PIAHSProceedings of the International Association of Hydrological SciencesPIAHSProc. IAHS2199-899XCopernicus GmbHGöttingen, Germany10.5194/piahs-369-87-2015Evolution of low flows in Czechia revisitedLedvinkaO.ledvinka@chmi.czhttps://orcid.org/0000-0002-0203-7064Hydrological Database & Water Balance, Czech Hydrometeorological Institute, Na Sabatce 2050/17, 143 06 Prague 412, Czech RepublicInstitute of Applied Mathematics and Information Technologies, Faculty of Science, Charles University in Prague, Albertov 6, 128 43 Prague 2, Czech RepublicO. Ledvinka (ledvinka@chmi.cz)11June2015369369879514April201514April2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://piahs.copernicus.org/articles/369/87/2015/piahs-369-87-2015.htmlThe full text article is available as a PDF file from https://piahs.copernicus.org/articles/369/87/2015/piahs-369-87-2015.pdf
Although a nationwide study focusing on the evolution of low flows in
Czechia was conducted in the past, a need for the revision of the results
has arisen. By means of the trend analysis, which specifically considers the
presence of significant serial correlation at the first lag, the former
study highlighted areas where 7-day low flows increase or decrease. However,
taking into account only the lag-one autoregressive process might still have
led to the detection of so-called pseudo-trends because, besides short-term
persistence, also long-term persistence may adversely influence the variance
of the test statistic when the independence among data is required.
Therefore, one should carefully investigate the presence of persistence in
time series. Before the trend analysis itself, the authors' previous studies
aimed at the discrimination between short memory processes and long memory
processes employing jointly the Phillips–Perron test and the
Kwiatkowski–Phillips–Schmidt–Shin test. This analysis was accompanied by the
Hurst exponent estimation. Here, the subsequent identification of trends is
carried out using three modifications of the Mann–Kendall test that allow
different kinds of persistence. These include the
Bayley–Hammersley–Matalas–Langbein–Lettenmaier equivalent sample size
approach, the trend-free pre-whitening approach and a block bootstrap with
automatic selection of the block length, which was applied for the first
time in hydrology. The general results are similar to those presented in the
former study on trends. Nevertheless, the divergent minimum discharges
evolution in the western part of Czechia is now much clear. Moreover, no
significant increasing trend in series incorporating Julian days was found.
Introduction
In recent decades, time series analysis applications in climatology and
hydrology have become more and more important. This is due to the fact that
many measured hydrometeorological variables are stored in nationwide and
even in world databases just as time series. This actually offers the
possibility to confirm on empirical basis whether climate change takes place
at all or whether humans significantly contribute to it. Climatologists, for
instance, found out that human activities cause changes in spatial
distribution of extreme precipitation patterns practically all around the
globe (see e.g. Min et al., 2011). In hydrology, responses of water
resources (in terms of various measured and derived characteristics) to
climate change are studied intensively. Very often, this topic is addressed
through the trend component identification which can be well documented by
numerous scientific papers published until today. Even if one concentrated
only on one year, the list of such literature would be huge. To name some,
in 2012 the relationships among trends in river flow and precipitation were
investigated by Chen et al. (2012), Ehsanzadeh et al. (2012), Liu et al. (2012),
Lorenzo-Lacruz et al. (2012), Shifteh Some'e et al. (2012), Steffens
and Franz (2012), Tabari et al. (2012) or Wagesho et al. (2012). Since the
mid-1990s, the role of serial correlation (autocorrelation) frequently
present in hydrometeorological series has been accented. It has been shown
that the serial correlation adversely affects the variance of test statistic
when the application of the test itself requires independent data
(e.g. Hamed and Rao, 1998; Kulkarni and von Storch, 1995; von Storch, 1999; Yue
and Wang, 2002, 2004; Yue et al., 2002, 2003). To overcome this downside,
several modifications of trend tests were proposed that, almost exclusively,
related to the influence of lag-one serial correlation on the Mann–Kendall
(MK) test that is well established in hydrology. A comprehensive overview of
the modifications can be found in Khaliq et al. (2009b).
Czech hydrologists employed the MK test in many cases as well (e.g. Kliment
and Matoušková, 2009; Kliment et al., 2011; Matoušková et
al., 2011), but they adopted its modifications with considerable delay
(Benčoková et al., 2011; Fiala et al., 2010; Vajskebr et al., 2013;
Vlnas and Fiala, 2010). In particular, Fiala et al. (2010) assessed trends
in hydrological drought indicators derived from mean daily discharges
recorded at water-gauging stations spread over the entire territory of
Czechia. Although not explicitly stated, they used the
Bayley–Hammersley–Matalas–Langbein–Lettenmaier equivalent sample size
(BHMLLESS) approach to correct the variance of the MK test. The BHMLLESS–MK
test accounts only for lag-one autoregressive (AR(1)) processes that are
known to be the representatives of so-called short-term persistence (STP).
However, Ehsanzadeh and Adamowski (2010), Khaliq et al. (2008, 2009a) or
Khaliq and Sushama (2012) pointed out that the series relative to
hydrological drought such as low flows or their timings may also be governed
by a stochastic process revealing long-term persistence (LTP). Through Monte
Carlo simulations, for example, Hamed (2008) showed the MK tests may still
overestimate the number of significant trends in hydrological series when
ignoring the assumption of LTP.
In Czechia, some regions have experienced shortages of water (drying up
rivers in the northwest were reported in Novický et al., 2010) and they
are envisaged to continue in the future according to several studies (see
e.g. Hanel et al., 2013). In spring 2014 just after the mild winter almost
without snow, there were even problems with water supply in the city of
Hradec Králové east of Prague. Therefore, sound reassessment of
trends in low flows is of high importance in Czechia, especially under the
consideration of above mentioned LTP effects which has been lacking since
Fiala et al. (2010) published their results.
To this purpose, Ledvinka (2014, 2015) first performed an analysis
regarding the scaling of low flows in Czechia. He, according to the
recommendation given in Khaliq et al. (2008), estimated the Hurst exponent H
via the fitting of fractionally integrated autoregressive-moving average
(FARIMA) models to the series investigated by Fiala et al. (2010). Instead
of studying the distribution of H for the purpose of getting the confidence
limits, he rather utilised so-called unit root tests to decide if the value
of H was significant or not. Finally, he concluded that especially southwest
and northeast parts of Czechia have some evidence of scaling in the series
composed of low flows and deficit volumes and their durations. The
exceptions might be the winter low flows and the series containing timings
of low flows. The reassessment of trend significances itself was left just
for the present paper. The main goals were following: (1) apply other
modifications of the MK test accounting for STP and LTP more correctly and
(2) by a simple comparison of the results of trend analyses try to examine
if the block bootstrap with the automatic selection of block length
(hereinafter ABBS) could be used properly in combination with the MK test
when detecting trends in drought-related series probably contaminated with LTP.
Data and the area of interest
As the area of interest (Czechia) was the same as in Fiala et al. (2010),
the reader who needs the detailed physical-geographical description of the
country is kindly referred to their paper. At this point, it should be
repeated that the climate of Czechia is influenced by terrain features to a
large extent. Among them the altitude determines what prevailing climate
drivers stand behind the advent of hydrological drought (i.e. long-term lack
of rains mixed with intense evapotranspiration or snowfall mixed with the
temperature below the freezing point) and thus belongs to the most important
here. This was the rationale for dividing the whole data set into two
subsets during all the analyses – Group 1 (G1) consisting predominantly of
the data from mountain stations and Group 2 (G2) containing the data mainly
from stations located in lowlands. Similarly, the seasonality of low flows,
associated with climate drivers as well, necessitated the separation of the
year into the summer period (from April to November) and the winter period
(from December to March). It is important to note that the start of the
period corresponding to one year was set to 1 April when the majority of
hydrographs in Czechia shows maxima due to the snow melting and the
probability of low flow occurrences is very low. To designate the station as
a mountain station, the number of annual 7-day low flows derived (see later)
had to occur at least 12 times in the winter period.
The data set subjected to the analyses was identical to that accessed by
Fiala et al. (2010). It consisted of mean daily discharges recorded at
144 water-gauging stations that pertain to the nationwide monitoring network
maintained by the Czech Hydrometeorological Institute (CHMI). The original
series were mostly uninterrupted. In few cases several figures were missing
but the gaps were successfully filled in with the help of the data from
neighbouring stations on the same river. After all, each of the discharge
series was complete and covered the period starting on 1 November 1960 and
ending on 31 October 2005.
From the series mentioned above, the series of following characteristics
indicating water scarcity were derived
annual 7-day low flow (Qmin A) along with the Julian day (JDA) of its occurrence;
summer 7-day low flow (Qmin S) along with the Julian day (JDS) of its occurrence;
winter 7-day low flow (Qmin W) along with the Julian day (JDW) of its occurrence;
annual deficit volume delimited based on the quantile Q330d (V330) and annual number of days with discharge under this quantile (D330);
annual deficit volume delimited based on the quantile Q355d (V355) and annual number of days with discharge under this quantile (D355).
In the case of winter low flow series, there were 45 entries, in all other
cases 44 entries. Note that the 7-day low flow was defined as the minimum
7-day moving average falling into the prescribed period. The definitions of
deficit volumes here somewhat differed from the study of Fiala et al. (2010)
since the threshold levels were derived with respect to COSMT (2014) that
requires these quantiles being computed using long-term flow duration curves
(here the period 1961–2005) and not as the averages of quantiles belonging
to each year. On the other hand, the deficit volumes, similarly as in Fiala
et al. (2010), were simple sums of mean daily discharges under the selected thresholds.
MethodsEvaluation of scaling properties of low flow series
To decide where in Czechia, and whether at all, some LTP stochastic
processes may govern the series of low flows or other series connected with
hydrological drought, an assessment in spirit similar to the procedure
outlined in Fatichi et al. (2009) was carried out. Due to the short lengths
of series, the heuristic approaches and also the approaches based on the
wavelet analysis failed when estimating the Hurst exponent H. Therefore, H was
estimated rather as the fractional differencing parameter d in FARIMA(p, d,
q) models iteratively fitted to the series using the “forecast” R package
(Hyndman and Khandakar, 2008). Because Khaliq et al. (2008) reported
suitability of FARIMA(0, d, 0) models as regards low flows in Canada, they
were thought to be appropriate for low flows in Czechia as well; so only the
parameter d was estimated, to which the value of 0.5 was added to get the
estimate of H. The analysis was accompanied by a joint application of two
unit root tests. These were the Phillips–Perron (PP) test (Phillips and
Perron, 1988) and the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test
(Kwiatkowski et al., 1992). These tests help one distinguish up to three
kinds of stochastic processes: (1) stationary processes (including STP
processes) about a deterministic trend, (2) unit root processes (including
random walks) and (3) other nonstationary processes (maybe LTP processes).
Particularly the third category was of great importance here because it
highlighted the values of H significantly higher than 0.5. More details
(e.g. necessary formulae) concerning this procedure can be found in Ledvinka (2015).
It remains to mention that only the 0.05 significance level was used at this point.
The original MK test
Kendall (1938) suggested a nonparametric statistic that was intended to
measure the bivariate correlation. Mann (1945) took advantage of Kendall's
characteristic and developed a test against trend that should be
distribution-free and insensitive to outliers. At the turn of the 1970s and
the 1980s, hydrologists dealing mainly with water quality advised using this
test due to its ability to cope with missing values including censored data
(see e.g. Hirsch et al., 1982).
Let Xt={x1, x2, …, xT} denotes the series
investigated. The procedure starts with the computation of the MK statistic SS=∑∀i<jsgnXj-Xi
where sgn(θ) stands for the sign function. Kendall (1970) points
out that for the discrete distribution of S, which has zero
mean under the null hypothesis, one can apply the approximation by the
standard Gaussian distribution if the length of a series T is greater than 10.
Then the standardized MK statistic Z can be calculated as
Z=S-1var(S)ifS>00ifS=0S+1var(S)ifS<0
with
var(S)=T(T-1)(2T+5)-∑m=1Ttmm(m-1)(2m+5)18
accounting for the number of ties tm of extent m (e.g. Yue et al., 2002).
After that, the p values of the test may be compared with quantiles
corresponding to the prescribed significance levels α.
This test has been accompanied by Sen's nonparametric approach to get the
estimate of linear trend magnitude according to (Sen, 1968)
b=medXj-Xij-i,∀i<j.
Its confidence limits can be acquired by a process outlined in Wagesho et al. (2012).
The BHMLLESS–MK test
Unfortunately, the presence of persistence in time series violates the
assumption of independence among data required before the application of the
original MK test. By means of Monte Carlo simulations Kulkarni and von
Storch (1995) studied the influence of STP on the MK test and found out that
the test detected significant trends more often than it should. In fact, as
shown by Yue et al. (2002), STP alters the variance of the test statistic –
positive serial correlation inflates it (and causes more frequent rejection
of the null hypothesis) while negative serial correlation has exactly the
opposite effect. To deal with this downside, several researchers suggested,
for instance, so-called variance corrections based on the computation of
equivalent (effective) sample sizes (e.g. Hamed and Rao, 1998; Yue and Wang, 2004).
Here, the author adopted very similar approach to that used in Fiala et al. (2010).
The difference was the lag-one serial correlation coefficient
r1 was computed from detrended series because the estimate of r1
might be affected by the trend (see Yue and Wang, 2004). Thus, the first
step in this procedure is detrending to gain the series XtDXtD=Xt-bt.
with b derived using Eq. (4). Then, assuming that the detrended series is
centred (i.e. its mean equals zero), one can estimate the population serial
correlation ρ1 as follows (e.g. Cowpertwait and Metcalfe, 2009)
r1=covXtD,Xt+1DvarXtD=1T∑t=1T-1XtD⋅Xt+1D1T∑t=1TXtD2=∑t=1T-1XtD⋅Xt+1D∑t=1TXtD2.
If r1 proves to be different from zero (see e.g. Anderson, 1942, for
testing it), the variance expressed by Eq. (3) is modified according to
(Matalas and Langbein, 1962)
var∗(S)=var(S)⋅1+2⋅r1T+1-T⋅r12+(T-1)⋅r1T⋅r1-12.
Finally, the standardized MK statistic Z can be quantified by introducing
var∗(S) in place of var(S) into Eq. (2). Otherwise, the BHMLLESS–MK
test is identical to the original one.
The trend-free pre-whitening MK (TFPW–MK) test
Pre-whitening, in general, represents another group of approaches intended
to eliminate the effect of STP. As the name suggests, the main purpose of
these procedures is to get a time series that is no longer contaminated with
a serial dependence and thus to allow the application of the original test
to such a pre-whitened series without any detriment. During the initial
pre-whitening approaches (see e.g. Kulkarni and von Storch, 1995), the
lag-one serial correlation ρ1 was estimated directly from data.
Later, it was proven that a coexistence of trend and STP (in terms of an
AR(1) process) has negative impact on the estimation of ρ1.
Therefore, Yue et al. (2002) proposed a modification of the MK test, the
first step of which is the removal of trend component as in Eq. (5). Next,
the estimation of ρ1 (as in Eq. 6) takes place followed by the
elimination of the AR(1) related to it. Having a detrended and centred time
series XtD we get
et=XtD-r1Xt-1D,t=2,3,…,T.
The estimate of white noise et is further blended with the term bt from
Eq. (5). The final (pre-whitened) product Yt of length n=T- 1 can then
be viewed as
Yt=et+bt,t=2,3,…,T
and it can be subjected to the MK test using Eqs. (1)–(3).
The ABBS–MK test
Besides mentioned in previous subsections, also computer-aided resampling
techniques are widely used in hydrology. Among them, mainly various
bootstrap utilisations were performed when addressing trends (see e.g. Abdul
Aziz and Burn, 2006; Burn and Hag Elnur, 2002; Burn and Hesch, 2007; Burn,
2008; Cunderlik and Burn, 2002; Douglas et al., 2000; Rivard et al., 2009;
Yue et al., 2003). A special kind referred to as the block bootstrap (BBS)
is advised if there is STP present in a time series (Khaliq et al., 2009b).
Khaliq and Sushama (2012) or Khaliq et al. (2008, 2009a) applied it even to
the low flow series in Canada.
The most challenging task in BBS is probably the step when one has to
determine the block length so that the investigated series could be treated
as a sequence of independent blocks. Until today, hydrologists based their
decisions regarding the block sizes almost solely on inspections of
autocorrelation functions. The present paper, however, is somewhat
innovative because the blocks were searched automatically (hence ABBS) by an
objective approach described in Politis and White (2004) and later revised
in Patton et al. (2009). The blocks then underwent the bootstrapping of time
series incorporated in the “boot” R package (Canty and Ripley, 2014). In
particular, from 1000 ensembles the bootstrap distributions of the MK
statistic S (from Eq. 1) were acquired for each series. Subsequently, the
confidence intervals were derived using the percentile method (for details
see Davison and Hinkley, 1997). Finally, if the test statistic corresponding
to the original permutation fell outside the confidence limits, the
alternative hypothesis that there is a trend was accepted.
Whereas the BBS technique is recommended for time series with STP, to a
large extent, the ABBS technique was hoped to be an appropriate alternative
working even with series contaminated with LTP due to its objective
treatment of correlation structures. Here, as far as the author knows,
subjecting the low flow characteristics from Czechia to the ABBS–MK test may
be the first examination of this method in global hydrology.
Numbers of significant trends in drought-related series in Czechia
during the period 1961–2005 according to various modifications of the
Mann–Kendall test at the 0.1 and 0.05 levels. Figures in parentheses refer
to the numbers associated with Hurst exponents significant at α= 0.05.
For series designations see Sect. 2.
Kendall's rank correlations between FARIMA(0, d, 0) Hurst exponent
estimates and p values of selected trend tests for all stations together and
separately for mountain (G1) and lowland (G2) water-gauging stations as
regards low flows in Czechia during the period 1961–2005. For series
designations see Sect. 2.
Spatial distribution of trends in summer (April–November)
7-day low flows (Qmin S) in Czechia during the period 1961–2005 (black
symbols – ABBS–MK test, red symbols – BHMLLESS–MK test).
Spatial distribution of trends in annual deficit volumes
V330(a) and trends in annual numbers of days with discharge under the
quantile Q330d(b) in Czechia during the period 1961–2005 (black
symbols – ABBS–MK test, red symbols – BHMLLESS–MK test).
Spatial distribution of trends in annual deficit volumes
V355(a) and trends in annual numbers of days with discharge under the
quantile Q355d(b) in Czechia during the period 1961–2005 (black
symbols – ABBS–MK test, red symbols – BHMLLESS–MK test).
Results and discussion
First of all, it must be stressed that a lot of materials (namely maps and
tables) were produced and thus there is no chance to present them all. For
comparison with the results appearing in Fiala et al. (2010), the spatial
distribution of trends in five indicators (Qmin S, V330, D330,
V355 and D355) can be seen in Figs. 1–3. The maps
corresponding to the ABBS–MK test were chosen as fundamental since the
method was newly proposed here. Alongside the symbols for this method, also
the symbols for the BHMLLESS–MK test were incorporated to emphasize
disparities. Table 1 provides information on the numbers of trends
discovered after the utilisation of all the three modifications described in
Sect. 3. In parentheses there are highlighted the numbers of Hurst exponents
significantly greater than 0.5 linked to each category of trend.
It is evident from Table 1, that only few places in Czechia experience some
changes in drought-related characteristics investigated. Regarding
significant changes, namely decreases in V355 at mountain stations can
be identified. They are rather due to decreases in D355 because trends
in 7-day low flows are not so apparent. On the contrary, at lowland stations
rather increases of deficit volumes can be seen. This closely coincides with
the course of both series D330 and D355, but also with the decreases
of 7-day low flows (apparently due to evapotranspiraion). The JD series show
constant shifts towards earlier times in the year regardless if they belong
to the mountain or lowland group of stations. This might be a result of
earlier thawing, which in turn may cause more frequent flooding in winter
months while weak groundwater recharges in summer months. Interestingly,
this phenomenon is more evident in lowlands.
As for the spatial distribution of significant trends in summer 7-day low
flows (Fig. 1), the territory of Czechia could be divided into the western
an eastern part. Whereas the eastern part (especially the Upper Morava River
basin and the catchments of left-hand tributaries of the Elbe River)
experiences important drops, the western part (the Upper Jizera River basin)
reveals upward trends. Annual and winter minima have analogous pattern (not
shown). Figure 2 depicts the spatial distribution of trends in series
V330 and D330. Both parts actually show the logical opposite pattern
to that of 7-day low flows. Figure 3 (for V355 and D355) provides
essential information on balanced difference in numbers of upward and
downward trends. Specifically, the trends in the southwest (Šumava Mts.)
strengthen when looking at the discharges under the lower quantile
Q355d. Although the general results here are very similar to those
presented in Fiala et al. (2010), the divergent minimum discharges evolution
in the western part of Czechia is now much clear. In addition, no
significant increasing trend in series incorporating Julian days was found here.
Table 1 also simply facilitates the assessment of uncertainty associated
with possible scaling of time series. The Hurst phenomenon is more likely in
the series of deficit volumes and the series of Julian days. Evidently,
rather the decreasing trends might be triggered by a fluctuating behaviour.
Surprisingly, the new ABBS–MK test shows similar uncertainty as the
BHMLLESS–MK test in terms of falsely detected trends in drought-related
characteristics. The TFPW–MK test proved to be much better from this viewpoint.
Experimentally, the Kendall rank correlation coefficients (Kendall, 1970)
were computed for the purpose of verification of the negative association
between the estimates of the Hurst exponent and the p values of selected
trend tests that was emphasized in Khaliq et al. (2009a) or in Khaliq and
Sushama (2012). From Table 2 it is apparent that the relationship applies
even to the series of low flows and other linked indicators in Czechia.
Concerning the JDW series, the scaling influence on trend detection might
be of high importance (see the boldface figures in Table 2). However, as
mentioned in previous studies (Ledvinka, 2014, 2015), the Hurst exponents
related to winter series are low or insignificant.
On the other hand, one should bear in mind that the series here are really
short (44–45 years) and it is very difficult to estimate the Hurst exponent
and to make inferences about LTP under these circumstances. If, for
instance, the series started 15 years earlier, they would involve two of the
most drastic hydrological droughts in the Czech history (1947 and 1953/1954)
which would flip the direction of discovered trends at most places (see
Vlnas and Fiala, 2010).
Conclusions
A reassessment of trends in series of 7-day low flows and other related
characteristics in Czechia during the period 1961–2005 was carried out. For
this purpose three modifications of the MK test were used. One of them, the
ABBS–MK test, was proposed here and utilised for the first time in hydrology
hoping that it would be able to capture the correlation structure typical
not only for STP but also for LTP. The findings are very similar to those
published in Fiala et al. (2010) or Vlnas and Fiala (2010). However, the
present paper substantially improved the understanding of the trends: (1) it
underlined a contrast between increasing deficit volumes in the eastern part
of the country and decreasing ones in the western part, and (2) showed no
upward trend in the series of Julian days, probably due to earlier thawing.
The ABBS–MK test proved to be as good as the BHMLLESS–MK test that accounts
for STP only. Nevertheless, the careful examination of its functionality
requires longer series ideally generated by an LTP stochastic process so it
could be compared to other methods recommended for testing such series
(e.g. Cohn and Lins, 2005; Hamed, 2008). In the future, the investigation should
additionally concern the sites where significant trends and Hurst exponents
occurred together.
Acknowledgements
The author would like to thank the Technology Agency of the Czech Republic
for its financial support under the grant no. TA02020320. Also, personal
communications with Theodor Fiala and Radek Vlnas are greatly appreciated.
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