Analyzing Bank Filtration by Deconvoluting Time
Series of Electric Conductivity
by Olaf A. Cirpka1, Michael N. Fienen3, Markus Hofer2, Eduard Hoehn2, Aronne Tessarini2,
Rolf Kipfer2, and Peter K. Kitanidis3
Abstract
Knowing the travel-time distributions from infiltrating rivers to pumping wells is important in the management of alluvial aquifers. Commonly, travel-time distributions are determined by releasing a tracer pulse into the
river and measuring the breakthrough curve in the wells. As an alternative, one may measure signals of a timevarying natural tracer in the river and in adjacent wells and infer the travel-time distributions by deconvolution.
Traditionally this is done by fitting a parametric function such as the solution of the one-dimensional advectiondispersion equation to the data. By choosing a certain parameterization, it is impossible to determine features of
the travel-time distribution that do not follow the general shape of the parameterization, i.e., multiple peaks. We
present a method to determine travel-time distributions by nonparametric deconvolution of electric-conductivity
time series. Smoothness of the inferred transfer function is achieved by a geostatistical approach, in which the
transfer function is assumed as a second-order intrinsic random time variable. Nonnegativity is enforced by the
method of Lagrange multipliers. We present an approach to directly compute the best nonnegative estimate and to
generate sets of plausible solutions. We show how the smoothness of the transfer function can be estimated from
the data. The approach is applied to electric-conductivity measurements taken at River Thur, Switzerland, and
five wells in the adjacent aquifer, but the method can also be applied to other time-varying natural tracers such as
temperature. At our field site, electric-conductivity fluctuations appear to be an excellent natural tracer.
Introduction
In Switzerland, about 40% of the drinking water is
produced by active pumping from alluvial aquifers
(SVGW 2002). Most pumping wells have been placed
close to the rivers, so that a large fraction of the extracted
water is thought to consist of freshly infiltrated river
water, characterized by ground water residence times of
1Corresponding author: Swiss Federal Institute of Aquatic
Science and Technology (Eawag), Department of Water Resources
and Drinking Water, Uberlandstr. 133, 8600 Dübendorf,
Switzerland; 41-44-823 5455; fax 141-44-823 521; Olaf.
Cirpka@Eawag.CH
2Swiss Federal Institute of Aquatic Science and Technology
(Eawag), Überlandstrasse 133, 8600 Dübendorf, Switzerland.
3Department of Civil and Environmental Engineering, Stanford
University, Stanford, CA 94305-4020.
Received June 2006, accepted November 2006.
Copyright ª 2007 The Author(s)
Journal compilation ª 2007 National Ground Water Association.
doi: 10.1111/j.1745-6584.2006.00293.x
318
a few days. Since many rivers in the area receive treated
sewage, there is the potential risk that the infiltrated water
contains contaminants and may pollute the wells. In several countries, zones for the protection of wells are
defined by means of travel times (e.g., 10 d in Switzerland [BUWAL 2004] and 50 d in Germany [DVGW
1995]). Within these zones, handling of potentially
hazardous chemicals is prohibited and restrictions apply
to agriculture and land use. Most agencies are concerned
when travel times between potentially contaminated rivers
and wells fall below the standards set for the delineation of
protection zones.
The standard technique to determine the travel-time
distribution from a river to a well is to add a pulse of an
easy-to-measure conservative tracer such as a fluorescent
dye into the river and observe the breakthrough curve in
the well (e.g., Davis et al. 1980; Lin et al. 2003; Käss
2004). From the breakthrough curve, the fraction of
freshly infiltrated water in the extracted water, the time of
first arrival, the mean arrival time, and the spread of the
Vol. 45, No. 3—GROUND WATER—May–June 2007 (pages 318–328)
travel-time distribution can be determined. The results,
however, hold only for the hydraulic conditions during
the test. To determine the travel-time distribution under
different hydrological conditions, a repetition of the tracer
test would be necessary. Also, the strong dilution in the
river requires adding large amounts of the tracer.
An alternative to adding a tracer is the analysis of
natural tracers that are already present in the water. In
order to determine travel times, natural time-varying
signals are needed. Silliman and Booth (1993) used temperature fluctuations in a river and the adjacent aquifer to
characterize the hydraulic exchange. Silliman et al.
(1995) constructed an analytical solution of temperature
in the aquifer for arbitrary temperature signals in the infiltrating river based on the one-dimensional (1D) solution
of the advection-dispersion equation for steplike injection.
Sheets et al. (2002) analyzed hydraulic heads, temperature, and electric conductivity at a wellfield in Cincinnati,
Ohio, by cross-correlation methods. Constantz et al.
(2003) compared results of a test with an injected tracer
with the analysis of natural temperature fluctuations,
using the 1D advective-dispersive model.
Although temperature has been established as an
easy-to-measure natural tracer for river–ground water
interaction (Anderson 2005), it has several disadvantages
in comparison to concentrations. Temperature signals are
retarded by a factor depending on the porosity and the
mineral composition of the medium; they are more
smoothed by diffusion-like processes than concentration
data; and the main thermal signals, namely, the diurnal
and seasonal fluctuations, are not unique to rivers, making it difficult to distinguish river-borne ground water
from water of other shallow origin (Hoehn and Cirpka 2006).
In the current study, we use electric-conductivity
fluctuations as natural tracer. The techniques presented
apply also to time series of temperature or other conservative quantities in the water. The basic problem is to
characterize the response of the measured signal in the
aquifer to changes of the signal observed in the river. In
contrast to the pulse-like tracer test, the measured time
series in the river exhibits continuous fluctuations, so that
the measured signal in the ground water well reflects
both the river signal and the transport process between
the river and the well. The main objective is to extract
from these data the characteristics exclusively of the
transport process. For this purpose, three classical techniques have been applied: cross correlation of the time
series, calibration of an advective-dispersive model, and
parametric deconvolution.
In cross correlation, the correlation coefficient of the
two time series is computed after shifting and potentially
smoothing one of the two data sets (e.g., Sheets et al.
2002). The time shift with maximum correlation coefficient is considered to represent the characteristic time
of transfer. If the input data had been smoothed, the
optimum degree of smoothing is an indicator of diffusive
processes. The cross-correlation coefficient quantifies to
which extent the variability observed at one observation
point is linearly related to that of the other observation
point (after shifting and smoothing one of the two time
series). It is useful also to compute the corresponding
coefficients of linear regression, namely, the slope and the
intercept.
Mathematically, cross-correlating smoothed and shifted
time series can be interpreted as applying a parametric
transfer function to the time series after subtracting their
mean values. The shape of the transfer function is defined
by that of the smoothing function, and the integral of
the transfer function is given by the slope of linear regression. Identifying the optimal smoothing function is
a particular challenge.
Analyzing time series of concentration or temperature by fitting advective-dispersive models appears to be
a more physical approach than cross-correlating the data.
In many applications, 1D transport with uniform and
time-invariant coefficients is assumed (see the examples
given in the review of Anderson [2005]). The transport
model is identical to using the inverse Gaussian distribution as parametric transfer function (Kreft and Zuber
1978). The nonuniformity of real aquifers and the multidimensional nature of solute transport pose major difficulties of this model, so that even if the measurements
were taken at points the breakthrough curves of tracer
tests with pulse-like injection would differ from the analytical solution. Since wells are typically screened over
a range of depths, the breakthrough curves are also
affected by sampling an ensemble of streamlines, which
may result in long tails or multiple peaks. The problem
could be addressed by using multidimensional advectivedispersive models. However, the calibration of such
models requires data with high spatial resolution that is
missing in most practical applications.
For the case of time-invariant transport, the breakthrough curve at an observation well can be computed
by convolution of the river signal with an appropriate
transfer function (e.g., Jury 1982). The transfer function
describes the response to a unit pulse. It may be worth
noting that identifying the transfer function for a given
pair of input and output signals is mathematically identical to identifying the input signal for a known transfer
function and output signal. The latter task has been tackled, for instance, in the reconstruction of contaminant
release history (Skaggs and Kabala 1994; Snodgrass and
Kitanidis 1997). As stated previously, the solution of the
1D advection-dispersion equation with uniform, constant
coefficients for a pulse injection is a particular parametric
transfer function, in this case defined by the two parameters velocity and dispersion coefficient. Other parametric
models may be more appropriate to capture typical properties of breakthrough curves (Luo et al. 2006). Choosing
the most appropriate parametric model is equivalent to
identifying the optimal smoothing function in the crosscorrelation method.
The major difficulty of parametric deconvolution is
that it restricts the inferred transfer function to a predefined shape. As an example, the true transfer function
may be multimodal because the well samples water from
two distinct layers. Nonetheless, it is possible to unwittingly fit a unimodal parametric transfer function to the
data so that major characteristics of the process may be
lost (Fienen and Kitanidis 2006). In nonparametric deconvolution, the transfer function is free to adjust to the data
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
319
since it does not have to conform to a preselected shape.
In discrete form, this implies that the number of estimated
parameters is identical to the number of time steps
discretizing the transfer function. In order to achieve reasonable estimates, it typically is necessary to enforce
a certain smoothness of the transfer function, which can
be done by Tikhonov regularization (Skaggs and Kabala
1994) or a geostatistical model (Fienen and Kitanidis
2006). Identifying the optimal degree of smoothness from
the data is challenging. Secondary peaks of the transfer
function disappear when a tight smoothness criterion is
used. Therefore, using a too tight criterion results in loss
of information, whereas a too loose criterion may lead to
erroneous peaks mapping noise of the two time series onto
each other. As a final constraint, one has to ensure that
the transfer function is physically reasonable. In particular,
it must not contain negative entries.
As discussed by Box and Jenkins (1970), the transfer
function can also be expressed as the sum of autoregressive (AR) terms, where the output signal at a given
time depends linearly on its own previous values, and the
so-called moving-average (MA) terms, expressing the
dependency on the input time series. This approach, denoted ARMA, has hardly been applied to subsurface
transport (Bidwell 1995). If we used the ARMA approach
rather than straightforward convolution, we would have to
face the same challenges, namely, determining the degree
of smoothness (which is related to the number of AR
terms considered) and ensuring that the reconstructed
transfer function is nonnegative. The latter would require
determining the coefficients by constrained optimization
(Bidwell 1995).
In the present study, we present a nonparametric
deconvolution method using a geostatistical smoothness
criterion and Lagrange multipliers to enforce nonnegativity.
We provide two schemes: in the first, we directly compute the best nonnegative estimate, whereas we generate
multiple plausible solutions in the second. The developed
methods are applied to time series of electric conductivity
measured at a test site at River Thur, Switzerland.
Theory
We assume that solutes undergo linear, time-invariant
transport from the river to the observation wells. Under
these conditions, the superposition principle holds, and
the response y(t) of concentration in an observation well
to any signal x(t) in the river can be computed by convolution (e.g., Jury 1982):
Z T
yðtÞ ¼
gðsÞxðt 2 sÞ ds
ð1Þ
0
in which g(s) is the transfer function, also denoted as the
impulse-response function or Green’s function, and T is
the time interval over which it is defined. The transfer
function describes the response of the system to a pulselike stimulus, that is, the breakthrough curve that would
result from an ideal tracer test in which the signal x(t) in
the river is a Dirac pulse. The transfer function can also
be interpreted as the probability density function of solute
320
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
travel times. According to Equation 1, the output function
y at time t depends on the input x from t 2 T to t. Thus, if
the input function x(t) varies continuously in time and
measurements are available only for t 0, the output
function y(t) is properly defined only for t T.
After uniform temporal discretization, we deal with
an nx 3 1 vector x and an ny 3 1 vector y of measurements in the river and observation well, respectively, and
an ng 3 1 vector g of discrete values of the transfer function. The vector g of transfer function values must be considerably shorter than the vector y of observations, i.e., ng
< ny . Now, the convolution integral, Equation 1, becomes
a matrix-vector product:
y ¼ Xg
ð2Þ
Xij ¼ txi2j11
ð3Þ
with
in which the ny 3 ng matrix X, which has Toeplitz structure, is made of the shifted input signal x times the time
increment t.
The purpose of the present paper is to infer the transfer function g from measured time series x and y, in
which both signals continuously vary in time. A least
squares solution can be derived but exhibits strong fluctuations because it is sensitive to the noise in the time series
data. For regularization, we introduce a measure of
smoothness in g(s) as prior knowledge. We do this by
assuming that g(s) is a second-order intrinsic random
time variable, following a linear semivariogram:
gðsÞ ¼ b 1 g9ðsÞ
ð4Þ
E½g9ðsÞ ¼ 0 "s
ð5Þ
1
2
E ðg9ðs1hÞ2g9ðsÞÞ ¼ cg ðhÞ ¼ hjhj
2
ð6Þ
in which b is the uniform mean value that is unknown
a priori; g9(s) is the deviation from the mean; cg(h) is the
semivariogram function, here assumed to follow the
linear model; h is the time distance; and h is the slope of
the linear variogram cg(h).
Now, the conditional probability density function
p(g|x,y) of the discretized transfer function g, given the
data x and y, can be computed by applying Bayes’ theorem:
pðgjx; yÞ ¼
pðyjx; gÞpðgÞ
pðyÞ
ð7Þ
in which p(y|x,g) is the likelihood of the output y given
the input x and the transfer function g; p(g) is the prior
probability density of g, parameterized by the autocorrelated model described previously; and p(y) is a scalar that
does not depend on g.
Assuming multi-Gaussian probability density functions (pdf) for the likelihood and the prior terms and expressing g by its mean b and deviation g9, the negative
logarithm of p(g|x,y) becomes:
Lðg9;bjx; yÞ ¼
ðy 2 Xðg9 1 ubÞÞ ðy 2 Xðg9 1 ubÞÞ
r2ep
2 g9T G21
gg g9 1 const
ð8Þ
in which u is a ng 3 1 vector of unit entries and r2ep is the
variance of the deviations between y and the best model
outcome. r2ep quantifies the epistemic error, which consists of measurement errors, possible numerical errors,
and erroneous conceptual assumptions. We assume that
the epistemic error is identical for all elements of y. Ggg is
the ng 3 ng matrix of discrete semivariogram values,
obtained for all pairs of elements in g; i.e., element (i,j)
of Ggg is defined by gg(i, j) ¼ cg(|ti2tj|). The negative
posterior log-likelihood L(g9,b|x,y) is the objective function in the direct computation of the most likely value, or
best estimate.
The solution minimizing L(g9,b|x,y) may lead to
negative elements of g ¼ g91ub. Representing the
response to a pulse-related tracer test, the transfer function must not be negative because that would imply negative concentrations in the observation well during the
tracer test. Thus, we have to enforce nonnegativity:
g ¼ g9 1 ub 0
ð9Þ
In the application to contaminant-source identification, Snodgrass and Kitanidis (1997) ensured nonnegativity
by applying a power-law transformation of the unknown,
which includes the log-transformation as a special case. In
this approach, it is impossible to get elements of zero in g.
Nonnegativity can also be ensured by using a prior distribution p(g) with zero probability density in the negative
range (Fienen and Kitanidis 2006). Then, the objective
function of Equation 8 is not valid. Fienen and Kitanidis
(2006) assumed a reflected Gaussian distribution of the
parameters and applied a Markov chain Monte Carlo
method with Gibbs sampler, originally derived for source
identification (Michalak and Kitanidis 2003), to the deconvolution problem. The latter approach has high computational costs, which makes the method ill-suited for
problems in which several hundred parameters need to be
identified.
In the present study, we use a different approach. We
keep Equation 8 and account for the nonnegativity
constraints by the method of Lagrange multipliers (e.g.,
Vogel 2002, chap. 9).
Direct Computation of Best Nonnegative Estimate
Equation 9 is an inequality constraint. Only when
a certain element of g becomes negative in the estimation
procedure, the constraint has to be activated. We may
express this by:
Hðg9 1 ubÞ ¼ 0
ð10Þ
in which H is a nt 3 ng selection matrix with nt being the
number of active constraints:
1; if gj is affected by the i -th constraint
Hij ¼
0; otherwise
ð11Þ
Then, the constrained minimization of L(g9,b|x,y)
consists of solving the following system of linear
equations:
32 3
2
1 T
1 T
21
T
g9
X
X
2
G
X
Xu
H
gg
2
76 7
6 r2ep
r
ep
76 7
6
76 7
6 1
1 T T
6
T T
T 76 b 7
u
X
Xu
u
6 2 u X X
t 7
54 5
4 rep
r2ep
t
H
ut
0
3
2
1 T
X y
7
6 r2ep
7
6
7
6 1
¼6
ð12Þ
T T 7
6 2 u X y7
5
4 rep
0
in which t is the nt 3 1 vector of Lagrange multipliers
and ut is a nt 3 1 vector of unit entries. Activation and
deactivation of the constraints depend on the behavior of
the solution. The following rules apply:
gj , 0/add constraint for element j
ti 0/keep constraint i
ti > 0/remove constraint i
Since solute transport cannot be instantaneous, we
require that the first element of g, representing g(s ¼ 0),
is always zero. That is, the Lagrange multiplier for the
first element is always active, regardless of the data.
The practical implementation requires an iterative
procedure, starting without constraints in the first iteration. Depending on the current estimate, constraints are
added, kept, or removed, and the estimation is repeated
until the selection of constraints stops changing between
two iterations. In our applications, typically, between 5
and 10 iterations were necessary to reach convergence.
Generation of Plausible Solutions
Since convolution is a linear operation, exact expressions for the uncertainty of the estimated parameter
vector g would be possible if the constraints were of the
equality type. Because the constraints are inequalities,
however, the overall optimization problem is effectively
nonlinear. This is expressed in the necessity to identify
the set of active constraints by an iterative procedure.
In order to evaluate the uncertainty of the estimate,
in addition to obtaining the most likely value of g by
minimizing the negative log-likelihood L(g9,b|x,y), we
also generate multiple plausible solutions, each exhibiting
variability consistent with the variogram cg(h) and meeting the measurements y with the required accuracy expressed by the epistemic error rep. We do this by the
method of smallest possible modification (e.g., Journel
and Huijbregts 1978).
In each plausible solution, we express the parameter
vector g as the sum of three contributions:
g ¼ gu9 1 gc9 1 ub 0
ð13Þ
in which gu9 is an unconditional realization of the fluctuations in the transfer function, gc9 is a smooth correction
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
321
term, and b is the unknown mean value. Here, gu9 is isomorphic to g9, that is, its expected value is zero, and its
generalized covariance function is 2cg(h). The unconditional realization gu9 is generated by multiplying the
inverse Cholesky decomposition of 2Ggg with a vector of
uncorrelated random numbers drawn from a standard normal distribution (e.g., Rubin 2003, section 3.6.1). For the
computation of each plausible solution, we also generate
a random deviation ey from the measured data y. The elements of ey have expected values of zero and variances of
r2ep . In the conditioning, gu9 remains fixed, whereas g9c and
b are estimated by minimizing the following objective
function:
W¼
ðy1ey 2Xðgu91gc91ubÞÞðy1ey 2Xðgu91g91ubÞÞ
c
r2ep
2g9cT G21
gg gc9
ð14Þ
subject to the nonnegativity constraints. For a given unconditional realization g9;
u the optimal set of g9c and b is found
by solving the following system of linear equations:
32 3
2
1 T
1 T
21
T
g9c
X
X
2
G
X
Xu
H
gg
2
76 7
6 r2ep
r
ep
76 7
6
76 7
6 1
1 T T
6
T T
T 76 b 7
u
X
Xu
u
6 2 u X X
t 7
54 5
4 rep
r2ep
t
H
ut
0
3
2
1 T
X ðy 1 ey 2 Xg9Þ
u
7
6 r2ep
7
6
7
6 1
ð15Þ
¼6
7
6 2 uT XT ðy 1 ey 2 Xgu9Þ 7
5
4 rep
2 Hg9u
in which the constraints are added, kept, and removed
according to the same rules as applied in the direct estimate of the most likely value of g.
In our application, we generate 1000 plausible solutions. The resulting conditional statistical distribution of
g is analyzed for each element of g, resulting in characteristic percentiles of the distribution P(g(s)), which provides approximate confidence intervals.
Estimation of Structural Parameters
The optimization problem posed depends on two structural parameters: the variance r2ep quantifying the epistemic
error and the slope h of the variogram cg(h) quantifying the
smoothness of the transfer function g(s). In principle, these
parameters can be derived from the measurements y by
maximizing their posterior marginal pdf (e.g., Hoeksema
and Kitanidis 1984; Kitanidis 1986). Evaluating the posterior marginal pdf may be complicated in nonlinear estimation problems. Also, the approach involves inversing
matrices of the order ny, which is the number of elements in
y. In our application, ny is larger than 10,000, which prohibits applying the mentioned approaches. Instead, we follow
a simplified approximate procedure.
For estimating the epistemic error, assumed identical
for all measurements, we enforce that the sum of squared
residuals, weighted by their variance r2ep , meets its expected value (see Press et al. 1992, equation 15.1.6):
322
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
r2ep ¼
ðy 2 Xðg9 1 ubÞÞ ðy 2 Xðg9 1 ubÞÞ
ny 2 n g 1 nt 2 1
ð16Þ
in which ny 2 ng 1 nt 2 1 corresponds to the degrees of
freedom in estimating g from y, subject to nt active constraints. This approach is well accepted in optimization of
overdetermined problems.
Estimating the slope h of the variogram cg(h) in a rigorous way is conceptually and computationally a more
demanding task than estimating r2ep . In many practical applications, it may be sufficient to vary h manually and
pick a value at which major features of the transfer function, such as larger peaks, are recovered whereas smallscale fluctuations are smoothed.
In the present study, we estimate h by the expectationmaximization method (McLachlan and Krishnan 1997).
The general idea is to find the slope h that best describes the
variability in the set of plausible solutions. For a discretized
random variable following a linear semivariogram, it can be
shown that the probability density function of the entire vector can be computed by considering the squared difference
between two neighboring elements (Michalak and Kitanidis
2003). In our analysis, we consider only the nonzero entries
of our estimated vector g. The vector of nonzero elements in
the j-th solution is denoted gnz,j. The remaining elements in
g are fixed to zero and affect the probability density of gnz, j.
The log-probability density ln p(gnz,j|h) of the nonzero elements for given h and Lagrange mulitpliers is:
nt; j 2 ng
ng 2 1
lnðsÞ
lnpðgnz; j jhÞ¼
lnð4phÞ2
2
2
nX
t;j 21
1
lnðs0i11; j 2 s0i;j Þ
2
i¼1
!
nX
g 21
ðgi11 2gi Þ2
2
1 const
4hðsi11 2si Þ
i¼1
1
ð17Þ
in which nt,j is the number of Lagrange multipliers in the
j-th solution, s is the time-shift increment of the full
vector g, and s0i;j is the time shift associated with the i-th
zero element in the j-th solution. The constant in Equation 17 does not depend on h. Given a set of solutions, an
updated value of h, fitting all plausible solutions best, is
computed by minimizing:
nr
X
ð18Þ
uðhÞ ¼ 2
ln pðgnz; j jhÞ
j¼1
The minimization is done by the Melder-Nead
simplex algorithm as implemented in Matlab (Lagarias
et al. 1998). After updating h, a new set of solutions has
to be generated. With the new set, h is updated again.
The procedure is repeated until the relative change of
h between two sets of solutions is less than 1%.
Application to Experimental Data from a
Site at River Thur
Description of the Study Site
We analyzed data collected from the capture zone of
a pumping well producing drinking water for about
30,000 people in the city of Frauenfeld, Canton Thurgau,
Switzerland. The site is located in the floodplain of the
central Thur valley, which is about 30 km long and 2 km
wide. During the Pleistocene, glaciers cut into the underlying tertiary bedrock. The retreating glacier left a lake
behind, filling the valley with fine lacustrine sediments,
acting now as aquitard. These clays are overlain by an
approximately 10-m-thick layer of bedded glaciofluvial
gravel and sand, forming a productive aquifer. From
pumping tests, the mean hydraulic conductivity of this
material is known to range between 1 3 1023 m/s and
5 3 1023 m/s. The top layer of approximately 2 m thickness consists of alluvial loam. At our study site, the aquifer
may be considered as semiconfined.
The upper catchment of River Thur is prealpine with
Mount Säntis (2502 m above sea level) being the highest
point. In the central Thur valley (altitude ~ 400 m above
sea level), the river was channelized in the 1890s. At our
site, River Thur flows from east to west along the northern edge of the valley. The main channel is about 40 to
45 m wide. There is no overbank on the right-hand side,
whereas the left overbank is about 130 m wide, defined
by a levee, behind which a side channel was installed to
capture discharge from tributaries and drain agricultural
land. The pumping well is located outside the overbank,
with a distance to the levee of approximately 80 m. It is
a horizontal well with nine arms, typically producing
approximately 9000 m3/d.
In the fall of 2003, four monitoring wells (PVC pipes
with 4.5 inch diameter) were instrumented in the capture
zone of the pumping well. Figure 1 gives a schematic
overview of the field site. Monitoring wells 1 and 2 are
located directly at the upper edge of the main channel,
with well 1 being screened over 2 m at the bottom of the
aquifer, and well 2 at the uppermost 2 m of the ground
water (depth to water table, 2 to 4 m). The fully screened
monitoring well 3 is placed in the middle of the overbank,
on the presumed streamline leading from monitoring
wells 1 and 2 to the pumping well. Monitoring well 4 is
also fully screened; this well was installed outside of the
overbank, close to a local upwelling spring feeding
a small riverine, denoted Giessen. At the study site, River
Thur infiltrates year-round. Ground water is assumed to
be stratified, with a local flow component on top that is
caused by the infiltration of river water and a regional
flow component at lower sections that follow the main
direction of the valley. The overall direction of flow is
from northeast to southwest.
The hydrological regime of River Thur has alpine
characteristic: precipitation in the upper catchment causes
rapid fluctuations of the water level along the entire river
course. These fluctuations are not dampened by natural or
anthropogenic reservoirs. Base flow is highest during snowmelt in the spring, but flow peaks can occur at any time of
the year in response to rainfall in the upper catchment.
Associated with changes of discharge are fluctuations of electric conductivity during high river flow. Rain
water contains lower concentrations of total dissolved
solids than soil water and ground water or effluents of
waste water treatment plants. As a consequence, electric
conductivity in River Thur decreases during peak flows
(Figure 2). The conductivity signal is propagated into the
aquifer and can be used as a natural tracer.
The monitoring wells, the pumping wells, and a
station in River Thur have been equipped with data
loggers, recording hydraulic head, temperature, and
Figure 1. Overview over the test site. Northing and easting are in meters according to the Swiss terrestrial reference system.
Gray lines: contours of hydraulic head (0.1 m/line) resulting from two-dimensional simulation of ground water flow.
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
323
A: Electric Conductivity
700
σ [µS/cm]
600
500
400
300
200
01−01−04
01−07−04
01−01−05
01−07−05
B: Hydraulic Head
h [m a.s.l.]
396
395
electric-conductivity signal measured in the river (gray
lines). All wells react to fluctuations of electric conductivity in the river. The fastest and strongest response is
observed in monitoring well 2, which is close to the river
and screened in the uppermost part of the aquifer. The
electric-conductivity signal in the other wells is more
dampened. It is also apparent that the electric conductivities measured in the wells differ systematically from
those in the river in the springtimes of 2004 and 2005.
We attribute this to snowmelt water in the river, which is
not in chemical equilibrium with the sediment material.
Upon infiltration, this water partially dissolves minerals,
leading to increased concentrations of total dissolved
solids and thus to increased values of electric conductivity; that is, electric conductivity is not always a conservative tracer.
394
393
392
01−01−04
01−07−04
01−01−05
01−07−05
Figure 2. Electric conductivity and hydraulic heads in River
Thur at our experimental site during the study period.
electric conductivity at time intervals of 1 h. The time
series begin on November 20, 2003, and end at different times, depending on the well. All stations cover at
least the time period until April 21, 2005. Figure 3
shows the time series of electric conductivity in the
wells. For comparison, the plots are underlain by the
Preparation and Analysis of the Data
As can be seen from Figure 3, the discrepancies in
electric conductivity between ground water and the river
appear only in an annually repeating period in the spring.
High-frequency components are less affected by the fact
that electric conductivity is a nonconservative tracer. In
order to eliminate the effects of springtime discrepancies,
we remove the seasonal trend from all time series. For
this purpose, we fit a uniform mean as well as sine and
cosine functions with frequencies 1/year, 2/year, 3/year,
and 4/year to the data by standard least square fitting.
These trend signals are subtracted from the original data.
The remaining time series still contain characteristic
high-frequency signals that can be related to each other.
B: Monitoring Well 2
700
600
600
σ[µS/cm]
σ[µS/cm]
A: Monitoring Well 1
700
500
400
500
400
300
300
200
01−01−04 01−07−04 01−01−05 01−07−05
200
01−01−04 01−07−04 01−01−05 01−07−05
D: Monitoring Well 4
700
600
600
σ[µS/cm]
σ[µS/cm]
C: Monitoring Well 3
700
500
400
500
400
300
300
200
01−01−04 01−07−04 01−01−05 01−07−05
200
01−01−04 01−07−04 01−01−05 01−07−05
E: Pumping Well
700
σ[µS/cm]
600
500
400
300
200
01−01−04 01−07−04 01−01−05 01−07−05
Figure 3. Time series of electric conductivity in the monitoring wells and the pumping well. Gray lines: river data; black
lines: well data.
324
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
In the procedure of estimating the transfer function,
we generate sets of 1000 solutions. In order to estimate
the slope h of the semivariogram cg(h), we have to analyze the set of solutions, update h, and generate a new set
of plausible solutions. Typically, this requires a few iterations. To speed up the process, we start with sets of 100
solutions until the estimates of h change only by a few
percent from one iteration to the next and continue with
sets of 1000 solutions. The program is implemented in
Matlab.
Results of Deconvolution
Figure 4 shows the estimates of the transfer functions
g(s) for all monitoring wells and the pumping well. The
solid lines indicate mean values ÆgðsÞjx; yæ of 1000 plausible solutions. The crosses denote the directly computed
best nonnegative estimate g^ðsÞ. The gray area marks the
range bounded by the 16 and 84 percentiles of the statistical distributions given by the set of plausible solutions.
That is, 16% of all plausible solutions fall below the gray
area and another 16% above. We choose these limits
because they mark deviations by 61 standard deviation
in the case of a Gaussian distribution. The dotted lines
indicate the minimum and maximum values obtained in
all solutions.
The most striking result of Figure 4 is the good
agreement between the mean ÆgðsÞjx; yæ of the plausible
solutions and the best estimate g^ðsÞ: We attribute this to
the fact that the estimation problem is almost linear.
Nonlinearity is exclusively introduced by the nonnegativity of the transfer function. For times at which the
best estimate g^ðsÞ is clearly larger than zero, only a few
plausible solutions give values of zero. The main differences between the best estimate g^ðsÞ and the mean of the
solutions ÆgðsÞjx; yæ can be found in regions where the
best estimate requires an active constraint. Here, the mean
ÆgðsÞjx; yæ of the plausible solutions typically is nonzero.
These periods include early times of the transfer functions
for monitoring wells 3 and 4 and the pumping well.
The mentioned differences may be significant for water
resources management because they concern the earliest
breakthrough of solutes in the observation point. According to the scheme to directly compute the best estimate,
there is a definite time before which no breakthrough is
possible. Using linearized uncertainty propagation, the
associated uncertainty would be zero, since at these times
Lagrange multipliers are active. The plausible solutions,
by contrast, reveal a finite probability of earlier breakthrough. The latter might be important for probabilistic
risk assessment because the concentration of a degrading
contaminant decreases with travel time. That is, a finite
probability of early breakthrough for a pulse of a conservative compound is associated with an increased probability of contaminant breakthrough.
Table 1 contains the identified structural parameters,
and Table 2 lists characteristic values for the directly
computed best estimates g^ðsÞ of the transfer functions.
Monitoring well 2 reacts the fastest to conductivity
B: Monitoring Well 2
1
0.8
g [1/d]
g [1/d]
A: Monitoring Well 1
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
0
1
2
τ [d]
4
5
6
7
τ [d]
C: Monitoring Well 3
D: Monitoring Well 4
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.05
0.04
g [1/d]
g [1/d]
3
0.03
0.02
0.01
0
0
5
10
15
20
τ [d]
0
10
20
30
40
50
60
τ [d]
E: Pumping Well
0.03
g [1/d]
mean of plausible solutions
0.02
16% − 84% probability
min, max
0.01
best nonnegative estimate
0
0
5
10
15
20
25
30
35
40
τ [d]
Figure 4. Estimates of the transfer functions g(s) for all monitoring wells and the pumping well. Solid line: mean of 1000
conditional realizations; gray area: range between 16 and 84 percentiles of the conditional statistical distributions; dotted
lines: minimum and maximum values obtained in all realizations; crosses: best nonnegative estimate.
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
325
Table 1
Structural Parameters Estimated for the Data Sets
Epistemic
Error rep (lS/cm)
h of Transfer
Function (d23)
14.5
20.1
20.3
21.4
19.5
3.24 3 1024
4.76 3 1022
1.03 3 1024
4.89 3 1026
1.33 3 1026
Monitoring well 1
Monitoring well 2
Monitoring well 3
Monitoring well 4
Pumping well
changes in the river. The peak occurs already after 10 h
and the center of the breakthrough curve after 1.5 d. Monitoring well 2 is also the well with the highest zeroth
moment m0 ðg^Þ of the transfer function. Ninety-four percent of the fluctuations in River Thur is recovered in this
monitoring well. This is a direct indication that almost all
water passing the monitoring well is freshly infiltrated
river water. Monitoring well 1 has the same distance to
the river as monitoring well 2 but is screened at greater
depth. Well 1 reacts considerably slower than well 2
because the path length from the river to the well is
longer. We assume that the fluctuating water level of the
river determines to what extent the water passing monitoring well 1 at a given time is freshly infiltrated river
water. For both monitoring wells 1 and 2, it is not possible to determine a time of first breakthrough; that is, the
response of the wells to electric-conductivity fluctuations
starts immediately.
The transfer functions determined for monitoring
wells 3 and 4, as well as for the pumping well, are delayed. According to the directly computed best estimate
g^ðsÞ, the times of first and maximum breakthrough
increase with increasing distance to the river (Table 2).
Considering the plausible solutions, the probability of
earlier breakthrough is fairly high, even though only
small values of the transfer function are estimated at these
times. The transfer function for monitoring well 4 extends
over the longest time, exhibiting a distinct tail and a small
secondary peak at a travel time of about 50 d (Figure 4).
As noted previously, monitoring well 4 is located in the
direct vicinity of a small spring, which indicates a local
anomaly of the hydraulic properties.
The computed zeroth moments of the transfer functions are 60%, 68%, and 26% for monitoring wells 3 and
4 and the pumping well, respectively. For perfectly conservative transport, the zeroth moments may be interpreted as recovery rate. Values considerably smaller than
100% could indicate mixing with water of other origin.
However, the missing 40% and 32% in monitoring wells
3 and 4, respectively, may also partially be in the truncated tails of the transfer functions. The possibility of the
latter becomes clear if one considers the tails of the plausible solutions rather than that of the best estimate g^ðsÞ.
Finally, one may speculate to what extent the incomplete
recovery of electric-conductivity fluctuations is attributed
to hydrogeochemical reactions of the infiltrating river
water with the mineral surfaces of the aquifer.
For comparison, Table 2 lists also the time shift tcorr
with highest correlation coefficient when the same data
sets are analyzed by cross correlation. In these evaluations, the river signals without seasonal trend are smoothed by a rectangular filter function and cross-correlated to
the well signals without seasonal trend. The optimal filter
width is individually identified for each well (data not
shown). The mean travel times tcorr estimated by crosscorrelation fall between the peak time tmod ðg^Þ and
the mean travel time tc ðg^Þ according to nonparametric
deconvolution. The differences in the mean travel times
obtained by cross correlation and nonparametric deconvolution indicate that the rectangular filter chosen for cross
correlation is not optimal.
The uncertainty of all transfer functions, expressed
by the error bounds (lines expressing 16% and 84% probability of exceedance), is fairly small. This is caused by
the small values of h, obtained by the expectation-maximization method (Table 1). Small values of h imply
smooth estimates of the transfer function. However, the
low uncertainty in g(s) comes with epistemic errors rep in
the range of 20 lS/cm (Table 1). As explained in the Theory section, the latter includes the model error. That is,
the uncertainty in g(s) is conditional on the validity of the
transfer function concept, including the assumption of
time-invariant transport. In order to accept the concept,
a fairly large discrepancy between simulated and measured time curves of electric conductivity must be accepted.
Discussion and Conclusions
We have presented a method to infer nonparametric
transfer functions from measured time series, guaranteeing
Table 2
Characteristics of the Directly Determined Best Nonnegative Estimates of Transfer Functions g^ðsÞ
Monitoring well 1
Monitoring well 2
Monitoring well 3
Monitoring well 4
Pumping well
T (d)
ng
m0 ð^
gÞ
14
7
20
60
40
336
168
480
1440
960
0.51
0.94
0.60
0.68
0.26
t0 ð^
gÞ
tc ð^
gÞ
tmod ð^
gÞ
rð^
gÞ
instant.
instant.
1d6h
4 d 22 h
7 d 17 h
3 d 18 h
1 d 11 h
6d2h
21 d 3 h
18 d 2 h
1 d 18 h
8h
4d3h
10 d 7 h
15 d 12 h
3 d 16 h
1 d 13 h
3 d 22 h
15 d 7 h
10 d 0 h
tcorr
3d7h
1d2h
5 d 14 h
14 d 2 h
17 d 10 h
T, truncation time; ng, number of estimated values; m0 ð^
gÞ, zeroth moment (recovery rate); t0 ð^
gÞ, time of first breakthrough; tc ð^
gÞ, center of gravity; tmodaf ; ð^
gÞ, peak
time; rð^
gÞ, standard deviation of determined travel-time distribution (measure of spread); tcorr, optimum time shift of cross-correlation method; instant., instantaneous.
326
O.A. Cirpka et al. GROUND WATER 45, no. 3: 318–328
smoothness and nonnegativity by introducing secondorder intrinsic autocorrelated behavior of the transfer
function as prior knowledge and applying Lagrange multipliers. We have applied the method to time series of
electric conductivity in a river (input signal) and in wells
in the corresponding alluvial aquifer (output signals). In
this context, the inferred transfer functions may be interpreted as the response to a tracer test with pulse-like
injection or as the probability density function of solute
travel time. The physical interpretation as breakthrough
curve and the statistical interpretation as probability density function exemplify the necessity of nonnegativity
because neither negative concentrations nor negative
probability densities are permitted.
In contrast to cross-correlation methods, deconvolution yields a full distribution of travel times rather than
a single optimal time-shift value. Such distributions may
be used to predict the breakthrough of contaminant in
case of an accidental spill into the river. They may also
be used to quantify the risk that a degrading river-borne
compound reaches a production well.
Choosing a nonparametric transfer function gives the
opportunity to identify special features of the travel-time
distribution like the secondary peak of the transfer function for monitoring well 4 or the plateau-like shape identified for the transfer function of monitoring well 1. Such
features would be lost in parametric deconvolution.
Strictly speaking, the convolution principle holds
only for time-invariant processes. As seen in Figure 2,
the hydraulic head in River Thur is fairly dynamic. The
heads in the wells react very fast to head changes in River
Thur. Nonetheless, the hydraulic gradient and therefore
the velocity of ground water are not constant. Even so, we
could obtain good results with a model that neglects such
fluctuations. This is so because the time scale of velocity
fluctuations is smaller than the typical travel times to the
wells. That is, fluctuations of velocity are already partially averaged out while the signal is propagated from
the river to the wells.
In the current study, we have applied the deconvolution to the full data sets of electric conductivity measured
in River Thur and five wells. If different types of behavior at different time periods can be observed, one may
subdivide the time series into several fractions and
perform the analysis for each of them. A prerequisite for
the latter approach is that the considered fractions are
significantly longer than the duration of the inferred
transfer function.
At our field site, electric conductivity has been
proven a valuable and easy-to-measure natural tracer.
Presumably because of hydrogeochemical reactions of
the infiltrating river water with the mineral matrix, electric conductivity is not fully conservative, but the discrepancies between the river and ground water data sets could
be eliminated by removing the seasonal trend. The
method of time series analysis presented here does also
apply to temperature data, such as those measured at our
site (data not shown). Electric conductivity, however, has
the advantage that it propagates faster than temperature
and is less smoothed. For applications at other sites, we
recommend measuring both temperature and electric
conductivity at time intervals of 1 h or smaller. If the
river does not exhibit significant fluctuations in electric
conductivity, or the electric-conductivity signals in the
wells are too strongly disturbed by reactive processes, the
temperature series may be used. In other cases, we prefer
analyzing the electric-conductivity data. Further improvement in natural-tracer analysis could be achieved by considering time series of truly conservative quantities such
as sodium and chloride ions, which, unfortunately, are
more difficult to measure continuously by probes.
Acknowledgments
This research was made possible by a grant from
Canton Thurgau, Switzerland. We are indebted to Romeo
Favero, Andreas Scholtis, and Marco Baumann of Amt
für Umwelt, Canton Thurgau, for installing, equipping,
and maintaining the observation wells. We thank Mark
Bakker and a second reviewer for their constructive
remarks in helping improve the paper.
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