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Water Emerg Contam Nanoplastics 2022;1:12. 10.20517/wecn.2022.07 © The Author(s) 2022.
Open Access Research Article

Data-based interpretation of emerging contaminants occurrence in rivers using a simple advection-reaction model

1IDAEA, CSIC, Jordi Girona 18-26, Barcelona 08034, Spain.

2ICRA-CERCA, Catalan Institute for Water Research, Scientific and Technologic Park of the UdG, Girona 17003, Spain.

Correspondence to: Prof. Antonio Ginebreda, IDAEA, CSIC, Jordi Girona 18-26, Barcelona 08034, Spain. E-mail:

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    Academic Editor: Teresa A.P. Rocha-Santos | Copy Editor: Haixia Wang | Production Editor: Haixia Wang

    © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License (, which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.


    Our knowledge of the river’s qualitative status generally relies on discrete spatial and temporal observations organized under what is commonly known as a “monitoring network”. Network performance is constrained by its spatial - temporal resolution, which is severely limited by the costs associated with the whole sampling and analytical process. Alternatively, modeling allows predicting the spatial - temporal variable profile at any resolution at affordable computing costs. However, it involves high uncertainty in the parameterization and requires experimental validation as well. Here, we aimed at reconciling monitoring and modeling, deriving simple steady-state advection-reaction (reactive-transport) models from monitoring data. They are based on graph-theoretical concepts, notably the use of the Laplacian matrix, which captures the river network topology, the interaction between adjacent sites, and the advection process between them. The local reactive process is described by a first-order decay reaction. The application of these models provided relevant information about the variables monitored, such as the local dynamics, the distance of the site’s influence, the degree of synchronization, or the external input/output to the system, which is useful for both scientific and management purposes.

    The model was tested in the Llobregat River (NE Spain) basin, with 70 emerging contaminants of different classes (pharmaceuticals, pesticides, perfluorinated substances, endocrine disruptors, and drugs of abuse). The monitoring network included 14 sites (7 in the mainstream, 4 in the Cardener, and 3 in the Anoia tributaries) and was monitored in 2 campaigns. These models can help water managers to optimize the design of river monitoring networks, a key aspect of environmental regulations.


    Rivers are net receivers of both point and diffuse pollution, such as nutrients, metals, emerging pollutants, etc.[1], which are considered one of the main causes of freshwater biodiversity impairment[2,3]. Many chemical pollutants are not environmentally persistent; rather, they change due to multiple biotic (biodegradation) and abiotic processes (sorption, photolysis, hydrolysis, etc.), giving rise to additional transformation products. Rivers extend more or less continuously through space and time under the influence of their catchment area. However, only a few variables can be measured with the highest resolution in time (i.e., online sensors) or space (i.e., remote sensing), and none in both dimensions[4]. Therefore, our knowledge of the river’s qualitative status relies on discrete spatial and temporal observations of a set of physical, chemical, or biological parameters, organized under what is commonly known as a “monitoring network” [Figure 1]. Monitoring networks are typically constituted by several sites deployed throughout the river basin area, which are sampled at a certain time-frequency. This is a current practice used for either research or management purposes and has given rise to large databases. Indeed, monitoring networks are a key aspect of the implementation of environmental regulations such as the EU Water Framework Directive (Directive 2000/60/EC). The accuracy of the “picture” obtained depends on both the spatial and temporal “resolution” of the network used. However, it is constrained by the economic cost of the whole sampling and analytical process, often expensive, as is the case of emerging contaminants. To cope with the inherent limitations of discrete monitoring networks, dynamic modeling of chemical fate and transport processes was raised as a complementary alternative[5]. While both approaches - monitoring and modeling - have their respective pros and cons, there is a growing interest in the latter due to the increasing development and affordability of computation and information techniques compared to costly monitoring campaigns[5,6]. Modeling approaches have been mostly focused on the prediction of environmental concentrations of pollutants. Existing models include GREAT-ER[7,8], PhATE[9], LF2000-WQX[10,11], and MERLIN-EXPO[12], among others[13-16]. However, advanced modeling is not exempt from limitations. They are “data-hungry” and require combining a hydrological model with that of the physical - chemical processes taking place in the river, whose parameterization is often subjected to high uncertainty[17] that depends on local conditions[18-20]. Moreover, they ultimately need to be validated through experimental measurements. Altogether, monitoring and modeling must be regarded as complementary tools that should be advantageously used together[5,21].

    Figure 1. (A) Map of the Llobregat River basin showing the sampling sites. (B) Network graph representation of the area studied. The distances (km) between adjacent sites are indicated in blue.

    Although the use of spatial (topological) relationships in the study of rivers is not new[21-27], here, we explore new possibilities for exploitation of experimental data available from river monitoring networks, and their interpretation in the light of simple reactive-transport (or advection-reaction) that can be readily derived therefrom. To do so, we make use of some graph-theoretical concepts recently used in the development of models (oscillation models) applied in other domains such as social networks (Internet networks, etc.)[28,29].

    The proposed advection-reaction model approach was tested using an available dataset that includes concentrations of ca. 70 emerging contaminants belonging to several chemical families and sampled alongside the Llobregat River (NE Spain) during 2010-2011 as part of the project SCARCE[30].


    Study area

    The Llobregat River (NE Spain)[31] is 156 km long and covers a catchment area of about 4957 km2 [Figure 1]. From the hydrologic point of view, the Llobregat is a typical Mediterranean river, its flow being characterized by high variability, which is closely controlled by seasonal rainfall. The mean annual precipitation is 3330 Hm3, and it has an annual average discharge of 693 Hm3. Its watershed is heavily populated with more than three million inhabitants, mostly in the lowest part located in the surroundings of the city of Barcelona. Together with its two main tributaries, River Cardener and River Anoia, the Llobregat is subjected to heavy anthropogenic pressure, receiving extensive urban and industrial wastewater discharges (137 Hm3/year; 92% coming from the wastewater treatment plants), which constitutes a significant part of its natural flow. Overall, 48% of these point sources are located in the studied area. Furthermore, in the middle part of the basin, it receives brine leachates from natural salt formations and mining operations, which have caused an increase in water salinity downstream. The Llobregat is thus an illustrative example of overexploited river.

    Advection-reaction model based on network theoretical concepts applicable to a river network of discrete monitoring measurements

    Network theory concepts

    The river studied can be described as a set of spatially distributed connected sites (nodes) in which we carry out measurements of a variable x (e.g., the concentration of a contaminant) at a certain time, which constitute the monitoring network [Figure 1]. Its structure is conveniently described[28,29] by a graph G(E, V), where V = {1, …, n} denotes the set of n nodes representing the monitoring sites and E is the set of edges between nodes. The network structure of nodes and links is captured by the adjacency matrix A. It is an n × n square defined as:

    We define the degree di of node i (i = 1, ..., n) as:

    where denotes the set of nodes adjacent (connected) to node i. The weighted degree matrix D can be thus defined as:

    The so-called graph Laplacian matrix is defined as:

    The elements of L are given by:

    The Laplacian matrix L is real, symmetric, and semi-definite positive, with all the eigenvalues nonnegative, being always the first eigenvalue λ1 = 0 the smallest one and its associated eigenvector v1= (1, 1, 1...1)T. This vector corresponds to the fully synchronized state, in which the variable studied x has the same value in all the nodes (i.e., x1=x2=…xi…= xn-1= xn).

    The network advection-reaction model

    The model provides a simple and general description capturing the network dynamics. Briefly, let the state of node i be xi so that x := (x1, .... , xn)T is the n-dimensional state vector of measurements of variable x for all the nodes. It is assumed that the time evolution of xi can be described by the following simple kinetic equation:

    where the first term on the right side of the equation reflects a first-order decay process with a rate constant k and the second term is an advection process between the sites connected, characterized by a mean advection velocity (see comment below). The terms δi are local inputs or sinks at each site i. We assume that, for a given compound, the rate constants k and the advection velocities v are equal throughout all the space (i.e., alongside the n measurement sites), although they may change at each sampling time. These are common assumptions in many models.

    It can be written in the following compact vector form:

    Assuming that the measurements correspond to a stationary state ( = 0), rearranging the above equation, and dividing both sides by k, it becomes:

    Numerically, the parameter could be assimilated to the slope of the regression line of Lx over x with the intercept set equal to 0. Its calculation can be readily done by ordinary least squares (OLS), and the vector ε = δ/k of errors (ε = x - ·Lx) captures the local input/output (under the OLS assumptions).

    It can be shown that the inverse of the OLS calculated slope is equal to:

    Noting that the right-hand expression in the above equation is the Rayleigh quotient of the Laplacian matrix L (, for any vector x, ρ is bounded between its minimum and maximum eigenvalues, so that:

    Furthermore, expanding x in terms of the normalized eigenvectors u of L, and considering that ci = uiTx (graph Fourier transform of x), we have the following expression that quantifies the contribution of the different eigenstates λi to ρ:

    with Equation (6). This allows defining an entropy S as:

    Entropy S provides an insight into how the ρ is allocated among the different eigenstates. In turn, the terms capture [Equation (5)] the respective contribution of each eigenstate i. For our purposes, the first one is particularly relevant, namely , the eigenstate associated with the first eigenvalue λ1 = 0, which quantifies the weight of the synchronized state (note, however, that the contribution of the synchronized state to ρ is zero since λ1 = 0).

    Definition of the “weighted” Laplacian for practical use

    Without loss of generality, the foregoing definitions of the adjacency and Laplacian matrices can be extended to “weighted” analogs, i.e., Aij = wij, that better capture real problems. Here, the weight associated with the edge (i, j) wij of the aquifer weighted adjacency matrix elements was set equal to the inverse of the distance rij between connected sites i and j: wij = 1/rij (dimension L−1). Defined in this way, it is worth noting that the terms (xi - xj)/rij in Equation (1) can be regarded as the discrete counterparts of a gradient ∂x/∂r. Hence, the term has the dimension L and can be interpreted as a characteristic length (referred to hereafter as ℓ). In addition, may be interpreted as a measure of the relative relevance of the two processes involved, i.e., advection (between neighbor nodes) and decay (local at each node). Likely, the vector ε = δ/k has the dimension of concentration [Equation (3)] and provides information on the local inputs/output of the x variable.


    All the calculations were performed using Excel (Microdsoft®) and the R environment (version 4.1.0). Measurements below the limit of detection (< LOD) and the limit of quantification (< LOQ) were set equal to 0 for calculation purposes.

    Monitoring of emerging contaminants

    The sampling of water for chemical characterization was performed at the end of the summer period in campaigns carried out in autumn for two consecutive years, namely 2010 (C1) and 2011 (C2). Sampling sites (14) were distributed in the Llobregat mainstream (seven sites), and the Cardener and Anoia tributaries (four and three sites, respectively) [Figure 1]. At each site, grab water samples were taken for chemical analyses of the organic micropollutants. In total, 199 organic micropollutants were measured using previously published analytical methods based on gas chromatography-tandem mass spectrometry or liquid chromatography - tandem mass spectrometry for pesticides[32], pharmaceuticals[33], endocrine-disrupting chemicals (EDCs) and related compounds (hormones, plasticizers, alkylphenols, parabens, phosphate flame retardants, anticorrosion agents, and bactericides)[34], perfluorinated compounds[35,36], and drugs of abuse[37].

    To avoid an excessive number of “0 values” (i.e., not detected or not quantified), the above-described methods were applied to compounds having detection frequencies equal to or greater than 20% in each of the two sampling campaigns performed, which resulted in a final selection of 71 compounds out of the 199 monitored [Supplementary Table 1].


    Monitoring information (limits of detection, detection frequency, and mean and maximum concentrations) corresponding to the selected compounds and the two campaigns are collected in Supplementary Table 1. Compounds were grouped per class as follows: drugs of abuse (5), endocrine disruptors (13), perfluorinated compounds (6), pesticides (6), and pharmaceuticals (37). A full description and discussion of monitoring results[32-37] and their associated ecotoxicological risk[38,39], as well as their relationship with the receiving ecosystems[1,40,41], was previously published.

    Advection-reaction network models

    The application of the above-described advection-reaction model (Section “Study area”) to each of the 71 compounds selected provided as main quantitative outputs the following information: (a) characteristic distances (ℓ) [Equation (3)]; (b) entropy [Equation (6)]; (c) local input/outputs [Equation (3)]; and (d) the contribution of the different eigenstates [Equation (5)]. Although these quantitative indicators are not fully independent and exhibit some quantitative relationships (see below), they provide somewhat complementary information useful for different interpretation purposes. Whereas the characteristic length (ℓ = ) gives an idea of the distance through which the advection process is effective compared to the decay process, its inverse [ ρ = xTLx/xTx, Equation (4)] is often referred to in graph theory as the “network energy”. Finally, its spectral decomposition in terms of the eigenvectors and eigenvalues of L allows us to describe it as a superposition (sum) of the different eigenstates (modes) contained in L [Equation (5)]. Among the latter, the contribution of the fully synchronized state, in which all sites have the same concentration for the variable under study, is particularly relevant, since it corresponds to the equilibrium state in which the concentration of the measured variable is the same in all sites. Finally, the relative contribution of each eigenstate to ρ (“network energy”) is conveniently captured by an entropy [Equation (6)] (the parallelism of the foregoing definitions with the corresponding thermodynamic concepts is rather evident). The statistical distributions of characteristic lengths (ℓ), entropies, and synchronized state contribution are shown in Figure 2A-C, respectively, for the different families of compounds and sampling campaigns. Characteristic lengths of the whole compound set were between 123 and 2.3 km, with a median of 16.4 km. This value is of the same order as the distance between adjacent monitoring sites (mean, 20.6 km; max, 55.2; min, 2.3 km; median, 18.2 km), meaning that the monitoring network fits the requirements reasonably well. In that sense, the characteristic lengths provide a relevant insight for the optimization of monitoring networks, whose separation between sites should ideally take into consideration the range of lengths of the set of compounds monitored (notably those with the shortest ones). In general, ℓ values in campaign C1 were larger than those in C2 for all the compound classes [Figure 2A], with more pronounced values and differences between campaigns corresponding to pesticides and perfluorinated compounds. Such small but perceptible differences could possibly be explained through corresponding variations in hydrology. Thus, for instance, whereas the autumn of 2010 was characterized by intense precipitation which resulted in a high flow, the autumn of 2011 was dry and the river flows were comparatively lower. High flows give rise to larger linear velocities and, hence, longer characteristic lengths, which tend to increase the contribution of the advection process, thus improving the river connectivity. This pattern (and its hydrological explanation) is also reflected in the increasing contribution of the synchronized state [Figure 2C]. It is worth noting that, in a hypothetical fully synchronized (i.e., 100%) situation, ρ = 0 [Equation (3)], and its inverse, the characteristic length ℓ, would thus be infinite. In that sense, the contribution of the synchronized state is typically higher in C1 than in C2, with maximum differences occurring in the case of pesticides.

    Figure 2. Boxplots showing the distribution of the values per sampling campaign (C1, 2010; C2, 2011) for the different compound classes: (A) characteristic length (km); (B) entropy; and (C) synchronized state contribution. The upper and lower bounds correspond to maximum and minimum values; boxes are limited by the 75th and 25th quartiles; and dots correspond to outliers.

    Entropy values for the different compound classes are given in Figure 2B. They show the lowest values for pesticides and appear systematically lower in C1 than in C2. Entropy captures the system “complexity”, providing a quantitative measure of how the different eigenstates of the Laplacian matrix contribute to the description of the system. Entropy takes its maximum value when all the states are equally allocated. High entropy values thus reflect a heterogenous hydrological environment and would be indicative of a “fragmented” situation, as happens under water scarcity, typical of Mediterranean rivers[42].

    As mentioned above, the three properties studied exhibit related trends, which are depicted in Figure 3A and B. Overall, the characteristic length has a negative correlation with entropy (Pearson correlation = -0.748, Figure 3A), while synchronization shows a positive one (Pearson correlation = 0.561, Figure 3B).

    Figure 3. Scatter plots showing the relationships between characteristic length and entropy as well as synchronized state of all compounds studied in the two campaigns (n = 142): (A) characteristic length (km) vs. entropy; and (B) characteristic length (km) vs. synchronized state contribution.

    Further analysis of the processes involved can be achieved considering the respective contributions of the different eigenstates of the Laplacian matrix L to Equations (5) and (6). Specifically, the contribution of the synchronized state (i.e., equal values of measured pollutant x in all nodes) vs. higher states is relevant, considering that a fully synchronized state would have ρ = 0 [Equation (5)], and thus an infinite characteristic length. In that sense, as one may expect, synchronization was aligned with a characteristic length [Figure 2A]. Focusing on the individual compounds, the highest values and differences between campaigns of characteristic lengths [Figure 4A] were found in some pesticides (chlorpyriphos, imazalil, and diazinon), perfluorinated compounds (PFOA and PFPeA), and pharmaceuticals (non-steroidal antiinflammatories naproxen, ibuprofen, ketoprofen, and meloxicam), with notably higher values in the 2010 campaign (C1).

    Figure 4. Value distribution per compound along the two sampling campaigns monitored (C1, 2010; C2, 2011): (A) characteristic length (km); (B) entropy; and (C) contribution of the synchronized state (%).

    Contrastingly, entropy values were generally higher in 2011 (C2) than in 2010 (C1) [Figure 4B], with the largest differences corresponding to the group of pesticides (isoproturon, imazalil, diazinon, and chlorpyriphos), while the synchronization showed a mixed situation [Figure 4C].

    Relevant information regarding the input/output pollution load throughout the basin can be obtained from the vector ε = δ/k of Equation (3), whose components’ signs provide insight into each sampling site’s behavior, indicating if it is a net receiver (positive) or a sink (negative) of pollution. Figure 5 shows some examples, including the total pollution per site along with the two campaigns C1 and C2 (i.e., the variable x was the aggregated sum of all pollutants measured at a site) [Figure 5D], as well as some individual compounds representative of the different sources of pollution, namely the anticorrosion agent 1H-benzotriazole (industrial), the insecticide chlorpyriphos (agriculture), and the pharmaceutical diclofenac (urban) (Figure 5A-C, respectively). Concerning the absolute input/output quantities, the observed order was: sum of all pollutants > 1H-benzotriazole > diclofenac > chlorpyriphos. Each one exceeded the next by circa one order of magnitude. Both campaigns showed similar input/output profiles for chlorpyriphos and the sum of all pollutants, while the industrial and urban representatives showed slightly higher input values in C2 than in C1, which is consistent with the hydrology (i.e., lower flow dilution in the former campaign). Input/output patterns per site were roughly similar for the sum of all pollutants, 1H-benzotriazole, and diclofenac, which is consistent with their respective contribution to the total pollutant load. They highlighted some sites such as ANO3 and LLO3 where pollution depletion exceeded the pollution received (negative sign), while others such as ANO1, CAR4, or LLO6 (positive sign) exhibited the opposite behavior, thus indicating that the reception of external sources of pollution exceeded their elimination capacity. The latter two are well-known polluted sites by both industry and urban population sources, located close to the Manresa and Barcelona metropolitan areas, respectively. Some other sites, mainly located in the relatively clean upper course of the Cardener tributary (CAR1, CAR2, and CAR3), have a “neutral” behavior, characterized by null values. Contrastingly, the insecticide chlorpyriphos exhibited a somewhat different pattern, spread along the mid and lower river basin, with the largest inputs observed in ANO2, CAR3, LLO3, and LLO6.

    Figure 5. Net input/output per site along the two sampling campaigns monitored (C1, 2010; C2, 2011) at each sampling site of: (A) 1H-benzotriazole (anticorrosion agent, industrial use); (B) chlorpyriphos (insecticide, agriculture use); (C) diclofenac (pharmaceutical, urban use); and (D) total pollution load (expressed as the sum of all pollutants analyzed).


    River basin quality status assessment is largely determined through discrete measurements of different variables carried out at certain sites located alongside the catchment and specified time frequencies, under what is usually known as a “monitoring network”. These are currently used for both research and water management purposes and have given rise to large datasets. Network performance is constrained by its spatial-temporal resolution, which, in turn, is severely limited by the, often expensive, costs associated with the whole sampling and analytical process, as in the case of emerging contaminants. Alternatively, modeling allows predicting the spatial-temporal variable profile at any resolution at an affordable computing cost. However, it involves high uncertainty in the parameterization and requires, in the end, experimental validation.

    In the present short contribution, we aimed at reconciling both approaches - monitoring and modeling - through the interpretation of the experimental data under the framework of a simple advection-reaction (i.e., reactive-transport) model based on graph-theoretical concepts, notably the use of the Laplacian matrix, which adequately captures the river network topology and the interaction between adjacent sites. Whereas the Laplacian matrix has been previously used in spatial geographic analysis (e.g., the so-called “Geary coefficient”[43], a widely used spatial correlation index, can be expressed in terms of the Laplacian matrix[44]), here we used it for the first time in two novel aspects: (a) to capture the advection process taking place between adjacent sites in the context of an advection-reaction model; and (b) the definition of an entropy that quantifies the contribution of the different eigenstates (modes) of the Laplacian to the measured variable.

    Despite their obvious limitations, the approach presented here is straightforward and can provide useful information regarding the dynamics of the variables monitored, such as the local dynamics, the influence of the neighbor sites, the degree of synchronization, or the external input/output to the system. Even though the methods and results presented in this study were specifically concerned with emerging contaminants, they can be easily extended to any other site-measured variables as well, such as nutrients, inorganic constituents, heavy metals, or even biological parameters such chlorophyll, dissolved enzymes (alkaline phosphatase), etc.

    Finally, since monitoring for water quality assessment is a key aspect in the implementation of environmental legislation, such as the Water Framework Directive (Directive 2000/60/EC), the presented modeling approach can provide valuable insights for water managers engaged in the proper and consistent design of river monitoring networks, whether surveillance, operational, or investigative, with potential practical consequences on the optimization of economic costs.



    This work has been supported by the Spanish Ministry of Science and Innovation (MCIN).

    Authors’ contributions

    Involved in the conception: Ginebreda A, Barceló D

    Involved in the design, data treatment and interpretation: Ginebreda A

    Writing of manuscript and critical reading: Ginebreda A, Barceló D

    Availability of data and materials

    Data are available at the Supplementary Material [Supplementary Table 1].

    Financial support and sponsorship

    This work has been financed by the Spanish Ministry of Science and Innovation (MCIN), through Project SINERGIA included in the Severo Ochoa Grant CEX2018-000794-S funded by MCIN/AEI/ 10.13039/501100011033.

    Conflicts of interest

    All authors declared that there are no conflicts of interest.

    Ethical approval and consent to participate

    Not applicable.

    Consent for publication

    Not applicable.


    © The Author(s) 2022


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    Cite This Article

    OAE Style

    Ginebreda A, Barceló D. Data-based interpretation of emerging contaminants occurrence in rivers using a simple advection-reaction model. Water Emerg Contam Nanoplastics 2022;1:12.

    AMA Style

    Ginebreda A, Barceló D. Data-based interpretation of emerging contaminants occurrence in rivers using a simple advection-reaction model. Water Emerging Contaminants & Nanoplastics. 2022; 1(3):12.

    Chicago/Turabian Style

    Ginebreda, Antonio, Damià Barceló. 2022. "Data-based interpretation of emerging contaminants occurrence in rivers using a simple advection-reaction model" Water Emerging Contaminants & Nanoplastics. 1, no.3: 12.

    ACS Style

    Ginebreda, A.; Barceló D. Data-based interpretation of emerging contaminants occurrence in rivers using a simple advection-reaction model. Water. Emerg. Contam. Nanoplastics. 20221, 12.




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