Introduction
Large-scale distributed hydrological and groundwater models are used
extensively for water resources management and research. We use large scale
to refer to models in the scale of 100 to 1000 km2 or
larger. Examples are water resources management in water-scarce regions (Gräbe et al., 2012;
Laronne Ben-Itzhak and Gvirtzman, 2005), groundwater depletion
(Scanlon et al., 2012), contamination (Li and Merchant, 2013;
Mukherjee et al., 2007), agricultural impacts on hydrogeological systems
(Rossman and Zlotnik, 2013), and well-capture zone delineation (Moutsopoulos et al.,
2007; Selle et al., 2013).
Such models are typically distributed, highly parameterized, and depend on
data availability to sufficiently represent the modeled systems. Model
parameterization includes, for example, the saturated and unsaturated zone
hydraulic properties, land use distribution and properties, and stream bed
configuration and properties. Hydrological forcing data such as
precipitation and temperature are also required. Parameters are estimated
through calibration, which requires hydrological observation data commonly
in the form of groundwater hydraulic heads and stream discharges.
Calibration data should be temporally and spatially representative for the
modeled system, and so should validation data sets.
One of the main challenges in modeling large-scale hydrogeological systems
is data scarcity (Refsgaard et al., 2010; Zhou et al., 2014). Uncertainty inherent in distributed
hydrological models is well known (Beven, 1989). Incorrect
system representation due to lack of data contributes to this uncertainty,
but the most important source of uncertainty in distributed groundwater
models is incorrect representation of geological structures (Refsgaard
et al., 2012; Seifert et al., 2012; Zhou et al., 2014). In this paper, we
refer to a 3-D subsurface model that delineates the structure of the
hydraulic conductivity (K) field as a hydrostratigraphic model.
Lithological borehole logs are the fundamental data source for constructing
hydrostratigraphic models. The modeling process is often cognitive, but
two-point geostatistical (He et al., 2013; Strebelle, 2002) and multiple-point statistical
(e.g. Park et al., 2013) methods are also used.
Geostatistical methods have the advantage of uncertainty estimation.
Spatially inconsistent sampling pattern and scarcity make lithological
borehole logs alone insufficient to capture local-scale geological
structures relevant for simulation of groundwater flow and contaminant
transport. Cognitive methods have the advantage of using information from
geological maps to assist interpretation of larger scale geological features.
Airborne electromagnetic (AEM) data are unique with respect to good spatial
coverage and high resolution. AEM is the only technique that can provide
subsurface information with a resolution down to ∼ 25 m in the
horizontal and ∼ 5 m in the vertical at regional scales
(Schamper et al., 2014). Geological
structures and heterogeneity, which spatially scarce borehole lithology data
may overlook, are well resolved in AEM data. Geophysical data and especially
AEM data are commonly used to support lithological borehole information in
geological mapping and modeling (Bosch
et al., 2009; Burschil et al., 2012; Høyer et al., 2011; Jørgensen et
al., 2010; Jørgensen et al., 2013; Steinmetz et al., 2014). Furthermore, multiple-point statistical methods are applied to invert geophysical data, where a
priori geological information is incorporated through training images
(e.g. Caers and Hoffman, 2006; Lange et al., 2012; Lochbuhler et al., 2015).
Although uncertainty of the estimated structures is available from the
inversion, multiple-point statistical methods are applied at scales smaller
than large-scale hydrological models. He et
al. (2014) used transition probabilities (two-point statistics) to
integrate AEM data with borehole lithological data.
Current practice for cognitive hydrostratigraphic and geological model
generation faces a number of challenges: structures that control groundwater
flow may be overlooked in the manual 3-D modeling process; geological models
are subjective, and different geological models may result in very different
hydrological predictions; structural uncertainty inherent in the model
building process cannot be quantified. Currently there is no standardized
way of integrating high-resolution AEM into hydrogeological models.
Sequential, joint and coupled hydrogeophysical inversion methods, as defined
by Ferré et al. (2009), have been developed and used
extensively in hydrological and groundwater research. In sequential
inversion, hydrological and geophysical models and inversions are set up and
performed separately (e.g. Binley
et al., 2001; Kemna et al., 2002). In joint inversion, hydrological and
geophysical models are set up separately but hydrological and geophysical
parameters are estimated simultaneously through a joint objective function
(e.g. Hyndman and Gorelick, 1996; Hyndman et al., 1994; Linde et al., 2006; Vilhelmsen et
al., 2014). In coupled inversion only one model is set up, the hydrological
and the geophysical data are evaluated by comparison to translated simulated
hydrological states (e.g. Hinnell et al., 2010; Kowalsky et al., 2005). The methods have been applied
to capture hydrological processes or estimate aquifer properties and
structures from geophysical data. Hydrogeophysical inversion addresses
hydrogeological property estimation or delineation of hydrogeological
structures. In the context of large-scale groundwater models studies, Dam
and Christensen (2003) and Herckenrath et al. (2013) translated between
hydraulic conductivity and electrical resistivity to estimate hydraulic
conductivity parameters of the subsurface in a joint hydrogeophysical
inversion framework. Petrophysical relationships, however, are uncertain,
partly because of unknown physical relationship between geophysical and
hydrological parameter space. The relationship may vary within and/or
between field sites depending on given conditions and cannot be determined a
priori. For electrical resistivity versus hydraulic conductivity,
relationships suggesting both positive and negative correlation have been
found (Purvance and Andricevic, 2000). Herckenrath et al. (2013) concluded
that sequential hydrogeophysical inversion was preferred over joint
hydrogeophysical inversion due to the uncertainty associated with the
petrophysical relationship. Structural inversions are often performed as
purely geophysical inversions, where subsurface structures (that mimic
geological or hydrogeological features) are favored during inversion by
choosing appropriate regularization terms. An example is the layered and
laterally constrained inversion developed by Auken and
Christiansen (2004), which respects vertically sharp and laterally smooth
boundaries found in sedimentary geology. Joint geophysical inversions have
been used extensively to delineate subsurface hydrogeological structures
under the assumption that multiple geophysical data sets carry information
about the same structural features of the subsurface (Christiansen et al.,
2007; Gallardo, 2003; Haber and Oldenburg, 1997) but examples of successful
joint hydrogeophysical inversion at larger scales are rare.
As a response to lack of global petrophysical relationships, clustering
algorithms as an extension to structural inversion methods have been applied
in geophysics (Bedrosian et al., 2007; Doetsch et al., 2010). Fuzzy c-means and k-means clustering algorithms
have been used with sequential inversion schemes (Paasche et al., 2006; Triantafilis and Buchanan, 2009)
and joint inversion schemes (Di Giuseppe et al., 2014; Paasche and Tronicke, 2007). These studies have
focused on the structural information contained in geophysical information,
and hydrogeological or geological parameters of the subsurface are assumed
uniform within the delineated zones. This approach corresponds well with the
common practice in groundwater modeling where degrees of freedom of the
subsurface are reduced by zoning the subsurface.
We present an objective and semi-automatic method to model large-scale
hydrostratigraphy from geophysical resistivity and lithological data. The
method is a novel sequential hydrogeophysical inversion for integration of
AEM data into the hydrological modeling process. Hydrostratigraphic
structures and parameters are determined sequentially by
geophysical/lithological and hydrological data, respectively.
As shown in Fig. 1, the 3-D subsurface zonation is
completed in two parts: (1) a hydrostratigraphic cluster modeling part, and
(2) a hydrological modeling part. In part 1 the hydrostratigraphic
structures are delineated (see Fig. 2c) through k-means cluster analysis on resistivity data (see
Fig. 2a) and clay-fraction values (see Fig. 2b). To obtain clay-fraction values,
resistivity data are translated into clay-fraction values by inverting for
the parameters of a spatially variable translator function (this is the
petrophysical relationship) (Foged et al., 2014).
The cluster analysis is performed on the principal components of normalized
resistivity data and clay-fraction values. In part 2 the K of each zone in the hydrostratigraphic cluster model is
estimated in a hydrological model calibration using observations of
hydraulic head and stream discharge. The zones identified in the cluster
analysis are assumed to have uniform hydrogeological properties, and thus
form the hydrostratigraphic model.
The method is applied to a Danish case study, for which details and results
are presented in the following sections.
Materials and methods
Study area
The Norsminde study area is located on the eastern coast of Jutland, Denmark,
and covers a land surface area of 154 km2.
Figure 3 shows a map of the area delineating the study
area boundary, streams, and hydrological data. An overview of the
geophysical and lithological data can be found in Foged et al. (2014). Within 5–7 km from the sea,
the land is flat and rises only to 5–10 m a.s.l. (above sea level). Further to
the west, the land ascends into an up-folded end moraine at elevations
between 50 and 100 m a.s.l. The town of Odder with approximately
20 000 inhabitants is located at the edge of the flat terrain in the middle
of the model domain.
Workflow of the two main parts in the method. Top grey box:
hydrostratigraphic cluster modeling using the structural information
carried in the geophysical data and lithological information. Lower box in
bold: hydrological calibration where hydraulic properties of the
hydrostratigraphic zones are estimated using hydrological data.
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Palaeogene, Neogene, and Quaternary deposits characterize the area. The
Palaeogene deposits are thick clays, and define the lower geological
boundary. Neogene marine clays interbedded with alluvial sands overlay the
Palaeogene deposits in the elevated northern and western parts of the model
domain. Quaternary deposits are glacial meltwater sediments and tills found
throughout the domain. The west–east striking Boulstrup tunnel valley (2 km by 14 km)
incises the Palaeogene clay in the south (Jørgensen and Sandersen,
2006). The unconsolidated fill materials are meltwater sand and gravel, clay
tills, and water-laid silt/clay.
Groundwater is abstracted for the drinking water supply, mainly from tunnel
valley deposits and the elevated southwestern part of the domain. The
groundwater resource is abstracted from 66 abstraction wells, with a total
production of 18 000–26 000 m3 yr-1, excluding smaller private wells.
Maximum annual abstraction from one well is 12 400 m3 yr-1. Actual
pumping variation among the 66 wells and inter-annual variation of pumping
rates are unknown. Abstraction is planned locally by water works and only
information about permissible annual rates has been obtained for this study.
Northwest–southeast profiles (vertical exaggeration × 5), location is marked
in Fig. 3. (a) Resistivity model, (b) clay-fraction model, and
(c) hydrostratigraphic cluster model for the 5-cluster case.
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Map of the Norsminde study area. The map shows the location of the three
discharge gauging stations (blue triangles) along the main river, hydraulic
head observations for the calibration period (red dots) and the validation
period (black crosses), and abstraction wells (stars). The black dashed line
delineates the model domain of the hydrological model.
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Groundwater hydraulic heads are available from 132 wells at various depths;
see Fig. 3 for the spatial distribution.
Hydraulic head data are collected from the Danish national geological and
hydrological database Jupiter (GEUS, n.d.).
Average annual precipitation is 840 mm yr-1 for the years 1990–2011. Most of
the area is tile drained. The catchment is drained by a network of
24 streams; the main stream is gauged at the three stations 270035, 270002, and 270003
(see Fig. 3). Streams vary from ditch-like
channels to meter wide streams. Low and high flows, respectively, are on the
order of 0.05–0.5 and 0.5–5 m3 s-1. Daily stream discharge data
are available from three gauging stations. Discharges are calculated from
mean daily water table measurements and translated with QH curves, which are
available from approximately monthly discharge measurements.
Time-domain electromagnetic (EM) data collected through ground and airborne
surveys are available for most of the study area. The AEM survey covers
2000 line kilometers, equivalent to 106 770 1-D models and was carried out with
the SkyTEM101 system (Schamper et al.,
2014). Lithological information is available at approximately 700 boreholes.
The borehole descriptions are from the Danish Jupiter database
(GEUS, n.d.) and the level of detail and quality varies from
detailed lithological description at 1 m intervals to more simple sand, clay,
till descriptions at layer interfaces. A thorough description of EM data
collection and processing and lithological borehole information can be found
in Foged et al. (2014).
Hydrostratigraphic model
Geophysical and lithological data are used to zone the subsurface.
Geophysical data consist of resistivity values determined from the inversion of
airborne and ground-based electromagnetic data. Lithological information is
represented in clay-fraction values determined through inversion within the
clay-fraction concept (CF concept). Zonation is performed in 3-D.
The CF concept is formulated as a least-squares inversion problem to
determine the parameters of a petrophysical relationship (in the inversion
this is the forward model) that translates geophysical resistivities into
clay-fraction values. The concept is described in detail in
Foged et al. (2014) and Christiansen et al. (2014), and
only a brief introduction is given here. The inversion minimizes the
difference between observed clay fraction as determined from borehole
lithological logs (in the inversion this is the data) and translated clay
fraction as determined from geophysical resistivity values (in the inversion
this is the forward data). Clay fraction expresses relative accumulated
thickness of clay material over an interval. In this context clay refers to
material described as clay in lithological logs, and not clay minerals. Clay
definitions include, among others, clay till, marl clay, mica clay, and
silty clay. In the CF inversion, the translator function is a heuristic
two-parameter function defined on a regular 3-D grid that is constrained
vertically and horizontally. Discretization is 1000 m in the horizontal and
4 m in the vertical. The translator function is a scaled inverse error
function (see Eq. (1) and Fig. 4).
W(ρ)=0.5⋅erfcK⋅2ρ-mup-mlowmup-mlow,K=erfc-1(0.05)
mlow and mup are the model parameters of the translator function,
W(ρ), that translates resistivity, ρ, into clay fraction. K
scales the error function so that W(ρ) equal 0.025 and 0.975 for
resistivity values equal to mlow and mup, respectively (see
Fig. 4). The parameters of the translator function
vary throughout the 3-D grid. The objective function, with a data misfit term
and vertical and horizontal regularization term, is minimized iteratively.
The regularization constraint is a measure of weighted-squared difference
between mlow and mup at neighboring grid nodes, where the
weighting is the regularization constraint. The final parameters of the
translator function translate geophysical resistivity values into CF values.
An experimental semi-variogram is estimated from the simulated CF values,
and 2-D block kriging is used to obtain a 3-D CF model. The resolution
difference between lithological borehole data and AEM data is discussed in
Foged et al. (2014).
Delineation of subsurface structures is performed as a k-means cluster
analysis on geophysical resistivities and clay-fraction values. Information
contained in clay-fraction values is to some extent duplicated in the
geophysical resistivity values. Heterogeneity captured in the resistivity
data, however, is simplified in the translation to clay fraction; for
example,
till and Palaeogene clay have, respectively, medium and low resistivity
values,
while the clay fraction for both materials is 1.
k-means clustering is a well-known cluster analysis that finds groups in
multivariate data based on a measure of similarity between cluster members
(Wu, 2012). Similarity is defined as the minimum of squared Euclidean distances
between each cluster member and cluster centroid, summed over all cluster
members. The number of clusters that the data are divided into is
defined by the user. We use the k-means analysis implementation in MATLAB
R2013a, which uses a two-phase search, batch, and sequential, to minimize the
risk of reaching a local minimum.
The translator function is the petrophysical relationship used in the
CF inversion. The parameters mlow and mup are varied to move the
translator function along the resistivity axis.
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Because clay-fraction values are correlated with geophysical resistivities,
k-means clustering is performed on principal components (PCs) of the original
variables. Principal components analysis (PCA) is an orthogonal
transformation based on data variances (Hotelling, 1933). PCA
thus finds uncorrelated linear combinations of original data while obtaining
maximum variance of the linear combinations (Härdle and Simar,
2012). The uncorrelated PCs are a useful representation of the original
variables as input to a k-means cluster analysis. Original variables must be
weighted and scaled prior to PCA, as PCA is scale sensitive, and the lack of
explicit physical meaning of the PCs makes weighting difficult. Clay-fraction values are unchanged as they range between 0 and 1. The normalized
resistivity values are calculated as ρnorm = logρ-logρminlogρmax-logρmin.
Where ρmin and ρmax is minimum and maximum resistivity values, respectively.
Eleven hydrostratigraphic cluster models consisting of 1–11 zones are set up and calibrated.
Hydrological model
Hydrological data are used to parameterize the structures of the
hydrostratigraphic model. Stream discharges and groundwater hydraulic heads
are used as observation data in the hydrological calibration.
The hydrological model is set up using MIKE SHE (Abbott et al., 1986; Graham and Butts,
2005), which is a physically based hydrological model code simulating
evapotranspiration, the unsaturated zone, overland flow, and saturated flow,
while stream discharge is simulated by coupling with the MIKE 11 routing
model code.
Hydrological model parameterization
The model has a horizontal discretization of 100 m × 100 m, and a vertical
discretization of 5 m following topography. The uppermost layer is 10 m
thick for numerical stability, which is not expected to negatively impact
river discharge as this is largely controlled by drainage. Because the model
represents a catchment, all land boundaries are defined as no-flow boundary
conditions following topographical highs. Constant head boundary conditions
are defined for sea boundaries, and the model domain extends 500 m into
the sea. Model grid cells 10 m below the Palaeogene clay surface have
been de-activated, due to the computational burden.
The unsaturated zone and evapotranspiration (ET) are modeled using the
two-layer water balance method developed to represent recharge and ET to/from
the groundwater in shallow aquifer systems (Yan and Smith, 1994). The reference
evapotranspiration is calculated using Makkink's formula
(Makkink, 1957). Soil water characteristics of the five soil
types and the associated 250 m grid product are developed and described by
Borgesen and Schaap (2005) and Greve et al. (2007),
respectively. Land use data are obtained from the DK-model2009, for which
root-depth-dependent vegetation types were developed (Højberg et al., 2010).
Stream discharge is routed using the kinematic wave equation. The stream
network is modified from the DK-model2009 (Højberg et al.,
2010) by adding additional calculation points and cross sections.
Groundwater interaction with streams is simulated using a conductance
parameter between aquifer and stream. Overland flow is simulated using the
Saint-Venant equations (DHI, 2012, 267–281). Manning number
and overland storage depth is 5 m1/3 s-1 and 10 mm, respectively.
Drainage parameters, drain time constant (s-1) and drain depth (m) are
uniform in space and time. Parameterization of spatial variable drain time
constant relies on direct drainage flow measurements, and
Hansen et al. (2013) found little variability in the
estimated time constants and no justification for a spatial variability
judging from eight hydrological performance criteria. Drain depth is 1 m
below terrain.
Saturated flow is modeled as anisotropic Darcy flow, xy to z anisotropy being
restricted to the orientation of the computational model grid
(DHI, 2012). A vertical anisotropy of 1/10 is assumed. The
saturated zone is parameterized with the cluster models. The lower boundary
of the saturated zone is defined by the surface of the Palaeogene clay,
available in 100 m grid, and has a fixed horizontal K of 10-10 m s-1.
Specific yield and specific storage are fixed at 0.15 and 5 × 10-5 m-1 for the entire domain.
Hydrological model calibration
Forward models are run from 1990 to 2003; the years 1990–1994 serve as warm
up period (this was found sufficient to obtain stable conditions); the
calibration period is from 2000 to 2003 and the validation period is from
1995 to 1999.
Composite-scaled sensitivity values of selected parameters in the
hydrological model. Sensitivities are shown for head and discharge
observation separately. The two top plots show average, minimum, and maximum
sensitivity of the 11 hydrostratigraphic cluster models. The two lower plots
show sensitivity of subsurface parameters given a 5-cluster model. Kh is
horizontal hydraulic conductivity and Kv is vertical hydraulic conductivity.
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Composite-scaled sensitivities (Hill and Tiedeman, 2007) were calculated based
on local sensitivity analyses. Figure 5 shows
calculated sensitivity for selected model parameters. Sensitivities of the
parameters, which are shared by the 11 cluster models, are calculated for
each cluster model. The top panel in Fig. 5 shows
sensitivities of the shared parameters. The bars indicate the mean value of
these sensitivities, and the error bars mark the minimum and maximum value
of these sensitivities. The lower panel in Fig. 5
shows subsurface parameters for the 5-cluster model.
The following parameters are a part of the model calibration:
The root-depth scaling factor, which was found sensitive (see Fig. 5,
top panel). Because root-depth values vary inter-annually and between crop types,
root-depth sensitivity was determined by a root-depth scaling factor, which
scales all root-depth values.
The drain time constant. Especially considering discharge observations,
the model shows sensitivity towards this parameter. Stream hydrograph peaks are
controlled by the drainage time constant (Stisen et al., 2011; Vazquez et al., 2008).
The river leakage coefficient.
The horizontal hydraulic conductivities of all zones of the 11 hydrostratigraphic
cluster models. Figure 5 shows sensitivity to K of the zones of the
5-cluster model. K of the zones is unknown; hence all K values have been calibrated.
Vertical K values are tied to horizontal K with an anisotropy factor of 10. Initial
horizontal K values are 10-4, 10-6, or 10-8 m s-1 depending
on the mean clay-fraction value of a zone.
Storage parameters were set to a priori values and not calibrated.
Calibration is performed using the Marquardt–Levenberg local search
optimization implemented in the parameter estimation software, PEST (Doherty, 2005). Observations
are 632 hydraulic heads from 132 well filters and daily stream discharge
time series from three gauging stations (see Fig. 3). Observation variances are estimated, and, in the absence of information,
observation errors were assumed to be uncorrelated. Objective functions for
head and discharge have been scaled to balance contributions to the total
objective function.
The aggregated objective function, Φ, shown in Eq. (2) is the sum of the
scaled objective function for head and discharge. The subjective
weight, ws, was determined through trial and error by starting numerous
calibration runs; ws was chosen to be 0.8.
Φ=ws∑i=1Nhhsim,i-hobs,iσi2+1-ws∑i=1Nqqsim,i-qobs,iσi2
Hydraulic head observation errors are determined according to the guidelines
following Henriksen et al. (2003). They suggest an error
budget approach that accounts for contributions from (1) the measurement
(e.g. with dip meter), (2) inaccuracy in vertical referencing of wells,
(3) interpolation between computational nodes to observation well location, and
(4) heterogeneity that is not represented in the lumped computational grid.
The total error expresses the expected uncertainty between observation and
corresponding simulation. The approach for estimating these uncertainties
can be found in Appendix A. Total errors amount to 0.95, 1.4, and 2.2 m.
Uncertainty of stream discharges is mainly due to translation from water
stages to discharge (daily mean discharges). Uncertainties originate from
infrequent calibration of rating curve, ice forming on streams, and
especially stream bank vegetation (Raaschou, 1991). Errors can
be as large as 50 %. Blicher (1991) estimated errors of
5 and 10 % on the water stage measurement and rating curve,
respectively. In cases of very low streamflows (1 L s-1),
Christensen et al. (1998) assigned a
standard deviation of 200 % while flow of 50 and 5–10 L s-1 are assigned
standard deviations of 5 and 25 %, respectively. We have assigned an
error of 20 % to all stream discharge observations.
Weighted RMSE of hydrological performance of hydrostratigraphic models
consisting of 1 to 11 clusters. Data are shown for all calibration
observations. Blue lines are mean standard deviation on log(K) values.
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Results and discussion
First, we show results for the hydrological performance of
11 hydrostratigraphic cluster models consisting of 1–11 zones. Second, details
of the cluster analysis for the case of a 5-cluster hydrostratigraphy are
shown. Finally, the cluster model hydrological performance is benchmarked
with comparable hydrological models.
Calibration and validation of hydrological model
Figure 6 shows the weighted root mean square error (RMSE) of model
performances for a hydrostratigraphic cluster model consisting of 1 to 11 zones,
head and discharge, respectively, is shown in Fig. 6a and b.
The 1-cluster model is a homogeneous representation of the subsurface
resulting in a uniform K field. The 1-cluster model represents a situation
where we have no information about the subsurface. Increasing the number of
clusters to represent the subsurface successively adds more information from
geophysical and lithological data to the calibration problem. The weights
used to calculate weighted RMSE are the same weights as used in Eq. (2).
Head and discharge contribute by approximately two-thirds and one-third of the total
objective function. From the 1-cluster to the 2-cluster model, weighted RMSE
for discharge is reduced by more than a factor 2. No significant improvement
of the fit to discharge data is observed for more than 2 clusters. Fit to
head data improve almost by a factor of 2 from the 1-cluster to the
2-cluster model. Improvement of the fit to head data continues up to the
5-cluster representation of the subsurface. Improvements are a factor of 3
from the 1-cluster to the 5-cluster model. Beyond the 5-cluster model, the
fit to head observations stagnates. The 7-cluster and 9-cluster
hydrostratigraphic models perform worse than the 3-cluster model. The 8-,
10-, and 11-cluster models obtain an equally good or better fit to head
data compared to the 5-cluster model.
2000–2003 calibration and 1995–1999 validation period performance statistics
for the 11 hydrostratigraphic cluster models consisting of 1–11 clusters.
The top row shows RMSE and the bottom row shows ME.
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The blue lines in Fig. 6 illustrate mean standard
deviation on log(K) values of the cluster models based on the
post-calibration standard deviation of log(K) for each K zone. Beyond the 4-
and 5-cluster models, the precision of the estimated K values decrease. The
mean standard deviations on log(K) for the 4- and 5-cluster models are 0.12
and 0.15. The corresponding widths of the 95 % confidence intervals are
between 15 and 90 % of the estimated K value for 3 out of 4 zones and
3 out of 5 zones, respectively. Beyond the 5-cluster model, mean standard
deviations on log(K) are between 0.17 and 0.27, and corresponding width of
the 95 % confidence intervals are largely above 100 % for all but two zones.
With the combined information from weighted RMSE values and standard
deviation on log(K) we are able to address over-parameterization. The
results indicate that we obtain good fit to observations without
over-parameterization with a 3- to 5-cluster hydrostratigraphic model.
In this paper, we have discussed the performance of the cluster models as a
measure of fit to hydraulic head and stream discharge observations.
Hydrological models are typically used to predict transport, groundwater
age, and capture zones, which are sensitive to geological features. It is
likely that the optimal number of clusters is different for these
applications. An analysis, as is presented here for head and discharge, for
predictive application is more difficult because observations are often unavailable.
The hydrostratigraphic models are constructed under the assumption that
subsurface structures governing groundwater flow can be captured by
structural information contained in clay-fraction values (derived from
lithological borehole data) and geophysical resistivity values. If this is
true, an asymptotic improvement of the data fit would be expected for
increasing cluster numbers. However, as shown in
Fig. 6, this is not strictly the case: weighted
RMSE of the 7-cluster and 9-cluster models is higher than weighted RMSE of
the 3-cluster, 6-cluster, and 8-cluster models. The likely
explanation is that the increasing number of clusters not only corresponds to
pure cluster sub-division but also to relocation of cluster interfaces in
the 3-D model space. We expect the difference in hydrological performance to
be due to changes in interface configuration.
It is well known that an unsupervised k-means clustering algorithm does not
result in a unique solution, due to choice of initial (and unknown) cluster
centroids. We have sampled the solution spaces (200 samples) of the eleven
cluster models. Clustering the principal components of geophysical
resistivity data and clay-fraction values into 1 to 5 clusters gives unique
solutions. Clustering the principal components of geophysical resistivity
data and clay-fraction values into 6 to 11 clusters results in three or more
solutions. However, the non-unique solutions have different objective
functions (squared Euclidean distance between points and centroids). In all
cases the cluster model with the lowest objective function was chosen as the
best solution.
Histograms of (a) logarithmic geophysical resistivity values and (b) clay-fraction values. Cluster memberships of the values are identified by shades
of grey and the histograms thus show how resistivity values and clay-fraction values are represented in the clusters. The histograms are shown as
percentage of total number of data values.
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Figure 7 shows RMSE and mean errors for calibration
and validation periods for all eleven cluster models. Data used to calculate the
statistics are a temporally split sample from 35 wells, which have
observations both in the calibration and validation period, and the
discharge is for stations 270002 and 270003.
The cluster models perform similarly in the periods 2000–2003 and 1995–1999. With
respect to RMSE, Fig. 7a, for head the validation
period is approximately 10 % worse than the calibration period. RMSE for
discharge (Fig. 7b) is lower in the validation,
approximately one-third of the calibration values. Mean errors for head (Fig. 7c) are lower and higher, respectively. The
hydrological models analyzed in this study generally under-simulate the
average discharge.
Data cloud of geophysical resistivity values and clay-fraction values.
Dotted black lines indicate cluster interfaces and cluster are labeled with
numbers. The cloud color represents bin-wise data density (300 bins), which
are shown in logarithmic scale.
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The cluster model
Figure 8 presents histograms of clay-fraction
values and resistivity values and how the values are represented in the five
clusters, which was chosen to be the optimal number. Counts are shown as
percentages of the total number of pixels in the domain. The histograms in
Fig. 8 show that the clay fraction attribute
separates high resistivity/low clay fraction (sandy sediments) from other
high-resistivity portions of the domain, while the resistivity attribute
separates low resistivity/high clay fraction (clayey sediments) from other
high clay-fraction portions. High resistivity/low clay-fraction values are
represented by clusters 1, 3, and 4, and low resistivity/high clay fraction
are represented by clusters 2 and 5 (see Fig. 8a).
Figure 9 shows the data cloud that forms the basis
of the clustering. The data cloud is binned into 300 bins in each dimension
and the color of the cloud shows the bin-wise data density. We see that
cluster boundaries appear as straight lines in the attribute space. Values
with a low resistivity and corresponding high clay fraction, mainly clusters 2
and 5, populate more than half of the domain. Clay is expected to dominate
this part of the domain.
The results of the cluster analysis are presented with respect to
geophysical resistivity and clay-fraction values, while the cluster analysis
is performed on the PC of geophysical resistivity and
clay-fraction values. The first PC explains the information where the two
original variables, log resistivity and clay fraction, are inversely
correlated. This corresponds to the situation where a clay fraction of 1
coincides with a low resistivity value, and vice versa for clay-fraction
values of 0 and high resistivities. This is the information that we expect,
i.e. our understanding of how geophysical resistivities relate to
lithological information as represented by the translator function (Eq. 1)
(defined under the assumption that variation in geophysical resistivities
with respect to lithological information depends on the presence of clay
materials). Thus, the first principal component is the “clay” information in
the geophysical resistivities. The second PC is less straight forward to
interpret. Ideally, the second PC represents the data pairs where the
resistivity response is not dominated or explained by lithological clay
material. This might reflect a situation where a low resistivity value – and
its associated low clay-fraction value – is a result of a sandy material
with a high pore-water electrical conductivity due to elevated dissolved ion
concentrations. The second PC can also be a result of the
CF conceptualization. Clay till, categorized as clay in the CF inversion,
can have electrical resistivities up to 60 Ωm (Jørgensen et al.,
2005; Sandersen et al., 2009), which will yield a high clay fraction
coinciding with a relatively high geophysical resistivity.
Calibration and validation statistics for the temporally split sample
consisting of observations from 35 wells, which have observations both in
the calibration and validation period, and discharge stations 270003 and 270002.
5-cluster model
Weighted
RMSE
ME
RMSE (–)
Calibration
head (m)
1.63
1.99
-0.79
2000–2003
discharge (m3 s-1)
0.338
0.278
-0.0107
Validation
head (m)
1.85
2.24
-0.981
1995–1999
discharge (m3 s-1)
0.524
0.203
-0.0354
Performance statistics of four Danish hydrological models that are
comparable to the Norsminde model. All models are set up using MIKE SHE.
Study
RMSE
ME
Horizontal
Model
Comment
(m)
(m)
discretization
size
5-cluster model
1.99
-0.79
100 m
156 km2
Stisen et al. (2011)
3.9
1.2
500 m
3500 km2
Mean of calibration using seven different calibration setups
Seifert et al. (2012)
3.03–6.34
-1.17–0.605
200 m
465 km2
Min and max of calibration of six different geological models
He et al. (2015)
4.85
–
100 m
101 km2
Mean using borehole-based geology
Madsen (2003)
1.08
0.19
–
440 km2
Balanced Pareto optimum
Observed and simulated stream discharge at stations 270003 (top row panels)
and 270002 (bottom row panels) from the 1995–1999 validation period. To the left stream
discharge hydrographs are shown and to the right scatter plots of observed
vs. simulated values. In the scatter plots the dotted and dashed red lines
mark misfits of 20 and 50 %, respectively.
![]()
Electromagnetic methods are sensitive to the electrical resistivity of the
formation, which is commonly dominated by clay-mineral content, dissolved
ions in the pore water and saturation. Groundwater quality data are available
at numerous sites in the domain. Pore-water electrical conductivity (EC)
values were gathered from the coast and inland following the Boulstrup tunnel
valley. From the coast to 12 km inland values are stable around
50–70 mS m-1 at 28 wells with varying filter depths. Four outliers with EC
ranging between 120 and 250 mS m-1 were identified at various locations and
depths. No trend due to salinity from the coast was identified. In theory,
variations in formation electrical resistivity that are not due to lithological
changes will implicitly be taken into account by spatial variation of the
translator function in the CF inversion. If there is a region in the
modeled domain where the electromagnetic signal, as well as the resulting
resistivity value, is affected by pore-water salinity (low resistivity value
is due to salinity and not clay content) and there is available borehole
information, the parameters of the translator function will adjust to obtain
lower values in order to translate a low resistivity value to a low clay-fraction value.
Benchmarking hydrological performance
Table 1 shows RMSE and mean error (ME) for head and discharge
based on the 5-cluster model. Weighted RMSE for discharge is below 1,
indicating that discharge is over fitted. The standard deviation of
discharge is 20 % of the observation, which is a conservative definition.
As presented in the methods, section errors may vary between 5 and 50 %.
The 1995–1999 hydrograph and scatter plot in Fig. 10 for the 270002 gauging station show good fit to data. Peak and low flows
are fitted, but baseflow recession is generally not matched very well. At
gauging station 270003, the model fails to capture dynamics and relative
magnitudes of the observations. Peak as well as low flows are
under-simulated, which is clearly demonstrated in the scatter plot for
station 270003 in Fig. 10. With respect to head,
the model under-simulates in the elevated parts of the domain (head above
50 m) (see Fig. 11). The head values below 20 m
represent the Boulstrup tunnel valley, where head is fitted the best. With
weighted RMSE for head of 1.63 and 1.85 the model is almost 2 SD (standard
deviations) from fitting head data. Assuming head observation error estimates
are correct, this indicates model deficiencies such as structural errors
and/or forcing data errors.
Scatter plot of observed and simulated heads values from the 1995–1999
validation period. Dashed lines mark misfits larger than 10 m and dotted
lines mark misfits larger than 5 m.
![]()
Distributed head results for the validation period 1995–1999; (a) 5-cluster
model simulated hydraulic head at 27 July 1997 at 0 m a.m.s.l; (b) Errors
(observed–simulated) between observed and simulated head.
![]()
Figure 12a–b show distributed head results.
Generally hydraulic head in the tunnel valley is disconnected from the
elevated terrain (Fig. 12a), and groundwater
overall flows towards the sea. Figure 12b shows
errors (obs–sim) between observed and simulated heads for 1995–1999. The
largest errors are found in the southeastern part of the domain, where
discharge station 270003, with the worst fit, is located (see Fig. 10, top row panels).
We have compared the hydrological performance of the Norsminde model based
on the 5-cluster hydrostratigraphic model with similar Danish hydrological
models. We have chosen Danish models due to comparability with respect to
data density and quality, and hydrostratigraphy. The model performances are
compared based on RMSE and ME of simulated heads; see
Table 2, as these statistics are reported in the
studies. The horizontal discretization of the models is 100, 200, and
500 m, and the models cover between 202 and 3500 km2. We can
see that the 5-cluster model is comparable with the other models.
Advantages and limitations
We have presented a method for automatic generation of hydrostratigraphic
models from AEM and lithological data for groundwater model applications.
Other automatic methods of integrating AEM data into geological models are
geostatistical methods presented by, for example, Gunnink et al. (2012), using
artificial neural networks, or He et al. (2014), using transition probabilities.
The risk of misinterpretation of AEM data, due to effects of saturation,
water quality, depth and material dependent resolution, and vertical
shielding, are higher with an automatic approach compared to a cognitive
approach, as these effects may be identified by a geologist during the
modeling process. AEM data can be integrated into geological models using
cognitive methods, for example, as presented by Jørgensen et al. (2013), who provide an
insightful discussion of the pros and cons of automatic versus cognitive
geological modeling from AEM data.
Geological knowledge, which can be incorporated into cognitive geological models (Royse,
2010; Scharling et al., 2009; Sharpe et al., 2007), cannot be included in
automatically generated models. Geological knowledge may identify
continuity/discontinuity of geological layers, or discriminate between materials
based on stratigraphy or depositional environment. For regional-scale
groundwater flow, characterization of sedimentation patterns and sequences
may not be relevant, but at smaller scales this information is valuable for
transport modeling.
The hydrostratigraphic cluster model presented in this paper does not
represent a lithological model, but has the advantage of incorporating close
to all the structural information contained in the large AEM data sets in a
fast and well-documented way. This is not possible in practice for cognitive
methods due to spatial complexity and the large amount of AEM data. For
hydrological applications hydrostratigraphic model uncertainty, and the
resulting hydrological prediction uncertainty, has great value. We believe
that the cluster model approach presented in this paper can be extended to
address structural uncertainty and its impact on hydrological predictions.
Cognitive geological model uncertainty is difficult to quantify.
The CF model is to some degree influenced by smoothing resulting from the
AEM data inversion and CF inversion, and the finial kriging of CF values to
a regular grid. Smoothing effects causing resistivity transition zones are
inconsistent with our understanding of geological interfaces. In future
studies different geophysical inversion schemes will be compared to evaluate
the effect of smoothing on the final cluster model. This work will partly
evaluate how the smooth transition zones impact hydrological results. We
expect the geological interfaces to lie in the transition zones, but the
exact location is unknown. We will address this problem by generating
several cluster models that identify zonal divides at different locations in
the transition zones. Hereby hydrological uncertainty as a result of the
transition zones may also be assessed.