Corresponding author: R. C. Naranjo, U.S. Geological Survey, 2730 N. Deer Run Road, Carson City, NV 89701, USA. (firstname.lastname@example.org)
 Biochemical reactions that occur in the hyporheic zone are highly dependent on the time solutes that are in contact with sediments of the riverbed. In this investigation, we developed a 2-D longitudinal flow and solute-transport model to estimate the spatial distribution of mean residence time in the hyporheic zone. The flow model was calibrated using observations of temperature and pressure, and the mean residence times were simulated using the age-mass approach for steady-state flow conditions. The approach used in this investigation includes the mixing of different ages and flow paths of water through advection and dispersion. Uncertainty of flow and transport parameters was evaluated using standard Monte Carlo and the generalized likelihood uncertainty estimation method. Results of parameter estimation support the presence of a low-permeable zone in the riffle area that induced horizontal flow at a shallow depth within the riffle area. This establishes shallow and localized flow paths and limits deep vertical exchange. For the optimal model, mean residence times were found to be relatively long (9–40.0 days). The uncertainty of hydraulic conductivity resulted in a mean interquartile range (IQR) of 13 days across all piezometers and was reduced by 24% with the inclusion of temperature and pressure observations. To a lesser extent, uncertainty in streambed porosity and dispersivity resulted in a mean IQR of 2.2 and 4.7 days, respectively. Alternative conceptual models demonstrate the importance of accounting for the spatial distribution of hydraulic conductivity in simulating mean residence times in a riffle-pool sequence.
 The mixing zone between river and groundwater flow systems, often referred to as the hyporheic zone, plays an important role in riverine ecosystems. Water entering the riverbed exchanges energy and solutes with groundwater and returns to the channel with a different chemical signature, depending on the solute mixing, chemical reactions, and residence time. Residence time of solutes is highly variable in space and time and is dependent on physical and hydraulic parameters of the river and underlying groundwater system [Vaux, 1968; Woessener, 2000; Haggerty et al., 2002]. Estimating spatially distributed exchange rates, flow paths, and residence times is important toward understanding the biogeochemical processes in the hyporheic zone [e.g., Triska et al., 1993; Brunke and Gonser, 1997; Dahm et al., 1998; Zarnetske et al., 2011]
 The spatial distribution of groundwater age is typically estimated using a multidimensional, spatially distributed groundwater flow and transport model, referred to herein as a distributed model. Distributed models are useful for understanding nutrient processing in rivers, such as the effects of river morphology, sediment heterogeneity, and chemical mixing on nutrient fate and transport. Distributed models provide considerable advantages to 1-D modeling approaches as they can account for mixing of flow paths and their influence on mean residence times. Estimating residence times with distributed models has been performed through particle tracking [Wroblicky et al., 1998; Storey et al., 2003; Kasahara and Wondzell, 2003; Salehin et al., 2004; Cardenas et al., 2004] and through solute-transport methods using the advection-dispersion equation (ADE) [Goode, 1996; Varni and Carrera, 1996; Woessener, 2000; Lautz and Siegel, 2006; Sawyer and Cardenas, 2009; Jiang et al., 2010]. Simulation of solute transport using the ADE has the advantage over advection-only particle tracking methods for estimating residence time because they include mixing of water from different sources through dispersion and diffusion. A novel approach for the direct simulation of the spatial distribution of mean groundwater age is the ADE redefined for groundwater age or “age-mass” [Goode, 1996]. Mean age is analogous to the solute concentration calculated by the ADE with a distributed source of unit strength. Thus, the simulated “concentration” at each point in the model domain at steady state is equal to the mean age and represents the combined effects of advection, dispersion, and diffusion [Goode, 1996; Varni and Carrera, 1998]. Several studies have used direct modeling of groundwater age with the inclusion of dispersion and diffusion in aquifer systems [e.g., Goode, 1996; Varni and Carrera, 1998; Engesgaard and Molson, 1998; Bethke and Johnson, 2002], reservoirs [e.g., Cornation and Perrochet, 2006], for use in inverse model calibration [Varni and Carrera, 1998; Weissmann et al., 2002; Ginn et al., 2009], and understanding biogeochemical processes [Ginn, 1999; Gomez et al., 2012]. Bethke and Johnson  compared the age-mass concept to the piston flow approach to simulate travel time in a regional aquifer system including aquitards. They found that for an aquifer system that contains a significant amount of aquitard material relative to aquifer material, groundwater age will be significantly older than estimates of water age using piston flow assumptions. This has important implications for understanding residence times in the hyporheic zone where layering is present due to colmation and other fluvial processes that restrict hyporheic flow [Naranjo et al., 2012]. River sediment is often heterogeneous and can consist of both fine and coarse-grained deposits that control the distribution of flow paths, mixing of young and old water, and the rate of biogeochemical processes. Inverse model calibration with sediment temperature and pressure observations can be used to characterize the spatial distribution of hydraulic conductivity and minimize uncertainty in residence times. The simulation of residence times can be used to support interpretation of the relationship between age, pore-water chemistry, and biogeochemical reactions [e.g., Triska et al., 1993; Findlay, 1995; Zarnetske et al., 2011].
 Previous studies have highlighted the effects of riverbed heterogeneity on hyporheic flow paths and residences times at the 10 m scale [Salehin et al., 2004; Marion et al., 2008; Sawyer and Cardenas, 2009] to the 100 to 2000 m scale [Fleckenstein et al., 2006; Frei et al., 2009; Jiang et al., 2010]. Heterogeneity within the riverbed has been shown to control vertical exchange and residence times by limiting the vertical depth over which exchange occurs and deflecting flows upward. Vaux  used simulation modeling to demonstrate that riverbed heterogeneity significantly affected hyporheic flow paths and illustrated that flow paths can be deflected into or out of different configurations of sediment heterogeneity. Representation of riverbed heterogeneity in models is difficult because it requires extensive data collection that is often impractical to collect in large river systems. Heterogeneous representations of streambeds can be developed using interpolation between observations [Cardenas et al., 2004], zonal representations parameterized using inverse modeling [Naranjo et al., 2012], and through geostatistics [e.g., Salehin et al., 2004; Packman et al., 2006; Fleckenstein et al., 2006]. The formation of sediment structure such as layered deposits has been shown to strongly influence depth of exchange even with the presence of high-permeable sediments near the surface [Packman et al., 2006; Marion et al., 2008], as well as the direction of seepage between streams and groundwater [Niswonger and Fogg, 2008].
 Simulated hyporheic flow on the basis of distributed models can have great uncertainty because they require parameterization of both the flow and transport equations. Inverse modeling techniques used for heat as a tracer studies are improving for estimating the most uncertain model parameters, specifically hydraulic conductivity [e.g., Niswonger and Rupp, 2000; Niswonger et al., 2005; Constantz, 2008; Naranjo et al., 2012], which provide new opportunities for developing distributed models of hyporheic flow in river systems. The use of distributed solute-transport modeling to derive travel times will be uncertain due to errors in hydraulic and transport parameters, boundary conditions, and errors in the conceptual model used to construct the simulation model. Despite the recent controversy (briefly discussed herein), the generalized likelihood uncertainty estimation (GLUE) method has been extensively used in hydrologic models to address predictive uncertainty on the basis of the equifinality concept [Beven and Freer, 2001; Beven, 2006]. The GLUE method remains an attractive choice for addressing parameter uncertainty in heat as a tracer studies due to the measurement of continuous time series data used in model calibration. Furthermore, by comparing results from standard Monte Carlo and GLUE methods, the reduction in uncertainty provided by observation data (i.e., temperature and pressure) can be quantified. Thus, the value of observation data can be evaluated on the basis of reductions in uncertainty for a particular model result, such as residence time.
 Models generally have large uncertainty that stem from the original conceptual model and assignment of hydraulic properties, observational errors, and inadequate representation of the boundary conditions. Equifinality asserts that multiple combinations of model parameters exist, each producing reasonable model performance, which can be addressed through uncertainty analysis. In groundwater model applications, the GLUE method has been used for capture-zone analysis [Feyen et al., 2001; Morse et al., 2003], stochastic-transport flow models [Hassan et al., 2008], and conceptual-model uncertainty [Rojas et al., 2010; Reeves et al., 2010]. These studies stress the importance of evaluating multiple parameter sets and conceptual models to improve the understanding of hydrological systems.
 In this paper, we expand on the work of Naranjo et al.  for the purpose of estimating mean residence times and evaluating the effect of parametric uncertainty on solute-transport simulations. The impetus of this work is to develop an approach for estimating model parameters using global methods for simulating mean residence times and their uncertainties. In the present work, residence times are of interest in how they relate to nutrient chemistry sampled from the hyporheic zone. Further, there are few investigations that evaluate 2-D longitudinal hyporheic flow paths and residence times derived from field investigations. The flow model used in this effort was extensively calibrated using measurements of riverbed temperatures and pressure. The value of the observation data was quantified through reductions in the standard deviation of the posterior distributions of residence time using the GLUE approach.
 Simulated mean and standard deviation of the spatially distributed residence times with fixed-flow and -transport parameters (base case) was compared to (1) uncertain flow parameters using the GLUE method with fixed-transport parameters and (2) fixed-flow parameters and uncertain-transport parameters using standard Monte Carlo sampling. Uncertainty in the vertical to horizontal anisotropy (Kz/Kx) was evaluated with respect to uncertainty in the mean residence time. Anisotropy was evaluated to better understand its effects on longitudinal hyporheic flow, assuming that typical fluvial processes result in layered sediment structure and strongly anisotropic hydraulic conductivity. Three alternative conceptual models were used to evaluate the differences in mean residence times given (1) a two-layer heterogeneous model with anisotropy (CM2), (2) a single-layer homogenous model with anisotropy (CM3), and (3) a single-layer homogenous model with isotropic conditions (CM4). The results of the conceptual models illustrate the need to account for heterogeneity and anisotropy in simulations of hyporheic flow.
 Simulations presented herein involved the estimation of mean residence times and flow paths using a distributed, 2-D longitudinal flow and transport model. The model represents a riffle-pool sequence and the associated underlying hyporheic zone. Here we apply the age-mass approach [Goode, 1996] in simulations of the hyporheic zone using a 2-D flow and solute-transport model. Relevance of the work lies in demonstrating the use of solute-transport simulations for estimating mean residence times in a riffle-pool sequence where exchange rates and flow paths were derived from heat as tracer method. Importantly, we show that calibration of hydraulic conductivity on the basis of temperature and pressure significantly reduces the uncertainty in estimates of residence time. However, after calibration, hydraulic conductivity remains the largest contributor of uncertainty in estimates of residence time relative to transport parameters. We demonstrate the use of a rigorously calibrated flow model to estimate the uncertainty in hydraulic conductivity, porosity, and dispersivity in simulating spatially distributed mean residence times.
2. Site Description
Naranjo et al.  provide full detail regarding the 165 m length riffle-pool sequence of the lower Truckee River located in northwestern Nevada (Figure 1). The Truckee River drains from a basin area of 8000 km2 and encompasses a part of the eastern slope of the Sierra Nevada and the westernmost part of the basin and range physiographic provinces. The river flows over a distance of approximately 184 km from Lake Tahoe to where it terminates at Pyramid Lake. The study site is located at a losing section of the lower Truckee River at Little Nixon, located 8 km from the river's terminus at Pyramid Lake. The channel is approximately 20 m wide at the top of the reach and 15 m wide in the lower section. The bed material consists of medium size cobbles and gravels in the riffle and coarse sands to fine silts in the pool area. The discharge is largely controlled by reservoirs and agricultural diversions, and during the period of this investigation, the low flows were approximately 5.0 m3/s. At this discharge rate, depths ranged between 0.20 m in the riffle and 1.3 m in the pool. The mean-monthly flow ranges between 4.81 and 34.55 m3/s at the U.S. Geological Survey (USGS) gage 4.5 km upriver of the study area (USGS 10351700 Truckee RV NR Nixon NV).
 The study area was instrumented with 16 in-river piezometers, and 4 shallow-riparian monitoring wells positioned along the riffle-pool sequence. Riverbed temperatures and pressure measurements were monitored during transient-flow conditions to develop a heat and flow model of the center of the channel [Naranjo et al., 2012]. We expand on the previous flow modeling effort along the 2-D longitudinal direction to simulate solute transport and residence time (Figure 1). The riverbed topography, river bank, and monitoring points were surveyed with a total station and referenced to a local control point. The observations were divided into the shallow-upper zone and deep-lower zone, denoted by prefix “s” and “d” in subsequent figures and tables. We use the same 2-D longitudinal transect to simulate the residence times for low-flow conditions of 5.0 m3/s. Because the river flow in this study area is highly controlled from upgradient reservoirs resulting in long periods of steady flow, thus simulating steady flow conditions is appropriate for this system.
3.1. Flow and Solute Transport
 This study uses a dual-phased approach toward simulating hyporheic flow and estimating travel times in a riffle-pool sequence. In phase 1, a variably saturated two-dimensional heat (VS2DH) flow model was calibrated using transient observations of stage, sediment pressure, and temperatures [Naranjo et al., 2012]. The uniform random-sampling (URS) [Duan et al., 1992; Wagener and Kollat, 2007] approach was used to estimate hydraulic conductivity (vertical and horizontal) and thermal parameters (thermal conductivity and heat capacity) on the basis of qualitative and quantitative examination of the solution space. The URS method for model calibration is implemented by specifying a region of the parameter space that is considered feasible for the study site, as determined from prior knowledge and values reported in the literature. Hydraulic parameters derived from the least-error solution, defined as “optimal” herein, were determined on the basis of the minimum-error solution, as assessed using observations of sediment temperatures and pressures.
 In phase 2, a solute-transport model given flow parameters used in phase 1 was used to simulate mean residence times and evaluate parametric uncertainty. The solute-transport model used in this study is the USGS variably saturated 2-D heat flow and transport (VS2DT) model [Healy, 1990; Healy and Ronan, 1996; Hsieh et al., 2000]. VS2DT model solves the 2-D Richard's equation coupled to the ADE using the finite-difference method. The inflow boundary on the right side of the model was extended 70 m beyond the area of interest to avoid boundary edge effects (Figure 2). The riverbed section of the model was assigned a continuous solute-boundary condition and a spatially variable specified head interpolated from observed measurements. Although the flow model was calibrated to transient stage and temperature conditions, the estimated mean residence times were made using steady flow conditions at the upper boundary with a temporally changing solute-boundary condition. The residence time in the context of this study reflects the mean age of water from different flow paths, not the full distribution of all the residence times in space as presented by transient simulations [e.g., Varni and Carrera, 1998; Ginn, 1999; Gomez et al., 2012]. Further, the breakthrough curves were evaluated at different locations most distant from the solute-boundary conditions to ensure that steady-state concentrations were reached. The mean age of water corresponds to the age since the start of the simulation calculated as the difference between elapsed simulation time and solute unit concentration for steady flow conditions [Goode, 1996; Varni and Carrera, 1998]. Thus, any location within the model domain will represent a mean age that reflects the given flow path the solute has traveled (with mixing) since the entry into the subsurface. Initial concentrations within the model domain were specified as zero; however, these conditions do not affect concentrations at steady state. Specified head at the top boundary (representing the depth of flowing water in the river) was defined on the basis of the spatially variable river depth corresponding to a flow of 5.0 m3/s. Grid dimensions were set at uniform spacing of a 0.10 m in the vertical direction and 3.0 m in the horizontal direction. The lower boundary condition was defined as a spatially variable specified head derived from interpolated head observations. The furthest right and left vertical boundaries were defined as specified head with a vertical hydraulic gradient of −0.01 m/m based on the average values at nearby piezometers (P2 and P15).
 Transport of solutes in the hyporheic zone is typically dominated by advection and dispersion, and to lesser extent diffusion. Molecular diffusion can play a significant role in low-permeable environments or heterogeneous multilayered aquitard systems. Following preliminary sensitivity analysis, we found molecular diffusion to be insignificant in the evaluation of mean residence time. Therefore, molecular diffusion was not included in the parametric uncertainty analysis.
 The relative importance of dispersion and advection is quantified by calculating the dimensionless-solute Peclet number: Pe = (ΔL/α), where ΔL is the grid-cell dimension and α is the dispersivity. If Pe values are small (<1.0), diffusion is important. Values of the solute-dispersion coefficient for both the horizontal and vertical dimensions were assigned by calculating the Pe number to refine the upper and lower bounds of the Monte Carlo sampling range to avoid erroneous results from numerical dispersion. The upper range in dispersion was calculated on the condition that the Pe number is less than two. The lower bound was assumed to be 1.5 m, which is one half the horizontal-grid dimension. Therefore, the longitudinal dispersivity (αL) is varied from 1.5 to 10.0 m, and the transverse dispersivity is 0.1 of αL.
3.2. Monte Carlo Uncertainty
 Depth decay of porosity and hydraulic conductivity can occur in large scale (>1 km) systems [Jiang et al., 2010]; here we assign ranges to hydraulic and transport parameters given the three-zone model conceptualization of a riffle-pool scale (80 m) and calibration results of Naranjo et al. . Uncertainty of porosity and dispersivity was evaluated separately by performing 1000 Monte Carlo realizations and calculating the mean, standard deviation, and coefficient of variation of residence time for both parameters. Transport simulations were run until spatially distributed solute breakthrough curves reach steady-state concentrations, and subsequently, steady-state residence times were achieved. Parameter ranges used in the VS2DT model are shown in Table 1. In each zone, we allowed flow and transport parameters to vary according to the ranges in the literature for each textural grouping and prior information [Naranjo et al., 2012] given a uniform probability distributions for 1000 realizations. Optimal-flow parameters defined in Table 2 were used to simulate distributed residence times and as a comparison of residence-time uncertainty for transport parameters, porosity, and dispersivity.
Table 1. Summary of Parameter Ranges Used in the VS2DT Model for Solute-Transport Simulations
 Traditional uncertainty analysis evaluates model uncertainty on the basis of probability distributions of input parameters, which are equally weighted without consideration of model behavior. The GLUE method expands on the Monte Carlo method through the application of weights according to a chosen likelihood measure. The GLUE method involves Monte Carlo sampling from the feasible parameter space using uniform distributions and calculation of likelihood measures. The likelihood measure is formed on the basis of flow model error represented by the root-mean-square error (RMSE) between the measured and simulated sediment temperature and pressure. The GLUE method was developed to address the concept of equifinality of models where multiple parameter sets can provide reasonable model results, and no single, unique optimal model exists [Beven and Freer, 2001].
 The GLUE method has received criticism for being incoherent due to the subjective selection of likelihood and threshold measures for conditioning the model predictions which can lead to wide uncertainty bounds [Montanari, 2005; Mantovan and Tondini, 2006; Stedinger et al., 2008]. They argue for the use of formal Bayesian approaches to predictive uncertainty and caution over the selection of the likelihood measure that does not account for sample size. In the general form, the GLUE method does not require the model to be calibrated which can lead to overestimation of the predictive uncertainty and the subjective definition of the likelihood measure, leading to incoherent results and inconsistencies with the formal Bayesian approach. The GLUE method also does not attempt to identify the maximum likelihood estimate and explicitly consider different types of error. However, the Bayesian approach may lead to overconditioning models when the likelihood function overestimates the real information content in the presence of input and model structural errors [Beven et al., 2008].
 In a recent evaluation, Vrugt et al.  compared the results of the GLUE and formal Bayesian uncertainty methods and reported the similar predictive uncertainty results given the different assumptions regarding the nature of statistical properties of the residuals (Bayesian) and representation of error. In our evaluation, model performance and the likelihood function are derived from continuous observations of pressure and temperature that is based on the RMSE objective function that accounts for the total number of observations or length of observations. In this application, we are cautious not to draw conclusions on the equifinality of model parameters; rather we express the uncertainty based on the differences in the mean and standard deviations of the predicted residence times weighted by the model performance.
 The method of applying weights to realizations is done by application of Bayes' theory
where L[R(Ii)|O] is the posterior likelihood of model realization R(Ii) that is the function of the set of input parameters Ii, L[O|R(Ii)] is the likelihood measure calculated with the set of observed variables O, Lo[R(Ii)] is the prior likelihood for realization R(Ii), and k is the number of realizations.
 The GLUE method of assigning weights to each realization is given by the likelihood measure
where N is the total number of observations values, σ1 is the simulated temperature or pressure, σ2 is the observed temperature or pressure, and M is a shape factor. The objective function, from which the likelihood is estimated, uses the RMSE. A single RMSE was computed from 1000 realizations that combined the error between the simulated and measured hourly temperature and head-pressure time series for all observation locations [Naranjo et al., 2012]. The temperature and pressure observations were combined for a single objective function describing the overall error.
 A value of zero for M is equivalent to having equal weight applied to all realizations similar to traditional Monte Carlo method, and values greater than zero accentuate the weight to the better performing models [Beven and Binley, 1992; Freer et al., 1996]. Multiple values of M are typically selected to show the overall sensitivity of model predictions [Hassan et al., 2008]. Another option for determining M is to manually adjust the value to maximize the number of observed data that occur within the 95% confidence interval estimated by the GLUE method. Hassan et al.  suggests a threshold limit be defined by the objective function, such that only the most acceptable parameter combinations are used in the uncertainty analysis. However, this approach may also underestimate the uncertainty by removing model realizations from the analysis [Beven and Freer, 2001]. For this modeling effort, we did not employ a threshold RMSE to remove models with poor or unrealistic results from the GLUE methodology. We evaluated the uncertainty with an M value of one and six. We applied weights to each realization based on the likelihood measure and calculated the mean, standard deviation, and the coefficient of variation of residence times.
 The weights in the GLUE method are derived from the error between the measured and simulated temperatures and measured and simulated pressures where greater weight is given to better performing model simulation. The unconditioned estimates of residence time are provided as comparisons to show the relative uncertainty given Monte Carlo sampling of porosity and dispersion parameters. The GLUE method was not used for the solute-transport parameters because measured porosity, dispersion, and solute data were not available for comparison. Uncertainty in residence time due to hydraulic conductivity was evaluated by applying GLUE weights with fixed porosity and dispersivity at the median values of the defined range in Table 1.
4.1. Optimal-Flow Parameters and Mean Residence Time
 Figure 3 shows likelihood values for the shallow and deep observations given ranges in hydraulic conductivity in the horizontal (Kx) and vertical (Kz) directions. The shape of the distribution of scatter points indicates the degree of uncertainty that is also affected, to a small degree, by the shape factor, M. The peak of the distribution corresponds to values that have the greatest likelihood and are well-defined parameters. Distributions that lack a distinct peak indicate that there are many equally likely sets of parameters, indicating that the parameter values are nonunique. Kz exhibited more distinct peaks (M = 1), suggesting greater sensitivity and uniqueness relative to Kx. For M = 6, both Kx and Kz parameters in the shallow and deep locations have distinct peaks. Comparisons of objective functions at individual to grouped observations within each zone show greater identifiability for optimal values [Naranjo et al., 2012].
 Exchange between the river and hyporheic zone is characterized by long horizontal flow paths and spatially discontinuous, short-mixing zones in both the riffle and pool areas. Figure 4 shows the spatial variation of vertical and horizontal velocities and estimated residence times for the optimal set of parameters in Table 2. Upwelling locations are shown with black arrows in Figure 4a. The mean downward velocity is 6.8 × 10−3 m/h across the riverbed interface, where the velocity variability increases with distance from the low-permeability zone. The vertical velocity is shown to be the greatest near the top of the riverbed and decreases uniformly with depth. Sawyer and Cardenas  also show this to be the case for homogenous and heterogeneous sediments. The spatial distribution of the horizontal velocity field shows greater variation where the convergence of flow paths occur (Figure 4b). The mean horizontal velocity at the riverbed interface is 1.92 × 10−2 m/h. The spatially distributed mean residence time within the modeled area of interest varies between 9 and 40 days for the optimal model (Figure 4c). The mean residence times at the shallow and deep temperature observations were 20.6 and 23.2 days, respectively. Results illustrate the relationship between flow paths and the spatial variability of residence times that are controlled by riverbed topography and heterogeneity in hydraulic conductivity.
4.2. Monte Carlo Uncertainty
 Figure 5 shows the distribution statistics (mean, range, quartiles) of residence times at each temperature observation (shallow and deep), given the uncertainty in hydraulic conductivity, dispersivity, and porosity. Each figure corresponds to 1000 realizations varying one parameter for each realization, while holding the other parameters constant. Parameter ranges are provided in Table 1. In the shallow (0.15–0.30 m) and deep (0.7–1.0) zones beneath the riverbed, the mean residence time was 20.9 and 23.5 days, respectively (Figure 5a). The relatively small increase in residence time with depth reflects the horizontally dominant flow paths that are accentuated by the low-permeable deposit beneath the riverbed.
 Uncertainty in hydraulic conductivity corresponds to the greatest overall residence-time uncertainty. In the shallow zone, the average interquartile range (IQR; 75–25 percentile) was 12.3 days. In the deep zone, the average IQR was only slightly longer at 13.7 days. The effect of dispersivity on residence time is smaller than the effect of hydraulic conductivity (Figure 5b). Uncertainty in dispersivity in the shallow zone resulted in a range of 3.0 days in the average IQR in the shallow zone. The average IQR range was slightly less (1.8 days) in the deep zone. Uncertainty in porosity resulted in an average IQR range of 4.5 and 4.8 days in the deep the shallow and deep zones, respectively. These results indicate that on the basis of the Monte Carlo analysis uncertainty in hydraulic conductivity produces the greatest uncertainty in the residence time relative to dispersivity and porosity. This results in no surprise given that large range in hydraulic conductivity for streambed material and that essentially no prior information about values of hydraulic conductivity were used for this uncertainty (i.e., uniform distributions were assumed). Additionally, uncertainty is not correlated to depth because the horizontal flux rates are much larger than the vertical flux rates. Finally, despite rigorous calibration of the hydraulic conductivity, its effect on residence uncertainty far outweighed uncertainty caused by solute parameters. This is true without any data to define the transport parameters beyond ranges published in the literature. However, methods like heat tracing can significantly reduce uncertainty in hydraulic conductivity for riverbed sediments as shown the following section using the GLUE method.
4.3. GLUE Uncertainty
 The estimated distribution of mean residence time given uncertainty in hydraulic conductivity using the GLUE method and weight of M = 6 is shown in Figure 6. Ensemble mean, standard deviation, and coefficient of variation were calculated using 1000 Monte Carlo realizations on the basis of the GLUE method. For these Monte Carlo realizations, porosity and dispersivity were set to the median value (Table 1). The effect of including uncertainty using the GLUE method on the mean residence-time distribution can be seen by comparing Figures 4c and 6a. The ensemble mean (M = 6) travel time in Figure 6a differs by 8%–17% from the distribution calculated on the basis of a single realization with the optimal hydraulic conductivity parameters. Consistent with the Monte Carlo uncertainty results, the spatial distribution of the standard deviation (Figure 6b) and the coefficient of variation (Figure 6c) show that hydraulic conductivity produces the largest amount uncertainty in mean residence time, despite the expansive set of the temperature and pressure data used to constrain the model, as implemented through the likelihood function. However, there is a significant reduction in uncertainty provided by temperature and pressure observations relative to uncertainty prior to incorporating these data (i.e., Monte Carlo uncertainty) [Naranjo et al., 2012]. Overall, the coefficient of variation is spatially similar across the three zones 35%–40%, with 45% variation occurring upgradient of an area of vertical exchange (Figure 6c).
4.4. Comparison of Methods
 Results comparing the mean and standard deviation of residence times estimated at each location and for each observation type (i.e., pressure and temperature) are summarized in Figure 7. Also shown are the effects of the different uncertainty methods and for different model parameters (i.e., single realization with optimal parameters, Monte Carlo realizations with variable hydraulic conductivity, porosity, and dispersivity, GLUE M = 1 and M = 6). There is a close agreement simulated between the GLUE-weighted-mean residence times and Monte Carlo realizations with variable porosity and dispersivity (Figures 7a and 7b) with the greatest differences in standard deviation. There is a 12% difference in mean residence time between the GLUE (M = 6) and the results from the optimal solution in both the shallow and deep observations that reflects the influence of equifinality on residence times. The uncertainty of hydraulic conductivity is clearly present given the standard deviation of mean residence times for both shallow and deep observations (Figures 7c and 7d). With increasing size of M, the mean and standard deviation decrease as it adds more weight to the better performing results. As shown by comparing the reductions in standard deviation resulting from the Monte Carlo and GLUE methods (M = 6) for hydraulic conductivity, model calibration using the pressure and temperature data reduced the uncertainty by an average of 24% in residence-time estimates (Figures 7c and 7d). Calibration of the model using pressure and temperature data reduced the average standard deviation of the mean residence time in shallow zone by 14% and in the deep zone by 34%. Thus, the results of the GLUE parametric uncertainty approach demonstrated that the pressure and temperature observation data were very useful for reducing uncertainty in mean residence time.
4.5. Additional Conceptual Models
 Solute transport is directly related to flow paths that are controlled by contrasts in hydraulic conductivity, anisotropy, and pressure distributed across the riverbed. To evaluate the effects of layered heterogeneity and vertical anisotropy on the simulation of mean residence time, three additional conceptual models (CM2–CM4) were constructed along with the original three-zone model (CM1). For each simulation, the parameters assigned to each zone correspond to the calibrated values in Table 2. Results shown in Figure 8a are repeated from Figure 4c for side-by-side comparison to the other conceptual models (CM2–CM4). The effect of removing the low-permeability zone (comparison between CM1 and CM2) leads a general decrease in residence times (Figure 8b). In CM1, the anisotropy was assigned a value of 0.04 in the lower zone, an equivalent of horizontal conductivity (Kx) being 25 times the vertical (Kz). Anisotropy of the lower zone in model CM2 exhibits great control on the estimated residence times throughout the model domain. Using the higher anisotropy from the shallow zone (Kz/Kx = 0.44) throughout the model domain, an increase by an order of magnitude resulted in more flow entering the deeper zone and subsequently reduced mean residence times by an order of magnitude (Figure 8c). CM3 corresponds to a single-layer model with an anisotropy ratio of 0.44, an equivalent of Kx being 2.27 times Kz. Residence times decreased across the model domain to a range of 1.3–3.5 days. Thus, residence time is controlled by the hydraulic conductivity and anisotropy of the lower zone. Residence times decreased another order of magnitude for the isotropic conditions and resulted in less variability (0.41–1.5 days) throughout the model domain and older water beneath the pool (Figure 8d). These findings support the inclusion of layered heterogeneity and anisotropy inferred from temperature and pressure data into solute-transport simulations to derive mean residence times. Further, simplified conceptualizations (1-D vertical only, homogenous, isotropic) of riverbed sediments have been reported to result in errors in estimating seepage in streams with strong horizontal components [Lautz, 2010; Shanafield et al., 2011].
 Vertical profiles of the mean residence time along the riffle-pool sequence for each conceptual model are shown in Figure 9. All four conceptual models reveal an increasing age with depth with slight variations according to location along the riffle-pool sequence and proximity to the location of the low-permeable material. In the three-layer model (CM1), the mean residence times are lower in the pool and where the low-permeability zone is absent (Figure 9b). Transport simulations without the low-permeable zone (CM2) reduced the mean residence time in the riffle locations (P2, P5, P8) by 10%–20% (Figure 9b), suggesting that the low-permeable zone has smaller influence on residence-time distributions relative to the low vertical anisotropy Kz/Kx = 0.04 in the lower zone. The age of the water flowing in the horizontal direction appears to be controlling age with relatively longer mean residence times at a reference depth of 2.0 m with CM1 mean residence times of 18–32 days and CM2 mean residence times of 15–24 days. The one-layer model (CM3) with anisotropic conditions (Kz/Kx = 0.44) resulted in greater depth of vertical flux and subsequently reduced residence times of 1.7–5.1 days at a reference depth of 2 m. The one-layer model with isotropic conditions and hydraulic conductivity of layer 1 (CM4) resulted in residence times of 0.7–2.0 days.
 We used observations of temperature and pressure to estimate the coarse spatial distribution of hydraulic conductivity, which is also the most sensitive and uncertain parameter in transport simulations of residence times. Our results indicate that reductions in uncertainty can be obtained by simulation of residence times through conditional processing with GLUE and comparison to standard Monte Carlo estimates of uncertainty. The simulation of age-mass with temperature-based numerical models presents unique opportunities for understanding multidimensional processes using heat as tracer investigations of the hyporheic zone.
5.1. VS2DT Application of Age-Mass
 This work emphasizes the consideration of 2-D longitudinal spatially distributed residence times to develop a better understanding of the linkages between surface-water and groundwater exchange, hyporheic flow, and biogeochemical processes in riffle-pool type riverine systems. It is necessary to quantify the spatial distribution of residence times to place measurements of nutrients, such as reactive nitrogen and oxygen, into context with the hyporheic flow system. We have shown spatially variable mean residence time beneath a riffle-pool system does not conform to simple conceptual models of hyporheic flow through a riffle-pool sequence. Rather, combining the vertical and horizontal mixing behavior with the spatially distributed solute-transport model, we have been able include the influx of surface water on the age of horizontal hyporheic flow paths which varies with the presence of low-permeable layers or layered heterogeneity, anisotropy, and influenced by changes in bed form topography. Thus, longitudinal hyporheic flow paths beneath river systems do not flow undisturbed or unaffected by vertical mixing typically assumed with residence times derived by advection-only particle tracking methods. Accounting for the mixing of different age inflows into the hyporheic zone has important considerations for biogeochemical processes such as nitrification and denitrification that are directly related to the length of time a solute spends in the hyporheic zone and the oxygen concentrations [Jones et al., 1995; Zarnetske et al., 2011]. Simulating the variability of exchange direction, as well as the flow paths and residence times along a flow path, can lead to improved estimation of variable nutrient sources or sinks termed hot spots/hot moments [McClain et al., 2003].
 Multidimensional approaches, such as the 2-D longitudinal simulations presented herein, are needed to adequately address important biological and ecological changes that occur in our riverine systems. One of the limitations of our modeling approach is not accounting for the third dimension and estimating the residence time given time-varying flow in the river. However, field data indicated that the third dimension (transverse to the river) was a subordinate direction for the flow of water relative to the flow of water in the vertical and longitudinal directions. In this study, we estimated mean residence times given steady flow conditions due to reservoir controls on river flow within the watershed. Further advancements could be the application of transient condition effects on mean residence times through the use of a 3-D model to examine the relationships between flow (stage) and channel morphology and heterogeneity.
 Our results indicate that the age-mass method [Goode, 1996; Varni and Carrera, 1998] or through simulation using solute-transport simulations is ideal for investigations where solute-tracer experiments are impractical given low hyporheic exchange rates, presence of low-permeability layers, long residence times (days-weeks), and for comparative analysis of nutrient concentrations such as nitrate and dissolved oxygen. The estimation of the age distribution can be easily implemented within calibrated temperature-based models (VS2DH/VS2DT) given the ability to estimate the spatial distribution of hydraulic conductivity. Further, the use of environmental tracers can further constrain flow and transport parameters in the simulation of age distributions [Sanford, 2011; Massoudieh and Ginn, 2011; Engdahl et al., 2013].
5.2. Effect of Layered Heterogeneity on Residence Time
 Distributed modeling of the riffle-pool sequence on the Truckee River indicate that heterogeneity and anisotropy of the riverbed have a strong influence on the exchange depth, flow paths, and the distribution of residence time along a hyporheic flow path. Water exchange across the riverbed interface at the study site is highly controlled the presence of a low-permeable sediment layer and reduced permeability in the deeper subsurface, which is commonplace in riverbed systems. These results are congruent with investigations of hyporheic flow through stratified and heterogeneous deposits [Salehin et al., 2004; Packman et al., 2006; Marion et al., 2008]. The permeability and porosity have been shown to decrease exponentially as a function of depth from the surface due to deposition and compaction of fluvial sediments [Sawyer and Cardenas, 2009; Cardenas and Jiang, 2010] and have implications and for nitrogen reactions [Bardini et al., 2013]. Contrary to work of Sawyer and Cardenas  and subsequent work by Bardini et al. , our results indicate that streambed layering and anisotropy beneath riffle-pool sequences have reduced the overall permeability and the depth of exchange and thus increased the mean residence times relative to the homogenous isotropic conditions. The large contrast in permeability (3 orders of magnitude) between the low-permeable material and anisotropy of the deeper deposits significantly limits the influx of newer water into the hyporheic zone. In the original simulations by Sawyer and Cardenas , the transport simulations that derived the residence times in the heterogeneous and homogenous model comparisons were based on correlation length scale of less than 30 cm, and a no-flow boundary at the base of the model (given the vertical distance of the model of less than 1 m) limits the transferability of those results to reach scale field investigations. Notwithstanding, the work presented herein shows that for a largely losing section of the river and short distances (<1 m) beneath the riverbed, a strong horizontal component driven by layering and anisotropy exists and controls the vertical exchange and mean residence times. By extension, the strong effects of layered heterogeneity on residence time would likely have a strong influence on nutrient chemistry in the hyporheic zone. Relatively long residence times are atypical for shallow hyporheic flow (<1.0 m), and low-permeable sediments in this system largely control the rate and depth of exchange. Results from the hypothetical simulations of a homogeneous isotropic condition indicate a two order of magnitude reduction in the residence time compared with simulations of anisotropic heterogeneous conditions of the three-zone model.
5.3. Parametric and Conceptual Uncertainty
 Using global methods such as URS for flow model calibration and the GLUE method to address parametric uncertainty have shown to be an effective and robust approach to simulating mean residence times in the hyporheic zone. The uncertainty of hydraulic conductivity was evaluated using both the Monte Carlo and GLUE methods constrained by model performance given measurements of riverbed temperatures and pressures. The results of the GLUE method (M = 6) produced a lower mean and standard deviation of residence time relative to the standard Monte Carlo method due to conditioning using weights and a shape factor that allows the estimates to be controlled by better performing models. Reduction in uncertainty provided by observations of sediment pressure and temperature was significant (i.e., 24% reductions in standard deviations of residence time), indicating that inexpensive measurements of sediment temperature and pressure can significantly improve models used to estimate residence time in the hyporheic zone. Furthermore, these results demonstrate the valuable information that is provided by the GLUE method combined with the standard Monte Carlo method.
 Hydraulic conductivity was the most uncertain and sensitive parameter, including its spatial variability and directional dependence. It is well understood that the spatial distribution of hydraulic conductivity is a large source of uncertainty in groundwater modeling [e.g., Freeze, 1975; Dagan, 1985; Gelhar, 1993]. This study uses the constrained parameter ranges from Naranjo et al.  and also found the hydraulic conductivity to contribute the greatest amount of uncertainty in estimates of mean residence times. Sediment porosity contributed to residence-time uncertainty, but to a lesser extent. Mechanical dispersion contributed the least amount of residence-time uncertainty in comparison to hydraulic conductivity and porosity for hyporheic flow due to the moderately high vertical and horizontal seepage velocities (10−4 to 10−2 m/h). These results are consistent with other studies focused on sediment hydraulic conductivity and heterogeneity and to a lesser extent, on characterizing sediment porosity and its spatial and temporal variability [Elliott and Brooks, 1997; Marion et al., 2008; Sawyer and Cardenas, 2009].
 The flow and transport model estimated moderately long residence times, in the order of days for shallow hyporheic flow of less than 0.2 m depth. The overall mean residence times were comparable between uncertainty approaches, within 10% between the GLUE and Monte Carlo realizations of uncertainty for hydraulic conductivity, porosity, and dispersivity with all realizations, resulting in long range in residence times (9–40 days) through the modeled area of interest. This suggests after identifying “optimal” hydraulic parameter values through the calibration process, the uncertainty in mean residence due to transport parameters is relatively low.
 We demonstrate that heterogeneity and anisotropy have a strong influence on mean residence time in the hyporheic zone using the age-mass approach. Recent studies have reported residence times are similar for homogenous versus heterogeneous riverbeds due to the reduction of hydraulic conductivity and porosity with depth [Sawyer and Cardenas, 2009; Jiang et al., 2010; Bardini et al., 2013]. Results presented herein using numerical simulations are contrary to these findings and demonstrate the effects of layered heterogeneity and anisotropy in controlling the advective exchange of flow and subsequently, the age of the water as it arrives at various locations within the model domain. The residence times estimated the solute-transport model account for mixing of different waters and hyporheic flow paths that is important for understanding biogeochemical processes along flow paths [e.g., Zarnetske et al., 2011].
 In this study area, horizontal flow dominates the hyporheic flow paths in the riffle area, and changes in bed form slope and the pressure at the surface are the primary driver for exchange into the riverbed. In the pool (P10, P12, and P15), where the low-permeable zone is absent, a greater vertical exchange is present, and subsequently, the mean residence time is reduced. These findings have important implications for biologically mediated redox reactions controlled by oxygen transport from the river into the subsurface. The results of these numerical experiments indicate that riverbed layering and anisotropy will limit the vertical transport of flow into the hyporheic zone, and subsequently, the transport of oxygen from the river. Long residence times support reductions in dissolved oxygen and nitrate through redox reactions and can be an important for nitrogen cycling in rivers.
 The mean residence times were estimated from a parsimonious 2-D flow and transport model given zonal representations of the hydraulic and thermal properties of the riverbed. The hydraulic conductivity was estimated using global parameter estimation techniques using the observations of temperature and pressure at points beneath the river. Uncertainty in the zonal representations of the original conceptual model and observation errors were not addressed in this investigation. The work of Irvine et al.  investigated errors associated with zonal characterization and found that homogenous equivalents reproduced infiltration fluxes if the river-aquifer system is connected. Because there is incomplete knowledge of the true nature of the hydraulic conductivity field beyond where our observational data are located, the existence of error in the estimation of flow and transport where locations of observations are distant from each other is highly likely. A greater density of temperature and pressure measurements to infer the hydraulic conductivity field could be avoided with spatial calibration techniques described by Doherty  and Keeting et al.  but with great computational challenges.
 Distributed flow and transport models provide a tool for simulating hyporheic flow processes; however, parameterization of models for accurate simulation of flow and chemical fate and transport in real systems remains a challenge. Naranjo et al.  demonstrated the use of a calibration approach for developing a longitudinal 2-D model of the hyporheic zone using continuous measurements of temperature and pressure to estimate the hydraulic conductivity field using a multiobjective calibration approach. Transport parameters such as porosity and dispersivity are not easily measured and are often treated constant in simulating transport in the hyporheic zone. In this study, we reduced the uncertainty of hydraulic conductivity given our constraints and performance of the flow model and assigned ranges in porosity and dispersion given the assumed texture and scale of our model domain. This method has several advantages for estimating hydraulic conductivity over commonly used approaches such as 1-D vertical flow models. For the latter approach, there is no means of evaluating the entire hyporheic flow field in a cohesive and continuous manner, and thus, residence-time distributions cannot be developed. Our simulations illustrate the need to account for the mixing of young and old water along a riffle-pool sequence where layering and anisotropy of riverbed sediments limit the rate and depth of exchange. In the approach presented herein, model performance was assessed for all measurement points in space simultaneously to provide robust constraints on model parameters and estimate residence-time uncertainty.
 Support for the first author was provided by the U.S. EPA Landscape Ecology Branch, Las Vegas. Support for the third author was provided by the U.S. Geological Survey's (USGS) Groundwater Resources Program through the Office of Groundwater. The authors are thankful for the technical reviews by Daniel Goode, David Prudic, Bethany Neilson, Ted Endreny, Audrey Sawyer, and one anonymous reviewer who provided valuable comments on this manuscript.