Source identification in river pollution problems: A geostatistical approach

Transcription

1 WTER RESOURCES RESERCH, VOL. 41, W723, doi:1.129/24wr3754, 25 Source identification in river pollution problems: geostatistical approach Fulvio Boano, Roberto Revelli, and Luca Ridolfi Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Turin, Italy Received 2 October 24; revised 2 May 25; accepted 4 May 25; published 23 July 25. [1] The geostatistical method, formulated in the groundwater field, has been applied to identify contaminant sources in river pollution problems. The problem consists of recovering a contaminant source at a known location from a finite number of concentration measurements. It is an ill-posed problem, whose solution is nonunique and cannot be determined through standard techniques. The presence of dead zones has been considered because of their relevant influence on transport processes. The possibility of linear decay reactions in the main stream and in the dead zones has also been included. pplications to field data show that the method is efficient in recovering the release history of an arbitrarily distributed source as well as multiple independent point sources, given a finite number of observed concentrations at one or several downstream points. The influence of the degree of dispersion on the observations is stressed. Finally, the effects of measurement errors and of the number of measurement points have been investigated. Citation: Boano, F., R. Revelli, and L. Ridolfi (25), Source identification in river pollution problems: geostatistical approach, Water Resour. Res., 41, W723, doi:1.129/24wr Introduction [2] Over recent years, the problem of river contamination has led to the need to have evaluation tools that are able to predict the fate of pollutants, either accidentally or intentionally introduced into streams. This interest has led to the development of mathematical models that are essentially based on the advection-dispersion equation (DE) derived from Taylor s theory of dispersion [Taylor, 1954]. Mention can be made of dead zone models [e.g., Bencala and Walters, 1983; Hart, 1995; Czernuszenko and Rowinski, 1997; Schmid, 22], models that describe exchanges with the hyporheic zone [e.g., Elliott and Brooks, 1997; Wörman, 2; Jonsson et al., 23] (see also Packman et al. [2, and references therein], Wörman et al. [22, and references therein], and Jonsson et al. [24, and references therein] for more physically based models), and stochastic models [e.g., Finney et al., 1982; Zielinski, 1991; Stijnen et al., 23; Revelli and Ridolfi, 24]. These studies have actually made it easier to solve the direct problem, that is, to forecast the concentrations that result from a known contaminant input. [3] However, the corresponding inverse problem, i.e., the characterization of a contaminant source from a series of concentration measurements, can also be interesting. For example, if a plume due to an unknown release is observed, it might be desirable to recover the source. From a mathematical point of view, this kind of question belongs to the class of ill-posed problems, whose solutions are not characterized by the usual existence, uniqueness and/or stability properties [Sun, 1994]. In particular, while the existence of the solution (i.e., the true source) is assured, the considered Copyright 25 by the merican Geophysical Union /5/24WR3754 W723 inverse problem is ill posed for two reasons. First, only a small number of concentration measurements are usually available. Thus one is interested in estimating the values of the source at a number of instants (n) which is much greater than the number of available observed concentrations (m). If the DE is discretized with respect to time, a system with fewer equations (observations) than unknowns (source values) is obtained. Therefore, since usually n m, multiple solutions that are consistent with the observed concentrations can be found. The second reason for the ill posedness of the problem is the irreversibility of the dispersion phenomena that progressively smooth the contaminant plume, thus reducing the amount of information that can be inferred from the observations. This makes the solution unstable, in the sense that small data errors do not correspond to small errors in the solution, as might be expected. Since observations are always affected by experimental errors, any recovered solution will always be irremediably distorted. [4] The source identification problem has been extensively studied in groundwater literature, where several sound methods for solving this problem can be found [see Snodgrass and Kitanidis, 1997; Michalak and Kitanidis, 24, and references therein]. In the river pollution field, a number of studies [Bloss et al., 1985; Piasecki and Katopodes, 1997; Piasecki and Sanders, 22] have focused on the control of an effluent discharge in order to prevent the violation of water quality standards in an estuary. In this case the concentration is known at certain target points and the desired source is found with an optimization procedure. Besides allocation problems, this method could also be applied to situations of unknown and/or unplanned discharges, where the main objective is the characterization of the source from some observed concentrations. The limit of the optimization procedure is that the source cannot be 1of13

2 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 estimated at a number of instants that is greater than the number of observations, while it might be desirable to increase the time resolution when only a few observations are available. Moreover, no information is gained about other possible sources that are consistent with the observations. [5] In this work a procedure is described for the identification of an arbitrarily distributed source, given a finite number of observed concentrations at different times at one or several downstream fixed points. The procedure herein proposed is an adaptation of an established technique for source identification problems, known as the geostatistical method. It has successfully been applied to the groundwater pollution problem of recovering the release history of a point source from a longitudinal profile of concentration measurements made at the same instant [Kitanidis, 1996; Snodgrass and Kitanidis, 1997]. The method relies on a probabilistic description of both the observations and the unknown source, and estimates a release function that is consistent with the observations. The covariance of the source is also estimated, so that (1) information on the family of multiple solutions can be derived and (2) the crucial influence of experimental errors is considered. [6] In order to describe the typical behavior of a chemical in real rivers, we take into account the presence of the socalled dead zones, which are zones of relatively immobile water that can temporarily trap part of the pollutant, eventually releasing it after a certain time. These trapping zones, which include dunes on the river bed and irregularities in the banks, can notably influence the fate of a contaminant in a stream, leading to long tails in the concentration distributions and thus explaining one of the main discrepancies between DE predictions and observed pollution events [Nordin and Troutman, 198; Hart, 1995; Schmid, 1995, 22; Czernuszenko and Rowinski, 1997]. nonconservative pollutant is considered, with the possibility of linear decay reactions occurring at different rates in the main stream and in the dead zones. [7] The method is developed for two types of sources. In the first, the source term can be written as the product of a time-dependent part, f(t), and a space-dependent one, g(x), the latter considered to be known, while the second type refers to the case of multiple independent point sources. These two forms are quite flexible and they allow a huge variety of real conditions to be described. The influence of multiple measurement points on the recovered source has also been investigated. Finally, the method is applied to field observation in order to demonstrate its suitability for real contamination problems. 2. Mathematical Problem [8] Let us consider a turbulent stream flowing in a onedimensional natural channel and, in order to include the existence of the dead zones in the DE, let us adopt the transient storage, or dead zone, model [see, e.g., Bencala and Walters, 1983; Czernuszenko and Rowinski, 1997; Schmid, 22]. mong the models used to describe the influence of dead zones, it is the best known, due to its intuitive physical meaning and its formal clarity. [9] ccording to this model, a river can be divided into two different parts: the main stream, where the contaminant is subject to both advection and dispersion, as for the classical DE, and a storage zone with negligible flow and dispersion; the two parts exchange pollutant according to first-order kinetics. The possibility of decay reactions with distinct rates in either zone has also been included, thus taking into account possibly different hydrodynamical and biochemical conditions in the two parts of the stream [Schmid, 1995]. The model can be mathematically expressed using the transient storage equations (TSE), that is, the system of equations [Hays et al., 1966; Bencala and ¼ 2 þ et 1 ðc s CÞ lc þ sx; ð tþ ¼ T 1 ðc C s Þ l s C s ð2þ where C(x, t) and C s (x, t) are the concentrations in the main stream and in the storage zone respectively, U is the crosssectionally averaged velocity of the stream, D is the dispersion coefficient, e is the ratio of the storage zone to the main stream volume per unit length, T is the mean residence time in the storage zone, l and l s are the two decay coefficients, and s(x, t) is the source term. ll the model parameters are considered to be known. [1] Our aim is to retrieve the unknown source function s(x, t), given the measured time evolution of concentration C(x Ml, t* i ), i =1,..., m, atp fixed points at location x Ml, l = 1,..., p, downstream to the contaminant release. In this work we deal with the following two mathematical forms of the source term and sx; ð tþ ¼ M tot gx ðþfðþ t sx; ð tþ ¼ XR r¼1 ð M tot Þ r dðx x r Þf r ðþ: t where M tot is the discharged contaminant total mass, is the (known) cross-sectional area of the main stream, d(x) isthe Dirac delta function, x r is the rth point source position, (M tot ) r are the correspondent discharged contaminant total masses, f r (t) and f(t) are the release histories, and g(x) isthe source spatial distribution. These functions are normalized, that is, R þ1 f r (t)dt = R þ1 f(t)dt = R þ1 1 g(x) dx = 1. They can be interpreted as the fraction of contaminant mass M tot that is released at time t and position x respectively. Moreover, the f r (t) are assumed to be independent. The first type of source (i.e., equation (3)) is quite general and manages to describe a number of actual situations relative to distributed contamination sources, e.g., the case of agricultural fertilizers leaching through soil and eventually reaching a stream, while form (4) describes the presence of R point inputs, e.g., due to different sewer discharges along the river. Since the case of unknown source locations is beyond the scope of this work, it has been considered that the locations x r, or the spatial distribution g(x) are known, and the problem of source identification is thus reduced to ð3þ ð4þ 2of13

3 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 the recovering of the release histories (M tot ) r f r (t), or M tot f(t). 3. Transfer Functions [11] In order to apply the geostatistical method, an explicit relationship between the release history and the observed concentration is needed. One approach to the problem is that of the dynamic systems theory [Schweppe, 1973; Jury et al., 1986; Sposito et al., 1986], which has been applied to model the river as an input-output system. In this section the Laplace transform of the transfer function is derived from the storage equations (1) (2). The point source and the distributed source cases are discussed separately Point Source [12] Without any loss of generality, let us focus on a point source located at x =, namely g(x) = d(x) (or, equivalently, R = 1 with x 1 = ), with a single measurement point (p =1) at x M1 = x M. In the interval x 2 [, x M ], the river can be described as a linear system with a single input (the source history f(t) atx = ) and a single output (the observed concentration C(x M, t) at location x M ). Because of the linearity of the system (1) (2), the output signal is proportional to the input according to the convolution integral Cx ð M ; t Þ ¼ M tot f ðtþkx ð M ; t tþdt ð5þ where k(x M, t) is the transfer function at x = x M, which is defined as the response of the system to a unitary impulse, i.e., for M tot / f(t) =d(t). The transfer function is thus the solution of the system of equations (1) (2) for s(x, t) =M tot / d(x) d(t) together with the initial conditions C(x, )= C s (x, ) =. This system is ¼ 2 þ et 1 ðc s CÞ lc ¼ T 1 ðc C s Þ l s C s ð7þ with the initial conditions C(x, )=M tot / d(x) and C s (x, ) =. [13] The application of the Laplace transform (with respect to time) ~C(x, s) = R þ1 C(x, t) e st dt to system (6) (7) and its initial conditions leads to the expression [Schmid, 1995] with ~kðx; sþ ¼ 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp U 2 þ 4DEðÞ s x 2D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii U U 2 þ 4DEðÞ s Es ðþ¼sþlþet 1 s þ l s s þ l s þ T 1 which represents the transformed transfer function. ð8þ ð9þ [14] It is then possible to numerically evaluate the transfer function k(x, t) atx = x M and to apply the geostatistical method. n efficient numerical algorithm for the Laplace inversion, proposed by [Bellman et al., 1966], is described in ppendix. [15] It should be noticed that the solution given by (8) represents the transfer function of the source, that is, how an external source leads to the observable C(x, t). lternatively, one could be interested in evaluating the concentration history C(, t) at the injection point. In this case, equations (6) and (7) should be solved with a Dirac delta d(t) asthe boundary condition instead of the initial condition d(x), and this would lead to ~kðx; s h Þ ¼ exp x 2D pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii U U 2 þ 4DEðÞ s : ð1þ [16] This transfer function represents how a boundary condition in x = is transferred downstream. The two processes are physically quite different, the former consisting of a contaminant flux that enters the stream from the outside and the latter representing a contaminant concentration which was already in the river and enters the examined reach through the upstream section (compare the transfer functions used by Schmid [1995] and Czernuszenko and Rowinski [1997]). Since we are interested in recovering the release history (i.e., the contaminant flux) that caused the observed concentration, only equation (8) will be used in this work Spatially Distributed Source [17] We are now interested in finding a relationship between the release history and the observed concentrations similar to equation (5) for the case of a spatially distributed source, with the distribution function g(x) considered to be known. The distribution g(x) could be imagined as a series of infinite point sources, each giving an infinitesimal contribution to the observations. The measured concentration can be interpreted as the sum of all these contributions, and the following integral is obtained: Cx; ð t Þ ¼ M tot Z x gðhþ f ðtþkx ð h; t tþdtdh ð11þ which, after changing the order of integration, becomes Cx; ð t where Þ ¼ M tot ¼ M tot f ðtþ Z x gðhþkx ð h; t tþdhdt f ðtþk* ðx; t tþdt ð12þ k* ðx; tþ ¼ Z x gðhþkx ð h; tþdh ð13þ is the new transfer function for a source distributed according to g(x). Since only the Laplace transform of the transfer function k(x, t), i.e., equation (8), is known, it is more straightforward to evaluate the transform of k*(x, t), namely ~k* ðx; sþ ¼ Z x gðhþ ~ kx ð h; sþdh ð14þ 3of13

4 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 and then to determine its inverse (at x = x M ) using the inversion algorithm. The computation of (14) can be performed analytically for some simple distributions g(x) or it can be evaluated numerically using classical quadrature formulas. 4. Geostatistical Method [18] It was previously stated that the identification of the contaminant source s(x, t) from a finite number of observed concentrations C(x Ml, t* i ) is an ill-posed problem, even when its spatial distribution is considered to be known. This means that there are many solutions that are compatible with the measurements, and it is therefore impossible to determine the true value of the source. Moreover, the inevitable presence of measurement errors increases the implicit uncertainty of the problem, since small errors in the observations can lead to large errors in the solution. [19] In the geostatistical approach, a probabilistic description of the involved quantities is adopted: the source time distribution f(t) is interpreted as a stochastic process and its sample functions represent possible release histories, among which we search for the ones that are consistent with the observations. This approach is developed in the next section for a product-type source and a single measurement point, and it is then extended to the cases of multiple point sources and multiple measurement points. The implementation of the nonnegativity constraint on the release history is also discussed Single Measurement Point [2] The geostatistical approach to contaminant source identification is summarized in this section. Further explanations are given by Kitanidis [1995, 1996] concerning the method and by Snodgrass and Kitanidis [1997] for its application to the source characterization problem. [21] If the time domain is discretized in n instants t j,the observations can be related to the source by the general expression z ¼ hf ðþþv ð15þ where z =[C(x M, t* 1 )... C(x M, t* m )] T is an [m 1] random vector of the observations at times t* i, f = M tot [f(t 1 )... f(t n )] T is an [n 1] random vector of the discretized release history, h is the model function and v =[v 1... v m ] T is an [m 1] random vector that represents the measurement errors. The unknown total contaminant mass M tot is thus absorbed in the release history vector f, which is no longer normalized. The mass M tot can eventually be evaluated as the area between the vector f and the time axis. It should be noticed that we are dealing with the more realistic case of n m, that is, with more unknowns than measurements, and that the observation instants t* i can be distinct from the discretized instants t j. [22] From a comparison of equation (15) with (5) and (12) it is possible to see that it is natural to assume the model function h(f) to be linear. Equation (15) can therefore be replaced with the linear relation z ¼ Hf þ v ð16þ where H is an [m n] matrix, whose generic element is Hði; jþ ¼ ( 1 k x M ; t i * t j for t i * > t j for t i * t j : ð17þ where k(x, t) is the transfer function given by the inversion of (8) or (14). One drawback of equation (16) is that the application of the geostatistical method to a linear model allows the release history to take on negative values, which are not theoretically acceptable from a physical point of view. Equation (16) can be modified to enforce nonnegative results; this is described in section 4.4. [23] The release history f and the error v and, consequently, the observation z are all regarded as random vectors. This reflects (1) the fact that multiple release histories are consistent with the observations and (2) the presence of unknown errors. [24] It is now necessary to make some assumptions to remove the ill posedness of the problem. ccording to the basic principle of inference, the simplest model structure that is physically plausible and which is consistent with the data is chosen. It is therefore assumed that the stochastic vector f has Gaussian joint distributions, with mean and covariance E½Š¼Xb f h E ðf XbÞðf Xb Þ T i ¼ QðÞ q ð18þ ð19þ where X is an [n 1] unit vector, b is the unknown mean, and q is a vector of unknown structural parameters. The error v is also Gaussian, with a zero mean and covariance matrix R = s R 2 I, where I is the [m m] identity matrix. In this work, two types of models for the [n n] covariance matrix Q have been used, that is, the linear model Q t i t j jq ¼ s 2 b 2 t i t j where q T =[s, b], and the Gaussian model Q t i t j jq " ¼ s 2 exp t 2 # i t j l 2 ð2þ ð21þ where q T =[s, l]. fter the application of the method, the model must be validated to verify the properness of the chosen covariance function. Both (2) and (21) have proved to be suitable for the examined problem. Validation tests on the orthonormal residuals are discussed by Kitanidis [1997]. See also Snodgrass and Kitanidis [1997] for other possible choices for the statistics of f. [25] It is then possible to write the pdf s pðfjb; qþ / jqj 1=2 exp 1 f Xb 2 ð ÞT Q 1 ðf XbÞ pðzjfþ / jrj 1=2 exp 1 2 ð z Hf ÞT R 1 ðz HfÞ ð22þ : ð23þ 4of13

5 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 [26] The geostatistical method, which consists of two steps, can now be applied: first, the principle of maximum likelihood is used to estimate the structural parameters q from the data; then the release history is determined by means of the kriging estimator. [27] In the first step, also known as structural analysis, the observation pdf conditional on q is determined as Z pðzjqþ ¼ b Z ¼ b Z pðzjb; qþdb ¼ Z f b Z f pðz; fjb; qþdfdb pðzjfþpðfjb; qþdfdb: ð24þ [28] This expression is maximized with respect to q, which is equivalent to minimizing LðÞ¼ ln q ½pðzjqÞŠ / 1 2 ln j2j XT H T 2 1 HX 1 þ 2 zt 7 z ð25þ where 2 ¼ HQH T þ R ð26þ 7 ¼ HX X T H T 2 1 1X HX T H T 2 1 : ð27þ [29] It is worth stressing that the minimization of equation (25) is a well-posed problem, since it corresponds to fitting a small number of parameters Q to a greater number of observations z. [3] The problem can be solved iteratively using the Gauss-Newton method, with the addition of a line search [Press et al., 1992] to improve the convergence rate. If the lth estimate ^q l is known, the next estimate can be found as ^qlþ1 ¼ ^q l rf 1 g where r is a multiplicative factor for the line search, and g i ¼ Tr i i F ij ¼ j 1 2 zt 7 i ð28þ ð29þ ¼ Tr 7 : ð3þ j [31] The routine eventually converges to the final estimate ^q of the structural parameters. The matrix F 1 also represents an estimate of the covariance matrix of q. [32] The second step of the geostatistical method is the kriging, a linear estimator ^f ¼ +z ð31þ which is unbiased and minimizes the estimate error variance, that is, min ^f E ^f f ¼ ð32þ h E ^f f ^f f T i : ð33þ [33] fter updating the matrices Q and 2 with the estimated structural parameters ^q, the following system has to be solved 2 HX + T ðhxþ T M ¼ HQ X T ð34þ where + is an [n m] matrix of coefficients and M is a [1 n] vector of multipliers. This system is equivalent to equations (32) and (33). [34] The application of the kriging equations (34) corresponds to discarding, from the ensemble of the release histories, those that are not consistent with the observations z. The mean of the release history is then estimated by equation (31), while its covariance matrix V can be evaluated as V ¼ XM þ Q QH T + T : ð35þ The confidence interval of 95% can be calculated as (^f i ± 2 V ii ). [35] From the observation of equations (26), (34) and (35), it can be noticed that the variance that is obtained with the geostatistical approach implicitly depends on the error magnitude s R 2. It should be recalled that this is a crucial aspect because of the effects of the errors on the estimated release history due to the nonstability of the solution Independent Point Sources [36] The extension of the geostatistical method to the case of multiple independent point sources is here described [see also Butera and Tanda, 23]. We have restricted ourselves to the case of two independent point sources f 1 (t) and f 2 (t) located at x = x 1 and x = x 2. However, the method is easily extendable to a greater number of sources. [37] The linearity of the TSE (1) and (2), with respect to the concentration C(x, t), allows us to write the solution ð Cx ð M ; tþ ¼ M totþ 1 ð þ M totþ 2 f 1 ðtþkx ð M x 1 ; t tþdt f 2 ðtþkx ð M x 2 ; t tþdt: ð36þ [38] s in the previous cases, the time domain is then discretized in n instants t j, thus forming the system of equations where and H k ði; jþ ¼ z ¼ Hf þ v ð37þ f ¼ f 1 ; H ¼ ½H f 1 H 2 Š ð38þ 2 ( 1 k x M x k ; t i * t j for t i * > t j for t i * t j : ð39þ 5of13

6 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 [39] Because of the independence of the sources, the [2n 2n] covariance matrix of the source processes takes the block diagonal form Q ¼ Q 1 ð4þ Q 2 where the structural parameters are now q T =[s 1, l 1, s 2, l 2 ]. [4] The geostatistical method could now be applied using the same steps as in section 4.1. However, it is worth noting that the problem of recovering two different sources cannot be theoretically solved. This is because this problem is indeterminate: the observed plume is composed of the sum of the contributions of the two sources, and it is not possible to univocally divide an observed concentration into two parts without further information being available. Roughly speaking, this situation corresponds to an equation with two unknowns, which has infinite solutions. [41] In order to solve the problem, more measurement points are needed. n example of two independent sources and two measurement points is solved in section 5. The second measurement point might seem useless, since the observations at the second point can be computed from those at the first point solving the TSE, and thus the second measurement point would not add any independent information. However, the reason for the success of the method lies in the transfer functions: due to its parabolic form, the TSE implicitly assumes an infinite diffusion velocity for the contaminant (as does the DE) [Fischer et al., 1979], therefore both measurement points should immediately show the effects of the two sources and, coherently, the transfer functions should be nonnegative in each point. However, as the tails of the transfer functions have very small values, they are numerically rounded to zero. These zeros can make one of the integrals in equation (36) vanish, so that only the contribution of the nearest source is sensed and the indeterminateness is removed. This situation corresponds to numerically introducing a finite diffusion velocity, which is exactly what happens in an actual river Multiple Measurement Points [42] The method can easily be extended to the case of p distinct measurement points located at x = x Ml, l =1,..., p. In this section, the product-type source is considered; the modifications for the case of multiple point sources are straightforward. [43] It is clearly possible to formulate an expression analogous to (5) or (12), that is, Cx ð Ml ; t Þ ¼ M tot f ðtþkx ð Ml ; t tþdt: ð41þ If C(x Ml, t) and f(t) are discretized at the instants t* i and t j, respectively, equation (41) becomes z l ¼ H l f þ v ð42þ and H l ði; jþ ¼ ( 1 k x M l ; t i * t j for t i * > t j for t i * t j : [44] The following expression can now be written where z ¼ Hf þ v z T ¼ z 1...z p ð44þ ð45þ ð46þ H T ¼ H 1...H p : ð47þ [45] Equation (45) can be considered a generalization of (16), and it corresponds to a system of p m equations in n unknowns. It should be noted that the Laplace inversion algorithm must be performed once per measure point, since H i depends on the transfer function k(x Ml, t), l =1...p. The effects of varying m and p are investigated in section Nonnegativity Constraint [46] The adopted model function (16) does not include any information about the nonnegativity of the release history f. This constraint can be imposed by working with the transformed variable [Snodgrass and Kitanidis [1997] ~f ¼ a f 1=a 1 ð48þ where a is a positive parameter. The release history f is enforced to be nonnegative as long as ~f + a >. Provided this condition is respected, it is a good choice to adopt a small value for a, since this will improve the convergence rate. Other transformations for the variable can also be chosen. [47] The method is now applied to the transformed variable, which is Gaussian distributed with the mean and covariance given by (18) and (19). Basic relationship (15) is replaced by z ¼ ~h ~f þ v ð49þ where the new model function is ~h (~f) =h[((~f + a)/a) a ]. Since the model is no longer linear, the solution is obtained via successive iterations. [48] First (l = ), the model is linearized about a starting value ~f l, thus finding and ~H l j f¼~f l ~z l ¼ z ~h ~f l Þþ ~H l ~f l ð5þ ð51þ where z T l ¼ ½Cx ð Ml ; t 1 * Þ... Cx ð Ml ; t m * ÞŠ ð43þ 6of13 ~2 ¼ ~H l Q ~H T l þ R ð52þ 1X ~7 ¼ ~2 1 ~2 1 ~H l X X T ~H T l ~2 1 T ~H l X ~H T l ~2 1 : ð53þ

7 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 while the evaluation of the covariance is not trivial. piece of information on the uncertainty of the recovered solution can be extracted from the confidence intervals. These can roughlyp be ffiffiffiffiffiestimated by evaluating the transformed bounds ~ f i ±2 ~V ii and then applying the inversion relationship (56). lternatively, the marginal pdf of each component of f can be computed as pf ð i Þ ¼ p ~ d f ~ f i i ¼ f 1=a 1 p i df ffiffiffiffiffiffiffiffiffiffi i 2p~V ii 2 a f 1=a i 1 ~ f i exp4 2~V ii 7 5; ð57þ from which the variance of f i can be inferred. Figure 1. Point source case, Gaussian covariance function. (a) Observations at x M = 3 km; (b) true and estimated release history, with 95% confidence interval. [49] Then, structural analysis and kriging are iteratively performed. t each step, the structural parameters are estimated according to (28), but replacing H, z, 2 and 7 with equations (5), (51), (52) and (53). Then, Q is updated with the new parameters and the kriging system (34) is solved. gain, expressions (5) (53) must be used instead of the original variables. The result is the new (transformed) release history ~f lþ1 ¼ +~z l ; ð54þ which is used in the next iteration. Structural analysis and kriging are repeated until convergence is reached. The transformed variable ~f is then Gaussian distributed, with mean given by (54) and covariance ~V ¼ XM þ Q þ Q ~H T l +T : ð55þ [5] s a consequence of the transformation (48), the release history f is not Gaussian distributed. Its mean value can be evaluated as ^f ¼ ~ a f þ a ; ð56þ a 5. pplications 5.1. Synthetic Examples [51] In this section the method is applied to some hypothetical contamination events in order to show its potential. We examine the case of the introduction of a nonconservative pollutant into a small plain river with U =.7 m/s, = 1 m 2, D =2m 2 /s, and l =51 6 s 1, and where the dead zones have e =.1, T = 3 s and l s =1 5 s 1. The release history is expressed by the function [see Snodgrass and Kitanidis, 1997] M tot fðþ¼ t exp 1 t ð Þ2 þ :4 exp 1 t ð Þ2 þ :6 exp 1 t ð Þ2 ; the measurements are made every 3 s, while the release history is recovered with a time step Dt = 3 s. If the function is integrated, the value M tot = 148 g is found. random error component (null mean and s R =1 5 g/m 3 )is added to each simulated observation, whose order of magnitude is 1 2 g/m 3. In the examples, a sufficient distance is considered between the source and the measurement point to allow the dispersion phenomena to considerably smooth the main features of the release history. This situation notably increases the difficulties in the source identification problem, as less information can be inferred from the measurements. [52] The first example refers to a point source placed at x = with the measurement point located at x M = 3 km. s can be seen from Figure 1a, the measured concentration curve exhibits a single-peaked shape, since the dispersive phenomena have already dissipated the two originally distinct peaks. Figure 1b shows the true and the recovered release histories, which were obtained from the application of the method with a Gaussian covariance matrix and the nonnegative constraint. confidence interval of 95% is also reported. Even though the presence of two peaks is not evident in the observations, the recovered solution agrees very well with the true history, the two main peaks having being estimated acceptably. More important, the true solution almost always lies within the confidence interval, the main difference being in correspondence to the main and the minimum peaks, which are in fact very narrow. The 7of13

8 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 Figure 2. Point source case, linear covariance function. True and estimated release history, with 95% confidence interval (a) without and (b) with the nonnegativity constraint. estimated mass of contaminant is evaluated to be ^M tot = 1484 ± 28 g, which is very close to the actual value (148 g). The minor peak at t =.65 h was not recovered by the method, due to the dispersion phenomena which had already smoothed it at the considered distance; this result confirms the importance of the degree of dispersion in the source identification problem [Skaggs and Kabala, 1994]. [53] Figure 2a shows the results of the application of the method to an analogous case but with the adoption of a linear covariance function. The source has satisfactorily been recovered, but the confidence interval has become wider. This means that there are more release histories that are consistent with the observations in the set of sample functions defined by the linear covariance function than in the Gaussian one. This is also reflected in the increased uncertainty of the estimated mass ^M tot = 1485 ± 516 g, whose mean value is however very similar to the true value. The introduction of the nonnegative constraint has lead to the solution presented in Figure 2b. It is clear that the constraint has removed the tail oscillations and has reduced the correspondent confidence interval, but the solution has generally become worse, as can be deduced from the increased uncertainty. The estimated mass ^M tot = 1484 ± 515 g has not significantly changed from the previous case. [54] The next example deals with the distributed source with g(x)=(x 2 x 1 ) 1 J(x x 1 ) J(x 2 x), where J(x) is the unitary step function; in this case x 1 = 5 m and x 2 = 5 m have been chosen. This kind of input can well represent a contaminant flow that passes through an aquifer into a stream and which is derived from an abuse of fertilizers in a cultivated zone. The observations have not been reported here because they appear to be very similar to those in Figure 1a. The results that obtained with the use of the Gaussian covariance function and the nonnegativity constraint are represented in Figure 3a. It can be seen that the recovered history still agrees well with the real one, even though the main peak has been slightly underestimated. The estimated total mass ^M tot = 1487 ± 35 g is again almost identical to the real value. It can be noted that quite different sources tend to generate similar observations, therefore it can be stated that the concentrations C(x M, t) observed at different times are not very sensitive to variations in the spatial distributions of the source, while they are greatly influenced by the release history. [55] In order to explore the effect of multiple measurement points, a second observation point was added at x = 4 km. It was decided to double the sampling interval to 6 s, thus leaving the total number of observations unchanged. The recovered release history is reported in Figure 3b. While the contaminant mass has been estimated correctly ( ^M tot = 1487 ± 4 g), the release history has become slightly worse than the one shown in Figure 3a, as can be seen from the increased uncertainty of the estimated mass. This is due to the fact that the contaminant observed at the second observation point has spent a longer time in the stream, thus allowing the dispersion phenomena to reduce the information associated with the measurements even more. These considerations suggest that, in this case, it would be opportune to make the observations as close to the source as possible. However, the one-dimensional TSE (as well as the DE) does not apply if the contaminant is not wellmixed through the cross-sectional area, which only happens after an initial mixing length downstream to the source [Fischer et al., 1979]. Provided this mixing length is respected, the reduction of the distance between the source and the measurement point would then improve the estimate precision. [56] The last example deals with two independent point sources located at x 1 = m and x 2 = 5 m, respectively, with two measurement points at x M1 = 3 km and x M2 = 3.5 km. The second source has actually discharged the contaminant mass (M tot ) 2 = 119 g. The observations are presented in Figures 4a and 4b, while the recovered histories are shown in Figures 5a and 5b, respectively. Gaussian covariance function and the nonnegative constraint have been adopted. s anticipated in section 4.2, the method has efficiently managed to recover the two sources. The estimated masses are ( ^M tot ) 1 = 14 ± 23 g and ( ^M tot ) 2 = 123 ± 24 g, respectively. It has also been verified (the examples have not been reported) that in the case of a single measurement point, the method fails to recover the two sources, unless a much greater number of observations 8of13

9 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 precision of the source estimate [Butera and Tanda, 23]. The model parameter values that have been used in this application are those given by Bencala and Walters [1983], i.e., =.4 m 2, U =.3 m/s, D =.1 m 2 /s, e = 1.75 and T = 4375 s. They were estimated by visually determining the set of parameters which yielded the best-fit to the concentration data [Bencala and Walters, 1983, p. 72]. Such an approach involves the subjective judgment of the experimenter, and it can therefore be used as a benchmark to test the performance of the geostatistical method with roughly estimated parameters. [59] The second problem concerns the magnitude of the experimental errors that affect the observations. The impact of these errors on the retrieved solution has already been discussed in the previous sections. Figure 6a shows the observed chloride concentration at 619 meters downstream to the injection point. This situation is realistic of actual errors that can be observed in field measurements. The error variance s R 2 =1 1 g 2 /m 6 was used. Figure 6b shows the results obtained with a linear covariance function. The injection rate and duration have been correctly estimated. However, the estimated solution shows an anticipation, with respect to the actual release history, and the abrupt discontinuities of the real solution have also been replaced by Figure 3. Distributed source case. True and estimated release history, with 95% confidence interval, for (a) single measurement and (b) two measurements. than that of the unknowns is chosen, thus turning the illposed problem into a well-posed one pplication to Field Data [57] The method has also been applied to actual stream observations in order to test its behavior in coping with real situations. In this section two examples are presented, relative to two different typologies of rivers, a smaller mountain stream and a wider plain river. [58] First, the method was applied to the observed concentrations from a chloride injection which was carried out in the Uvas Creek, California [Bencala and Walters, 1983], a small mountain stream with a low discharge (Q =.13 m 3 /s during the test). The tracer was released at a constant rate of.1 g/s for three hours, and its concentration was measured 619 meters downstream to the injection point. This reach was characterized by a high-porosity gravel-and-cobble bed and many low-velocity pools that could act as dead zones. The first practical issue that has to be dealt with in a field experiment is the calibration of the model, i.e., the determination of the model parameter values. This is a nontrivial step in source recovering, since the uncertainty in the values of the parameters affects the Figure 4. Two point sources. Observations at (a) x M1 = 3 km and (b) x M2 = 3.5 km. 9of13

10 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 for a considerable distance along the stream, approximately corresponding to a travel time of 13 h. During this time, the dye pulse was subject to the effects of dispersion, whose order of magnitude is a thousand times higher than that of the Uvas Creek. These considerations suggest that one consequence of the greater spatial and temporal scales is the larger amount of dispersion experienced by the tracer cloud. [61] Figure 7a shows the observed concentrations at x M = km. The value s R =1 3 g/m 3 has been adopted for the experimental errors. Figure 7b shows the results obtained with the use of a Gaussian covariance function and the nonnegativity constraint. The recovered source exhibits an excellent agreement with the true one, and the tracer mass ^M tot = 58. ± 28.7 kg has also been estimated very well. It should be noticed that the method has failed to recover the source when the nonnegativity constraint was not introduced. This is due to the peculiar shape of the release history (i.e., a Dirac pulse), which is almost always equal to zero. In this case, the information that the tails of the source are nonnegative are necessary to correctly solve the inverse problem. [62] It can be concluded that the geostatistical method can efficiently be applied to both smaller mountain streams and Figure 5. Two point sources. True and estimated release history at (a) x 1 = m and (b) x 2 = 5 m. milder slopes. These differences are due to the uncertainties introduced by the model calibration and the experimental errors. It should be noticed that both the actual release history and the recovered one are consistent with the observations, since the nonuniqueness of the solution is a characteristic feature of this kind of ill-posed problems. Nonetheless, the total contaminant mass ^M tot = 198 ± 573 g is almost the same as the true value (19 g). Despite the described inaccuracies, the recovered source is satisfactory for most practical purposes. s in the example of Figure 2b, the introduction of the nonnegativity constraint has not brought substantial improvements. [6] The method was also applied to field data from a study on dispersion in the Missouri River [Yotsukura et al., 197]. During this test, a total mass M tot = 54.4 kg of rhodamine WT was quickly released into the full width of the river. The dye concentration was then measured km downstream to the release section. The values of the parameters were taken from Seo and Cheong [21], whose moment-based calibration method gave U = 1.38 m/s, D = m 2 /s, e =.1 and T = s. This example was intended to test how the method performs when dealing with greater spatial and temporal scales. The tracer travelled Figure 6. Uvas Creek tracer test. (a) Observations at x M = 619 m; (b) true and estimated release history, with 95% confidence interval. 1 of 13

11 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 Figure 7. Missouri River tracer test. (a) Observations at x M = 65 km; (b) true and estimated release history, with 95% confidence interval. wider rivers. The validation tests have confirmed the suitability of both the linear and the Gaussian covariance models for actual river pollution problems. 6. Conclusions [63] The geostatistical method is a technique that has fruitfully been applied in the groundwater context for the characterization of a contaminant source. n application of this method to a river pollution problem has been described in this work, both for a spatially distributed source and for several independent point sources. The geostatistical approach is particularly suitable for this problem, as it gives a probabilistic description of the possible sources that are compatible with the observations, and it allows the release history to be estimated at a greater number of instants than the number of observations. Moreover, the estimated covariance considers the influence of experimental errors on the retrieved release history. [64] The efficiency of this procedure has been illustrated in the reported examples; in particular, the applications to field data have shown that the method can deal with both small and large spatial and temporal scales. The estimation 11 of 13 of the model parameter values must be carefully addressed, since erroneous values of the parameters lead to systematic errors in the recovered solution. However, satisfactory results have been obtained even with roughly estimated parameters. [65] covariance function must be chosen to apply the method. This corresponds to choosing an ensemble of possible release histories, among which the ones that are consistent with the observations are searched for. The covariance model needs to be validated in order to verify its properness. Both the Gaussian and the linear models have proved to be appropriate for the examined cases. Moreover, it was observed that the introduction of the nonnegativity constraint on the release history does not always improve the estimate of the solution. In particular, the use of the constraint with a linear covariance function has led to an increased uncertainty of the recovered solution. This behavior was observed in all the examined cases. However, the information of the nonnegativity of the solution has proved to be essential in the case of a sudden release of contaminant, when the greatest part of the release history is equal to zero. [66] The influence of the dispersion experienced by the observed contaminant has also been emphasized. The dispersion phenomena tend to progressively diminish the amount of information that is associated with the observations, and this effect increases with the distance between the measurement point and the source and also with the magnitude of the experimental errors. [67] Finally, the method can deal with observations made at different measurement points. This is fundamental in the case of multiple point sources, since it is not possible to correctly recover two sources when there is a single measurement point, unless a huge number of observations is available. However, the observations made at a second measurement point provide the independent information that is needed to solve the inverse problem. ppendix : lgorithm for the Numerical Inversion of the Laplace Transform [68] The algorithm that was used for the inversion of equation (8) or (14) is here summarized. Bellman et al. [1966] should be consulted for further details. The procedure can be used with any sufficiently regular function, i.e. without strong oscillations, which is usually the case of the transfer functions encountered in river pollution problems. [69] Let us consider the integral Z þ1 e st kt ðþdt ¼ ~ ks ðþ ð1þ which is the definition of the Laplace transform. The new variable z = e t is introduced to turn the improper integral (1) into another one defined on the finite interval [, 1], that is, Z 1 z s 1 gz ðþdz¼ ~ ks ðþ ð2þ where g(z) =k( ln z) is the new unknown and ~ k(s) is given by equation (8) or (14). The Gaussian quadrature formula is

12 W723 BONO ET L.: SOURCE IDENTIFICTION IN RIVER POLLUTION PROBLEMS W723 then used, and the integral (2) can be approximated by the sum X N i¼1 w i z s 1 i gz ð i Þ ¼ ~ ks ðþ ð3þ where z i, i =1...N are the roots of the N-degree Legendre polynomial defined in [,1], and w i are the correspondent weights. [7] The only unknowns in equation (3) are the N values of g(z) at the nodes z i, i =1... N. N values of the complex parameter s can be chosen to write a system of N equations with N unknowns, which can be solved with standard techniques. The simplest choice is to take s =1,2... N, thus avoiding the presence of complex numbers. [71] The points t i = ln z i tend to gather near small values of t due to the logarithmic transformation, and it is opportune to modify equation (3) to obtain t i that cover a wider time interval. If it is recalled that L½kat ð ÞŠ ¼ 1 a ~ k s a it is possible to recast equation (3) as X N i¼1 w i z s 1 i ð4þ kð a ln z i Þ ¼ 1 a ~ k s : ð5þ a [72] n appropriate value of the scaling factor a is chosen, and equation (5) can be used to build a new system, as described for equation (3), to be solved for the values of the transfer function at the new points t i = a ln z i. [73] It should be noted that even though the evaluation of the transfer function can be a computationally onerous task, it has to be done only once per each measurement point x M. s this is a basic step for the resolution of the inverse problem, it would be opportune to spend some time on this operation. It is important to choose a so as to have an adequate number of points in correspondence to the peak of k(t), therefore different values of a would probably have to be tried. The reduction in oscillations on the inverse transform tails, which were here eliminated by setting the corresponding values equal to zero, is also a symptom of a good choice of a. [74] cknowledgments. The authors would like to thank Luciano Pandolfi for his valuable hints on inverse problems and three anonymous reviewers for their useful comments and advice. This research was financed in part by the CRC Foundation (Fondazione Cassa di Risparmio di Cuneo). References Bellman, R. E., R. E. Kalaba, and J. Lockett (1966), Numerical Inversion of the Laplace Transform, Modern nal. Comput. Methods Sci. Math., vol. 4, Elsevier, New York. Bencala, K. E., and R.. Walters (1983), Simulation of solute transport in a mountain pool-and-riffle stream: transient storage model, Water Resour. Res., 19(3), Bloss, S., J. Woolrath, and W. Zielke (1985), Optimal control of discharge into unsteady flows, in 21st International ssociation of Hydraulic Research Congress, pp , Inst. of Eng., Canberra, CT. 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