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An R function for the estimation of trend signifcance under the scaling hypothesis- application in PET parametric annual time series


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We present an R function for testing the significant trend of time series. Te function calculates trend significance using a modified Mann-Kendall test, which takes into account the well-known physical behavior of the Hurst-Kolmogorov dynamics. Te function is tested at 10 stations in Greece, with approximately 50 years of PET data with the use of a recent parametric approach. A significant downward trend was detected at two stations. Te R software is now suitable for extensive use in several fields of the scientific community, allowing a physical consistent of a trend analysis.
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Open Water Journal
Volume 4
Issue 1 Open Water Science and Soware Article 6
An R function for the estimation of trend
signicance under the scaling hypothesis-
application in PET parametric annual time series
Aristoteles Tegos
National Technical University of Athens,
Hristos Tyralis
Demetris Koutsoyiannis
National Technical University of Athens,
Khaled Hamed
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Tegos, Aristoteles; Tyralis, Hristos; Koutsoyiannis, Demetris; and Hamed, Khaled (2017) "An R function for the estimation of trend
signi=cance under the scaling hypothesis- application in PET parametric annual time series," Open Water Journal: Vol. 4 : Iss. 1 , Article
Available at: h>p://
An R function for the estimation of trend signicance
under the scaling hypothesis- application in PET
parametric annual time series
Soware Introduction
Aristoteles Tegos1*, Hristos Tyralis2, Demetris Koutsoyiannis3, Khaled H.
1,2,3Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical
University of Athens, Iroon Polytechniou 5, 15780 Zografou, Greece
4Department of Irrigation and Hydraulics, Faculty of Engineering, Cairo University
*Corresponding Author:
We present an R function for testing the signicant trend of time series. e function calculates trend signicance using
a modied Mann-Kendall test, which takes into account the well-known physical behavior of the Hurst-Kolmogorov
dynamics. e function is tested at 10 stations in Greece, with approximately 50 years of PET data with the use of a
recent parametric approach. A signicant downward trend was detected at two stations. e R soware is now suit-
able for extensive use in several elds of the scientic community, allowing a physical consistent of a trend analysis.
Hurst; Potential evapotranspiration; Parametric model; R soware; Trend analysis
1.0 Introduction
Trend estimation in hydro-climatic time series has
focused the attention of the scientic community (Sen,
2013). Many studies have examined the trend of precip-
itation, streamow, groundwater regime, temperature,
potential evapotranspiration both at annual and seasonal
scales (Markonis et al. 2016, Stevens et al. 2016, Panda et
al. 2012, Arora et al. 2005, Kumar et al. 2010). Specically,
trend estimation in potential or actual evapotranspiration
pay the attention of the researchers (Gocic and Trajkovic,
2014, Mo et al. 2015, Tabari et al., 2011). Generally, the
trend results are mixed across dierent climatic regions,
as Tabari et al (2011) found a positive trend for 70% of 20
Iranian meteorological stations during the period 1996-
2005, but Gocic and Trajkovic (2014) calculated a signif-
icant increasing downward trend in 70% of 12 Serbian
meteorological stations (study period 1980-2010). Finally,
Mo et al. 2015, by investigating the areal evapotranspira-
tion in China for the period 1981-2010 with remote sens-
ing data, observed an increasing trend from the 1980s to
the mid-1990s, followed by a decreasing trend. For the
examination of physical variability, the Mann-Kendall
under the independence assumption has been pro-
67Open Water
posed as a standard statistical measure for ethe valuation
and quantication of trends (Ahn and Palmer, 2015).
Furthermore, dierent Mann–Kendall statistical
methodologies have been developed and proposed,
namely the Mann–Kendall under the Markovian behav-
ior assumption aer trend-free pre-whitening, the
Mann–Kendall with complete autocorrelation struc-
ture and the Mann–Kendall under the long-term per-
sistence assumption (Kumar et al. 2009). e latter
test, proposed by Hamed (2008), oers a consistent
framework to consider the Hurst phenomenon, which
is observed in many climatological and hydrological
processes, resulting in the increase of physical variabil-
ity (Koutsoyiannis 2003; Koutsoyiannis and Montanari
2007). Hurst coecient was rst introduced by engineer
Harold Hurst during the design of the Aswan reservoir
(Sutclie et al. 2016) and plays a signicant role in the
hydrological variability (O’Connell et al, 2016). Its pres-
ence in large measured hydrometeorological samples is
ubiquitous (Iliopoulou et al. 2016) Comparative analy-
sis of dierent trend model shows signicant dierences
in the totally results (Hamed 2008, Kumar et al. 2009)
and thus a physical consistent framework is needed.
is study presents an R function embedded in an
automatic and user-friendly environment fol-
lowing modern views of water resources mod-
eling tools (Guo et al. 2016, Turner and Ganelli
2016). e package implements the modied
Hamed’s (2008) framework and the procedure is
tested in annual parametric PET time series from
10 sites in Greece, which cover the period 1950-2000.
2. Materials and methods
2.1 e parametric-PET model
e parametric model was rst introduced by
Koutsoyiannis and Xanthopoulos in 1998, mainly as
a framework to ll and extend PET time series and
included the calibration of the parametric model in
a Penman-Monteith time series. Later, Tegos et al.
(2009, 2013, 2015) implemented the model at point
and regional scale in Greece territory and com-
pared the results with Hargraves and Oudin models.
Recently, an extended and comparative analysis in
dierent climatic regimes were made which included
the development of the model in the well-known
CIMIS network (California, U.S.A) as well in European
stations (Tegos et al. 2015). e results of the imple-
mentation were quite satisfactory and the frame-
work allowed consistent monthly and annual PET
estimation at point and especially in regional scale.
Another key conclusion was the better agreement
with Penman-Monteith measured samples against
other world-recognized radiation-based models such
as Hargreaves, Oudin, Jensen and McGuiness. e
most recently application was the daily and monthly
implementation of the model for the PET mapping
in an irrigated plain of Greece (Malamos et al., 2015).
e mathematical expression of the paramet-
ric model for every time step is the following:
where, PET potential evapotranspiration (mm)
Ra (KJm-2) is the extraterrestrial radiation, a
(KgKj-1), b (Kgm-2) and c (C-1) are the calibrated
parameters, while T (C) is the mean air temperature.
e extraterrestrial radiation Ra, for each
day of the year and for dierent latitudes is esti-
mated from the solar constant, the solar decli-
nation and the time of the year by the formula:
where Ra (MJm-2d-1) extraterrestrial radiation, Gsc
solar constant = 0.0820 MJm-2min-1, dr inverse rel-
ative distance Earth-Sun, ωs (rad) sunset hour
angle, φ latitude (rad), δ solar declination (rad).
2.2 Mann-Kendall test under the scaling hypothesis
e Mann-Kendall test under the scaling hypothesis
consists of three consecutive hypothesis tests, namely O
(Original MK test), H (Hurst Parameter test) and M. e
mathematic background and framework are presented
from Hamed (2008). Let H0i denote the null hypothesis
Gsc drssin(φ)sin(δ)+cos(φ)cos(d)sin(ωs)](2)
Ra =
Open Water 68
of each test and let H1i denote the alternative hypoth-
esis, where i = O, H, M denotes the step of the Mann-
Kendall test under the scaling hypothesis. We dene:
H0O: No trend under the independence
H1O: Signicant trend under the indepen-
dence assumption.
H0H: No signicant LTP.
H1H: Signicant LTP.
H0M: No trend under LTP assumption.
H1M: Signicant trend exists under LTP
en the three steps of the test are summa-
rized by the following sequences
{H0O}: No trend.
{H1O}: Possible signicant trend. Proceed to
step H.
{H1O, H0H}: Signicant trend exists.
{H1O, H1H}: Possible LTP eect. Proceed to
step M.
{H1O, H1H, H0M}: No trend.
{H1O, H1H, H1M}: Signicant trend exists.
Hurst coecient can be dened by a simple pow-
er-law relationship of its standard deviation:
where σ σ(1) and H is the entropy production
in logarithmic time (Koutsoyiannis 2011), and the
parameter ranges between 0 and 1. For values H
> 0.5, the process exhibits long-term persistence,
while for H < 0.5 the process is anti-persistent.
For the test implementation, we used the R func-
tion MannKendallLTP from the HKprocess R pack-
age (Tyralis, 2015). e R function computes the
p-value in each step of the test. If the p-value is
higher than a predened signicance level α (e.g.
α = 0.05), then we cannot reject H0. A p-value less
than or equal to α gives evidence that H1 is true.
2.3 Study area and procedures
Ten meteorological stations (National
Meteorological Services of Greece) well-distrib-
uted over Greece were used. Table 1 presents the list
of the meteorological stations used in our study.
Stations φ(o) z (m)
Heraklion 35.20 39
Ioannina 39.42 484
Kavala 40.54 63
Kerkyra 39.37 2
Kozani 40.18 626
Larissa 39.39 74
Lemnos 39.54 17
Methoni 36.50 34
Skyros 38.54 5
Tripoli 37.32 663
Based on our previous study (Tegos et al. 2013) the
parametric model was calibrated and tested in monthly
time step for the period 1968-1989. For the purposes of
this study, monthly air temperature data for the period
1950-2000 were collected and the parametric model was
applied to the total length. Finally, every monthly time
series was aggregated into annual step with the use of
the HYDROGNOMON soware (Kozanis et al. 2010).
3. Results
Table 2 presents the results of our analysis. In seven
out of the ten stations tested, no trends were found
under the independence assumption. e estimate of
the Hurst parameter for annual PET time series var-
ies in the range from 0.43 to 0.76. Out of the three
stations that had signicant trends under the inde-
pendence assumption, only two stations (Ioannina,
Limnos) showed a signicant downward trend.
In Figure 1, we present the PET at Ioannina. In Table 2,
we observe a signicant trend under the independence
assumption. is assumption is valid. At Kerkyra (see
Figure 2) we do not observe any signicant trend. At
Larissa (Figure 3), we nd a signicant trend under the
independence assumption; however, this trend is not
signicant under the long-term dependence assump-
tion. Finally, we observe a signicant trend under a valid
independence assumption at Limnos (see Figure 4).
Table 1. Meteorological stations with their latitude (φο) and
elevation (z).
69Open Water
4. Discussions and conclusions
We present an R function that implements the Mann-
Kendall test under the long-term persistence hypothe-
sis. e test applied and tested in annual time series of
PET estimated from a recent parametric approach. e
parametric model estimation allows the consistent esti-
mation of the PET with minimal data requirements and
it’s useful for climatic studies when crucial hydrome-
terological data are missing (wind velocity, relative
humidity, extraterrestrial radiation). e results of
our preliminary case study analysis show that in seven
cases, no signicant trend was detected under the inde-
pendence assumption. In one case, no signicant trend
was detected under the long-term persistence assump-
tion, while the trend was signicant under the inde-
pendence assumption. In the remaining two cases, we
found a signicant downward trend under both the
independence and the long-term persistence assump-
tions. In summary, an R function is ready and user-
friendly for use in other eld of water resources studies.
e authors wish to kindly acknowledge one anon-
ymous reviewer for his/her constructive suggestions
which improved earlier version of this manuscript.
Appendix A Supplementary material
Supplementary data and code for reproducing the
analysis of this paper as well as additional Figures,
associated with the present study but not included
here for brevity, are available as supporting material in
Appendix A.
Soware Availability
Name of soware: MannKendallLTP R function in
the HKprocess R package
Developers: Hristos Tyralis
Contact: Hristos Tyralis, Athens 10443, contact:
Year rst available: 2016
Required soware: R (≥ 3.2.3)
Cost: Free. e function is public available in
the R package at
Table 2. Summary results of the application of the Mann-Kendall modied test to the PET data. e Hurst parameter was estimated
using the maximum likelihood estimator (Tyralis and Koutsoyiannis 2011). e trend identication is performed for a predened level
α = 0.05 in each step.
Staons Hurst pa-
2-sided p-value
(Step O)
of H, 2-sided
p-value (Step
all LTP 2-sided
p-value (Step M)
Trend idenca-
Heraklion 0.67 0.31 {H0O}, no trend
Ioannina 0.58 0.05 0.27 {H1O,H0H}, trend
Kavala 0.76 0.63 {H0O}, no trend
Kerkyra 0.71 0.90 {H0O}, no trend
Kozani 0.63 0.31 {H0O}, no trend
Larissa 0.76 0.04 0.00 0.42 {H1O,H1H,HOM} no
Lemnos 0.74 0.00 0.26 {H1O,H0H}, trend
Methoni 0.69 0.06 {H0O}, no trend
Skyros 0.46 0.40 {H0O}, no trend
Tripoli 0.43 0.46 {H0O}, no trend
Open Water 70
Figure 1. Annual PET at Ioannina
Figure 2. Annual PET at Kerkyra
Figure 3. Annual PET at Larissa
71Open Water
Ahn, K. H., & Palmer, R. N. (2015). Trend and Variability in
Observed Hydrological Extremes in the United States. Journal
of Hydrologic Engineering, 21(2), 04015061.
Arora, Manohar, N. K. Goel, and Pratap Singh. “Evaluation of
temperature trends over India.”Hydrological sciences journal
50.1 (2005).
Gocic, Milan, and Slavisa Trajkovic. “Analysis of trends in refer-
ence evapotranspiration data in a humid climate.Hydrologi-
cal Sciences Journal 59.1 (2014): 165-180.
Guo, D., Westra, S., & Maier, H. R. (2016). An R package for
modelling actual, potential and reference evapotranspiration.
Environmental Modelling & Soware, 78, 216-224
Hamed KH (2008) Trend detection in hydrologic data: e
Mann-Kendall trend test under the scaling hypothesis. Journal
of Hydrology 349(3-4):350-363.
Iliopoulou, T., Papalexiou, S. M., Markonis, Y., & Koutsoyiannis,
D. (2016). Revisiting long-range dependence in annual precip-
itation. Journal of Hydrology.
Kozanis, S., Christodes, A., Mamassis, N., Efstratiadis, A. and
Koutsoyiannis, D., 2010, May. Hydrognomon–open source
soware for the analysis of hydrological data. In EGU General
Assembly Conference Abstracts (Vol. 12, p. 12419).
Koutsoyiannis, D., 2003. Climate change, the Hurst phenom-
enon, and hydrological statistics. Hydrological Sciences-
Journal-des Sciences Hydrologiques 48 (1).
Koutsoyiannis, D., Montanari, A., 2007. Statistical analysis of
hydroclimatic time series: uncertainty and insights. Water
Resources Research 43, W05429.
Koutsoyiannis D. 2011. Hurst–Kolmogorov dynamics as a
result of extremal entropy production. Physica A: Statistical
Mechanics and its Applications, 390(8): 1424-1432
Koutsoyiannis, D., and  Xanthopoulos. “Engineering hydrolo-
gy.”Edition 3 (1999): 418.
Kumar, S., Merwade, V., Kam, J., & urner, K. (2009).
Streamow trends in Indiana: eects of long term persistence,
precipitation and subsurface drains. Journal of Hydrology,
374(1), 171-183.
Markonis, Y., Batelis, S. C., Dimakos, Y., Moschou, E., &
Koutsoyiannis, D. (2016). Temporal and spatial variability of
rainfall over Greece. eoretical and Applied Climatology,
Mo, X., Liu, S., Lin, Z., Wang, S., & Hu, S. (2015). Trends in land
surface evapotranspiration across China with remotely sensed
NDVI and climatological data for 1981–2010.Hydrological
Sciences Journal, 60(12), 2163-2177.
O’Connell, P. E., Koutsoyiannis, D., Lins, H. F., Markonis, Y.,
Montanari, A., & Cohn, T. (2016). e scientic legacy of
Harold Edwin Hurst (1880–1978). Hydrological Sciences
Journal, 1-20.
Panda, D. K., A. Mishra, and A. Kumar. “Quantication
of trends in groundwater levels of Gujarat in western
India.”Hydrological Sciences Journal57.7 (2012): 1325-1336.
Stevens, Andrew J., Derek Clarke, and Robert J. Nicholls. “Trends
in reported ooding in the UK: 1884–2013.”Hydrological
Sciences Journal 61.1 (2016): 50-63.
Sutclie, J., Hurst, S., Awadallah, A. G., Brown, E., & Hamed,
K. (2016). Harold Edwin Hurst: the Nile and Egypt, past and
future. Hydrological Sciences Journal, 1-14.
Tabari, H., Maro, S., Aeini, A., Talaee, P. H., & Mohammadi,
K. (2011). Trend analysis of reference evapotranspiration in
the western half of Iran. Agricultural and Forest Meteorology,
151(2), 128-136.
Tegos, A., Efstratiadis, A., & Koutsoyiannis, D. (2013). A para-
metric model for potential evapotranspiration estimation
based on a simplied formulation of the Penman-Monteith
equation. Evapotranspiration-an overview. InTech, Rijeka,
Croatia, 143-165.
Tegos, A., Malamos, N., & Koutsoyiannis, D. (2015). A parsimo-
nious regional parametric evapotranspiration model based on
a simplication of the Penman–Monteith formula. Journal of
Hydrology, 524, 708-717.
Tegos, A., Efstratiadis, A., Malamos, N., Mamassis, N., &
Koutsoyiannis, D. (2015). Evaluation of a parametric approach
for estimating potential evapotranspiration across dierent
climates. Agriculture and Agricultural Science Procedia, 4,
Tegos, A., N. Mamassis, and D. Koutsoyiannis. “Estimation
of potential evapotranspiration with minimal data depen-
dence.”EGU General Assembly Conference Abstracts. Vol. 11.
Turner, S. W. D., & Galelli, S. (2016). Water supply sensitivity
to climate change: An R package for implementing reser-
voir storage analysis in global and regional impact studies.
Environmental Modelling & Soware, 76, 13-19.
Tyralis H, Koutsoyiannis D (2011) Simultaneous estimation of
the parameters of the Hurst-Kolmogorov stochastic process.
Stochastic Environmental Research and Risk Assessment
Tyralis H. (2016). HKprocess: Hurst-Kolmogorov process.
R package version 0.0-2.
... While Mann-Kendall trend tests used here accounted for increased uncertainty of trend results due to serial correlation, they did not account for uncertainty that may occur due to scaling, or longer-term climatic variability (Hamed, 2008). To test whether scaling was an issue for streamflow and precipitation variables in the URGB, a modified Mann-Kendall trend test that accounts for scaling was applied using the MannKendallLTP function in the R package HKprocess (Tegos et al., 2017;Tyralis, 2016). For streamflow, baseflow, and runoff, uncertainty in the trend results increased when scaling was taken into account (Table S7), particularly for baseflow. ...
Understanding how changing climatic conditions affect streamflow volume and timing is critical for effective water management. In the Rio Grande Basin of the southwest U.S., decreasing snowpack, increasing minimum temperatures, and decreasing streamflow have been observed in recent decades, but the effects of hydroclimatic changes on baseflow, or groundwater discharge to streams, have not been investigated. In this study, we determine how trends in precipitation, snowpack accumulation, and snowmelt rate relate to total streamflow, baseflow, and the hydrologic partitioning of baseflow and runoff at 12 sites in the Upper Rio Grande Basin (URGB) during 1980 to 2015. Total streamflow was partitioned into baseflow and runoff components at a daily time step using conductivity-mass-balance hydrograph separation. Trends in annual total streamflow, baseflow, runoff, baseflow index, precipitation, snowmelt rate, and peak snow water equivalent (SWE) were evaluated from 1980 to 2015 using the non-parametric Mann-Kendall trend test. Results indicate that baseflow forms a large component of total streamflow, contributing an average of 49% of total discharge upstream of Albuquerque, NM. During 1980 to 2015, decreasing trends in total streamflow occurred at 9 of 12 sites and were almost always associated with decreases in baseflow, suggesting that baseflow volumes can respond to changing climatic and anthropogenic conditions within decades. Decreasing snowmelt rates were more frequently associated with decreases in baseflow and total streamflow than were decreases in precipitation and peak SWE, highlighting the importance of snowmelt rate as a process controlling streamflow generation. If snow accumulation and snowmelt rates continue to decrease in the future, results indicate that total streamflow and baseflow volumes will decline, and that baseflow will become a larger fraction of total streamflow in the URGB.
... In light of concerns for intensification of hydrological extremes due to anthropogenic forcing, the investigation of clustering receives additional interest (Ntegeka and Willems, 2008;Tye et al., 2018;Merz et al., 2016;Serinaldi and Kilsby, 2018), as attribution of trends to an external deterministic forcing presupposes that at least the presence of natural inherent variability has been beforehand properly accounted for. In this respect, increasing evidence reporting the presence of persistence in various hydroclimatic variables (Hurst, 1951; A c c e p t e d M a n u s c r i p t 3 Koutsoyiannis, 2003;Montanari, 2003;Markonis and Koutsoyiannis, 2016;O'Connell et al., 2016;Iliopoulou et al., 2016;Tegos et al., 2017;Dimitriadis, 2017) gives rise to the question of whether or not, and to what extent a regular behaviour of the extremes originating from persistent processes could be misinterpreted as a result of an anthropogenic cause. ...
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Clustering of extremes is critical for hydrological design and risk management and challenges the popular assumption of independence of extremes. We investigate the links between clustering of extremes and long-term persistence, else Hurst-Kolmogorov (HK) dynamics, in the parent process exploring the possibility of inferring the latter from the former. We find that (a) identifiability of persistence from maxima depends foremost on the choice of the threshold for extremes, the skewness and kurtosis of the parent process, and less on sample size; and (b) existing indices for inferring dependence from series of extremes are downward biased when applied to non-Gaussian processes. We devise a probabilistic index based on the probability of occurrence of peak-over-threshold events across multiple scales, which can reveal clustering, linking it to the persistence of the parent process. Its application shows that rainfall extremes may exhibit noteworthy departures from independence and consistency with an HK model.
... The We used the test implementation in the R package HKprocess (Tyralis, 2016) with a predefined significance level α = 0.05 for all steps. For more details on the algorithm and its implementation using the R package HKprocess the interested reader is referred to Tegos et al. (2017). Furthermore, we estimated the trends of the annual time series with the fitting of a linear model. ...
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... The model performance was satisfying as the proposed framework provides consistent monthly PET estimates at point and especially at the regional scale. The most recent application was the daily and monthly implementation of the model for the PET mapping in an irrigated plain of Greece [29] and the investigation of trend analysis in Greece through the development of an R-script tool [35]. ...
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We present and validate a global parametric model of potential evapotranspiration (PET) with two parameters that are estimated through calibration, using as explanatory variables temperature and extraterrestrial radiation. The model is tested over the globe, taking advantage of the Food and Agriculture Organization (FAO CLIMWAT) database that provides monthly averaged values of meteorological inputs at 4300 locations worldwide. A preliminary analysis of these data allows for explaining the major drivers of PET over the globe and across seasons. The model calibration against the given Penman-Monteith values was carried out through an automatic optimization procedure. For the evaluation of the model, we present global maps of optimized model parameters and associated performance metrics, and also contrast its performance against the well-known Hargreaves-Samani method. Also, we use interpolated values of the optimized parameters to validate the predictive capacity of our model against monthly meteorological time series, at several stations worldwide. The results are very encouraging, since even with the use of abstract climatic information for model calibration and the use of interpolated parameters as local predictors, the model generally ensures reliable PET estimations. Exceptions are mainly attributed to irregular interactions between temperature and extraterrestrial radiation, as well as because the associated processes are influenced by additional drivers, e.g., relative humidity and wind speed. However, the analysis of the residuals shows that the model is consistent in terms of parameters estimation and model validation. The parameter maps allow for the direct use of the model wherever in the world, providing PET estimates in case of missing data, that can be further improved even with a short term acquisition of meteorological data.
... • Estimation of trends and their significance using the Mann-Kendall test under the LTP assumption (MKt-LTP, Hamed 2008, Tegos et al. 2017). ...
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The stochastic structures of potential evaporation and evapotranspiration (PEV and PET or ETo) are analyzed using the ERA5 hourly reanalysis data and the Penman–Monteith model applied to the well-known CIMIS network. The latter includes high-quality ground meteorological samples with long lengths and simultaneous measurements of monthly incoming shortwave radiation, temperature, relative humidity, and wind speed. It is found that both the PEV and PET processes exhibit a moderate long-range dependence structure with a Hurst parameter of 0.64 and 0.69, respectively. Additionally, it is noted that their marginal structures are found to be light-tailed when estimated through the Pareto–Burr–Feller distribution function. Both results are consistent with the global-scale hydrological-cycle path, determined by all the above variables and rainfall, in terms of the marginal and dependence structures. Finally, it is discussed how the existence of, even moderate, long-range dependence can increase the variability and uncertainty of both processes and, thus, limit their predictability.
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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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The aim of the Ph.D thesis is the foundation of a new temperature-based model since simplified PET estimation proves very useful in absence of a complete data set. In this respect, the Parametric model is presented based on a simplified formulation of the well-established Penman-Monteith expression, which only requires mean daily or monthly temperature data. The model was applied at both global and local regions and the outcomes of this new approach are very encouraging, as indicated by the substantially high validation scores of the proposed approach across all examined data sets. In general, the parametric model outperforms well-established methods of the everyday practice. A second analysis which was examined as part of this thesis is related to which spatial techniques is the optimal in order to transform the point scale estimate in regional. A thorough analysis of different geostatistical model was carried out (Kriging, IDW, NN, BSS) and it can be concluded that the IDW even is the most simplify geostatistical model, it can be produce consistent spatial PET results. Another part of the thesis was the development of an R function for testing the trend significance of time series. The function calculates the trend significance using a modified Mann- Kendall test, which takes into account the well-known physical behavior of the Hurst-Kolmogorov dynamics. The function is tested in 10 stations in Greece, with approximately 50 years of PET data with the use of a recent parametric model. Finally, a number of hydrological, agronomist and climatologist applications are presented for lighting the robustness of the new Parametric approach in multidiscipline areas.
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Methods to make inference about the Hurst-Kolmogorov and the AR(1) process.
Conference Paper
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Potential evapotranspiration (PET) is key input in water resources, agricultural and environmental modelling. For many decades, numerous approaches have been proposed for the consistent estimation of PET at several time scales of interest. The most recognized is the Penman-Monteith formula, which is yet difficult to apply in data-scarce areas, since it requires simultaneous observations of four meteorological variables (temperature, sunshine duration, humidity, wind velocity). For this reason, parsimonious models with minimum input data requirements are strongly preferred. Typically, these have been developed and tested for specific hydroclimatic conditions, but when they are applied in different regimes they provide much less reliable (and in some cases misleading) estimates. Therefore, it is essential to develop generic methods that remain parsimonious, in terms of input data and parameterization, yet they also allow for some kind of local adjustment of their parameters, through calibration. In this study we present a recent parametric formula, based on a simplified formulation of the original Penman-Monteith expression, which only requires mean daily or monthly temperature data. The method is evaluated using meteorological records from different areas worldwide, at both the daily and monthly time scales. The outcomes of this extended analysis are very encouraging, as indicated by the substantially high validation scores of the proposed approach across all examined data sets. In general, the parametric model outperforms well-established methods of the everyday practice, since it ensures optimal approximation of PET.
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Recent studies have showed that there is a significant decrease in rainfall over Greece during the last half of the pervious century, following an overall decrease of the precipitation at the eastern Mediterranean. However, during the last decade an increase in rainfall was observed in most regions of the country, contrary to the general circulation climate models forecasts. An updated high-resolution dataset of monthly sums and annual daily maxima records derived from 136 stations during the period 1940–2012 allowed us to present some new evidence for the observed change and its statistical significance. The statistical framework used to determine the significance of the slopes in annual rain was not limited to the time independency assumption (Mann-Kendall test), but we also investigated the effect of short- and long-term persistence through Monte Carlo simulation. Our findings show that (a) change occurs in different scales; most regions show a decline since 1950, an increase since 1980 and remain stable during the last 15 years; (b) the significance of the observed decline is highly dependent to the statistical assumptions used; there are indications that the Mann-Kendall test may be the least suitable method; and (c) change in time is strongly linked with the change in space; for scales below 40 years, relatively close regions may develop even opposite trends, while in larger scales change is more uniform.
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Long-range dependence (LRD), the so-called Hurst–Kolmogorov behaviour, is considered to be an intrinsic characteristic of most natural processes. This behaviour manifests itself by the prevalence of slowly decaying autocorrelation function and questions the Markov assumption, often habitually employed in time series analysis. Herein, we investigate the dependence structure of annual rainfall using a large set, comprising more than a thousand stations worldwide of length 100 years or more, as well as a smaller number of paleoclimatic reconstructions covering the last 12,000 years. Our findings suggest weak long-term persistence for instrumental data (average H = 0.59), which becomes stronger with scale, i.e. in the paleoclimatic reconstructions (average H = 0.75).
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Evapotranspiration (ET) is a vital component of the hydrological cycle and there are a large number of alternative models for representing ET processes. However, implementing ET models in a consistent manner is difficult due to the significant diversity in process representations, assumptions, nomenclature, terminology, units and data requirements. An R package is therefore introduced to estimate actual, potential and reference ET using 17 well-known models. Data input is flexible, and customized data checking and pre-processing methods are provided. Results are presented as summary text and plots. Comparisons of alternative ET estimates can be visualized for multiple models, and alternative input data sets. The ET estimates also can be exported for further analysis, and used as input to rainfall-runoff models.
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Evaporation can be viewed both as energy (heat) exchange and an aerodynamic process. According to the energy balance approach, the net radiation at the Earth’s surface (Rn = Sn – Ln, where Sn and Ln are the shortwave—solar—and longwave—earth—radiation, respectively) is mainly transformed to latent heat flux, Λ, and sensible heat flux to the air, H. The evaporation rate, expressed in terms of mass per unit area and time (e.g. kg/m²/d), is given by the ratio E΄ := Λ / λ, where λ is the latent heat of vaporization, with typical value 2460 kJ/kg. By ignoring fluxes of lower importance, such as soil heat flux, the heat balance equation is solved for evaporation, yielding: where b := H / Λ is the co-called Bowen ratio. The estimation of b requires the measurement of temperature at two levels (surface and atmosphere), as well as the measurement of humidity at the atmosphere. On the other hand, the estimation of the net radiation Rn is based on a radiation balance approach to determine the components Sn and Ln. Typical input data required (in addition to latitude and time of the year), are solar radiation (direct and diffuse, or, in absence of them, sunshine duration data or cloud cover observations), temperature and relative humidity. The net radiation also depends of surface properties (i.e. albedo) and topographical characteristics, in terms of slope, aspect and shadowing. Recent studies proved that the impacts of topography are important at all spatial scales, although they are usually neglected in calculations [23]. where Δ is the slope of vapor pressure/temperature curve at equilibrium temperature (hPa/K), γ is a psychrometrcic coefficient, with typical value 0.67 hPa/K, and D is the vapor pressure deficit of the air (hPa), defined as the difference between the saturation vapor pressure ea and the actual vapor pressure es, which are functions of temperature and relative humidity. We remind that (2) estimates the evaporation rate (mass per unit area per day), which is expressed in terms of equivalent water depth by dividing by the water density ρ (1000 kg/m³). Next we will use symbols Ε΄ for evaporation rates, and E := Ε΄ / ρ for equivalent depths per unit time. In this context, FAO proposed the application of the Penman–Monteith method for the hypothetical reference crop, thus introducing the concept of reference evapotranspiration. With standardized height for wind speed, temperature and humidity measurements at 2.0 m and the crop height of 0.12 m, the aerodynamic and surface resistances become ra = 208 / u2 (where u2 is the wind velocity, in m/s) and rs = 70 s/m. The experts of FAO suggested using the Penman–Monteith method as the standard for reference evapotranspiration and advised on procedures for calculation of the various meteorological inputs and parameters [7].
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Evapotranspiration is a key hydrometeorological process and its estimation is important in many fields of hydrological and agricultural sciences. Simplified estimation proves very useful in absence of a complete data set. In this respect, a parametric model based on simplification of the Penman-Monteith formulation is presented. The basic idea of the parametric model is the replacement of some of the variables and constants that are used in the standard Penman-Monteith model by regionally varying parameters, which are estimated through calibration. The model is implemented in various climates on monthly time step (USA, Germany, Spain) and compared on the same basis with four radiation-based methods (Jensen-Haise, McGuiness and Bordne, Hargreaves and Oudin) and two temperature-based (Thornthwaite and Blaney-Criddle). The methodology yields very good results with high efficiency indexes, outperforming the other models. Finally, a spatial analysis including the correlation of parameters with latitude and elevation together with their regionalization through three common spatial interpolation techniques along with a recent approach (Bilinear Surface Smoothing), is performed. Also, the model is validated against Penman-Monteith estimates in eleven stations of the well-known CIMIS network. The total framework which includes the development, the implementation, the comparison and the mapping of parameters illustrates a new parsimonious and high efficiency methodology in the assessment of potential evapotranspiration field.
Whilst there are numerous global and regional studies of climate impacts on water resources, relatively few authors have incorporated reservoir storage into their earth system models. Consequently, such studies are unlikely to provide coherent estimates of how changes in climate might affect water supplies globally. This short communication describes an R package, named reservoir, which has been designed for rapid and easy routing of runoff data through storage. The package comprises tools for capacity design, release policy optimisation and performance analysis-allowing users to specify realistic reservoirs and then assess performance in terms of meeting water delivery targets. We demonstrate some of the capabilities of reservoir using 271 runoff records from the Global Runoff Data Centre. The package is freely available through the Comprehensive R Archive Network (CRAN).
Identification of changes in hydrological extremes plays an important role in water resources management. This study investigates various properties in hydrological extremes including annual maximum daily flow (high flow) and annual minimum 7-day flow (low flow) in 252 unimpaired streamflow gauges in the continental United States. Three statistical methodologies are performed: the nonparametric Mann-Kendall test is used to evaluate temporal trends; the non-parametric change point Pettitt test, to identify abrupt shifts; and the quantile perturbation method, to assess temporal variability. In the results of trend analysis, low flow exhibited some spatial patterns compared to high flow. In addition, low flow has increased over 64 years (1950-2013) in many stations. The Pettitt test indicates that low flows have experienced more significant abrupt changes compared to high flow. The results of the quantile perturbation method confirm that many significant increasing trends obtained during the 64 years of low flow are attributable to infrequent low flow events in the period before 1970. These findings highlight that various statistical approaches complement each other and must be simultaneously applied to hydrological time series.
H.E. Hurst spent some sixty years studying the Nile for the Egyptian government, and laid the foundation for a monumental set of hydrological records and investigations. His studies of the size of over-year reservoirs needed to maintain a given yield from Nile flows showed that this was greater than that based on random series. This finding, known as the Hurst phenomenon, was confirmed by other natural series and led to important advances in practical and theoretical statistics. His work led to the design of the Aswan High Dam and to continued research in Egypt.