3.1. Statistical Relationships Derived From Temporal Moment Analysis
 Information regarding longitudinal mixing and exchange processes can be found in the normalized central moments (moments about the mean). Figure 1a shows that the variance scales in a nonlinear (non-Fickian) form with the mean travel time. If dispersion processes in streams were Fickian, the regression presented in Figure 1a would have a slope of ∼1.0, still preserving a scatter pattern that would be associated with the magnitudes of the dispersion coefficient for each experiment (i.e., different intercepts). Non-Fickian dispersion processes have been widely observed in stream ecosystems [e.g., Fischer, 1967; Nordin and Sabol, 1974; Nordin and Troutman, 1980; Bencala and Walters, 1983, and references therein], and in heterogeneous porous media [e.g., Rao et al., 1980; Haggerty and Gorelick, 1995; Dentz and Tartakovsky, 2006]. A non-Fickian behavior is, broadly defined, the result of the presence of multiscale heterogeneities that cannot be integrated into a singular dispersion coefficient [Neuman and Tartakovsky, 2009]. To date, several approaches have been proposed to better represent non-Fickian transport, which are largely based on the conceptualization of TS processes and/or the definition of smaller representative elementary volumes, where local homogeneities can be integrated in space and time.
 We also correlated versus and versus . Figure 2a suggests that solute transport data have a small range in their coefficient of skewness ( , equation (7)). The coefficient of skewness is a measure of asymmetry, i.e., when the data is perfectly symmetrical (no tailing), but it is known that solute transport experiences tailing effects due to surface and hyporheic TS, regardless of the type of stream ecosystem. For the 98 tracer tests (384 BTCs), (95% confidence bounds). In Figure 2b, we show that the product is a quasi-linear estimator of ( ). This result, although not representing a predefined statistical descriptor on its own, will be later used to define objective functions for predictive solute transport models (see section 3.3.). Not unexpectedly, based on the results from Figure 1, is a much weaker predictor of the ratio ( , results not shown), suggesting that a satisfactory bottom-up estimation of normalized central moments is restricted to one level at most.
3.2. Observed Scale Invariance in Streams and Solute Transport Models
 Nordin and Sabol  first reported observations revealing persistent skewness (longitudinally) from Eulerian observations of solute transport time distributions. Nordin and Troutman  investigated the performance of the Fickian-type diffusion equation (advection dispersion equation (ADE)), and the inclusion of dead zone processes (i.e., TS model (TSM)) to account for the persistence of skewness, concluding that “…the observed data deviate consistently from the theory in that the skewness of the observed concentration distributions decreases much more slowly than the Fickian theory predicts,” and that although the inclusion of dead zones “…yields a theoretical skewness coefficient [ ] considerably larger than that given by the ordinary Fickian diffusion equation,” “…the skewness of the observed concentrations does not appear to be decreasing as rapidly as the theory predicts.” The skewness of BTCs also do not begin with values as high as those predicted by the TSM (cf. Nordin and Troutman, 1980, Figure 3).
 The work by van Mazijk  reported that tracer experiments conducted to develop the River Rhine alarm model also showed time distributions with persistent along the extensive reach studied (100 km < L < 1000 km; ; cf. van Mazijk, 2002, Figure 6), i.e., . These observations justified the use of the Chatwin-approximation (Edgeworth series) [Chatwin, 1980] to predict solute concentrations in space and time, by fixing for the whole river. Further tracer experiments in the River Rhine ( ) supported the existence of a persistent [van Mazijk and Veling, 2005].
 Schmid  investigated the conditions under which the TSM could represent the persistence of skewness in solute transport processes. Schmid  examined the case of a slug injection into a uniform channel and concluded that a small parametric region (a loop right bounded by ; cf. Schmid [2002, Figure 1]) could generate a nondecreasing . However, this condition was hypothetical and does not play a major role in practice. Such conditions, if they exist, would be logically inconsistent because tailing effects would be inversely proportional to TS. Schmid  also examined a more general scenario with a time-varying concentration distribution as an upstream boundary condition, the division of long reaches into hydraulically uniform subreaches and a routing procedure to link temporal moments at both ends of the subreaches. This analysis suggested that “…the TS model has the potential to explain persistent or growing temporal skewness coefficients, if applied to a sequence of subreaches with respective parameter sets different from each other.” However, predicting solute transport meeting these conditions is rather impractical.
 If a transport theory is to be capable of scaling and predicting solute transport processes, it will have a persistent and statistically constant . Our observations of being statistically constant for widely different hydrodynamic conditions suggest that is not only persistent for a given stream (with distance traveled downstream), but can also be used to scale and predict solute transport processes across ecosystems. At a minimum, a persistent value of is a test that a theory of solute transport must pass.
 We used the theoretical temporal moments of three models commonly used for the analysis of in-stream solute transport (ADE, TSM, and the aggregated dead zone model (ADZM)) to calculate their theoretical . If these models were systematically capable of representing the scale-invariant patterns observed in our meta-analysis, the parameters would be self-consistent when describing . The model equations and the theoretical temporal moments and (calculated for an impulse-type boundary condition, e.g., Cunningham and Roberts ) are shown below, along with the consequences of the invariance of on the model parameters. We also included in our analysis (see section 3.2.4) three additional transport models less commonly used to describe solute transport in streams, but that have been used in groundwater systems.
3.2.1. Advection Dispersion Equation
where [ML−3] is the concentration of the solute in the main channel; [L3T−1] the discharge; [L2] the cross-sectional area of the main channel; [LT−2] the dispersion coefficient; [L] the reach length; [T] time; [T] is the conservative mean travel time; the Peclet number; and the mean velocity in the main channel [LT−1].
 Equation (9) suggests that if is constant, the Peclet number should also be constant. This implies that, under steady-state flow conditions, the dispersion coefficient must scale linearly with the distance traveled. This violates the assumption of spatially uniform coefficients. Therefore, the ADE with spatially uniform coefficients is incapable of representing the experimental observations. Dispersion coefficients scaling with distance have been widely observed in porous media [e.g., Pickens and Grisak, 1981; Silliman and Simpson, 1987, Pachepsky et al., 2000, and references therein]. Note that the ADE with constant coefficients predicts BTCs with longitudinally decreasing skewness ( ), becoming asymptotically Gaussian (i.e., ).
3.2.2. Transient Storage Model
(10a) (10b) (11)
where [ML−3] is the concentration of the solute in the storage zone; [L2] is the cross-sectional area of the storage zone; [T−1] is the mass-exchange rate coefficient between the main channel and the storage zone; and . Other variables are as defined for the ADE. The TSM in equation (10a) is the same presented by Bencala and Walters  and Runkel  for a reach without lateral inputs, with a slightly different definition of . Note that when .
 If dispersion effects were assumed negligible [e.g., Wörman, 2000; Schmid, 2002], in equation (11) would simplify to
 Using the value found in our meta-analysis, the mean residence time in the storage zones ( ) normalized by scale linearly with travel time ( ), i.e.,
 Equations (11) and (12) suggest that the standard TSM generates BTCs with longitudinally decreasing skewness ( ), becoming asymptotically Gaussian (i.e., ). The physical meaning of the parameters describing is unclear unless dispersion is assumed negligible ( ). In this case, equation (13) suggests that the TSM model parameters are not independent and that their ratio grows with distance traveled. This analysis supports the results of other studies showing problems of equifinality for the TSM [e.g., Wagner and Harvey, 1997; Wagener et al., 2002; Camacho and González-Pinzón, 2008; C. Kelleher et al., Stream characteristics govern the importance of transient storage processes, submitted to Water Resources Research, 2012]. Equations (11) and (13) suggest that the physical meaning of the TSM parameters is limited, and that relationships between TSM parameters and biogeochemical processing may be site dependent (as was discussed in section 1) or even experiment dependent.
3.2.3. Aggregated Dead Zone Model
where [T] is the lumped ADZ residence time parameter representing the component of the overall reach travel time associated with dispersion; [ML−3] is the known concentration at the input or upstream location; and [T] is the time delay describing solute advection due to bulk flow movement.
 Equation (14) describes the mass balance of an imperfectly mixed system (ADZ representative volume), where a solute undergoes pure advection, followed by dispersion in a lumped active mixing volume [Lees et al., 2000]. In the ADZM, the distance implicitly appears in the model description through the time parameters. Note that when , the mean travel time ( ) could be written as . In equation (15), the parameter represents the number of identical ADZ elements serially connected ( for a single ADZ representative volume) to route the upstream boundary condition. The serial ADZM, although capable of representing a persistent , would require the specification of the nonphysical parameter . More complex ADZM structures can be defined under the database mechanistic approach [e.g., Young, 1998], but we restricted our discussion to those that have been more commonly used in stream solute transport modeling [Young and Wallis, 1993; Lees et al., 2000; Camacho and González-Pinzón, 2008; Romanowicz et al., 2013].
3.2.4. Alternative Solute Transport Models
 Similar sets of calculations also show that the multirate mass transfer (MRMT) model [Haggerty and Gorelick, 1995; Haggerty et al., 2002] (Appendix A) and a decoupled continuous time random walk (dCTRW) model [e.g., Dentz and Berkowitz, 2003; Dentz et al., 2004; Boano et al., 2007] (Appendix B) are equally incompatible with observations of persistent skewness. The in both of these models also scales as .
 We also explored a Lévy-flight dynamics model (LFDM) (Appendix C) [e.g., Shlesingerm et al., 1982; Pachepsky et al., 1997, 2000; Sokolov, 2000], which describes the motion of particles behaving similarly to Brownian motion, but allowing occasional clusters of large jumps (significant deviations from the mean). Lévy-flight models have constant transition times, combined with transition length distributions that are characterized by power-law behaviors for large distances. Therefore, such models represent processes characterized by large velocities for long transitions and low velocities for short transitions, and would account for transport in the continuum of river and storage, with the high velocities present in the stream. We were able to generate an LDFM with persistent for a Lévy distribution parameter (this is different from the mass-exchange rate coefficient used in the TSM and MRMT model, (cf. (C2) and (C31)). However, gives an inconsistent scaling of the variance with distance, i.e., (cf. (C25)). Furthermore, this distribution parameter would imply a velocity distribution in the stream that scales as at large velocities, which does not appear realistic.
3.3. Use of Moments Scaling Properties to Predict Solute Transport
 While the models contain an error that needs correction, it may be possible (in the meantime) to adjust the parameters in a way that is predictive of field behavior. In this section, we use the regressions from the temporal moment analysis (section 3.1.) to predict solute transport. We provide the parameterization of the TSM, ADZM, and two probability distributions. We then provide an example using data from tracer experiments that were conducted in the River Brock, River Conder, River Dunsop, and River Ou Beck in the United Kingdom [Young and Wallis, 1993, pp. 160–165]. The first three rivers are natural, and River Ou Beck is a concrete urban channel.
 The methodology requires an independent estimation of the mean travel time ( ). One way to do this is to regress against discharge ( ) using a power law or an inverse relationship in [Young and Wallis, 1993; Wallis et al., 1989; Pilgrim, 1977; Calkins and Dunne, 1970]. Once is estimated, the results from our temporal moment analysis can be used to constrain predictive (forward) simulations of solute transport models. We exemplify this methodology using the experiments by Young and Wallis , which were not used in the previous moment analysis, because they show the technique to estimate mean travel times from discharge.
3.3.1. Predicted Solute Transport With Classic Solute Transport Models
 The parameters of solute transport models can be determined by matching theoretical and experimental moments. Here, we show how the empirical scaling relationships described in section 3.1 can be used to direct the search of the parameters of the TSM and the ADZM in predictive simulations.
126.96.36.199. Predicted Solute Transport With TSM
 We used the empirical relationships derived for versus and versus (Figure 2) to match the theoretical moment equations presented by Czernuszenko and Rowinski . These theoretical equations have been developed for a general upstream boundary condition with tracer distribution . The parameters for the TSM are those defined by Bencala and Walters  and Runkel .
(16) (17) (18)
 We have eight variables, i.e., the dispersion coefficient , ( ), the mass-transfer rate , the length of the reach , the discharge ( ), and the normalized central moments , , . We have five equations: three for the theoretical moments (equations (16)-(18)) and two empirical relationships (derived from Figure 2). To balance the degrees of freedom ( ), we therefore need to specify three ( ) variables, namely , , and . We used a Newton-Raphson algorithm to solve for the five unknowns by minimizing the objective function ( ) shown in equation (19). We estimated the mean travel time as: , with , and randomly varied the regression coefficients of our meta-analysis within the 95% confidence bounds.
 In the optimization routine, we allowed the TSM parameters to vary within ranges typically found in similar streams, i.e., (m2/s), (m2), (m2), (s−1). Once the system of equations was optimized for each random set of estimated mean travel time and fitting coefficients (n = 1000), we ran a forward simulation using the optimum parameters. Results from the Monte Carlo simulations are presented in Figure 3 and Tables 3 and 4. We used the Nash–Sutcliffe model efficiency coefficient ( ) [Nash and Sutcliffle, 1970] to estimate the goodness of fit of the predictions, i.e., how well the plot of observed versus simulated data fits a 1:1 line.
Table 3. Best Parameter Sets From 1000 Monte Carlo Simulations Using Empirical Relationships Derived From Normalized Central Moment Meta-Analysis (n = 384 BTCs) and the Moment-Matching Techniquea
|Brock||4.5 × 10−1||128||2.33||1.31 × 10−2||9.77||0.96||218.01||0.98|
|Conder||1.0||116||2.20||8.12 × 10−3||8.08||0.99||151.95||0.97|
|Dunsop||5.4 × 10−1||130||1.33||1.45 × 10−2||7.89||0.98||332.55||1.00|
|Ou Beck||3.5 × 10−2||127||0.67||4.40 × 10−3||8.92||1.00||135.95||0.76|
Table 4. List of Estimated Parameters and Prediction Efficiencies for Each Predictive Model Exploreda
|TSM|| , , , b, b||0.74–0.96||0.71–0.99||0.39–0.99||0.26–1.00|
|Gumbel dist.|| ||0.39–0.96||0.45–0.95||0.38–0.99||0.18–0.77|
|Lognormal dist.|| ||0.42–0.94||0.47–0.92||0.45–0.97||0.18–0.74|
3.3.2. Predicted Solute Transport With Probability Distributions
 Time series described by probability distributions can be used to predict solute transport processes. Here, we show how the empirical scaling relationships described in section 3.1 can be used to estimate the temporal moments of two probability distributions and then to perform predictive simulations.
188.8.131.52. Predicted Solute Transport With Gumbel Distribution
 We chose the Gumbel (Extreme Value I) probability distribution because of its constant , which closely agrees with the empirical relationships derived from our meta-analysis ( ). This distribution is typically used to describe hydrologic events pertaining to extremes [Brutsaert, 2005]. The concentration distribution of a solute BTC using this distribution takes the form:
where and are the location (mode) and scale parameters, respectively. Note that these parameters, and those of any other probability distribution, have no direct physical interpretation.
 The use of probability distributions requires the explicit definition of moments beyond the mean travel time, i.e., variance and in some cases the skewness. Therefore, we would need to use empirical relationships such as those derived in Figure 1, even though . In our predictive analysis, we used , with to estimate the uncertainty of , and , with , as it was suggested by our meta-analysis (i.e., , , regression not shown in Figure 1). The results obtained are presented in Figure 5 and Table 4.
184.108.40.206. Predicted Solute Transport With Lognormal Distribution
 A random variable described by a lognormal distribution comes from the product of n variables, each with its own arbitrary density function with finite mean and variance. This distribution has been widely used in hydrologic modeling of flood volumes and peak discharges, duration curves for daily streamflow, and rainfall intensity-duration data [Chow, 1954; Stendinger, 1980]. Applications in solute transport suggested that the solute velocity, saturated hydraulic conductivity, and dispersion coefficient are lognormally distributed [Rogowski, 1972; Van De Pol et al., 1977; Russo and Bresler, 1981]. The concentration distribution of a solute BTC with this distribution takes the form:
where and are the mean and the standard deviation of . In our predictive analysis, we followed the same procedure described for the Gumbel distribution. The results obtained are presented in Figure 6 and Table 4.
3.3.3. Analysis of Predictive Solute Transport Modeling
 In our predictive analyses, we used two classic models (TSM and ADZM) and hypothesized that these models could adequately predict solute transport if the results of our meta-analysis were defined as objective functions to minimize the differences between the theoretical and empirical temporal moments. Our main goal therefore was to fix a constant regardless of the longitudinal positioning. The predictive results presented in Figures 3 and 4 and Tables 3 and 4 show that this approach required only basic information (i.e., , , and an estimation of the mean travel time) to adequately predict the behavior of the solute plumes traveling downstream. For the TSM (four parameters), the best predictions in the uncertainty analysis had for the four rivers. For the ADZM (two parameters), the best predictions had for all natural rivers, and for the concrete channel. Although satisfactory results can be achieved with this predictive methodology, it is important to bear in mind that good fittings do not necessarily come from adequate interpretations of mechanistic processes and, therefore, the physical meaning of the parameters should not be taken literally in both inverse (used for calibration) and forward (predictive) simulations.
 Besides from predicting solute transport with classic models, we explored the use of probability distributions. We developed predictive models through the parameterization of the Gumbel and lognormal probability distributions, using the results from our meta-analyses and performing uncertainty estimations. The results of our predictive simulations can be summarized as (Table 4): (1) the Gumbel distribution ( ) yielded better predictions when the distributions were parameterized with the observed and , suggesting that is a consistent pattern derived from our meta-analysis and (2) estimating the variance ( ) of the distributions from the mean travel time ( ) can be highly uncertain, and it is explicitly required for using probability distributions in predictive mode; therefore, uncertainty analysis must be always included. Importantly, the parameters of these distributions do not have direct physical meaning, and this has two main consequences: (1) solute transport understanding cannot be mechanistically advanced and (2) erroneous parametric interpretations from physically based, but poorly constrained models are explicitly avoided.
 In summary, we found that the regressions from our meta-analysis can be used to adequately predict solute transport processes using either transport models (fixing ) or probability distributions. We consider this a transitional methodology (“a patch solution”) between our current understanding and an improved transport theory that better represents the experimental results.
3.4. Implications for Scale-Invariant Patterns
 Other experimental findings reveal intriguing similarities to the scale-invariant patterns that we have highlighted here. These include the linear relationship between cross-sectional maximum and mean velocities [Chiu and Said, 1995; Xia, 1997; Chiu and Tung, 2002], and the relatively constant behavior of the dispersive fraction (a parameter derived from the ADZM) in alluvial and headwater streams [Young and Wallis, 1993; González-Pinzón, 2008]. These observations suggest that stream cross sections establish and tend to maintain a quasi-equilibrium entropic state by adjusting the channel characteristics, i.e., erodible channels adjust their geomorphic characteristics with discharge (bedform and type of sediment transported, slope, alignment, etc.) and nonerodible channels adjust their velocity distributions by changing the maximum velocity and flow depths [Chiu and Said, 1995; Chiu and Tung, 2002]. An improved solute transport theory should address these observed scale-invariant hydrodynamic patterns and explore the physical meaning of the persistence of skewness, which perhaps could be based on principles of thermodynamics and fluid dynamics.
 The coefficient of skewness of the classic solute transport models discussed in section 3.2 shows that Fickian dispersion is inconsistent with the experimental results. The inclusion of macroscopic Fickian dispersion generates a system where the variance of a dispersing solute grows linearly with the distance traveled, generating skewed distributions that later become asymptotically Gaussian [Fisher et al., 1979; Nordin and Troutman, 1980]. This behavior is independent of the assumption of hydraulically uniform stream reaches, suggesting that a revised dispersion approach would be needed unless other mechanisms included in the transport theory (e.g., TS) were capable of counteracting the ever decreasing skewness represented by Fickian dispersion.
 Although we have not yet investigated scale-invariant behaviors of temporal distributions in processes other than solute transport, we predict that similar patterns can be derived from meta-analysis of flow routing BTCs. We ground this prediction in the fact that the conservative tracers used in our analyses have marked up how water flowed through the different stream ecosystems considered, experiencing similar physical characteristics and processes involved in flow routing (i.e., shear effects, heterogeneity and anisotropy, and dual-domain mass transfer). Regardless of the adequacy of current transport and flow routing modeling approaches, clear similarities appear when comparing the BTCs of these hydrologic processes, and the temporal moments of (for example) the ADZM and those of the Nash cascade [Nash, 1960] and the Linear (and Multilinear) Discrete (Lag) Cascade channel routing models [O'Connor, 1976; Perumal, 1994; Camacho and Lees, 1999]. If similar patterns were found with respect to the persistence of skewness in solute transport and flow routing, this could be advantageously used to better understand, scale, and predict solute transport processes under flow dynamic conditions, which is a problem that still remains largely unresolved [Runkel and Restrepo, 1993; Graf, 1995; Zhang and Aral, 2004].