dataesr/scientific-paragraphs-categorization huggingface dataset:
Botter, 2012). Numerical solutions are usually based on the method of characteristics (see. Benettin & Bertuzzo, 2018; Harman, 2015; Harman & Fei Xu, 2022 , for open software implementations), but in certain cases, it may be desirable to uncouple time and age numerically, depending on data requirements and simulation needs  (Rodriguez & Klaus, 2019). Solution to the age balance equation allows one to simulate, at any time, the rank storage S T and, through the SAS functions, the outflow age distributions: 𝑃𝑃𝑄𝑄(𝑇𝑇 𝑇 𝑇𝑇) = Ω𝑄𝑄 (𝑆𝑆𝑇𝑇 (𝑇𝑇 𝑇 𝑇𝑇)𝑇 𝑇𝑇) (17) 𝑝𝑝𝑄𝑄(𝑇𝑇 𝑇 𝑇𝑇) = 𝜕𝜕𝜕𝜕𝑄𝑄 𝜕𝜕𝑇𝑇 = 𝜕𝜕Ω𝑄𝑄 (𝑆𝑆𝑇𝑇 (𝑇𝑇 𝑇 𝑇𝑇)𝑇 𝑇𝑇) 𝜕𝜕𝑇𝑇 ( 18 ) Equation 18 can be used to feed the time-variant convolution integral (Equation  8 ) and compute tracer concentration in each outflow. Thus, all this theory that started from a mass balance equation ultimately leads to a problem that can be solved numerically and can be confronted with tracer data.
On SAS Functions The SAS function can be seen as a statistical summary of the transport behavior of a hydrologic system. It quantifies the release of waters of different ages from the storage to an outflow  (Rinaldo et al. , 2015). This function may have three main conceptual shapes (Figure  5 ): preferential release of relatively younger storage volumes (decreasing function), uniform release of all volumes (constant function), or preferential release of older storage volumes (increasing function). These behaviors can be modeled through probability distributions such as the beta (van der Velde et al. , 2012), power function  (Queloz, Carraro, et al. , 2015)  or gamma  (Harman, 2015)  distributions. By making the parameters of these distributions vary over time, the SAS functions can also be made time variable.
The SAS functions can be parametrized as a function of the rank storage S T ("rank" SAS functions, rSAS), or as a function of the normalized rank storage S T /S(t) ("fractional" SAS functions, fSAS). The main difference is that the function domain is the variable interval [0, S(t)] for rSAS functions and the fixed interval [0, 1] for fSAS functions. While there are practical differences in using one parameterization over the other  (Harman, 2015) , and both approaches are used in the literature, these different approaches are not in contrast with each other and they fundamentally express the same transport behaviors. The particular case of uniform storage selection implies that the water storage is uniformly (or randomly) sampled by an outflow (Benettin, van der Velde, van der Zee,  Rinaldo, & Botter, 2013).
The mixing coefficient (C M ) quantifies the fraction of input water that is "mixed" with the resident water, while the remaining fraction (1 -C M ) bypasses the storage entirely. In spite of the differences in terminology and the practical procedure to calculate the TTDs, the method is functionally and mathematically equivalent to a piece-wise linear SAS-function  (Hrachowitz et al. , 2016). Estimating SAS functions quantitatively (i. e. , not just their shape but also the actual value that they may take) in a real-world catchment is perhaps the greatest challenge in SAS approaches. SAS functions cannot be measured directly (except under special circumstances, see  Harman & Kim, 2014) , they may not be regular, and they likely change over time. Current approaches to estimate SAS functions rely on the calibration of SAS parameters to tracer data.
This faces the issues of model parameter calibration through a convolution integral, which is only partly alleviated by the use of formal calibration techniques. The use of different SAS functions may induce different model performances and water age estimates and only few studies addressed this issue explicitly  (Borriero et al. , 2022). Moreover, there is great uncertainty related to the SAS function of ET fluxes because of the lack of tracer data in ET (see Sections 4. 2 and 4. 5 for a discussion), which makes it difficult to evaluate the convolution integral for ET. This certainly influences the estimated SAS function of Q. However, one can use the water age balance to test different assumptions on the SAS functions of ET and see how/if they impact the estimated SAS function of Q (and related transit times, see  Visser et al.
, 2019). More generally, estimating the "true" SAS function of a real-world catchment may be difficult, but one can formulate competing hypotheses (e. g. , is a given SAS function more plausible than random sampling? Is it necessary to use time-variant SAS functions to adequately reproduce a given tracer time series?) and use formal model comparison methods to test them. The last 15 yr of research provide some guidance about the shapes that these functions are expected to take (see Section 4. 2).
Practical Ways to Apply the Age Balance Equation The water age balance can be applied to any hydrologic control volume as long as input/output fluxes and SAS functions are provided. We can differentiate two practical ways to construct catchment-scale transport models using this approach. They are distinguished on the basis of whether they partition the watershed into conceptual compartments (sometimes referred to as "buckets") or not (see  Benettin, Soulsby, et al. , 2017 , Figure  1 ). Approaches that do compartmentalize the catchment can be seen as an extension of conceptual hydrologic models, where a water age balance is solved for each compartment. The alternative is a more "pure" application of the SAS approach, in which the entire catchment is treated as a single control volume, and internal compartmentalization is replaced by a catchment-scale SAS function.
The use of multiple conceptual compartments provides great model flexibility. The simplest case is that of two compartments, typically interpreted as soil and groundwater storage, but many more compartments can be introduced such as canopy interception, hillslope and riparian zone compartments (see  Hrachowitz et al. , 2013). Such an approach requires knowledge of the internal fluxes from/to all compartments. As these usually cannot be measured, they need to be modeled as well and so these models need to integrate both flow and transport processes. This application also requires a suitable SAS function for each storage component, such that the number of parameters quickly increases with model complexity. Thus, a careful testing procedure is generally required to limit parameter uncertainty.
Following progress in our conceptual understanding of transport processes, we expect that storages preferentially release relatively young water to streamflow (see Section 4. 2). Various studies explicitly modeled an unsaturated root zone compartment with preferential release of younger ages, using either piece-wise linear (e. g. ,  Cain et al. , 2019; Fenicia et al. , 2010; Hrachowitz et al. , 2015 Hrachowitz et al. , , 2013;; McMillan et al. , 2012)  or beta distributions  (Hrachowitz et al. , 2021). Berkowitz and Zehe (2020)  argue that the preferential release of younger water can be highly relevant also in the groundwater environment and should therefore be reflected by the use of suitable SAS functions. Yet, many studies  (Benettin, Bailey, et al. , 2015; Rodriguez et al. , 2018) , opted for a simpler uniform sampling scheme (often termed complete or uniform mixing in those studies) in each storage component.
While physically unrealistic, this simplifying assumption provided satisfactory results in terms of tracer simulation with a much-reduced computational effort. Indeed, it can be shown  (Benettin, Kirchner, Rinaldo, & Botter, 2015)  that, in well-mixed cases or those with constant mixing coefficients, TTDs are not needed to compute tracer concentrations (they can be computed a posteriori) and computational times are drastically reduced. In the "pure" SAS application (e. g. ,  Lapides et al. , 2022; Lutz et al. , 2017; van   Harman, 2015). This implementation has the advantage that inferences about transport processes are not artifacts of the watershed compartmentalization. However, they may still contain artifacts of the chosen SAS function and its coupling to state variables. There is no general rule as to when a single control volume is better than a catchment compartmentalization and the choice usually depends on whether one wants to model water fluxes along with tracer transport or not.
A somewhat hybrid and spatially distributed approach has been proposed by  Nguyen et al. (2021) ;  Nguyen et al. (2022)  through the mHM-SAS model. In such a model, water flow and transport in the top soil layers is simulated using a grid-based model under a random sampling assumption, while transport in the subsurface component (either spatially distributed or not) is tracked through a SAS function approach. The mHM-SAS model has been tested and applied to nitrate circulation in mesoscale catchments  (Nguyen et al. , 2021 (Nguyen et al. , , 2022)).
Other Applications of the Age Balance While the theory described here addresses the water age balance equation in the general time-variant case, there are similar equations that can be used for different purposes. The water age balance equations can be solved at steady state (i. e. , with water fluxes, storages, and age distributions that are constant over time) to provide unique, long-term relationships between storage and flux age distributions  (Berghuijs & Kirchner, 2017; Bolin & Rodhe, 1973; Harman, 2015). In case the decoupling between flow and transport is not necessary, Equation 10 can be treated as a simple population balance and it takes the form of the McKendrick-Von Foerster (MKVF) equation (see  Lewis & Nir, 1978; Trucco, 1965). The MKVF equation makes use of the "loss function" λ, which is functionally equivalent to the SAS function, but it is not bound to nonlinear constraints like Equation  14.
New Approaches Based on Spatially Distributed Models and Water Age Tracking While this review does not specifically address the variety of existing spatially distributed transport models, it is worth mentioning that such models can be calibrated on tracer data and coupled to water age tracking algorithms to provide catchment-scale water age distributions. Physically based models, in particular, may help identify the physical controls on the shape of lumped functions such as TTDs and SAS functions. A simple 1-D advection-dispersion model was used by  Kirchner et al. (2001)  as a mechanism to explain fractal scaling in stream tracer concentrations and to identify the Péclet (Pe) numbers that produce TTDs similar to gamma distributions with shape parameter α < 1. 1-D advection and dispersion was similarly used by  Benettin, Rinaldo, and Botter (2013)  to relate the shape of a SAS function to the Pe number.
Other examples of age distributions and SAS functions computed from process-based models and tracer data include: stationary two-dimensional groundwater flow problems (van der Velde et al. , 2012), non-stationary 1-D and 2-D flow through unsaturated porous media  (Asadollahi et al. , 2020; Pangle et al. , 2017; Sprenger et al. , 2018) , non-stationary groundwater flow  (Kaandorp et al. , 2021) , and fully coupled catchment-scale flow and transport models  (Kim et al. , 2022; Kuppel et al. , 2018; Remondi et al. , 2018; Smith, Tetzlaff, & Soulsby, 2020; Wilusz et al. , 2020; Yang et al. , 2018). Recently,  Kim and Harman (2022)  used hydraulic groundwater theory to derive analytical expressions for the TTD and SAS functions of hillslope subsurface flow under steady recharge. These approaches usually compute TTDs through virtual flux tracking or particle tracking.
Extending these methods to multiple time lags directly yields (ensemble-averaged) TTDs, which again can be estimated for different ensembles of time steps, to map out how TTDs respond to catchment conditions and external forcing. TTD shapes are not assumed a priori, but instead estimated directly from the data. Both new water fractions and TTDs can be expressed as "backward" fractions of streamflow, or "forward" fractions of precipitation, with or without volume-weighting. The calculations are conceptually straightforward but may be somewhat tricky to implement correctly, so user-friendly scripts are available for both R and Matlab  (Kirchner & Knapp, 2020). Perhaps most importantly, the methods have been extensively benchmark tested  (Kirchner, 2019) , and demonstrated with chloride and isotope data from Plynlimon, in a proof-of-concept illustration of how F new values and TTDs vary seasonally, with flow regime, and under varying precipitation intensity  (Knapp et al.
3. 3) and using "ensemble" approaches like ensemble hydrograph separation. It is also largely mitigated by considering TTDs in flow-weighted time  (Kim & Troch, 2020). SAS functions are not affected by the external variability because they aim to describe transport processes independently of water flow rates. Streamflow age distributions typically have significant contributions from both a narrow range of young (days to months) waters and from a very large range of old (>1-2 yr) waters (see Figure  2  and  Benettin, Bailey, et al. , 2017; Kirchner et al. , 2000). This means that, for many environmental tracers like chloride and water stable isotopes, the bulk tracer signature of old water is almost constant over time, while the tracer signature of young water is variable, reflecting event and seasonal variability in the tracer inputs.
The transit time studies carried out in the past few years provide insight into the relationships between storage and outflow age distributions. The general expectation in catchments is that "Streams are generally younger than the water storage that they drain"  (Berghuijs & Allen, 2019; Berghuijs & Kirchner, 2017). This behavior originates from subsurface velocity contrasts (induced by subsurface heterogeneity) and results in SAS functions that generally decrease with age. Declining hydraulic conductivity with depth also results in the preferential release of younger water because a large portion of young water is stored above old water  (Ameli et al. , 2016; Kim & Harman, 2022). Modeling work based on tracer data has usually confirmed this expectation, at least under wet conditions  (Benettin, Kirchner, Rinaldo, & Botter, 2015; Benettin, Soulsby, et al.
The systematic release of older storage to streamflow (which corresponds to a SAS function that increases with age) appears to be unlikely in catchments, but it may occur in soils and hillslopes when flow advection is significantly larger than dispersion (high Péclet numbers, similar to piston flow). This behavior was identified in large experimental soil columns  (Asadollahi et al. , 2020; Queloz, Carraro, et al. , 2015)  and in some northern latitude soils  (Sprenger et al. , 2018) , although in this case, the low-frequency (monthly) sampling strategy could not exclude the occurrence of short-term preferential flow. The SAS functions estimated for the Landscape Evolution Observatory hillslopes also indicate that those hillslopes mainly drain the older stored water  (Kim et al. , 2022) , due to the high hillslope Péclet number and the convergent topography.
, 2019) , as also suggested by  Thaw et al. (2021)  using stable and radioactive isotope data. But plant strategies are complex and experimental work has shown that in many cases individual trees may take up relatively old water  (Allen et al. , 2019; Evaristo et al. , 2019)  stored in smaller soil pores, including, in some extreme cases, water several decades old (Z. Q. Zhang et al. , 2017). The use of large-scale water balances can provide a lower bound to distributed ET age and it showed that ET must be at least several months old across large areas of the western continental United States  (Hahm et al. , 2022). Models that attempted to determine the SAS function of ET solely based on stream tracer data typically resulted in uncertain ET age estimates  (Asadollahi et al. , 2020). The lack of tracer data in evaporation and transpiration fluxes is also the reason why SAS approaches have not yet been used to separate the transit times of evaporation from those of transpiration, although these fluxes are likely to sample different water age pools.
Overview of Existing Applications The Supporting Information (Text S1-S2 in Supporting Information S1, Tables  S1-S2 , Figures  S1-S4  in Supporting Information S1) describes over 80 applied studies that estimated water age distributions using post-2006 methods based on tracer data. As much of the theory was consolidated between 2011 and 2016, the number of applications started to grow around 2015 (Figure  S1  in Supporting Information S1). Most applications are based on SAS approaches, well/partially mixed compartments and spatially distributed models with virtual tracer tracking. In some cases, multiple approaches are combined together. Applications were usually carried out on relatively small catchments between 0. 5 and 5 km 2 (Figure  S3  in Supporting Information S1), with just a few catchments larger than 100 km 2.
Advantages and Disadvantages of the Different Approaches The approaches introduced in Section 3 differ in terms of purpose, data requirements, and types of output they can provide (Table  1 ). For simplicity, the approaches are here classified as: pre-2006, including the standard and flow-weighted lumped convolution approaches; data-based, including the young water fraction estimated from tracer cycle damping and ensemble hydrograph separation; and SAS-based approaches, making use of the water age balance coupled to a time-variant or time-invariant SAS function. The pre-2006 and the data-based approaches generally aim to infer the long-term average streamflow age distributions or some portions or statistics of this distribution. Data-based approaches can also compute these outputs for different periods or flow regimes (if enough data is available).
By design, these approaches are not meant to characterize TTDs during any particular storm event and should not be used for that purpose. SAS approaches Note. The usefulness of an approach for given data set properties is marked with "Yes/No" if there is general evidence that the approach is suitable/unsuitable, while it is marked as "Maybe" when there is no general guideline and additional factors have to be taken into consideration to choose that approach (see Section 4. 5).
The young water fraction simply requires that the TTD's shape is not piston-flow-like, which is reasonable in catchments (but not in soil columns), while the ensemble hydrograph separation requires no particular assumption. SAS approaches currently assume that the shape (either time-variant or not) of the SAS functions, rather than the shape of the streamflow TTD, can be approximated by an analytical function (though approaches for relaxing this assumption are in development  (Harman, 2019b) ). There is, however, a difference between what these approaches aim to estimate and how they can do it in practice. Data-based approaches make more parsimonious assumptions but depend more on the characteristics of the tracer data. For example, young water fraction calculations can become uncertain if the tracer input does not exhibit a clear seasonal cycle.
Modeling approaches (both pre-2006 and SAS-based) need to estimate parameters that cannot be measured and they must be calibrated on available tracer data. They also need to assume a-priori some functional form (either for the TTD or for the SAS function and its possible coupling with state variables). While pre-2006 approaches use the average system state to fit the tracer data measured under varying hydrologic conditions, SAS approaches can take at least part of the dynamic system conditions into account. All the approaches need tracer data in a catchment's input (precipitation and snowmelt) and output (at least streamflow). Methods based on convolutions or mass balances (including SAS approaches) require continuous input tracer data, meaning that any data gaps must be filled. This requirement does not apply to young water fractions or the ensemble hydrograph separation approach.
Additionally, the flow-weighted time approach requires a discharge time series, while SAS approaches require an estimate of all terms in the water balance (e. g. , precipitation, evapotranspiration, streamflow). While precipitation and streamflow are often measured in catchments, ET estimates are more data-demanding and so some hydrological modeling is required. Data-based approaches do not require estimating ET fluxes. They only need streamflow data to compute flow-weighted statistics or when focusing on different hydrologic periods/regimes. However, since the number of observations available is typically fixed, there is a trade-off between the number of distinct hydrologic regimes that can be investigated and the number of tracer observations available for each regime (which affects the uncertainty in ensemble-averaged TTD estimates).
g. , large storage variability), the pre-2006 approaches are expected to be less reliable. When high-frequency (daily or sub-daily) tracer data is available but just for short periods (single events or seasons) the long-term average TTD cannot be estimated and SAS approaches may account for the variability of water age (and its effect on the measured tracer concentration) during these short periods. One may wonder whether, moving from pre-2006 convolution approaches to SAS approaches, we simply shifted the problem from estimating the parameters of a TTD to estimating the parameters of a SAS function. While in terms of procedure (calibration on tracer data) the approaches may appear similar, the goals and especially the outputs are very different. Contrary to what is sometimes thought, SAS approaches do not need higher-resolution tracer data.
SAS models can leverage high-frequency data if available and use it to estimate the variability of age distributions, otherwise, they can simply run at lower frequency (and provide lower-frequency variability) or even at hydrologic steady state. They are fundamentally more robust because they can explicitly account for the presence of ET (rather than using an "effective precipitation") and the variability of the water storage, and they always conserve mass. SAS approaches are generally expected to fit tracer data better than standard convolution approaches not because they have more parameters but because they are structurally more suited to deal with hydrologic variability. A barrier to the use of SAS approaches may be the more involved model implementation, but open code in different programming languages already exists (see Section 3.
2. 2). Despite the differences among all these approaches, comparisons among them can be useful to test the consistency of the estimated water ages. The instantaneous age distributions estimated by SAS approaches are typically jagged and irregular, but when these are averaged out, they typically take a smooth shape that can be compared to steady state or data-based approaches. For example, the marginal TTD computed by  Benettin, Kirchner, et al. (2015)  at Plynlimon was shown to be very similar to a gamma distribution, as previously found for Plynlimon by  Kirchner et al. (2000) ;  Kirchner et al. (2001)  using spectral methods. Spectral analyses may also support model-data consistency. For example, the 1/f fractal scaling observed in many tracer time series (e. g. ,  Aubert et al. , 2014; Kirchner et al.
SAS approaches embed the recharge assumption into the water age balance and go beyond the concept of effective precipitation because they allow the ET fluxes to have a full age distribution instead of being just made of event water. Thus, both these challenges are integrated into the water age balance and are now replaced by the selection of the right SAS functions. Ensemble hydrograph separation does not require estimating ET fluxes (there is no need for a recharge assumption) and it does not require selecting a TTD, as the TTD is estimated directly from the tracer data fluctuations. Thus, these challenges have been solved or transformed through the new approaches. The model evaluation challenge (Equation  6 ) has seen significant progress. Model parameter estimation has evolved markedly through the extended use of calibration approaches (see.