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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 [3,4,21–31],

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)

i−ˆ

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=(i−1)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)

i−ˆ

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(1−H)/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(1−H)/M+λk/q+e−k/q−1

(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 :=(p−q+1)Eh(F(x))p−q(x−µ)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 :=(p−q+1)Eh(2F(x)−1)p−q(x−µ)qi(9)

for

p≥q

, 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

n−1

n

∑

i=p

p−1

n

Γ(n−p+2)

Γ(n)

Γ(i)

Γ(i−p+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)=1−P11+ζξ((x−d)/λ)ζ−1

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+(|x−d|/a)2b(c−1)/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

((x−d)/a)2

<< 1,

a=σ√2

,b= 1, c= 0, d=

µ

,

and

λ0=

1

/σ√2π

, 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

(|x−d|/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(Td−T

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 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.

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

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 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 N≥60 Years (1st Row), 30 < N< 60 (2nd Row) and N≤30 (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 N≤30 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 [164–169].

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 (H≤0.75).

(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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