Conference PaperPDF Available

Smart identification system of surface water contamination by an innovative biosensor network

Authors:

Abstract

2) Novaetech Srl-INAF spin-off company, Italy (3) Promete Srl-CNR spin-off company, Italy (4) Abstract The presence of some organic and inorganic contaminants in surface water is a longstanding problem in many countries. Contaminants may reach surface waters, accidentally or intentionally; via agricultural activities or from sewage discharges and they may represent a risk for human, animal and vegetable health. The recent development of ICT technologies and innovative sensors can offer novel opportunities to mitigate these risks helping the relevant authorities to discover the cause of the contamination. In particular, the availability on the market of wireless solutions to connect a large on-line sensor network, coupled with the simulation software to analyze big data in almost real-time, offer opportunities to develop early-warning systems for the protection of surface waters. In this paper, a first preliminary study to design a smart integrated system able to localize one or more emission sources of contaminant is presented. It is considered a scenario in which each biosensor is able to detect the position, magnitude, lifespan and time of the contamination source in a reach enclosed between two biosensors. A possible sensor suitable for the realization of such a smart monitoring system might be a biosensor based on a Quartz Crystal Microbalances (QCM) functionalized with antibodies through a novel immobilization technique, the Photonic Immobilization Technique (PIT). In fact, this biosensor has the high sensitivity typical of QCM coupled to the intrinsic specificity warranted by the antibodies. The biosensors based on QCMs are very cheap and, consequently, could be installed in many sections of a river thus implementing a large monitoring network. 1. INTRODUCTION Improperly managed agricultural activities may impact surface waters as rivers, lakes and other waterbodies. Pesticides are among the most hazardous environmental contaminants owing to their mobility and long-term effects on living organisms. Since in water environment they undergo to various transformations, which may convert them into substances of even greater toxicity, the quick detection of these pollutants together with the early identification of contamination sources is a key aspect in water quality monitoring. To minimize the risks associated with the use of the pesticides, huge public resources have been allocated to develop and implement policies based on substantial scientific evidence, thus reaching the strict regulatory framework for the potential exposure of humans to pesticide residues that is currently in force in the European Union. Directive 2008/105/EC established a list of 33 priority substances requiring special attention in regard to water protection of which one-third are pesticides (Economic European Communities, 2008). Directive 2006/118/EC on the protection of groundwater against pollution and deterioration set the maximum for a specific and for total pesticides. Among the techniques currently used for the determination of pesticides, the most reliable are the chromatography and capillary electrophoresis [1]. The main drawback of these methods is that they
1
Smart identification system of surface water contamination by an
innovative biosensor network
A. Di Nardo1,5*, G.F. Santonastaso1,5, R. Battaglia2,5, D. Musmarra2,5, F.P. Tuccinardi3,5,
F. Castaldo3, B. Della Ventura4,5, M. Iervolino1, R. Velotta4,5
(1) Department of Civil Engineering, Design, Building and Environment, Second University of Naples,
Aversa, Italy
(2) Novaetech Srl - INAF spin-off company, Italy
(3) Promete Srl - CNR spin-off company, Italy
(4) Department of Physics, University of Naples “Federico II”, Italy
(5)Action Group CTRL+SWAN of European Innovative Partnership on Water, EU;
*Corresponding author: E-mail: armando.dinardo@unina2.it, Tel +39 0815010202, Fax: +39 0815037370
Abstract
The presence of some organic and inorganic contaminants in surface water is a longstanding
problem in many countries. Contaminants may reach surface waters, accidentally or intentionally;
via agricultural activities or from sewage discharges and they may represent a risk for human,
animal and vegetable health. The recent development of ICT technologies and innovative sensors
can offer novel opportunities to mitigate these risks helping the relevant authorities to discover the
cause of the contamination. In particular, the availability on the market of wireless solutions to
connect a large on-line sensor network, coupled with the simulation software to analyze big data in
almost real-time, offer opportunities to develop early-warning systems for the protection of surface
waters. In this paper, a first preliminary study to design a smart integrated system able to localize
one or more emission sources of contaminant is presented. It is considered a scenario in which each
biosensor is able to detect the position, magnitude, lifespan and time of the contamination source in
a reach enclosed between two biosensors. A possible sensor suitable for the realization of such a
smart monitoring system might be a biosensor based on a Quartz Crystal Microbalances (QCM)
functionalized with antibodies through a novel immobilization technique, the Photonic
Immobilization Technique (PIT). In fact, this biosensor has the high sensitivity typical of QCM
coupled to the intrinsic specificity warranted by the antibodies. The biosensors based on QCMs are
very cheap and, consequently, could be installed in many sections of a river thus implementing a
large monitoring network.
Keywords: Smart water network; surface water; biosensors; contamination; pesticides, water protection.
1. INTRODUCTION
Improperly managed agricultural activities may impact surface waters as rivers, lakes and other
waterbodies. Pesticides are among the most hazardous environmental contaminants owing to their
mobility and long-term effects on living organisms. Since in water environment they undergo to
various transformations, which may convert them into substances of even greater toxicity, the quick
detection of these pollutants together with the early identification of contamination sources is a key
aspect in water quality monitoring.
To minimize the risks associated with the use of the pesticides, huge public resources have been
allocated to develop and implement policies based on substantial scientific evidence, thus reaching
the strict regulatory framework for the potential exposure of humans to pesticide residues that is
currently in force in the European Union. Directive 2008/105/EC established a list of 33 priority
substances requiring special attention in regard to water protection of which one-third are pesticides
(Economic European Communities, 2008). Directive 2006/118/EC on the protection of groundwater
against pollution and deterioration set the maximum for a specific and for total pesticides.
Among the techniques currently used for the determination of pesticides, the most reliable are the
chromatography and capillary electrophoresis [1]. The main drawback of these methods is that they
2
require sophisticated laboratory equipment and as such they are inherently off-line, thereby being
unable to provide the desirable quick response. If we also consider that such analyses are costly and
require expert personnel we can easily conclude that these techniques are unsuitable for the
realization of monitoring networks.
On the other side, the growing number of samples that have to be analyzed in monitoring studies
demands for an high-throughput via fully automated techniques that utilize easy-to-use equipment
to carry out operations with a low requirement of chemicals if not requiring sample preparation, in
order to substantially simplify the analytical process and minimize handling errors. At the same
time, these techniques should enable the pesticide identification and its quantification with high
accuracy and precision.
To accomplish these broad goals, the recent development of systems for monitoring carried out with
biosensors - Early Warning Systems (EWS) – seems highly promising in order to provide an
indispensable tool for real-time signaling of critical situations. Among the suitable sensors for an
early warning network, Quartz Crystal Microbalances (QCMs) stand out for their limited cost,
robustness and easy readout. Since QCM is essentially a scale in its nature, its functionalization is
necessary in order to make it specific against a pollutant.
To this aim the recently Photonic Immobilization Technique (PIT) [2] offers a simple and effective
way to tether antibodies hooking-upside to the electrode so that their variable part (i.e. the antibody
region capable to recognize the antigen) is exposed to the solution. PIT is based on the UV
irradiation of the antibodies and on the subsequent formation of free thiols which form a stable
bond with the gold. This technique has been successfully demonstrated for contaminants of
environmental interest such as parathion [3] as well as for toxic or undesired compounds in food
like patulin [4] and gliadin. According to the guidelines set forth in European directives related to
water control, these sensors would be ideal candidates for a monitoring sensor network, designed
for the control of the presence of local and diffuse contamination in a warning systems designed to
ensure safety in case of critical events, thus implementing effective policies for environmental
management and health protection.
Once the issue of the appropriate sensor is addressed and data with spatial and temporal distribution
are achieved from the monitoring network, there is still the need of a careful analysis based on the
use of simulation methods able to model diffusion processes starting from the knowledge of the
evolutionary dynamics, the dilution times and the disposal of investigated contaminants.
For instance, in the agricultural application area, the pesticides enter surface waters mainly through
surface runoff so that the resulting concentration in the destination waters is highly variable both
spatially and temporally. Characterizing these fluctuations and their associated risks requires the
implementation of extensive monitoring programs designed in accordance to the relevant scale and
the dissipation - convection processes. Many factors need to be considered for modeling the
pesticide space-time evolution to localize the contamination source: timing and amount of chemical
use, specific land-management practices such as irrigation and drainage, geographic distribution of
natural features.
The identification of the contamination source regards the inverse source problem [5, 6] and, only
in some limited cases it is possible to find solution analitycally [7, 8, 9].
In this paper, a numerical approach, based on heuristic optimization algorithm developed in
MATLAB, is used in a first theoretical application. Specifically a smart monitoring of a river reach
by an innovative biosensors network, able to identify contamination source, is presented.
Specifically, the idea consists in the integration of a smart identification system in each biosensor
able to detect the position, magnitude, lifespan and time of the local contamination to allow the
localization of source in a reach enclosed between by two biosensors.
The spreading of the contaminant in the river stream is described by a one-dimensional
convective/diffusive transport equation. The river is divided into a number of reaches, each
monitored by upstream and downstream biosensors, where flow is assumed to be steady with
constant cross section and velocity.
3
2. SMART IDENTIFICATION SYSTEM BY INNOVATIVE SENSOR NETWORK
The methodology proposed is based on the integration of on-line innovative sensors in a smart
network able to identify the most likely position X, magnitude C*, lifespan δ and injection time T of
a river reach bounded upstream and downstream by two biosensors, where an impulsive discharge
of contaminant occurs.
In the networks for water early-warning, the sensors are the eyes of the system: a dense grid of low-
cost sensors gives more protection than just a few complex stations. The threat of accidental
contamination of water systems is not new, but in the past few years considerable effort is
underway to develop mathematical algorithms in support of contamination events in water
distribution networks. In particular, least squares formulations are applied to early warning systems,
minimizing the difference between measured and simulated concentration predictions, in an attempt
to locate the position X, magnitude C*, lifespan δ and time T. The issue concerning monitoring of
rivers was not so deeply investigated.
In this paper, we analyze inversion methods to determine the optimal parameterization to recover
the injection location along the river reach. The physical processes accounted for in the transport
model are the convection and the diffusion due to hydrodynamic dispersion. We apply regularized
least squares formulations, constrained by chemical transport dynamics, for identifying source
locations from sparse sensor measurements. The continuous measurement of the targeted
contaminant, provided by the biosensors, is analyzed so that the location of the contamination
source can be identified. Of course, the higher the number of biosensors the better the accuracy in
detecting the contamination source.
The smart sensor network is conceived as a Smart Identification System (SIS) able to provide to the
environmental inspectors some precious information about contamination.
2.1 Innovative Biosensors
The piezoelectric sensors are based on the Quartz-Crystal Microbalance (QCM) produced by
Novaetech srl and functionalized at the Department of Physics of the University of Naples
“Federico II”. These two AG partners are applying a novel antibody immobilization technique,
Photonic Immobilization Technique, PIT, [2] to the gold surfaces of a Quartz Crystal
Microbalances (QCMs), thus achieving an immunosensor with the high sensitivity typical of QCM
coupled to the intrinsic specificity warranted by the antibodies. The resulting device has been
optimized for the fluidic circuit thus achieving a “miniaturized” QCM (see Figure 1(a)) with
positive effects on stability and robustness.
In Figure 1(b) the typical output for a single measurement is illustrated, from which it is evident that
only few minutes are required to achieve the response from the device. The limit of detection
(LOD) for Parathion of such a device is approximately 60 nM (nano Molari) [3]. Several other light
analytes have been tested (see e.g. [4] for the patulin) showing that the LOD is always in the range
10-100 nM (approximately few µg/kg for these molecules). Leveraging on these promising results
and given the ample range of contaminants detectable with such a method, the research now aims at
the detection of bacteria in water.
Figure 1, illustrates the biosensor: (a) QCM realized by Novaetech srl connected to the solution to
analyze. The output can be read in remote. (b) Typical immunosensor output with the steps
reported. Phosphate-buffered saline (abbreviated PBS) is a buffer solution commonly used in
biological research, whereas Bovine Serum Albumine (BSA) is used to check that the whole gold
surface is well covered by antibodies.
The final wash ensures one that the sensor is ready for the measurement. The three steps are carried
out once for all and the sensor is ready for measurements. The change of approximately 80 Hz is
observed when a solution of 1.7 µM of Parathion is conveyed to the QCM. A limit of detection
(LOD) of approximately 60 nM is achieved with such immunosensor.
4
Figure 1. Miniaturized QCM (a) and typical output (b)
2.1 Sensor network
Sensor network is realised by placing some smart biosensors Bn along the river, illustrated
schematically with an indefinite line in the Figure 2. In particular, a river reach is defined by two
biosensors Bs and Bs+1. Given the number of biosensors, it is possible to define each reach and limit
the space where the contamination may have occurred in terms of some identification variables:
position X, magnitude C*, lifespan δ and injection time T. The identification process is achieved
comparing measures in Bs and Bs+1 vs. simulated values using a suitable iterative heuristic procedure
based on the optimization method proposed by [10].
Figure 2. Identification variables and network sector.
2.2 Mono-dimensional diffusion model
The transport of a conservative pollutant dissolved in the flowing water can be described as an
advection-dispersion process and the 1-D model equation for a non-prismatic open channel can be
written as follows [11]:
( ) ( )
!
"
#
$
%
&
+
=
x
C
DA
x
CQ
x
CA
t
(1)
where A is the river cross-section area, C the average solute concentration, D the dispersion
coefficient, Q is the volumetric flow rate, t is the time and x is the streamwise coordinate. For
steady uniform flow, when A=cost, Q=cost and D=constant, Equation (1) simplifies into:
2
2
x
C
D
x
C
U
t
C
+
=
(2)
Furthermore to take account the emission of pollutant in the channel it can be assumed that the
mixing of fluid with the injected pollutant is complete and instantaneous. Thus the downstream
concentration can be computed as follows:
pollutant
pollutantpollutant1
QQ
QCQC
Cx
x+
+
=
(3)
Where the Cx is the pollutant concentration in emission downstream section, Cx-1 is the upstream
pollutant concentration, Qpollutant and Cpollutant are the flow rate and the concentration of pollutant,
5
respectively. The partial differential equations PDEs (2) has been solved numerically by the finite-
difference method [12] proposed by [13], with the following boundary conditions:
!
"
!
#
$
==
==
Lx
x
C
xC
0
0co nst
(4)
2.3 Contamination emission scenarios
The identification algorithm is based on the use of an input parameterized emission that can
represent some possible contamination typologies produced by industrial or agricultural processes.
As illustrated in the Figure 3, in this preliminary work, the shape of contamination emission is
simplified as a triangular input parameterized to change the concentration magnitude C*,the
positioning X, lifespan δ and time delay T.
Specifically two Contamination Scenarios (CS) have been investigated: CS1) a triangular
pollutograph in a clean reach (i.e. null concentration from upstream); CS2) a triangular pollutograph
over a constant upstream concentration Cin (measured by biosensor Bs).
S1)
Figure 3. Two Input Emission scenarios.
2.2 Contaminant identification algorithm
The identification process is based on a real-time processing driven by analysis of the measure of
each couple biosensors in the network, Bs and Bs+1 . The identification of position X, magnitude C*,
lifespan δ and time delay T of the contamination is carried out by an optimization algorithm.
Specifically, in the reasonable hypothesis of mass conservation in the channel, without spills or
phase transformation during the flow, the total mass of contaminant injected in the reach is
measured by the downstream biosensors Bs+1. Based on this information, it is possible to define the
parameter C* for a triangular emission, which discharges at distance X the same total mass
measured by the downstream sensor Bs+1 as a function of lifespan δ with a time delay T:
=
m
0
m
*dtC
2
C
δ
δ
(5)
where δm and Cm are the measured lifespan and concentration of pollutant in Bs+1. The identification
process is carried out minimizing the following objective function (FO):
( )
=
=
ma x
0
2
)()(
t
t
im tCtCFO
(6)
where tmax is the total time of simulation, Ci(t) is the pollutant concentration at time t computed in
the downstream section by numerical solution of equation (2), and Cm(t) is the measured
concentration at time t.
6
For the minimization of FO (6) a modified Nelder-Mead algorithm was used, starting form an initial
solution provided by the user. The initial solution has been selected by sampling the solution space
in terms of injection position X and time T with a sampling frequency of ΔX =
1000
1+ss BB
in the
range [Bs; Bs+1] and ΔT = Tm/1000 for the starting time in the range [0; tmax], while the lifespan has
been assumed equal to
3
m
δ
, and taking the sample with the minimum FO value. Starting from the
initial solution, the optimization algorithm has been repeated, computing equation (2), until the FO
becomes smaller than
001.0=
ε
using the solution of each previous iteration as the initial point for
the next one. The identification process is schematically illustrated in the flow chart of Figure 4.
Figure 4. Flow chart of SIS
3. RESULTS
The methodology has been tested on a synthetic example reach with the following features: a length
of 10 km, a flow rate Q=20 m3/sec, mean velocity v=0.7 m/sec and dispersion coefficient D=10
m2/sec in compliance with hydraulic properties of a typical Italian alluvial river.
Then, the parameters of theoretical inputs of contamination (Cin, Cpollutant, δ, X, T) assumed as
“measured concentration” (MC) in two simulated scenarios, are reported in Table 1. Furthermore,
the volume of contaminant inserted in the both scenarios is equal to W= 10 m3.
Table 1. Contamination parameters
In the Table 2, the identification results are reported, showing the effectiveness of SIS with very
low values of FO and practically the correct value of the injection point (with a maximum error of
50 m) of the concentration magnitude C*, the positioning X, lifespan δ and time delay T for both
scenarios considered.
CS1
CS2
Cin [mg/l]
0.00
0.20
Cpollutant [mg/l]
5 ·103
5 ·103
δ [sec]
900
900
X [m]
3000
6000
T [sec]
3600
500
W [m3]
10
10
7
a)
b)
c)
d)
Figure 5. Measured Concentration in Bs and Bs+1 respectively for CS1 (a)-(b) and CS2 (c)-(d).
Table 2. Identification results for SC1 and SC2
The effectiveness of SIS is also confirmed by the comparison illustrated in the Figure 5 in which
with continuous line is represented the Measured Concentration (MC) vs. Contamination Scenario 1
(CS1) and Measured Concentration (MC) vs. Contamination Scenario 2 (CS2).
a)
b)
Figure 6. MC vs. CS1(a) and MC vs. CS1(b).
Magnitude C*
[mg/l]
Lifespan δ
[sec]
Positioning X
[m]
Time T
[sec]
FO
Scenario 1
2.769
903.744
3018.189
3619.230
3.566·10-4
Scenario 1
2.768
904.488
6018.008
518.965
9.516·10-4
8
In the present example, in which in the Figure 6 the lines are practically overlapping, the
identification system is therefore able to detect with reasonable accuracy the features of the
contamination: further research will be devoted to investigate the effect on the identification of
potential source of disturbance such as measuring errors, uncertainties in the flow parameters,
unknown shape of the pollutograph.
4. CONCLUSIONS
A preliminary setup of a smart integrated system for the identification of contamination sources in
rivers has been presented. The monitoring system is based on Quartz Crystal Microbalances
(QCMs) biosensor functionalized with antibodies through the novel Photonic Immobilization
Technique (PIT). The continuously measured concentration in the downstream reach of a river is
used to detect the contamination occurrence. Once the contamination has occurred, an identification
algorithm is run in order to detect the position, the magnitude, the lifespan and the time delay of the
pollutant discharged into the river reach, by coupling a simplified transport model with a heuristic
optimization technique. The procedure has been applied to a synthetic test-case in the absence or in
presence of an upstream pollutant concentration, with encouraging results about the detection
accuracy.
References
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... The serious problem of river water contamination is even more frequent and essentially due to industrialization, rapid population growth and increase in urbanization [1], but also to the use of chemical agents in agricultural [2]. These contamination events can occur also accidentally, for example, during the interruption of service of a treatment plant but, unfortunately, in some cases, are intentionally caused through non-authorized discharges of urban wastewaters. ...
... The recent development of biosensors [4,5], based on different technologies, as on Luminescent bacteria, combined with the quantitative measurement of chemical parameters applying standard spectrophotometric/fluorimetric methods, Quartz Crystal Microbalances (QCMs), Fluorimetric methods using specific organic liquid reagents, Nanofiber materials with electrochemical techniques, Optofluidic jet waveguide, Microwave resonator, Electrochemical, Bio-based electrodes, etc. [5] seems highly promising and able to provide devices for real-time monitoring of critical situations also with low costs [2]. These technologies can be arranged in a sensor network for water early-warning in which the sensors are the eyes of the system. ...
... The methodology proposed is based on the comparison between the analytical solution and a numerical algorithm in order to identify the most likely position X, magnitude C * , lifespan δ and injection time T of the impulsive contamination in a river reach equipped with two innovative biosensors, upstream and downstream, as reported in the Figure 1. As already proposed in [2], the sensor network can be realised by placing an array of B n smart biosensors along the river, illustrated schematically with an indefinite line in the Figure 1. In particular, the river can be split into several reaches, generally bounded by two biosensors B s and B s+1 . ...
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The identification of the source of a contamination in a river, either accidental or intentional, is of extreme interest for the technical and scientific communities, because it is the preliminary step towards risk mitigation for human, animal and vegetable health. Recently, the development of innovative wireless sensors, coupled with computational tools appropriate to analyse big data almost in real-time, offers novel opportunities to develop early-warning systems for the protection of surface waters. Within this framework, it is possible to use exact solutions of the equations describing the conservation of contaminant mass to identify the location where contamination took place, if the geometry, the hydraulic parameters and the boundary conditions are sufficiently simple. On the other hand, for more general conditions these equations can be solved only numerically. In both cases, an inverse problem can be formulated coupling the appropriate solution method to heuristic optimization algorithms that provide an optimal solution for the magnitude, the shape and the position of the contamination sources. In this paper, a novel identification algorithm, based on a parametric method for the contamination inputs, a genetic algorithm for optimization problem and a statistical approach to define the best solution, is proposed. As far as the mass conservation equation is concerned, the response to a rectangular contamination pattern is computed by both the exact solution based on the convolution of the pulse response and the finite difference Cranck-Nicolson discretization of the governing equations. These procedures have been applied on a hypothetical case study of Volturno river in South of Italy, close to the town of Caserta, modelled as one-dimensional flow. The comparison between the two approaches, exact and numerical, to the governing equation solution provides interesting results in terms of effectiveness, time responses and model limitations, useful to implement an identification network of smart sensors.
... The serious problem of river contamination is more and more frequent and essentially due to industrialization, rapid population growth and increase in urbanization [1], but also to the use of chemical agents in agricultural practices [2]. These contamination events can also occur accidentally, for example, during the interruption of service of a treatment plant but, unfortunately, in some cases, they occur intentionally as a consequence of non-authorized discharges of wastewaters. ...
... The recent development of biosensors [4][5][6][7][8][9][10][11], based on different technologies -such as on luminescent bacteria, combined with the quantitative measurement of chemical parameters applying standard spectrophotometric/fluorimetric methods, quartz crystal microbalances, fluorimetric methods using specific organic liquid reagents, nanofiber materials with electrochemical techniques, optofluidic jet waveguide, microwave resonator, electrochemical, bio-based electrodes, etc -seems highly promising and able to provide devices for real-time monitoring of critical situations also at low costs [2]. These technologies can be arranged in a sensor network for water early warning, overcoming most of the operational difficulties described above, in which the sensors are the "eyes" of the system. ...
... Starting from the preliminary results reported in [2], in this paper, a novel identification algorithm, based on a parametric method for the contamination inputs, a suitable mathematical model for the contaminant mass conservation and transport, a genetic algorithm for optimization problem and a statistical approach to define the best solution, is proposed. ...
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Fast and accurate identification of unknown pollution sources is a crucial and challenging task in water resources management, in which the characteristics of unknown pollution sources should be determined from temporal or spatial distribution of concentration data at limited observation points. This field of research is currently in the stage of rapid development, but is not thorough yet. A review of the literature suggested that, one of the main gaps lies in the extension of methods to more realistic cases. Several methods were introduced to solve the source identification problem with novel variations and extensions, however most of them had never been tested in real-world applications and the underlying transport processes was greatly simplified. Most recently the paramount importance of the practical aspect of the pollution source identification problem is more appreciated. More particularly, there have been a growing interest to consider complex real-world scenarios, reduce associated computational cost and quantify the uncertainty in retrieved source characteristics. However, the dissemination of research results to practice has been slow. This paper reviews some studies in the context of pollution source identification in surface and groundwater. The current issues, limitations, and suggestions for future studies are also discussed.
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Studies on pollution source identification in water resources date back over three decades. These studies use inverse solution of the transport equation for recovering the release history and/or the location of the pollutant sources. Each of these studies has its own advantages and limitations in accordance with the methods employed and the complexity of the solutions. Moreover, conducted studies on pollution source identification in surface water resources (e.g. rivers) are fewer in number compared to those on groundwater resources. In nearly all previous related studies on rivers, researchers have often developed their own numerical forward models, but these models have never taken into account the complexity of the problem in real conditions, such as river topography, real flow conditions, etc. Therefore, compared to commercial models, these models have many drawbacks and limitations. Thus, it would be desirable to provide a framework that can be employed to solve the transport equation in the inverse model using available software packages. In this study, the inverse solution of the transport equation for recovering release histories of multiple pollutant sources in rivers is achieved by the help of ready software packages via a feasible framework. The proposed framework can be readily applied in one-, two-, and three-dimensional problems. The underlying concept of the proposed framework is the linearity of the governing equation, i.e. transport equation. Furthermore, the present study presents the conditions and rules for the arrangement (number and location) of the measurement points of the pollutant concentration under various conditions. The proper arrangement of the measurement points is significant, since it solves the non-uniqueness problem of the inverse model. The most important factor affecting the arrangement of the measurement points is the flow pattern. In this study, it is suggested that the complexity of the flow pattern may lead to practical arrangements for the control points. The proposed procedure was verified by the use of three series of data in one- and two-dimensional domains under real conditions of flow and topography by employing well-known existing software packages. In each test case, a proper and practical arrangement was proposed for the measurement points of the concentration-time curve. The suggested arrangement resulted in the correct operation of the inverse model. The proposed inverse model showed a good capability with a reasonable percentage of errors of the recovering release rate of pollutant sources. Accordingly, it can be conveniently used in cases where forward models are readily available.
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We consider the problem of identification of a pollution source in a river. The mathematical model is a one-dimensional linear advection-dispersion-reaction equation L[u](x,t)=F(x,t),0<x<ℓ,0<t<T, where L[u](x,t)=∂ t u(x,t)-D∂ xx u(x,t)+V∂ x u(x,t)+Ru(x,t) with the right-hand side spatially supported at a point (the source) and a time-dependent intensity, both unknown. Assuming that the source becomes inactive after the time T * , we prove that it can be identified by recording the evolution of the concentration at two points, one of which is strategic.
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This book introduces the reader to the area of inverse problems. A relatively new branch of Applied Mathematics, the study of inverse problems is of vital interest to many areas of science and technology such as geophysical exploration, system identification, nondestructive testing and ultrasonic tomography. The aim of this book is twofold: in the first part, the reader is exposed to the basic notions and difficulties encountered with ill-posed problems. Basic properties of regularization methods for linear ill-posed problems are studied by means of several simple analytical and numerical examples.The second part of the book presents two special nonlinear inverse problems in detail - the inverse spectral problem and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness and continuous dependence on parameters. Then some theoretical results as well as numerical procedures for the inverse problems are discussed. The choice of material and its presentation in the book are new, thus making it particularly suitable for graduate students. Basic knowledge of real analysis is assumed.
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In this paper we consider an inverse problem in which we seek to deter- mine an unknown source or reaction term in a reaction-diffusion equation from overspecified data measured on the boundary of the spatial region where the equation applies. The analysis is based on the observation that the overspecified data depends monotonically on the unknown source term in the equation. Here we use this monotonicity to establish a type of unicity result for the inverse problem. The attractiveness of the monotonicity methods illustrated here lies in their versatility. Surveying the rapidly expanding literature devoted to the topic of inverse problems, one finds a considerable variety of ad hoc approaches. Methods which apply to a general class of problems seem to be few. Monotonicity methods, however, apply to a variety of inverse problems involving partial differential equations of parabolic or elliptic type. This paper is organized as follows. In Section 1 we formulate the direct initial boundary value problem (IBVP) and state assumptions on the data and a priori assumptions on the unknown source term. Several properties of the solution to the direct problem are derived. In Section 2 the principal monotonicity estimates are established and in Section 3 these are applied to the inverse problem to prove a type of unicity result for the solution to the inverse problem. In particular, it is a Corollary of Theorem 3.2 that in the class of analytic source terms, the solution of the inverse problem is unique. The problem of identifying an unknown source term in a heat equation 155
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