Statistical detection of trends in hydrometeorological
time series is a crucial task when revealing how river systems react to
environmental and human-induced changes. It was shown that the
autocorrelation structure of a series influences the power of parametric and
nonparametric trend tests. While the order of short-memory processes can be
sufficiently captured by AR(I)MA models, the determination of the Hurst
exponent, which describes the long memory, is still challenging, considering
that the available methods partially give different results. In the Elbe
River basin, Europe, several studies focusing on the detection (or
description) of long-term persistence were performed. However, different
lengths of series and different methods were used. The aim of the present
work is to gather the results gained in various parts of the Elbe basin in
Central Europe and to compare them with our estimation of the Hurst exponent
using six discharge series observed in selected subbasins. Instead of the
dependence of the exponent on the catchment area suggested by the theory of
aggregated short-memory processes, we rather found a relationship between
this parameter and the series length. As the theory is not supported by our
findings, we suppose that the Hurst phenomenon is caused by a complex
interplay of low-frequency climate variability and catchment processes.
Experiments based on distributed water balance models should be the further
research objective, ideally under the umbrella of mutual international
projects.
Introduction
The question of the existence and cause for trends in hydrometeorological
(HM) time series has already been posed since the 19th century (e.g.
v. Berg, 1867). In the course of the 20th century, predominantly
nonparametric trend tests have been employed to analyze HM series.
Theoretical works have shown (often through Monte Carlo experiments; e.g.
Yue et al., 2002a) that these classical tests, although powerful in many
cases (e.g. Yue et al., 2002b), are in fact sensitive to the violation of
the assumption of independence in time series. Because HM series, as well as
many other geophysical series, often reveal autocorrelation which is
incompatible with that assumption, many modifications of trend tests came
into being. This especially applies to the assumption of short-term
persistence (STP) in HM time series that can be addressed by AR(I)MA models
(or their variants) before utilizing trend tests themselves. However,
building on others, Hamed (2008) showed that also long-term persistence
(LTP) can adversely affect trend analyses through influencing the variance
of test statistics.
Bearing the afore-mentioned facts in mind, we focus here specifically on the
Elbe River basin located in Europe for which the decision on the presence of
LTP in various HM series is very important for predicting the future
development of discharge response to climate change, especially in
connection with better estimating the uncertainty limits, which should be
beneficial for future activities related to the Elbe and its tributaries,
such as river navigation, water supply, hydropower production, irrigation,
and so on. As there has been neither a closing result about the causes of
LTP in HM series nor a single superior approach to the estimation of its
parameter H, we offer a concise overview of the state of the discussion and
put our own results into the context outlined by the outcomes of previous
studies.
Theoretical background
The Hurst phenomenon (along with LTP) is well known among today's
hydrologists. Therefore, we only stress some important points and possible
misconceptions. An overview of the history of investigating LTP can be found
in Beran (1994), Graves et al. (2017) or Montanari (2003). Hurst analysed
the rescaled range R (see Sect. 3.1) as a measure of reservoir capacity and
found that for many runoff series Eq. (1) applies:
R/S∝NH,
with standard deviation S, sample size Nand the parameter H taking a mean value
of 0.73 which differs clearly from the theoretical value 0.5 (Montanari,
2003). This is equivalent to the fact that the series autocorrelation
function does not decay exponentially, but follows a power law and diverges.
A large part of the amazement, these findings have caused, originates from
the assumptions that runoff can be characterized by a stationary process for
which H about 0.5 is expected. Mandelbrot and Van Ness (1968) and Mandelbrot
and Wallis (1968) proposed a statistical model based on stationarity that
should account for LTP: the fractional Brownian motion (fBm) model, which
brought two other terms: infinity and self-similarity. Unwittingly, this
model maybe pushed the search for the comprehension of the Hurst effect
towards an unfavourable direction. The Gaussian distributed random process
underlying runoff was already questioned by Louis M. Laushely in the
discussion part of Hurst's seminal paper (Hurst, 1951). Even Hurst himself
mentioned shifting mean as a possible reason for the phenomenon (Montanari,
2003). Eventually, Mandelbrot and Wallis (1968) stated that models
underestimate the complexity of hydrological fluctuations and considered
fluctuations on all timescales: macrometeorological, anthropological,
climatological and paleoclimatological (Mandelbrot and Wallis, 1969). The
fBm model was therefore never supposed to provide an explanation for LTP
discovered in runoff series, but rather to be a tool able to simulate these
series in a satisfactory manner. Although Mandelbrot and Wallis (1969)
explained how the term “infinity” should be understood, a number of
hydrologists refused to use the fBm approach together with its spooky
“infinite memory” and considered LTP a mysterious intrinsic property of
runoff series whose stochastic behaviour should be distinguished from
deterministic trends (e.g. Kantelhardt et al., 2001). The concept was
already questioned early by others. For example, Klemeš (1974) wondered
“by what sort of physical mechanism” a measured value should be dependent
on all previous values. Later, the operational school drew a lot of
criticism. Klemeš (1978) stated that it failed “to see that the
statistical and stochastic properties of hydrologic processes have definite
physical causes”. Mesa and Poveda (1993) wrote that “the so-called Hurst
effect and other related anomalities are probably the result of a mixture of
scales more than infinite memory. Knowledge of those scales is a more
fundamental issue from a physical viewpoint”. Beran (1994) demanded models
“that can be explained by the physical mechanism that generates the data”.
One example of such a model is the concept of storage cascade mechanisms,
which was proposed by Klemeš (1974), and in which Mudelsee (2007) saw
the explanation for the Hurst phenomenon. However, Klemeš (1974) also
postulated that LTP observed in hydrological series is caused by a complex
interplay of different processes on different scales. This postulate, as
Montanari (2003) stated, can nowadays have a broad acceptation within the
hydrological community.
There are different approaches to quantify the strength of LTP through the
parameter H called the Hurst exponent (or coefficient). In the following
section, we briefly describe the two methods that have been applied in the
Elbe River basin and also in our case study.
Estimation methods
We used (1) rescaled range (R/S) analysis which is based on the works of
Hurst (1951) and Mandelbrot and Wallis (1969), and (2) detrended fluctuation
analysis (DFA) which was proposed by Peng et al. (1994). With regard to
validity and performance, Willems and Min (2016) compared functions of the
programming language R which have implemented R/S or DFA. Inspired by the
review, we used the package “pracma” (Borchers, 2019) for R/S and the
package “nonlinearTseries” (Garcia, 2020) for DFA.
Rescaled range analysis
The adjusted range (R) is defined as the difference between the most positive
and the most negative departures from the mean cumulated (discharge) curve,
which is normalized by the sample standard deviation (S) to get the rescaled
adjusted range statistic R/S(n). The procedure can be formalized as in Eq. (2):
R/Sn=max1≤k≤n{Yk-knYn}-min1≤k≤n{Yk-knYn}1n∑k=1nXk-1nYn2,
where Xk is a sequence of random variables and Yn the nth partial
sum X1+…+Xn. The parameter H can then be estimated by the
slope of a double logarithmic plot of R/S(n) against n/2 (for details see e.g.
Graves et al., 2017).
Detrended fluctuation analysis
The idea behind DFA is to remove trends prior estimating the scaling
behaviour. It is also based on the sequence of partial sums Yn which is
divided into segments of equal length s. For each segment, a polynomial
regression model is fitted to the part of the time series and subtracted
from it. Subsequently, the variance σ is calculated for each
segment. The fluctuation function is the square root of the average variance
subject to the segment width: F(s)=12Ns∑t=12Nsσt,s.
Similar to R/S analysis, H can be estimated by the slope of a double
logarithmic plot of F(s) against s.
Overview of studies performed in the Elbe basin
Mudelsee (2007) fitted an ARFIMA(1, d, 0) model to the data of six
water-gauging stations in the Elbe basin and analyzed if there was a
correlation between catchment area and LTP. Although Mudelsee (2007) pointed
out that, using a permutation test, a positive relationship was found also
in the case of the Elbe basin, looking at the figures presented in that
paper, one could argue that such a finding might be questionable.
Marković and Koch (2005) analyzed LTP in monthly high flows at the gauge
Dresden (Elbe) and compared six different estimation methods which were
applied to the raw data and to four time series derived by various filtering
techniques. The comparison of methods, such as R/S analysis or the level of
zero crossings approach, resulted in a wide range of H estimates.
Kantelhardt et al. (2006) analyzed daily runoff series of the gauge Dresden,
covering the period 1852–2002. Through DFA, they found that H= 0.80.
Markovic and Koch (2014) determined H from discharge records of the gauges
Neu Darchau (on the Elbe) and Niederstriegis (Mulde). Analyzing raw data,
they found that for Neu Darchau H= 0.80 and for Niederstriegis
H= 0.63. After filtering the series (i.e. subtracting annual and
low-frequency components), they observed considerable decreases of H. They
concluded that LTP in discharge was caused by seasonal and decadal
variability.
Regarding the Czech part of the Elbe basin, mainly Ledvinka (2014, 2015a, b) carried out several calculations that incorporated also a number of
water-gauging stations relevant to this study. Three series of mean daily
discharge from the Ore Mts. (i.e. one of the headwater areas of the Elbe
basin) were subjected to wavelet and R/S analyses, and to the aggregated
variance method. Also, maximum likelihood estimation of the fractional
differencing parameter for ARFIMA models was conducted. Unexpectedly, quite
high values of H were detected. Furthermore, it was found that many series
involving indices related to hydrological drought in Czechia reveal high
values of H as well, which was confirmed by unit root tests. On the other
hand, precipitation series in Czechia for the period 1961–2012 showed no
LTP (Ledvinka, 2015c). The differences between discharge and climate
variables from this point of view, using selected stations within the Czech
part of the Elbe basin, were highlighted also in Ledvinka and Jedlicka (2018).
Location of six water-gauging stations selected for the
case study within the Elbe River basin.
Closer look at the catchments represented by investigated
gauges, river network within the catchments and elevation conditions in the
case study area.
Case study
For our own analyses, we selected six discharge time series representing
small tributaries in the central part of the Elbe basin close to the borders
between Germany and Czechia. All gauges are located within a mountain range
(see Figs. 1–2): Aue 1 (on the Schwarzwasser), Rothenthal (on the
Natzschung) and Dohna (on the Müglitz) belong to the Ore Mts.;
Elbersdorf (on the Wesenitz), Porschdorf 1 (on the Lachsbach) and
Kirnitzschtal (on the Kirnitzsch) belong to the Elbe Sandstone Mts. The
period, for which all gauges have complete mean daily discharge data, spans
from 1929 to 2005. The gauges Kirnitzschtal and Porschdorf 1 had, at the
time of writing, the longest time series (i.e. 1911–2017). The Hurst
exponent was calculated using the R/S and DFA methods based on mean monthly
discharge time series which were deseasonalized using the method described
in Hipel and McLeod (1994) and implemented in the R package
“deseasonalize” (McLeod and Gweon, 2013). The results are summarized in
Table 1. For all gauges, we found that H>0.5, which means that,
very likely, the series contain LTP. DFA gives higher values for H and shows
a higher variability in comparison to R/S. Figure 3 shows a plot of H against
catchment area. Based on the series of Porschdorf 1, Fig. 4 depicts how the
estimates of H depend on varying sample size. It is obvious that for both
applied methods, H decreases with increasing time series length.
Estimated H exponents for deseasonalized monthly mean discharge time
series (1929–2005).
The values of the Hurst exponent H against catchment area.
Dependence of the estimates of the Hurst exponent
H on the length of analyzed series (shown for the
Porschdorf 1 gauge).
Discussion and conclusions
The quantification of LTP in HM series is a difficult task, especially due
to the fact that many estimation methods give different results regarding
the magnitude of the LTP parameter H. This is evident also from our case
study where only two methods were chosen (see Figs. 3–4). As shown in Fig. 4, also the length of available time series influences the estimation of
H. Regarding the obvious tendency for H to decrease in Fig. 4, we must state
that our series are still too short to give a clear answer to the question
about the general behaviour of H for very long time series. Both convergence
to the value of 0.5 as well as to the value signifying LTP might be
possible. The issue of the time series length was demonstrated very clearly
for precipitation by Markonis and Koutsoyiannis (2016) who compared the
behaviour of instrumental records with that of much longer time series
derived from proxy data. However, when dealing with changes or trends in
hydrological regimes, LTP is a key player since the outcomes of various
tests strongly depend on the correct setting of null hypotheses. As the
instrumental records are usually short, one should keep in mind the issues
mentioned above before constructing models for practical purposes such as
setting the lifetime of water works.
Overall, it is believed that LTP is stronger in discharge time series than
in other HM series. Today, many hydrologists suggest that LTP is inherited
by discharge from the low-frequency climate variability that may be
reflected, for instance, in precipitation. The discrepancy between the
behaviour of discharge and precipitation could be explained by physical
processes similar to filtering (Milly and Wetherald, 2002). The theory that
only aggregated short-memory processes are the cause for LTP is somewhat
contradictory to our findings in the Elbe basin because the small catchments
here reveal relatively high values of the parameter H as well. Moreover, Fig. 3 shows that there is no such dependence suggested by Mudelsee (2007).
Further research into this topic should focus on the low-frequency
variability of driving forces of runoff and their possible amplification by
the catchment system. Distributed water balance models can be a tool for
suitable experiments because studying time series of components, such as
soil moisture, groundwater recharge, evapotranspiration and others, might
give valuable answers regarding interactions and thresholds occurring in the
Elbe River basin, many of which may still not be fully understood.
As already stated in several past papers (e.g. Koutsoyiannis, 2003, 2006;
Koutsoyiannis and Montanari, 2007), not considering LTP properly may in
practice lead to underestimation of process variability that, moreover,
needs to be distinguished from change caused by external drivers. Especially
in the fields of hydrology and water management, this is a very difficult
but important task for designing water works. Therefore, the research into
this topic should definitely continue.
Data availability
Demanded files with data and geodata from Saxony can be currently selected and downloaded using the iDA information system (https://www.umwelt.sachsen.de/umwelt/infosysteme/ida/, last access: 15 July 2020, LfULG, 2019). Individual tiles of digital elevation model for Fig. 2 were selected and retrieved using the USGS EarthExplorer Portal (https://earthexplorer.usgs.gov/, last access: 15 July 2020, USGS, 2020). Both sources require user's registration first, which is for free.
Author contributions
TR worked on the theoretical part, the case study (including the table and
figures apart from Fig. 2) and the literature overview regarding Germany. OL
worked on the rest of the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Special issue statement
This article is part of the special issue “Hydrological processes and water security in a changing world”. It is a result of the 8th Global FRIEND–Water Conference: Hydrological Processes and Water Security in a Changing World, Beijing, China, 6–9 November 2018.
Acknowledgements
We would like to acknowledge Petra Walther (LfULG Sachsen) for providing
suitable time series from the Saxon part of the Elbe basin.
References
Beran, J.: Statistics for Long-Memory Processes, Chapman & Hall, New
York, 1994.Borchers, H. W.: pracma: Practical Numerical Math Functions, R package version 2.2.9, available at: https://CRAN.R-project.org/package=pracma (last access: 21 April 2020), 2019.Garcia, C. A.: nonlinearTseries: Nonlinear Time Series Analysis, R package version 0.2.8, available at: https://CRAN.R-project.org/package=nonlinearTseries (last access: 21 April 2020), 2020.Graves, T., Gramacy, R., Watkins, N., and Franzke, C.: A brief history of
long memory: Hurst, Mandelbrot and the road to ARFIMA, 1951–1980, Entropy,
19, 437, 10.3390/e19090437, 2017.Hamed, K. H.: Trend detection in hydrologic data: the Mann–Kendall trend
test under the scaling hypothesis, J. Hydrol., 349, 350–363,
10.1016/j.jhydrol.2007.11.009, 2008.
Hipel, K. W. and McLeod, A. I.: Time Series Modelling of Water Resources and
Environmental Systems, Elsevier, Amsterdam, 1994.
Hurst, H. E.: Long term storage capacity of reservoirs (with discussions),
Trans. Am. Soc. Civ. Eng., 116, 770–808, 1951.Kantelhardt, J. W., Koscielny-Bunde, E., Rego, H. H., Havlin, S., and Bunde,
A.: Detecting long-range correlations with detrended fluctuation analysis,
Phys. Stat. Mech. Its Appl., 295, 441–454,
10.1016/S0378-4371(01)00144-3, 2001.Kantelhardt, J. W., Koscielny-Bunde, E., Rybski, D., Braun, P., Bunde, A.,
and Havlin, S.: Long-term persistence and multifractality of precipitation
and river runoff records, J. Geophys. Res.-Atmos., 111, D01106,
10.1029/2005JD005881, 2006.Klemeš, V.: The Hurst phenomenon: a puzzle?, Water Resour. Res., 10,
675–688, 10.1029/WR010i004p00675, 1974.
Klemeš, V.: Physically based stochastic hydrologic analysis, in Advances
in Hydroscience, vol. 11, edited by: Chow, V. T., Academic
Press, New York, 285–356, 1978.Koutsoyiannis, D.: Climate change, the Hurst phenomenon, and hydrological
statistics, Hydrol. Sci. J., 48, 3–24,
10.1623/hysj.48.1.3.43481, 2003.Koutsoyiannis, D.: Nonstationarity versus scaling in hydrology, J.
Hydrol., 324, 239–254, 10.1016/j.jhydrol.2005.09.022, 2006.Koutsoyiannis, D. and Montanari, A.: Statistical analysis of hydroclimatic
time series: Uncertainty and insights, Water Resour. Res., 43, W05429,
10.1029/2006WR005592, 2007.
Ledvinka, O.: Are there nonstationarities and the Hurst phenomenon in
discharge series within the Ore Mountains region?, in: Hydrology of a Small
Basin 2014, vol. 1, edited by: Brych, K. and Tesař, M., Czech
Hydrometeorological Institute, Praha, 287–295, 2014.Ledvinka, O.: Evolution of low flows in Czechia revisited, Proc. IAHS, 369, 87–95, 10.5194/piahs-369-87-2015, 2015a.Ledvinka, O.: Scaling of low flows in Czechia – an initial assessment, Proc. IAHS, 366, 188–189, 10.5194/piahs-366-188-2015, 2015b.
Ledvinka, O.: Nonstationarities in technical precipitation series in
Czechia, Acta Hydrol. Slovaca, 16, 199–207, 2015c.Ledvinka, O. and Jedlicka, M.: Freely available daily hydrometeorological
data from Czechia: further insights, edited by: Kaźmierczak, B., Kutyłowska, M., Piekarska, K., and Jadwiszczak, P., E3S Web Conf., 44, 00093, 10.1051/e3sconf/20184400093, 2018.LfULG: interdisziplinäre Daten und Auswertungen, available at: https://www.umwelt.sachsen.de/umwelt/infosysteme/ida/ (last access: 15 July 2020), 2019.Mandelbrot, B. B. and Van Ness, J. W.: Fractional Brownian motions,
fractional noises and applications, SIAM Rev., 10, 422–437,
10.1137/1010093, 1968.
Mandelbrot, B. B. and Wallis, J. R.: Noah, Joseph, and operational
hydrology, Water Resour. Res., 4, 909–918, 10.1029/WR004i005p00909,
1968.Mandelbrot, B. B. and Wallis, J. R.: Some long-run properties of geophysical
records, Water Resour. Res., 5, 321–340, 10.1029/WR005i002p00321,
1969.Markonis, Y. and Koutsoyiannis, D.: Scale-dependence of persistence in
precipitation records, Nat. Clim. Change, 6, 399–401,
10.1038/nclimate2894, 2016.Marković, D. and Koch, M.: Sensitivity of Hurst parameter estimation to
periodic signals in time series and filtering approaches, Geophys. Res.
Lett., 32, L17401, 10.1029/2005GL024069, 2005.Markovic, D. and Koch, M.: Long-term variations and temporal scaling of
hydroclimatic time series with focus on the German part of the Elbe River
Basin, Hydrol. Process., 28, 2202–2211, 10.1002/hyp.9783, 2014.
McLeod, A. I. and Gweon, H.: Optimal deseasonalization for monthly and daily
geophysical time series, J. Environ. Stat., 4, 1–11, 2013.Mesa, O. J. and Poveda, G.: The Hurst effect: The scale of fluctuation
approach, Water Resour. Res., 29, 3995–4002, 10.1029/93WR01686,
1993.Milly, P. C. D. and Wetherald, R. T.: Macroscale water fluxes 3. Effects of
land processes on variability of monthly river discharge, Water Resour.
Res., 38, 1235, 10.1029/2001WR000761, 2002.
Montanari, A.: Long-range dependence in hydrology, in: Theory and
Applications of Long-Range Dependence, edited by: Doukhan, P., Oppenheim, G.,
and Taqqu, M. S., Birkhäuser, Boston, Massachusetts, 461–472, 2003.Mudelsee, M.: Long memory of rivers from spatial aggregation, Water Resour.
Res., 43, W01202, 10.1029/2006WR005721, 2007.Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E., and
Goldberger, A. L.: Mosaic organization of DNA nucleotides, Phys. Rev. E,
49, 1685–1689, 10.1103/PhysRevE.49.1685, 1994.USGS: EarthExplorer Portal, available at: https://earthexplorer.usgs.gov/, last access: 15 July 2020.
v. Berg: Wie kann man die Verminderung des Wasserstandes in der Elbe
erklären?, Wiss. Beil. Leipz. Ztg., 13 June 1867.
Willems, W. and Min, A.: Praxisgerechte Detektion von Trends unter
Berücksichtigung von Kurz- und Langzeit-Autokorrelationsstrukturen,
IAWG, Ottobrunn, 2016.Yue, S., Pilon, P., Phinney, B., and Cavadias, G.: The influence of
autocorrelation on the ability to detect trend in hydrological series,
Hydrol. Process., 16, 1807–1829, 10.1002/hyp.1095, 2002a.Yue, S., Pilon, P., and Cavadias, G.: Power of the Mann–Kendall and
Spearman's rho tests for detecting monotonic trends in hydrological series,
J. Hydrol., 259, 254–271, 10.1016/S0022-1694(01)00594-7, 2002b.