Knowledge about the strength and travel times of hyporheic exchange is vital to predict reactive transport and biogeochemical cycling in streams. In this study, we outline how to perform and analyze stream tracer tests using pulse injections of fluorescein as conservative and resazurin as reactive tracer, which is selectively transformed to resorufin when exposed to metabolically active zones, presumably located in the hyporheic zone. We present steps of preliminary data analysis and apply a conceptually simple mathematical model of the tracer tests to separate effects of in-stream transport from hyporheic exchange processes. To overcome the dependence of common parameter estimation schemes on the initial guess, we derive posterior parameter probability density functions using an adaptive Markov chain Monte Carlo scheme. By this, we can identify maximum-likelihood parameter values of in-stream transport, strength of hyporheic exchange, distribution of hyporheic travel times as well as sorption and reactivity coefficients of the hyporheic zone. We demonstrate the approach by a tracer experiment at River Goldersbach in southern Germany (60 L/s discharge). In-stream breakthrough curves were recorded with online fluorometers and jointly fitted to simulations of a one-dimensional reactive transport model assuming an exponential hyporheic travel-time distribution. The findings show that the additional analysis of resazurin not only improved the physical basis of the modeling, but was crucial to differentiate between surface transport and hyporheic transient storage of stream solutes. Parameter uncertainties were usually small and could not explain parameter variability between adjacent monitoring stations. The latter as well as a systematic underestimation of the tailing are due to structural errors of the model, particularly the exponential hyporheic travel-time distribution. Mean hyporheic travel times were in the range of 12 min, suggesting that small streambed structures dominate hyporheic exchange at the study site.
 The hyporheic zone is the transition zone between streams and aquifers. A certain fraction of the river water enters the hyporheic zone, stays there over a distribution of times, and returns to the stream. This process, denoted hyporheic exchange, plays a vital role concerning retention, turnover, and filtering of contaminants and nutrients in river water, and is important in the forming of habitats for a variety of organisms [Boulton et al., 1998; Robertson and Wood, 2010; Runkel, 2007; Wondzell, 2011]. Hyporheic exchange flows are most commonly induced by morphological features, such as steps, dams, pools, or riffles. The strength of hyporheic exchange and the travel time of hyporheic water primarily depend on the bathymetry of the river bed, the pressure distributions on the bed surface, and the hydraulic conductivity of the stream bed [Cardenas et al., 2008; Elliott and Brooks, 1997; Kasahara and Wondzell, 2003].
 In many studies, the exchange of solutes between natural rivers and the hyporheic zone has been conceptualized by the transient-storage model, in which an active main channel with rapid flow is in contact with adjacent zones of immobile water, and the solute exchange between those two compartments is driven by the concentration difference [Bencala and Walters, 1983; Castro and Hornberger, 1991; De Smedt et al., 2005; Runkel, 1998]. In the simplest models, these immobile transient-storage zones are lumped to a single zone with apparent physical properties [Bencala and Walters, 1983; Engelhardt et al., 2011; Hart, 1995; Wörman, 1998; Zaramella et al., 2003]. However, water retention may take place in both the river itself (e.g., by pools and dead zones) and in the hyporheic zone [Phanikumar et al., 2007]. In case of different subsets of transient storage with substantially different physical properties, considering only a single immobile zone may fail to adequately describe solute transport [Choi et al., 2000; Gooseff and McGlynn, 2005; Kazezyilmaz-Alhan and Medina, 2006]. For this reason, additional immobile zones have been incorporated into the models which improved the model outcome [Briggs et al., 2009; Gooseff et al., 2004; Harvey et al., 2005]. Harvey et al.  used an information theory index that gives an objective criterion whether the improvement of the model fit justifies the introduction of additional parameters describing the second storage zone. Introducing multiple immobile zones may also be chosen as a parameterization of mass exchange with a single hyporheic zone exhibiting a distribution of travel times that does not follow a simple parametric distribution [Haggerty et al., 2002].
 Stream-tracer tests are considered a reliable method to assess parameters describing transport in streams [Wagner and Harvey, 1997]. In such tests, a compound is added into the stream and its concentration over time (denoted as breakthrough curve (BTC)) is measured at a location further downstream. However, interpreting conservative tracer tests with regard to hyporheic exchange may be challenging because in-stream mixing (dispersion), in-stream transient storage (e.g., by pools or oxbows), and hyporheic exchange have similar retention effects on solutes, so that the effects of these processes on the BTC can hardly be separated in the analysis of the conservative-tracer data [De Smedt et al., 2005; Harvey et al., 1996]. Fitting models of stream-tracer transport to measured breakthrough curves can already be nonunique for the simplest single transient-storage model with first-order mass exchange, which may be manifested by the dependence of the estimated parameters on initial guesses. The problem of ambiguity becomes even more prevalent when fitting more complicated transient-storage models with multiple storage zones or multiparameter residence-time distributions, which puts the practical applicability of these methods into question.
Haggerty et al.  proposed to use the weakly fluorescent dye resazurin as a stream tracer, which undergoes irreversible degradation to the more strongly fluorescent dye resorufin predominantly within the hyporheic zone. The traditional application of resazurin is within cell viability tests because the resazurin-to-resorufin reduction is coupled to the oxidation of the electron carrier NADH to NAD+ by enzymes which are available in all viable cells [O'Brien et al., 2000]. Outside of cells, the reduction rates of resazurin are negligible. Due to the small number of planktonic bacteria, the reduction and decay rates in river water are very small, and therefore negligible with respect to typical tracer test durations. However, these rates are significantly higher when the tracer is exposed to conditions with relatively high microbial activity as can be found in the hyporheic zone [Gonzalez-Pinzon et al., 2012]. Thus, jointly analyzing BTCs of a conservative tracer and the reactive tracer resazurin may help separating effects of hyporheic exchange and in-stream mixing on conservative tracer BTCs alone and yield parameter values quantifying hyporheic exchange and metabolic activity of the stream bed. Previous studies showed that the use of this compound provided reliable information for characterizing hyporheic transient-storage zones that are metabolically active [Argerich et al., 2011; Haggerty et al., 2009].
 In this paper, we present a coupled analysis of the stream-tracers fluorescein and resazurin to infer parameters describing characteristics of in-stream transport as well as hyporheic exchange and metabolic activity of the hyporheic zone. To our knowledge, this study is the first comprehensive joint analysis of a resazurin and fluorescein tracer test where pulse injections of the tracers and online fluorometry were used to measure the breakthrough-curves at different stream locations. In contrast to previous studies, online fluorometry of conservative and reactive tracers at multiple stations allowed to obtain high frequency and high quality data series which are unaffected by sample transport and storage and can be used to better distinguish hyporheic exchange from in-stream processes.
 Our own experience in estimating the parameters of transient-storage models by traditional gradient-based parameter-estimation methods, such as the Levenberg-Marquardt approach, was that the parameter estimates strongly depended on the initial guess. This is a clear indication of local minima in the objective function. Also, the traditional approach of estimating the parameter uncertainty by linearized uncertainty propagation [e.g., Press et al., 1992, chap. 15] may be unreliable under such conditions. As an alternative, we thus use a gradient-free ensemble approach based on the Markov chain Monte Carlo (MCMC) method for parameter identification [Schoups and Vrugt, 2010; ter Braak and Vrugt, 2008; Wöhling and Vrugt, 2011], which results in the entire distribution of the parameters, conditioned on the measurements, and thus provides a comprehensive uncertainty analysis.
 The main objectives of the present study are (1) to assess the suitability of the improved experimental approach of combined conservative and reactive stream-tracer tests with pulse injection to derive the bulk transport characteristics of a stream including hyporheic exchange, (2) to quantify parameter uncertainties, model errors and parameter correlations associated with common model approaches, and (3) to evaluate the benefit gained by combining conservative and reactive tracers, joint analysis of multiple stations, and an increase in model complexity (e.g., by adding a second storage zone).
 Although we use improved methods for the tracer experiment and subsequent parameter identification, the data analyses in this paper is still based on commonly applied model approaches including simple parametric transfer functions, first-order reaction kinetics, and equilibrium sorption of the reactive tracer. The reason for this is to retain comparability with previous studies and to quantify the structural model error associated with this type of model approach. In a companion paper [Liao et al., 2013], we present a more elaborate analysis that does not rely on a specific parametric shape of the hyporheic travel-time distribution (see also Liao and Cirpka ) and considers two-site sorption of the reactive tracer and its daughter compound within the hyporheic zone. The latter approach, however, is not accessible to the comprehensive MCMC-based uncertainty analysis applied in the present contribution.
2. Materials and Methods
2.1. Description of the Field Site
 A fluorescent-tracer test where fluorescein and resazurin were combined as conservative and reactive tracers, respectively, was conducted in River Goldersbach which is located in southwestern Germany, approximately 30 km south of Stuttgart. This third-order stream drains an area of about 70 km2. The watershed is characterized by a humid, temperate climate, a hilly morphology with altitudes ranging from 360 to 580 m a.s.l., and vegetation that is dominated by mixed forest. The geology of the area is dominated by interbedded sandstone and marlstone layers of the Upper Triassic (“Keuper”) at the hillslopes and quaternary sediments formed from these formations in the valleys.
 The stream section for the tracer tests has a total length of 1210 m and is located approximately 3 km upstream of the confluences of River Goldersbach with River Ammer and subsequently River Neckar within the city of Tübingen. The river flows along a highly meandering course through a narrow floodplain that is flanked by moderately steep hillslopes on both sides. The river morphology is characterized by riffle-pool sequences with abundant point bars. The river bed consists of medium to coarse sand with frequent boulders that cover the bed rock with a thickness of up to several decimeters. The mean physicochemical properties of the stream water during the tracer experiment are listed in Table 1.
Table 1. Mean Physicochemical Properties of River Goldersbach at Monitoring Station 1 During the Experiment
Standard deviations of the respective measurements (n ≥ 150).
 In approximately 15 L of stream water, 8.70 × 10−3 moles of fluorescein and 1.13 × 10−1 moles of resazurin were premixed. The solution was added as a pulse and spread out over the entire width of the stream to reduce the distance until which complete cross-sectional mixing of the tracers was reached. Both tracers are subject to photodegradation when exposed to sunlight [Gaigalas et al., 2004; Haggerty et al., 2008]. Therefore, the experiment was started after sunset and completed before sunrise.
 The BTCs of the tracers fluorescein, resazurin, and resorufin as well as the temperature and turbidity of the river water were measured online at three monitoring stations denoted MS1, MS2, and MS3 at 830, 1075, and 1210 m distance downstream of the injection site (Figure 1), using portable field fluorometers of the type GGUN FL-30 [Lemke et al., 2013; Schnegg, 2002; Schnegg and Flynn, 2002] with a temporal resolution of 10 s. The devices were placed in the thalweg of the river to ensure an optimal oncoming flow. The pH of the river water was monitored at MS2. Resorufin was measured for consistency check of the mass recovery but not included in the model-based analysis of the present study. The calibration of the online fluorometers was performed with stream water directly before the experiment to account for the strong dependence of the fluorescence intensity of the dyes on the pH value and temperature of the river water. Multiparameter probes that measured and recorded temperature and absolute water pressure with a temporal resolution of 5 min were placed at every monitoring station. A barometric pressure probe yielded the data to convert the absolute pressure to stream water depths. The latter was used to check whether discharge had varied during the course of the experiment. All probes were operative over a duration of at least 24 h starting approximately 6 h prior to tracer injection.
 Data evaluation after the tracer experiment included removal of sporadic data outliers by applying a moving median with a window size of 50 s to all BTCs. During the tracer test, the pH exceeded the value of 8 at all times. Above this value, the fluorescence intensities of the tracers used in the experiment do not depend on pH, so no corrections were necessary in this respect. Corrections of tracer concentrations due to temperature fluctuations during the experiment were performed following the procedures described by Lemke et al. .
2.3. Governing Equations
 For the analysis of the acquired data, we adopt a simplified version of the conceptual model for streams undergoing hyporheic exchange described by Liao and Cirpka , among others. In the model, solutes are assumed to undergo one-dimensional advection and dispersion in the main channel, supplemented by storage within the hyporheic zone, which is described as a specific volume flux of river water entering the hyporheic zone, staying there over a distribution of times and reentering the stream at the same point. This model is mathematically equivalent to other transient-storage models [Bencala and Walters, 1983; Haggerty et al., 2000; Wörman, 1998; among others]. The governing equation describing these processes reads as:
in which ci [M/L3] is the concentration of compound i, which is given as molar concentration throughout the paper, t [T] is time after the injection of the tracer, v [L/T] is the velocity of the stream water, x [L] is the distance downstream of the injection point, D [L2/T] is the longitudinal dispersion coefficient, qhe [T−1] is the discharge undergoing hyporheic exchange per volume of stream water, gi(τ) [T−1] is the hyporheic travel-time distribution of compound i, τ [T] is the time that a water particle resides in the hyporheic zone, and λi [T−1] is the first-order reaction rate coefficient of compound i. In the following i = flu stands for fluorescein and i = raz for resazurin. Equation (1) is subject to an initial condition of ci = 0 for all x and a boundary condition representing a pulse injection into the flux at x = 0:
in which Mi [M] is the injected mass (or number of moles) of compound i, A [L2] is the cross-sectional area of the stream, and δ(·) is the Dirac delta function (with inverse units of the argument). In the model, we assume that all coefficients introduced in equation (1) are uniform over the length of the subreach and that the concentrations are uniformly distributed over the cross section of the stream. Moreover, the model adheres to a zero-value auxiliary condition at the infinite-distance limit:
 In the hyporheic zone, the reactive tracer resazurin is subject to reduction to the daughter compound resorufin as well as decay to other compounds. Although to date little is known about the kinetics of these two reaction processes, they have previously been successfully described with first-order rate coefficients [Gonzalez-Pinzon et al., 2012; Haggerty et al., 2008, 2009; Stanaway et al., 2012]. Analogously to Argerich et al. , we thus describe the total removal of resazurin in the hyporheic zone as a consolidated first-order rate coefficient λraz. Also, we account for sorption of resazurin within the hyporheic zone by implementing a retardation factor Rraz into the hyporheic travel-time distribution. For reasons of continuity, we decided to closely follow the concepts regarding sorption properties in the previous works cited above and likewise employed a simple one-site equilibrium sorption concept. By this, we accept that this choice may pose an oversimplification in certain settings where retardation of the tracers is dominated by kinetic processes. In case of fluorescein the rate coefficient λflu = 0 and retardation Rflu = 1 because fluorescein is assumed to be an ideal tracer under the given pH conditions.
 Different parametric models have been suggested in the literature for the hyporheic travel-time distribution (g(τ) in equation (1)). Haggerty et al. [2000, 2002] used a truncated power-law in multirate mass-transfer studies, while Wörman et al.  showed that a log-normal model was adequate to describe solute transport in a 30 km reach in Sweden. However, most commonly an exponential distribution of hyporheic travel times is used as it is implemented in the standard single transient-storage model OTIS with linear exchange [Bencala and Walters, 1983; Briggs et al., 2010; Claessens et al., 2010; Laenen and Bencala, 2001; Wondzell, 2006]. We decided to adapt this concept for our study, because its mathematical description is straightforward and also because it can be fully described with a single parameter. Thus, we arrived at the following hyporheic travel-time distribution:
in which k [T−1] is the inverse of the mean hyporheic travel time and R [-] is the retardation factor within the hyporheic zone.
 Equations (1)-(3) can be transferred into the Laplace domain where the time derivative becomes a multiplication with the complex frequency s [T−1], resulting in the following ordinary differential equation:
in which and are the Laplace transforms of ci(t) and gi(τ), respectively. In the Laplace domain, the boundary and auxiliary conditions become:
 Equations (5)-(7) have the following analytical solution:
 The backtransformation of equation (8) to the time domain is done numerically following the approach of De Hoog et al. .
2.4. Preliminary Data Analysis
 Prior to fitting the model to the data, the recorded tracer BTCs were cut at early and late times when the concentrations had fallen below the limit of quantification for the tracers. The preliminary data analysis is based on the temporal moments of the BTCs that provide useful information on tracer mass balance and river discharge. Moreover, the moments can be used to derive approximate values of the in-stream transport parameters. The raw temporal moments of the tracer concentrations were calculated as follows:
in which indices a and i denote the order of the moment and the compound, respectively. The second central moment is computed by
 In order to test the plausibility of the tracer experiment, we calculated the river discharge Q [L3/T] at every monitoring station from the zeroth temporal moment μ0,flu of the conservative tracer according to the tracer dilution method:
in which Minj,flu [M] is the number of fluorescein moles injected. This expression assumes a complete recovery of the conservative tracer fluorescein, which usually is not the case in real tracer tests. Hence, an apparent slight increase in the computed discharge downstream the injection point might be caused either by a real inflow of water and/or by incomplete recovery of the tracer.
 Data plausibility can be further examined by calculating the recovery rate of the reactive tracers resazurin and resorufin rec by
where M stands for the number of moles of tracer measured in the stream and Minj for the number of moles injected. The recovery rate rec of the reactive tracers must not exceed 100%, because this would imply that either resazurin was produced within the stream or resorufin was produced by a process different than resazurin degradation, which is not possible. Resazurin and resorufin have unequal molar masses, so that recovery rate calculations have to be done with molar concentrations in order to obtain a resazurin-to-resorufin stoichiometry of 1:1.
2.5. Parameter Inference Using Bayes Theorem
 A Bayesian approach was adopted in this study to quantify model parameters and their uncertainty. Let us consider the model f that simulates the response with length n using a vector of m model parameters, : Y = f(u). Further, we consider that denotes a vector with the observed data. We then combine the data likelihood, , with a prior distribution p(u) by Bayes theorem to infer the posterior probability density function of the model parameter vector u:
 We assume measurement errors, σflu = 0.35 μmol m−3 and σraz = 2.23 µmol m−3 for fluorescein and resazurin concentrations, respectively, which accounts for the higher uncertainty in the measurements of resazurin due to its lower quantum yield. If we further assume the error residuals to be uncorrelated and normally distributed with constant variance, the data likelihood function can be calculated as
where yj(u) and , j = 1, … , n denote the model predicted values and their corresponding observations, respectively. In our approach, we prefer to use the log-likelihood function, of equation (16) rather than :
 In the parameter inference scheme, we follow a three-step approach. In the first step, only fluorescein data are fitted for each monitoring station individually using the log likelihood function of equation (17) with . These runs are subsequently referred to as DF1, DF2, and DF3 for the stations MS1, MS2, and MS3, respectively.
 In the second step, the BTCs of fluorescein and resazurin are jointly fitted at each monitoring station. We use an aggregated likelihood function for the joint fitting:
where, yflu,j(u) and yraz,j(u) are the model predicted values for fluorescein and resazurin, respectively, and (j = 1, … , nflu) and (j = 1, … , nraz) are the corresponding observations. The parameter vector utilized in equation (18) is . The corresponding runs are subsequently referred to as DJ1, DJ2, and DJ3 for the stations MS1, MS2, and MS3, respectively.
 In the third step, we attempt to obtain a single set of parameters that describes both BTCs of fluorescein at MS1 and MS2 simultaneously as well as a set of parameters that can describe the BTCs of fluorescein and resazurin simultaneously at these two monitoring stations. The corresponding runs are subsequently referred to as DF12 and DJ12, respectively.
 The prior distribution, p(u) was assumed to be uniform with the following parameter ranges: v = [0.01 … 0.5]; D = [0.01 … 5.0]; qhe = [1 × 10−5 … 1 × 10−3]; k = [1 × 10−6 … 1 × 10−2]; λraz = [1 × 10−6 … 1 × 10−2]; and Rraz = [1 … 3]. The choice of these ranges was guided by the preliminary data analysis (v and D) and previously published values [Haggerty et al., 2008, 2009].
 To generate samples from the posterior distribution, we use the differential evolution adaptive metropolis (DREAMZS) adaptive MCMC scheme with sampling from an archive of past states to generate candidate points in each of N parallel Markov chains. The N chains define a population which corresponds to an N × m matrix, with each chain as a row. The states of the individual chains are independent at any generation after DREAMZS has become independent of its initial value. After this burn-in period, the convergence of a DREAMZS run can thus be monitored with the statistic of Gelman and Rubin . In all calculations reported herein, we use N = 10 and selected the last 10,000 accepted samples after convergence was observed in all chains for the calculation of the posterior parameter pdfs. All other algorithmic parameters are set to their recommended values [Vrugt et al., 2009]. For more details on the parameter inference scheme, please refer to ter Braak and Vrugt , Wöhling and Vrugt , Schoups and Vrugt , and Wöhling et al. .
3.1. Tracer Breakthrough Curves
 The BTCs of the tracers fluorescein, resazurin, and resorufin measured at the three monitoring stations are shown in Figure 2. Because unequal numbers of moles were injected for fluorescein and resazurin, respectively, the tracer concentrations were normalized by the number of injected moles to allow for a better comparability of the respective BTCs. The turbidity of the river water never exceeded five nephelometric turbidity units (NTU) during the experiment, indicating good conditions for the separation of tracer signals [see Lemke et al., 2013]. We injected the tracers significantly upstream of our monitoring stations following the suggestions in Barsby  to allow for complete lateral mixing.
 The zeroth moments of all tracers at each monitoring station were calculated using equation (11) and are listed in Table 2. The zeroth moment of fluorescein slightly decreased from MS1 to MS3 which may be the result of either an increase of discharge (when substituted into equation (13)), dilution by incoming groundwater that is balanced by gross losses of the stream [Payn et al., 2009], or an incomplete recovery of the injected fluorescein mass. However, there are no surface tributaries along the investigated subreach and the data from the pressure loggers showed that the river stage was nearly constant over the whole duration of the experiment (Table 1). Thus, the cause for the decrease of the zeroth moment ultimately remains unclear. However, the estimated zeroth moment only decreased by about 7% (between MS1 and MS3), and we therefore believe that no significant errors are introduced to the analysis of the BTCs.
Table 2. Results of Moment Analysis, Recovery Rate of Resazurin and Resorufin, and the Estimated Discharge for the Three Monitoring Stations
μ0,i, zeroth temporal moment of compound i, normalized by the number of injected moles of fluorescein for i = flu and normalized for the injected moles of resazurin for i = raz and i = rru; rec, recovery rate estimated from equation (14); Q, discharge estimated from equation (13).
Distance to injection point
 With distance from the injection point, the zeroth moment of resazurin decreased while the zeroth moment of resorufin increased. This is in agreement with our understanding of the system, because with distance and time resazurin is increasingly transformed to resorufin. At MS3, 18% of the resazurin has reacted to its daughter compound resorufin. The calculated recovery of the reactive tracers according to equation (14) shows a loss of about 24% (1-rec) at MS3, which is already normalized by the zeroth moment of fluorescein and thus unaffected by dilution. This indicates that (1) resazurin and/or resorufin partially reacted to metabolites that are not detectable with our fluorometers [e.g. Haggerty et al., 2009], (2) a part of the tracers are sorbed at sediment surfaces over time periods exceeding the duration of our tracer test (as reported in Argerich et al. ) or (3) a combination of both options. However, the reasons for the tracer mass loss is irrelevant for our model because none of these processes take place in the stream water and the depletion of resazurin in the hyporheic zone is captured by the integral parameter λraz.
3.2. Parameter Estimation Using Different Tracer and Model Constraints
 Figure 3 shows the simulations of fluorescein BTCs with the maximum-likelihood parameter set from runs DF1, DF2, and DF3 (fit of fluorescein only) and Figure 4 the simulations of fluorescein and resazurin with the maximum-likelihood parameter set from run DJ1, DJ2, and DJ3 (joint fit of fluorescein and resazurin) at the individual monitoring stations, respectively. The figures also include the measurements, the bandwidth of model results due to parameter uncertainty, and the remaining model uncertainty.
 By combining the reactive and the conservative tracer in the aggregated likelihood function (equation (18)), we expected to better distinguish the contribution of hyporheic exchange from overall transient-storage processes. Therefore, we will focus on the joint analysis of fluorescein and resazurin which provides the maximum-likelihood parameters v, D, k and qhe as well as a retardation factor Rraz and the first-order loss term λraz to account for the reactive properties of resazurin (Rflu = 1 and λflu = 0 by definition for the conservative tracer).
 Table 3 lists the maximum-likelihood parameter values of the runs DF1, DF2, DF3, DJ1, DJ2, and DJ3 together with the standard deviation of the posterior parameter pdf. In all cases, the parameter uncertainties are extremely small, with k having the largest coefficient of variation on the order of 10%. This implies that a clear global minimum of the objective function can be identified by the robust MCMC-based estimation scheme, whereas gradient-based parameter estimation methods applied to the same problem showed the tendency to get stuck in local minima (results not shown). It may be noted that the differences between the parameter values obtained by fitting the breakthrough curves of individual monitoring stations differ beyond the uncertainty range of the individual parameters. This already indicates model uncertainty, because the underlying equations are for uniform coefficients over the entire length of the reach.
Table 3. Summary of the Derived Maximum-Likelihood Parameter Values, Their Uncertainty, and the Associated Agreement Between Simulations and Measurements
MS1 + MS2
v, mean in-stream velocity; D, longitudinal dispersion coefficient; qhe, discharge undergoing hyporheic exchange per volume of stream water; k, inverse of mean hyporheic travel time; λraz, first-order rate constant for transformation of resazurin to resorufin and other (undetected) compounds; Rraz, Retardation factor of resazurin. The uncertainties for each maximum-likelihood parameter were calculated as one standard deviation of the respective posterior probability density function of the respective parameter derived from the DREAMZS model; RMSE, root-mean-square error of the simulations with the maximum-likelihood values of the parameters; nRMSE, RMSE normalized for the mean concentration of the respective measurements.
Fit of Fluorescein Only
v (10−2 m/s)
D (10−1 m2/s)
qhe (10−4 s−1)
k (10−3 s−1)
Joint Fit of Fluorescein and Resazurin
v (10−2 m/s)
D (10−1 m2/s)
qhe (10−4 s−1)
k (10−3 s−1)
λraz (10−4 s−1)
RMSE Flu (µmol/m3)
RMSE Raz (µmol/m3)
 To quantify the agreement between measured and simulated BTCs, we computed the root-mean-square error (RMSE) for each BTC and list the corresponding values in Table 3. Because the concentrations of fluorescein and resazurin differ significantly, we also normalized the given RMSE by the mean observed concentration in the corresponding BTC, denoted nRMSE. For the joint fits of monitoring stations MS1 and MS2 (DF12 and DJ12), nRMSE values were computed for each BTC first and subsequently averaged over the two monitoring stations. These values are also listed in Table 3.
 The maximum-likelihood simulations of fluorescein and resazurin generally match the measured data for rising and peak concentrations very well. This is supported by nRMSE values of 0.12 and lower. However, the model underestimates the measured concentrations systematically when concentrations of fluorescein drop below 5 µmol/m3 (≈2 µg/L) in the BTC tails, regardless if fluorescein was analyzed alone or jointly together with resazurin. The limit of quantification of fluorescein for the given conditions during the experiment was estimated to be 0.265 µmol/m3 (≈0.1 µg/L).
 The variation of the parameters k, λ, and R between the three monitoring stations is generally small whereas v, D, and qhe show higher variability, regardless if the BTCs were analyzed jointly or if only fluorescein was considered (see Table 3). The parameter values found for MS3 deviate significantly from those found for the other two monitoring stations in all cases, which is mostly the reason for the high variability of the respective parameter ranges.
 Figures 3 and 4 include bands that describe the effects of parameter and model uncertainties on the modeled BTCs. The parameter uncertainty is calculated as the 95% confidence interval of the posterior parameter distributions whereas the model uncertainties are calculated by two standard deviations of the error between the maximum-likelihood simulation and the measured BTC. In all our runs, the parameter uncertainty bands are very small due to the small variances in the posterior parameter distributions, and, therefore, the bands appear to be lines in Figures 3 and 4. When both tracers are analyzed together (runs DJ1, DJ2, and DJ3), the parameter uncertainties of fluorescein are smaller whereas the model uncertainties are bigger compared to the analysis of fluorescein alone (runs DF1, DF2, and DF3). Thus, the additional consideration of the reactive tracer resazurin in the analysis results in a narrowing of the posterior parameter distributions obtained from the DREAMZS calculations and a slight decrease of the fit for fluorescein (nRMSE from 0.10 to 0.14).
 As mentioned above, the uncertainty ranges of the parameters are much smaller than the variability of the parameters between the different monitoring stations. In particular, the deviation of MS3 from the other two stations cannot be explained by the parameter uncertainty derived from the maximum-likelihood approach. However, there is no evidence of significant morphological or hydrological differences between MS2 and MS3. Although the analysis of moments provided reasonable values for MS3, we cannot exclude technical problems during BTC recording. Thus, MS3 is not considered in the following analysis.
 In the third step, we tried to find a maximum-likelihood parameter set that could describe the BTCs of fluorescein at MS1 and MS2 (run DF12) and another maximum-likelihood parameter set that could describe both the BTCs of fluorescein and resazurin at these two stations simultaneously (run DJ12).
 Figure 5 shows that the simulations with the maximum-likelihood values match the measured data of both tracers at both monitoring stations very well (nRMSE for fluorescein and resazurin are 0.18 and 0.11, respectively). Most maximum-likelihood parameter values of run DF12 lie within the range that is span by the respective values of MS1 and MS2 (runs DF1 and DF2) (see Table 3). For the joint analysis (run DJ12), the parameter values for D (0.28 m2 s−1) and qhe (3.1 × 10−4 s−1) were close to the mean value of the two monitoring stations (model runs DJ1 and DJ2) while the values for v (8.9 × 10−2 m s−1), k (1.4 × 10−3 s−1), λraz (1.1×10−4 s−1), and Rraz (1.1) were near or slightly outside the range of the single monitoring stations.
 Parameter values significantly differ between the results of the exclusive analysis of fluorescein and the joint analysis with resazurin. When compared to the joint analysis of both tracers, the analysis of fluorescein alone finds values for D that are up to 1.7 times higher while values for qhe and k are lower by factors of up to 2.1 and 1.6, respectively.
 The high quality of the presented simulations, as indicated by low normalized RMSE values, show that the improved experimental and modeling approach provide a robust and consistent set of bulk parameters describing in-stream transport and hyporheic exchange processes. This is supported by the narrow parameter uncertainties derived from the maximum-likelihood approach suggesting reliable identification of the model parameters. The low parameter uncertainty is partly due to the high temporal resolution of the measurements (10 s intervals) using the recently introduced field fluorometer [Lemke et al., 2013]. The online measurements significantly increased the quality of the BTC simulations compared to data obtained by manual sampling [Gooseff and McGlynn, 2005; Wagner and Harvey, 1997].
 In the previous field applications of the resazurin-resorufin system, the reactive tracer was continuously injected [Argerich et al., 2011; Gonzalez-Pinzon et al., 2012; Haggerty et al., 2009], whereas we used a pulse injection of both the conservative and reactive tracers. Analyzing the transient data from the pulse test allowed to determine the relevant transport parameters (v, D, qhe, k, R), which hardly influence steady-state concentrations, so that also for continuous injection most information of transport parameters are in the transient parts of the BTCs. In addition, the pulse injection has a few practical advantages over continuous-injection schemes: (1) significantly less tracer mass is needed, which reduces costs of the expensive reactive tracer and simplifies obtaining permits from regulators; (2) the field setup is considerably simpler because neither carboys, nor injection pumps are needed; and (3) the problem of maintaining a constant concentration of the tracers during injection is avoided. However, if determining an effective transformation rate is the key objective of the reactive-tracer test, most likely more reliable information is gained from measurements under steady-state transport conditions. Also, concentrations in a pulse-tracer test drop fairly quickly so that large hyporheic travel times may remain unnoticed.
 For the parameter estimation process, we used the DREAMZS code that samples the global parameter space and thus is able to find posterior parameter distributions and maximum-likelihood parameter sets that yield the best attainable fit to the measured data with the given model. By this, we overcame a typical problem of gradient-based search algorithms of running into local minima, so that the values of the estimated parameters depend on their respective initial guesses. The approach also allowed unconditionally quantifying uncertainties of the different model parameters. Table 3 shows that the uncertainties of the maximum-likelihood parameters are very small in each case. As a consequence, possible correlations of parameter uncertainties have no consequence on the quality of the parameter determination. Moreover, the mismatch between model fits and observed BTCs cannot be explained by uncertainties of the maximum-likelihood parameters but are a result of the model error due to structural deficits of the model approach. The model error, derived from the deviation between the maximum-likelihood simulation and the measured BTC, is large for the tailings, where the simulations generally underestimate the observations. This behavior of the exponential travel-time distribution is well known from previous studies [e.g., Cardenas et al., 2008; Haggerty et al., 2002] and is part of the structural errors of the model.
 Although the data collection at the different monitoring stations was entirely independent, the parameters obtained from individual BTC simulations agreed well between stations MS1 and MS2 (model runs DF1 and DF2) as well as with the values of the simulation with the maximum-likelihood parameter set from run DF12. The same holds for the simultaneous analysis of both tracers at monitoring stations MS1 and MS2 (model run DJ12). Here, we still get a decent fit and obtain parameters that largely correspond to the values found by the analysis of the two tracers at individual monitoring stations (model runs DJ1 and DJ2), indicating that the estimated parameter values are representative for the entire reach under investigation. These findings may also serve as a plausibility check of the parameters, because the distance between MS1 and MS2 is much shorter than their distance from the injection point. Since all parameters represent averaged values between the points of injection and observation, a significant change in parameter values over a short distance requires large changes in stream discharge or morphology over that distance. Such a short-range change in the maximum-likelihood parameters was found for MS3, which could not be related to drastic changes in the stream characteristics (slope, discharge, or general morphology) between MS2 and MS3 and, therefore, was attributed to technical problems with the fluorometer at MS3.
 By contrast, the joint analysis of the conservative and the reactive tracer (model runs DJ1, DJ2, and DJ12) yielded substantially different values for the parameters describing hyporheic exchange and in-stream transport compared to those obtained from the analysis of the conservative tracer alone (model runs DF1, DF2, and DF12). When analyzing both tracers jointly, the values found for qhe and k were generally higher while the values for D were generally smaller compared with the results of the analysis of fluorescein alone. In-stream dispersion processes and hyporheic exchange with small residence times may, to certain extent, have similar effects on the measured tracer BTCs. Thus, we believe that parts of the supposed in-stream mixing processes actually are to be assigned to a more pronounced hyporheic exchange, and that the parameters found by the joint analysis of both tracers describe the real processes in the stream in a more elaborate way. This finding is in accordance with the understanding of dispersion: With increasing travel distance the spread of a conservative-tracer pulse caused by transient-storage mechanisms increasingly resembles the spread caused by Fickian dispersion, deteriorating the capability to separate these processes by the analysis of conservative-tracer tests alone. A more decent fitting of the BTC's late tailing only slightly changes (∼30%) the parameter values for qhe and mean hyporheic travel time (k−1) found here, as was demonstrated by the companion study by Liao et al.  where the same BTCs as used in this paper were fitted using a nonparametric travel-time distribution. At the investigated reach of River Goldersbach, pools only represent a small fraction of the stream morphology. This suggests that the parameters found by the analysis of a single conservative and those obtained by the joint analysis of a conservative tracer and resazurin will be even more distinct when investigating streams with a relatively high portion of pools or other stagnant zones.
 The mean hyporheic travel time estimated with the joint analysis of both tracers is in the range of about 12 min, implying that there is also a high contribution of hyporheic water returning to the stream after even shorter travel times. These relatively short travel times give evidence that the observed BTCs are predominantly affected by hyporheic exchange at small-scale bedforms, such as ripples and dunes. This may challenge the assumption of equilibrium sorption underlying the use of a retardation factor in equation (4). The influence of nonequilibrium sorption as well as the incorporation of nonparametric travel-time distributions is studied in the companion paper by Liao et al. , containing rather complex mathematical expressions. Moreover, the in-stream dispersion coefficients may take account for small fractions of the hyporheic exchange so that very short hyporheic travel times have to be handled with caution since they could just as well be the result of dispersion processes.
 Values for qhe reported in the literature (the equivalent to the parameter α in the OTIS-model of Runkel ) show a huge variability over at least 2 orders of magnitude for different rivers. For example, Gooseff et al.  found values of 0.98 × 10−4 s−1 in a medium sized stream in Oregon, USA, while Wondzell  found values as high as 22 × 10−4 s−1 in the same area, but in a stream with a considerably lower discharge. Laenen and Bencala  showed that this parameter can also vary significantly when analyzing the same subreach multiple times. The values for qhe found in our study (1.2–4.7 × 10−4 s−1) lie well within this range of previously published parameters. Converting the values of qhe to characteristic exchange lengths using the stream velocity yields a value range of 250–350 m for the distance after a complete exchange of stream water with the hyporheic zone has occurred. This suggests a high potential for solute transformations for the investigated stream reach, provided metabolic activities are present in the respective sediments.
 In addition to parameters reflecting physical transport, we also obtained quantities of reactive-tracer transformation and sorption. These parameters are related to the metabolic activity of the stream bed [e.g., Gonzalez-Pinzon et al., 2012] and most likely sorption to other organic compounds within the hyporheic zone. The mean depletion rate of resazurin in the hyporheic zone λraz from all monitoring stations is 1.1 × 10−4 s−1. This is lower than the reported values of 4.3 × 10−4 s−1 by Haggerty et al. , 2.9 × 10−4 s−1 by Haggerty et al. , and 5.3 × 10−4 s−1 by Argerich et al. . The mean water temperature during our experiment was similar to or even higher than the stream temperatures reported in the publications cited above, and River Goldersbach typically shows elevated DOC concentrations (2–3 mg/L during baseflow and > 4 mg/L during storm flow), which suggests an enhanced potential of microbial metabolic activity in the hyporheic zone. However, Gonzalez-Pinzon et al.  showed that the reaction rates for resazurin strongly depend on the species of the microbes, so we believe that in spite of advantageous conditions the comparably low reaction rates can be attributed to the composition of the microbial community in the tested subreach.
 We have chosen the exponential distribution of hyporheic travel times for our model due to its simplicity and acceptable goodness of the fits. However, in our case this model systematically underestimates the late tailing of the BTCs. As this is the case for both the conservative and the reactive tracer, the discrepancy in the tailing behavior cannot be explained by sorption or decay, but rather by the shape of the hyporheic travel-time distribution as it has been reported in numerous other studies before [e.g., Cardenas et al., 2008; Haggerty et al., 2002]. These late arrival times of the tracers are of special interest when studying the effects of slower biogeochemical transformations within the hyporheic zone on the overall behavior of the stream.
 Assuming an exponential distribution of storage residence times is identical to applying the conventionally used single transient-storage model with linear mass exchange between the mobile and immobile zones [Runkel, 1998]. This model has been extended to account for two distinct stream water retention mechanisms by incorporating a second transient-storage zone [Briggs et al., 2009; Choi et al., 2000; Gooseff et al., 2004; Harvey et al., 2005]. Thus, we tested if the incorporation of a second storage zone would improve our model fits, and if so, whether or not this improvement would justify adding two parameters in the model by means of the models selection criterion introduced by Harvey et al. . The results indicated that the integration of a second storage zone generally yielded a slightly improved fit to the measured data. Also, the models selection criterion indicated that the two storage-zone model has a higher information content and should, therefore, be used. However, the most commonly mentioned disadvantage of the exponential model, namely the misfit of the late tracer arrival times, could not be eliminated. Furthermore, the model-selection criterion chosen by Harvey et al.  might not be appropriate in our application with a tremendous number of measured data points (>3000 for fluorescein) that essentially invalidates the penalty term for the number of parameters. Thus, we do not believe that introducing a second storage zone with first-order exchange to the stream is a real improvement for the River Goldersbach data and do not show the results obtained for this model. Depending on the objective of the study, more complicated models (e.g., accounting for nonparametric travel-time distributions or two-site sorption as done by Liao et al.  might be required to successfully capture also the tracer arrival at later times.
 In this study, we have presented the joint tracer test of a conservative and a reactive tracer introduced as a pulse into River Goldersbach, Germany. The mathematical analysis of the high-frequency tracer concentration data allowed obtaining spatially consistent parameters that describe solute transport within the stream as well as exchange with the hyporheic zone. The uncertainties of the respective parameters are very small in all cases, so that the estimated parameter values are independent of possible parameter correlations. We could furthermore show that the incorporation of the reactive tracer resazurin significantly improved the understanding of the hyporheic-exchange processes compared to the conventional technique of using a single conservative tracer alone. The joint analysis of both the conservative and the reactive tracer reveals that the strength of the hyporheic exchange is in fact significantly higher whereas in-stream mixing processes (basically dispersion and exchange with in-stream transient storage zones) and the mean travel time of hyporheic water are smaller compared to what can be concluded from the information of the conservative tracer only. However, the model uncertainties are relatively large and the model based on exponential hyporheic travel-time distributions is unable to entirely describe late tracer arrivals, regardless if one or two transient-storage mechanisms are considered. This suggests that the model cannot sufficiently reflect all occurring processes in the stream. Thus, more complicated models might be needed to also account for unconventional hyporheic travel-time distributions and/or kinetic sorption mechanisms. However, the simultaneous simulations of fluorescein and resazurin with the parametric travel-time distribution proposed here already provide a reasonable estimate of the magnitude and time scale of hyporheic exchange processes, and are computationally much more accessible than more elaborate schemes.
 While online measurements of conservative tracers are common in conservative stream-tracer tests, this is the first field application where this technique was applied to measure the concentrations of the tracers fluorescein, resazurin, and its reaction product resorufin simultaneously and directly in the stream with an accuracy that is comparable to laboratory spectrofluorometers [Lemke et al., 2013]. By this, errors that are possibly affiliated to manual sampling (mix up of the samples, sample contamination, altering of the tracer concentrations in the samples due to long transport/storage times) could be avoided.
 Using a global search algorithm based on Markov chain Monte Carlo sampling has definitely paid off in the current application. Already fitting the comparatively simple single transient-storage model with a total of six parameters applied here has the tendency to get stuck in local minima when using standard parameter estimation schemes. Thus, refitting the model with different initial guesses leads to different parameter estimates which in turn might lead to the erroneous interpretation that the parameter uncertainty is high, whereas the global population-based search algorithm indicated a rather narrow band of posterior parameter values. Applying such techniques to computationally more demanding models, including multiple storage zones or unconventional hyporheic travel-time distributions, where parameter ambiguity may even be more pronounced, necessarily requires much larger computational capabilities, which might become a limiting factor because the chosen MCMC approach utilized here already requires ten thousands of model runs.
 We are grateful to Roy Haggerty at Oregon State University for his advice on the resazurin/resorufin tracer system. This work was supported by a grant from the Ministry of Science, Research and Arts of Baden-Württemberg (AZ Zu 33-721.3-2), the Helmholtz Centre for Environmental Research, Leipzig (UFZ), the German Academic Exchange Service (DAAD), and the National Science Foundation grant EAR 08-38338 to Roy Haggerty at Oregon State University. We also thank Martin Gritsch, Julia Knapp, Rémi Lavergne, and Dijie Liao for their active assistance in the fieldwork.