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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 GmbH</publisher-name>
<publisher-loc>Göttingen, Germany</publisher-loc>
</publisher>
</journal-meta>

    <article-meta>
      <article-id pub-id-type="doi">10.5194/hess-19-4327-2015</article-id><title-group><article-title>Regional analysis of groundwater droughts <?xmltex \hack{\newline}?> using hydrograph classification</article-title>
      </title-group><?xmltex \runningtitle{Regional analysis of groundwater droughts using hydrograph classification}?><?xmltex \runningauthor{J.~P.~Bloomfield et al.}?>
      <contrib-group>
        <contrib contrib-type="author" corresp="yes" rid="aff1">
          <name><surname>Bloomfield</surname><given-names>J. P.</given-names></name>
          <email>jpb@bgs.ac.uk</email>
        <ext-link>https://orcid.org/0000-0002-5730-1723</ext-link></contrib>
        <contrib contrib-type="author" corresp="no" rid="aff2">
          <name><surname>Marchant</surname><given-names>B. P.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff2">
          <name><surname>Bricker</surname><given-names>S. H.</given-names></name>
          
        </contrib>
        <contrib contrib-type="author" corresp="no" rid="aff3">
          <name><surname>Morgan</surname><given-names>R. B.</given-names></name>
          
        </contrib>
        <aff id="aff1"><label>1</label><institution>British Geological Survey, Wallingford, UK</institution>
        </aff>
        <aff id="aff2"><label>2</label><institution>British Geological Survey, Keyworth, UK</institution>
        </aff>
        <aff id="aff3"><label>3</label><institution>Environment Agency, Lincoln, UK</institution>
        </aff>
      </contrib-group>
      <author-notes><corresp id="corr1">J. P. Bloomfield (jpb@bgs.ac.uk)</corresp></author-notes><pub-date><day>28</day><month>October</month><year>2015</year></pub-date>
      
      <volume>19</volume>
      <issue>10</issue>
      <fpage>4327</fpage><lpage>4344</lpage>
      <history>
        <date date-type="received"><day>7</day><month>May</month><year>2015</year></date>
           <date date-type="rev-request"><day>8</day><month>June</month><year>2015</year></date>
           <date date-type="rev-recd"><day>15</day><month>September</month><year>2015</year></date>
           <date date-type="accepted"><day>13</day><month>October</month><year>2015</year></date>
      </history>
      <permissions>
<license license-type="open-access">
<license-p>This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/3.0/">http://creativecommons.org/licenses/by/3.0/</ext-link></license-p>
</license>
</permissions><self-uri xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015.html">This article is available from https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015.html</self-uri>
<self-uri xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015.pdf">The full text article is available as a PDF file from https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015.pdf</self-uri>


      <abstract>
    <p>Groundwater drought is a spatially and temporally variable phenomenon. Here
we describe the development of a method to regionally analyse and quantify
groundwater drought. The method uses a cluster analysis technique
(non-hierarchical <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means) to classify standardised groundwater level
hydrographs (the standardised groundwater level index, SGI) prior to
analysis of their groundwater drought characteristics, and has been tested
using 74 groundwater level time series from Lincolnshire, UK. Using the test
data set, six clusters of hydrographs have been identified. For each cluster
a correlation can be established between the mean SGI and a mean
standardised precipitation index (SPI), where each cluster is associated
with a different SPI accumulation period. Based on a comparison of SPI time
series for each cluster and for the study area as a whole, it is inferred
that the clusters are independent of the driving meteorology and are
primarily a function of catchment and hydrogeological factors. This
inference is supported by the observation that the majority of sites in each
cluster are associated with one of the principal aquifers in the study
region. The groundwater drought characteristics of the three largest
clusters, which constitute <inline-formula><mml:math display="inline"><mml:mo>∼</mml:mo></mml:math></inline-formula> 80 % of the sites, have been
analysed. There are differences in the distributions of drought duration,
magnitude and intensity of groundwater drought events between the three
clusters as a function of autocorrelation of the mean SGI time series for
each cluster. In addition, there are differences between the clusters in
their response to three major multi-annual droughts that occurred during the
analysis period. For example, sites in the cluster with the longest SGI
autocorrelation experience the greatest-magnitude droughts and are the
slowest to recover from major droughts, with groundwater drought conditions
typically persisting at least 6 months longer than at sites in the other
clusters. Membership of the clusters is shown to be related to unsaturated
zone thickness at individual boreholes. This last observation emphasises the
importance of catchment and aquifer characteristics as (non-trivial)
controls on groundwater drought hydrographs. The method of analysis is
flexible and can be adapted to a wide range of hydrogeological settings
while enabling a consistent approach to the quantification of regional
differences in response of groundwater to meteorological drought.</p>
  </abstract>
    </article-meta>
  </front>
<body>
      

<sec id="Ch1.S1" sec-type="intro">
  <title>Introduction</title>
      <p>Groundwater drought is a type of hydrological drought characterised by
sustained low groundwater levels, reduced base flow and reduced flows to
springs and groundwater-fed rivers and wetlands (Van Lanen and Peters, 2000;
Tallaksen and Van Lanen, 2004; Mishra and Singh, 2010; Van Loon, 2015). Like
other hydrological aspects of drought, groundwater droughts are not a simple
function of meteorological drivers. The impact of droughts on regional
groundwater resources can vary in space and time. This is because the
response of groundwater systems to meteorological droughts, through changes
in groundwater levels and baseflow to groundwater-supported rivers, is
influenced by spatial variations in intrinsic catchment and aquifer
characteristics and processes. These include highly non-linear unsaturated
zone processes, recharge, and saturated groundwater storage, flow and
discharge over a range of spatial and temporal scales (Tallaksen et al., 2009;
Bloomfield and Marchant, 2013; Van Lanen et al., 2013; Van Loon and Laaha, 2015).</p>
      <p>In order to improve the design and operation of groundwater drought
monitoring networks, the analysis and interpretation of data from such
networks, and, more generally, water resource management at the onset,
during and after episodes of groundwater drought, there is a need for a much
better understanding of the heterogeneous spatio-temporal response of
aquifers to major meteorological droughts (Bloomfield and Marchant, 2013).
This includes the need for robust methods to systematically characterise and
quantify the heterogeneous response of groundwater to meteorological
droughts at a regional scale prior to investigation and attribution of the
causes of any heterogeneous response. Despite extensive work on the regional
analysis of meteorological and other hydrological droughts, to date there
has been no systematic investigation of heterogeneities in groundwater
droughts at the regional scale. This paper describes the application of one
such suite of methods to regionally analyse groundwater level hydrographs
and to assess variations in the spatial response of groundwater to
meteorological droughts using a case study from the UK.</p>
<sec id="Ch1.S1.SS1">
  <title>Controls on spatial heterogeneity in groundwater drought</title>
      <p>A few previous studies have presented evidence for the spatially
heterogeneous response of groundwater to meteorological droughts. To help
develop an optimal monitoring network for groundwater resources under
drought conditions, Chang and Teoh (1995) described the heterogeneous
response of groundwater levels at 13 observation boreholes to meteorological
droughts across a basin in Ohio, USA, although they did not investigate the
hydrogeological causes of the heterogeneity. Van Lanen (2005) and Van Lanen
and Tallaksen (2007) observed that drought characteristics derived from
groundwater levels have “spatial effects” and noted that these spatial
effects on groundwater drought are an important consideration when
monitoring droughts using groundwater levels. Van Lanen and Tallaksen (2007)
compared modelled groundwater recharge and discharge for a humid continental
climate (Missouri, USA) and a tropical savannah climate (Guinea) for quick-
and slow-responding catchments and showed that both climatology and the
responsiveness of the catchment as defined by the aquifer characteristics
have an influence on drought generation. Peters et al. (2006) investigated
the propagation and spatial distribution of aspects of modelled groundwater
drought, including recharge, groundwater level and groundwater discharge in
the Pang catchment in the UK. They found that short droughts in groundwater
levels were most severe near streams and were attenuated with distance from
the streams, that longer periods of below-average recharge had more effect on
suppressing groundwater levels on interfluves near groundwater divides, and
that droughts in groundwater discharge are more attenuated upstream and less
so downstream in the catchment. Tallaksen et al. (2009) also modelled the
spatio-temporal response of the Pang catchment to drought events and found
large differences between the spatio-temporal response of groundwater
recharge, level and discharge and the driving meteorological droughts, where
droughts in groundwater recharge and levels were found to cover relatively
small areas but last longer than the meteorological droughts.</p>
      <p>Mendicino et al. (2008) developed a groundwater resource index for drought
monitoring and forecasting based on a simple distributed run-off/water
balance model, and they evaluated the use of the index in three catchments in
southern Italy. They found that the groundwater resource index was highly
spatially variable and related it to variations in hydraulic conductivity
across the catchments. Using a newly developed groundwater drought index,
the standardised groundwater level index (SGI), Bloomfield and Marchant (2013)
also investigated hydrogeological controls on groundwater drought.
Based on 14 observation boreholes in different catchments across England,
UK, they showed that groundwater drought duration depended on the
autocorrelation structure of SGI time series. This was in turn inferred to
be a function both of spatially varying recharge processes and of saturated
flow processes within the local aquifer systems.</p>
</sec>
<sec id="Ch1.S1.SS2">
  <title>Regional analysis of groundwater drought</title>
      <p>There has been significant work on the regional analysis of meteorological
and other hydrological droughts. Cluster analysis (CA), principal component
analysis (PCA) or some combination of both techniques has been used
extensively by meteorologists and hydrologists to investigate the
spatio-temporal distribution of hydrological variables, including drought
indices (e.g. Klugman, 1978; Karl and Koscienly, 1982; Eder et al., 1987;
Stahl and Demuth, 1999, 2001; Lana et al., 2001; Bonaccorso et al., 2003;
Vincente-Serrano, 2006; Vicente-Serrano and Cuadrat-Prats, 2007; Raziel et
al., 2008; Santos et al., 2010; Fleig et al., 2011; Hannaford et al., 2011;
Lorenzo-Lacruz et al., 2013).</p>
      <p>Although not previously applied to groundwater drought, CA and/or PCA
techniques have been used to classify groundwater level hydrographs for a
range of purposes. Winter et al. (2000) classified groundwater hydrographs
from three small lake-dominated catchments to investigate groundwater
recharge and differences in the hydrographs as a function of the geology of
the catchments. Similarly, Moon et al. (2004) applied PCA to 66 groundwater
level hydrographs from South Korea to characterise the spatial variability
in groundwater recharge. Upton and Jackson (2011) used CA and PCA (following
a methodology developed by Hannah et al., 2000) with 52 groundwater level
hydrographs from the Pang and Lambourn catchments in the UK to produce
regional or “master” hydrographs for modelling the spatial distribution of
groundwater flooding.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F1" specific-use="star"><caption><p>Case study area (left panel) and simplified geology map (right panel) showing
locations of the observation boreholes. Cross section (bottom panel) illustrating
the stratigraphy–depth relationships between the three major aquifers in
the study region: the Lincolnshire Limestone, the Spilsby Sandstone and the Chalk.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f01.jpg"/>

        </fig>

      <p><?xmltex \hack{\newpage}?>Here we present the first systematic regional analysis of groundwater
droughts using a case study from Lincolnshire, UK. The case study consists
of 74 groundwater hydrographs from an area of approximately 8000 km<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula>
that includes three regionally important aquifers, the Lincolnshire
Limestone, the Chalk and the Spilsby Sandstone aquifers, each with
contrasting aquifer characteristics (Sect. 2). The groundwater hydrographs
have been normalised using the SGI
technique of Bloomfield and Marchant (2013), and groups or clusters of
similar groundwater hydrographs have been identified using CA, where
hydrogeologically meaningful clusters are identified by explicitly searching
for groups of hydrographs that can be explained by a posteriori knowledge of the
groundwater system (Sect. 4.2). The drought characteristics of the
clusters have been quantified in terms of drought event duration, magnitude
and intensity, and the impact of three major, multi-annual droughts on
the SGI time series has been investigated (Sect. 4.4). Controls on the
groundwater drought response in each of the clusters have been explored and
the results briefly discussed in terms of the implications for monitoring
and managing groundwater droughts (Sect. 5).</p>
</sec>
</sec>
<sec id="Ch1.S2">
  <title>The case study</title>
      <p>The case study area of Lincolnshire is situated in the east of England, UK.
It is bounded by the North Sea to the east, the Wash estuary to the south
and the Humber estuary to the north (Fig. 1). The area is predominantly
rural with highly productive agricultural and horticultural land, fens and
estuarine wetlands. Lincoln, Boston and Scunthorpe are the principal small
conurbations in the study area. The land is generally flat and low-lying,
typically less than 30 m a.s.l. (above sea level), apart from the Chalk of
the Lincolnshire Wolds and the Lincolnshire Limestone outcrop, which form
northwest–southeast-trending escarpments that reach elevations of
approximately 150 and 70 m a.s.l. respectively.</p><?xmltex \hack{\newpage}?>
<sec id="Ch1.S2.SS1">
  <title>Hydrometeorology and drought history</title>
      <p>As a first-order approximation, it is assumed that the broad meteorological
drought history of the study area is spatially homogeneous. This assumption
means that any relative differences in drought histories between sites or
clusters need to be explained in terms of catchment or hydrogeological
factors, rather than differences in the drought climatology. This assumption
is tested as part of the analysis of correlations between precipitation and
regional groundwater levels (see Sect. 4.2). It is also supported by the
observations that the whole study area is governed by the same broad
climatic patterns, i.e. rain-bearing low-pressure systems from the Atlantic
and high-pressure systems leading to a lack of rainfall, with only small
variation in annual precipitation across the region (Marsh and Hannaford,
2008). The assumption is also consistent with the previously documented
spatial coherence of major hydrological (surface water) droughts in the UK
(Hannaford et al., 2011; Fleig et al., 2011; Folland et al., 2015) where the
current study area falls within a homogeneous drought region (“region 4”
of Hannaford et al. (2011), “region GB4” of Fleig et al. (2012) and Kingston et
al. (2013), and the “English Lowlands” of Folland et al. (2015)), although it
is noted that the effects of landscape processes can cause heterogeneous
meteorological signals to become attenuated (Van Loon, 2015).</p>
      <p>Mean annual rainfall varies across the study area from about 600 to 700 mm
(Marsh and Hannaford, 2008). The groundwater hydrographs used in the study
have been analysed from 1983 to 2012. During this period, three multi-annual
episodes of drought have previously been documented by Marsh et al. (2007,
2013), Kendon et al. (2013), Parry and Marsh (2013) and Folland et al. (2015) as
follows: 1988 to 1992, 1995 to 1997 and 2010 to 2012 respectively. All are known to have
been major drought events causing reduced surface flows and suppressed
groundwater levels throughout large areas of central, eastern and southern
UK as well as over parts of northwestern Europe (Lloyd-Hughes and Saunders,
2002; Lloyd-Hughes et al., 2010; Hannaford et al., 2011; Fleig et al., 2012;
Kingston et al., 2013).</p>
</sec>
<sec id="Ch1.S2.SS2">
  <title>Geology and hydrogeology</title>
      <p>The study area consists of a sequence of Jurassic and Cretaceous aquifers
separated by low-permeability clay and shale units. The whole sequence
generally dips gently eastwards, and where each of the aquifer units passes
under an overlying low-permeability formation they typically become
confined. The whole sequence is unconformably overlain by Quaternary
superficial deposits. Figure 1 shows the distribution of the three main
aquifers in the region – the Jurassic Lincolnshire Limestone; the Lower
Cretaceous–Upper Jurassic Spilsby Sandstone; and the Upper Cretaceous Chalk
– and includes a schematic cross section of the hydrostratigraphy of the study
area. These aquifers are hydrogeologically distinct from each other, and two
of them, the Lincolnshire Limestone and the Chalk, have previously documented
spatial variability. Below we summarise these features as they inform the
heuristic rules used in Sect. 3.2.2 to guide the selection of clusters as
part of the CA.</p>
      <p>The Lincolnshire Limestone Formation is an oolitic limestone with
fine-grained, micritic and peloidal units (Allen et al., 1997), and it is up to
40 m thick at outcrop in the west. It dips and thins to the east, where it
becomes confined and eventually pinches out down-dip. Maximum unsaturated
zone thickness is up to about 45 m towards the southwest of the outcrop.
Groundwater movement is almost entirely by fracture flow along
well-developed bedding plane fractures and joints. Abstraction takes place
mainly from the region immediately to the east of the outcrop. It has highly
variable transmissivities and storage coefficients typical of a fractured
limestone. Allen et al. (1997) have reported a wide range of transmissivity
values for the Lincolnshire Limestone with an interquartile range of 260 to
2260 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and a geometric mean of 660 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, with
slightly higher transmissivities being reported from the south of the
region, and a very wide range of storage coefficients from 2 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">7</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> to 0.58.</p>
      <p>The Spilsby Sandstone aquifer is up to about 30 m thick, consisting of a
variably, but often poorly cemented, pebbly quartz sandstone with alternating
thin clays and marls (Whitehead and Lawrence, 2006). It outcrops along the
foot of the Wolds escarpment (Fig. 1), where it is associated with springs
and maximum unsaturated zone thickness is about 30m. It dips to the east and
away from outcrop and is generally confined by clays above and below
(Fig. 1). Jones et al. (2000) reported transmissivity values in the range
130 to 170 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and a geometric mean of
140 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>, with storage coefficients ranging from 1 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> to 1 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and with a
geometric mean of 4 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>.</p>
      <p>The Chalk is a microporous fractured limestone (Bloomfield et al., 1995).
Storage and transmissivity are controlled by local sub-karstic development
of the fracture network (Bloomfield, 1996; Maurice et al., 2006). The Chalk
group reaches a thickness of over 250 m. Groundwater flows from the recharge
areas in the west, eastward down-dip towards and into the confined Chalk to
the east. The Chalk bedrock surface was significantly altered during the
Ipswichian interglacial of the Quaternary. As a result of glacial activity a
cliff line and wave-cut platform were eroded into the Chalk (Fig. 1). The
Chalk to the east of the palaeo-cliff line is now buried beneath a covering
of till, sand and gravel superficial deposits (Whitehead and Lawrence,
2006). Maximum unsaturated zone thickness occurs towards the northwest of
the Chalk outcrop and is about 60 m, contrasting with the relatively thin
unsaturated zone to the east of the palaeo-cliff line. Allen et al. (1997)
and Whitehead and Lawrence (2006) have reported that transmissivity values
differ between the northern and southern Chalk in Lincolnshire. In the
northern part of the region, transmissivity has an interquartile range of
1020 to 6070 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> with a geometric mean of 2350 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>,
whereas in the southern area, in the region of the eroded
Chalk, transmissivity is slightly reduced and has an interquartile range of
850 to 3010 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> with a geometric mean of 1380 m<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> day<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>.
Similarly, Allen et al. (1997) report storage coefficients
with an interquartile range of 3.5 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">5</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> to 1.5 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and with a
geometric mean of 2 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> for the northern Chalk and 6.1 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">5</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> to
2.7 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula> and with a geometric mean of 1.5 <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> 10<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>
for the southern Chalk.</p>
      <p>The Quaternary superficial deposits in the study area comprise
glaciofluvial sand and gravels and tills; peat; tidal flat deposits; river
terrace sands and gravels; and overlying alluvium. The Lincolnshire
Limestone Formation and the western part of the Chalk outcrop are largely
absent of superficial cover.</p>
</sec>
</sec>
<sec id="Ch1.S3">
  <title>Data and methods</title>
<sec id="Ch1.S3.SS1">
  <title>Data</title>
      <p>Groundwater level data for the 74 observation boreholes (Fig. 1) has been
provided by the Environment Agency from their groundwater level monitoring
network database (Environment Agency, 2014). Prior to the study none of the
sites were believed to be significantly impacted by abstraction, although all
three regional aquifers are used for public water supply, abstractions for
agricultural irrigation and industrial use (Allen et al., 1997; Whitehead
and Lawrence, 2006). Where observation boreholes penetrate both the Chalk
and underlying Spilsby Sandstone aquifer, the boreholes are completed with
screens so that they monitor water levels in only one of the two aquifers.
Groundwater levels have been recorded over a range of frequencies, but
typically at weekly to monthly time steps. Based on the raw groundwater
level data, mean monthly groundwater levels have been estimated. If no
observations were available for a given month, then a linear interpolation
was used to estimate the monthly groundwater levels following the method
described by Bloomfield and Marchant (2013).</p>
      <p>Precipitation data have been taken from the Centre for Ecology and
Hydrology's Continuous Estimation of River Flows (CERF) 1 km gridded
precipitation data set (Keller et al., 2005; Dore et al., 2012; Bloomfield
and Marchant, 2013). CERF daily gridded precipitation data are generated from
rain gauge data held in the UK Met Office national precipitation monitoring
network. A triangular planes methodology is used to produce a daily
1 km<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> grid based on a weighted average (inverse distance) of the three
nearest rain gauges. Daily rainfall is then summed to give total monthly
gridded rainfall. The precipitation data that are used with each groundwater
level observation site are the monthly total for the CERF 1 km<inline-formula><mml:math display="inline"><mml:msup><mml:mi/><mml:mn mathvariant="normal">2</mml:mn></mml:msup></mml:math></inline-formula> grid
square that contains the given groundwater observation borehole.</p><?xmltex \hack{\newpage}?>
</sec>
<sec id="Ch1.S3.SS2">
  <title>Methods</title>
<sec id="Ch1.S3.SS2.SSS1">
  <title>Hydrograph normalisation using the SGI method</title>
      <p>The groundwater level hydrographs have been normalised to the SGI of Bloomfield and Marchant (2013). This is a
non-parametric normalisation of data that assigns a value to the monthly
groundwater levels based on their rank within groundwater levels for a given
month from a given hydrograph. The normal scores transform is undertaken by
applying the inverse normal cumulative distribution function to <inline-formula><mml:math display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula> equally
spaced <inline-formula><mml:math 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 ranging from 1/(2<inline-formula><mml:math display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula>) to 1 <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula> 1/(2<inline-formula><mml:math display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula>). The values that
result are the SGI values. They are then re-ordered such that the largest
SGI value is assigned to the <inline-formula><mml:math display="inline"><mml:mi>i</mml:mi></mml:math></inline-formula> for which <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is largest, the
second-largest SGI value is assigned to the <inline-formula><mml:math display="inline"><mml:mi>i</mml:mi></mml:math></inline-formula> for which <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is second largest and
so on. In summary, for each of the 74 study sites, normalised indices are
estimated from the groundwater level data for each calendar month using the
normal scores transform. These normalised indices are then merged to form a
continuous SGI. Precipitation records for each site have also been
normalised. At each site a version of the standardised precipitation index (SPI)
after McKee et al. (1993) has been estimated for precipitation
accumulation periods of 1, 2, …, 36 months. For consistency between
groundwater and precipitation indices, SPIs are estimated using the normal
scores transform applied to accumulated precipitation data for each calendar month.</p>
</sec>
<sec id="Ch1.S3.SS2.SSS2">
  <title>Cluster analysis</title>
      <p>Cluster analysis attempts to identify clusters of similar individuals
amongst a multivariate data set. In the context of this paper CA is used to
form clusters of groundwater level hydrographs which exhibit similar
fluctuations in their SGI time series. A wide range of CA algorithms exist.
They are most coarsely distinguished according to whether or not they assume
that the resultant clusters are hierarchical. Given the wide variety of
algorithms, it is difficult to decide upon the best approach to cluster a
particular data set. Webster and Oliver (1990) stress that this decision is
rather subjective, although previous studies that have used CA to cluster
hydrographs have typically justified their choice of algorithm by claiming
that some produce more physically interpretable groupings. For example,
Hannah et al. (2000) used the agglomerative hierarchical average linkage
algorithm as they thought it was more interpretable than alternatives such
as the centroid and Ward's clustering procedures. Webster and Oliver (1990)
recommend that multiple clustering algorithms should be applied and expert
knowledge of the system being investigated used to decide which set of
clusters is most relevant. In this paper we adapt this approach by applying
one hierarchical and one non-hierarchical method.</p>
      <p>Hierarchical classifiers require a measure of the similarity (or
dissimilarity) between each pair of individuals. Common examples include the
Euclidean distance or the correlation between the measurements of the
individuals. The pairwise similarities between <inline-formula><mml:math display="inline"><mml:mi>s</mml:mi></mml:math></inline-formula> individuals are expressed
in a <inline-formula><mml:math display="inline"><mml:mi>s</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>×</mml:mo></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi>s</mml:mi></mml:math></inline-formula> matrix <inline-formula><mml:math display="inline"><mml:mi mathvariant="bold">B</mml:mi></mml:math></inline-formula>. A mathematical criterion is then used to
allocate the individuals to different clusters in a manner that maximises
the similarity between the individuals within the groups whilst minimising
the similarity between individuals in different clusters. For our
hierarchical clusters we measure the similarity between groundwater level
hydrographs by the correlation matrix of their SGI time series and then
apply the agglomerative hierarchical complete-linkage strategy (Webster and
Oliver, 1990) to merge the boreholes into clusters.</p>
      <p>We also apply the commonly used non-hierarchical <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering
algorithm. It is widely used in spatial analysis studies; for example,
Santos et al. (2010), Raziei et al. (2012) and Sadri et al. (2014) have all
used the <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering algorithm to investigate the regional
characteristics of droughts. The approach partitions the individuals into a
specified number of clusters. A numerical optimisation routine is used to
select the partitioning which maximises the similarity between each
individual and the centroid of the cluster in which it is contained. Again
there is flexibility in the choice of similarity measure and the manner in
which the centroid of a cluster is calculated. We use the squared Euclidean
distance between the vectors of time series observations from each site to
assess similarity and define the centroid of a cluster as the
multi-dimensional mean of the time series within the cluster.</p>
      <p>Clustering methods do not produce a unique partitioning of a given data set
on their own, and for both the hierarchical and non-hierarchical approaches
there remains the issue of deciding upon the optimal number of clusters.
This can be achieved by asking an expert on the system in question to
compare the attributes of clusterings consisting of a different number of
groups. Here we use a rule-based approach to help identify the number of
clusters based on knowledge of the general hydrogeology of the study area.
Bloomfield and Marchant (2013) have previously shown that groundwater
drought characteristics are a function of unsaturated zone thickness in
fractured aquifers such as the Lincolnshire Limestone and Chalk aquifers,
and that when a broader range of aquifer types are considered groundwater
drought characteristics are also a function of the hydraulic diffusivity of
aquifers. Here we use these observations and knowledge of the spatial
variation in these features across the three aquifers in the study area
(Sect. 2.2) to design rules to aid in the selection of clusters. The rules
adopted for the current study are to identify the smallest number of
clusters that (i) broadly resolve the spatial distribution of the three
aquifers across the study region; (ii) distinguish more
than one region of the Lincolnshire Limestone, given the previously documented N–S
variation in aquifer properties and unsaturated zone thickness across the
Lincolnshire Limestone aquifer (Allen et al., 1997); and (iii) distinguish more than one region of the Chalk,
given variations in aquifer properties and unsaturated zone thickness across the Chalk aquifer
both N–S and across the buried cliff line (Allen et al., 1997). Note that this set of rules
is specific to the current study; however, for any given study area the
target number of classes and hence the rules used can be adapted to reflect
the regional hydrogeology and in particular any knowledge of heterogeneity
in the aquifer systems under investigation. However, mathematical criteria
can also be used as a guide to clustering. We also calculate the RMSSD, the
square root of sum of the squared Euclidean distance between each individual
and the centroid of the group to which it is allocated. In combination with
expert judgement related to the system under consideration, it is common
practice to inform the choice of the number of clusters using plots of RMSSD
versus cluster number. Since RMSSD decreases non-linearly as the number of
clusters increases, a cluster number is selected associated with a decrease
in the rate of RMSSD decline.</p>
</sec>
<sec id="Ch1.S3.SS2.SSS3">
  <title>Autocorrelation structure of the SGI time series</title>
      <p>Bloomfield and Marchant (2013) demonstrated the importance of the
autocorrelation structure of SGI time series for groundwater drought studies
by establishing a relationship between the range of significant
autocorrelation in the SGI series, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, and corresponding SPI. They
showed that <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> scales linearly with <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is the
SPI accumulation period which leads to the strongest correlation between SGI
and SPI. Both <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> are also used here to characterise and
quantify groundwater droughts within each of the clusters of groundwater
hydrographs and have been estimated as follows.</p>
      <p>If the mean SGI for a borehole is denoted by <inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mtext>SGI</mml:mtext><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:math></inline-formula>, then the
<inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>th sample autocovariance coefficient is defined to be

                  <disp-formula id="Ch1.E1" content-type="numbered"><mml:math display="block"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn mathvariant="normal">1</mml:mn><mml:mi>n</mml:mi></mml:mfrac><mml:munderover><mml:mo movablelimits="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfenced open="{" close="}"><mml:mtext>SGI</mml:mtext><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mtext>SGI</mml:mtext><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:mfenced><mml:mfenced open="{" close="}"><mml:mtext>SGI</mml:mtext><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>-</mml:mo><mml:mover accent="true"><mml:mtext>SGI</mml:mtext><mml:mo mathvariant="normal">‾</mml:mo></mml:mover></mml:mfenced></mml:mrow></mml:math></disp-formula>

            and the <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>th sample autocorrelation coefficient is

                  <disp-formula id="Ch1.E2" content-type="numbered"><mml:math display="block"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>

            where <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mn mathvariant="normal">0</mml:mn></mml:msub></mml:mrow></mml:math></inline-formula> reduces to the population variance function (see Eq. (1) when
<inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 0). The correlogram is a plot of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> against <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>. If there is no
correlation between the SGI(<inline-formula><mml:math display="inline"><mml:mi>i</mml:mi></mml:math></inline-formula>) observed <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> months
apart and if the SGI values are normally distributed, then <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is
approximately normally distributed, with mean 0 and variance 1<inline-formula><mml:math display="inline"><mml:mrow><mml:mo>/</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula>.
Therefore values of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> with magnitude greater than 2<inline-formula><mml:math display="inline"><mml:mrow><mml:mo>/</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:mrow></mml:math></inline-formula> indicate
significant correlation at approximately the 5 % level. We define the
range of significant temporal correlation of a SGI time series to be the
largest <inline-formula><mml:math display="inline"><mml:mi>m</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, for which <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>&gt;</mml:mo></mml:math></inline-formula> 2<inline-formula><mml:math display="inline"><mml:mrow><mml:mo>/</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:mrow></mml:math></inline-formula> for all <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>≤</mml:mo></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi>m</mml:mi></mml:math></inline-formula>. Since all
of our groundwater records are of <inline-formula><mml:math display="inline"><mml:mi>n</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 355 months, the threshold on <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> is
equal to 0.11. To estimate <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, Pearson correlation coefficients are
calculated between SGI and SPI with accumulation periods of <inline-formula><mml:math display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 1,
2, …, 36 months,
and the accumulation period associated with the maximum correlation gives <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>.</p><?xmltex \hack{\newpage}?>
</sec>
</sec>
</sec>
<sec id="Ch1.S4">
  <title>Results</title>
<sec id="Ch1.S4.SS1">
  <title>Identification of regional droughts from average SPI and SGI time series</title>
      <p>Before undertaking the regional drought analysis, the correlation between
mean SPI and SGI for the entire region, based on all 74 sites, was
investigated and the large-scale drought history of the study area were defined.</p>
      <p>Figure 2a is a heat map showing the correlation coefficient between SPI for
precipitation accumulation periods <inline-formula><mml:math display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 1 to 36 months and SGI for lags
between SPI and SGI of 0 to 5 months based on average values of SPI and SGI
for all 74 sites. Dark blue denotes zero correlation and dark red a perfect
correlation. Figure 2a shows that there is a good correlation between SPI
and SGI. The strongest correlation (0.84, denoted by the closed black circle
in Fig. 2a) is for a precipitation accumulation period (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>) of
12 months (SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula>) with no lag between the SGI and SPI time series. This is
consistent with the observations of Bloomfield and Marchant (2013), who
previously reported <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for a variety of groundwater hydrographs from
the UK with an average of 13 months, and Folland et al. (2015), who reported a
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> of 12 months for aggregated time series representing the English
Lowlands. Figure 2b and c, the average SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula> and SGI time series
respectively, have similar features. For example, episodes of high
groundwater levels in 1983, 1994, 2002, and 2008 correspond with high values
of SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula>. Three episodes of regionally significant groundwater drought
associated with prolonged low groundwater levels from October 1988 to
November 1993, May 1995 to February 1998, and from August 2010 to August 2012
correspond closely with episodes of meteorological drought in the
SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula> time series and are consistent with those identified by previous
studies (Lloyd-Hughes and Saunders, 2002; Marsh et al., 2007, 2013; Kendon et al.,
2013; Hannaford et al., 2011; Parry and Marsh, 2013; Folland et al., 2015).
It is inferred from these observations that the large-scale drought history
of the study area is represented well by the average SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula> and SGI time series.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F2"><caption><p><bold>(a)</bold> SPI–SGI correlation as a heat map, <bold>(b)</bold> mean SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula> time
series and <bold>(c)</bold> mean SGI time series for all 74 hydrographs.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f02.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F3" specific-use="star"><caption><p><bold>(a)</bold> Cluster dendrogram for hierarchical classification (<inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 6) of
SGI time series, <bold>(b)</bold> map showing the distribution of sites by clusters based
on hierarchical classification (<inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 6), and <bold>(c)</bold> map showing the distribution
of sites by clusters formed by <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering (<inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 6).</p></caption>
          <?xmltex \igopts{width=398.338583pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f03.jpg"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F4"><caption><p>RMSSD as a function of the number of clusters for the hierarchical
and non-hierarchical <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering algorithms and for a three-fold
classification based on geology alone.</p></caption>
          <?xmltex \igopts{width=170.716535pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f04.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F5"><caption><p>Heat maps of Pearson correlation between SGI and SPI for <inline-formula><mml:math display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 1 to
36 months and for lags up to 5 months. Maximum correlation is denoted by the
closed black circles.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f05.png"/>

        </fig>

</sec>
<sec id="Ch1.S4.SS2">
  <title>Regional analysis of the SGI hydrographs</title>
      <p>CA has been used to analyse the heterogeneous response of groundwater to
droughts across the study region. Clustering has been undertaken using both
an agglomerative hierarchical complete-linkage algorithm and a
non-hierarchical <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering algorithm, and the resulting clusters
searched for those that are hydrogeologically meaningful and that can be
explained by known features of the catchment and groundwater systems. Figure 3a
is a dendrogram that fully illustrates the level of similarity between
individuals within the clusters formed by the hierarchical clustering. The
number of clusters is controlled through the threshold on the distance
between groups. For example, a threshold of 0.62 leads to the six clusters
shown in Fig. 3b. Figure 3c is an equivalent map showing the distribution of
sites by clusters formed by <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering for <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 6.</p>
      <p>Figure 3b and c show that the spatial distribution of sites as a function
of the clusters formed by the hierarchical and non-hierarchical approaches
are broadly similar, so the choice of clustering algorithm is based on a
plot of RMSSD against number of clusters. Figure 4 shows that the RMSSD for
the <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering is systematically lower than that for the hierarchical
clustering algorithm where there are three clusters or more, so we have
chosen to use the non-hierarchical <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering approach. Note also that
both clustering algorithms are better than a clustering scheme based solely
on the three classes of aquifer (e.g. Lincolnshire Limestone, Chalk and
Spilsby Sandstone). However, an optimal number of <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-mean clusters is not
clearly evident in Fig. 4. After careful inspection of the clusters formed
by a range of <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering classes and a consideration of the
study-specific clustering rules described in Sect. 3.2.2, <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 6 was selected.
Based on <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering where <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>=</mml:mo></mml:math></inline-formula> 6, Fig. 3c shows the distribution of
sites between the six clusters (cluster 1 to cluster 6, or CL1–CL6).</p>
      <p>It can be seen from Fig. 3c that the resulting <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters have a degree
of spatial coherency. We have previously assumed that such spatial
correlations in the SGI time series are primarily a function of catchment
and hydrogeological factors and not a consequence of heterogeneity in the
driving meteorology. Here we test if this is the case, prior to further
exploration of the features of each cluster, by investigating if
precipitation associated with each cluster is substantially different from
regional average precipitation. To do this, we first need to identify a
representative accumulation period, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, for precipitation for each cluster.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F6"><caption><p>Mean SPI times series for each of the <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters based on
the accumulation period <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for each cluster. The black line is
SPI based on gridded precipitation series for sites in a given cluster and
the red line is SPI for the mean rainfall across the whole study area based
on the different aggregation periods, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, for each cluster.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f06.png"/>

        </fig>

      <p>Figure 5 is a set of heat maps, similar to Fig. 2a, showing the correlation
between SPI for precipitation accumulation periods, <inline-formula><mml:math display="inline"><mml:mi>q</mml:mi></mml:math></inline-formula>, 1 to 36 months, and
SGI for lags between SPI and SGI time series of 0 to 5 months for each of
the six clusters. Dark blue denotes zero correlation, and dark red a perfect
correlation, with the strongest correlation for each cluster marked by the
closed black circle. Table 1 gives <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for each cluster and also gives
the maximum associated correlation coefficient. In all cases except CL2,
the maximum correlation between SPI and SGI is found where there is no lag
between the two time series. For CL2 it is found at a lag of 1 month. The
highest correlations are for CL2, CL4 and CL1 at 0.86, 0.82 and 0.74
respectively. The correlations for CL3 and CL5 are moderate (0.36 and 0.53),
and for CL6 there is effectively no correlation (0.09). This is consistent
with the observations made in Sect. 4.3 below that linear trends in CL3
and CL5 appear to affect the SGI time series and that the SGI hydrograph for
CL6 appears to be anomalous, departing from the mean regional SGI and SPI
signals. Values of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for CL1 to CL5 from Fig. 5 are 4, 16, 15, 9, and
17 months respectively. Based on these, Fig. 6 shows SPI time series for
each cluster, where black lines are the mean SPI for the cluster and the red
lines are average SPI across the study area based on the same
cluster-specific <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>. Since Fig. 6 illustrates that the two SPI time
series for each cluster are similar, we infer that heterogeneity in the
driving meteorology across the study region, or at least between the
clusters as defined here, does not play an important role in the clustering
process and that membership of clusters is dominated by catchment or
hydrogeological factors.</p>
</sec>
<sec id="Ch1.S4.SS3">
  <title>Characteristic features of the SGI hydrograph clusters</title>
      <p>Figure 7 shows the mean SGI time series for each cluster. Two main
qualitative observations can be made regarding the SGI hydrographs. Five of
the six clusters have a similar overall form to the mean SGI hydrograph for
the region (Fig. 2c) showing common patterns of low (and high) groundwater
level stand. However, CL6 appears to be an exception with a different
overall form to the SGI hydrograph – it also exhibits an anomalous step
change in SGI from drought to high groundwater level stand over an 8-month
period from May 1990 to December 1990. Secondly, two of the clusters,
CL3 and CL5, appear to show declining linear trends in SGI, making direct
comparison of drought histories between these and other clusters problematic.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F7"><caption><p>Mean SGI time series for each of the six <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f07.png"/>

        </fig>

      <p>Bloomfield and Marchant (2013) have previously shown that <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, a
measure of the significant autocorrelation length of SGI time series,
relates to features of groundwater drought. A similar analysis of
autocorrelation structure of SGI time series for each cluster is presented
here. Figure 8 shows autocorrelation plots for SGI hydrographs for each of
the six clusters. In each figure the pale grey lines are autocorrelation
plots for individual sites and the solid black line is the autocorrelation
plot for the mean SGI time series for the cluster, with the horizontal dashed
line indicating the significant level of autocorrelation based on the record
length. Based on these plots, values of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for the mean SGI time
series for each cluster are given in Table 1. Values of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for CL3,
CL5 and CL6 are anomalously large, consistent with the anomalous features of
these SGI hydrographs described above. For the remaining clusters, Fig. 8
and Table 1 show that CL1 has the shortest autocorrelation of 15 months. In
comparison, CL2 has an autocorrelation of 23 months and CL4 is intermediate
at 18 months.</p>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T1" specific-use="star"><caption><p>Summary of features of the six <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="9">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="center"/>
     <oasis:colspec colnum="3" colname="col3" align="center"/>
     <oasis:colspec colnum="4" colname="col4" align="center"/>
     <oasis:colspec colnum="5" colname="col5" align="center"/>
     <oasis:colspec colnum="6" colname="col6" align="left"/>
     <oasis:colspec colnum="7" colname="col7" align="center"/>
     <oasis:colspec colnum="8" colname="col8" align="center"/>
     <oasis:colspec colnum="9" colname="col9" align="center"/>
     <oasis:thead>
       <oasis:row>  
         <oasis:entry colname="col1">Cluster</oasis:entry>  
         <oasis:entry rowsep="1" namest="col2" nameend="col5">Number of sites </oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry rowsep="1" namest="col7" nameend="col9">Statistic </oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2">Total</oasis:entry>  
         <oasis:entry colname="col3">Lincolnshire</oasis:entry>  
         <oasis:entry colname="col4">Spilsby</oasis:entry>  
         <oasis:entry colname="col5">Chalk</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">SPI–SGI</oasis:entry>  
         <oasis:entry colname="col8">Representative</oasis:entry>  
         <oasis:entry colname="col9">Autocorrelation</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"/>  
         <oasis:entry colname="col3">Limestone</oasis:entry>  
         <oasis:entry colname="col4">Sandstone</oasis:entry>  
         <oasis:entry colname="col5"/>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">maximum</oasis:entry>  
         <oasis:entry colname="col8">accumulation</oasis:entry>  
         <oasis:entry colname="col9">range, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"/>  
         <oasis:entry colname="col3"/>  
         <oasis:entry colname="col4"/>  
         <oasis:entry colname="col5"/>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">correlation</oasis:entry>  
         <oasis:entry colname="col8">period, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col9">(months)</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"/>  
         <oasis:entry colname="col3"/>  
         <oasis:entry colname="col4"/>  
         <oasis:entry colname="col5"/>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7"/>  
         <oasis:entry colname="col8">(months)</oasis:entry>  
         <oasis:entry colname="col9"/>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>  
         <oasis:entry colname="col1">CL1</oasis:entry>  
         <oasis:entry colname="col2">13</oasis:entry>  
         <oasis:entry colname="col3">13</oasis:entry>  
         <oasis:entry colname="col4">0</oasis:entry>  
         <oasis:entry colname="col5">0</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">0.74</oasis:entry>  
         <oasis:entry colname="col8">4</oasis:entry>  
         <oasis:entry colname="col9">15</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">CL2</oasis:entry>  
         <oasis:entry colname="col2">23</oasis:entry>  
         <oasis:entry colname="col3">2</oasis:entry>  
         <oasis:entry colname="col4">0</oasis:entry>  
         <oasis:entry colname="col5">21</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">0.86</oasis:entry>  
         <oasis:entry colname="col8">16</oasis:entry>  
         <oasis:entry colname="col9">23</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">CL3</oasis:entry>  
         <oasis:entry colname="col2">6</oasis:entry>  
         <oasis:entry colname="col3">2</oasis:entry>  
         <oasis:entry colname="col4">4</oasis:entry>  
         <oasis:entry colname="col5">0</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">0.36</oasis:entry>  
         <oasis:entry colname="col8">15</oasis:entry>  
         <oasis:entry colname="col9">60</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">CL4</oasis:entry>  
         <oasis:entry colname="col2">24</oasis:entry>  
         <oasis:entry colname="col3">19</oasis:entry>  
         <oasis:entry colname="col4">0</oasis:entry>  
         <oasis:entry colname="col5">5</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">0.82</oasis:entry>  
         <oasis:entry colname="col8">9</oasis:entry>  
         <oasis:entry colname="col9">18</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">CL5</oasis:entry>  
         <oasis:entry colname="col2">5</oasis:entry>  
         <oasis:entry colname="col3">0</oasis:entry>  
         <oasis:entry colname="col4">0</oasis:entry>  
         <oasis:entry colname="col5">5</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">0.53</oasis:entry>  
         <oasis:entry colname="col8">17</oasis:entry>  
         <oasis:entry colname="col9">28</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1">CL6</oasis:entry>  
         <oasis:entry colname="col2">3</oasis:entry>  
         <oasis:entry colname="col3">0</oasis:entry>  
         <oasis:entry colname="col4">3</oasis:entry>  
         <oasis:entry colname="col5">0</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7">0.09</oasis:entry>  
         <oasis:entry colname="col8">–</oasis:entry>  
         <oasis:entry colname="col9">–</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">Total</oasis:entry>  
         <oasis:entry colname="col2">74</oasis:entry>  
         <oasis:entry colname="col3">36</oasis:entry>  
         <oasis:entry colname="col4">7</oasis:entry>  
         <oasis:entry colname="col5">31</oasis:entry>  
         <oasis:entry colname="col6"/>  
         <oasis:entry colname="col7"/>  
         <oasis:entry colname="col8"/>  
         <oasis:entry colname="col9"/>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup></oasis:table></table-wrap>

      <p>These contrasting characteristics between the clusters can be seen clearly
in Fig. 9a, which illustrates SGI time series for all sites within each
cluster, grouped in their respective clusters and presented in the form of
a heat map where low values of SGI (associated with drought conditions) are
in shades of green to red (increasing drought intensity) and episodes of
high groundwater level stand are in shades of green to blue (increasingly high
groundwater levels). The three major episodes of drought can be seen clearly
in the heat maps for CL1, CL2 and CL4 but are obscured by the trends in CL3
and CL5 and absent in CL6. The degree of coherency of individual SGI time
series within each cluster also appears to be consistent with differences in
autocorrelation between the clusters. Figure 9b is a heat map of the
cross-correlation coefficients for all the individual SGI time series
ordered as a function of the six clusters, where dark red denotes high
correlations and dark blue denotes low correlations. Sites within CL1 and
CL4, i.e. clusters with moderate or short autocorrelation, show relatively low
levels of internal coherency compared with sites in CL2 with relatively long
autocorrelation that are highly correlated.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F8"><caption><p>Correlograms for each of the mean SGI time series (bold) and
individual site time series (grey) for each of the six <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters
showing variation in the autocorrelation function (ACF) for lags up to 60 months.</p></caption>
          <?xmltex \igopts{width=236.157874pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f08.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F9" specific-use="star"><caption><p>Heat maps showing <bold>(a)</bold> SGI varying with time for all 74 sites as
a function of the six <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters and <bold>(b)</bold> correlations between
all pairs of sites sorted as a function of the six <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f09.jpg"/>

        </fig>

      <p>Based on the above, the following is a summary of the features of each cluster:
<list list-type="bullet"><list-item><p>CL1 is dominated by sites from the northern parts of the Lincolnshire
Limestone. The mean SGI time series of CL1 has a relatively short autocorrelation
(<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> of 15 months), and within the cluster SGI hydrographs are relatively variable.</p></list-item><list-item><p>CL2 is dominated by sites from the northern part of the Chalk. The
cluster has the longest mean SGI autocorrelation (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> of 23 months), and
hydrographs within CL2 are highly correlated, indicating a high degree of
coherency in groundwater levels across the northern part of the Chalk in the study area.</p></list-item><list-item><p>CL3 is a relatively small cluster of six sites, four of which are from
the confined Spilsby Sandstone and two from the Lincolnshire Limestone. The
main feature of the cluster is a trend in decreasing SGI across the observational
record. This trend is consistent with a previous water balance assessment for
the Spilsby Sandstone (Whitehead and Lawrence, 2006), where annual groundwater
deficits have been reported. The sites in this cluster are inferred to be
possibly variably impacted by long-term abstraction. Given this inference and
the small size of the cluster of sites, CL3 is not included in the subsequent
analysis of groundwater droughts.</p></list-item><list-item><p>CL4 is dominated by sites from the southern Lincolnshire Limestone and
also includes five unconfined sites on the southern Chalk and one site located
in the northern Lincolnshire Limestone. It has a moderate autocorrelation,
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, of 18 months. Individual SGI hydrographs within the cluster show a
moderate degree of coherency.</p></list-item><list-item><p>CL5 is a small cluster of five sites all from the southeastern Chalk to
the east of the palaeo-wave-cut platform, and they are the five sites closest to the
coast. It has a moderately long autocorrelation, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, of 28 months that
may be affected by an apparent weak trend in declining SGI – there is only a
weak correlation between SPI and SGI. Given the small size of the cluster and
the apparent trend in mean SGI, CL5 is not included in the subsequent analysis
of groundwater droughts.</p></list-item><list-item><p>CL6 consists of three SGI hydrographs from the confined Spilsby Sandstone
aquifer. The hydrographs are characterised by an anomalous step change in SGI
from drought to high groundwater level stand over an 8-month period from
May 1990 to December 1990. The mean SGI hydrograph shows no correlation with
the other five clusters, and there is no correlation between SPI and SGI within
the cluster. All three sites are within a radius of about 3 km of a public
water supply borehole, and it is inferred that groundwater levels may be
influenced by abstraction. So, as with CL3 and CL5, this very small cluster is
not included in the subsequent analysis of groundwater droughts.</p></list-item></list></p>
</sec>
<sec id="Ch1.S4.SS4">
  <title>Analysis of droughts using the hydrographs from CL1, 2 and 4</title>
      <p>Clusters CL1, CL2 and CL4 consist of 61 of the 74 hydrographs analysed. Here
the characteristics of groundwater droughts in these clusters are quantified,
and the response of the clusters to three major drought episodes is investigated.</p>
      <p>The duration, magnitude and mean intensity of groundwater drought events
have been investigated based on an analysis of the SGI hydrographs where,
following the convention of McKee et al. (1993), negative values of SGI
denote drought conditions (note, however, that the current convention of the
World Meteorological Organization for SPI refers to drought conditions where
SPI is continuously negative and reaches and intensity of <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.0 or less and
that negative values between 0 and <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1 are classified as near normal and
simply indicate less than a median precipitation; World Meteorological
Organization, 2012). Groundwater drought duration, D, is taken to be the
total number of consecutive months where SGI is negative. Groundwater
drought magnitude, M, is taken to be the total cumulative value of monthly
SGI for a given drought event, and mean drought intensity, I, is given by M / D.
Summary drought statistics for CL1, CL2 and CL4 are given in Table 2.</p>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T2"><caption><p>Summary of drought event statistics for clusters C1, C2 and C4.</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="4">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="left"/>
     <oasis:colspec colnum="3" colname="col3" align="left"/>
     <oasis:colspec colnum="4" colname="col4" align="left"/>
     <oasis:thead>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2">CL1</oasis:entry>  
         <oasis:entry colname="col3">CL2</oasis:entry>  
         <oasis:entry colname="col4">CL4</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>  
         <oasis:entry colname="col1">Number of drought events</oasis:entry>  
         <oasis:entry colname="col2">39</oasis:entry>  
         <oasis:entry colname="col3">15</oasis:entry>  
         <oasis:entry colname="col4">18</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">Mean duration (months)</oasis:entry>  
         <oasis:entry colname="col2">4.6</oasis:entry>  
         <oasis:entry colname="col3">11.3</oasis:entry>  
         <oasis:entry colname="col4">9.1</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">Maximum duration (months)</oasis:entry>  
         <oasis:entry colname="col2">27</oasis:entry>  
         <oasis:entry colname="col3">61</oasis:entry>  
         <oasis:entry colname="col4">49</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">Mean event magnitude</oasis:entry>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>2.9</oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>7.9</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>6.6</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">Mean event intensity</oasis:entry>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.43</oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.28</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.4</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">Maximum event intensity</oasis:entry>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.1</oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.05</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.13</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">No. of events where <inline-formula><mml:math display="inline"><mml:mi>I</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>&lt;</mml:mo></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1</oasis:entry>  
         <oasis:entry colname="col2">3</oasis:entry>  
         <oasis:entry colname="col3">2</oasis:entry>  
         <oasis:entry colname="col4">2</oasis:entry>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup></oasis:table></table-wrap>

<?xmltex \floatpos{t}?><table-wrap id="Ch1.T3" specific-use="star"><caption><p>Summary of the 1988–93, 1995–98 and 2011–12 drought events for
clusters CL1, CL2 and CL4 (where <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>
denote indices for drought event duration, magnitude and intensity respectively).</p></caption><oasis:table frame="topbot"><oasis:tgroup cols="6">
     <oasis:colspec colnum="1" colname="col1" align="left"/>
     <oasis:colspec colnum="2" colname="col2" align="left"/>
     <oasis:colspec colnum="3" colname="col3" align="left"/>
     <oasis:colspec colnum="4" colname="col4" align="left"/>
     <oasis:colspec colnum="5" colname="col5" align="left"/>
     <oasis:colspec colnum="6" colname="col6" align="left"/>
     <oasis:thead>
       <oasis:row>  
         <oasis:entry colname="col1">Drought</oasis:entry>  
         <oasis:entry colname="col2">Drought</oasis:entry>  
         <oasis:entry colname="col3">Regional</oasis:entry>  
         <oasis:entry colname="col4">Mean SGI</oasis:entry>  
         <oasis:entry colname="col5">Mean SGI</oasis:entry>  
         <oasis:entry colname="col6">Mean SGI</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1">episode</oasis:entry>  
         <oasis:entry colname="col2">index</oasis:entry>  
         <oasis:entry colname="col3">SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col4">CL1</oasis:entry>  
         <oasis:entry colname="col5">CL2</oasis:entry>  
         <oasis:entry colname="col6">CL4</oasis:entry>
       </oasis:row>
     </oasis:thead>
     <oasis:tbody>
       <oasis:row>  
         <oasis:entry colname="col1">1988 to 1993</oasis:entry>  
         <oasis:entry colname="col2">Start date</oasis:entry>  
         <oasis:entry colname="col3">Dec 1988</oasis:entry>  
         <oasis:entry colname="col4">Oct 1988</oasis:entry>  
         <oasis:entry colname="col5">Nov 1988</oasis:entry>  
         <oasis:entry colname="col6">Oct 1988</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2">End date</oasis:entry>  
         <oasis:entry colname="col3">Oct 1992</oasis:entry>  
         <oasis:entry colname="col4">May 1993</oasis:entry>  
         <oasis:entry colname="col5">Nov 1993</oasis:entry>  
         <oasis:entry colname="col6">May 1993</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3">47</oasis:entry>  
         <oasis:entry colname="col4">56</oasis:entry>  
         <oasis:entry colname="col5">61</oasis:entry>  
         <oasis:entry colname="col6">56</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>56.8</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>37</oasis:entry>  
         <oasis:entry colname="col5"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>63.6</oasis:entry>  
         <oasis:entry colname="col6"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>41.6</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.2</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.7</oasis:entry>  
         <oasis:entry colname="col5"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.0</oasis:entry>  
         <oasis:entry colname="col6"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.7</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">1995 to 1998</oasis:entry>  
         <oasis:entry colname="col2">Start date</oasis:entry>  
         <oasis:entry colname="col3">May 1995</oasis:entry>  
         <oasis:entry colname="col4">May 1995</oasis:entry>  
         <oasis:entry colname="col5">Aug 1995</oasis:entry>  
         <oasis:entry colname="col6">Jul 1995</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2">End date</oasis:entry>  
         <oasis:entry colname="col3">Oct 1997</oasis:entry>  
         <oasis:entry colname="col4">Jul 1997</oasis:entry>  
         <oasis:entry colname="col5">Feb 1998</oasis:entry>  
         <oasis:entry colname="col6">Aug 1997</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3">30</oasis:entry>  
         <oasis:entry colname="col4">27</oasis:entry>  
         <oasis:entry colname="col5">31</oasis:entry>  
         <oasis:entry colname="col6">26</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>34.3</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>18.7</oasis:entry>  
         <oasis:entry colname="col5"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>32.4</oasis:entry>  
         <oasis:entry colname="col6"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>29.3</oasis:entry>
       </oasis:row>
       <oasis:row rowsep="1">  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.1</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.7</oasis:entry>  
         <oasis:entry colname="col5"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.0</oasis:entry>  
         <oasis:entry colname="col6"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.1</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1">2010 to 2012</oasis:entry>  
         <oasis:entry colname="col2">Start date</oasis:entry>  
         <oasis:entry colname="col3">Jan 2011</oasis:entry>  
         <oasis:entry colname="col4">May 2011</oasis:entry>  
         <oasis:entry colname="col5">Jan 2011</oasis:entry>  
         <oasis:entry colname="col6">Jul 2010</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2">End date</oasis:entry>  
         <oasis:entry colname="col3">Apr 2012</oasis:entry>  
         <oasis:entry colname="col4">May 2012</oasis:entry>  
         <oasis:entry colname="col5">Aug 2012</oasis:entry>  
         <oasis:entry colname="col6">May 2012</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3">16</oasis:entry>  
         <oasis:entry colname="col4">13</oasis:entry>  
         <oasis:entry colname="col5">20</oasis:entry>  
         <oasis:entry colname="col6">23</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>16.1</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>13.9</oasis:entry>  
         <oasis:entry colname="col5"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>11.7</oasis:entry>  
         <oasis:entry colname="col6"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>21</oasis:entry>
       </oasis:row>
       <oasis:row>  
         <oasis:entry colname="col1"/>  
         <oasis:entry colname="col2"><inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mtext>event</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula></oasis:entry>  
         <oasis:entry colname="col3"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.0</oasis:entry>  
         <oasis:entry colname="col4"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.1</oasis:entry>  
         <oasis:entry colname="col5"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.6</oasis:entry>  
         <oasis:entry colname="col6"><inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.9</oasis:entry>
       </oasis:row>
     </oasis:tbody>
   </oasis:tgroup></oasis:table></table-wrap>

      <p>Table 2 shows that there are differences in the character of the groundwater
drought events in the SGI hydrographs for clusters CL1, CL2 and CL3. For
example, CL1 has more than twice the number of drought episodes
(39 episodes) as CL2 (15 episodes), and the average and maximum duration of
droughts in CL1 (4.6 and 27 months respectively) are less than half those of
CL2 (11.3 and 61 months). The mean drought event magnitude in CL1 (<inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>2.9) is
less than half that in CL2 (<inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>7.9), and the mean drought event intensity in
CL1 (<inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.43) is almost twice that of CL2 (<inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>0.28). In all cases, the drought
event statistics for CL4 fall between those for CL1 and CL2. In summary, CL1
exhibits shorter but generally more intense drought episodes compared with
CL2, with CL4 drought events being of intermediate character. These relative
drought phenomena are a consequence of the degree of autocorrelation in the
respective SGI time series, where CL1 has a relatively short autocorrelation
compared with relatively long autocorrelation for CL2. This observation is
consistent with previous site-specific and modelling studies that noted a
similar relationship between the “flashiness” or responsiveness of the
groundwater system to meteorological divers and the number of droughts,
where quickly responding groundwater systems typically experience more
droughts than more slowly responding catchments (Peters et al., 2003; Van
Loon and Van Lanen, 2012; Van Lanen et al. 2013).</p>
      <p>There is a strong relationship between drought duration and magnitude for
all three clusters (Fig. 10), where longer episodes of groundwater drought
are associated with droughts of greater magnitude. However, there is no such
regular or simple relationship between drought duration and intensity.
Maximum drought intensity is similar for all three clusters – for CL1, CL2
and CL4 it is <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.10, <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.05 and <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.13 respectively (Table 2 and Fig. 11) – and
is associated with two of the major drought events, i.e. with the latter
part of the 1988–1993 drought for CL2 and the 2010–2012 drought for
CL1 and CL4. Figure 11 shows the empirical distribution of D, M and I for
clusters CL1, CL2 and CL4. Drought duration (Fig. 11) in all three clusters
is highly positively skewed with many short drought events and relatively
few long drought events. As previously noted, the longest duration droughts
are associated with CL2, the cluster with the longest autocorrelation in the
SGI time series. These observations are consistent with those of Hisdal and
Tallaksen (2003), Tallaksen et al. (2009) and Fleig et al. (2011), who have
also described strongly skewed distributions of hydrological drought durations.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F10"><caption><p>Drought magnitude versus drought duration for sites in clusters CL1, CL2 and CL4.</p></caption>
          <?xmltex \igopts{width=170.716535pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f10.png"/>

        </fig>

      <?xmltex \floatpos{t}?><fig id="Ch1.F11" specific-use="star"><caption><p>Empirical distribution of <bold>(a)</bold> drought duration, <bold>(b)</bold> drought
magnitude, and <bold>(c)</bold> drought intensity for clusters CL1, CL2 and CL4.</p></caption>
          <?xmltex \igopts{width=341.433071pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f11.png"/>

        </fig>

      <p>Three major, multi-annual droughts have already been described from the
regional (Fig. 2) and the cluster-specific (Figs. 7 and 9a) SGI time series.
Table 3 summarises differences in the relationships between the driving
meteorology and the drought characteristics of each cluster for the three
major droughts. Each of the major drought episodes has been quantified
using drought characteristics as applied to SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula> and SGI for each of
the clusters.</p>
      <p>The 1988–1993 event was the longest of the three major droughts and
consequently had the greatest drought magnitude. The groundwater and
meteorological droughts started approximately contemporaneously in the winter
of 1988. In CL2 the drought was continuous with negative SGI from November 1988
to November 1993, whereas in CL4 there were two short breaks in the
drought and numerous breaks in the drought in CL1. In CL2 there was a
gradual intensification in the drought magnitude across the event, peaking
in June 1992 at an SGI of <inline-formula><mml:math display="inline"><mml:mo>-</mml:mo></mml:math></inline-formula>1.85 (4 months after the peak SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula>
meteorological drought). In contrast, not only were there short breaks in
the drought in CL1 and CL4 but there were approximately annual cycles of
drought intensification and decline over the 4-year period – these were
particularly pronounced in CL4. This is seen in Fig. 9a, where between 1988
and 1993 the drought status of CL4 is designated by the red tones in the
heat map, but these tones show a series of approximately annual
variations giving the appearance of vertical stripes during that period and
within that cluster. However, the most pronounced differences in response to
major droughts between clusters CL1, CL2 and CL4 is in the timing of the end
of drought. Groundwater drought conditions ended in CL1 and CL4 in May 1993,
7 months after the end of the meteorological drought, but this was still
6 months before the groundwater drought ended in CL2 (Fig. 9a).</p>
      <p>The 1995–1997 drought, although shorter than the 1988–1993 drought,
followed a similar pattern, with groundwater drought starting approximately
contemporaneously with the meteorological drought. Although it was a
continuous event for all three clusters (there were no breaks in the drought
for CL1 and CL4), CL1 and CL4 again show approximately annual
intensifications and declines in drought status during the episode. Such
approximately annual changes in drought status are not seen in CL2. The 1995–1997
drought had the greatest magnitude in CL2 due to the prolonged end
to the drought in this cluster, with groundwater drought in CL1 and CL4
finishing approximately contemporaneously with the meteorological drought
but 6 months later in CL2. The 2011–2012 drought was much shorter than
the other two multi-annual droughts, lasting just over a year starting
relatively abruptly in early 2012 and finished abruptly in CL1 and CL4 in
May 2012 in response to an unusual episode of spring recharge (Parry et al., 2013).
The groundwater drought in CL2 again finished relatively late, this
time about 3 months later, in August 2012. The relatively short delay in
the breaking of the groundwater drought in CL2 compared with CL1 and CL4
probably reflects the relatively smaller groundwater drought deficit
accumulated due to the shorter duration and lower magnitude of the drought
compared with the 1988–1993 and 1995–1998 drought episodes.</p><?xmltex \hack{\newpage}?>
</sec>
</sec>
<sec id="Ch1.S5">
  <title>Discussion</title>
      <p>The results of the regional analysis of droughts based on cluster analysis
are consistent with current conceptualisations of the dynamics of drought in
hydrological systems. Propagation of drought through catchments and in
particular through the groundwater compartment is well documented (Peters et
al., 2003, 2006; Tallaksen et al., 2006), and four components of drought
propagation are recognised, i.e. pooling, attenuation, lag and lengthening,
three of which (attenuation, lag and lengthening) are associated with
modifications of drought signals in groundwater (Van Loon, 2015).
Attenuation results in smoothing of the maximum drought anomaly, lag
describes the delay in the onset of the drought signal as it passes through
the hydrological cycle (for example, see Figs. 3a and 4 of Van Loon,
2015), and lengthening extends the period of drought. Considering Table 3,
which summarises the three multi-annual droughts, and comparing event
magnitude for SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula>, CL1, CL2 and CL4, there is, as would
be expected, evidence of a general attenuation of the SPI drought signal in
the three clusters compared with SPI<inline-formula><mml:math display="inline"><mml:msub><mml:mi/><mml:mn>12</mml:mn></mml:msub></mml:math></inline-formula>. Lagging of the multi-annual
groundwater droughts behind meteorological droughts is not so easy to
quantify unambiguously. Clearly the nature and degree of the lag is
sensitive to the rainfall accumulation period used to define the
meteorological drought index most closely correlated with SGI. In the
present case, accumulation periods of 4, 16, and 9 months are required for
CL1, 2 and 4 respectively to achieve optimal correlation between the SPI and
SGI time series. Finally, the results of the present study strongly support
the concept of lengthening of groundwater drought relative to meteorological
drought (Van Loon, 2015). The results demonstrate that lengthening is most
pronounced following longer and deeper groundwater droughts. They serve to
emphasise that there can be significant differences in the lengthening
response between different clusters, even within with the same aquifer. It
also appears that the degree of lengthening may also be related to SGI
autocorrelation (the greatest degree of lengthening is observed in cluster
CL2 associated with the largest SGI autocorrelation, <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>).</p>
      <p><?xmltex \hack{\newpage}?>The results of the regional analysis add to our current understanding of the
controls on groundwater droughts. Bloomfield and Marchant (2013)
investigated how unsaturated zone thickness and the hydraulic diffusivity of
aquifers may relate to <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>. Using 14 SGI time series from four
different aquifers around the UK (including one site from the Lincolnshire
Limestone and nine sites on the Chalk, although none from the present study),
they found that <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> was broadly an inverse function of log hydraulic
diffusivity, <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>log⁡</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mtext>diff</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> (where <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mtext>diff</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> is given by <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>T</mml:mi><mml:mo>/</mml:mo><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula> and where <inline-formula><mml:math display="inline"><mml:mi>T</mml:mi></mml:math></inline-formula> is
aquifer transmissivity and <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi></mml:math></inline-formula> is specific storage of the aquifer). But
they also noted that when fractured aquifers – such as the Lincolnshire
Limestone and the Chalk, which have similarly high hydraulic diffusivities –
are specifically considered there is no clear relationship between
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math display="inline"><mml:mrow><mml:mi>log⁡</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mtext>diff</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>. However, they did find a positive relationship
between unsaturated zone thickness and <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for fractured aquifers such
as the Chalk and Lincolnshire Limestone. Based on this observation, they
proposed that unsaturated zone drainage and recharge processes were an
important contributory factor in determining autocorrelation or “memory” in
groundwater level hydrographs and by inference an influential factor on
groundwater drought characteristics, particularly in fracture aquifer
systems. Here we investigate if a similar relationship between <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> and
unsaturated zone thickness holds for CL1, CL2 and CL4, clusters dominated by
fractured aquifers.</p>
      <p>Figure 12 shows box plots of unsaturated zone thickness for CL1, CL2 and CL4
as a function of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> for each cluster (where unsaturated zone thickness
is taken as the mean depth to groundwater recorded for sites in each cluster
over the study period). In addition, corresponding observations for 10
boreholes in fractured aquifers from Bloomfield and Marchant (2013) are also
shown for reference. The results of the present study are consistent with
those of Bloomfield and Marchant (2013, Fig. 13a) and show increasing mean
unsaturated zone thickness with increasing cluster <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>; increasing
variability in unsaturated zone thickness with increasing cluster
<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>; and increasing maximum unsaturated zone thickness with increasing
cluster <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>. Bloomfield and Marchant (2013) previously noted that such
observations are consistent with the findings of Peters et al. (2005), since
unsaturated zone thickness is a function of distance to streams. However, in
the present study area (Fig. 1) surface drainage is virtually absent from
the northern Lincolnshire Limestone that dominates CL1 and is limited over
both the Chalk (CL2) and the southern Lincolnshire Limestone (CL4). Instead
we postulate that unsaturated zone thickness, and hence <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>, is
affected by more general catchment characteristics such as extent of
outcrop, topography, intrinsic aquifer characteristics and aquifer thickness, which all influence, through unsaturated zone drainage and saturated flow
processes, the overall shape of the piezometric surface in the aquifers. For
example, of the three aquifers in the study region the Chalk has the most
extensive outcrop; it is the thickest aquifer, up to 5 times thicker than
the Lincolnshire Limestone; it forms hills up to <inline-formula><mml:math display="inline"><mml:mo>∼</mml:mo></mml:math></inline-formula> 150 m a.s.l., compared
to hills about 70 m a.s.l. across the southern Lincolnshire Limestone;
and it is associated (CL2) with the largest <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula> and the longest and
largest magnitude droughts. As such, the relationships between unsaturated
zone thickness, SGI autocorrelation and hence groundwater drought
characteristics are not trivial and appear to reflect a number of
fundamental catchment properties and processes that effect groundwater level
dynamics and hence groundwater drought phenomena.</p>

      <?xmltex \floatpos{t}?><fig id="Ch1.F12"><caption><p>SGI autocorrelation (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mtext>max</mml:mtext></mml:msub></mml:mrow></mml:math></inline-formula>) as a function of unsaturated zone thickness.</p></caption>
        <?xmltex \igopts{width=170.716535pt}?><graphic xlink:href="https://hess.copernicus.org/articles/19/4327/2015/hess-19-4327-2015-f12.png"/>

      </fig>

      <p>Although clustering of groundwater hydrographs is not novel in itself
(Winter, 2000; Moon et al., 2004; Upton and Jackson, 2011), this is the first
time these techniques have been systematically applied to investigate
groundwater droughts. The approach described is generic and widely
applicable, and here we briefly highlight some of the methodological
considerations, and implications for monitoring and prediction of
groundwater droughts. The <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering has been performed on the
complete SGI hydrographs, including periods of relatively high groundwater
level stand, even though the aim of the hydrograph classification has been
to investigate regional variations in groundwater droughts. Yet the
resulting clusters have been shown to effectively identify distinct regional
groundwater drought responses across the study area. For example, they
reflect the major drought history across the study region (Figs. 2 and 7)
and identify spatially coherent hydrographs that are consistent with
know hydrogeological differences across the study area (Figs. 3c and 9a).
Eltahir and Yeh (1999) investigated the asymmetry of groundwater
hydrographs to high and low groundwater level stands and noted that
“droughts leave a significantly more persistent signature on groundwater
hydrology than floods”. They inferred that this phenomenon was because
discharge of groundwater to streams is an efficient dissipation mechanism
for wet anomalies and that this discharge is often strongly non-linear. This
may explain, at least in part, why the hydrograph classification scheme
based on full hydrographs provides such a good basis for analysis of the
heterogeneous response of groundwater to drought at the regional scale.
However, there is potential for future work to investigate if the hydrograph
classification can be improved by focussing on, or giving more weight to,
episodes of drought in the SGI time series.</p>
      <p>In addition to identifying three clusters of SGI hydrographs – CL1, CL2 and
CL4 – that exhibit different characteristic responses to meteorological
drivers, the <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering also identified three relatively small
clusters of SGI hydrographs – CL3, CL5 and CL6 – where there were
trends in the SGI time series; temporal anomalies expressed as anomalous
phase relationships between cluster SGI and the regional SGI time series; or
relatively poor coherency in SGI time series with a given cluster. In these
three clusters it has been inferred that hydrographs may have been variably
impacted by anthropogenic factors, such as groundwater abstraction. Although
the CA was not specifically designed to identify anthropogenically impacted
groundwater hydrographs, the classification scheme could be used to that end
since it can differentiate between clusters showing trends superimposed on
the regional signals (e.g. CL3 and CL5) and clusters with anomalous phase
relationships with the regional signal (e.g. CL6). The presence of a trend
in a cluster of hydrographs may be indicative of an anthropogenic impact,
for example from unsustainable abstraction (declining trend) or from
groundwater rebound (rising trend). Where there is limited prior information
regarding groundwater withdrawals across a region, a not uncommon situation
in areas where abstraction is not highly regulated, cluster analysis could
be used, either as it has been in the present study based on a set of
heuristic rules to identify a suitable number of clusters or in an
exploratory manner. If it is used in a more exploratory manner, either
hierarchical or non-hierarchical clustering could be undertaken and then
clusters searched to identify spatially coherent clusters that show
significant downward trends in hydrographs (where significance of trends in
a cluster could be tested and quantified using standard tests, such as
Mann–Kendall and Sen's slope estimates). Any spatial coherence in clusters
exhibiting downward trends may be taken as indicating the presence of
potentially unsustainable abstraction. For the purposes of a study where the
stationarity of the data is important, if trends in individual hydrographs
are already known then either these hydrographs can be removed from an
analysis or the trends could be identified and removed prior to standardisation
and clustering of the hydrographs.</p>
      <p>It has been shown that there can be pronounced differences in the
characteristics of multi-annual drought episodes between aquifers within a
region (Fig. 9a). During multi-annual droughts some clusters temporarily go
out of drought conditions while others will continually show deepening
drought conditions over 2 or more years, and some clusters stay in
groundwater drought for many months after groundwater (and meteorological)
drought has ceased in other clusters. If observations such as these or
similar ones can be made for a region, they may have important implications for
monitoring groundwater droughts and water resource management in
multi-aquifer (cluster) systems. For example, at the end of a drought, sites
in more quickly responding clusters may act as leading indicators of the end
of groundwater drought at sites in more slowly responding clusters. In
addition to the implications for groundwater monitoring particularly during
long droughts, if there is sufficient understanding of regional variations
in groundwater responses (i.e. relative differences in the timing and
intensity of groundwater drought between different aquifers in a region or
between sub-regions within an aquifer), then this understanding could be
used to inform appropriate groundwater water resource management strategies
and so may enable some of the worst impacts of the groundwater drought to be mitigated.</p>
      <p>More generally we see a range of possible benefits to clustering groundwater
hydrographs. For example, “sentinel” boreholes within each cluster, those
that are closest to the mean behaviour of a group, could be identified and
used as indicative of the groundwater response of a wider area. Missing data
is a common issue with groundwater hydrographs, and clustering techniques
could potentially be used to identify suitable boreholes from which
groundwater levels could be infilled. However, more importantly, clustering
could be used in combination with groundwater models to aid the prediction
of groundwater droughts. A range of techniques can be used to model
groundwater hydrographs at a site, i.e. non-distributed groundwater models,
including statistical models (Ahn, 2000; Bloomfield et al., 2003), artificial
neural network models (Sreekanth et al., 2009) and “black-box” models (Mackay
et al., 2014). The hydrograph cluster analysis could be used in combination
with any of these techniques for groundwater drought forecasting. For
example, forecasts of groundwater levels 1 to 3 months out are currently
undertaken in the UK for selected sites using a black-box, lumped-parameter
model (Jackson et al., 2013; Mackay et al., 2014; Hydrological Outlooks, 2015)
driven by probabilistic estimates of future rainfall. Regional inferences of
future groundwater levels are then based on qualitative interpretations of
the individual sites. Applying similar modelling systems to mean cluster
hydrographs that are representative of spatially coherent regions of
groundwater drought response instead of individual site-specific hydrographs
could enable more rigorous forecasts of the spatial distribution of
groundwater drought.</p>
</sec>
<sec id="Ch1.S6" sec-type="conclusions">
  <title>Conclusions</title>
      <p>Cluster analysis when applied to SGI time series of consistent length
for multiple sites across a region has been shown to provide a robust
approach to the regional analysis of groundwater droughts. In the present
study an agglomerative hierarchical complete-linkage strategy and a
<inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering strategy were tested. The <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clustering was found to
be most suitable. However, for any given case study a range of
non-hierarchical algorithms and hierarchical classification schemes should
be explored to see which is most appropriate.</p>
      <p>A heuristic, rule-based approach was found useful in guiding the selection
of the optimal number of clusters, where the rules applied prior knowledge
of the hydrogeology of the study area, including information related to
spatial variations in catchment and aquifer characteristics. For the present
case study, both non-hierarchical algorithms and hierarchical classification
schemes provide better clustering of SGI time series than a simple
three-fold classification simply based on geology alone, with the <inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means
clustering providing the best clustering. Membership of the resulting
<inline-formula><mml:math display="inline"><mml:mi>k</mml:mi></mml:math></inline-formula>-means clusters is shown to be dominated by hydrogeological factors, and the
effect of heterogeneity in precipitation over the study area on cluster
composition is inferred to be negligible.</p>
      <p>The clusters successfully discriminate different responses to groundwater
drought, both in terms of drought metrics for the complete time series and
with respect to the detailed response of sites in each cluster during
episodes of major multi-annual drought. Groundwater drought
characteristics can be linked, through the autocorrelation structure of
cluster hydrographs, to the distribution of unsaturated zone thickness. This
reflects the role of a range of catchment and aquifer properties and
processes that influence groundwater level dynamics, including topography,
aquifer thickness and extent of outcrop, unsaturated zone drainage
characteristics and saturated groundwater flow.</p>
      <p>This approach to groundwater hydrograph clustering is flexible, can be
applied in a wide range of hydrogeological settings where suitable
hydrographs are available, and enables spatially variable responses of
groundwater to drought to be quantified.</p>
</sec>

      
      </body>
    <back><ack><title>Acknowledgements</title><p>We would like to thank Henry Holbrook for help in preparation of the
figures. The work described has been funded by the British Geological Survey
(Natural Environment Research Council), and this paper is published with the
permission of the Executive Director of the British Geological Survey
(Natural Environment Research Council). <?xmltex \hack{\newline}?><?xmltex \hack{\newline}?>
Edited by: H. Cloke</p></ack><ref-list>
    <title>References</title>

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