Corresponding author: V. Cvetkovic, Department of Land and Water Resources Engineering, Royal Institute of Technology, Brinellvägen 28, 10044 Stockholm, Sweden. (firstname.lastname@example.org)
 A Lagrangian framework for material transport along hydrological pathways is presented and consequences of statistically stationary space-time flow velocity variations on advective transport are investigated. The two specific questions addressed in this work are: How do temporal fluctuations affect forward and backward water travel time distributions when combined with spatial variability? and Can mass transfer processes be quantified using conditional probabilities in spatially and temporally variable flow? Space-time trajectories are studied for generic conditions of flow, with fully ergodic or only spatially ergodic velocity. It is shown that forward and backward distributions of advective water travel time coincide for statistically stationary space-time variations. Temporal variability alters the statistical structure of the Lagrangian velocity fluctuations. Once this is accounted for, integration of the memory function with the travel time distribution is applicable for quantifying retention. Further work is needed to better understand the statistical structure of space-time velocity variability in hydrological transport, as well as its impact on tracer retention and attenuation.
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 In this work we focus on water and solute transport along the different hydrological pathways that connect a recharge with a discharge location of a catchment. With mean flow resolved, our main task is to incorporate random effects under conditions where both spatial and temporal variations are significant along the hydrological pathways through the catchment.
2. Problem Formulation and Assumptions
 A catchment is viewed as a three-dimensional flow system (Figure 1) driven by prescribed boundary conditions. In general, the system consists of components such as rivers and streams, lakes and wetlands, and soil water and groundwater flowing through unconsolidated and consolidated (fractured) porous media. We define a “hydrological unit” (HU) as any one of the components with at least the mean properties specified. The boundary conditions for the mean flow are assumed known, and an average streamline/trajectory pattern in a catchment can be resolved, e.g., by numerical simulation based on estimated hydraulic properties of the HUs. The effect of spatial heterogeneity within HUs at, or below, a scale resolvable by simulations, can be accounted for analytically [Persson and Destouni, 2009].
 Two key assumptions of this work are summarized as follows: First, we consider water and advective solute transport along hydrological pathways (trajectories) that are controlled by average boundary conditions; hence pathway geometry is approximately constant with velocity that may vary in space and time. Second, the solute concentrations are relatively low such that retention is linear in a generalized sense; this implies that superposition for tracer concentration or flux is applicable.
 If the velocity temporal fluctuations are primarily controlled by the boundary conditions, it is convenient to express as separable, i.e., , where and are approximated as independent; this approximation will be used in the following for generic illustrations. In cases where the separation does not hold, v(x, t) (inferred from measurements and/or simulations) would be used directly.
 Water entering a catchment at any given recharge location (Figure 1) will be partitioned between different transport pathways (e.g., red and blue trajectories in Figure 2). Temporal fluctuations will, in general, be different along the different pathways, with the red trajectory example expected to have stronger temporal fluctuations than the blue trajectory (Figure 2). Depending on prevailing boundary conditions and internal catchment structure, more or less of the water flow and solute mass will be partitioned between different pathways [Bosson et al., 2012]. The two main questions addressed are: How do the temporal fluctuations affect forward and backward water travel time distributions when combined with spatial variability? and What are the conditions for the applicability of mass transfer processes using conditional probabilities in spatially and temporally variable flow?
 In sections 3 and 4 we first summarize the Lagrangian framework and then quantify water (or ideal tracer) travel time along trajectories. Retention is discussed in section 5. Finally, in section 6 our results are set in a broader context, and main conclusions drawn in section 7.
3. Lagrangian Framework
 Consider a catchment in three dimensions with a known structure and specified boundary conditions for flow. A typical trajectory/streamline transects N “hydrological units” (HUs) which are specified, such as a stream, lake, wetland, unsaturated subsurface flow, and shallow and deep groundwater flow in porous media as well as fracture zones, fractured rock, etc. (Figure 2). Note that here streams would be considered in a classical sense of drainage networks, i.e., as essentially known one-dimensional channels with a spatially and temporally varying velocity.
 Particles injected at a recharge point a with support end up at a discharge location b, depending on the flow/trajectory pattern. Let denote the time a water (or ideal tracer) particle takes along a trajectory from a to b in a single HU, where t0 is the starting time. In reality we typically have many particles injected at different times, at different a, which exit at different or the same discharge locations (Figure 1).
 A water particle is injected at as . The particle travels through the system with time and is then discharged at b at time t = T where . We refer to as the water travel time, whereas T is referred to as the “arrival time.” For steady state flow, it is typical to set and travel and arrival times coincide, i.e., .
 The normalized discharge for an ideal tracer particle starting (or entering) at a at time and arriving at (or exiting) the system at time is obtained by the convolution,
 Let a total amount of ideal tracer mass be recharged over a given elementary area at a, with a rate [M/L2T]; we have
 The ideal tracer discharge at b is
where is the forward WTT probability density function (pdf) conditioned on the starting time t0 at a; the corresponding expression for a nonideal tracer will be discussed in section 5.
 In the general case, both and t0 are random, where is dependent on t0. There are different pdfs that may be of interest using variables , and t0. First, is the pdf of t0 at a if t0 cannot be specified deterministically. Then we may consider a joint pdf for instance, or . We may also consider conditional pdfs such as the backward WTT pdf , where travel times are conditioned on exit time T, or the forward pdf where travel times are conditioned on starting time t0. This work addresses the conditional backward and forward pdfs and , respectively, in cases where temporal variations of the flow are significant. Note that if a catchment is conceptualized as a mixed reactor, conditional pdfs and can be directly related, provided that temporal variations of water storage in the catchment are known [Botter et al., 2010].
 The flow field in a catchment is assumed continuous such that at an arbitrary x within the catchment, a streamline can always be found. At an arbitrary time t, one can trace as the water particle age, and as water particle life expectancy [Nauman, 1969]; the special case is with .
4. Water Travel Time
 The basic kinematic relationship of a trajectory is well known [Dagan, 1984],
 If the reverse flow relative to the mean velocity from a to b is negligible, a trajectory can be projected onto a suitably defined mean flow direction, for instance, x of . In that case, the travel (residence) time between a and b, is,
where and a is the starting position; the boundary condition is either or , i.e., either the starting time t0 is known, or the arrival time is known. For the separable case we write,
where W(x) is a Lagrangian velocity along a trajectory parameterized with the distance [Cvetkovic et al., 1996; Gotovac et al., 2009]. If (1) is solved with we have the “forward” solution that is the basis for computing , and if solved with then the “backward” solution is obtained as the basis for computing .
 The conditional pdfs and will be site-specific for any given catchment and boundary conditions. Here we wish to investigate their basic properties under idealized flow conditions. A generic setup is defined in Appendix A (Figures A1 and A2), and is suitable for exploring the forward and backward travel time distributions. The magnitude of the temporal fluctuations are assumed uniform along the domain; in reality, these fluctuations can be expected to diminish with transport distance [Destouni, 1991; Foussereau et al., 2001]. In Appendix A, the temporal aggregation unit for the data-based temporal fluctuations is daily; however, in our illustrations we shall leave the spatial and temporal units arbitrary (unspecified), depending on the units of the mean velocity U0 which (unless specified otherwise) is set to unity.
4.2. Forward and Backward Distributions
Figure 3 illustrates forward and backward WTT distributions obtained from 500 trajectory realizations. In Figure 3a, fully ergodic conditions are assumed, i.e., 500 realizations of V(x) and 500 realizations of . The simulated forward and backward cumulative distribution functions (CDFs, dotted curves) are compared to the analytical form (solid curves) assuming the solution of the advection-dispersion equation (ADE), or the tempered one-sided stable (TOSS) density with in (B5) (Appendix B). A steady state flow case (with 500 identical realizations of V[x]) is also shown (black). The ADE model provides a close representation for the transient case roughly after the 10th percentile; for percentiles lower than 10, deviation is obvious between the simulated (symbol) and analytical (solid) curves. Similar comparison for the illustrated steady state case is much closer over the entire range (black symbols and curves). Random temporal variations may shift the CDF relative to the steady state case, depending, for instance, on how the mean velocity is defined (see Appendix A), or whether temporal fluctuations decrease with transport distance [Destouni, 1991; Foussereau et al., 2001], etc. The forward and backward CDFs of WTT almost coincide (compare red and blue curves in Figure 3a).
 Next, a more realistic case of the data-based (top-right diagram in Figure A1 of Appendix A) is explored, where is a “single realization” identical for all 500 spatially random fields V(x). The black symbol curve in Figure 3b shows the forward CDF, starting at time , whereas the colored curves show the backward CDF as a function of four chosen arrival time (T) values (8000, 7500, 7000, 6500, 6000). Clearly, there are visible differences in the backward CDF for the different arrival times, and also compared to the forward CDF (symbols), since the temporal variations are now a single realization (in this case using data-based ). However, the differences are relatively modest indicating that even in single realizations forward and backward CDFs may be quite comparable if longer periods of time are considered.
 For statistically stationary temporal variations and fully ergodic conditions, the distribution of is independent of t0 (forward density), and the distribution of is independent of T (backward density), i.e.,
where for the water “life expectancy” (forward) density is defined, and for the water “age” (backward) density is defined, using the terminology of Nauman . Written out in words this is, the water age and life expectancy pdfs are equivalent for the same length scale. Note that the three-dimensional vectors a, b, and x have been suitably projected onto the mean flow direction which is assumed for simplicity parallel to the x-axis, and is part of the trajectory between a and b. For nonergodic conditions of temporal variations, where a single realization of is considered, (2) holds approximately, provided that the fluctuations are statistically stationary and one is considering sufficiently large WTT relative to the integral scale of the temporal fluctuations.
4.3. Temporal Moments
Equation (2) is further explored based on the first two moments. Specifically, water particle transport is considered for the synthetic , assuming both a single realization and an ensemble of (ergodic conditions). The mean this illustration is assumed to be , and the backward first temporal moment is computed for different arrival (exit) times T.
 The mean for the backward density with T in the range 10,000–20,000 with intervals of 1000, is illustrated in Figure 4a; the fully ergodic case is shown with the thin red lines and the single realization case of with the thin black lines. The mean for the forward density with is also included as the thick blue line, solid for the ergodic case and dashed for the nonergodic case.
 First, moment maintains in all cases an approximately linear dependence on distance (Figure 4a). For the nonergodic case, exhibits more dispersion for different T (black thin lines), compared to the ergodic case (red thin lines). Only 100 realizations are used for the “ergodic” case; convergence toward a more compact linear dependence of with distance, relative to the single-realization case is clearly seen by comparing the red and black thin lines in Figure 4a. Note that for the pathway between a = 0 and b, the thin lines are the mean water age for different arrival times T; mean groundwater age is typically addressed for entire aquifers in steady state flows [e.g., Ginn et al., 2009].
 The variance for the same cases is shown in Figure 4b. Here the spreading for different T is clearly more significant, in particular for the single realization of ; at the end of the domain, the spreading of for different T is almost 50%. For full variability, the dispersion is lower: the red curves for the 10 values of T differ around 20%. Further increase in the number of trajectories would reduce this difference in the curves obtained with different T. In spite of the variations visible in Figure 4b, the overall convergence of the travel time variance for statistically stationary spatial and temporal variations, is toward a consistent linear dependence. For single realizations of , (2) is one possible approximation. Note that in Figure 3b, the single realization is obtained with the data-based (Figure A1, Appendix A), whereas in Figure 4 the single realization case is based on the synthetic (Figure A2, Appendix A). Clearly, further work is needed to better understand the implications of (2) for different cases of temporal flow variations.
4.4. Segmented Trajectories
 To quantify hydrological transport through different subsystems of a whole catchment, now consider segmented trajectories representing N different HUs as illustrated schematically in Figure 2. If the HUs have different physical properties, independence of WTT between HUs is a reasonable assumption. Here we wish to address the issue of independence more fundamentally, by considering HUs as geometrical entities with identical flow properties. To convolute WTTs across N different segments, we need to identify under what conditions for spatially and temporally varying flows, the advective water travel times in each segment can be considered to be approximately independent. The answer will depend on the correlation structure of the Lagrangian velocity W(x), a result of “transforming” the fields V(x) and through . In the following, segmented trajectories are explored for the generic space-time velocity field (Appendix A), by computing the convolution, as well as by using the expression for macrodispersion (B5) (Appendix B).
 In Figure 5, forward CDFs and complementary cumulative distribution functions (CCDF = 1 – CDF) are compared with N = 4 segments, for a transient and steady state random velocity; in the transient case, the fully ergodic synthetic is used. The blue curve is obtained by computing the travel time moments over each segment, then using these to compute an analytical ADE CDF/CCDF for each segment; the red curve is computed as the ADE CDF/CCDF with travel time moments obtained from simulated trajectories for the entire domain. With the four-segment example used for illustration in Figure 5, there is a slight difference between the blue and red curves, visible both for the transient and the steady state case.
 An additional test is to use the derived macrodispersion expression (B5) for a different number of segments. In Table 1 we compare the WTT coefficient of variation (equation (B5) in Appendix B) computed segment-wise, with computed for the entire domain (N = 1); identical values would then indicate total independence, as well as consistency with the ADE model. Note that each realization of V(x) covers 20 log-integral scales over the entire domain of 1000; temporal variations tend to further increase the integral scale of relative to that of V(x) (Figure A2 of Appendix A, bottom diagram).
Table 1. Comparison of WTT Coefficient of Variation (B5) for a Segmented Trajectory With Different Segments N, for Transient and Steady-State Cases
Table 1 shows the increasing deviation between computed over an entire domain (first row with N = 1), and computed using (B5) for the same domain divided in N equal segments. For N = 2 and N = 4, the segmented case agrees well with calculated for the entire domain, the difference being up to a few percent. For N = 10 the deviation is more significant, ∼20%, indicating that the independence of segments is questionable. The deviation increases with further reduction of segment (or HU) size (increase of N). Note that the deviation in between N = 1 and N = 10 does not differ for transient and steady state cases, although the actual value of is lower for the illustrated steady state case.
5. Solute Retention
 All solutes (tracers) transported through catchments will be subject to some degree of retention, the minimum being molecular diffusion into stagnant water, whether in the sediment, or intragranular porosity of rocks. Transport of some tracers may be significantly delayed due to sorption combined with diffusion. The main interest for most applications is to compute the tracer residence time, or tracer discharge at specified locations b.
 Under steady state flow, irrespective of the time spent in the immobile regions, a tracer particle is subject to the same advective transport when mobile at any given . In such a case, the methodology using conditional probabilities for relating water and tracer travel times [Villermaux, 1974; Cvetkovic and Dagan, 1994] is applicable. The main question explored in this section, is to what extent the same methodology would be applicable for both spatially and temporally variable flow.
 Tracer particles are injected with associated (dissolving) “water particles” simultaneously and under identical conditions [Cvetkovic and Dagan, 1994; Cvetkovic and Haggerty, 2002] at a at time t0. Consider a trajectory between a and b (Figure A1, Appendix A), again projected onto the mean flow direction, here the x-axis. Let the trajectory between a and b of length x be divided into N equal segments. For simplicity, we use the same notation N for the number of segments as in section 4, where reference was made to HUs. Here N is chosen such that, given the temporal fluctuations for the sorptive (reactive) problem at hand, the selected ensures that velocity temporal changes over scale are relatively small.
 The transport of water and tracer particles is conceptualized as a time domain random walk, the sketch of which is shown in Figure 6. If denotes the WTT across an arbitrary , and denotes the retention time along the same segment for the tracer particle, then is a particle residence time for segment . Let denote the density of . Based on the mass balance for advective transport across , we can write in the Laplace domain as [Cvetkovic and Dagan, 1994]
where ( ) denotes the Laplace transform. In other words, for a given segment with , the solute retention time is drawn from a distribution dependent on and a set of retention parameters denoted as a vector P. The parameters P are assumed to be known and approximately independent of time (diffusion constants, sorption coefficients, intra-aggregate porosity, etc.). The function g is referred to as the “memory function” quantifying the kinetics of mass transfer.
 for the selected scale will, in the general case, be correlated between segments. For transport where retention is significant, however, the retention time will be independent between segments; hence, may also be considered independent. In such a case, the tracer residence time for the entire trajectory is and its density can be obtained by convolution,
where is referred to as the “virtual” water travel time between a and b. The “virtual” aspect of can be explained as follows.
 The time domain random walk of the transport implies that at each segment we have a random water travel time and a random retention time (Figure 6). The retention time is the delay due to the immobilization of the tracer particle within the segment . When the tracer particle is mobilized again, its retention in the next segment will be conditioned on the water particle travel time for that next segment. This means the tracer particle now follows a trajectory with another starting time t0, which in principle is obtained by backward tracking, as illustrated in Figure 6 for five trajectories. Thus, the “virtual” aspect is due to the fact that no actual water particle is transported between a and b during time .
 A metaphor for the tracer particle under these conditions would be a passenger traveling by train between stations. At each station, the passenger disembarks and spends some time (retention due to immobilization); the passenger boards new trains at successive stations with different starting times t0 and a new space-time trajectory. Note that depending on g, t0 for each segment may be very different. For steady state flow, trajectories are independent of t0, thus irrespective of how large is (i.e., how long a passenger waits at a station), the virtual (aggregated) water travel time is always equal to the water travel time , i.e., .
 For approximately stationary temporal fluctuations, (2) is valid and ; then we write,
i.e., the virtual WTT pdf is approximately equal to the actual WTT pdf, .
5.2. Residence Time and Discharge
 Let denote the pdf of tracer residence time for HU “i,” conditioned on the WTT in the same HU; in line with (3), we have,
 In other words, for a given of HU “i,” tracer residence time is drawn from a distribution dependent on a set of retention parameters denoted as a vector for HU “i.”
 To compute an unconditional tracer residence time pdf which accounts for the variability of , we ensemble average to obtain by virtue of the Laplace transform definition:
where is the Laplace transform of the WTT pdf for the HU “i” and (4) has been used. Furthermore, h in (6) is the unconditional pdf of tracer particle residence time for pulse injection, in effect a “transfer function” for the HU “i.”
 The mass discharge of a nonideal, interacting tracer is obtained by,
which reduces to the expression for an ideal tracer (water) with . Instantaneous injection of mass implies . For steady state flow and , the above equation reduces to the well-known expression [e.g., Cvetkovic and Dagan, 1994].
 Typically, the temporal variations are approximately stationary for longer timescales, similar to the example used in this work. When considering past events, e.g., for model verification, a single realization of the temporal variations will be available (a nonergodic case, or only a spatially ergodic case). For predictions of future transport over the decadal timescale, one will typically use a synthetic replica based on past statistics, where considering an ensemble is relevant (an ergodic case). If temporal trends are present but are relatively mild and the transport is considered over sufficiently long times, (2) and (4) may be applicable. Only in cases where abrupt changes take place over a relatively short period, such that the mean velocity exhibits a stepwise form, (2) and (4) will not be applicable. In cases where (4) is not applicable, a recently developed time domain random walk model for transient flow (J. O. Selroos, H. Cheng, S. Painter, and P. Vidstrand, Radionuclide transport during glacial cycles: Comparison of two different approaches for representing flow transients, submitted to Journal of Physics and Chemistry of the Earth, 2012) can be used for incorporating retention and possibly degradation, where each water particle is associated with an ensemble of tracer particles that are traced probabilistically, as they leave and return to the immobile zone due to mass transfer.
 The dispersion of a tracer in a catchment may be due to different effects. First, tracer advection will be subject to spatial and temporal flow variability. Second, even if injected over a small area, tracer particles may take different pathways, from the shallow to the deep subsurface, to surface flow; although the overall length scale of these pathways may be comparable, the hydraulic properties of the “hydrological units” are generally very different. Third, if injected over larger areas, tracer particles discharged at the same location will have different pathway lengths, which will significantly contribute to overall dispersion. All of these effects combined together may be referred to as “hydrological dispersion.” Finally, tracer particles will, in the general case, be subject to some type of mass transfer as well as possible reactive transformation.
 Given the complexity and variety of conditions involved, diverse conceptual approaches have been used to address hydrological dispersion. A similar approach to ours was presented for studying dispersion in a drainage network [Rinaldo et al., 1991, 2006; Rinaldo and Rodriguez-Iturbe, 1996]. In these studies, dispersion is accounted for on two levels: within an individual stream reach (hydrodynamic dispersion) and between different stream reaches within a stream network (geomorphological dispersion); the solution of the advection-dispersion equation for a given stream reach is convoluted to quantify the network effect [e.g., Rinaldo et al., 1991]. Moreover, the routing through serial stream reaches is aggregated into multiple stream pathways of varying length, which when combined, result in the dispersion on the stream network scale [Rinaldo et al., 1991, 2006].
 Our present work also relates to recent studies of Darracq et al. , Destouni et al. , and Persson et al. , where subsurface flow and transport to and through the stream network is considered, with a focus on each single pathway between a recharge and discharge location. The pathways are of arbitrary length and transect a given number of “stream reaches” as well as other hydrological units, combining surface and subsurface flow.
 One means of incorporating varying pathway lengths is by integrating over varying distances from a finite recharge area [Destouni and Graham, 1995]. Let a total amount of solute mass M0 be released over a given recharge area, A, starting at t0. We then have,
 In real catchments, a given discharge location typically collects water from many recharge locations or spatially distributed recharge, whereby many trajectories converge to a few discharge locations (Figure 1). Consider a finite recharge area A that is discharged into a discharge location at b. If the injected solute over area A is discharged at b, the normalized solute discharge for a noninteracting (ideal) tracer is,
 Hydrological dispersion has traditionally been analyzed using the “instantaneous unit hydrograph,” for instance, by routing instantaneous inflow through a series of linear reservoirs which leads to a gamma distribution for the entire catchment [Nash, 1959]. An alternative theory based on Boltzmann statistics was proposed by Lienhard  who derived a particular form of the generalized gamma distribution for the unit hydrograph. Using a similar methodology, the result of Lienhard  was extended by Rinaldo and Marani  to obtain a mass response function as a generalized gamma distribution.
 With hydrological pathways distributed over entire catchment areas, the responses tend to exhibit exponential features characteristic of mixed reactors; hence, the exponential as well as the gamma distribution have been used in a number of works to represent catchments as one or a few mixed reservoirs [e.g., Haitjema, 1995; Rodhe et al., 1996; McGuire et al., 2005]. Note that the TOSS includes both the ADE model used, e.g., by Rinaldo et al.  as well as the gamma and exponential densities [e.g., McGuire et al., 2005]; it can be shown that TOSS can reasonably well approximate even the generalized gamma density by a suitable choice of the exponent in (B5).
 Taking advantage of analogous conceptualizations in chemical engineering [Nauman, 1969; Niemi, 1977], the mixed reservoir has recently been used to systematically address the issue of forward and backward travel time densities in catchment transport [Botter et al., 2010, 2011; Rinaldo et al., 2011]. In the present work, we also address the forward and backward travel time densities from a Lagrangian trajectory perspective, explicitly accounting for both spatial and temporal fluctuations that are illustrated for a generic setup (Appendix A). Using TOSS with the Lagrangian approach to address the forward and backward travel time densities, this work further bridges the gap between the pathway [e.g., Rinaldo et al., 1991; Destouni et al., 2010; Persson et al., 2011] and the classical mixed reactor approaches (of which Botter et al.  is conceptually the most advanced) of hydrological transport.
 Finally, a few notes on the water age distribution. As it has been emphasized, the backward probability density function of conditioned on the time of arrival T at a specified location x, , is the water age density. A general theory of the water age for transient flow in a continuum advection-dispersion framework was given by Ginn . Recently, Cornaton  presented an analytical solution for an abrupt change of the mean velocity using the one-dimensional formulation of Ginn . In Figure 7, our simulated is compared to the corresponding cumulative distribution function obtained from the analytical solution of Cornaton ; the analytical solution and the details of the illustrated case are given in Appendix C. The close comparison between the two results demonstrates consistency between a Lagrangian particle tracking approach and the continuum approach to computations of the water age distribution.
7. Summary and Conclusions
 If a tracer is applied over the entire catchment area, then given the wide range of pathway lengths, the tracer breakthrough at the exit appears consistent with a mixed reactor. Hence, catchments have frequently been conceptualized as internally fully mixed, in particular, when quantifying for instance transport of environmental isotopes.
 However, pathways in a catchment will typically transect different surface and subsurface hydrological units, usually with different biogeochemical properties; approximating transport as internally mixed may in such cases seriously limit the extent to which process-based descriptions of mass transfer and biogeochemistry can be applied. Furthermore, in cases where solutes (e.g., pollutants) are injected over limited areas, and where mass transfer or biogeochemical processes are of prime interest, the Lagrangian conceptualization of different hydrological pathways is most appropriate.
 The main concluding points of this work are summarized as follows:
 For spatially and temporally varying velocity that are statistically stationary, (2) holds for fully ergodic conditions; if flow is only spatially ergodic, i.e., a single realization of temporal variations is considered, then (2) holds approximately, provided that the mean WTT is sufficiently large relative to the integral scale of temporal fluctuations.
 If significant, temporal variability will affect the statistical structure of the Lagrangian velocity fluctuations; once this effect is accounted for, convolution is applicable provided that transition times between predefined segments are approximately independent.
 Macrodispersion for a single hydrological pathway can be related to dispersion in underlying independent HUs by using the derived equation (B5); this expression was shown applicable for a wide range of dispersion processes, from the ADE model to anomalous transport, using the TOSS density (B5).
 The memory function can be applied within a LaSAR framework for quantifying exchange kinetics, provided that the temporal fluctuations are statistically stationary such that (4) is applicable. In the case of abrupt velocity changes in time, or strong trends, a particle-based time domain random walk (TDRW) approach [Painter et al., 2008] could be suitable as an approximate computational approach, if developed further; work in this direction is currently in progress.
 Depending on the distribution of the temporal fluctuations, temporal moments may differ from the steady state case with identical realizations of spatially variable velocity; compared to the ADE, temporal fluctuations may emphasize more strongly early arrival (low percentiles of WTT).
 The backward WTT density obtained by particle tracking and the water age distribution for transient flow obtained using a continuum advection-dispersion approach [Ginn, 1999; Cornaton, 2012] are consistent.
 A generally important task is to accurately quantify flow partitioning and mass delivery fractions through different pathways that originate from recharge locations (source inputs) of interest [Darracq et al., 2010; Destouni et al., 2010]. Flow partitioning and mass delivery fractions indicate how much of the solute mass is potentially deliverable along pathways with significant temporal variations. For pathways that carry much of the water flow and solute mass in a typical catchment, especially through the subsurface, temporal variations may be relatively small and a steady state assumption sufficient [Destouni, 1991; Foussereau et al., 2001; Russo and Fiori, 2008]. Shallower pathways capture shorter-term fluctuations [Foussereau et al., 2001; Fiori and Russo, 2008], while the full spectrum of subsurface transport pathways may reflect longer-term trends and their changes [Darracq et al., 2008]; better understanding of transport partitioning and temporal velocity variations in the deeper pathways of this spectrum are still needed. Change in the flow and transport partitioning characteristics are particularly important when studying flow and transport under climate change [Bosson et al., 2012].
 Another aspect to explore are the internal sources that release solute along entire pathways. The solute may then be geogenic (e.g., dissolved inorganic carbon (DIC) and dissolved organic carbon (DOC) [Lyon et al., 2010]) or anthropogenic from long-term accumulation [e.g., Eriksson and Destouni, 1997; Botter et al., 2005; Darracq et al., 2008]; these results can, in principle, be incorporated into the proposed methodology, either directly using the analytical LaSAR formulation, or using a particle-based time domain random walk. [Painter et al., 2008].
Appendix A:: Generic Example
 Consider an idealized space-time velocity field of the type,
where Y and Z are log-normally distributed with zero mean, and variances and , respectively. The log-integral scales for Y and Z are IY and IZ, respectively, where the correlation structure is assumed to be a negative exponential. The mean velocity U0 is set to unity (unless specified otherwise), and will be left with arbitrary units; consequently, all temporal and spatial quantities will be left with arbitrary units. For illustration, we shall assume and in all cases, with the spatial domain being 20 IY = 1000.
 Two basic forms of are to be considered. One is a normalized water discharge record over 23 yr (8400 d) from an ∼300 km2 Swedish catchment; the monthly averaged is illustrated in the top right side of Figure A1; the normalization is such that =1 for the data-based . The distribution of is approximated as lognormal (the top left side of Figure A1). In the illustrations, a synthetic will be used (in addition to the “single realization” data-based ), for representing fully ergodic transport conditions as given by (A1); in this case and .
 By solving (1) with the forward or backward tracking boundary condition, an example of 50 trajectories over an x = 1000 scale is illustrated in Figure A2, using the data-based in the interval (0, 4000) for all 50 realizations of V(x). The backward tracking is done from . A single realization of V(x) is also illustrated in Figure A2 (the thin gray line in the bottom diagram) for which the trajectory is noted with a thick blue curved line in the main diagram. For the same trajectory, the Lagrangian velocity is shown in the bottom diagram as the blue thick line. The dashed line in the bottom side of Figure A2 is the Lagrangian velocity following only temporal variations, computed as , where is obtained with full variability. The right diagram in Figure A2 illustrates the data-based (used in the calculations of all of the illustrated trajectories), as well as a single realization of the synthetic, lognormal . It is clearly seen how the temporal variations alter the local trajectory slopes, consistent with fluctuations of the Lagrangian velocity shown in the bottom diagram (blue line).
 The mean advective WTT for a steady state flow field obtained by setting Z = 0 in (A1) (i.e., ) is where VH is the harmonic mean velocity . The mean advective WTT with the flow governed by (A1) with Y = 0 (i.e., ) is computed by , where . Finally, the mean advective WTT for a flow field defined by (A1) is well approximated by , implying and if . Note that the steady state example in Figure 3b is obtained by setting Z = 0 in (A1), hence the shift of the CDF toward later times.
Appendix B:: Dispersion Along Segmented Trajectories
 Let denote the mean and the coefficient of variation of the WTT for a segment “i”; note that this segment may be a HU or any given discretized (segmented) trajectory. The WTT is random due to space-time velocity variations.
 Let denote the pdf of with , and let denote the pdf of . Assuming that and are given, we wish to quantify dispersion for the N number of segments.
 It has been shown recently that the tempered one-sided stable (TOSS) density is a general WTT distribution for hydrological transport [Cvetkovic, 2011a]. Using the TOSS density, we write for segment “i”:
where s is the Laplace transform variable, and the parameters ai and ci are defined by,
ai being the cut-off rate, ci the scaling coefficient, and an exponent for segment “i.” Modifying the solution given, e.g., by Hughes , a convergent series expansion yields
 Assuming that are independent, the WTT density for the entire system composed of N consecutive segments is obtained by convolution,
Depending on , ai, and ci, a wide range of dispersive phenomena may be captured. Only if the exponent and ai are constant for all segments is f an infinitely divisible distribution with , i.e., c scales as a sum over all segments.
 Using (B5), we find that the coefficient of variation for WTT of the entire system is,
Equation (B5) demonstrates that macrodispersion as quantified by the WTT coefficient of variation , is a function of mean velocities and dispersion properties of individual segments but does not depend on the nature of transport as defined by the TOSS exponent .
Equation (B5) is valid if the segments are chosen such that WTT between segments can be considered approximately independent; hence, we can use (B5) as test criteria for segment independence, as summarized in Table 1 (columns two and three).
Appendix C:: Analytical Solution for Water Age Density
 We summarize here the analytical solution for water age density presented by Cornaton  and used in Figure 7.
 The governing equation for the water age distribution is based on the theory presented by Ginn ; in one-dimensional simplest form it writes,
where is the water age, or the travel time of a water particle entering the system at time t0 and exiting at t = T; hence, , as noted earlier. U in (C1) is the mean velocity and the dispersivity.
 In line with Cornaton , we consider a specific system where the mean flow velocity changes abruptly at , from to for . Boundary conditions are summarized as follows:
where , and is known, in this case assumed as an inverse Gaussian distribution, equivalent to the TOSS density (B4) (with , and ).
 The solution of the above system has been obtained in the Laplace domain by Cornaton  as,
with defined in (C5), the cumulative distribution function of the water age that is shown in Figure 7 is computed by numerical inversion of .
 For an illustration example of Figure 7, and (similar to what was used by Cornaton  for illustration). In this example, the abrupt change takes place at , for a domain of x = 1000 with a dispersivity of . In the particle tracking simulations, we have a spatially variable velocity with a log-integral scale of 50, and a log-standard deviation of 0.6; note that the first-order result yields . The overall setup is as summarized in Appendix A. The black curve in Figure 7 is the initial (steady state) inverse Gaussian distribution .
starting (entrance) location for trajectory; a specified recharge location
end (exit) location for a trajectory; a specified discharge location
arbitrary area of a catchment
forward pdf of water travel time conditioned on the starting time t0 from a specified recharge location for a hydrological pathway; corresponds to in Figure 1 of Botter et al. 
backward pdf of water travel time conditioned on the arrival time T at a specified discharge location for a hydrological pathway; this is referred to as water age pdf, i.e., how old the water is since entrance/recharge; corresponds to in Figure 1 of Botter et al. 
forward pdf of water travel time, from a to x, independent of the staring time, t0
backward pdf of water travel time from x to b, independent of the exit time, T
memory function for retention processes
unconditional tracer residence time distribution; here we use the notion “residence time” for tracer particles which in the general case includes retention in addition to advection
tracer discharge at a specified discharge location for a single hydrological pathway (“point source”)
tracer discharge at a specified discharge location for a finite source area “A”
number of segments or hydrological units
spatially dependent vector of retention parameters
Laplace transform variable
starting time for a water or tracer particle along a trajectory
time of arrival of a water or tracer particle at a specified discharge location; in some cases it is convenient to use t and T interchangeably
water travel time obtained by solution of (1) for specified boundary conditions
three-dimensional space-time velocity vector with components [ ]
x component of the spatially variable velocity vector V; thus, V and vx are equivalent
Lagrangian velocity in the x direction computed as
position vector with components ( )
equation of the trajectory starting at a; assumption of a steady state flow pattern implies that X does not change in time, i.e., is independent of t0
conditional pdf for tracer particle residence time that in general includes retention
release rate at injection location
normalized temporal variation function (see Figure A1).
 Critical comments by Andrea Rinaldo (EPFL, Switzerland) and Tim Ginn (UCD, USA) on the original version of the manuscript have significantly helped to improve this work; their effort is sincerely appreciated. The authors gratefully acknowledge the financial support by the Swedish Nuclear Fuel and Waste Management Co. (SKB), the NOVA Research and Development Center, Oskarshamn (Sweden), and the Swedish Research Council (VR, project 2009–3221).