the first four moments, and in the secondorder dependence structure. Stochastic similarities are also detected among the examined processes, forming a specific hierarchy among their marginal and dependence structures, similar to the one in the hydrological cycle. Finally, similarities are also traced to the isotropic and nearly Gaussian turbulence, as analyzed through extensive lab recordings of grid turbulence and of turbulent buoyant jet along the axis, which resembles the turbulent shear and buoyant regime that dominates and drives the hydrological-cycle processes in the boundary layer. The results are found to be consistent with other studies in literature such as solar radiation, ocean waves, and evaporation, and they can be also justified by the principle of maximum entropy. Therefore, they allow for the development of a universal stochastic view of the hydrological-cycle under the Hurst–Kolmogorov dynamics, with marginal structures extending from nearly Gaussian to Pareto-type tail behavior, and with dependence structures exhibiting roughness (fractal) behavior at small scales, long-term persistence at large scales, and a transient behavior at intermediate scales.
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hydrology
Article
A Global-Scale Investigation of Stochastic Similarities in
Marginal Distribution and Dependence Structure of Key
Hydrological-Cycle Processes
Panayiotis Dimitriadis * , Demetris Koutsoyiannis , Theano Iliopoulou and Panos Papanicolaou


Citation: Dimitriadis,P.;Koutsoyiannis,
D.; Iliopoulou, T.; Papanicolaou,P. A
Global-Scale Investigation of Stochastic
Similarities in Marginal Distribution
and Dependence Structure of Key
Hydrological-Cycle Processes.
Hydrology 2021,8, 59. https://doi.org/
10.3390/hydrology8020059
Academic Editors: Nicholas Dercas
and Minxue He
Received: 31 December 2020
Accepted: 15 March 2021
Published: 31 March 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
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iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Department of Water Resources and Environmental Engineering, School of Civil Engineering,
National Technical University of Athens, Heroon Polytechneiou 5, 15880 Zographou, Greece;
dk@itia.ntua.gr (D.K.); tiliopoulou@hydro.ntua.gr (T.I.); panospap@hydro.ntua.gr (P.P.)
*Correspondence: pandim@itia.ntua.gr
Abstract:
To seek stochastic analogies in key processes related to the hydrological cycle, an extended
collection of several billions of data values from hundred thousands of worldwide stations is used
in this work. The examined processes are the near-surface hourly temperature, dew point, relative
humidity, sea level pressure, and atmospheric wind speed, as well as the hourly/daily streamflow
and precipitation. Through the use of robust stochastic metrics such as the K-moments and a second-
order climacogram (i.e., variance of the averaged process vs. scale), it is found that several stochastic
similarities exist in both the marginal structure, in terms of the first four moments, and in the second-
order dependence structure. Stochastic similarities are also detected among the examined processes,
forming a specific hierarchy among their marginal and dependence structures, similar to the one
in the hydrological cycle. Finally, similarities are also traced to the isotropic and nearly Gaussian
turbulence, as analyzed through extensive lab recordings of grid turbulence and of turbulent buoyant
jet along the axis, which resembles the turbulent shear and buoyant regime that dominates and
drives the hydrological-cycle processes in the boundary layer. The results are found to be consistent
with other studies in literature such as solar radiation, ocean waves, and evaporation, and they can
be also justified by the principle of maximum entropy. Therefore, they allow for the development
of a universal stochastic view of the hydrological-cycle under the Hurst–Kolmogorov dynamics,
with marginal structures extending from nearly Gaussian to Pareto-type tail behavior, and with
dependence structures exhibiting roughness (fractal) behavior at small scales, long-term persistence
at large scales, and a transient behavior at intermediate scales.
Keywords:
stochastic similarities; hydrological cycle; global scale; heavy-tail; long-term persistence;
fractal behavior; Hurst–Kolmogorov dynamics
1. Introduction
Stochastic approaches have been preferable over the deterministic ones when the
identification and simulation of the complicated fluctuations observed in geophysical
processes are the tasks at hand. In recent times, we characterize a process as complex if it is
difficult to analyze or explain in a simple manner.
The origin of the word “complex” comes from Latin but has been re-borrowed from
ancient Greek (originated from the verb “
συµπλ´
εκω
”) and is attributed to “a whole com-
prised of parts”. It constitutes the Latin preposition “com” or “cum”, which is related to
the Greek preposition “
συν
” and is used, usually at the beginning of a word, to declare
union, ensemble, etc., as well as the Latin verb “plectere”, which comes from the Greek
verb “πλ´
εκω” meaning “weave”, “twine”, etc.
Climate dynamics is characterized by high complexity since it involves the spatio-
temporal evolution of numerous geophysical variables (i.e., multivariate stochastic pro-
cesses) interacting with each other in a nonlinear way, forming, among others, the hydro-
Hydrology 2021,8, 59. https://doi.org/10.3390/hydrology8020059 https://www.mdpi.com/journal/hydrology
Hydrology 2021,8, 59 2 of 26
logical cycle. Nevertheless, even if we could determine a set of physical laws that describe
in full detail the complexity of climate dynamics, it would be impossible to combine the
equations for the purpose of predictability due to the existence of chaos, i.e., a nonpredic-
tive sensitivity to initial conditions. For example, consider the analysis of Poincare [
1
] for
the three-body problem, where chaotic behavior emerges from the equations of classical
mechanics when studying the interacting gravitational forces among three bodies (e.g.,
planets). Similar behavior came into sight from Lorenz [
2
] while applying a simplified
set of equations for the analysis of atmospheric dynamics. Lorenz came across the idea
that nonlinear dynamical systems may have limited predictability, whose time horizon for
weather prediction was estimated to be around two weeks, even if the model is perfect and
even if the initial conditions are known almost perfectly. Later on, numerous methodolo-
gies were initiated not for predicting the exact outcome of a nonlinear system, which as we
already explained may be impossible, but rather for estimating the limits of this prediction
through the alternative approach of stochastic analysis.
However, a detailed stochastic analysis of such complicated fluctuations would de-
mand high availability of observational records at both small (e.g., hourly or even finer)
and large scales (e.g., several decades)—a rare case in the hydrological-cycle processes.
In this work, we use robust stochastic metrics to increase the extracted information from
time series in order to seek similarities among hydrological-cycle processes in terms of the
second-order statistics. In particular, for the analysis of the marginal structure, we imple-
ment the central knowable moments (K-moments) that enable more reliable estimation
from data since they substitute higher-order deviations from the mean by the use of the
probability distribution function [
3
]. The second-order dependence structure (commonly
expressed through the autocovariance function) is estimated through the climacogram (i.e.,
variance of the averaged process vs. scale) and other climacogram-based metrics (similar to
the commonly applied autocorrelation function and power spectrum). They have all been
shown to exhibit a smaller estimation bias in both large scales and in a variety of Markovian
and long-term persistent (LTP) processes, as well as other advantages, compared to the
aforementioned common metrics [4].
In seeking a broad and comprehensive investigation of the stochastic similarities
among the hydrological-cycle processes, we implement the above metrics to a global-scale
network of stations, as well as to the key processes of the hydrological cycle. To be specific,
we study the near-surface hourly temperature, relative humidity, sea level pressure, and
wind speed (extracted from the ISD database), as well as the hourly and daily stream-
flow (extracted from the USGS and CAMELS databases), hourly and daily precipitation
(extracted from the HPD and the GHCN databases), and we also take into account other
processes related to the hydrological cycle, such as hourly solar radiation, ocean waves and
evaporation, as analyzed at global scale in other studies. In total, we extract and handle
approximately 50
×
10
10
data values from over 2
×
10
5
hydroclimatic global-scale stations
on a daily, hourly, and subhourly resolution, with some dating even back to 1800. Specifi-
cally, we transform to hourly resolution the subhourly timeseries of air temperature, dew
point, humidity, wind-speed, and sea level pressure (with around 15,000 active stations
each). Similarly, we transform to hourly resolution over 10,000 timeseries of streamflow
and 5,000 timeseries of precipitation located in the USA, and we merge them with over
600 USA streamflow and over 10
5
worldwide precipitation timeseries of daily resolution.
It is noted that after quality control, only the 10% of records are finally selected for the
analysis (Figure 1and Table 1).
Hydrology 2021,8, 59 3 of 26
Hydrology 2021, 8, x FOR PEER REVIEW 3 of 27
Figure 1. Locations of the selected stations for each hydrological-cycle process: (a) near-surface air-
temperature; (b) relative humidity and dew-point; (c) precipitation; (d) streamflow; (e) wind; and
(f) sea-level pressure.
Table 1. Mean Time Resolution of Each Source, Total Number of Selected Stations and Total Records for Each Hydrological-
Cycle Process, and from the Massive Set of Databases. The Humidity Time Series Are Extracted by the Combining the
Temperature and Dew-Point Datasets.
Near-Surface
Temperature Dew Point Humidity Sea Level
Pressure
Wind
Speed Precipitation Streamflow
Temporal resolu-
tion Hourly hourly hourly hourly hourly hourly/daily hourly/daily
Total number of
stations/time se-
ries
6613 5978 4025 4245 6503 93,904 1815
Total number of
records (×10
6
) 907.1 730.0 540.2 364.9 781.7 938.7 13.5
Time period 1938–today 1938–today 1940–today 1939–today 1939–today 1778–today 1900–today
Finally, the microscopic processes driving and generating the hydrometeorological
ones are governed by turbulent state. For example, the size of the drops, which is highly
Figure 1.
Locations of the selected stations for each hydrological-cycle process: (
a
) near-surface air-temperature; (
b
) relative
humidity and dew-point; (c) precipitation; (d) streamflow; (e) wind; and (f) sea-level pressure.
Table 1.
Mean Time Resolution of Each Source, Total Number of Selected Stations and Total Records for Each Hydrological-
Cycle Process, and from the Massive Set of Databases. The Humidity Time Series Are Extracted by the Combining the
Temperature and Dew-Point Datasets.
Near-Surface
Temperature Dew Point Humidity Sea Level
Pressure Wind Speed Precipitation Streamflow
Temporal
resolution Hourly hourly hourly hourly hourly
hourly/daily hourly/daily
Total number of
stations/time series 6613 5978 4025 4245 6503 93,904 1815
Total number of
data values (×106)907.1 730.0 540.2 364.9 781.7 938.7 13.5
Time period 1938–today 1938–today 1940–today 1939–today 1939–today 1778–today 1900–today
Hydrology 2021,8, 59 4 of 26
Finally, at the smallest time scales the processes driving and generating the hydrome-
teorological ones are governed by turbulent state. For example, the size of the drops, which
is highly linked to the form and intensity of precipitation events, is strongly affected by
the turbulent state of small-scale atmospheric wind [
5
]. In addition, in a physical basis,
the rain rate is found to be a function of the gradient level wind speed, the translational
velocity of the tropical cyclone, the surface drag coefficient, and the average temperature
and saturation ratio inside the tropical cyclone boundary layer [
6
]. Therefore, since the
above key processes are dominated and driven by the turbulent shear and buoyancy in
the nearly Gaussian atmospheric boundary layer [
7
9
], we investigate whether further
stochastic similarities exist with respect to such turbulence time series, and specifically, to
a huge grid-turbulence lab database (provided by the Johns Hopkins University) and to
laboratory records of turbulent buoyant jets along the jet axis, both of which are found to
be nearly Gaussian and isotropic.
The current study focuses on the Hurst–Kolmogorov dynamics, which characterizes
a geophysical process exhibiting the Hurst phenomenon [
10
] (or else LTP), with an arbi-
trary marginal distribution function and dependence structure extending in a continuous
manner from extremely fine scales with fractal behavior to very large scales with LTP. The
mathematical description of the Hurst phenomenon is attributed to Kolmogorov [
11
], who
developed it while studying turbulence, and it corresponded only to a power-law decay
autocorrelation function (e.g., [
12
]). To give credit to both contributing scientists and to
incorporate alternate short-range dependence with an arbitrary marginal structure, Kout-
soyiannis [
13
] named this general behavior as Hurst–Kolmogorov. In this work, similarities
are found to exist in both (a) the marginal structure, through the mixed Hurst–Kolmogorov
(e.g., Pareto–Burr–Feller for positively defined processes) probability density distribution
functions, which, depending on the selected shape and scale parameters, is applicable to
all the examined processes from (truncated) nearly Gaussian distributions to heavy-tail
Pareto ones; and (b) the second-order dependence structure, through a Hurst–Kolmogorov
generalized model, which is also expanded to include the observed curved behavior at the
intermediate scales. These similarities can be well described within the framework of the
maximum entropy and the Hurst–Kolmogorov dynamics (see definitions in [
14
18
]), and
can be implemented through the method of moments (e.g., explicit models), through nonlin-
ear transformation (e.g., copulas), or disaggregation (e.g., downscaling and pulse models)
schemes, as summarized for the field of
Hydrology [19,20] and beyond [3,4,2131],
preserv-
ing both the marginal and dependence structures for a vast range of scales, including double
periodic and intermittent behaviors (see the discussion in [18], and references therein).
In Section 2, we introduce the applied metrics and methodology, while in Section 3,
we present several statistical characteristics of the selected stations of the global databases,
as well as the results obtained from the analysis with a focus on the marginal and the
dependence structures. Finally, in Sections 4and 5, we summarize our findings, and
we discuss how they may allow for the development of a uniting stochastic view of the
hydrological-cycle processes under the Hurst–Kolmogorov dynamics, expanding from
nearly Gaussian to Pareto-type tail behavior, and from fractal and intermittent behavior at
small scales to LTP behavior at large scales.
2. Methodology
In this section, we define the central K-moments and the climacogram-based metrics
that are used for the investigation of the stochastic similarities of the key hydrological-
cycle processes.
Hydrology 2021,8, 59 5 of 26
2.1. Dependence Structure Metrics
We focus on the dependence structure of second-order statistics (see the discussion
in [
3
,
4
,
26
,
32
,
33
]), which is commonly estimated through the autocovariance function, c(h),
where his the continuous time lag in time units, or through the power spectrum, whose
definition is based on the autocovariance function, i.e.,
s(w)
:
=
2
R
c(h)cos(2πwh)dh
,
where
w
is the continuous frequency in inverse time units. A common estimator of the
latter is the so-called periodogram, which is often used as a preferable estimator over the
autocovariance classical estimator [
34
,
35
]. Since these two estimators are based only on
the domain of lag and frequency, Beran [
36
] attempted to implement a similar method to
the scale domain, by plotting in logarithmic axes the variance of the accumulated [
37
] or
else aggregated [
38
] process vs. scale (a method often misnamed as “aggregated-variance”;
however, variance is not aggregated but rather scale [
39
]). This attempt was inspired
by the empirical work of Smith [
40
] in agricultural crops and by other similar works
in other fields [
41
,
42
], but only to be later abandoned as a bad estimator of the long-
term persistence mainly due to the large estimation bias [
35
,
43
]. The alternative estimator
of the rescaled/range (R/S), which was introduced by Hurst [
10
] for the identification
of long-term persistence, was also deemed to exhibit similar issues as the aggregated-
variance method [
35
,
44
,
45
] or even worse [
19
,
46
]. In addition, since a single name for this
method did not exist (as, for example, for the periodogram or the correlogram methods),
Koutsoyiannis [
13
] coined the term climacogram to emphasize the graphical representation
and the link of the concept to scale (i.e., climax in Greek), so as not to be confused with
the already established term of scale(o)gram. It is noted that the climacogram is explicitly
linked to the autocovariance c(h), i.e.,
c(h)=1
2d2h2γ(h)/dh2
, and thus, also to the
power spectrum [32].
Finally, the identification of the stochastic structure in the scale domain is revisited
much later and compared to the lag domain (i.e., through the correlogram estimator of
the autocorrelation function) and the frequency domain (i.e., through the periodogram
estimator of the power spectrum) but after theoretically defining the expressions, estima-
tors, and bias expressions for all three metrics [
20
]. Specifically, after properly adjusting
the climacogram for estimation bias ([
15
,
26
,
47
,
48
]), and the autocovariance and power
spectrum for estimation and discretization bias [
4
], and thus, after eliminating the main
criticized limitations of the three classical estimators, Dimitriadis and Koutsoyiannis [
4
]
show that the climacogram outperforms the other two estimators. Hence, a stochastic
process with short-term fractal and long-term persistent behaviors should be preferred to
be analyzed in the scale domain instead of in the lag and frequency domains.
Therefore, based on the scientific boost, the climacogram (and not the other two met-
rics) was found to be adequate for the identification and model building of a stochastic pro-
cess. Since then, interest in the scale domain and the climacogram estimator has increased,
and the climacogram has been implemented in education material [
49
], and has been
used to identify the LTP behaviour in various scientific studies, such as 2D precipitation
fields [
50
], multidimensional spatiotemporal domain [
51
], paleoclimatic temperature [
52
]
and precipitation [
53
,
54
], Bayesian statistical models of rainfall and temperature [
55
],
higher-order moments of skewness and kurtosis vs. scale in grid turbulence [
26
], annual
precipitation [
56
], water demand [
57
], daily river flows [
58
], precipitation and temperature
for a bivariate drought analysis [
59
], wind and solar energy [
60
], water-energy nexus [
61
],
solar radiation [
62
], wave height and period [
63
], daily streamflow [
64
], and monthly tem-
perature and precipitation ([
65
,
66
]), annual streamflow ([
30
,
66
]), ecosystem variability [
67
],
2D rock formations [
68
], urban streamflows [
69
], global temperature and wind of resolution
spanning 10 orders of magnitude from ms to several decades [
70
], disaggregation schemes
from daily to hourly rainfall and runoff [
71
], hourly wind and daily precipitation [
26
], fine
scale precipitation [
3
,
22
,
72
78
], fine scale wind [
26
,
70
], 2D landscapes [
79
,
80
], flood risk
assessment [
81
], bridge scour [
82
], art [
83
85
], spatiotemporal evolution of clustering [
86
],
alternative statistical moments (e.g., L-moments, K-moments; [
3
,
87
]), comparison of multi-
Hydrology 2021,8, 59 6 of 26
source data [
88
], daily extreme global temperature [
89
], hourly offshore and coastal wind
for energy assessment [90], and in many other applications [3,18].
The second-order climacogram, in terms of the variance, is defined as:
γ(k):=VarZk
0x(y)dy/k2(1)
where
k
is the scale in units of
x
, and similarly, the climacogram-based spectrum (CBS), or
else called climacospectrum, is defined as follows [3]:
ζ(k):=k(γ(k)γ(2k))
ln 2 (2)
The estimators of the above climacogram-based metrics can be expressed through the
estimator of the climacogram, i.e.,
ˆ
γ(k)=1
[n/κ]1
[n/κ]
i=1x(κ)
iˆ
µ2(3)
where
κ=k/
is the dimensionless scale,
is the time resolution of the process, [
n/κ
] is
the integer part of n/
κ
, and
x(κ)
i
is the i-th element of the averaged sample of the process at
scale κ, i.e.,
x(κ)
i=1
κ
iκ
j=(i1)κ+1
xj(4)
Although the above two metrics (i.e., climacogram and climacospectrum) contain the
exact same information, their estimators exhibit a lower estimation bias in different range
of scales. Particularly here, we use the climacogram for the identification of the dependence
structure behavior at large scales (or else Hurst behavior; [
10
,
37
,
43
,
91
]), and the CBS for
the investigation of small scales characterized by the fractal behavior [
92
95
], while for the
intermediate scales, both metrics are used (e.g., see [75,76]).
Since stations may have different lengths and recording resolutions, we adjust the
climacogram Varx(κ)
iat resolution for the estimation bias as follows [26]:
ˆ
γ(κ)=1
[n/κ]
[n/κ]
i=1x(κ)
iˆ
µ2+γ([n/κ]κ)(5)
where
γ(k)
is the climacogram model, which for the examined processes is shown to be
well described by a Hurst–Kolmogorov (HK) dynamics model, expressed as:
γ(k)=λ
1+(k/q)2M(1H)/M(6)
where
λ
is the variance of the the instantaneous process; qis a scale parameter in units
of the scale kthat affects the behavior of the intermediate scales of the process; Mis a
dimensionless shape parameter that describes the fractal behavior of the process at small
scales; and His the Hurst parameter indicative of the long-term persistence of the process,
i.e., for 0.5 < H< 1 the process exhibits LTP behavior, while for 0 < H< 0.5 an anti-persistent
behavior, and for H= 0.5 a white-noise behavior.
Hydrology 2021,8, 59 7 of 26
Another model that is found to adequately simulate the observed convex-shaped
behavior at the intermediate scales (see empirical evidence in all examined processes below)
can be expressed as:
γ(k)=λ
21+(k/q)2M(1H)/M+λk/q+ek/q1
(k/q)2(7)
where the second expression corresponds to a continuous-time Markov model. A more
generalized model includes the replacement of the second Markovian model with an
HK model [4].
The standardized climacogram and CBS, which are used here for the identification
of the dependence structure behavior, can be easily estimated by dividing them with the
sample variance of the process, i.e.,
ˆ
γ(k)/ˆ
γ(1)
and
ˆ
ζ(k)/ˆ
γ(1)
, respectively. An in-depth
analysis of other climacogram expressions for the HK dynamics can be found in [3].
2.2. Marginal Structure Metrics
The investigation of the marginal structure of the key hydrological-cycle processes
is based here on the estimation of high-order moments, which can be indicative of the
intermittence and tail behavior of the process (e.g., see several applications up to the 6th
moment-order in [
26
]). Therefore, we focus our investigation on the similarities presented
in the skewness and kurtosis coefficients, based on the unbiased estimators of the stan-
dardized central-moments (C-moments), as well as on the standardized L-moments, i.e.,
L-skewness and L-kurtosis, and the corresponding K-moments, which are described below.
Koutsoyiannis [
3
] has extensively illustrated that classical moments beyond the second
order are unknowable, especially in the presence of LTP, in the sense that their estimation
from data is not feasible. To tackle this issue, the so-called (knowable) K-moments were
introduced, in which the higher-order estimations are implemented by the use of the
probability distribution function. In particular, the central K-moments that are used for
estimation here, are invariant to a shift of origin and they are homogeneous in terms of
multiplication by a scalar, and thus, they enable more reliable estimations. The central
K-moments are defined as:
Kpq :=(pq+1)Eh(F(x))pq(xµ)qi(8)
Koutsoyiannis [
86
] also introduced the hyper-central K-moments, which can be esti-
mated directly from the central K-moments, but they present several advantages over the
latter in model fitting and inference from data, and can be defined as:
K+
pq :=(pq+1)Eh(2F(x)1)pq(xµ)qi(9)
for
pq
, where pand qare the moment orders,
x
is the stochastic process of interest
with marginal probability distribution
F(x)
(it is noted that underlined quantities denote
random variables), and µis the process mean.
The hyper-central K-skewness and K-kurtosis that are used here for illustrating the
behavior of the marginal structure can be expressed through the central K-moments,
respectively, as:
HCKs=K+
32
K+
22
=2K32
K22 2 (10)
and
HCKk=K+
42
K+
22
=4K42
K22 6K32
K22 +3 (11)
Hydrology 2021,8, 59 8 of 26
A sample estimator of the central K-moments for
q=
2, which is used here and is
approximately adjusted for bias, is derived as:
ˆ
Kp2=n
n1
n
i=p
p1
n
Γ(np+2)
Γ(n)
Γ(i)
Γ(ip+2)(x(i:n)ˆ
µ)2(12)
where
n
is the length of the sample,
ˆ
µ=n
i=1xi/n
is the estimator of the mean, and
x(i:n)
is
the sample of the process rearranged in ascending order.
For comparison, we show results from the classical unbiased estimators of the skew-
ness and kurtosis coefficients, denoted here as C-skewness and C-kurtosis, respectively.
Moreover, we show results for the classical L-moments by Hosking [
96
], but they are
estimated here through the hyper-central K-moments for comparison, respectively, as:
LCKs=K+
31
K+
21
=2K31
K21 3=λ3
λ2(13)
and
LCKk=K+
41
K+
21
=4K41
K21 8K31
K21 +6=4
5
λ4
λ2+6
5(14)
where
λ2
,
λ3
, and
λ4
, are the well-known L-moments of the 2nd, 3rd, and 4th order,
respectively. We note that the commonly used L-kurtosis, i.e.,
λ4/λ2
, is linearly related to
the modified version of the L-kurtosis shown in the latter equation.
The investigation here focuses on the marginal moments instead on the probability
distribution itself due to the large estimation bias entailed in the estimators of the latter.
Instead of directly fitting a distribution function to data, one may estimate a small number of
moments from data equal to the number of parameters in the target probability distribution
function, and then estimate the parameters of the distribution from the values of the
moments. A flexible probability distribution density function that can simulate processes
with a lower threshold, such as the ones examined here, is the truncated-mixed Pareto–
Burr–Feller (PBF) distribution, as named after (a) the engineer Pareto, who discovered
the family of power-type distributions while working on the size distribution of incomes
in a society [
97
,
98
]; (b) to Burr [
99
] who identified and analyzed this distribution family
(but without giving a justification), first proposed as an algebraic form by Bierens de
Haan; and (c) to Feller [
100
] who linked it to the Beta function and distribution through a
linear power transformation, which was further analyzed and summarized by Arnold and
Press [
101
]. The truncated-mixed PBF distribution is used here to test whether its limits on
the skewness–kurtosis frame plot can adequately simulate the observed values estimated
through the K-moments, by including a nonzero probability at a lower-truncation value.
The mixed PBF can be expressed as follows (e.g., [3]):
F(x)=1P11+ζξ((xd)/λ)ζ1
ξζ (15)
where
ζ
and
ξ
are shape parameters,
λ
is a scale parameter, and
P1=
1
P{x=0}
. Then,
the truncated mixed PBF distribution is constructed by dividing the above expressions by
F(xu)F(xl)
, where
xu
and
xl
are the upper and lower limits of the truncation. Specifically,
xl=
0 for temperature (if expressed in kelvin degrees), dew point, humidity, wind speed,
pressure, streamflow, and precipitation, and
xul =
1 for humidity, while for the dew point
xuis set at the temperature value, and for the rest of the processes at potential infinity.
The PBF distribution can also be viewed from the perspective of the generalized HK
model defined in the previous section, i.e.,
f(x)=λ01+(|xd|/a)2b(c1)/b
, where a
is a scale parameter, band care shape parameters, dis a position parameter, and
λ0
can
be estimated so that
Rf(x)=
1. Similarly to the PBF, this expression contains a variety
Hydrology 2021,8, 59 9 of 26
of distributions from Gaussian (e.g., for
((xd)/a)2
<< 1,
a=σ2
,b= 1, c= 0, d=
µ
,
and
λ0=
1
/σ2π
, where
µ
and
σ
are the mean and standard deviation of the process)—
such as the temperature, atmospheric pressure, or grid turbulence and turbulent buoyant
jet processes—to the Pareto-tail distribution (e.g., for
(|xd|/a)2b
>> 1,
a=p2p1/(p1+1)
,
c= (1 p1)/2,
and
λ0=p1
, where
p1
and
p2
are the shape and scale parameters of the
Pareto distribution)—such as the wind speed, streamflow, precipitation, and ocean wave
processes—while it can be easily truncated between absolute zero and temperature for the
dew point, between solar radiation on the ground and on the top of the atmosphere for the
clearness index, and between 0 and 1 for the relative humidity processes.
2.3. Global-Scale Data Extraction and Processing
The selected global-scale databases for the analysis after quality control are the ISD
database [
102
111
] for near-surface hourly temperature, dew point, sea level pressure,
and wind speed; the CAMELS database [
112
115
] for the daily streamflow; the USGS
database [
116
] for the hourly streamflow; the GHCN database [
117
121
] for the daily
precipitation; and the HPD database (https://www.ncei.noaa.gov/, lastly accessed on
15 December 2020
) for the hourly precipitation. From the contained stations in each
database, we selected stations with more than 5 years of full records (see Table 1for
more information on the selected stations, and Figure 1for the visualization of their
locations), and we discarded recordings that have quality flags. In addition, for other
streamflow-related processes, we analyzed and discussed one of the largest daily discharge
time series of the river Po [
122
] and the largest annual time series of the stage at the river
Nile (e.g., [123]).
The relative humidity is estimated from the hourly temperature and dew point
values as [124]:
U=T
Td5.06
e24.921T0(TdT
TdT)(16)
where
U
is the relative humidity (%);
T
and
Td
are the near-surface temperature and the
dew point expressed in kelvin degrees, respectively; and
T0=
273.16
K
is the triple point
of water.
To mitigate the effect that the periodicity of hydrological-cycle processes, prominent
both in the diurnal and seasonal cycles (e.g., [
125
132
]), exerts on their modelling, we
apply a double standardization on the processes with hourly resolution and a seasonal
standardization on the ones with daily resolution. In particular, we subtract the mean from
each periodicity cycle, and we divide with its standard deviation. In this way, although the
effect of the double periodicity cannot be entirely removed, we diminish the propagation of
its effect on the dependence structure. Moreover, to increase the available records in each
cycle and the estimation accuracy, and without much affecting the dependence structure,
we standardize each process based on a 6 hourly
×
3 monthly resolution of the cycles,
i.e., the diurnal periodicity is divided into 24/6 = 4 cycles, and the seasonal periodicity is
divided also in 12/3 = 4 cycles.
In addition, the marginal and dependence structures of other key hydrological-cycle
processes were included in the analysis, and the results from previous studies of global-
scale stations or from the longest available samples worldwide were discussed. Specifically,
these processes are the potential evapotranspiration [
133
,
134
], the solar radiation clearness
index [62], and the ocean wind-wave height and period processes [135].
Finally, for the investigation of the stochastic similarities between the above key
hydrological-cycle processes and the isotropic and nearly Gaussian turbulence, which
resemble the turbulent shear and buoyancy in the atmospheric boundary layer, we show
the results from a massive grid-turbulence database that includes 40 time series, each
with 36
×
10
6
records of longitudinal wind velocity along the flow direction, all measured
by X-wire probes placed downstream of the grid and in different positions, and with
a sampling temporal resolution of 25
µ
s [
136
]. To shift from the spatial to the temporal
domain [
137
], we apply a standardization to all time series by subtracting their mean
Hydrology 2021,8, 59 10 of 26
and dividing by their standard deviation (see more information and results in [
76
]). In
this manner, we may directly estimate the expected marginal and temporal dependence
structure by combining the estimations from all the time series, approximately as if the
same experiment was performed multiple times at the same position. For the buoyancy
behavior, we discuss the results from several studies of Papanicolaou and List [
138
,
139
] and
Dimitriadis et al. [
18
,
140
,
141
], where more than 10 time series of horizontal and vertical
positively buoyant thermal jets of temperature concentration, recorded with the laser-
induced-fluorescence technique, and with a 30 ms resolution, various nozzle diameters,
discharges, initial and ambient temperature, and of more than 10
4
sample length each,
were analyzed.
3. Results
The investigation of the marginal structure of the key hydrological-cycle processes is
based here on the estimation of moments, and especially of high-order, which are indicative
of the tail behavior. In this section, we present the results from the global-scale analysis, in
terms of the second-order statistics, and particularly, the plots of the mean vs. standard
deviation, of the skewness vs. kurtosis estimated from the C-, L-, K-moments, and of the
climacogram and the climacospectrum (mean, and the 5% and 95% quantiles). Note that
for the estimation of the model parameters adjusting for bias, we use scales up to 10% of
the observation period for the climacogram (and thus, 5% for the CBS), in order not to use
variance estimates from a sample size less than 10 [4].
In Figures 28, which show the climacogram and climacospectrum for each process,
we observed similarities in the shape of the dependence structure spanning from strong
correlations at the small scales to a power-type behavior at large scales, with a convex-shape
at the intermediate scales.
Hydrology 2021, 8, x FOR PEER REVIEW 10 of 27
al. [18,140,141], where more than 10 time series of horizontal and vertical positively buoy-
ant thermal jets of temperature concentration, recorded with the laser-induced-fluores-
cence technique, and with a 30 ms resolution, various nozzle diameters, discharges, initial
and ambient temperature, and of more than 104 sample length each, were analyzed.
3. Results
The investigation of the marginal structure of the key hydrological-cycle processes is
based here on the estimation of moments, and especially of high-order, which are indica-
tive of the tail behavior. In this section, we present the results from the global-scale anal-
ysis, in terms of the second-order statistics, and particularly, the plots of the mean vs.
standard deviation, of the skewness vs. kurtosis estimated from the C-, L-, K-moments,
and of the climacogram and the climacospectrum (mean, and the 5% and 95% quantiles).
Note that for the estimation of the model parameters adjusting for bias, we use up to the
10% of the maximum scales of the climacogram (and thus, 5% of the CBS), in order not to
use variance estimates from a sample length less than 10% of the original length [4].
In Figures 2–8, which show the climacogram and climacospectrum for each process,
we observed similarities in the shape of the dependence structure spanning from strong
correlations at the small scales to a power-type behavior at large scales, with a convex-
shape at the intermediate scales.
(a) (b)
(c) (d)
Figure 2. Near-surface air temperature: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean,
and 5% and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Figure 2.
Near-surface air temperature: (
a
) hourly sample (Boston, USA); (
b
) climacogram and climacospectrum (mean,
and 5% and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Hydrology 2021,8, 59 11 of 26
Hydrology 2021, 8, x FOR PEER REVIEW 11 of 27
(a) (b)
(c) (d)
Figure 3. Near-surface dew point: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean, and 5%
and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
(a) (b)
(c) (d)
Figure 4. Near-surface relative humidity: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean,
and 5% and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Figure 3.
Near-surface dew point: (
a
) hourly sample (Boston, USA); (
b
) climacogram and climacospectrum (mean, and 5%
and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Hydrology 2021, 8, x FOR PEER REVIEW 11 of 27
(a) (b)
(c) (d)
Figure 3. Near-surface dew point: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean, and 5%
and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
(a) (b)
(c) (d)
Figure 4. Near-surface relative humidity: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean,
and 5% and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Figure 4.
Near-surface relative humidity: (
a
) hourly sample (Boston, USA); (
b
) climacogram and climacospectrum (mean,
and 5% and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Hydrology 2021,8, 59 12 of 26
Hydrology 2021, 8, x FOR PEER REVIEW 12 of 27
(a) (b)
(c) (d)
Figure 5. Sea level pressure: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean, and 5% and
95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
(a) (b)
(c) (d)
Figure 6. Wind speed: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean, and 5% and 95%
quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Figure 5.
Sea level pressure: (
a
) hourly sample (Boston, USA); (
b
) climacogram and climacospectrum (mean, and 5% and
95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Hydrology 2021, 8, x FOR PEER REVIEW 12 of 27
(a) (b)
(c) (d)
Figure 5. Sea level pressure: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean, and 5% and
95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
(a) (b)
(c) (d)
Figure 6. Wind speed: (a) hourly sample (Boston, USA); (b) climacogram and climacospectrum (mean, and 5% and 95%
quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Figure 6.
Wind speed: (
a
) hourly sample (Boston, USA); (
b
) climacogram and climacospectrum (mean, and 5% and 95%
quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Hydrology 2021,8, 59 13 of 26
Figure 7.
Streamflow: (
a
) hourly sample (Potomac River, USA); (
b
) climacogram and climacospectrum (mean, and 5% and
95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Hydrology 2021, 8, x FOR PEER REVIEW 13 of 27
(a) (b)
(c) (d)
Figure 7. Streamflow: (a) hourly sample (Potomac River, USA); (b) climacogram and climacospectrum (mean, and 5% and
95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
(a) (b)
(c) (d)
Figure 8. Precipitation: (a) hourly sample (Potomac River, USA); (b) climacogram and climacospectrum (mean, and 5%
and 95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Figure 8.
Precipitation: (
a
) hourly sample (Potomac River, USA); (
b
) climacogram and climacospectrum (mean, and 5% and
95% quantiles); (c) mean vs. standard deviation; (d) C-skewness vs. C-kurtosis.
Hydrology 2021,8, 59 14 of 26
Although similarities may also be traced in the skewness–kurtosis plots, the more
robust estimation of the L-skewness vs. L-kurtosis (Figures 9and 10) allows for more
powerful comparisons. Still, the more advanced K-skewness and K-kurtosis estimations
enable an even clearer view of the detected pattern in Figures 9and 11. Specifically, the
K-skewness increases from the dew point to the humidity, to the temperature, sea level
pressure, and grid turbulence, followed by a weaker increase of the K-kurtosis, while for the
wind speed, streamflow, and precipitation processes, both the K-skewness and K-kurtosis
highly increase. In addition, we observe how all sample K-skewness and K-kurtosis fall into
the range of the mixed PBF distribution only by the analysis through K-moments. Finally, in
Figures 12 and 13, we observed that the dew point, relative humidity, temperature, and sea
level pressure, along with grid turbulence, exhibit a stronger structure at the intermediate
and large scales, followed by the wind speed, streamflow, and precipitation.
Therefore, this hierarchy of the key hydrological-cycle processes may be described by
a transformation of the nearly Gaussian processes (i.e., dew point, humidity, temperature,
and sea level pressure) with a stronger LTP dependence at large scales (H> 0.75) to the
Pareto-type processes (i.e., wind speed, streamflow, and precipitation) with a weaker
dependence LTP at larger scales (H
0.75). Interestingly, the same hierarchy is observed in
a simplified cycle of the energy exchange among hydrological processes trough turbulent
mixing, with the dew point, relative humidity, temperature, and sea level pressure, feeding
the wind speed, while triggering precipitation, whose energy is then temporally stored in
rivers and soil, and finally returned to the former processes through evaporation.
For the rest of the processes included in the current analysis, it was found that the
clearness index of the solar radiation [
62
] and the ocean waves (height and period; [
135
]),
as analyzed from the global databases mentioned above, exhibit a marginal structure
similar to the sea level pressure and wind speed, respectively. In addition, they exhibit an
HK dependence structure with H
0.8 and q
30 h, and H
0.9 and q
10 h, for the
wave height and period, respectively, and H= 0.83 and q= 2 h for the clearness index of
the solar radiation, while both having a fractal behavior with roughness (M< 0.5). The
standardized temperature concentration of the turbulent buoyant jet along the axis [
141
], is
found to exhibit similar behavior as in the grid turbulence with a nearly Gaussian marginal
structure, and a rough (M< 0.5, at small scales) and strong LTP (H> 0.6, at the jet-like area
and
H> 0.9
at the plume-like area) dependence structure. Finally, the evapotranspiration
process was also found to exhibit a weak LTP behavior (H
0.6) and a similar marginal
structure to temperature and solar radiation [133,134].
Hydrology 2021, 8, x FOR PEER REVIEW 14 of 28
Although similarities may also be traced in the skewness–kurtosis plots, the more
robust estimation of the L-skewness vs. L-kurtosis (Figures 9 and 10) allows for more pow-
erful comparisons. Still, the more advanced K-skewness and K-kurtosis estimations ena-
ble an even clearer view of the detected pattern in Figures 9 and 11. Specifically, the K-
skewness increases from the dew point to the humidity, to the temperature, sea level pres-
sure, and grid turbulence, followed by a weaker increase of the K-kurtosis, while for the
wind speed, streamflow, and precipitation processes, both the K-skewness and K-kurtosis
highly increase. In addition, we observe how all sample K-skewness and K-kurtosis fall
into the range of the mixed PBF distribution only by the analysis through K-moments.
Finally, in Figures 12 and 13, we observed that the dew point, relative humidity, temper-
ature, and sea level pressure, along with grid turbulence, exhibit a stronger structure at
the intermediate and large scales, followed by the wind speed, streamflow, and precipita-
tion.
Therefore, this hierarchy of the key hydrological-cycle processes may be described
by a transformation of the (truncated) nearly Gaussian processes (i.e., dew point, humid-
ity, temperature, and sea level pressure) with a stronger LTP dependence at large scales
(H > 0.75) to the Pareto-type processes (i.e., wind speed, streamflow, and precipitation)
with a weaker dependence LTP at larger scales (H 0.75). Interestingly, the same hierar-
chy was observed in a simplified cycle of the energy exchange among hydrological pro-
cesses trough turbulent mixing, with the dew point, relative humidity, temperature, and
sea level pressure, feeding the wind speed, while triggering precipitation, whose energy
is then temporally stored in rivers and soil, and finally returned to the former processes
through evaporation.
Figure 9. L-skewness vs. L-kurtosis, and K-skewness vs. K-kurtosis estimated through the hyper-central K-moments, for
the key hydrological-cycle and the grid-turbulence processes.
Figure 9.
L-skewness vs. L-kurtosis, and K-skewness vs. K-kurtosis estimated through the hyper-central K-moments, for
the key hydrological-cycle and the grid-turbulence processes.
Hydrology 2021,8, 59 15 of 26
Hydrology 2021, 8, x FOR PEER REVIEW 15 of 27
Figure 10. L-skewness vs. L-kurtosis (modified, i.e., 4/5×/+6/5), estimated through the hyper-central K-moments,
for the key hydrological-cycle and the grid-turbulence processes, and the empirically calculated limits of the mixed Pareto-
Burr-Feller distribution for different probabilities of zero values.
Figure 11. K-skewness vs. K-kurtosis, estimated through the hyper-central K-moments, for the key hydrological-cycle and
the grid-turbulence processes, and the empirically calculated limits of the mixed Pareto-Burr-Feller distribution for prob-
abilities of zero values at 25% and 75%. The mean values of the K-skewness and K-kurtosis for each process are depicted
by the square markers with the x-symbol inside.
Figure 10.
L-skewness vs. L-kurtosis (modified, i.e., 4/5
×λ4
/
λ2
+ 6/5), estimated through the
hyper-central K-moments, for the key hydrological-cycle and the grid-turbulence processes, and the
empirically calculated limits of the mixed Pareto-Burr-Feller distribution for different probabilities of
zero values.
Hydrology 2021, 8, x FOR PEER REVIEW 15 of 27
Figure 10. L-skewness vs. L-kurtosis (modified, i.e., 4/5×/+6/5), estimated through the hyper-central K-moments,
for the key hydrological-cycle and the grid-turbulence processes, and the empirically calculated limits of the mixed Pareto-
Burr-Feller distribution for different probabilities of zero values.
Figure 11. K-skewness vs. K-kurtosis, estimated through the hyper-central K-moments, for the key hydrological-cycle and
the grid-turbulence processes, and the empirically calculated limits of the mixed Pareto-Burr-Feller distribution for prob-
abilities of zero values at 25% and 75%. The mean values of the K-skewness and K-kurtosis for each process are depicted
by the square markers with the x-symbol inside.
Figure 11.
K-skewness vs. K-kurtosis, estimated through the hyper-central K-moments, for the key
hydrological-cycle and the grid-turbulence processes, and the empirically calculated limits of the
mixed Pareto-Burr-Feller distribution for probabilities of zero values at 25% and 75%. The mean
values of the K-skewness and K-kurtosis for each process are depicted by the square markers with
the x-symbol inside.
Hydrology 2021,8, 59 16 of 26
Hydrology 2021, 8, x FOR PEER REVIEW 16 of 27
Figure 12. The mean climacogram for the key hydrological-cycle processes and the grid-turbulence process accustomed
for illustration to a 25 μs-scale rather than to the hourly-scale. Dashed and continuous lines at streamflow and precipitation
correspond to the hourly and daily stations.
Figure 13. The mean climacospectrum for the key hydrological-cycle processes and the grid-turbulence process accus-
tomed for illustration to a 25 μs-scale rather than to the hourly scale. Dashed and continuous lines in streamflow and
precipitation correspond to the hourly and daily stations.
For the rest of the processes included in the current analysis, it was found that the
clearness index of the solar radiation index [62] and the ocean waves (height and period;
[135]), as analyzed from the global databases mentioned above, exhibit a marginal struc-
ture similar to the sea level pressure and wind speed, respectively. In addition, they ex-
hibit an HK dependence structure with H 0.8 and q 30 h, and H 0.9 and q 10 h, for
the wave height and period, respectively, and H = 0.83 and q = 2 h for the clearness index
of the solar radiation, while both have a fractal behavior with roughness (M < 0.5). The
standardized temperature concentration of the turbulent buoyant jet along the axis [141],
is found to exhibit similar behavior as in the grid turbulence with a nearly Gaussian mar-
ginal structure, and a rough (M < 0.5, at small scales) and strong LTP (H > 0.6, at the jet-
like area and H > 0.9 at the plume-like area) dependence structure. Finally, the evapotran-
spiration process was also found to exhibit a weak LTP behavior (H 0.6) and a similar
marginal structure to temperature and solar radiation [133,134].
Figure 12.
The mean climacogram for the key hydrological-cycle processes and the grid-turbulence process accustomed for
illustration to a 25
µ
s-scale rather than to the hourly-scale. Dashed and continuous lines at streamflow and precipitation
correspond to the hourly and daily stations.
Hydrology 2021, 8, x FOR PEER REVIEW 16 of 27
Figure 12. The mean climacogram for the key hydrological-cycle processes and the grid-turbulence process accustomed
for illustration to a 25 μs-scale rather than to the hourly-scale. Dashed and continuous lines at streamflow and precipitation
correspond to the hourly and daily stations.
Figure 13. The mean climacospectrum for the key hydrological-cycle processes and the grid-turbulence process accus-
tomed for illustration to a 25 μs-scale rather than to the hourly scale. Dashed and continuous lines in streamflow and
precipitation correspond to the hourly and daily stations.
For the rest of the processes included in the current analysis, it was found that the
clearness index of the solar radiation index [62] and the ocean waves (height and period;
[135]), as analyzed from the global databases mentioned above, exhibit a marginal struc-
ture similar to the sea level pressure and wind speed, respectively. In addition, they ex-
hibit an HK dependence structure with H 0.8 and q 30 h, and H 0.9 and q 10 h, for
the wave height and period, respectively, and H = 0.83 and q = 2 h for the clearness index
of the solar radiation, while both have a fractal behavior with roughness (M < 0.5). The
standardized temperature concentration of the turbulent buoyant jet along the axis [141],
is found to exhibit similar behavior as in the grid turbulence with a nearly Gaussian mar-
ginal structure, and a rough (M < 0.5, at small scales) and strong LTP (H > 0.6, at the jet-
like area and H > 0.9 at the plume-like area) dependence structure. Finally, the evapotran-
spiration process was also found to exhibit a weak LTP behavior (H 0.6) and a similar
marginal structure to temperature and solar radiation [133,134].
Figure 13.
The mean climacospectrum for the key hydrological-cycle processes and the grid-turbulence process accustomed
for illustration to a 25
µ
s-scale rather than to the hourly scale. Dashed and continuous lines in streamflow and precipitation
correspond to the hourly and daily stations.
Finally, the summary statistics from the global-scale analysis can be seen in Tables 2
and 3, where the mean from the C-, L-, and K-moments as well as the scale, fractal, Hurst,
and scale parameter of the expected second-order dependence structure, respectively, were
estimated and are discussed in more detail at the next section. It is noted that merger
of the time series of a different resolution, such as in streamflow and precipitation, was
performed after adjusting for bias (see more details from a similar global-scale analysis of
fine-resolution and large-scale temperature and wind in [70]).
Hydrology 2021,8, 59 17 of 26
Table 2.
Summary Statistics of the Mean Values of the C-, L-, and K-Moments (and Their Standard Deviation in Parentheses)
for Time Series Lengths of N60 Years (1st Row), 30 < N< 60 (2nd Row) and N30 (3rd Row).
Near-Surface
Temperature
Relative
Humidity Dew Point Sea Level
Pressure Wind Speed Streamflow Precipitation
Mean
14.6 (9.3) 0.68 (0.1) 8.3 (8.1) 1013.9 (3.3) 3.7 (1.2) 1498.7 * 2.3 (1.5)
12.6 (3.6) 0.72 (0.2) 9.0 (2.3) 1013.9 (187) 3.51 (0.9) 9.5 (1.5) 2.5 (1.9)
15.3 (3.1) 0.71 (0.1) 6.3 (1.9) 1014.1 (158) 3.53 (0.8) 7.6 (0.2) 2.8 (2.0)
Standard
deviation
8.2 (3.2) 0.2 (0.04) 8.0 (3.2) 7.1 (2.9) 2.4 (0.5) 1007.0 * 7.2 (4.0)
7.3 (2.1) 0.2 (0.04) 6.6 (1.8) 7.4 (1.6) 2.5 (0.6) 16.3 (2.2) 7.4 (4.8)
8.8 (1.9) 0.2 (0.04) 8.0 (1.6) 8.0 (1.3) 2.4 (0.5) 17.9 (0.5) 7.8 (5.0)
C-skewness 0.2 (0.3) 0.3 (0.5) 0.6 (0.4) 0.1 (0.3) 0.9 (0.5) 2.3 * 7.7 (3.8)
0.2 (0.2) 0.4 (0.2) 0.8 (0.2) 0.4 (0.2) 2.1 (0.5) 8.5 (0.7) 6.6 (3.7)
0.2 (0.1) 0.4 (0.1) 0.5 (0.1) 0.2 (0.2) 1.1 (0.4) 9.0 (0.2) 5.5 (3.1)
C-kurtosis
3.3 (0.6) 3.3 (0.7) 4.0 (1.5) 4.0 (2.7) 5.9 (3.3) 7.5 * 136 (218)
6.7 (3.4) 3.9 (0.9) 10.2 (3.4) 20.8 (8.0) 30.2 (10.8) 160.7 (17.6) 93 (115)
5.2 (3.0) 3.6 (0.7) 4.6 (1.5) 6.5 (6.1) 9.6 (7.6) 160.8 (3.2) 53 (50)
L-skewness 0.04 (0.05) 0.05 (0.09) 0.1 (0.06) 0.02 (0.04) 0.1 (0.07) 0.4 * 0.7 (0.1)
0.03 (0.01) 0.07 (0.03) 0.09 (0.02) 0.04 (0.01) 0.2 (0.04) 0.05 (0.04) 0.7 (0.3)
0.04 (0.01) 0.08 (0.02) 0.09 (0.02) 0.03 (0.01) 0.1 (0.03) 0.6 (0.01) 0.7 (0.3)
L-kurtosis
(modified)
1.3 (0.01) 1.3 (0.02) 1.3 (0.02) 1.3 (0.01) 1.3 (0.02) 1.4 * 1.6 (0.1)
1.3 (0.3) 1.3 (0.3) 1.3 (0.3) 1.3 (0.2) 1.3 (0.3) 1.5 (0.1) 1.6 (0.7)
1.3 (0.3) 1.3 (0.2) 1.3 (0.2) 1.3 (0.2) 1.3 (0.3) 1.6 (0.02) 1.6 (0.8)
K-skewness 0.1 (0.2) 0.1 (0.3) 0.3 (0.2) 0.07 (0.1) 0.4 (0.2) 1.5 * 1.7 (0.1)
0.1 (0.05) 0.2 (0.1) 0.3 (0.07) 0.1 (0.04) 0.6 (0.1) 1.6 (0.1) 1.7 (0.7)
0.1 (0.04) 0.2 (0.1) 0.3 (0.06) 0.1 (0.04) 0.5 (0.1) 1.6 (0.02) 1.7 (0.8)
K-kurtosis
2.1 (0.05) 2.1 (0.07) 2.1 (0.07) 2.1 (0.02) 2.1 (0.05) 2.1 * 2.7 (0.1)
2.1 (0.5) 2.1 (0.5) 2.1 (0.5) 2.2 (0.4) 2.2 (0.5) 2.7 (0.2) 2.6 (1.1)
2.1 (0.4) 2.1 (0.4) 2.1 (0.4) 2.1 (0.3) 2.1 (0.4) 2.7 (0.04) 2.6 (1.2)
The mean and standard deviation are in the units of the process, while the rest aredimensionless. Note that only the daily statistics from the
large daily sample of streamflow and precipitation timeseries are presented, since the hourly samples included timeseries of much shorter
length, while all statistic estimates correspond to a sample from at least 36 stations (for the N
60 case of sea level pressure) to at most
55,285 stations (for the N30 case of precipitation). * These values correspond to the daily discharge at the Po river dating back to 1900.
Table 3.
Summary Statistics of the Scale, Fractal and Hurst Parameters of the Second-Order De-
pendence Structure Adjusted for Bias Based on the Climacogram Estimation, with the 5% and
95% Quantiles in Parentheses, and for Each Key Hydrological-Cycle Process of Hourly Resolution.
q(h) Fractal Parameter (M) LTP Parameter (H)
Near-surface temperature
135.1 (9.2–323.1) 0.16 (0.01–0.22) 0.81 (0.61–0.82)
Relative humidity 17.4 (5.6–57.3) 0.23 (0.2–0.27) 0.83 (0.62–0.85)
Dew point 120.3 (16.4–213.2) 0.23 (0.15–0.46) 0.77 (0.58–0.79)
Sea level pressure 36.5 (10.0–67.2) 0.36 (0.25–0.55) 0.7 (0.53–0.77)
Wind speed 9.1 (0.1–25.9) 0.15 (0.07–0.3) 0.85 (0.69–0.86)
Streamflow 96.5 (16.8–533.1) 0.43 (0.2–0.46) 0.78 (0.67–0.86)
Precipitation 2.1 (0.1–3.0) 0.25 (0.18–0.67) 0.61 (0.52–0.69)
4. Discussion
The investigation of the uncertainty in the hydrological cycle is an important scientific
field, as recognized by the International Association of Hydrological Sciences (IAHS) by
launching the Panta Rhei research initiative for the Scientific Decade 2013–2022 [142].
Hydrology 2021,8, 59 18 of 26
An overall conclusion from this study is that the Hurst parameter is estimated sig-
nificantly above 0.5 for all processes (Table 3and Figures 12 and 13), indicating that the
observed uncertainty and climatic variability in the hydrological cycle may be caused by
the presence of the long-term persistent (LTP) behavior (see also discussion in [
143
,
144
]).
This is consistent with the universality of LTP behavior as confirmed in various other
studies (than the ones mentioned in Section 2.1) and statistical attributes in literature (for a
review see [
18
,
145
147
]), such as in global-scale key hydrometeorological processes [
3
,
18
],
trend analysis [
148
] and extremes [
149
,
150
], precipitation [
151
155
], streamflow [
155
157
],
turbulent jets [158,159] and grid turbulence [160,161].
The intermediate scale behavior of the dependence structure (Table 3and
Figure 13
),
which was found to be consistent with the K41 law of Kolmogorov [
162
,
163
] for all pro-
cesses, corresponds to a fractal parameter of M= 1/3 (i.e., roughness behavior of
M< 0.5).
For even smaller scales, the fractal parameter was estimated even lower or close-to-zero
for all the examined processes. However, for the more robust estimation of the fractal
parameter, additional data and of higher resolution are required. For example, in [
70
],
where high resolution samples (of 10 Hz) were applied to the temperature and wind speed,
the fractal parameter was estimated in both processes as M
1/3. It is noted that in the
multifractal analysis, the change of the dependence structure is viewed as a scale break,
and similar positions of the break is identified in the intermediate scales of streamflow
and precipitation in the analyses [164169].
In terms of the marginal structure, our results are consistent with other global-scale
studies in literature (besides the ones previously mentioned in Section 2.2) on nearly Gaus-
sian temperature [
89
], wind speed transition of extreme tail from Rayleigh to Weibull and
Pareto-type [
70
,
170
172
], sea level pressure [
173
], streamflow heavy tail [
157
,
174
], pre-
cipitation Pareto-type tail [
3
,
74
,
149
,
175
182
], evaporation [
183
,
184
], and nearly Gaussian
turbulent processes [185].
The high uncertainty of geophysical dynamics is linked to the power-law type of the
marginal distribution as well as of the dependence structure through empirical evidence
and physical justification. Although the above and other studies have focused on one or
a limited number of processes, in this study we analyze several key hydrological-cycle
processes, namely near-surface temperature, dew point, humidity, sea level pressure, atmo-
spheric wind speed, streamflow, precipitation, as well as other processes from previous
studies, such as solar radiation clearness index and ocean waves, where we trace stochas-
tic similarities in their marginal and dependence structures. Moreover, we find similar
stochastic structures in turbulent shear and buoyancy processes, as studied through labo-
ratory records of grid-turbulent wind speeds and temperature concentrations of buoyant
turbulent jets along the axis.
Specifically, a hierarchy emerges (a) for the marginal structure visualized through the
skewness–kurtosis plot, both estimated through the K-moments, while similar but weaker
empirical conclusions can be derived by the plot of the C-moments and L-moments; and
(b) for the dependence structure visualized through the climacogram and climacospec-
trum. This hierarchy starts with temperature, dew point, relative humidity, solar radiation
index, evaporation, and sea level pressure, all of which exhibit a stronger skewness over
kurtosis absolute ratio than the turbulent processes, wind speed, and ocean waves. All
the latter processes exhibit a stronger LTP behavior in the dependence structure, whereas
streamflow and precipitation present a weaker skewness over kurtosis absolute ratio and
an LTP behavior.
Interestingly, the same hierarchy is observed in the energy exchange among processes
in the hydrological cycle. It starts with the solar radiation, temperature, dew point (or
equivalently, relative humidity), and sea level pressure. These feed the wind speed and
ocean waves through nearly Gaussian and isotropic turbulent mixing in the boundary
layer. Next, they trigger precipitation, which then moves to streams and soil, and is finally
returned to the former processes through evaporation.
Hydrology 2021,8, 59 19 of 26
5. Conclusions
The major innovation of this study is the uniting view of the key hydrological-cycle
processes through the analysis of several billions of observations from hundred thousands
of stations by robust statistical metrics of (a) the K-moments, for the estimation of the
marginal structure of the first four moments, and of (b) the climacogram, for the estimation
of the second-order dependence structure. The key examined hydrological-cycle processes
are the near-surface temperature, dew point, humidity, sea level pressure, atmospheric
wind speed, streamflow, and precipitation, as well as other processes from previous studies,
such as shear and buoyant turbulent processes analyzed through small-scale laboratory
experiments, and solar radiation, and ocean waves. The main traced stochastic similarities
are as follows:
(1)
A hierarchy related to the hydrological cycle was identified with the dew point,
temperature, relative humidity, solar radiation, and sea level pressure all exhibiting
a lower skewness over kurtosis absolute ratio than the turbulent processes, wind
speed, and ocean waves, and with a stronger long-term persistence (LTP) behavior
in the dependence structure (H> 0.75), followed by streamflow and precipitation,
both of which exhibit a smaller skewness–kurtosis absolute ratio and a weaker LTP
behavior (H0.75).
(2)
All the examined processes can be adequately simulated by the truncated mixed-
PBF distribution, adjusting for probability dry and lower (or upper) truncation, in
terms of the first four moments, and ranging from (truncated) nearly Gaussian to
Pareto-type tails.
(3)
As the sample size increases, different records of the same process from several
locations converge to a smaller area of the nondimensionalized statistics (skewness–
kurtosis), indicating a common marginal behavior.
(4)
All the examined hydrological-cycle processes exhibit a similar dependence structure
that extends from the fractal behavior with roughness (M< 0.5) located at the small-
intermittent scales to the LTP behavior at large scales (H> 0.5), while both indicate
large uncertainty and high climatic variability.
(5) Finally, since the above empirical findings are consistent with previous studies and can
be justified by the principle of maximum entropy, they allow for a uniting stochastic
view of the hydrological-cycle processes under the Hurst–Kolmogorov (HK) dynamics
in terms of both the marginal and dependence structures.
Author Contributions:
Conceptualization, P.D., D.K., T.I., P.P.; Formal analysis, P.D., D.K., T.I., P.P.;
Methodology, P.D., D.K., T.I., P.P.; Resources, P.D. and T.I.; Supervision, D.K. and P.P.; Validation, P.D.,
D.K., T.I., P.P.; Visualization, P.D., D.K., T.I., P.P.; Writing—original draft, P.D.; Writing—review &
editing, D.K., T.I., P.P. All authors have read and agreed to the published version of the manuscript.
Funding:
This research is carried out/funded in the context of the project “Development of Stochastic
Methods for Extremes (ASMA): identification and simulation of dependence structures of extreme
hydrological events” (MIS 5049175) under the call for proposals “Researchers’ support with an
emphasis on young researchers—2nd Cycle”. The project is co-financed by Greece and the European
Union (European Social Fund; ESF) by the Operational Programme Human Resources Development,
Education, and Lifelong Learning 2014–2020.
Hydrology 2021, 8, x FOR PEER REVIEW 20 of 27
stochastic view of the hydrological-cycle processes under the Hurst–Kolmogorov
(HK) dynamics in terms of both the marginal and dependence structures.
Author Contributions: Conceptualization, P.D.; Formal analysis, P.D.; Methodology, P.D. and D.K.;
Resources, T.I.; Supervision, D.K. and P.P.; Validation, D.K.; Visualization, P.D.; Writing—original
draft, P.D.; Writing—review & editing, T.I. and P.P. All authors have read and agreed to the pub-
lished version of the manuscript.
Funding: This research is carried out/funded in the context of the project “Development of Stochas-
tic Methods for Extremes (ASMA): identification and simulation of dependence structures of ex-
treme hydrological events” (MIS 5049175) under the call for proposals “Researchers’ support with
an emphasis on young researchers—2nd Cycle”. The project is co-financed by Greece and the Euro-
pean Union (European Social Fund; ESF) by the Operational Programme Human Resources Devel-
opment, Education, and Lifelong Learning 2014–2020.
Acknowledgments: We thank the anonymous reviewers, and the academic and assistant editors’
fruitful comments and handling of the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
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Hydrology 2021,8, 59 20 of 26
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