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  <front>
    <journal-meta><journal-id journal-id-type="publisher">HESS</journal-id><journal-title-group>
    <journal-title>Hydrology and Earth System Sciences</journal-title>
    <abbrev-journal-title abbrev-type="publisher">HESS</abbrev-journal-title><abbrev-journal-title abbrev-type="nlm-ta">Hydrol. Earth Syst. Sci.</abbrev-journal-title>
  </journal-title-group><issn pub-type="epub">1607-7938</issn><publisher>
    <publisher-name>Copernicus Publications</publisher-name>
    <publisher-loc>Göttingen, Germany</publisher-loc>
  </publisher></journal-meta>
    <article-meta>
      <article-id pub-id-type="doi">10.5194/hess-22-6209-2018</article-id><title-group><article-title>An assessment of trends and potential future changes in groundwater-baseflow
drought based on catchment response times</article-title><alt-title>Potential future changes in groundwater-baseflow drought</alt-title>
      </title-group><?xmltex \runningtitle{Potential future changes in groundwater-baseflow drought}?><?xmltex \runningauthor{J. Hellwig and K. Stahl}?>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes" rid="aff1">
          <name><surname>Hellwig</surname><given-names>Jost</given-names></name>
          <email>jost.hellwig@hydrology.uni-freiburg.de</email>
        <ext-link>https://orcid.org/0000-0001-7331-7656</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff1">
          <name><surname>Stahl</surname><given-names>Kerstin</given-names></name>
          
        <ext-link>https://orcid.org/0000-0002-2159-9441</ext-link></contrib>
        <aff id="aff1"><institution>Faculty of Environment and Natural Resources, University of Freiburg,
79098 Freiburg, Germany</institution>
        </aff>
      </contrib-group>
      <author-notes><corresp id="corr1">Jost Hellwig (jost.hellwig@hydrology.uni-freiburg.de)</corresp></author-notes><pub-date><day>3</day><month>December</month><year>2018</year></pub-date>
      
      <volume>22</volume>
      <issue>12</issue>
      <fpage>6209</fpage><lpage>6224</lpage>
      <history>
        <date date-type="received"><day>4</day><month>May</month><year>2018</year></date>
           <date date-type="rev-request"><day>14</day><month>May</month><year>2018</year></date>
           <date date-type="rev-recd"><day>2</day><month>October</month><year>2018</year></date>
           <date date-type="accepted"><day>31</day><month>October</month><year>2018</year></date>
      </history>
      <permissions>
        
        
      <license license-type="open-access"><license-p>This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link></license-p></license></permissions><self-uri xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018.html">This article is available from https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018.html</self-uri><self-uri xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018.pdf">The full text article is available as a PDF file from https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018.pdf</self-uri>
      <abstract>
    <p id="d1e86">Drought is an important natural hazard with large impacts
on society. Changes in drought characteristics have been studied for
different parts of the hydrological cycle, but insights into changes of
groundwater resources are obscured due to the lack of long-term observations
and large heterogeneity of hydrogeological conditions. Moreover, predicted
future changes in precipitation are uncertain and have a lagged effect on
streamflow and groundwater. We investigated past changes and potential
future changes in catchment baseflow as a reflection of groundwater drought
for 338 headwater catchments across Germany based on catchments'
characteristic response times. First, baseflow dynamics as a proxy of
groundwater storage and outflow on a catchment scale were derived from
streamflow records and related to precipitation input. Second, past trends
in baseflow minima were calculated and attributed to climate and catchment
controls. Last, response times and the timing of yearly baseflow minima were
combined into estimates of the sensitivity to future precipitation changes.
Baseflow response times of the studied headwaters are heterogenous across
Germany, ranging from a few months to several years, and depend significantly
on hydrogeological conditions. Few significant trends were found in past
baseflow minima, and trends are highly dependent on the period of analysis.
Based on the assumption of a typical regional scenario of increasing winter
precipitation and decreasing summer precipitation, increases in hydrological
drought hazard or no changes are projected for most parts of Germany.
Catchments with longer response times can buffer interannual precipitation
shifts, whereas catchments with fractured rocks are sensitive to summer
precipitation decreases. These results urge for a surface water and groundwater
management based on local groundwater response to precipitation and help to
assess impacts of climate change on overall water supply.</p>
  </abstract>
    </article-meta>
  </front>
<body>
      

<sec id="Ch1.S1" sec-type="intro">
  <title>Introduction</title>
      <p id="d1e96">Drought is a natural phenomenon occurring in all compartments of the
hydrological cycle. Accordingly, it is classified into meteorological
drought, hydrological drought, agricultural drought, socio-economic drought
and groundwater drought (Mishra and Singh, 2010). Due to the large number of
people affected by drought and the high economic loss related to drought
events (EC, 2007), it is important to enhance the understanding of drought
processes considering projected changes in drought hazard, most importantly
due to climate change. Empirical analysis of monitored hydrological time
series remains an important tool with which to validate theory-based or model-derived
hypotheses on these changes, since projected future changes are often
uncertain. Particularly for the groundwater compartment there is a high
diversity in response to climate input (Eltahir and Yeh, 1999; Green et al.,
2011), making predictions even more difficult. Most of the empirical studies
on long-term trends in drought hazard have focused on meteorological
drought (e.g. Sheffield et al., 2012; Spinoni et al., 2017) and on
hydrological drought (e.g. Stahl et al., 2010; Laaha et al., 2017). For
groundwater drought, empirical trend analysis is difficult for two reasons:
(i) groundwater time series are usually short or influenced by abstractions,
and (ii) where long and natural time series are available, they only give
point information.</p>
      <?pagebreak page6210?><p id="d1e99">Some countries and states now display groundwater level anomalies at observation wells
as part of their drought monitoring (e.g. Switzerland:
<ext-link xlink:href="https://www.hydrodaten.admin.ch/de/messstationen_grundwasserzustand.html">https://www.hydrodaten.admin.ch/de</ext-link>, last access: 28 November 2018;
Bavaria: <uri>https://www.nid.bayern.de/grundwasser</uri>, last access: 28 November 2018).
Studies on groundwater drought have mainly used borehole time series for index
generation and statistical analysis (e.g. Bloomfield and Marchant, 2013;
Bloomfield et al., 2015; Kumar et al., 2016; Van Loon et al., 2017;
Heudorfer and Stahl, 2017; Leelaruban et al., 2017). The groundwater response
to drought and its changes over time are typically very site specific, and
observations are therefore hardly scalable from point to catchment scale
(Kumar et al., 2016). However, larger regional-scale estimates of groundwater
resources are relevant for effective groundwater management and adaptation to climate change. Model-based
approaches for groundwater drought analysis and estimation (e.g. Peters et
al., 2003, 2006; Tallaksen et al., 2009; Li and Rodell, 2015; Apurv et al.,
2017) are one common way to get information on the catchment scale. There
are also a few studies that analyse spring data (Fiorillo and Guadagno,
2012) or baseflow time series (Brutsaert, 2008, 2010; Fendeková and
Fendek, 2012), two flux variables that provide a more integrative measure of
the groundwater situation during drought.</p>
      <p id="d1e108">To overcome the difficulties related to borehole data, in this study we use a
baseflow approach to characterize groundwater drought on a catchment scale.
We analyse a large dataset of long baseflow time series in central Europe.
In this region groundwater is often used as drinking water, and aquifers act
as an important buffer to climatic variability. Most droughts start with a
deficit in precipitation, especially when precipitation falls as rain (Van
Loon and Van Lanen, 2012). For the propagation from a meteorological to a
groundwater drought different processes are relevant, i.e. attenuation,
delay and pooling (e.g. Peters et al., 2003; Tallaksen et al., 2009;
Heudorfer and Stahl, 2017). Therefore, the drought signal in groundwater
storage not only depends on current meteorological conditions, but also the
previous months are important. A catchment-specific timescale for this
dependence may be called the catchment's response time.
Response times have been analysed by correlations between groundwater depth
and time series of precipitation accumulated for different periods. Studies
have found that the response times for borehole water tables (Bloomfield et al.,
2015; Van Loon et al., 2017; Leelaruban et al., 2017) or spring
discharge (Fiorillo and Guadagno, 2012) vary strongly. Moreover, some
studies suggest time lags for the highest correlations between precipitation
and groundwater time series because of delayed groundwater response
(Bloomfield et al., 2015; Fiorillo and Guadagno, 2012). However, when
looking at monthly scales, this lag was always found to be quite small and
often non-existent (e.g. Haas and Birk, 2017).</p>
      <p id="d1e111">There are two approaches to identifying drought periods in a time series. The
climatological approach is based on anomalies and often used also in
hydrology to track the propagation of relative seasonal water deficits
through the hydrological cycle (e.g. Barker et al., 2016; Kumar et al.,
2016). The traditional hydrological approach is the “threshold level
approach”, which defines streamflow droughts as events below a certain
fixed limit and is therefore focused on actual low water availability (e.g.
Yevjevich, 1967; Peters et al., 2006; Tallaksen et al., 2009). In this work
we use the term drought according to the threshold level approach; thus we
consider drought events as periods of low baseflow in absolute terms. If
there is a distinct seasonal regime, droughts mostly occur in the dry
season.</p>
      <p id="d1e115">Recent work on central European low flows, i.e. periods when streamflow
mostly consists of baseflow, found that climate change is expected to alter
low flows (Marx et al., 2018; Forzieri et al., 2014; Van Vliet et al., 2015;
Gosling et al., 2017). However, the sign and magnitude of change in central
Europe are subject to model choice and emission scenario (Marx et al., 2018;
Forzieri et al., 2014). Those modelling studies were focused on large river
basins, and the change they predicted reflects strongly that of the
precipitation change. Marx et al. (2018) found a high correlation between
changes in annual precipitation sums and low flows. Stahl et al. (2012)
found that hindcasting summer low flow trends with large-scale models
suggests a too-homogenous spatial pattern of change compared to trends found
in observations. Together with difficulties of models to capture the
persistence of drought events in runoff generation found by Tallaksen and
Stahl (2014), it can be assumed that some large-scale models do not
necessarily resolve the heterogeneity of catchment storage and release for
the hydrological response on a headwater catchment scale. However, recent
drought events have demonstrated that especially headwaters are prone to
groundwater-related drought impacts like shortages in water supply (Van
Lanen et al., 2016). This coincides with findings that, independent from
elevation, groundwater is an important catchment storage (Staudinger et al.,
2017). Thus, predicting future changes in groundwater drought on a catchment
scale will be a prerequisite for effective drought management.</p>
      <p id="d1e118">Depending on the projected climate change, different scenarios of the future
development of natural baseflow can be expected (Fig. 1). If there is an
increase (decrease) in precipitation projected for the entire year, flow
during the dry season is also expected to increase (decrease). However, if a
seasonal shift of precipitation is expected, the future development of flow
during the dry season is not that straightforward. It will depend on the
timing of seasonal shift and dry period and on the catchment's characteristic response time to precipitation. Stölzle et al. (2014)
found that for baseflow drought changes in precipitation are especially
relevant during the recharge period, which is dependent on the hydrogeology
of the catchment.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F1"><caption><p id="d1e123">Schematic direction of change of natural baseflow in the dry
season under different climate change scenarios: <bold>(a)</bold> all seasons
becoming wetter, <bold>(b)</bold> all seasons becoming drier and <bold>(c)</bold> seasonal shift of precipitation.</p></caption>
        <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f01.png"/>

      </fig>

      <p id="d1e141">For many parts of central Europe climate projections indicate a seasonal
shift of precipitation to wetter winters and drier summers rather than a
consistent increase/decrease<?pagebreak page6211?> (Jacob et al., 2014), urging for statistical
tools to assess the prospective changes in baseflow. Knowledge of the
seasonal to multi-annual scale of the baseflow response to climatic
variation and extremes is therefore particularly important in central Europe
under this seasonally diverging expected climate change.</p>
      <p id="d1e144">This study aims to explain differences in drought trends by catchment
characteristics to allow for more accurate predictions under climate
projection uncertainty on a headwater scale. Firstly, past trends in baseflow
drought and catchment-relevant response times are analysed. Secondly, past
trends are attributed to climatic and catchment controls. Finally, based on
these statistics an estimate for future changes in baseflow drought valid
for all common emission scenarios and climate models is realized.</p>
</sec>
<sec id="Ch1.S2">
  <title>Study area and data</title>
      <p id="d1e153">The dataset used in this study is the same set of headwater catchments (that
is, all catchment areas are below 200 km<inline-formula><mml:math id="M1" display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula>) that were used in
Hellwig et al. (2018) to evaluate the representativeness of meteorological
grid data. It comprises streamflow data of daily resolution available from
the responsible environment agencies of the German federal states. The
lengths of available time series differ, so there is a trade-off between
selected period and the number of records with sufficient data. For this
study, we only used catchments with streamflow data of the period 1970 to
2009. Records with data gaps were not considered. Even though the dataset
consists of near-natural headwaters with minimal regulations, the
hydrographs and their double mass curves were visually screened for signs of
anthropogenic influence. In a natural catchment under relatively constant
conditions, the precipitation–streamflow relationship should be similar for
the entire observation period. Four of the gauges showed sudden changes in
the relationship and were subsequently removed. The final dataset consisted
of 338 unnested and independent gauges across Germany (Fig. 2).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F2"><caption><p id="d1e167">Location of gauges in Germany, catchments' dominant type of porosity
(derived from the German hydrogeological map) and mean annual precipitation
sums (calculated from European Climate Assessment and Dataset E-OBS).</p></caption>
        <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f02.png"/>

      </fig>

      <p id="d1e176">The selected catchments cover the flat lowland regions in the north of
Germany, the low mountain ranges in south-central Germany, and the
Alpine foothills and non-glacierized front range in the south. Precipitation
varies with highest annual precipitation sums in the Alpine south
(&gt; 2000 mm) and lowest sums in the northeast (&lt; 500 mm).
Climate in Germany is humid with slightly higher precipitation sums in
summer than in winter for most regions. Precipitation was analysed in the
form of monthly precipitation sums taken from the European Climate
Assessment and Dataset (Haylock et al., 2008), Version 13.1. According to
the procedure described in Hellwig et al. (2018), catchment-specific
precipitation was calculated as an area-weighted mean of the intersecting
grid cells. Hellwig et al. (2018) found that due to the low spatial
resolution of the meteorological dataset compared to catchment size there
are some biases towards products of higher resolution; however, correlations
between products were found to be very high.</p>
      <?pagebreak page6212?><p id="d1e179">Information on the hydrogeology of the catchments was taken from the digital
German hydrogeological map (Hydrogeologische Übersichtskarte von Deutschland <inline-formula><mml:math id="M2" display="inline"><mml:mrow><mml:mn mathvariant="normal">1</mml:mn><mml:mo>:</mml:mo><mml:mn mathvariant="normal">200.000</mml:mn></mml:mrow></mml:math></inline-formula>, Oberer Grundwasserleiter; BGR and SGD, 2016). The catchments were
classified according to the main type of porosity found in the underlying
geology, either “porous” for porous aquifers such as unconsolidated
alluvial and glacial fillings or “fractured” for fractured bedrock. Where
less than two-thirds of the catchment's area is covered by one of these types, the
class “mixed” was assigned, including catchments which are dominated by
other types of porosity like karst. Additional analysis revealed that other
reasonable thresholds for classifying the catchments do not change the results.
According to this classification, 80 headwater catchments (23.7 %) are
mainly “porous”, 170 catchments (50.3 %) are mainly “fractured” and 88 catchments (26.0 %) have “mixed” porosity (Fig. 2).</p>
      <p id="d1e195">The dominant land use was derived from CORINE Land Cover 2006 data
(available online from the German Environment Agency (<uri>https://www.umweltbundesamt.de</uri>,
last access: 28 November 2018). Each catchment is classified as agricultural
(53.6 %), urban (1.2 %) or forested (45.2 %).</p>
      <p id="d1e201">To differentiate the catchments regarding topography, the average height
above the catchment's outlet was calculated. This metric describes the
potential groundwater waterbody that can contribute to the flow at the
gauge and thus characterizes the potential storage. Elevation data (1 arcsec digital elevation model over Europe) were obtained from the European
Environment Agency (2013).</p>
</sec>
<sec id="Ch1.S3">
  <title>Methods</title>
<sec id="Ch1.S3.SS1">
  <title>Characterizing baseflow drought</title>
<sec id="Ch1.S3.SS1.SSS1">
  <title>Baseflow as a measure of groundwater storage at the catchment
scale</title>
      <p id="d1e220">Baseflow <inline-formula><mml:math id="M3" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is defined as the delayed component of streamflow <inline-formula><mml:math id="M4" display="inline"><mml:mi>Q</mml:mi></mml:math></inline-formula>, i.e.
flow originating from stored sources as opposed to flow originating from
event water (WMO, 2008). In groundwater-dominated catchments, where
groundwater is the main water storage, <inline-formula><mml:math id="M5" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is directly related to
groundwater outflow (see Appendix A for a detailed rationale). Therefore,
<inline-formula><mml:math id="M6" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> can be taken as an integrated measure of groundwater storage <inline-formula><mml:math id="M7" display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> as it
was done in the study of Fendeková and Fendek (2012) and others. It is
important to be aware that the unknown functional relationship between
<inline-formula><mml:math id="M8" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M9" display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> strongly differs between catchments and might be non-linear,
e.g. due to temporal changes in connectivity of groundwater and surface
water (Elthair and Yeh, 1999; Brunner et al., 2011). Hence, without further
knowledge about this functional relationship, the observation of <inline-formula><mml:math id="M10" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>
allows for conclusions about <inline-formula><mml:math id="M11" display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> solely on an ordinal scale.</p>
      <p id="d1e307">Despite the clear concept of baseflow, there is no universally valid way to
separate <inline-formula><mml:math id="M12" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> from time series of <inline-formula><mml:math id="M13" display="inline"><mml:mi>Q</mml:mi></mml:math></inline-formula>. Instead, a number of methods exist
(e.g. WMO, 2008; Lyne and Hollick, 1979; Nathan and McMahon, 1990; Eckhardt,
2005; Mei and Anagnostou, 2015) that differ especially in baseflow
estimation during high flow, when <inline-formula><mml:math id="M14" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is relatively uncertain. For the
type of extreme low-flow conditions considered in this study, most methods
consistently assume that <inline-formula><mml:math id="M15" display="inline"><mml:mi>Q</mml:mi></mml:math></inline-formula> is almost completely comprised of <inline-formula><mml:math id="M16" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. The
World Meteorological Organization recommends a baseflow separation method
(WMO, 2008) that was applied in this work using the package “lfstat”
(Koffler et al., 2016) in the programming language R (R Core Team, 2016). The
separation procedure consists of five steps: (i) divide the time series into
non-overlapping blocks of a certain length <inline-formula><mml:math id="M17" display="inline"><mml:mi>N</mml:mi></mml:math></inline-formula>, where <inline-formula><mml:math id="M18" display="inline"><mml:mi>N</mml:mi></mml:math></inline-formula> is recommended to be
5 days for rainfall regimes which have a fast streamflow response; (ii) select the minimum value for every block; (iii) compare each minimum value
with the adjacent ones (if the central value is <inline-formula><mml:math id="M19" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula> times smaller than the
adjacent values, then it becomes a turning point; <inline-formula><mml:math id="M20" display="inline"><mml:mi>a</mml:mi></mml:math></inline-formula> is recommended to be 0.9 for rainfall
regimes which have fast streamflow response); (iv) join the turning points by
straight lines; and (v) compute the baseflow for every day by linearly
interpolating between the turning points (if the computed value is higher
than observed flow, use the observation instead).</p>
</sec>
<sec id="Ch1.S3.SS1.SSS2">
  <title>Catchment-specific response times to precipitation</title>
      <p id="d1e392">To quantify the catchments' baseflow response times to
precipitation (<inline-formula><mml:math id="M21" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>), time series of very different characteristics need
to be related to each other. To enhance comparability over space, time and
different parts of the hydrological cycle, it is common to standardize time
series. Recent studies on drought have used either parametric (e.g. Fiorillo and
Guadagno, 2012; Barker et al., 2016) or non-parametric approaches (e.g.
Bloomfield and Marchant, 2013; Kumar et al., 2016) for standardization.
Parametric approaches like the Standardized Precipitation Index (SPI),
introduced by McKee et al. (1993), rely on the fitting of a theoretical
distribution. Since time series of precipitation (<inline-formula><mml:math id="M22" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula>) and <inline-formula><mml:math id="M23" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> have distinct
characteristics, entirely different distributions would have to be selected.
To ensure consistency in the way of standardization, in this study a
non-parametric approach was applied (Bloomfield and Marchant, 2013; Kumar et
al., 2016). All time series (<inline-formula><mml:math id="M24" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> in accumulation periods from 1 to 36 months,
<inline-formula><mml:math id="M25" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>) were standardized according to the procedure of the Standardized
Groundwater Index (SGI; Bloomfield and Marchant, 2013).</p>
      <p id="d1e442"><inline-formula><mml:math id="M26" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> (or <inline-formula><mml:math id="M27" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>) was transformed by using the inverse normal
cumulative distribution function to calculate <inline-formula><mml:math id="M28" display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula> equally spaced <inline-formula><mml:math id="M29" display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> values
between <inline-formula><mml:math id="M30" display="inline"><mml:mrow><mml:mn mathvariant="normal">1</mml:mn><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M31" display="inline"><mml:mrow><mml:mn mathvariant="normal">1</mml:mn><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="M32" display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula> is the number of observations. These
values were afterwards assigned to the time series by reordering them
according to the ranks of the original time series. Hence, the ranks of the
original time series and the new ones coincide, but the newly generated time
series are normally distributed. Standardization<?pagebreak page6213?> was done separately for all
12 months of the year to account for the regime.</p>
      <p id="d1e528">For all 338 catchments, correlation coefficients <inline-formula><mml:math id="M33" display="inline"><mml:mi>R</mml:mi></mml:math></inline-formula> between <inline-formula><mml:math id="M34" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M35" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> were
calculated:
              <disp-formula id="Ch1.E1" content-type="numbered"><mml:math id="M36" display="block"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">cov</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:mi mathvariant="normal">cov</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
            where <inline-formula><mml:math id="M37" display="inline"><mml:mi>x</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M38" display="inline"><mml:mi>y</mml:mi></mml:math></inline-formula> are <inline-formula><mml:math id="M39" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M40" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, respectively, with cov() the covariance and <inline-formula><mml:math id="M41" display="inline"><mml:mi mathvariant="italic">σ</mml:mi></mml:math></inline-formula> the
standard deviation of each time series, which equals one for standardized
time series. To derive the catchments' response times to precipitation,
precipitation accumulation periods from 1 to 36 months were tested as well
as time lags of these indices to baseflow from 0 to 10 months. For each
catchment, the combination of precipitation accumulation period and time
lag that has the highest correlation coefficient was selected as the
optimal accumulation period and the optimal time lag to describe the
<inline-formula><mml:math id="M42" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> response. Longer optimal accumulation periods indicate higher
groundwater attenuation of precipitation input, whereas higher time lags
represent a more delayed groundwater response to precipitation. To account
for seasonal variations in the response times, correlations were calculated
separately for all seasons (winter: DJF; spring: MAM; summer: JJA;
autumn: SON).</p>
</sec>
<sec id="Ch1.S3.SS1.SSS3">
  <title>Catchment-relevant recharge period</title>
      <p id="d1e663">Catchments analysed in this study follow a distinct regime in <inline-formula><mml:math id="M43" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. Thus,
droughts in <inline-formula><mml:math id="M44" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> are most likely to occur in a certain period of the year.
To derive the catchment-characteristic drought season, annual minimal
7-day <inline-formula><mml:math id="M45" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (<inline-formula><mml:math id="M46" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>) were calculated. The most frequent month of
<inline-formula><mml:math id="M47" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> was assigned as the month of highest baseflow drought hazard
(<inline-formula><mml:math id="M48" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>). Subsequently, the derived seasonal <inline-formula><mml:math id="M49" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> was used to determine
the months influencing <inline-formula><mml:math id="M50" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> in <inline-formula><mml:math id="M51" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. For example, if <inline-formula><mml:math id="M52" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> in a
catchment was October and <inline-formula><mml:math id="M53" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> for that catchment in autumn was 6
months, then the catchment-relevant recharge period would be May through
October.</p>
</sec>
</sec>
<sec id="Ch1.S3.SS2">
  <title>Detection of past baseflow drought trends</title>
      <p id="d1e822">Time series of <inline-formula><mml:math id="M54" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> for the time period of 1970–2009 were used to analyse
past trends in minimal baseflow. The analysis was carried out with the
non-parametric Mann–Kendall (MK) test, which is a common tool used to detect
monotonic increases or decreases in hydrological time series (e.g. used by
Douglas et al., 2000; Lins and Slack, 1999; Lorenzo-Lacruz et al., 2012;
Rennermalm et al., 2010; Asarian et al., 2016) and is robust to outliers. The
test was performed after pre-whitening the time series to account for the
influence of serial correlation (see Appendix B for calculation steps). A
significance level of <inline-formula><mml:math id="M55" display="inline"><mml:mrow><mml:mi mathvariant="italic">α</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.05</mml:mn></mml:mrow></mml:math></inline-formula> was used.</p>
      <p id="d1e851">Trends only give information on the period they are calculated for. Many
studies found that trends of a certain period are not part of a trend on
another timescale (e.g. Stahl et al., 2010; Hannaford and Buys, 2012;
Giuntoli et al., 2013; Hannaford et al., 2013). The analysis of trends for
multiple periods may help to assess whether observed trends are steady or
rather fluctuating. To evaluate the trends found for the period 1970–2009,
we additionally calculated trends over multiple periods for the five gauges
with longest continuous records. Three of these are in the porosity class
“fractured”, and one each in “porous” and “mixed”.</p>
</sec>
<sec id="Ch1.S3.SS3">
  <title>Attribution of baseflow drought trends</title>
      <p id="d1e860">Trends in <inline-formula><mml:math id="M56" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> were assumed to be driven by both climatic (climate
control) and catchment-specific (catchment control) factors. Thus, the
results were related to a set of potential predictors: (i) trends in <inline-formula><mml:math id="M57" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> for
the catchment-specific relevant recharge period, (ii) <inline-formula><mml:math id="M58" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> for the
catchment-specific <inline-formula><mml:math id="M59" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, (iii) <inline-formula><mml:math id="M60" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, (iv) dominant porosity of the
catchment, (v) catchment size (<inline-formula><mml:math id="M61" display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula>), (vi) dominant land use and (vii) potential
storage. Statistical significance of the factors was determined using the
correlation coefficient <inline-formula><mml:math id="M62" display="inline"><mml:mi>R</mml:mi></mml:math></inline-formula> (Eq. 1) for the continuous variables (trends in
<inline-formula><mml:math id="M63" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M64" display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula>, potential storage), Spearman's rank-based correlation coefficient for
the discrete variable (<inline-formula><mml:math id="M65" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>) and an analysis of variance (ANOVA) combined
with the post hoc Tukey test (for details see Appendix C) for the
nominally scaled ones (<inline-formula><mml:math id="M66" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, dominant porosity, dominant land use).</p>
</sec>
<sec id="Ch1.S3.SS4">
  <title>Estimate of drought sensitivity to expected future precipitation
change</title>
      <p id="d1e995">Since past trends derived from empirical trend analysis (e.g. MK) are solely
valid for the observation period, they cannot be extrapolated beyond the
period of data availability. Future drought predictions therefore mostly
rely on climate projections and process modelling. For Germany climate
projections indicate little to no changes of annual precipitation sums but
seasonal shifts to lower summer precipitation and higher winter
precipitation. However, the magnitude of the shifts differs considerably for
different projections (Zebisch et al., 2005; Jacob et al., 2012, 2014;
Hübener et al., 2017). A common approach to dealing with such uncertainties
is to use a range of possible trajectories to model hydrological change.
Instead of using uncertain quantitative inputs in forward modelling, here we
propose a more qualitative inverse approach. We assume that the general
direction of future development is the most important piece of information for future
groundwater management planning and formulate a qualitative test scenario of
the consistent direction of different projections of future precipitation
change. Thus, the approach is scenario-neutral regarding emission scenarios
and climate models.</p>
      <p id="d1e998">Trends in future baseflow drought hazard were assumed to depend not only on
precipitation changes but also on <inline-formula><mml:math id="M67" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M68" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> calculated for every
catchment. A particular test scenario can still have three possible outcomes
regarding future changes in baseflow drought hazard (Fig. 3): (i)<?pagebreak page6214?> increased
baseflow – there are relevant months with increasing precipitation but not
with decreasing precipitation; (ii) decreased baseflow – there are relevant months with
decreasing precipitation but not with increasing precipitation; or (iii) no change – either
there are relevant months with both increasing and decreasing precipitation
or there are no relevant months with any change.</p>
      <p id="d1e1030">The test scenario applied in this study is a decrease of precipitation in
summer (JJA) and an increase of precipitation in winter (DJF) with no change
in the annual precipitation sum. To derive the future change in <inline-formula><mml:math id="M69" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> for
a catchment, we analyse whether months with increasing or decreasing
precipitation belong to the catchment-relevant recharge period. For example,
for a catchment with a relevant recharge period from May to October we would
estimate an increased baseflow drought hazard, since there are months with
decreasing precipitation in the relevant recharge period (June, July and
August) but no months with increasing precipitation. This analysis was
carried out for all 338 catchments.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F3"><caption><p id="d1e1049">Potential directional changes in <inline-formula><mml:math id="M70" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> under a constant
seasonally diverging precipitation change scenario, depending on <inline-formula><mml:math id="M71" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and
<inline-formula><mml:math id="M72" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>: <bold>(a)</bold> increase, <bold>(b)</bold> decrease and <bold>(c)</bold> no change.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f03.png"/>

        </fig>

</sec>
</sec>
<sec id="Ch1.S4">
  <title>Results</title>
<sec id="Ch1.S4.SS1">
  <title>Past minimal baseflow trends</title>
      <p id="d1e1123"><inline-formula><mml:math id="M73" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> occurs in most parts of Germany in late summer or autumn with some
exceptional winter low-flow catchments in the mountainous regions (Fig. 4). Seasonal response times vary across Germany, ranging from short
subseasonal response times (1–3 months) to response times of over a year
(Fig. 4b). Altogether 22 out of 338 catchments (<inline-formula><mml:math id="M74" display="inline"><mml:mrow><mml:mo>∼</mml:mo><mml:mn mathvariant="normal">6.5</mml:mn></mml:mrow></mml:math></inline-formula> %)
have a time lag in the response, though these are mainly short. All five
catchments with a time lag above 1 month have response times longer than
12 months. According to the MK test, 40 out of the 338 catchments show a
significant trend of <inline-formula><mml:math id="M75" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> (Fig. 4c), corresponding to 11.8 %.
Assuming independence of the baseflow observations, this is slightly more
than the expected 5 % in case of no real trends. Hence, for the period of
1970–2009 there are some changes detectable for all
catchments' porosity classes. For <inline-formula><mml:math id="M76" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> there are even fewer
significant trends than would be expected by chance (Fig. 4d). However,
the results are skewed towards predominantly positive trends in <inline-formula><mml:math id="M77" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> during the
relevant recharge periods across Germany.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F4"><caption><p id="d1e1183">Patterns of <inline-formula><mml:math id="M78" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> across Germany. <bold>(a)</bold> Low-flow month <inline-formula><mml:math id="M79" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and
<bold>(b)</bold> seasonal response time <inline-formula><mml:math id="M80" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. <bold>(c)</bold> Trend of <inline-formula><mml:math id="M81" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> and <bold>(d)</bold> <inline-formula><mml:math id="M82" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> in the
relevant recharge period for 1970–2009.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f04.jpg"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F5" specific-use="star"><caption><p id="d1e1271"><inline-formula><mml:math id="M83" display="inline"><mml:mi>z</mml:mi></mml:math></inline-formula>-score values for all possible periods for the five catchments with
longest continuous records.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f05.png"/>

        </fig>

      <p id="d1e1287">The five selected gauges for a trend analysis on multiple periods all have a
negative <inline-formula><mml:math id="M84" display="inline"><mml:mi>z</mml:mi></mml:math></inline-formula> score in the original period: 1970–2009 (Fig. 5). However, in
all cases a significant positive trend could also have been observed if
another period had been chosen for analysis (Fig. 5). This reveals a high
influence of the observation period on the results of the trend analysis.
The high variability of trends is also independent of the porosity,
<inline-formula><mml:math id="M85" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M86" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>.</p><?xmltex \hack{\newpage}?>
</sec>
<?pagebreak page6215?><sec id="Ch1.S4.SS2">
  <title>Trend attribution</title>
      <p id="d1e1333">None of the factors tested was found to explain past trends of <inline-formula><mml:math id="M87" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> very
well. This lack of statistical relation coincides with the results that few
trends are significant and trend magnitude is highly dependent on the period
of analysis. However, there is a small but significant correlation between
the trends in <inline-formula><mml:math id="M88" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> for the catchment-relevant recharge period and trends in
<inline-formula><mml:math id="M89" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> (Fig. 6). Moreover, the ANOVA indicates a significant influence of
<inline-formula><mml:math id="M90" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> on the trends in <inline-formula><mml:math id="M91" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> even though Tukey's test does not
distinguish two groups significantly. The largest differences are between
July and November, so the season of low flow might be relevant for trend
magnitude in <inline-formula><mml:math id="M92" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> as well. The remaining factors (<inline-formula><mml:math id="M93" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, dominant
porosity, <inline-formula><mml:math id="M94" display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula>, dominant land use and potential storage) do not show any
relation to the trends in <inline-formula><mml:math id="M95" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> (Fig. 6).</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F6" specific-use="star"><caption><p id="d1e1452">Attribution of trends in <inline-formula><mml:math id="M96" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> to potential influencing factors.
<bold>(a)</bold> Trends in precipitation <inline-formula><mml:math id="M97" display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> during the relevant recharge period, <bold>(b)</bold> length
of response time <inline-formula><mml:math id="M98" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <bold>(c)</bold> <inline-formula><mml:math id="M99" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, <bold>(d)</bold> dominant porosity in the catchment
<bold>(e)</bold> catchment size <inline-formula><mml:math id="M100" display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula>, <bold>(f)</bold> dominant land use in the catchment and <bold>(g)</bold> potential
storage. Boxes represent the quartiles 0.25 and 0.75, and whiskers last to
the most extreme value within a maximum distance of a 1.5-fold interquartile
range.</p></caption>
          <?xmltex \igopts{width=312.980315pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f06.png"/>

        </fig>

</sec>
<sec id="Ch1.S4.SS3">
  <title>Potential future drought hazard</title>
      <p id="d1e1547">According to the test scenario, <inline-formula><mml:math id="M101" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> will decrease or not change for most
of Germany (Fig. 7). The only catchments with estimated increases of
<inline-formula><mml:math id="M102" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> are located in the mountainous regions of southeast Germany,
especially in the Alpine foothills in the catchments with annual low flows
in winter. However, in the catchments with annual low flows in winter
or spring, impacts might also be more severe for low flows in summer or autumn.
Additional tests revealed that, for 22 (27) out of 35 catchments with
<inline-formula><mml:math id="M103" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> in winter or spring, baseflows of the summer (autumn) season are
expected to decrease according to the test scenario.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F7"><caption><p id="d1e1598">Future changes in <inline-formula><mml:math id="M104" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> according to the test scenario.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f07.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F8"><caption><p id="d1e1623">Relationships of projected changes, <inline-formula><mml:math id="M105" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and catchment's dominant
porosity. <bold>(a)</bold> Distribution of <inline-formula><mml:math id="M106" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> for projected changes, <bold>(b)</bold> frequency of
catchments with a certain dominant porosity for different projected changes
and <bold>(c)</bold> distribution of <inline-formula><mml:math id="M107" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> for dominant porosities. Boxes represent the
quartiles 0.25 and 0.75, and whiskers last to the most extreme value within a
maximum distance of a 1.5-fold interquartile range. Red letters indicate
results of the post hoc Tukey test to distinguish
significantly different groups.</p></caption>
          <?xmltex \igopts{width=199.169291pt}?><graphic xlink:href="https://hess.copernicus.org/articles/22/6209/2018/hess-22-6209-2018-f08.png"/>

        </fig>

      <?pagebreak page6216?><p id="d1e1676">The changes in drought hazard are significantly related to the catchments'
response times <inline-formula><mml:math id="M108" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (Fig. 8). Moreover, there is a strong statistical
linkage to the dominant porosity (Fig. 8b). Catchments dominated by
fractured rocks more frequently show estimated decreases in <inline-formula><mml:math id="M109" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> than the
other classes. This coincides with a significant relationship between
<inline-formula><mml:math id="M110" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and the dominant porosity of the catchment (Fig. 8c). According to
Turkey's test, catchments with fractured rocks have shorter response times
than the other classes. Contrarily, catchments with porous aquifers or mixed
hydrogeology, which have long <inline-formula><mml:math id="M111" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, can more often compensate for summer
precipitation decreases with winter increases and are therefore more
frequently in the category “no change”.</p>
</sec>
</sec>
<sec id="Ch1.S5">
  <title>Discussion</title>
<sec id="Ch1.S5.SS1">
  <title>The catchment baseflow response to precipitation</title>
      <p id="d1e1739">Catchments' response times to precipitation were found to be
highly diverse across Germany, ranging from 1 month to 3 years. In
general, baseflow response times determined as a proxy of groundwater
response are rather short compared to other studies. Fiorillo and Guadagno (2012) found response times of 12 to 24 months for a karst region in
southern Italy and highest correlations for shorter precipitation accumulation periods when adding a short time lag. Bloomfield and Marchant (2013)
also found in 3 out of 14 cases a time lag for highest correlations. We
found time lags to be an exception, supporting the results of Kumar et al. (2016),
Barker et al. (2016) and Haas and Birk (2017). This indicates that,
in the headwater catchments studied, the delay of the groundwater baseflow
response to meteorological conditions<?pagebreak page6217?> may be shorter than 1 month and
therefore not detectable on the monthly scale, whereas the attenuation of
meteorological variability is clearly attributable to characteristic
precipitation accumulation periods.</p>
      <p id="d1e1742">The large differences of baseflow response times for different porosity
classes match the theoretical assumptions that baseflow strongly depends on
hydrogeological conditions. For the entire streamflow, differences were
found to be much smaller (not shown; compare e.g. Haslinger et al., 2014),
since other processes like overland flow are also important. The patterns of
<inline-formula><mml:math id="M112" display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi mathvariant="normal">R</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> which are related to the catchments' hydrogeology support the
assumption that baseflow can be used as a proxy of the groundwater situation
on a catchment scale. More detailed information on hydrogeology has the
potential to further improve the understanding of differences in baseflow
response. However, this information is not yet detailed enough for the scale
and distribution of headwater catchments. For example, two large-scale
estimates on hydraulic conductivity available for Germany (GLHYMPS from
Gleeson et al., 2014, and HÜK200 from BGR and SGD, 2016) were
considered and found to differ up to 6 orders of magnitude. Moreover,
information on superficial deposits that might be particularly important for
streamflow generation could enhance results but is not available
in detail yet.</p>
      <p id="d1e1756">Consistent with the work by Bloomfield et al. (2015), we found that
hydrogeology is a highly relevant factor for the explanation of different
groundwater baseflow response times. Kumar et al. (2016) did not find a
relationship between the hydraulic conductivity and groundwater response
time for boreholes. A possible reason for this different finding is that for
point data even small local influences, which are hard to determine, are
quite relevant (e.g. human influences), whereas baseflow reflects more the
overall situation within the catchment. Small influences may be negligible
at this scale, and the underlying influence of hydrogeology may be easier to detect.</p>
      <p id="d1e1759">In general, the results indicate that groundwater storage – represented by
baseflow – is vitally driven by precipitation on a catchment scale in the
relevant recharge period. However, the season of low flow is also expected
to have an influence: regimes with winter low flows in central Europe are
governed by snow storage during that season. Thus, not only precipitation
but also temperature is a major factor for that catchments. Moreover, it is
impossible to distinguish snowmelt from groundwater outflow during baseflow
separation. Therefore, a baseflow approach does not allow for conclusions on
groundwater storage in snow-dominated catchments.</p>
</sec>
<?pagebreak page6218?><sec id="Ch1.S5.SS2">
  <title>Past and potential future changes in low baseflow</title>
      <p id="d1e1768">The trend analysis revealed that the trend in baseflow minima is highly
dependent on the period it is calculated for. The observation of a trend in
<inline-formula><mml:math id="M113" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> thus is a poor indicator of future developments. Despite the
relatively narrow seasonality of low-baseflow timing (except for the alpine
areas), trends are highly variable across the region – a result of the
likewise variable response times. However, the correlation tests for
attribution of the trends revealed that precipitation during the
catchment-relevant recharge period is an important factor, which confirms
previous model-based findings for part of the region (Stölzle et al.,
2014). Thus, this catchment-relevant recharge period should be considered
for projections.</p>
      <p id="d1e1785">The method employed in this study provides a straightforward projection of
the probable future directions of changes in baseflow or
groundwater drought under climate change. We accounted for the uncertainty
in future climate projections by taking only concordant directions of
precipitation change in our approach. Contrary to future climate
projections, past trends in precipitation for the catchment-relevant
recharge period were found to be small and mostly positive. Similarly, Kopp
et al. (2018) found for southern Germany a high variability of annual
groundwater recharge without distinct trends. In the past, precipitation
trends have not been seasonally diverging, but climate models suggest that
changes will become more relevant in the second half of this century (e.g.
Jacob et al., 2012). As the magnitude of the trends differs for the climate
models, we did not quantify our scenario and thus did not quantify the
magnitude of future baseflow either.</p>
      <p id="d1e1788">Catchments' characteristic response times were assumed to remain constant.
Under a more extreme climate change however, changes in
catchments' responses (e.g. due to non-stationary response
times) cannot be excluded. Based on assumed precipitation change in the
catchments' respective recharge periods, decreasing <inline-formula><mml:math id="M114" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> (i.e. increased
groundwater drought hazard) were estimated for the majority of the
catchments. This is because of the timing of <inline-formula><mml:math id="M115" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>, which occurs
predominantly in summer/autumn. Only those catchments with long response
times can compensate for the decreases in summer precipitation, and only in the
Alpine foothills, where <inline-formula><mml:math id="M116" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is in winter, is a decrease in drought hazard
predicted. However, in this region winter precipitation is mainly snow
and therefore does not immediately contribute to groundwater recharge.
Moreover, air temperature and its changes over time – which are not
considered in the test scenario – are especially important for winter
low flows. Hence, uncertainty is quite high for these projections. If
response times are not assumed constant, further conclusions may be drawn. A
decrease of response times might be caused by decreased storage, e.g. due to
urbanization, or accelerated drying and increased evapotranspiration in
spring. Such changes would increase the drought hazard for most parts of
Germany due to the timing of <inline-formula><mml:math id="M117" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>. An increase in response times by the
opposite processes is possibly less likely but could compensate for the reduced
precipitation during summer for the catchments that are currently estimated
to have increased drought hazard.</p>
</sec>
</sec>
<sec id="Ch1.S6" sec-type="conclusions">
  <title>Conclusions</title>
      <p id="d1e1866">Climate change is expected to alter the hydrological drought hazard.
However, uncertainty of climate projections and even contrasting seasonal
changes impede a straightforward assessment of the prospective changes in
central Europe and elsewhere. Here we presented a statistical approach to
estimating the potential direction of future changes in hydrological drought
hazard. Past trends were found to be too variable to provide a consistent regional
picture of past and expected changes because of their high dependency on the
trend calculation period. But they did allow the attribution of
trends in baseflow to precipitation changes in catchment-specific recharge
periods. Based on that information, a more process-oriented approach was
developed, using the catchments' characteristic response times
to precipitation and the relevant recharge periods for projections valid for
all common emission scenarios and climate models. These projections are
efficient alternatives to ensemble projections and target the most important
information for management. Especially for regions where directions of
climate change are seasonally varying, they can provide valuable insights
into the basic changes of the system.</p>
      <p id="d1e1869">Catchments with short response times were found to have a high probability
for a decrease in baseflow minima and hence an increase in the groundwater
drought hazard, as seasonal changes cannot compensate for each other. However,
there is no homogeneous pattern of response times across central Europe, and
so predicted changes of groundwater drought hazard also vary regionally.
This urges for a regionally adapted groundwater management based on
the local catchment response times. As past events like the 2015 central
European drought have already caused groundwater-related drought impacts in
headwater regions, there is an urgent need for adaptation in catchments
facing even higher drought hazard in the future.</p>
      <p id="d1e1872">The diversity of response times, the dearth of long-time data on groundwater
storage, and the absence of distinct past trends in precipitation and hydrological
variables limit the potential to generalize the results. On the path to
extensive predictions of future groundwater drought hazard across central
Europe, further model-based work will be needed. Reproducing the
catchment-relevant response times with high-resolution large-scale models may be key
for an assessment of future changes and related implications for groundwater
management under various scenarios and for ungauged catchments.</p><?xmltex \hack{\newpage}?>
</sec>

      
      </body>
    <back><notes notes-type="dataavailability">

      <p id="d1e1880">Streamflow data are available on request for scientific purposes from the
responsible federal state agencies, i.e. the State Environmental Agency of
Baden-Württemberg, Bavaria, Brandenburg, Hesse, Mecklenburg-Western Pomerania, Lower Saxony, North Rhine-Westphalia,
Rhineland-Palatinate, Saarland, Saxony, Saxony-Anhalt, Schleswig-Holstein and Thuringia. Climate data
are available via the website of the European Climate Assessment and
Dataset (<uri>https://www.ecad.eu/</uri>, ECA and D, 2018); the German hydrogeological map (BGR &amp; SGD,
2016) is online available as well
(<uri>https://www.bgr.bund.de/DE/Themen/Wasser/Projekte/laufend/Beratung/Huek200/huek200_projektbeschr.html</uri>,
Bundesanstalt für Geowissenschaften und Rohstoffe, 2018).</p>
  </notes><?xmltex \hack{\clearpage}?><app-group>

<?pagebreak page6220?><app id="App1.Ch1.S1">
  <title>Mathematical relationship between baseflow and groundwater
storage</title>
      <p id="d1e1898">In groundwater-dominated catchments <inline-formula><mml:math id="M118" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is an integrated signal of
groundwater conditions in the entire area and mainly depends as a function
<inline-formula><mml:math id="M119" display="inline"><mml:mi>g</mml:mi></mml:math></inline-formula> on the hydraulic head <inline-formula><mml:math id="M120" display="inline"><mml:mi>H</mml:mi></mml:math></inline-formula> of the groundwater:

              <disp-formula id="App1.Ch1.E1" content-type="numbered"><mml:math id="M121" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mfenced open="(" close=")"><mml:mi>H</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>r</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">with</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>H</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mo>∀</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mi>H</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>

        where <inline-formula><mml:math id="M122" display="inline"><mml:mi>r</mml:mi></mml:math></inline-formula> is the additional flow from other stored sources. <inline-formula><mml:math id="M123" display="inline"><mml:mi>H</mml:mi></mml:math></inline-formula> is a
monotonic function <inline-formula><mml:math id="M124" display="inline"><mml:mi>f</mml:mi></mml:math></inline-formula> of groundwater storage <inline-formula><mml:math id="M125" display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula>:

              <disp-formula id="App1.Ch1.E2" content-type="numbered"><mml:math id="M126" display="block"><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mi>S</mml:mi></mml:mfenced><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">with</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mo>∀</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>S</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>

        Combining Eqs. (A1) and (A2) leads to

              <disp-formula id="App1.Ch1.E3" content-type="numbered"><mml:math id="M127" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mfenced close=")" open="("><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>S</mml:mi></mml:mrow></mml:mfenced><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mfenced open="(" close=")"><mml:mi>S</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>r</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">with</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mo>∀</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi>S</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>

        Because of the monotonic behaviour, function <inline-formula><mml:math id="M128" display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> is reversible:

              <disp-formula id="App1.Ch1.E4" content-type="numbered"><mml:math id="M129" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=")" open="("><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:mfenced><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">with</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mo>∀</mml:mo><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>r</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>

        Potential sources for <inline-formula><mml:math id="M130" display="inline"><mml:mi>r</mml:mi></mml:math></inline-formula> include snowmelt, interflow, lake outflow, sewage
discharge or other anthropogenic sources of discharge. Anthropogenic
influences can be assumed negligible, as catchments in this study were
selected as near-natural. Moreover, there are no big lakes or other surface
water storages in the catchments. Snowmelt remains an important factor that
might blur the true signal of groundwater storage. This is particularly
important for the catchments at higher elevations, e.g. in the Alps, and
must be considered for the interpretation of the results. However, in
catchments where these factors are of minor relevance compared to
groundwater outflow, Eq. (A4) simplifies to

              <disp-formula id="App1.Ch1.E5" content-type="numbered"><mml:math id="M131" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mi>S</mml:mi><mml:mo>≃</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup><mml:mfenced open="(" close=")"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:mfenced><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">with</mml:mi><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mo>∀</mml:mo><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:mo>;</mml:mo></mml:mrow></mml:math></disp-formula>

        i.e. higher baseflow indicates higher groundwater storage, and vice versa.
However, without further knowledge about <inline-formula><mml:math id="M132" display="inline"><mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>, the observation of
<inline-formula><mml:math id="M133" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mi mathvariant="normal">b</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> allows for conclusions about <inline-formula><mml:math id="M134" display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> solely on an ordinal scale.</p>
</app>

<app id="App1.Ch1.S2">
  <title>The Mann–Kendall trend test with pre-whitening</title>
      <p id="d1e2360">To calculate trends, the non-parametric Mann–Kendall (MK) test was applied.
However, the results are affected by serial correlation which increases the
type I error (i.e. rejecting the no-trend hypothesis although there is no
trend). To test for serial correlation in the data, we fitted an
autocorrelation AR(1) model to each <inline-formula><mml:math id="M135" display="inline"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant="normal">b</mml:mi><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> time series. The calculated
autocorrelation is significant on a level of <inline-formula><mml:math id="M136" display="inline"><mml:mrow><mml:mi mathvariant="italic">α</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.0</mml:mn></mml:mrow></mml:math></inline-formula>5 if absolute
autocorrelation is higher than <inline-formula><mml:math id="M137" display="inline"><mml:mrow><mml:mn mathvariant="normal">1.96</mml:mn><mml:mo>/</mml:mo><mml:mo>√</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula> (Douglas et al., 2000), where
<inline-formula><mml:math id="M138" display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula> is the length of the time series (in this case <inline-formula><mml:math id="M139" display="inline"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">40</mml:mn></mml:mrow></mml:math></inline-formula>). One hundred thirty (130) out of 338
time series showed significant serial correlation, thus requiring a
pre-processing before using MK. Kulkarni and Von Storch (1995) recommended
pre-whitening time series to allow for the MK. Since the sample size is
relatively small here and trend magnitude not too large, pre-whitening is not
expected to reduce the MK test power much in this case, but rather to reduce the
type I error (Bayazit and Önöz, 2007). Pre-whitening was done
according to other studies (Kulkarni and Von Storch, 1995; Douglas et al.,
2000) as in Eq. (A6):

              <disp-formula id="App1.Ch1.E6" content-type="numbered"><mml:math id="M140" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:msub><mml:mi>Y</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>

        where <inline-formula><mml:math id="M141" display="inline"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the pre-whitened time series at time step <inline-formula><mml:math id="M142" display="inline"><mml:mi>t</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math id="M143" display="inline"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the
original time series at time step <inline-formula><mml:math id="M144" display="inline"><mml:mi>t</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math id="M145" display="inline"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> is the serial correlation
determined with the AR(1) model. Pre-whitening reduced the serial
correlation in all time series to close to zero.</p>
      <p id="d1e2509">The MK test compares the number of concordant pairs in the data with the
number of discordant pairs. This gives the Kendall score <inline-formula><mml:math id="M146" display="inline"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> (Kendall,
1948):

              <disp-formula id="App1.Ch1.E7" content-type="numbered"><mml:math id="M147" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant="normal">sign</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mi mathvariant="normal">sign</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>

        where sign() is a function that returns the algebraic sign. For the
Mann–Kendall trend test that was applied here, <inline-formula><mml:math id="M148" display="inline"><mml:mi>Z</mml:mi></mml:math></inline-formula> is time. Therefore, the
second part of Eq. (A7) always equals <inline-formula><mml:math id="M149" display="inline"><mml:mrow><mml:mo>+</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:math></inline-formula>, and it simplifies to

              <disp-formula id="App1.Ch1.E8" content-type="numbered"><mml:math id="M150" display="block"><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:munder><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:mi mathvariant="normal">sign</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>

        In the case of no real trend, <inline-formula><mml:math id="M151" display="inline"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> has a mean value of zero; in the case of no
ties in the data, it has a standard deviation <inline-formula><mml:math id="M152" display="inline"><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> of (Kendall, 1948)

              <disp-formula id="App1.Ch1.E9" content-type="numbered"><mml:math id="M153" display="block"><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msqrt><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:mi>n</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:mfenced><mml:mo>(</mml:mo><mml:mn mathvariant="normal">2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn mathvariant="normal">5</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mn mathvariant="normal">18</mml:mn></mml:mfrac></mml:mstyle></mml:msqrt><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>

        Thus, <inline-formula><mml:math id="M154" display="inline"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> can be transformed into the standard normal distributed <inline-formula><mml:math id="M155" display="inline"><mml:mi>z</mml:mi></mml:math></inline-formula> score
by dividing by the standard deviation of <inline-formula><mml:math id="M156" display="inline"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>:

              <disp-formula id="App1.Ch1.E10" content-type="numbered"><mml:math id="M157" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mfenced open="{" close=""><mml:mtable class="array" columnalign="left"><mml:mtr><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">if</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>&gt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mn mathvariant="normal">0</mml:mn><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">if</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant="italic">σ</mml:mi><mml:mi mathvariant="normal">S</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mi mathvariant="normal">if</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace linebreak="nobreak" width="0.125em"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mspace width="0.125em" linebreak="nobreak"/><mml:msub><mml:mi>S</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo>&lt;</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>

        <inline-formula><mml:math id="M158" display="inline"><mml:mi>z</mml:mi></mml:math></inline-formula> was assumed to be significant if it was not within the interval <inline-formula><mml:math id="M159" display="inline"><mml:mrow><mml:mo>±</mml:mo><mml:mn mathvariant="normal">1.96</mml:mn></mml:mrow></mml:math></inline-formula>, i.e. a significance level of <inline-formula><mml:math id="M160" display="inline"><mml:mrow><mml:mi mathvariant="italic">α</mml:mi><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0.05</mml:mn></mml:mrow></mml:math></inline-formula>.</p>
</app>

<app id="App1.Ch1.S3">
  <title>ANOVA and Tukey's test</title>
      <?pagebreak page6221?><p id="d1e2902">To detect the influence of a categorical variable with multiple levels on a
numerical variable, an ANOVA was used in this work. The ANOVA compares
<inline-formula><mml:math id="M161" display="inline"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, the sum of squares explained by the categorical
variable's levels, with the sum of squares of the residuals
(<inline-formula><mml:math id="M162" display="inline"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>). The mean sums (<inline-formula><mml:math id="M163" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">Se</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math id="M164" display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">Sr</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>) are tested for significance using
the <inline-formula><mml:math id="M165" display="inline"><mml:mi>F</mml:mi></mml:math></inline-formula> test:

              <disp-formula id="App1.Ch1.E11" content-type="numbered"><mml:math id="M166" display="block"><mml:mrow><mml:mstyle displaystyle="true" class="stylechange"/><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">Se</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi mathvariant="normal">Sr</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi mathvariant="normal">fe</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi mathvariant="normal">r</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi mathvariant="normal">fr</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>

        where <inline-formula><mml:math id="M167" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> denotes the degrees of freedom (<inline-formula><mml:math id="M168" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi mathvariant="normal">fe</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>: number of levels minus
1; <inline-formula><mml:math id="M169" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi mathvariant="normal">fr</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>: amount of data minus <inline-formula><mml:math id="M170" display="inline"><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi mathvariant="normal">fe</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> minus 1). The test statistic
<inline-formula><mml:math id="M171" display="inline"><mml:mi>F</mml:mi></mml:math></inline-formula> follows the <inline-formula><mml:math id="M172" display="inline"><mml:mi>F</mml:mi></mml:math></inline-formula> distribution and was assumed to be significant for
<inline-formula><mml:math id="M173" display="inline"><mml:mi>p</mml:mi></mml:math></inline-formula> &lt; 0.05.</p>
      <p id="d1e3080">The ANOVA gives information about the general significance of the categorical
variable. An equally important piece of information is which of the categorical
variable's levels differ significantly regarding the target variable. This
information was obtained using the post hoc Tukey test. For this analysis
a pairwise <inline-formula><mml:math id="M174" display="inline"><mml:mi>t</mml:mi></mml:math></inline-formula> test between all levels is carried out:

              <disp-formula id="App1.Ch1.E12" content-type="numbered"><mml:math id="M175" display="block"><mml:mrow><mml:mstyle class="stylechange" displaystyle="true"/><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mfrac style="display"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover><mml:mn mathvariant="normal">1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow><mml:msqrt><mml:mrow><mml:mstyle displaystyle="false"><mml:mfrac style="text"><mml:mrow><mml:msubsup><mml:mi>s</mml:mi><mml:mn mathvariant="normal">1</mml:mn><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn mathvariant="normal">1</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>-</mml:mo><mml:mstyle displaystyle="false"><mml:mfrac style="text"><mml:mrow><mml:msubsup><mml:mi>s</mml:mi><mml:mn mathvariant="normal">2</mml:mn><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:msqrt></mml:mfrac></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>

        where <inline-formula><mml:math id="M176" display="inline"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo mathvariant="normal">‾</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M177" display="inline"><mml:mrow><mml:msubsup><mml:mi>s</mml:mi><mml:mi>i</mml:mi><mml:mn mathvariant="normal">2</mml:mn></mml:msubsup></mml:mrow></mml:math></inline-formula> are the mean and the
variance for the two levels, respectively, and <inline-formula><mml:math id="M178" display="inline"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is the amount of data per level.
Tukey's test accounts for the multiple-comparison problem (i.e. a higher
probability of getting significant results by chance due to a higher number
of comparisons) by using the studentized range distribution instead of the
<inline-formula><mml:math id="M179" display="inline"><mml:mi>t</mml:mi></mml:math></inline-formula> distribution. Again, the results were assumed to be significant for
<inline-formula><mml:math id="M180" display="inline"><mml:mrow><mml:mi>p</mml:mi><mml:mi mathvariant="italic">&lt;</mml:mi><mml:mspace width="0.125em" linebreak="nobreak"/><mml:mn mathvariant="normal">0.05</mml:mn></mml:mrow></mml:math></inline-formula>.</p><?xmltex \hack{\clearpage}?>
</app>
  </app-group><notes notes-type="authorcontribution">

      <p id="d1e3224">JH and KS designed the study. JH performed all analyses and prepared the manuscript.
Both authors revised the manuscript and approved the final version.</p>
  </notes><notes notes-type="competinginterests">

      <p id="d1e3230">The authors declare that they have no conflict of
interest.</p>
  </notes><ack><title>Acknowledgements</title><p id="d1e3236">We acknowledge the data providers at the federal states' agencies for the streamflow data, the E-OBS dataset from the EU-FP6 project
ENSEMBLES (<uri>https://ensembles-eu.metoffice.com</uri>, last access: 28 November 2018) and the data providers in the
ECA&amp;D project (<uri>https://www.ecad.eu</uri>, last access: 28 November 2018). The authors were funded by the DFG
(Jost Hellwig: project TrenDHy DFG STA632/4-1; Kerstin Stahl: Heisenberg programme STA632/3-1).
Benedikt Heudorfer provided helpful comments on the manuscript. We are grateful for the
comments of John Bloomfield and Spencer Sawaske, which helped to improve the quality of the manuscript. The article processing
charge was funded by the German Research Foundation (DFG) and the University of Freiburg in the funding programme Open Access Publishing.
<?xmltex \hack{\newline}?><?xmltex \hack{\newline}?>
Edited by: Stacey Archfield <?xmltex \hack{\newline}?>
Reviewed by: John Bloomfield and Spencer Sawaske</p></ack><ref-list>
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    <!--<article-title-html>An assessment of trends and potential future changes in groundwater-baseflow drought based on catchment response times</article-title-html>
<abstract-html><p>Drought is an important natural hazard with large impacts
on society. Changes in drought characteristics have been studied for
different parts of the hydrological cycle, but insights into changes of
groundwater resources are obscured due to the lack of long-term observations
and large heterogeneity of hydrogeological conditions. Moreover, predicted
future changes in precipitation are uncertain and have a lagged effect on
streamflow and groundwater. We investigated past changes and potential
future changes in catchment baseflow as a reflection of groundwater drought
for 338 headwater catchments across Germany based on catchments'
characteristic response times. First, baseflow dynamics as a proxy of
groundwater storage and outflow on a catchment scale were derived from
streamflow records and related to precipitation input. Second, past trends
in baseflow minima were calculated and attributed to climate and catchment
controls. Last, response times and the timing of yearly baseflow minima were
combined into estimates of the sensitivity to future precipitation changes.
Baseflow response times of the studied headwaters are heterogenous across
Germany, ranging from a few months to several years, and depend significantly
on hydrogeological conditions. Few significant trends were found in past
baseflow minima, and trends are highly dependent on the period of analysis.
Based on the assumption of a typical regional scenario of increasing winter
precipitation and decreasing summer precipitation, increases in hydrological
drought hazard or no changes are projected for most parts of Germany.
Catchments with longer response times can buffer interannual precipitation
shifts, whereas catchments with fractured rocks are sensitive to summer
precipitation decreases. These results urge for a surface water and groundwater
management based on local groundwater response to precipitation and help to
assess impacts of climate change on overall water supply.</p></abstract-html>
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