Discharge-dependent pollutant dispersion in rivers: Estimation of aggregated dead zone parameters with surrogate data

Transcription

1 WATER RESOURCES RESEARCH, VOL. 42,, doi:0.029/2005wr004008, 2006 Discharge-dependent pollutant dispersion in rivers: Estimation of aggregated dead zone parameters with surrogate data Paul Smith, Keith Beven, Jonathan Tawn, Sarka Blazkova, 2 and Ladislav Merta 3 Received 3 January 2005; revised 5 August 2005; accepted 7 January 2006; published 20 April [] Much has been done to mitigate the effects of intermittent discharges of pollutants; however, pollution incidents still occur, and the downstream transport and dispersion of pollutants must be predicted. The application of transient storage models based on the advection dispersion equation is often limited by the strong dependence of the parameters on changes in discharge. In this paper a methodology is outlined for estimating the parameters of the simple aggregated dead zone model using surrogate data derived from continuous water quality measurements such as conductivity, including a full treatment of the errors and prediction uncertainties within a Bayesian framework. This methodology is demonstrated in the prediction of a tracer experiment on a reach of the river Elbe in the Czech Republic. Citation: Smith, P., K. Beven, J. Tawn, S. Blazkova, and L. Merta (2006), Discharge-dependent pollutant dispersion in rivers: Estimation of aggregated dead zone parameters with surrogate data, Water Resour. Res., 42,, doi:0.029/2005wr Introduction [2] As a consequence of the integrated nature of human interactions with the environment, rivers are often used as a conduit for transporting waste. In recent years much has been done to mitigate the effects of such activity, but due to events such as accidents and process failures in factories large intermittent pollutant discharges still occur. Often the first step in mitigating the effects of such discharges is to model the affected reach of a river to predict the movement of the pollutant plume. The movement of the plume is a complex three-dimensional mixing process with the controls on mixing being dependant upon discharge as it affects the local velocity distributions. [3] It has been commonly assumed [Fischer et al., 979] that once cross-section mixing is complete, longitudinal dispersion due to vertical velocity shear is the most important mechanism controlling the shape of the pollutant cloud. This has led to the development of many process models based on Taylor s [Taylor, 954] one-dimensional advection-dispersion equation @C where A is the cross-sectional area, U the mean crosssectional velocity in direction x, K the longitudinal dispersion coefficient, C the mean cross-sectional concentration and t time. However, the ADE does not give good predictions of the long tails in the concentration curve that are commonly seen in tracing experiments and observations of dispersing pollutant. A common extension of the ADE Lancaster Environment Centre, Lancaster University, Lancaster, UK. 2 T. G. Masaryk Water Research Institute, Prague, Czech Republic. 3 Elbe River Authority, Hradec Králové, Czech Republic. Copyright 2006 by the American Geophysical Union /06/2005WR ðþ model (), but one with at least two additional parameters, is the inclusion of solute trapping in dead zones. The parameters of these transient storage (TS) models [Bencala and Walters, 983] can be estimated directly from plume data. Analytical methods for undertaking this estimation include moment matching [Schmid, 2003; Seo and Cheong, 200], routing methods [Cheong and Seo, 2003] and nonlinear least squares [Wagner and Gorelick, 986]. While there is little doubt that TS models are capable of fitting many plumes [Harvey and Wagner, 2000] the parameters of the model are all dependent upon discharge. This means that in predicting the movement of a plume at a discharge where observations have not been made, the parameters of the model must be predicted. Parameter prediction for the TS model has focused on linking parameters, such as the longitudinal dispersion coefficient to readily observable bulk parameters such as mean channel velocity and flow depth [Deng et al., 2002; Harvey and Wagner, 2000; Koussis and Rodriguez-Mirasol, 998; Seo and Cheong, 998; Wagner and Harvey, 997]. The inability to predict these parameters has been shown to limit the accuracy of the predictions of the pollutant plume, for example in the TS based Rhine alarm model [van Mazijk, 2002]. [4] An alternative, though related [Lees et al., 2000], data-based mechanistic approach was proposed by Beer and Young [Beer and Young, 983; Green et al., 994; Wallis et al., 987] following time series analysis of tracer data. The aggregated dead zone (ADZ) model can allow for flexibility of model structure as inferred from analysis of the data but its simplest two parameter form (three in the case of a nonconservative solute where loss can be assumed to be proportional to concentration) is usually adequate to reproduce the observed dispersion characteristics at a given discharge, including extended tailing. It has been found to be effective at modeling dispersion over a wide range of reach lengths and river sizes [Green, 998; Wallis et al., 989]. The parameters of the ADZ model can be related to of9

2 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS the timescales of the transport directly. Although these timescales change with discharge in a nonlinear way, empirical analysis suggests that the ratios of timescales (known in the ADZ model as the dispersive fraction, as defined below) seem to change little with discharge [Green, 998; Wallis et al., 989], something that was also noted in the simple scaled triangle model formulation of Kilpatrick and Taylor [986]. Thus the most important aspect of defining an ADZ model for a specified reach of channel is to relate one of the timescale parameters to discharge. In the past, this has been done by regression analysis of timescale parameters against discharge using data collected from multiple tracer experiments [Green et al., 994; Wallis et al., 989]. Tracer experiments are, however, expensive to carry out and are not often repeated at multiple discharges. [5] In this paper the ADZ technique is extended to derive the change of advective time delay with discharge from recognizable perturbations in continuously monitored data, such as conductivity, which are used as surrogate water quality data. This means, however, that the errors in the estimates of traveltimes that are derived from the surrogate data will be greater than for tracer data. Thus a methodology is developed to provide a more comprehensive treatment of the errors on the observed data. The proposed methodology encompasses an approach to the errors-in-variables problem, which assumes that every variable can have error or noise, and attempts to maximize the information gained from all sources of historical data. It is then demonstrated by the prediction of a single available tracer experiment, not used in deriving the traveltime relationships, on a reach of the river Elbe in the Czech Republic. Finally the tracer experiment data are incorporated into the calibration data set to investigate the effects on future predictions of pollutant transport. 2. ADZ Model [6] The aggregated dead zone (ADZ) model [Beer and Young, 983] is a simple one-dimensional, discrete time model for prediction of the movement and dispersion of a pollutant cloud moving downstream in a river network at a quasi-steady discharge. The simplest form of ADZ model allows for one flow pathway in a reach with the downstream output y k (indexed in time steps) being related to the upstream input u k by y k ¼ ay k þ bu kd where a, b and d are the model parameters. This formulation of the ADZ model is most suitable for reaches where initial mixing has taken place, there is a single channel and there are no significant tributaries. [7] Three event summary values, which characterize reach properties during an event, can be directly linked to the parameters in the first-order ADZ model using the following equations Dt DF a ¼ exp ð Þ t ; b ¼ SSG ð aþ; d ¼ int tdf Dt where Dt is the model time step, t is the advective time delay, DF is the dispersive fraction and SSG is the steady ð2þ state gain. For each event the t is estimated as the difference between the arrival time of a pollutant plume at the head and foot of the reach. The DF can be viewed as the effective fraction of the volume of the reach which acts as a dead zone in mixing a tracer or pollutant. A method of estimating DF for an event under steady state flow conditions is DF ¼ V e V ¼ t t t where V e is the effective mixing volume, V the volume participating in transport and t mean traveltime of the reach. The SSG summarizes the degree of conservation of the pollutant, a value less then one indicating degradation, absorption or other loss at a rate assumed proportional to the concentration. The SSG can be estimated from plume measurements by taking the ratio of the areas under the upstream and downstream mass flux curves, however this estimated value can also be influenced by unaccounted changes in discharge downstream. 3. Study Reach [8] The reach of the river Elbe studied; situated approximately 00km east of Praha; runs for 79.5km between the towns of Valy and Lysá in the Czech Republic (Figure ). The flow is controlled by several weir structures. Several tributaries add additional discharge which amounts to approximately 6% of the flow at Lysá. Modeling of this reach is motivated by the fact that an industrial effluent enters the Elbe 400 m upstream of Valy. Pollution incident prediction could be of operational importance. [9] For the study reach electrical conductivity data were available at both the upstream and downstream sites on a 0 minute time step for the period 995 to 999. Discharge data, recorded at Valy, were available for the same period at a similar resolution. Three pollutant events could be traced from Valy to Lysá. These allowed the estimation of DF, t and SSG over a range of discharges (Table ). To allow the estimation of DF the conductivity readings were converted to a chemical concentration equivalent based on Potassium Chloride. Potassium Chloride was selected due to the linear nature of the transformation over the range of conductivities observed. It was also possible to make further estimates of t by studying the response of continuous conductivity measurements to changes in discharge and selecting identifiable perturbations common to both upstream and downstream sites. [0] Data from a single tracer experiment at low flow are available [Dostal et al., 2000]. The tracer used was the nontoxic [Behrens et al., 200], light-stable, ph-independent and nonsorptive [Käß, 998] Amidorhodamine G. A two kilogram mass of tracer was instantaneously injected midstream at Němčice (Figure ), with the plume being measured (again midstream) at Přelouč and Čelákovice using Turner Designs 0 AU fluorometers running in continuous measurement mode. The injection site at Němčice is 26.9 km upstream of Přelouč. The distance between Přelouč and Čelákovice is km. The reach between Přelouč and Čelákovice corresponds closely to the main study reach running from Valy to Lysá. The mean observed discharge Q tracer during the 24 hour period before 2of9

3 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS Figure. Map showing main features of the river Elbe around Valy and Lysá. Along with the river network the map shows towns, weirs (angle marks), continuous water quality monitoring stations (diamonds), and the tracer injection point (square). the survey was very low (7.3 m 3 /s). This value is lower than any of the flows for which an advective time delay could be estimated from the electrical conductivity data. 4. Statistical Models [] The prediction of the movement of an unobserved pollutant plume requires the prediction of t, DF and SSG for the unobserved event given observed data about past events. The first step toward achieving prediction of t, DF, and SSG is to formulate a statistical model that represents the beliefs about them. The choice of statistical model needs to reflect all sources of variability in the observed data. The two main sources of variation are the underlying distribution of the true values of t, DF, SSG over pollution events and the (possibly significant) errors made when observing the true value. [2] No correlation was witnessed between t, DF, and SSG in the observed data. This may be partially due to the low number of samples of SSG and DF but is in keeping with past studies [Green, 998; Green et al., 994; Wallis et al., 989]. Because of this, t, DF, and SSG are treated as independent of each other and so can be statistically modeled individually. The timing of the events in the observed data series is also well spaced. As a result, observations of each of t, DF, or SSG can be treated as a series of values from independent random variables. The remainder of this section considers the formulation of a statistical model for each of t, DF, and SSG. Table. Summary Characteristics of Three Pollution Events in the River Elbe Reach Event Dispersive Fraction Input Output Steady State Gain Advective Time Delay, hours 24 hour Mean Discharge, m 3 /s : : : Dispersive Fraction [3] The value of DF is known to vary between events. This variation is driven by the physical nature of the catchment as the DF will depend upon in reach conditions, such as the operation of the weirs within the reach and the in-stream vegetation. These are unobserved variables in terms of the information available for the study so cannot be explicitly represented. The low number of data points (three) for DF meant that no meaningful relationship with discharge was witnessed in the observed data. There is, however, empirical evidence from elsewhere [Green, 998; Wallis et al., 989] that DF does not change significantly with discharge outside of flood events. [4] Consider then a basic belief about the statistical model for the DF. Let us propose that the true value of the ith observation of DF, denoted cdf i, is distributed normally with unknown mean m and known variance w 2. Then, under the assumption of the temporal independence of the observations, we can express n the joint probability o of the true values of DF; cdf ¼ cdf ;...; cdf n ; given the unknown m and denoted P( cdfjm) as P cdf m ¼ Yn pffiffiffiffiffi exp 2 cdf 2p w 2w 2 i m : ð3þ Let us propose also that we observe cdf i with an additive normal error with mean zero and known variance s 2. The probability of the observed data series DF, given the true values cdf is P DFDF c ¼ Yn pffiffiffiffiffi exp 2 2p s 2s 2 DF i ddf i : ð4þ The selection of P( cdfjm) andp(dfj cdf cdf) is motivated by the desire to approximate the distribution of DF in a parametric form Steady State Gain [5] The value of the SSG is known to vary between events. This variation is driven by variations in the flow that enters during the length of the reach as well as the reactivity 3of9

4 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS observed with error that is also assumed to be lognormal with unknown standard deviation s t so giving PðTj^T; s t Þ / s n t exp X n 2s 2 t ðlogðt i Þlogð^t i ÞÞ!: 2 ð6þ This selection is motivated by the fact that the shorter advective time delays correspond to more rapid transport conditions in the reach. The more rapid the transport of pollutant the less time is available for mixing of the pollutant. This leads to better definition in the observed rising concentration limb downstream, thus a more accurate estimation of the observed value of the advective time delay can be made. [8] To represent the underlying relationship between advective time delay and discharge the model ^t i ¼ a þ b^q i ð7þ Figure 2. Scatterplot of observed advective time delay t against the observed mean 24 hour discharge Q showing the increase in t as Q decreases. or volatility of the pollutant. The low number of data points for SSG meant that no meaningful relationship with either the main channel discharge, or that of its tributaries, was witnessed in the data. This leads to the proposition of a similar statistical model to that of the DF Advective Time Delay [6] Since the ADZ model assumes a steady discharge the estimates the observed values of t were related to the observed mean 24 hour discharge (Q) during the event. Figure 2 shows a clear relationship between Q and t. Although the observed data T and Q appear correlated it may be that observational error masks a stronger relationship between the true values of advective time delay (^T) and the true discharge values ( ^Q). In investigating this relationship it should be noted that there are errors on the observation of both variables as neither t or Q can be accurately measured. Both of these errors have to be treated explicitly to achieve unbiased estimation of any parameter of the model relating them [Jefferys, 980]. In keeping with the independence assumptions made earlier in this section the observational errors on ^T and on ^Q are treated as independent over different events. [7] The value ^Q i is observed with error that is assumed to be lognormal with unknown standard deviation s Q. This results in P Q ^Q; s Q / s n Q X n exp 2s 2 Q! 2 ðlogðq i Þ log ^Q i : ð5þ This selection is motivated by the fact that the variance of the observation of high flows is normally greater than that of lower flows. Also if the monitoring equipment is well sited, calibrated and maintained any bias in the observation will be small compared to the variance. The value ^t i is is proposed. While this is one of several models that could be used it has been found to be most satisfactory in other studies [Green et al., 994]. The parameters a and b of the model are unknown. Use of this model allows (6) to be expressed as in terms of the unknowns of the underlying relationship and the true discharges as follows: PðTj^T; s t Þ ¼ P Ta; b; ^Q; s t / s n t exp X n 2s 2 t!! logðt i Þ log a þ b^q 2 A: i 5. Statistical Inference Allowing for Errors in the Data [9] The prediction of the movement of an unobserved pollutant plume requires the prediction of the event summary variables given observed data about passed events. For a future summary variable ^k new (one of DF, SSG, or t) predicted given a vector of observed data K the predictive distribution is P(^k new jk); i.e., the probability of ^k new given K. When constructing predictive distributions there are two choices too be made, the statistical model and its inference. [20] The statistical model is chosen to represent the probability of sampling the observed data, denoted P(KjQ) where Q is a vector of unknowns. From section 4 P(KjQ) for DF is formed by multiplying equations (3) and (4) with the vector of unknowns being Q ={cdf, m}. For t P(KjQ) is the product of equations (5) and (8) with Q ={a, b, s Q, s t, ^Q}. [2] The second choice is the means of drawing inference about the unknowns, Q, in the statistical model. The two most common forms of drawing statistical inference about Q are maximum likelihood estimation (MLE) [Pawitan, 200] and Bayesian Inference [Lee, 2004]. In MLE P(KjQ) is maximized over Q by selection of an optimal parameter vector Q MLE. In this optimization Q is treated as a constant to be estimated. The predictive distribution is then approximated by P(^k new jq MLE ). The Bayesian inference methodology treats Q as a random variable, with a distribution P(Q) ð8þ 4of9

5 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS defined prior to the observation of data. Through the use of Bayes theorem P KQ PðQÞ P QK ¼ PðKÞ / P KQ PðQÞ the prior distribution P(Q) is updated using the information within the observations to give the posterior distribution P(QjK). The posterior distribution contains all the information learnt about the unkncowns Q. The predictive distribution P(^k new jk) is therefore given by Q ð9þ Z Pð^k new jkþ ¼ Pð^k new jqþpðqk j ÞdQ: ð0þ [22] In this paper we adopt a Bayesian approach. There are two primary reasons for this choice: P(Q) allows any information about Q not contained within the observed data to be represented in the analysis; and the predictive distribution allows for the full uncertainty of Q to be included in the prediction Step 2 [25] The posterior distribution of { cdf, m} given DF and the chosen statistical model can then be expressed, up to proportionality, as P cdf; mjdf / exp 2 f 2 ðm gþ2 þ Xn 2þ!! 2 cdf w 2 i m s 2 DF i cdf i ð2þ Manipulation of the expression (2) allows the unknown values cdf to be integrated out to give the posterior distribution of m approximately for 0 < m <as PðmjDFÞ / exp s 2 m m 2 s where m and s are given by m ¼ s ¼ s 2 þ! X n w 2 w 2 s 2 DF i þ g f 2 n w 2 þ f 2 n w 4 s 2 þ! w 2 : ð3þ 6. Prediction of a Pollutant Plume [23] For each of DF, SSG, and t the prediction of a pollutant plume consists of three steps: the selection of a prior distribution P(Q) for the unknowns; the combination of the selected P(Q) and the statistical model from section 4 to construct the predictive distribution in the form of (0); and the generation of a random sample from the predictive distribution. A sample from the joint predictive distribution of DF, SSG, and t is formed by combining the random samples from the independent predictive distributions. The predictive distribution of the ADZ output can then evaluated. 6.. Dispersive Fraction 6... Step [24] The resulting vector of unknowns from the statistical model specified in section 4 is Q ={cdf, m}. An independent prior is proposed for each element of cdf, denoted cdf i which is consistent with the earlier assumption of independence. The definition of DF limits the values cdf i can take to between zero and one. A choice of prior distribution then that represents ignorance about where in this range cdf i is, until an observation is made, cdf i U[0, ]. For computational simplicity the prior of m is specified as PðmÞ ¼ ffiffiffiffiffi p exp 2p f 2f 2 ðm g Þ2 ðþ where {g, f} are a known mean and standard deviation of the prior distribution of m. Recalling that the DF and hence m must lie in the interval [0, ] selecting g = 0.5 and f = 000 gives, in effect, a uniform prior over this interval. 5of9 Approximating P(mjDF)on0<m < is reasonable since the prior on cdf constrains m to within this interval. Since the posterior distribution of m is a proper probability distribution it is clear from (3) that the distribution is normal with mean M S and variance on the range 0 < m <. Limiting the S predicted value of the dispersive fraction cdf new to within the plausible interval can be achieved by introducing the indicator variable I cdf new 8 < cdf new 2 ½0; Š ¼ : 0 otherwise thus resulting in the predictive distribution Pð^k new jkþ P cdf new jdf Z / I cdf new m exp 2w 2 s 2 m m 2 dm: s 2 cdf new m ð4þ ð5þ Simplification of this formula is possible by integration out of the unknown term m to show that P cdf new jdf / I cdf new exp 2 S cdf new M! 2 ð6þ S where S ¼ w 2 w 4 w 2 þ s M ¼ mw 2 w 2 : þ s

6 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS So the predictive distribution of the dispersive fraction, cdf new given the observed data, is a normal distribution within in the plausible region Step 3 [26] Sampling from the predictive distribution can then be undertaken by generating values from a normal distribution with mean M S and variance and retaining in the sample S only those that lie in the interval [0, ] Steady Stage Gain [27] In section 4 the proposed model for SSG is identical to that of DF. The specification of the prior distributions and the construction of the resulting predictive distribution to follow in a similar fashion to section 6. so long as it is believed that the SSG lies between zero and one. This should always be the case where, as in the study reach, there is a single pollutant source and the pollutant decays within the reach Advective Time Delay Step [28] Priors have to be placed upon the unknowns {a, b, s Q, s t, ^Q}. Treating these unknowns as independent variables allows them to be considered individually. The true discharges {^Q i ; i =,...n} must be positive values. Since ^T can also be expected to be positive, and decreasing with increasing discharge, a and b can be expected to be positive values. Other than this there is no prior knowledge on these unknowns. To represent this a prior can be constructed by considering each of {a, b, s Q, s t, ^Q} to distributed as jxj where X N(0, 0 2 ). The variance value chosen (0 2 ) is selected to be suitably large compared to the values expected for each of the variables {a, b, ^Q i } which can be expected to be of order (0, 0 3, 0 2 ) respectively. The variance parameters are also confined to the positive real space. Care has to be taken to avoid placing too heavy a weight on the lower tail of these distributions which would be consistent with the belief that there is no observational error. To achieve this s Q and s t are given lognormal priors with a mean of 0.2 and variance Step 2 [29] The posterior distribution of the unknowns can then be expressed as P a; b; s Q ; s t ; ^Q jt; Q / P a; b; s Q ; s t ; ^QÞP Ta; b; s t ; ^QÞP Q ^Q; s Q! / s n t sn Q exp S t 2 s 2 þ S Q t s 2 þ Q 0 6 a 2 þ b 2 þ Xn ^Q 2 i!! þ ð c s t dþ 2 þ c s 2 Q d ð7þ where S Q ¼ Xn S t ¼ Xn 2 logðq i Þlog ^Q i!! logðt i Þlog a þ b^q 2 i and {c, d} {0.8, 2.02} are the results of mapping the quoted mean and variance of the lognormal distribution into the log space. [30] In prediction we wish to evaluate the probability of the new advective time delay (^t new ) given a new observed flow (^Q new ). Assuming the independence of this event from those already observed, the predictive distribution P(^t new jq new, T, Q) can be expressed as Pð^t new jq new ; T; QÞ Z b ¼ P ^Q new ¼ ^t new a Q new; s Q a;b;s t;s Q; ^Q P a; b; s t ; s Q ; ^Q jt; Q dadbdst ds Q d ^Q: ð8þ Step 3 [3] Because of the form of P(a, b, s t, s Q, ^QjT, Q) in (7) it is not be possible to evaluate the integral in (8) directly and so alternative methods are required to simulate a sample from the predictive distribution. Simulation of a sample from the predictive distribution can be achieved by a three stage process. The first stage is to simulate N samples Q ½Š i ¼ n a ½Š i ; b ½Š i ; s ½Š i Q ; s ½Š t i o ; i ¼ ;...; N from the posterior distribution. The second stage is to [i] simulate a sample ^Q new ; i =,...N of ^Q new given {^Q new, s [i] [i] Q } with log (^Q new ) following a normal distribution with mean log(q new ) and standard deviation s [i] Q. The third stage is to evaluate t ½Š new i ¼ a i ½Š þ b ½Š i ^Q ½Š i new [i] The values ^t new ; i =,...,N represent a sample from the predictive distribution of ^t new. [32] Many sampling techniques exist [Ripley, 987; Tanner, 993] for sampling from the posterior distribution but one of the most general, that will be used here, is the Markov chain Monte Carlo (MCMC) technique [Chib and Greenberg, 995; Gilks et al., 996] as this allows simulation from distributions defined up to proportionality only. This technique generates a Markov chain whose stationary distribution is the required posterior distribution P(QjT, Q), provided that P(QjT, Q) obeys some general regularity conditions [Roberts and Smith, 994] and that the method of proposing jumps can reach all values of Q for which P(QjT, Q) is nonzero. The construction of the Markov chain is defined as follows. Suppose at step i the Markov chain is in state Q [i], a new state Q [i+] is proposed with probability q(q [i], Q [i+] ). The new state is accepted if P Q ½iþŠ jt; Q q Q ½iþŠ ; Q ½Š i z ð9þ P Q ½Š i jt; Q q Q ½Š i ; Q ½iþŠ where z is an evaluation from a uniform distribution on the interval [0, ], otherwise Q [i+] = Q [i]. Repeated proposition and testing of new states results in a chain which converges to a sample from the posterior. Note that several techniques : 6of9

7 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS Figure 3. Plots of probability (y axis) of DF (x axis) showing the sensitivity of the predictive distribution generated using the basic belief model for various values for the variance of the underlying true distribution (w 2 ) and error distribution (s 2 ). for improving the efficiency of the algorithm were used, such as centering [Roberts and Sahu, 997] and adjustment of the variances of the proposal distributions so the rate of accepting jumps was close to optimal [Roberts and Rosenthal, 200] ADZ Output [33] The predictive distribution of the ADZ output is now evaluated. By combining the samples of the independent samples of ^T new cdf new, and dssg new using equation (2) a sample from the predictive distribution of the ADZ model parameters {a, b, d} is obtained. The ADZ model can then be run for each set of parameters. The results of these ADZ runs are a sample from the predictive distribution of the ADZ model output. 7. Prediction of Tracer Experiment [34] As noted in section 3 a single tracer experiment was available for the reach. The prediction of the movement of the tracer plume is used here to demonstrate the application of the methodology outlined in sections 4 and 6. Two different applications of the methodology are made and the results compared. The applications differ in the method used for predicting t. In case A, t is predicted with the assumption that the discharge is observed without noise, that is ^Q = Q and s Q = 0. This results in a statistical model consistent with those used elsewhere [Green, 998], however the statistical inference used for prediction here is Bayesian in contrast to the MLE approach used previously. Case B follows the procedure in section 6.3 exactly, thus giving proper recognition to the sources of variation within the data. The prediction of the DF was similar in both cases following the procedure in section 6.. The sensitivity to the variance of observation error (s 2 ) and underlying distribution (w 2 ) was tested. Figure 3 shows that so long as w is not small (which would contradict the choice of model) the predictive distribution of cdf new is not sensitive to the actual values of w and s chosen. Values of 0.5 and 0.05 were selected for w and s respectively. The tracing chemical, Amidorhodamine G, was selected to have minimal decay (section 3), thus differs significantly from the reactive pollutants for which the SSG estimates are valid for. Direct estimation of the SSG for the tracer experiment was impractical since the tail of the tracer plume may not be fully captured. Given the nature of the tracer and the low additional input from the tributaries during the period of the experiment a constant SSG value of one has been predicted for the tracer experiment. [35] The evolution of states of the MCMC sampling for case A indicated a well mixed chain that has reached its stationary distribution. The evolution of the states of the MCMC sampling for case B suggest the variance parame- 7of9

8 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS for the tracer (^t tracer ) is negligible compared to the other variances (i.e., ^t tracer = t tracer ), b can be expressed as b ¼ ^Q tracer ðt tracer aþ and eliminated from the regression equation. Thus the Bayesian inference allows the tracer data to be incorporated into the analysis without increasing the number of unknowns, hence it should provide helpful and additional information about the unknowns in the statistical model. [39] The evolution of the states of the MCMC chain indicates good mixing with a similar pattern in the variance to that found for case B in section 7. Figure 4 shows a summary of the predictive distribution of ^t new for a variety of hypothetical values of Q new with and without the inclusion of the tracer experiment. The effect of including the tracer data using this substitution is to decrease the variance of the predictive distribution of ^t new for Q new close to Q tracer with a smaller effect at higher discharges. Figure 4. Summaries of the predicted distributions of advective time delay for three observed flows and three models: case A (white), case B (light gray), and case B including the tracer result (dark gray). The box encompasses the inter quartile range of the distribution, while the tails extend to the 5th and 95th percentiles. ters are highly correlated, with the underlying model fitting either predicting the T accurately at the expense of the fit to Q, or vice versa. However the joint distribution of a and b appears unimodal. The mixing of the chain in the a and b parameters may be inefficient when s Q is larger then s t. This will result in a more correlated series of states, thus a more correlated sample. [36] A summary of a sample of ten thousand predicted values of ^t new for the tracer experiment produced by each of the cases is shown in Figure 4. In both cases the predictive distribution brackets the observed value of 20 hours. Comparison of cases A and B indicates that explicit representation of s Q broadens the variance of the predictive distribution showing that the recognition of known uncertainty in the estimates of ^Q translates into increased uncertainty in the predictions for ^t new. [37] Simulation of the ADZ model shows that the ninety percent interval of the predictive distribution brackets the observed tracer plume for both cases A and B (Figure 5). The lower variance of the samples of ^t new in case A is witnessed in a more temporally focused predictive distribution compared to case B. 9. Discussion [40] Prediction of the movement of the tracer plume shows that the methodology appears to give reasonable results, with the 90% predictive interval bracketing the observed plume. The explicit representation of the errors of the discharge observation within the prediction of the advective time delay (case B) clearly shows that case A under represents the known sources of error and so is likely to under represent the ADZ predictive interval. While this visually appears not to compromise the results when predicting the tracer plume, further study on other reaches would reveal whether this is typical of a general pattern. The realistic incorporation of the tracer experiment into the calibration data set indicates, as would be hoped, that the 8. Effects on Prediction of Incorporating the Tracer Results Into the Estimation of T [38] The effects of incorporating the tracer results in prediction were considered using the full conditional relationship outlined in case B. Incorporating the tracer experiment data will give a further {t, Q} pair. Incorporation of the additional {t, Q} pair into the regression may appear to indicate the addition of an extra unknown ^Q tracer. However, using the assumption that the error on the observed t tracer 8of9 Figure 5. Tracer experiment plume (points) and graphical summary of the ADZ predictive distributions. Lines, from top to bottom, represent 95th percentile, median, and 5th percentile of predictive distributions of case A (dotted line) and case B (solid line) at each time step respectively.

9 SMITH ET AL.: POLLUTANT DISPERSION IN RIVERS, ADZ PARAMETERS incorporation of high-quality data reduces the uncertainty in the predictions. Further experiments at higher discharges would also be valuable in constraining the uncertainties at higher flows. [4] There are limitations in the methodology. As with all one dimensional dispersion models, the ADZ approach is only applicable once the tracer or pollutant is well mixed with the flow. It has been shown that higher-order ADZ models can be used to represent the initial mixing [Green et al., 994] but the parameters of higher-order models will not be estimated easily from the type of surrogate quality variables that have been used successfully in this study. Uncertainty in the prediction remains significant, and in the case of an incident involving a particular reactive or volatile pollutant, would be compounded by uncertainty in the effective SSG. The uncertainty of the effective SSG of a pollutant would, in the study reach, be further increased by the fluctuating input from the tributaries. The extraction of the estimates of DF, SSG and t from the surrogate data (in the case study conductivity measurements) involves a large number of subjective choices about the interpretation of time series data. The expert judgment needed for this requires the modeler to be well acquainted with the reach being modeled. In the case study, estimation of DF requires the calculation of a mean traveltime between upstream and downstream perturbations. This was achieved by converting the surrogate conductivity data to a concentration. Here the conversion used was approximately linear but this may not always be the case and indicates that further research into the suitability of various types of surrogate data series may prove fruitful. As with all uncertainty analysis the results are conditional upon the statistical model and observational error model selected from amongst the many that are consistent with the modeler s prior assumptions about the operation of the system. However, the methodology may still be useful for the wide range of rivers for which only limited information on dispersion is available. [42] Acknowledgments. The NERC long-term research program on Uncertainty in Environmental Modeling (NER/L/200/00658) supports Paul Smith. The Czech Republic Ministry of Environment grants VaV/ 50/0/99 and VaV/650/5/03 supported Sarka Blazkova. The continuous water quality data were collected by the Elbe River Authority as part of an ongoing monitoring program. BfG Koblenz, Elbe River Authority and TGM WRI performed the tracer experiment. References Beer, T., and P. C. Young (983), Longitudinal dispersion in natural streams, J. Environ. Eng., 09, Behrens, H., et al. (200), Toxicological and ecotoxicological assessment of water tracers, Hydrogeol. J., 9, Bencala, K. E., and R. A. Walters (983), Simulation of solute transport in a mountain pool-and-riffle stream: A transient storage model, Water Resour. Res., 9, Cheong, T. S., and I. W. Seo (2003), Parameter estimation of the transient storage model by a routing method for river mixing processes, Water Resour. Res., 39(4), 074, doi:0.029/200wr Chib, S., and E. Greenberg (995), Understanding the Metropolis-Hastings algorithm, Am. Stat., 49, Deng, Z. Q., et al. (2002), Longitudinal dispersion coefficient in singlechannel streams, J. Hydraul. Eng., 28, Dostal, K., et al. (2000), Zpráva o 3.testovacím pokusu na Labi Němčice - Mělník, report, 5 pp., Povodi Labe, Hradec Králové, Czech Republic. Fischer, H. B., et al. (979), Mixing in Inland and Coastal Waters, Elsevier, New York. Gilks, W. R., et al. (Eds) (996), Markov Chain Monte Carlo in Practice, CRC Press, Boca Raton, Fla. Green, H. M. (998), Predictive uncertainty of the aggregated dead zone model for longitudinal dispersion, Ph.D. thesis, Lancaster Univ., Lancaster, U. K. Green, H. M., et al. (994), Pollution incident prediction with uncertainty, in Mixing and Transport in the Environment: A Memorial Volume for Catherine M. Allen (954-99), edited by K. Beven, et al., pp. 3 40, John Wiley, Hoboken, N. J. Harvey, J. W., and B. J. Wagner (2000), Quantifying hydrologic interactions between streams and their subsurface hyporheic zones, in Streams and Ground Waters, edited by J. B. Jones and P. J. Mulholland, pp. 4 45, Elsevier, New York. Jefferys, W. H. (980), On the method of least-squares, Astron. J., 85, Käß, W. (998), Tracing Technique in Geohydrology, 58 pp., A. A. Balkema, Brookfield, Vt. Kilpatrick, F. A., and K. R. Taylor (986), Generalization and applications of tracer dispersion data, Water Resour. Bull., 22, Koussis, A. D., and J. 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Gorelick (986), A statistical methodology for estimating transport parameters: Theory and applications to onedimensional advective-dispersive systems, Water Resour. Res., 22, Wagner, B. J., and J. W. Harvey (997), Experimental design for estimating parameters of rate-limited mass transfer: Analysis of stream tracer studies, Water Resour. Res., 33, Wallis, S. G., et al. (987), A microcomputer based fluorometric data logging and analysis system, Inst. Water Eng. Sci. J., 42, Wallis, S. G., et al. (989), Experimental investigation of the aggregated dead zone model for longitudinal solute transport in stream channels, Proc. Inst. Civ. Eng., Part 2, 87, 22. K. Beven, P. Smith, and J. Tawn, Lancaster Environment Centre, Lancaster University, Bailrigg, Lancaster LA 4YQ, UK. (p.j.smith@lancs. ac.uk) S. Blazkova, T. G. Masaryk Water Research Institute, Podbabska 30, Prague 6, Czech Republic. L. Merta, Povodi Labe, s.p., Vita Nejedleho 95, Hradec Králové, Czech Republic. 9of9