Introduction
Large-scale geological and groundwater models are used extensively to support
aquifer management. (Here “large-scale” refers to an area of tens to
thousands of square kilometers.) Determining the distribution of hydraulic
properties and the geometry and connectivity of the groundwater system is of
significant importance because these features control the flow paths
(Desbarats and Srivastava, 1991; Fogg et al., 1999; Weissmann and Fogg,
1999). Incorrect reconstruction of the geological structures has thus been
recognized as an important source of uncertainty when a groundwater model is
used to make predictions outside its calibration base (Refsgaard et al.,
2012; Seifert et al., 2012; Zhou et al., 2014). The data traditionally used
for structural mapping include lithological logs from boreholes, hydrological
data, and hydraulic testing results, but these data are often sparse and
unevenly distributed within an investigated domain. In these (very common)
cases, data scarcity becomes a major obstacle for structural mapping in
relation to large-scale groundwater modeling (Refsgaard et al., 2012; Zhou et
al., 2014).
Ground-based and airborne electromagnetic methods have shown great potential
for mapping geological structures (Jørgensen et al., 2003; Thomsen et al.,
2004; Abraham et al., 2012; Oldenborger et al., 2013; He et al., 2014; Munday
et al., 2015). For large-scale mapping, the airborne electromagnetic method
(AEM) is efficient and cost-effective, supplementing traditional data with
dense estimates of electrical resistivity which, in some environments, inform
about the lithology and thereby about the structure (Robinson et al., 2008;
Binley et al., 2015). AEM measurements can be made quickly over large areas,
and the resolution can be as fine as 25 m in the horizontal direction and
5 m in the vertical (Schamper et al., 2014), with a penetration depth of up
to several hundred meters (Siemon et al., 2009).
Various methods to incorporate resistivity estimates (hereafter referred to
as resistivity models) into groundwater model construction have been
reported. Manual and knowledge-driven approaches have been used to combine
geological, hydrological, and geophysical data with expert knowledge
(Jørgensen et al., 2013). However, the manual approach is subjective and
can be very time consuming and expensive to use when resistivity models from
large AEM surveys are to be incorporated into model construction.
Alternatively, more objective and cost-efficient geostatistical modeling
approaches (Carle and Fogg, 1996; Deutsch and Journel, 1998; Strebelle, 2002)
are available for generating models from a combination of borehole
information and AEM-determined resistivity models. For example: He et
al. (2014) used a transition probability indicator simulation approach (Carle
and Fogg, 1996), while Gunnink and Siemon (2015) used sequential indicator
simulation (Deutsch, 2006). Marker et al. (2015) used a deterministic
strategy for the integration of AEM resistivity models into the hydrological
modeling process.
The studies just mentioned all used sequential hydrogeophysical inversion
approaches (SHI, as defined by Ferré et al., 2009). In SHI the
geophysical data are inverted first and independently from the later
inversion of the hydrological data. For large-scale groundwater modeling,
Herckenrath et al. (2013) and Christensen et al. (2016) used both SHI and
joint hydrogeophysical inversion approaches (JHI; as defined by Ferré et
al., 2009). In JHI, the geophysical and hydrological data are inverted
jointly by linking the geophysical and hydrological models directly through
some of their parameters. The linking can, for example, be done by using an
Archie's law inspired petrophysical relationship (Archie, 1942) to translate
between the geophysical and hydrologic parameters.
In general, petrophysical relationships are difficult to establish because
such translation tends to be site-, scale- and facies-specific
(Chen et al., 2001; Hyndman and Tronicke, 2005; Slater,
2007) and uncertain (Mazáč et al., 1985;
Slater, 2007). The studies by
Herckenrath et al. (2013) and
Christensen et al. (2016) used a fixed petrophysical relationship throughout
the model domain. Better results can be obtained by using a spatially
variable relationship, which allows for local translation between hydraulic
conductivity and electrical resistivity, and by including the spatially
dependent petrophysical parameters in the optimization process
(Linde et al., 2006).
There are two other challenges for incorporating resistivity models into
large-scale groundwater modeling: differences in model discretization and
choice of geophysical regularization methodology. Groundwater models are
often discretized in a regular voxel grid, while the traditional resistivity
models are 1-D and placed at the respective sounding location. For airborne
surveys, for example, the resistivity models are normally located along the
flight lines (Christiansen et al., 2006). Such resistivity models therefore
need to be relocated to conform to the grid of the groundwater model. The
relocation will often be a subtle process where information can be lost. To
address this issue, Fiandaca et al. (2015) presented a geophysical modeling
approach referred to as “voxel inversion”, which decouples the geophysical
inversion model space from the geophysical measurement positions. This allows
estimation of a 3-D geophysical model that is discretized on the same voxel
grid as the groundwater model.
Traditionally, geophysical regularization includes horizontal and vertical
smoothing constraints (Constable et al., 1987) or is limited to a few-layer
inversion (Auken and Christiansen, 2004), whereas a groundwater system often
has sharp-layer or body boundaries. It has therefore been recognized,
e.g., by Day-Lewis (2005) and others,
that the regularization used to stabilize the geophysical inversion may not
reflect the actual hydrologic conditions unless it is chosen carefully. If,
for example, smooth regularization is used to estimate resistivity models in
a sharply layered system, it will produce a blurred resistivity distribution
from which one should be careful with inferring the spatial distribution of
hydraulic conductivity to be used in a groundwater model. In this case, it
would be better to use minimum gradient support regularization (Portniaguine
and Zhdanov, 1999; Blaschek et al., 2008; Vignoli et al., 2015) for the
geophysical inversion because the estimated resistivity distribution will
tend to consist of fewer, more sharply defined layer boundaries (vertically
and horizontally). However, it is often ignored that geophysical data can be
inverted using alternative regularization schemes, and to test whether the
alternative geophysical models affect the predictive capability of a
groundwater model.
Conceptual flowchart for the sequential hydrogeophysical inversion.
First step (box 1): geophysical inversion. Second step (box 2): groundwater
model calibration where shape factors of the petrophysical relationship are
estimated using hydrological data. Third step (box 3): the calibrated
groundwater model is used for predictive modeling.
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The main objective of the present study is to present a novel sequential
hydrogeophysical approach whereby a voxel-based 3-D resistivity model is used to
parameterize and calibrate a groundwater model. The model parameterization
methodology allows the calibration to compensate for errors in the
resistivity model. Furthermore, we will demonstrate that it is important for
groundwater flow simulations that the underlying resistivity model is
estimated using regularization constraints that conform well to the
geological environment. Finally, we analyze how groundwater model prediction
accuracy depends on the quality of the geophysical data that were used to
estimate the resistivity model. Section 2 of the paper presents the
methodology. Section 3 describes the synthetic test case used for our
demonstration purposes. Section 4 presents the results, while Sects. 5 and 6
present discussions and conclusions of the work, respectively.
Methodology
Conceptually, we define a translator function that describes the
petrophysical relationship between electrical resistivity and hydraulic
conductivity. The petrophysical relationship can vary horizontally and
vertically, thereby adapting to the local conditions in translation from the
geophysical model space to the hydrological model space. Through inversion,
the 3-D spatially dependent optimal parameters of the petrophysical
relationship are estimated for each layer interval, thereby covering the
entire 3-D model space.
Figure 1 provides a workflow for the method. First, the gathered airborne
electromagnetic (AEM) data from the survey area are inverted with smooth or
sharp horizontal and vertical constraints (Vignoli et al., 2015). This is
done by using a recently developed voxel inversion scheme which decouples the
geophysical model from the position of the acquired data (Fiandaca et al.,
2015). The geophysical model space thus corresponds to the full 3-D
hydrological model grid. Secondly, the geophysical voxel-based resistivity
model is used as input for the sequential hydrological inversion. The
geophysical model parameter (resistivity) is linked to the main investigated
parameter (hydraulic conductivity) through a petrophysical relationship that
has unknown shape factor values. The shape factor values are estimated
through a hydrological inversion which minimizes an objective function
describing the misfit between simulated groundwater model responses and
corresponding observed hydrological data. Finally, the calibrated groundwater
model can be used to make a set of relevant hydrologic predictions. The
various steps of the methodology are explained in more detail in the
following.
Geophysical voxel inversion
In the first step (Fig. 1, box 1), the AEM data undergo constrained
deterministic inversion using recently developed voxel inversion approaches.
This approach allows the geophysical model spaces to be spatially decoupled
from the geophysical measurement positions (Fiandaca et al., 2015). In most
inversion schemes, the forward and inverse formulations use the same model
discretization. In the voxel formulation, the two model discretizations are
decoupled. The voxel model space thus defines the geophysical properties on
the nodes of a regular 3-D grid.
For calculating the forward responses, a “virtual” 1-D model is built at
each sounding position. The “virtual” 1-D model is defined by a number of
layers, and layer thicknesses. The geophysical properties are interpolated
from the voxel model space into the layer centers of the virtual model that
is subsequently used to simulate the forward response for the corresponding
sounding.
The voxel inversion approach thus allows for inversion of AEM data into a
geophysical model defined on a 3-D regular grid, regardless of the sounding
positions. As a result, the geophysical inversion can be conducted using the
same grid as that defined for a 3-D groundwater model, thereby minimizing
scaling issues in the coupling of geophysical and hydrological models.
The general solution to the non-linear geophysical inversion problem can be
found in Auken et al. (2014). To stabilize the inverse problem, either of two
types of regularization methods can be applied. The first regularization
method is commonly referred to as smoothness-constrained inversion (Constable
et al., 1987). The smoothness-constrained inversion tends to reduce contrasts
and the resulting geophysical model may appear blurred. The reason for this
is found in its minimum-structure L2 norm inversion formalism (Constable et
al., 1987; Menke, 2012). Following the notation used by Vignoli et
al. (2015), this can be expressed as
mi-mj2/σi,j2,
where mi and mj are the constrained parameters and σi,j
defines the constraint strength. The penalization of structures is clearly
seen in Eq. (1), where (mi-mj)k2/σi,j2 is
proportional to the square of the value of the variation (mi-mj).
This implies that an increase in model parameter variation will always result
in a penalization in the stabilizer. The smoothness regularization thus
prevents reconstruction of sharp transitions.
The second regularization method is the minimum gradient support
(Portniaguine and Zhdanov, 1999; Blaschek et al., 2008; Vignoli et al.,
2015), which allows for large sharp vertical and horizontal model
transitions. The minimum gradient support regularization seeks to minimize
the spatial variations vertically and laterally by penalizing the vertical
and horizontal model gradients through the stabilizer expressed as (Vignoli
et al., 2015)
mi-mj2/σi,j2mi-mj2/σi,j2+1.
In Eq. (2), σi,j is a
parameter used to control the sharpness of the regularization constraints.
The stabilizer contribution to the objective function is thus one when
mi-mj≫σi,j and
zero when σi,j≫mi-mj. The minimum gradient support functional thus counts the number of
model variations larger than σi,j for the stabilizer
term of the objective function. This formalism allows sharp vertical and
horizontal model transitions, which are penalized excessively by the
smoothness-constrained inversion.
The geophysical voxel inversion is carried out on the logarithm of the
resistivity values (m=logρ), and the constraint
values are expressed in terms of constraint factors CFs, representing the
relative strength of the constraints (Auken et al., 2014). The actual values
of the constraint standard deviations σi,j of Eqs. (1) and (2) are
then computed as σi,j=logCFi,j.
For instance, a constraint factor value of CFi,j=1.9 gives
mi-mj2/σi,j2=1 in Eq. (1) when
ρi/ρj=1.9, i.e., when the resistivity values are 90 %
different (Vignoli et al., 2015).
Hydrological model parameterization
In the second step (Fig. 1, box 2), the 3-D distribution of electrical
resistivity values is linked to the hydrological parameters (i.e., hydraulic
conductivity) through a spatially varying petrophysical relationship. Shape
factors of this relationship are calibrated.
Linking hydraulic conductivity and electrical resistivity is not trivial
because the parameter values and the form of the petrophysical relationship
may vary dramatically between different types of environments. In addition,
there can be fundamental questions about how the effective properties
controlling electrical current flow are related to the effective properties
controlling fluid flow (Slater, 2007). The primary factors controlling this
relationship are porosity, pore water conductivity, tortuosity, grain size,
degree of saturation, amount of clay minerals, etc. (McNeill, 1980). The
simplest petrophysical relationship is the empirical relationship known as
Archie's law (Archie, 1942), which relates porosity, pore water conductivity,
and the degree of saturation to bulk electrical conductivity. However, this
type of relationship does not take the electrical surface conductance of clay
minerals into account. The Waxman and Smits model (Waxman and Smits, 1968)
combined with the dual-water model by Clavier et al. (1984) provides a basis
for establishing empirical relationships for shaly sand and sediments
containing clays (Revil and Cathles, 1999; Revil et al., 2012). For glacial
sedimentary environments, it is reported that clay has low electrical
resistivity and also low hydraulic conductivity, and sand has high electrical
resistivity and high hydraulic conductivity (Mazáč et al., 1985). For
these environments, it is common to use a power law relationship, which is
given some theoretical support by Purvance and
Andricevic (2000). The relationship is expressed as
K=α⋅ρβ,
where K is the hydraulic conductivity (m s-1), ρ is the
electrical resistivity (ohm m), and α and β are two empirical
shape factors. To compute K for each element in the groundwater
model grid, α and β need to be parameterized and estimated. We
suggest making the parameterization by pilot points placed in a regular grid
in each layer of the groundwater model (Certes and De Marsily, 1991; Doherty,
2003). Each pilot point holds a set of α and β parameters, and
kriging is used for spatial interpolation of α and β from the
pilot points to the model grid. This kind of parameterization creates smooth
transitions in the parameter fields and allows for variation in both the
horizontal and vertical directions of the ρ to K translation.
Hydraulic conductivity can thus be calculated by Eq. (3) for every element in
the groundwater model grid.
Hydrological inversion
The model parameters, α and β at the pilot points, are
calibrated by fitting the groundwater model to hydrological data. When the
number of model parameters is large compared to the number of observation
data, the minimization must be stabilized by regularization. The total
objective function to be minimized is therefore a balanced compromise
between a measurement term (Φm) and a regularization
term (Φr). The combined objective function has the form
Φtotal=Φm+μ⋅Φr=∑i=1dnωd,idobs,i-dsim,i2+μ⋅Φr
where Φtotal is the total objective function,
dobs,i and dsim,i are measured and equivalent
simulated data values, ωdi is a data-dependent weight, μ is a weight factor, and ϕr is a Tikhonov regularization
term. Here, ϕr is defined as preferred difference
regularization, where the preferred difference between neighboring parameter
values is set to zero. Φtotal is minimized iteratively, and
the regularization weight factor, μ, is calculated during the iteration
to ensure that Φm, the measurement part of the objective function,
becomes approximately equal to a user-specified target value (Doherty, 2010).
Synthetic example
For illustrative purposes, we use a 3-D synthetic system very similar to that
presented by Christensen et al. (2016). The only difference is that the
active part of the groundwater system only consists of 5 layers, whereas
Christensen et al. (2016) used a 20-layer model.
Groundwater reference system and hydrological data
The groundwater system is intended to mimic a glacial landscape and covers an
area that is 7000 m (N–S) by 5000 m (E–W). The geology of the system was
generated using T-PROGS (Carle, 1999) as having a horizontal discretization
of 25 m × 25 m and a vertical discretization of 10 m. The system
extends 50 m in the vertical direction, where it reaches impermeable clay
with a horizontal surface. The T-PROGS generated geology above the
impermeable clay consists of categorical deposits of sand, silt, and clay.
Within each of the three types of deposits, hydraulic conductivity, recharge,
and porosity were generated as horizontally correlated random fields using
FIELDGEN (Doherty, 2010). All boundaries of the domain were defined as having
no-flow conditions, except the southern boundary where hydraulic head was
defined as constant, h=0 m. The local recharge depends on the type of
sediment at the uppermost layer. Most groundwater discharges through the
southern boundary, but approximately 35 % discharges into a river running
north to south in the middle of the domain (Fig. 2). Groundwater flow was
simulated as confined steady-state flow employing MODFLOW-2000 (Harbaugh et
al., 2000) with the spatial discretization equal to the geological
discretization. Groundwater is pumped at a rate of 0.015 m3 s-1
from a well located at x=2487.5 m and y=1912.5 m and the well screens
the deepest 10 m of the groundwater system. In the following, this system is
called the reference system.
A map of locations of boreholes, a pumping well, pilot points, head
recovery prediction, and location of a geophysical cross section.
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Thirty-five boreholes are found within the domain (Fig. 2). Each borehole
contains a monitoring well that screens the deepest 10 m of sand registered
in the borehole. For each system realization, the hydraulic head in the 35
wells and the river discharge at the southern boundary were extracted from a
forward simulation made by MODFLOW-2000. The 35 simulated hydraulic head
values were contaminated by independent Gaussian error with zero mean and
0.1 m standard deviation. The river discharge was corrupted with independent
Gaussian error with zero mean and a standard deviation corresponding to
10 % of the true river discharge. The 36 contaminated values constitute
the hydrological data used for groundwater model calibration.
Geophysical reference system and data
The geophysical reference system was designed so that there is perfect
correlation between hydraulic conductivity and electrical resistivity. This
implies that a relationship between hydraulic conductivity and measured
electrical resistivity is likely to exist. The true relationship is of the
same form as Eq. (3), and it uses constant shape factor values α=1e-12 and β=4. This corresponds to conditions where clay has low
electrical resistivity and also low hydraulic conductivity, and sand has high
electrical resistivity and high hydraulic conductivity. The impermeable clay
at the base of the reference system was assigned a constant value of
5 ohm m.
The AEM data were simulated using AarhusInv (Auken et al., 2014) for a system
setup similar to a typical dual-moment SkyTEM-304 system (Sørensen and
Auken, 2004). The simulated survey consists of 35 E–W flight lines with
200 m spacing between the flight lines. AEM system responses were simulated
for every 25 m along the flight lines, giving a total of 6300 sounding
locations for both the transmitted high and low moments. AarhusInv is a 1-D
modeling code. To mimic the loss of resolution with layer depth we simulated
the responses using the 2-D logarithmic average resistivity of all model
cells inside the radius of the footprint at a given depth. To obtain the
geophysical data set, the simulated data were contaminated with noise
according to the noise model suggested by Auken et al. (2008):
Vresp=V⋅1+G0,1⋅SDuni2+VnoiseV21/2,
where Vresp is the perturbed synthetic data, V is the synthetic
noiseless data, G0,1 is standard Gaussian noise (with zero
mean and unit standard deviation), and SDuni2 is uniform noise
variance. Vnoise is the background noise contribution given by
Vnoise=b⋅t10-3-1/2,
where t is the gate center time in seconds, and b is the background noise
level at 1 ms. For the following analysis we generated geophysical data sets
with four levels of background noise, i.e., b equal to 1, 3, 5, and
10 nV m-2, respectively. The uniform standard deviation, which
accounts for instrument and other non-specified noise contributions, was set
to 3 % for dB/dt responses. After the data were perturbed with noise,
they were processed as a field data set (Auken et
al., 2009), resulting in an uneven number of gates per sounding. Figure 3
illustrates the resulting low and high-moment AEM sounding data,
respectively, for the different background noise levels.
AEM sounding data corrupted by four levels of background noise. The
value on top of each subplot corresponds to the noise level at 1 ms and to
the b value in Eq. 6. The black dashed curves indicate the
background noise levels, low and high-moment earth responses are illustrated
as red and blue error bars, respectively, and the black error bars illustrate
data which are removed by the data processing.
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Geophysical voxel inversion
The geophysical data were inverted by voxel inversion
(Fiandaca et al., 2015) using AarhusInv
(Auken et al., 2014). The voxel inversion was conducted in
two different ways: by using L2-norm “smooth” constraints, or by using
minimum gradient support “sharp” constraints (both implemented in
AarhusInv; Auken et al., 2014).
To avoid the influence of numerical discretization errors, the geophysical
voxel inversion uses the same spatial discretization as the reference system
and the groundwater model. For both smooth and sharp inversions, a 40 ohm m
uniform half-space was used as the starting model and spatial regularization
was applied using the same settings throughout all inversions. Considering
the small number of layers and the shallow discretization, it was unnecessary
to apply vertical constraints for any of the inversions. By contrast, depth-
and direction-dependent horizontal constraint factors were used for both
smooth and sharp inversions. The strength given to the horizontal constraints
is based on experience, keeping in mind that the constraint factors should
not prevent data fitting, but promote model consistency. Therefore, a few
experiments were made to “manually” tune the magnitude of the constraint
factors. Different values along the flight lines and perpendicular to them,
respectively, were found to give better results. This is a result of having
higher data density along the flight lines compared to the perpendicular
direction. In these synthetic tests (similar to what is done with field data
with analogous data density) the smooth regularization constraint factors of
CF =1.9 along the flight lines and CF =1.05 perpendicular to the flight
lines were used for the first layer.
In contrast to the conventional inversion of geophysical data, where the
vertical discretization of the geophysical model is normally characterized by
logarithmically increasing layer thicknesses, in this study fixed layer
thicknesses were used in the geophysical models. To account for the loss of
resolution with depth without increasing the layer thicknesses, the
horizontal constraint factors were set to decrease linearly with depth
(tighter bands for the deeper layers), resulting in constraint factors of 1.4
along the flight lines and 1.02 perpendicular to the flight lines for the
sixth layer.
The same directional and depth-dependent tuning used for smooth
regularization was also applied to the sharp inversion. In this case
constraint factors of CF =1.0625 along the flight lines and 1.01
perpendicular to the flight lines were used for the first layer, while
factors of CF =1.025 along the flight lines and CF =1.01 perpendicular
to the flight lines were used for the sixth layer. The smaller values of the
constraint factors in the sharp inversion are due to the different role that
the factors play in the regularization definition, as is evident when
comparing Eqs. (1) and (2). The difference in constraint values between
smooth and sharp inversion is analogous to what has been used in other
studies (e.g., Vignoli et al., 2015). All the constraint values used in this
study represent typical values working also in other applications, both for
synthetic and filed data.
Groundwater model parameterization and calibration
In the following, the groundwater model will be parameterized in two
different ways. Both approaches treat the shape factors between hydraulic
conductivity and resistivity, α and β, in a relationship
(Eq. 3), as spatially dependent parameters
to be estimated. The two parameterizations differ by the resistivity model
that is used to calculate the hydraulic conductivity field of the groundwater
model.
The first type of parameterization uses a resistivity model estimated by
smooth voxel inversion of AEM data collected with a background noise level
of 3 nV m-2. These models will be referred to as SHI-smooth-3.
The second type of parameterization uses a resistivity model estimated by
sharp voxel inversion of AEM data collected with a background noise level of
either 1, 3, 5, or 10 nV m-2. These models will be referred to as
SHI-sharp-1, SHI-sharp-3, SHI-sharp-5, and SHI-sharp-10, respectively.
The shape factors, α and β, of the petrophysical relationship
are parameterized by placing pilot points in a uniform grid, with five nodes
in the x direction and seven in the y direction. Hence, in total the
groundwater model is parameterized by
5 × 7 × 5 = 175 petrophysical relationships, each
having two parameters (the shape factors).
The parameter values are estimated by fitting the available hydrological data
consisting of the 35 observations of the hydraulic head and one river
discharge observation. Calibration is done by minimization of the total
objective function given by Eq. (4), where the measurement objective function
is computed as
Φm=nh-1∑i=1hnωhhobs,i-hsim,i2+nr-1∑i=1nrωrrobs,i-rsim,i2,
where nh and nr are the number of head and river
measurements, respectively; hobs and hsim are
observed and corresponding simulated hydraulic heads; robs and
rsim are observed and corresponding simulated river discharge;
and ωh and ωr are subjectively chosen
weights for head and discharge data, respectively. If a model is expected not
to have structural defects, then it would be ideal to choose the weights
ωh=σh-1 and ωr=σr-1, where σh and σr are the
standard deviations of measurement error for head and river measurements,
respectively. However, in this case (as in all real cases) the model has
structural errors that make the misfit between hydraulic head data and
equivalent simulated values much larger than what can be explained by
measurement error. In accordance with common groundwater modeling practice
(e.g., Christensen et al., 1998), we therefore conducted residual analysis
and a few experiments to estimate the magnitude of the total head error
(which is the sum of observation error and structural error). This indicated
that the standard deviation for the total error on the hydraulic head is
approximately 10⋅σh, while the total error for the
river discharge is totally dominated by measurement error. As weights we
therefore used ωh=10⋅σh-2=1.0 and ωr=σr-2=1.38⋅105, respectively. Using these weights, and
averaging over the 20 system realizations, gave a minimized objective
function value of ϕ‾m=2.5. This is close to the value of
2.0, which would be expected from (Eq. 7) if the weighting used reflects the
error magnitudes.
Calibration was performed using BeoPEST, a version of PEST
(Doherty, 2010) that allows the inversion to run in parallel using
multiple cores and computers.
It should be noted that for calibration and model prediction we applied the
recharge field and boundary conditions of the reference system.
Model predictions
In step 3 (Fig. 1, box), the calibrated groundwater model is used to make
predictions.
In the following synthetic demonstration study, the calibrated SHI-smooth and
SHI-sharp groundwater models are evaluated by comparing their simulated model
predictions with corresponding predictions simulated for the (synthetic and,
therefore, known) reference system. The former are called “model
predictions”; the latter are called “reference predictions”.
Prediction types 1 and 2 relate to steady-state flow when groundwater is
pumped from the well. This is also the condition for which the hydrologic
data used for calibration were sampled. Type 1 is the average age of the
groundwater pumped from the well. Type 2 is the size of the recharge area of
the pumping well. Both of these predictions differ in type from the
calibration data. For these model predictions, we used a homogeneous
porosity of 0.2 (the average value of the reference system porosity fields
is 0.184).
Prediction types 3 and 4 relate to a new stress situation long after pumping
from the well has ceased: type 3 is groundwater discharge into the stream,
and type 4 is head recovery for a well screening a layer northeast of the
pumping well (the location is shown in Fig. 2).
Reference and model prediction types 3 and 4 were simulated by MODFLOW-2000
(Harbaugh et al., 2000), while types 1 and 2 were simulated by forward
particle tracking using MODPATH version 5 (Pollock, 1994) and MODFLOW-2000
results.
The first two types of prediction are interesting from the perspectives of
protection and resource management of a well field, while the latter two are
relevant in the case of possible change in management practice resulting in a
new stress.
Evaluation of prediction performance
As said at the beginning of Sect. 2, steps 1–3 of the framework can be
repeated for a number of system realizations to provide consistent
statistical interference regarding the model prediction results. Here, 20
different reference system realizations were used. For each prediction, we
therefore have 20 corresponding sets of reference predictions and model
predictions that can be used to evaluate the performance of a calibrated
model with respect to that prediction. The performance is evaluated for the
SHI-smooth and SHI-sharp models, respectively, and it is done in the
following ways.
Prediction error characteristics are quantified by the mean absolute error
(MAE), the mean error (ME) following
MAE=1N∑i=1Nxi-ti,
ME=1N∑i=1Nxi-ti,
where xi is the model prediction of realization i, ti is
the reference prediction of realization i, and N=20 is the number
of system realizations. MAE measures how close the model
prediction tends to be to the reference prediction; ME measures the
tendency of positive or negative bias in the model prediction.
The figure shows an east–west cross section of resistivity for the
reference system (realization number 20), and inversion results for smooth
and sharp inversion, respectively. The last row shows a histogram of
resistivity for each layer. The black curve is the resistivity distribution
for the reference system, the red curve shows the resistivity distribution
for the smooth inversion, and finally the green curve shows the resistivity
distribution for the sharp inversion.
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Scatter plot of true and estimated electrical resistivity fields
for smooth geophysical inversion and sharp geophysical inversion for
different data qualities of the AEM data for model realization number 20. On
top of each window a Pearson correlation coefficient (PCC) is calculated.
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Results
Geophysical results
Figure 4 shows a representative cross section for 1 of the 20 system
realizations. Both geophysical models in Fig. 4 were inverted using data
perturbed with a background noise level of 3 nV m-2. Comparing the
geophysical model results with the reference model, we find that SHI-smooth-3
resolves the main features reasonably well for the upper layers. The main
discrepancy is found in the fifth layer, where the sand bodies are not
resolved. In general, the resistivity of the sand bodies (dark orange in the
reference system) is underestimated, and the transitions between the
categorical deposits are artificially smooth.
Figure 4 shows that SHI-sharp-3 resolves the sand
body in layer 5 much better than SHI-smooth-3. Moreover, the locations and
boundaries of the geological deposits tend to be less smeared out when using
the sharp constraints. Inspection of the histograms at the bottom of
Fig. 4 shows that the SHI-sharp-3 model tends to
produce resistivity distributions that are more similar to the reference
distributions than the SHI-smooth-3 model. This improvement could allow for
easier translation from electrical resistivity into hydraulic conductivity
and correspondingly more faithful representation of hydrogeologic structure
and connectivity.
Figure 5 shows voxel-by-voxel density plots of reference versus estimated
electrical resistivity for a SHI-smooth model and corresponding SHI-sharp
models. Pearson's correlation coefficient (PCC; Cooley and Naff, 1990) is
shown on top of the density plot for each layer. A comparison of the density
plots and the PCC values of the SHI-smooth-3 and SHI-sharp-3 models shows
that using sharp instead of smooth constraints improves the inverted
geophysical model. The improvement is seen most clearly for the sand
deposits.
For both the SHI-smooth and SHI-sharp models there is a strong correlation
between the electrical resistivity estimates and the true electrical
resistivities of the first layer, but the SHI-smooth model has weaker
correlation than the SHI-sharp models. For both types of models, the
correlation weakens with depth and background noise. The former is caused by
the resolution limitations of AEM data. However, the depth and resistivity of
the low-resistivity clay at the base of the model are well resolved by both
the SHI-smooth and SHI-sharp model inversions (results not shown).
Hydrological calibration results
The calibration results for the 20 different system realizations are shown in
Fig. 6. The figure shows that the measurement objective function value,
Φm, for most system realizations is close to 2.0. This is the case
for almost all of the SHI-sharp model realizations, even for large background
noise levels. For many of the realizations, the SHI-smooth model also fits
the data well; but, several realizations lead to higher misfit than desired.
This makes EΦm equal to 5.8 for the SHI-smooth-3
models, while it is 2.5 for the SHI-sharp-3 models. That is, the estimated
hydraulic conductivity field tends to be better for sharp models than for
smooth models.
Measurement objective function value obtained for the various
groundwater model calibration cases, while EΦm is the mean value across all 20 different system realizations. The
dashed line indicates the expected target value for the model calibrations.
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Parameter estimation
Figure 7 shows a cross section of the estimated K-, α- and β-fields for one of the system realizations. The two
columns show estimates for the SHI-smooth-3 and SHI-sharp-3 models. Figure 8
shows a density plot of the reference hydraulic conductivity distribution and
the estimated hydraulic conductivity distributions. The results in Figs. 7
and 8 are typical for all 20 system realizations.
East–west cross section for model realization number
20. (a) shows the parameter fields for the SHI-smooth-3
calibrated model. (b) shows the parameter fields for the SHI-sharp-3
calibrated model. The first row shows the reference K-field, the second row
shows the estimated K-field, and the third and fourth rows show shape
factors of the petrophysical relationship for α and β, respectively.
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From Fig. 7a and b it is seen that the estimated α and β parameter values
change smoothly in the horizontal direction but have sharp transitions in
the vertical direction. The second row of Fig. 7
shows the corresponding estimated K fields whose main features are
determined by the underlying resistivity models
(Fig. 4), but they are “corrected” during model
calibration to make the groundwater model fit the hydrological data.
For the SHI-smooth-3 model, α and β take compensatory roles,
particularly in the first layer. Here, the estimated α and β
values are higher than the shape factors of the true relationship that was
used to construct the geophysical reference system. This increases the
hydraulic conductivity in layer 1 to compensate for the too low hydraulic
conductivity (and resistivity, Fig. 4) in layer 2 and deeper layers. The
estimated α and β values are not sufficient to compensate for
the missing deep high-resistivity body in layer 5 of the SHI-smooth-3 model
(Fig. 4).
For the SHI-sharp-3 model, the estimated α and β parameter
values only vary slightly from the shape factor values of the true
relationship, except for layer 5 (Fig. 7b). This indicates that for the
shallower layers the sharp inversion of AEM data sufficiently resolves the
resistivity of features that are important for groundwater model calibration.
In layer 5 the estimate of shape factor β turns out to be fairly high
to compensate for the too low resistivity estimates in this layer (Fig. 4).
Scatter plot of true and estimated hydraulic conductivity fields for
smooth geophysical inversion and sharp geophysical inversion for different
data qualities of the AEM data for model realization number 20. On top of
each window the Pearson correlation coefficient (PCC) is calculated.
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Figure 8 shows voxel-by-voxel density plots of reference versus estimated
hydraulic conductivity for the SHI-smooth and SHI-sharp models. The results
confirm that the K field tends to be overestimated for the first
layer, in particular for the SHI-smooth-3 model. From the second layer and
deeper, the hydraulic conductivity values tend to be underestimated for sand
but overestimated for silt and clay. Moreover, the distributions of estimated
K smear out with depth. Judged by PCC values and visual inspection
of Fig. 8 (highlighting the connectivity of the K field), the hydraulic
conductivity field estimated for any SHI-sharp model is in better agreement
with the reference field than the field estimated by the SHI-smooth-3 model.
Model structural accuracy comparison for the groundwater model using
both smooth or sharp geophysical models and different background noise
levels. The results are averaged over the 20 system realizations. A value of
1.0 means that the model's hydraulic conductivity field is in good agreement
with the reference field; a value of 0.0 means no agreement (see the body
text for an exact definition of “structural accuracy”).
Layer 1
Layer 2
Layer 3
Layer 4
Layer 5
SHI-3 smooth
0.89
0.79
0.56
0.54
0.64
SHI-1 sharp
0.96
0.91
0.81
0.61
0.48
SHI-3 sharp
0.96
0.92
0.82
0.64
0.5
SHI-5 sharp
0.96
0.91
0.78
0.64
0.49
SHI-10 sharp
0.96
0.90
0.78
0.6
0.46
Model structural accuracy is quantified in Table 1 for both the SHI-smooth
and SHI-sharp models. Structural accuracy is calculated here as the fraction
of the total number of voxels for which the estimated log10-hydraulic
conductivity plus/minus 20 % contains the true log10-hydraulic
conductivity value of the reference model. The results are averaged over the
20 system realizations. From Table 1 it is seen that all SHI-sharp models
outperform the accuracy of the SHI-smooth models, except for layer 5. The
exception occurs because the SHI-smooth models are fairly good at estimating
the K distributions for silt and clays, but underestimate K
for sand (Fig. 8). In contrast, SHI-sharp models overestimate the K
distributions for silt and clays, but only slightly underestimate K
for sand (Fig. 8). Therefore, for layer 5, the model structural accuracy
appears to be better for SHI-smooth than for SHI-sharp models.
Scatter plots of calibrated model prediction versus the reference
model prediction using results from the 20 system realizations. The plots in
the first and second columns are the average groundwater age and recharge
area, respectively, of the pumping well. Column three is for head recovery
when pumping has stopped in the observation well shown in Fig. 10, and column
four is for groundwater discharge to the river after pumping has ceased; ME
and MAE are used to quantify the prediction error on the basis of the 20
realizations.
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Prediction results
For each of the 20 system realizations, the calibrated groundwater models
were used to make the model predictions described in Sect. 3.5. Figure 9
shows scatter plots of the reference prediction versus the calibrated model
prediction; each plotted point corresponds to a particular system realization
and the corresponding SHI-smooth-3 or SHI-sharp-3 model. The mean error (ME)
and mean absolute error (MAE) of the prediction are also given in Fig. 9.
Figure 10 shows a MAE contour map for head recovery predictions.
MAE contour map for head recovery prediction. (a) For
predictions using the SHI-smooth models. (b) For predictions using
the SHI-smooth models. (c) Difference between the maps shown
in (a) and (b). Red dots mark the location of the
observation well for the head recovery prediction shown in Fig. 9. The red
cross marks the location of the pumping well.
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Particle tracking predictions
The first column of Fig. 9 shows results for prediction of the average age of
the groundwater pumped from the pumping well. The scatter plot illustrates
that SHI-sharp models tend to overpredict average age. This is seen by the
majority of points plotting above the identity line as well as by the value
of ME = 32 (Fig. 9). The age prediction results are similar for the
SHI-smooth models, although the spread of points is larger than for
SHI-sharp-3 (e.g., quantified by the larger value of MAE). There are two
major explanations for these relatively “poor” predictive performances.
First, the calibrated K-fields underestimate the hydraulic conductivity of
sand deposits in the deeper layers (Fig. 8), which results in too slow
particle travel times at depth. Secondly, the reconstruction of the deepest
layers is too smooth for both the SHI-smooth and SHI-sharp models (Fig. 7)
and does not resolve the small-scale variability that controls the transport
of particles.
The second column of Fig. 9 reports results related to prediction of the
recharge area of the pumping well. The scatter plot shows that the SHI-smooth
models underpredict the recharge area. This happens because the smooth models
lead to estimation of hydraulic conductivities in the deepest layers that are
too low. This creates a deep cone of depression around the pumping well that
extends upward locally to reach the river bed. This induces a local discharge
of water from the stream through the groundwater system to the pumping well.
These models thus predict that a significant proportion of the pumping will
come from local discharge from the river. (This is compensated for by
increased model predicted groundwater discharge to other parts of the river.)
For the true, reference system used to generate the data, the river does not
lose water, and all water pumped from the well originates from groundwater
recharge.
The SHI-sharp models are better predictors of the recharge area, but these
models also tend to predict an area that is too small. These models also
predict local discharge from the river to the groundwater system, but to a
lesser degree than the SHI-smooth models. This is likely because the main
features of the reference system are better reconstructed by the SHI-sharp-3
models.
Head recovery and discharge predictions
The prediction of head recovery at an observation well (location shown in
Fig. 10) is done poorly by SHI-smooth-3 (Fig. 9). The predicted head recovery
is very small for most of these models because they tend to have too little
hydraulic connectivity between the deepest layers, the estimated hydraulic
conductivities are too low in the deep sand layers, and the simulated cone of
depression is therefore too deep and too local.
The SHI-sharp-3 models make less biased, fairly reasonable predictions of
the head recovery (Fig. 9) because they resolve
the variations of hydraulic conductivity at depth better than the
SHI-smooth-3 models. The superiority of SHI-sharp-3 models for recovery
prediction is also seen from the MAE contour maps in
Fig. 10. The MAE is seen to be spatially
dependent: it is largest at the pumping well, and smallest at the constant
head boundary to the south
The fourth column of Fig. 9 shows that both types
of models are good predictors of discharge to the river after cessation of
pumping. However, the SHI-sharp-3 model prediction is superior (its points
plot closer to the identity line). For SHI-smooth-3, the prediction tends to
be positively biased and more spread than for SHI-sharp-3.
Prediction error as function of data quality
In Fig. 11 MAE is used as a metric to
evaluate how the prediction performance of SHI-sharp models depends on the
level of background noise for the geophysical data. The noise levels were
kept unchanged for the hydrological data.
Figure 11 shows that the average age prediction made by SHI-sharp models is
nearly unaffected by the quality of the geophysical data. It is speculative,
but this result may be because this prediction is highly dependent on
small-scale variability in hydraulic conductivity and porosity that cannot be
resolved from any of the geophysical data sets. That is, even the highest
quality geophysical data are not highly informative, so reducing the data
quality further has little effect.
Prediction error as a function of the background noise on the
geophysical data. The black dots are the SHI-smooth models using a background
noise level of 3 nV m-2. The red dots are the SHI-sharp models as a
function of background noise level.
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It is different for the recharge area prediction (Fig. 11): MAE increases for
this by approximately 25 % when the level of background noise is
increased from 1 to 10 nV m-2. This happens because the variations of
resistivity (and thus hydraulic conductivity) are less well resolved when
using the poor-quality geophysical data.
The third and fourth rows of Fig. 11 show the
head recovery and river discharge prediction after cessation of the pumping
well. Head recovery and discharge predictions also tend to depend on the
quality of the geophysical data. The MAE increases by 17 % for
recovery prediction and 23 % for discharge prediction when the noise
level of the geophysical data increases from 1 to 10 nV m-2.
Discussion
Estimation of parameters in the petrophysical relation
Parameterizing the groundwater model by assuming a spatially dependent
petrophysical relationship between resistivity and hydraulic conductivity
makes it possible to use a resistivity voxel model for construction and
calibration of a groundwater model. Assuming that the relationship is
spatially dependent can account for two challenges: (i) there may be actual
changes in the petrophysical relationship within an investigated domain,
and (ii) there may be resolution limitations in the estimated resistivity model.
Challenge (i) is likely to be the rule for many environments, especially
sedimentary environments, where the formation resistivity is primarily
controlled by the pore water resistivity and the clay content. In the case of
spatial changes in pore water resistivity and/or the content of various clay
mineral content, the discrimination between clay and sands may be less clear
in the estimated resistivity values. For large-scale groundwater systems, the
variation of pore water resistivity (e.g., saline pore water) is expected to
vary smoothly, which would be accounted for by the spatially varying
petrophysical relationship. However, the procedure only works as applied here
if the underlying assumption that clay-rich deposits have lower electrical
resistivity than sand deposits is valid.
Challenge (ii) concerns the geophysical model resolution of the true
formation resistivity. EM methods are, by nature, more sensitive to deposits
of low electrical resistivity than to deposits of high resistivity, and
their vertical and horizontal resolutions decrease with depth. This
challenge affects the resistivity models estimated in the present synthetic
study. Spatially dependent shape factors can take a compensatory role for
the resolution issues of the estimated geophysical voxel model. The
calibrated shape factors may thus no longer have firm physical meaning
because they mainly act as correction parameters for absorbing structural
errors of the geophysical model. This is acceptable as long as the resulting
hydraulic conductivity values are reasonable. The idea of calibrating the
shape factors is related to the concept of compensatory parameters in highly
parameterized calibration described by Doherty and
Welter (2010) and by Doherty and Christensen (2011).
Finally, Auken et al. (2008) showed that using
borehole data as a priori information in the geophysical inversion improves
the reconstruction of the model features significantly. Estimation of
EM-based resistivity models should therefore, wherever possible, be
supported by borehole information to improve the decreasing spatial
resolution of the EM methods.
Geophysical inversion strategy and data quality
Inversion of AEM data using a 1-D geophysical model usually applies
smoothness constraints in order to regularize the inversion (Auken and
Christiansen, 2004; Viezzoli et al., 2008). Traditionally, the regularization
includes both lateral and vertical smoothing constraints (Constable et al.,
1987) or a few-layer parameterization (Auken et al., 2008). Inversion using
the former type of regularization produces smooth images with blurred
formation boundaries which can be problematic when it is important to resolve
structural connections in a complex geological system. The latter few-layer
inversion may also be prone to producing artifacts when used to map such
systems. Day-Lewis (2005) and others therefore recognized that regularization
used to stabilize the geophysical inversion can lead to artifacts that do not
reflect the actual hydrogeological conditions. Thoughtless use of such
results to construct groundwater models can have serious ramifications.
For the present case study, the number of vertical transitions is a great
challenge for the AEM method due to the principle of high-resistivity
equivalence. That is, it is difficult to resolve a high-resistivity layer
between two low-resistivity layers because the energy loss, and therefore the
sensitivity, is concentrated in the less resistive layers. This will result
in layer suppression, because the data sensitivity to the high-resistivity
layer is low (Christiansen et al., 2006). This effect is present for both the
smooth and sharp inversions, but in the sharp inversion the effect is less
fuzzy, and features, especially for the fifth layer, could be more clearly
reconstructed (Fig. 4). When the sensitivity of the AEM method is too low,
the regularization may make information migrate from areas with higher
measurement sensitivity (Vignoli et al., 2015). In contrast to the smooth
regularization scheme, the sharp regularization method is designed to
penalize smooth transitions, which eventually improves the reconstruction of
the deeper sand bodies in the present study. Therefore, for the studied case,
the sharp regularization methodology should be preferred over smooth
regularization, because the sharp constraints correspond better to the actual
structures of the reference system (sharp transitions between categorical
deposits; Fig. 4). Moreover, because the sharp regularization methodology
leads to improved reconstruction of subsurface structures, these models lead
to greater accuracy and improvement of most groundwater model predictions
(Fig. 9).
The groundwater system considered here is relatively shallow, at least as
seen from the perspective of the AEM system used in the demonstration
example. This is evident from the transmitted EM signal (Fig. 3). The
background noise primarily affects the last time gates
(10-4–10-3 s) of the low-moment and, only to a small degree, the
high-moment time gates (even for low-quality data). This implies that the
resolution of the AEM data is generally high for the upper layers. Therefore,
in the present case the upper layers of all the geophysical models (both
SHI-smooth and SHI-sharp) are well resolved and to a large extent unaffected
by AEM data quality (Fig. 5). However, the deep sand units are difficult to
resolve because they give only a weak signature in the AEM data (Figs. 3
and 5). This is particularly true for the poorest AEM data quality cases
where the late time gates for the low-moment measurements are disturbed by
background noise.
Summary and conclusion
We present a workflow for efficient construction and calibration of
large-scale groundwater models using a combination of airborne
electromagnetic (AEM) data and hydrological data. Other types of data could
be integrated as well following the same procedure. First, the AEM data are
inverted to form a 3-D geophysical model. Subsequently, the geophysical model
is translated into a 3-D model of hydraulic conductivity by using a spatially
dependent petrophysical relationship for which the shape parameters are
estimated by fitting the groundwater model to hydrological data. The
estimated shape factors of the petrophysical relationship primarily work as
translators between resistivity and hydraulic conductivity, but they can also
compensate for structural defects in the model.
The method is demonstrated for a synthetic case study where the subsurface
consists of categorical deposits with different geophysical and hydraulic
properties. The AEM data are inverted using both smooth and sharp
regularization constraints, resulting in two competitive geophysical models.
Furthermore, the influence of the AEM data quality is tested by inverting the
sharp geophysical models using data corrupted with four different levels of
background noise. The resulting groundwater models are each calibrated on the
basis of head and discharge data, and their predictive performance is tested
for four types of prediction beyond the calibration base. Predictions of a
pumping well's recharge area and groundwater age apply the same stress
situation as applied during hydrologic model calibration, while predictions
of head and stream discharge are done for a changed stress situation.
It is found that a geophysical model inverted with sharp constraints
(SHI-sharp) leads to a more accurate groundwater model than one that is
based on a geophysical model inverted with smooth constraints (SHI-smooth).
The SHI-sharp model leads to an estimated hydraulic conductivity field of
greater accuracy and to improvement of most groundwater model predictions.
The explanation is that the reference system (like many real hydrogeologic
systems) is characterized by sharp transitions between categorical deposits;
this is resolved better by the SHI-sharp resistivity model than by the
SHI-smooth model.
Finally, it is shown that prediction accuracy improves with AEM data quality
for predictions of recharge area, head change, and stream discharge, while
the accuracy appears to be unaffected for prediction of groundwater age,
which cannot be predicted accurately even with high-quality geophysical data.