I will first illustrate this approach with a simple example of a time-varying mixing model. Let us assume that we have measured tracer concentrations in streamflow, and in at least one contributing end-member, over an ensemble of time intervals j. The simplest possible mass balance for the water that makes up streamflow would be Qj=Qnewj+Qoldj, where Qnew represents the water flux in streamflow Q that originates from recent precipitation (or, potentially, any other end-member in which tracers can be measured) during time interval j. All other contributions to streamflow are lumped together as Qold. Conservative mixing implies that QjCQj=QnewjCnewj+QoldjColdj, where CQ and Cnew are the tracer concentrations in the stream and the new water, and Cold is the tracer signature of all other sources that contribute to streamflow. Combining Eqs. (4) and (5), we directly obtain CQj=FnewjCnewj+1-FnewjColdj, where Fnewj=Qnewj/Qj is the fractional contribution of Qnew to streamflow Q. Equation (6) can be rewritten as CQj-Coldj=FnewjCnewj-Coldj, which in turn could be rearranged as a conventional mixing model (Eq. 3), with the important difference that the new and old water concentrations are time-varying rather than constant. If we represent the old water composition using the streamwater concentration during the previous time step, Eq. (7) becomes CQj-CQj-1=FnewjCnewj-CQj-1.

The lagged concentration CQj-1 serves as a reference level for measuring fluctuations in precipitation and streamflow tracer concentrations and the correlations between them. Thus, it is not necessary that CQj-1 consists entirely of old water as defined in conventional hydrograph separations (i.e., groundwater or baseflow water). It is only necessary that CQj-1 contains no new water (that is, no precipitation that fell during time step j), and this condition is automatically met because CQj-1 is measured during the previous time step. The net effect of CQj-1 is to factor out the legacy effects of previous tracer inputs and to filter out long-term variations in CQ that could otherwise lead to spurious correlations with Cnew.

The ensemble hydrograph separation approach is based on the observation that Eq. (8) is almost equivalent to the conventional linear regression equation, yj=βxj+α+εj,yj=CQj-CQj-1,xj=Cnewj-CQj-1, where the intercept α and the error term εj can be viewed as subsuming any bias or random error introduced by measurement noise, evapoconcentration effects, and so forth. The analogy between Eqs. (9) and (8) suggests that it may be possible to estimate the average value of Fnewj from the regression slope of a scatterplot of the streamflow concentration CQj against the new water concentration Cnewj, both expressed relative to the lagged streamflow concentration CQj-1.

However, astute readers will notice an important difference between Eqs. (8) and (9): in Eq. (9), the regression slope β is a constant, whereas in Eq. (8) Fnewj varies from one time step to the next. It is not obvious how an estimate of the (constant) slope β will be related to the (non-constant) Fnewj or whether this relationship could be affected by the other variables in Eq. (8). The answer to this question can be derived analytically and tested using numerical experiments (see Appendix A). As explained in Appendix A, the regression slope in a scatterplot of CQj-CQj-1 versus Cnewj-CQj-1 (Fig. A1d) will closely approximate the average value of Fnewj (averaged over the ensemble of time steps j), under rather general conditions:

The slope of the relationship between Fnewj and Cnewj-CQj-1, times the mean of Cnewj-CQj-1, should be small compared to the average Fnew. This will usually be true for conservative tracers, for two reasons. First, because all streamflow is ultimately derived from new water, mass conservation implies that the mean of Cnewj-CQj-1 should usually be small. Second, unless there is a correlation between storm size and tracer concentration (not just between storm size and tracer variance), the slope of the relationship between Fnewj and Cnewj-CQj-1 should also be small. Thus the product of these two small terms should be small.

The least-squares solution of Eq. (9) can be expressed in several equivalent ways. For consistency with the analysis that will be developed in Sect. 4 below, I will use the following formulation, which is mathematically equivalent to those more commonly seen: Fnew=β^=covyj,xjvarxj, where β^ is the least-squares estimate of β, and Fnew is the average of the Fnewj over the ensemble of points j. Values of yj that lack a corresponding xj, or vice versa (due to sampling gaps, for example, or lack of precipitation), are omitted.

The uncertainty in Fnew, expressed as a standard error, can be written as SEFnew=SEβ^=sysx1-rxy2neff-2=β^neff-21rxy2-1, where sx and sy are the standard deviations of x and y, rxy is the correlation between them, and neff is the effective sample size, which can be adjusted to account for serial correlation in the residuals (Bayley and Hammersley, 1946; Brooks and Carruthers, 1953; Matalas and Langbein, 1962): neff≈nxy1+rsc1-rsc-2nxyrsc1-rscn1-rsc2-1, where nxy is the number of pairs of xj and yj, and rsc is the lag-1 serial correlation in the regression residuals yj-β^xj-α. For large nxy, Eq. (12) can be approximated as (Mitchell et al., 1966) neff≈nxy1-rsc1+rsc, where for all positive rsc, Eq. (13) is conservative (it underestimates neff from Eq. 12), and for rsc=0.5 and nxy>50, for example, Eqs. (12) and (13) differ by less than 3 %. If the scatterplot of yj=CQj-CQj-1 versus xj=Cnewj-CQj-1 contains outliers, a robust fitting technique such as iteratively reweighted least squares (IRLS) may yield more reliable estimates of Fnew than ordinary least-squares regression. However, the analyses presented here are based on outlier-free synthetic data generated from a benchmark model (see Sect. 3), so in this paper I have used conventional least squares (Eqs. 10–11) instead.

The meaning of the new water fraction Fnew depends on how the new water and streamwater are sampled. For example, if the new water concentrations Cnew are measured in daily bulk precipitation samples and the stream water concentrations CQ are measured in instantaneous grab samples taken at the end of each 24 h precipitation sampling period, then Fnew will estimate the average fraction of streamflow that is composed of precipitation from the preceding 24 h. If the sampling interval is weekly instead of daily, then Fnew will estimate the average fraction of streamflow that consists of precipitation from the preceding week. This will generally be larger than the Fnew calculated from daily sampling, for the obvious reason that on average more precipitation will have fallen during the previous week than during the previous 24 h, so this precipitation will comprise a larger fraction of streamflow. Also, if the weekly streamflow concentrations are measured in integrated composite samples rather than instantaneous grab samples, then Fnew will estimate the fraction of same-week precipitation in average weekly streamflow rather than in the instantaneous end-of-week streamflow. The general rule is: Fnew should generally estimate whatever new water has been sampled as Cnew, expressed as a fraction of whatever streamflow has been sampled as CQ.

In all of these cases, β^ from Eq. (10) estimates the average fraction of new water in streamflow during time steps with precipitation, because time steps without precipitation lack a new water tracer concentration Cnewj and thus must be left out from the regression in Eq. (9). Using Qp to denote discharge during periods with precipitation, we can represent this event new water fraction as QpFnew.

Periods without precipitation will inherently lack same-day (or same-week) precipitation in streamflow. Thus we can calculate the average fraction of new water in streamflow during all time steps, including those without precipitation, as QFnew=QpFnewnpn=β^npn, where QFnew is the new water fraction of all discharge, QpFnew is the new water fraction of discharge during time steps with precipitation (as estimated by the regression slope β^, from Eq. 10), and np/n is the fraction of time steps that have precipitation. The ratio np/n in Eq. (14) accounts for the fact that during time steps without rain, the new water contribution to streamflow is inherently zero. The same ratio is also used to estimate the uncertainty in QFnew: SEQFnew=npnSEβ^=QFnewneff-21rxy2-1.

The regression derived through Eqs. (4)–(9) gives each time interval j equal weight. As a result, β^ from Eq. (10) can be interpreted as estimating the time-weighted average new water fraction. Alternatively, one can estimate the volume-weighted new water fraction, β^*=∑j∈xyQjyj-y‾xy*xj-x¯xy*∑j∈xyQjxj-x‾xy*2, where x‾xy* and y¯xy* are the volume-weighted means of x and y (averaged over all j for which xj and yj are not missing), x‾xy*=∑j∈xyQjxj∑j∈xyQj,y‾xy*=∑j∈xyQjyj∑j∈xyQj, and the notation j∈xy indicates sums taken over all j for which xj and yj are not missing. Equations (16)–(17) yield the slope coefficient for linear regressions like Eq. (9), but with each point weighted by the discharge Qj. We can denote the weighted regression slope β^* as QpFnew*, the volume-weighted new water fraction of time intervals with precipitation, where the asterisk indicates volume-weighting.

If, instead, one wants to estimate the new water fraction in all discharge (during periods with and without precipitation), following the approach in Sect. 2.4 one simply rescales this regression slope by the sum of discharge during time steps with precipitation, divided by total discharge: QFnew*=QpFnew*Q‾pQ‾npn=β^*Q‾pQ‾npn, where QFnew* is the volume-weighted new water fraction of all discharge, QpFnew* is the fitted regression slope β^ from Eq. (16), Q‾p is the average discharge for time steps with precipitation, Q‾ is the average discharge for all time steps (including during rainless periods), and np/n is the fraction of time steps with rain.

Because the volume-weighting will typically be uneven, the effective sample size will typically be smaller than n; for example, in the extreme case that one sample had nearly all the weight and the other samples had nearly none, the effective sample size would be roughly 1 instead of nxy. Thus, uncertainty estimates for these volume-weighted new water fractions should take account of the unevenness of the weighting. One can account for uneven weighting by calculating the effective sample size, following Kish (1995), as neff=∑Qj(xy)2∑Qj(xy)2, where the notation Qj(xy) indicates discharge at time steps j for which pairs of xj and yj exist. Equation (19) evaluates to nxy (as it should) in the case of evenly weighted samples and declines toward 1 (as it should) if a single sample has much greater weight than the others. To obtain an estimate of the effective sample size that accounts for both serial correlation and uneven weighting, one can multiply the expressions in Eqs. (19) and (12) or (13). Combining these approaches, one can estimate the standard error of QFnew* as SEQFnew*=∑Qp∑QSEβ^*=QF*newneff-21rxy2-1,neff=∑Qjxy2∑Qjxy21-rsc1+rsc, where β^* is the fitted regression slope from Eq. (16).

One can also express the flux of new water as a fraction of precipitation rather than discharge. Recently, von Freyberg et al. (2018) have noted, in the context of conventional hydrograph separation, that expressing event water as a proportion of precipitation rather than discharge may lead to different insights into catchment storm response. Analogously, within the ensemble hydrograph separation framework we can estimate the new water fraction of precipitation, denoted PFnew, as PFnew=QpFnewQ‾pP‾p, where QpFnew is the new water fraction of discharge during time steps with precipitation (as estimated by the regression slope β^, from Eq. 10), and Q‾p and P‾p are the average discharge and precipitation during these time steps. An alternative strategy is to recast Eq. (8) by multiplying both sides by Qj/Pj, such that the Fnew on the right-hand side now expresses new water as a fraction of precipitation, QjPjCQj-CQj-1=QjPjFnewjCnewj-CQj-1=PFnewjCnewj-CQj-1.

This yields a linear regression similar to Eq. (9), but with yj rescaled, yj=βxj+α+εj,yj=QjPjCQj-CQj-1,xj=Cnewj-CQj-1, where the regression slope β^, which can be calculated from Eq. (10) with the new values yj, should approximate the average new water fraction of precipitation PFnew.

The approaches represented by Eqs. (21) and (22)–(23) are not equivalent. Equation (21) is based on the ad hoc assumption – which is verified by the benchmark tests in Sect. 3.3–3.5 – that the average of PFnewj (new water in streamflow, as a fraction of precipitation) should approximate the average Fnewj (new water in streamflow, as a fraction of discharge), rescaled by the ratio of average discharge Qpj to average precipitation Ppj. This is only an approximation, of course; it relies on the approximation that appears in the middle of the following chain of expressions: PFnew=PFnewp=FnewjQjPjp≈FnewpQjpPjp=QpFnewQ‾pP‾p, where the “p” subscripts on the angled brackets indicate averages taken only over time intervals with precipitation. Whether this is a good approximation will depend on how Pj, Qj, and Fnewj are distributed, and how they are correlated with one another. By contrast, the approach outlined in Eqs. (22)–(23) is based on the exact substitution of FnewjQj/Pj for PFnewj, which requires no approximations. The same substitution also leads to two other algebraically equivalent formulations of Eq. (22), CQj-CQj-1=PFnewjPjQjCnewj-CQj-1 and QjCQj-CQj-1=PFnewjPjCnewj-CQj-1.

But although Eqs. (22), (25), and (26) are algebraically equivalent, their statistical behavior is different when they are used as regression equations to estimate the average value of PFnew. The regression estimate of PFnew depends on the distributions of Pj, Qj, and Fnewj and their correlations with each other, and benchmark testing shows that Eq. (22) yields reasonably accurate estimates of PFnew, but Eqs. (25) and (26) do not. One can also note that the approach outlined in Eq. (21) – the other approach that is successful in benchmark tests – represents an ad hoc time averaging of Pj and Qj in Eq. (22), because it is formally equivalent to Q‾pP‾pCQj-CQj-1=PFnewjCnewj-CQj-1.

The precise interpretation of PFnew depends on how streamflow is sampled. If the streamflow tracer concentrations come from integrated composite samples over each day or week, then PFnew can be interpreted as the fraction of precipitation that becomes same-day or same-week streamflow. If the streamflow tracer concentrations instead come from instantaneous grab samples (as is more typical), then PFnew can be interpreted as the rate of new water discharge at that time (typically the end of the precipitation sampling interval), as a fraction of the average rate of precipitation. Adapting terminology from the literature of transit time distributions (TTDs), we can call PFnew the “forward” new water fraction because it represents the fraction of precipitation that will exit as streamflow soon (during the same time step), and call QpFnew and QFnew “backward” new water fractions because they represent the fraction of streamflow that entered the catchment a short time ago. Although the backward new water fraction of discharge comes in two forms (QpFnew or QFnew), depending on whether one includes or excludes rainless periods, the forward new water fraction PFnew can only be defined for time steps with precipitation (otherwise PFnew represents the ratio between zero new water and zero precipitation and thus is undefined).

Readers should keep in mind that although PFnew represents the fraction of precipitation that becomes same-day (or same-week) streamflow, different fractions of precipitation may leave the catchment the same day (or week) by other pathways, most notably by evapotranspiration. One could also estimate PFnew for water that leaves the catchment by evapotranspiration if one had tracer time series for evapotranspiration fluxes, but at present such time series are not available. Thus, to echo the principle outlined in Sect. 2.3 above, the new water fraction of precipitation does not represent the forward new water fraction for all possible pathways, but only whatever pathway has been sampled.

The new water fraction of precipitation as estimated by Eq. (21) is a time-weighted average, in which each day with precipitation counts equally. One may also want to estimate the volume-weighted new water fraction of precipitation, which we can denote as PFnew*, in keeping with the naming conventions used above. We can estimate PFnew* at least two different ways. The first method involves recognizing that we are seeking the ratio between the total volume of new water – that is, same-day precipitation reaching streamflow – and the total volume of precipitation. This will equal the volume-weighted new water fraction of discharge (total new water divided by total discharge, which has already been derived in Sect. 2.5 above), rescaled by the ratio of total discharge to total precipitation: PFnew*=QFnew*Q‾P‾=QpFnew*Q‾pP‾npn, where Q‾ and P‾ are the average rates of discharge and precipitation (averaged over all time steps), Q‾p is the average discharge on days with rain, and np/n is the fraction of time steps with rain. An alternative strategy, which yields nearly equivalent results in benchmark tests, precipitation-weights the regression for PFnew (Eq. 22), yielding β^*=∑j∈xyPjyj-y‾xy*xj-x‾xy*∑j∈xyPjxj-x‾xy*2,yj=QjPjCQj-CQj-1,xj=Cnewj-CQj-1, where x‾xy* and y‾xy* are the precipitation-weighted means of x and y (averaged over all j for which xj and yj are not missing), x‾xy*=∑j∈xyPjxj∑j∈xyPj,y‾xy*=∑j∈xyPjyj∑j∈xyPj, and where the regression slope β^* approximates the precipitation-weighted average forward new water fraction PFnew*.

To test the methods outlined in Sect. 2 above, I use synthetic data generated by a simple two-box lumped-parameter catchment model. This model is documented in greater detail in Kirchner (2016a) and will be described only briefly here. As shown in Fig. 1a, drainage L from the upper box is a power function of the storage Su within the box; a fraction η of this drainage flows directly to streamflow, and the complementary fraction 1-η recharges the lower box, which drains to streamflow at a rate Ql that is a power function of its storage Sl. The model's behavior is determined by five parameters: the equilibrium storage levels Su,ref and Sl,ref in the upper and lower boxes, their drainage exponents bu and bl, and the drainage partitioning coefficient η. For simplicity, evapotranspiration is not explicitly simulated; instead, the precipitation inputs can be considered to be effective precipitation, net of evapotranspiration losses. Discharge from both boxes is assumed to be non-age selective, meaning that discharge is taken proportionally from each part of the age distribution. Tracer concentrations and mean ages are tracked under the assumption that the boxes are each well mixed but also distinct from one another, so their tracer concentrations and water ages will differ. Water ages and tracer concentrations are also tracked in daily age bins up to an age of 70 days, and mean water ages are tracked in both the upper and lower boxes.

The model operates at a daily time step, with the storage evolution of the lower box calculated by a weighted combination of the partly implicit trapezoidal method (for greater accuracy) and the fully implicit backward Euler method (for guaranteed stability). Unlike in Kirchner (2016a), here the storage evolution of the upper box is calculated by forward Euler integration at 50 sub-daily time steps of 0.02 days (roughly 30 min) each. At this time step, forward Euler integration is stable across the entire parameter ranges used in this paper and is more accurate than daily time steps of trapezoidal or backward Euler integration (which are still adequate for the lower box, where storage volumes change more slowly). Following Kirchner (2016a), the model is driven with three different real-world daily rainfall time series, representing a range of climatic regimes: a humid maritime climate with frequent rainfall and moderate seasonality (Plynlimon, Wales; Köppen climate zone Cfb), a Mediterranean climate marked by wet winters and very dry summers (Smith River, California, USA; Köppen climate zone Csb), and a humid temperate climate with very little seasonal variation in average rainfall (Broad River, Georgia, USA; Köppen climate zone Cfa). Synthetic daily precipitation tracer (deuterium) concentrations are generated randomly from a normal distribution with a standard deviation of 20 ‰ and a lag-1 serial correlation of 0.5, superimposed on a seasonal cycle with an amplitude of 10 ‰. The model is initialized at the equilibrium storage levels Su,ref and Sl,ref, with age distributions and tracer concentrations corresponding to steady-state equilibrium values at the mean input fluxes of water and tracer. The model is then run for a 1-year spin-up period; the results reported here are from 5-year simulations following this spin-up period.

For the simulations shown here, the drainage exponents bu and bl are randomly chosen from uniform distributions of logarithms spanning the range of 1–20, and the partitioning coefficient η is randomly chosen from a uniform distribution ranging from 0.1 to 0.9. The reference storage levels Su,ref and Sl,ref are randomly chosen from a uniform distribution of logarithms spanning the ranges of 50–200 mm and 200–2000 mm, respectively. These parameter distributions encompass a wide range of possible behaviors, including both strong and damped response to rainfall inputs.

I illustrate the behavior of the model using two particular parameter sets, one that gives damped response to precipitation (Su,ref=100 mm, Sl,ref=1000 mm, bu=10, bl=3, and η=0.3) and one that gives a more rapid response (the same parameters, except η=0.8). These parameter values are not preferable to others in any particular way; they simply generate strongly contrasting streamflow and tracer responses that look plausible as examples of small catchment behavior. They can be interpreted as the behavior of two contrasting model catchments, which for simplicity (but with some linguistic imprecision) I will call the “damped catchment” and the “flashy catchment”, as shorthand for “model catchment with parameters giving more damped response” and “model catchment with parameters giving more flashy response”.

The model also simulates the sampling process and its associated errors. I assume that tracer concentrations cannot be measured when precipitation rates are below a threshold of Pthreshold=1 mm day-1, such that tracer samples below this threshold will be missing. I further assume that 5 % of all other precipitation tracer measurements, and 5 % of all streamflow tracer measurements, will be lost at random times due to sampling or analysis failures. I have also added Gaussian random errors (with a standard deviation of 1 ‰) to all tracer measurements.

Panels b–i of Fig. 1 show 2 years of simulated daily behavior driven by the Smith River daily precipitation record applied to the damped and flashy catchment parameter sets. The simulated stream discharge responds promptly to rainfall inputs, and unsurprisingly the discharge response is larger in the flashy catchment (Fig. 1b, c). The streamflow isotopic response is strongly damped in both catchments, with isotope ratios between events returning to a relatively stable baseline value composed mostly of discharge from the lower box (Fig. 1d, e). Like the stream discharge and the isotope tracer time series, the instantaneous new water fractions (determined by age tracking within the model) also exhibit complex nonstationary dynamics (Fig. 1f–i). Despite the complexity of the modeled time-series behavior, ensemble hydrograph separation (Eqs. 14, 18, 21, and 28) accurately predicts the averages of these new water fractions, both unweighted and time-weighted, as can be seen by comparing the dashed and solid lines (which sometimes overlap) in Fig. 1f–i.

It should be emphasized that the ensemble hydrograph separation and the benchmark model are completely independent of one another. The ensemble hydrograph separation does not know (or assume) anything about the internal workings of the benchmark model; it knows only the input and output water fluxes and their isotope signatures. This is crucial for it to work in the real world, where any particular assumptions about the processes driving runoff could potentially be violated. Likewise, the benchmark model is not designed to conform to the assumptions underlying the ensemble hydrograph separation method. It would be relatively trivial to model a tracer time series assuming that new water constituted a fixed fraction of discharge, and then demonstrate that this fraction can be retrieved from the tracer behavior. What Fig. 1 demonstrates is much less obvious, and more important: that even when the new water fraction is highly dynamic and nonstationary, an appropriate analysis of tracer behavior can accurately estimate its mean.

This result holds not just for the two parameter sets shown in Fig. 1, but throughout the parameter ranges that are tested in the benchmark model. The scatterplots shown in Fig. 2 show new water fractions estimated by ensemble hydrograph separation, compared to the true average new water fractions determined by age tracking in the benchmark model, for 1000 random parameter sets spanning the parameter ranges described in Sect. 3.1. Figure 2 shows that ensemble hydrograph separation yields reasonably accurate estimates of average event new water fractions (Fig. 2a, b), new water fractions of discharge (Fig. 2c) and precipitation (Fig. 2d), and volume-weighted new water fractions (Fig. 2e, f). Estimates derived from single years of data (Fig. 2b) understandably exhibit greater scatter than those derived from 5 years of data (Fig. 2a), but in all of the plots shown in Fig. 2 there is no evidence of significant bias (the data clouds cluster around the 1:1 lines). The scatter of the points around the 1:1 line generally agrees with the standard errors estimated from Eqs. (11), (15), and (20), suggesting that these uncertainty estimates are also reliable.

Mean transit times have often been estimated in the catchment hydrology literature, often under the assumption that they should also be correlated with other timescales of catchment transport and mixing as well. This naturally leads to the question, in the context of the present study, of whether there is a systematic relationship between mean transit times and new water fractions, such that they could potentially be predicted from one another. The benchmark model allows a direct test of this conjecture, because it tracks mean water ages as well as new water fractions. Figure 3a shows that, across the 1000 random parameter sets from Fig. 2, the relationship between new water fractions and mean transit times is a nearly perfect shotgun blast: mean transit times vary from about 40 to 400 days and new water fractions vary from nearly zero to nearly 0.1, with almost no correlation between them. Both of these quantities are estimated from age tracking in the benchmark model, so their lack of any systematic relationship does not arise from difficulties in estimating either of them from tracer data. It instead arises because the upper tails of transit time distributions (reflecting the amounts of streamflow with very old ages) exert strong influence on mean transit times, but have no effect on new water fractions (reflecting same-day streamflow).

I have recently proposed the “young water fraction”, the fraction of streamflow younger than about 2.3 months, as a more robust metric of water age than the mean transit time (Kirchner, 2016b). Figure 3b shows that, like the mean transit time, the young water fraction is also a poor predictor of the new water fraction, beyond the obvious constraint that new water (≤1 day old) must be a small fraction of young water (≤69 days old). The new water fraction will only be correlated with the young water fraction or mean transit time if the shape of the underlying transit time distribution is held constant, which is not the case for the 1000 random parameter sets considered here and is not likely to be true in real-world catchments either.

Many long-term water isotope time series have been sampled at weekly intervals. Can new water fractions be estimated reliably from such sparsely sampled records? To find out, I aggregated the benchmark model's daily time series to weekly intervals, volume-weighting the isotopic composition of precipitation to simulate the effects of weekly bulk precipitation sampling, and subsampling streamflow isotopes every seventh day to simulate weekly grab sampling. I then performed ensemble hydrograph separation on the aggregated weekly data, using the methods presented in Sect. 2.

Figure 4 shows the behavior of the benchmark model at weekly resolution for both the damped and flashy catchments. At the weekly timescale, the benchmark model exhibits complex nonstationary dynamics in discharge (panels a, b), water isotopes (panels c, d), and new water fractions (panels e, h). Nonetheless – and even though the weekly sampling timescale is much longer than the timescales of hydrologic response in the system – ensemble hydrograph separation yields reasonable estimates for the mean new water fractions of both precipitation and discharge (both unweighted and flow-weighted), as one can see by comparing the dashed and solid lines in Fig. 4e–h.

A comparison of Figs. 1 and 4 shows that the isotopic signature of precipitation is less variable among the weekly samples than among the daily samples, reflecting the fact that the weekly bulk samples of precipitation will inherently average over the sub-weekly variability in daily rainfall. By contrast, the weekly grab samples of streamflow lose all information about what is happening on shorter timescales. The new water fractions calculated from the weekly data are distinctly higher than those calculated from the daily data, owing to the fact that the definition of new water depends on the sampling frequency: the proportion of water ≤7 days old (new under weekly sampling) can never be less than the proportion ≤1 day old (new under daily sampling).

Figure 5 shows scatterplots comparing new water fractions estimated by ensemble hydrograph separation and those determined by age tracking in the benchmark model, analogous to Fig. 2 but for weekly instead of daily sampling. The weekly new water fractions are larger than the daily ones, for the reasons described above, and exhibit more scatter because they are based on fewer data points than their daily counterparts are. A small overestimation bias is visually evident in Fig. 2d and an even smaller underestimation bias is evident in Fig. 2c. These reservations notwithstanding, Fig. 5 shows that ensemble hydrograph separation can reliably predict new water fractions of both discharge and precipitation, with and without volume-weighting, based on weekly tracer samples.

Ensemble hydrograph separation does not require continuous data as input, so it can be used to estimate Fnew values for (potentially discontinuous) subsets of a time series that reflect conditions of particular interest. For example, if we split the time series shown in Fig. 1 into several discharge ranges, we can see that at higher flows, tracer fluctuations in the stream are more strongly correlated with tracer fluctuations in precipitation (Fig. 6a, b). Each of the regression slopes in Fig. 6a, b defines the event new water fraction QpFnew for the corresponding discharge range. Repeating this analysis for each 10 % interval of the discharge distribution (0th–10th percentile, 10th–20th percentile, etc.), plus the 95th–100th percentile, yields the profiles of QpFnew as functions of discharge, as shown by the blue dots in Fig. 6c–h. The green squares show the corresponding forward new water fractions PFnew for comparison. The light blue and light green lines show the corresponding true new water fractions determined by age tracking in the benchmark model.

If, instead, we split the time series shown in Fig. 1 into subsets reflecting ranges of precipitation rates rather than discharge, we obtain Fig. 7. Figure 7 is a counterpart to Fig. 6, but with QpFnew and PFnew plotted as functions of rainfall rates rather than discharge. The two figures exhibit broadly similar behavior. Unsurprisingly, new water fractions are higher at higher discharges and rainfall rates, because under these conditions a higher fraction of discharge comes from the upper box, which has younger water. Forward new water fractions are typically smaller than event new water fractions, because during storms the rainfall rate is higher than the streamflow rate, so the ratio between same-day streamflow and the total rainfall rate (PFnew) will necessarily be smaller than the ratio between same-day streamflow and the total streamflow rate (QpFnew). Exceptions to this rule arise when rainfall rates are lower than discharge rates, such as during periods of light rainfall while streamflow is still undergoing recession from previous heavy rain. Thus the green and blue curves cross over one another at the left-hand edges of Fig. 7c–h, whereas in Fig. 6c–h they do not.

Three conclusions can be drawn from Figs. 6 and 7. First, in these model catchments, new water fractions vary dramatically between low flows and high flows, and between low and high precipitation rates, with the event new water fraction QpFnew and the forward new water fraction PFnew diverging from one another more at higher flows and higher rainfall forcing. Second, different catchment parameters (different columns in Fig. 6) and different precipitation forcings (different rows in Fig. 6) yield different patterns in the relationships between the new water fractions QpFnew and PFnew on the one hand and precipitation and discharge on the other. And third, these patterns are accurately quantified by ensemble hydrograph separation, which matches the age-tracking results (shown by the solid lines) within the estimated standard errors in most cases.

Thus the patterns describing how new water fractions change with precipitation and discharge may be useful as signatures of catchment transport behavior and can be estimated directly from tracer time series using ensemble hydrograph separation. These observations raise the question of whether any of these signatures of behavior, as inferred from the patterns in these plots (if not the individual numerical values), might imply something useful about the characteristics of the catchments themselves, ideally in a way that is not substantially confounded by precipitation climatology. A comprehensive answer is not possible within the scope of this paper, since it focuses mostly on just two parameter sets and three precipitation records. But as a first approach, one can try superimposing the results in Figs. 6 and 7 on consistent axes (note that the axes in these figures' various panels differ from one another in order to show the full range of behavior). Doing so yields Fig. 8, which overlays the age-tracking results from Figs. 6c–h and 7c–h in its left- and right-hand panels, respectively. In Fig. 8, catchments with the damped and flashy parameter sets are denoted by green and blue curves, respectively, with different levels of brightness corresponding to the three different precipitation climatologies. The key question is: are there patterns in QpFnew or PFnew that clearly distinguish the flashy catchment from the damped catchment, regardless of the precipitation forcing? Figure 8a shows an example where this is not the case; instead, the two catchments' behaviors largely overlap in a tangle of blue and green lines. In the other three panels, however (and particularly for the trends in PFnew as a function of precipitation rates, as shown in Fig. 8d), the blue and green curves are relatively distinct from one another, but the different climatologies largely overlap for each catchment. This result suggests that these traces may be useful as diagnostic signatures of catchment characteristics, which are relatively insensitive to precipitation climatology. However, Fig. 8 can only be considered a preliminary indication of what might be possible, rather than a definitive demonstration.

The behavior summarized in Figs. 6–8 shows that, in general, new water fractions are functions of both catchment characteristics and precipitation climatology. Moreover, new water fractions will depend on the sequence of precipitation events, not just on their frequency distribution, because they will depend on antecedent wetness. Thus although the ensemble hydrograph separation approach does not require continuous data, and thus can be applied to time series with data gaps, any inferred new water fractions will obviously represent only the particular time intervals that are included in the analysis.

One implication of the forgoing considerations is that seasonal differences in storm size and frequency should also be reflected in seasonal variations in new water fractions. Figure 9a shows a scatterplot of tracer fluctuations in streamflow and precipitation, color-coded by season, for the flashy catchment simulation shown in Fig. 1. The regression lines (whose slopes define the event new water fractions QpFnew for the corresponding seasons) show that tracer concentrations in streamflow and precipitation are more tightly coupled in winter and spring than in summer and autumn. Panels b–d of Fig. 9 demonstrate large variations in the event new water fraction QpFnew, the new water fraction of discharge QFnew, and the forward new water fraction of precipitation PFnew from month to month, with a broad seasonal trend towards larger new water fractions in winter and spring. The month-to-month variations in the age-tracking results (the smooth curves) are usually quantified by the ensemble hydrograph separation estimates (the solid dots) within their calculated uncertainties (as shown by the error bars). Thus Fig. 9 suggests that ensemble hydrograph separation can be used to quantify how catchment transport behavior is shaped by seasonal patterns in precipitation forcing.

Any analysis based on water isotopes must deal with the potential effects of isotopic fractionation due to evaporation (e.g., Laudon et al., 2002; Taylor et al., 2002; Sprenger et al., 2017; Benettin et al., 2018). A detailed treatment of evaporative fractionation would necessarily be site-specific and thus beyond the scope of this paper. Nonetheless, it is possible to make a simple first estimate of how much evaporative fractionation could affect new water fractions estimated from ensemble hydrograph separation. The benchmark model does not explicitly simulate evapotranspiration and its effects on the catchment mass balance, but the issue to be addressed here is different: how much could evaporative fractionation alter the isotope values measured in streamflow, and how could this affect the resulting estimates of new water fractions?

To explore this question, I first adjusted the isotope values of infiltration entering the model in Fig. 1 to mimic the effects of seasonally varying evaporative fractionation. I assumed that evaporative fractionation was a sinusoidal function of the time of year, ranging from zero in midwinter to 20 ‰ in midsummer. Thus I assumed that evaporative fractionation effectively doubled the seasonal isotopic cycle in the water entering the model catchment (but not in the sampled rainfall itself, since any fractionation that occurs before the rainfall is sampled will not distort the ensemble hydrograph separation). I then calculated new water fractions based on the time series of sampled precipitation tracer concentrations and of streamflow tracer concentrations (altered by the lagged and mixed effects of evaporative fractionation), and compared these to the true new water fractions calculated by age tracking within the model.

The results are shown in Fig. 10, which compares 1000 Monte Carlo trials with evaporative fractionation (the blue dots) and another 1000 Monte Carlo trials without evaporative fractionation (the gray dots). One can see that, in these simulations, evaporative fractionation leads to a slight tendency to underestimate new water fractions. Nonetheless, the blue and gray dots largely overlap, and both generally follow the 1:1 lines. These results are reassuring, because the modeled fractionation effects were designed to be a worst-case scenario, in the following sense. Because ensemble hydrograph separation is based on patterns of fluctuations in precipitation and streamflow tracers, any fractionation process that created a constant offset between inputs and outputs would introduce no bias. For the same reason, any fractionation process that was uncorrelated to the input isotopic signature would also introduce no bias; thus, for example, the modeled seasonal fractionation cycle would have had no effect if there were no seasonal pattern in the precipitation isotopes themselves. But because the seasonal fractionation cycle is correlated with the seasonal pattern in the precipitation isotopes, it can potentially bias the resulting estimates of new water fractions. The fact that these biases are small, as shown in Fig. 10, suggests that ensemble hydrograph separation should yield realistic estimates of new water fractions, even with substantial confounding by evaporative fractionation.

I assume that catchment inputs and outputs are sampled at the same fixed time interval Δt and define the time that a parcel of water enters the catchment (via rainfall, snowmelt, etc.) as ti and the time that it exits via streamflow as tj. The lag interval between precipitation and streamflow is indexed as k=j-i. Pi is the rate that precipitation or snowmelt (net of evaporative losses) enters the catchment at time ti, and Qj is the rate of discharge that exits the catchment at time tj. CPi and CQj are the tracer concentrations in precipitation and streamflow, respectively. The water flux that enters as precipitation at time ti and leaves as streamflow k time steps later (at time tj=ti+k) is represented as qj+k. The sum of qjk over all lag times k (corresponding to all previous entry times i=j-k) is the total discharge Qj. Each of the qjk will be a fraction of the total precipitation falling at time ti=j-k and a (typically different) fraction of the total discharge at time tj. The fraction of discharge exiting at time tj that entered k time steps earlier is qjk/Qj, and the distribution of qjk/Qj over lag time k yields the transit time distribution conditioned on the exit time tj (also called the “backward” transit time distribution). The fraction of precipitation entering at time ti that subsequently leaves as streamflow k time steps later is qjk/Pj-k=qi+k,k/Pi, and the distribution of qi+k,k/Pi over lag time k yields the transit time distribution conditioned on the entry time ti (also called the “forward” transit time distribution).

In practice, precipitation fluxes are typically measured as averages over discrete time intervals, and tracer concentrations in precipitation are likewise volume-averaged over discrete intervals (such as a day or a week) during which the sample accumulates in the precipitation collector. By contrast, discharge fluxes are typically measured instantaneously, and discharge tracer concentrations are typically measured in instantaneous grab samples. In most of what follows, I will assume that Pi and CPi are averages over the interval ti-1<t≤ti, and Qj and CQj are instantaneous values at t=tj. However, in a few catchment studies, discharge concentrations have instead been measured in time-integrated samples. The analysis presented below is the same, whether the discharge tracer concentrations CQj are instantaneous at t=tj or are integrated over each time interval tj-1<t≤tj. The interpretation is slightly different, however, because the average lag time corresponding to a given lag interval k will depend on how precipitation and streamflow are sampled. Usually, streamwater samples are collected more or less instantaneously (grab sampling), and precipitation samples are integrated over the time interval that the sampler is open. A typical daily sampling scheme, for example, might involve collecting a precipitation sample at noon (which integrates precipitation that fell over the previous 24 h) and also collecting a grab sample of streamflow at noon. In this case, the average lag time between a raindrop falling as precipitation and being sampled in the same day's streamflow (i.e., k=0) would be 12 h, assuming that, on average, the probability of rainfall is independent of the time of day. Thus in this conventional sampling scheme, the average lag time will be (k+0.5)Δt, where Δt is the sampling interval. If, instead, the stream samples were daily composites, then (for example) the same-day raindrops appearing in the first hour's subsample of streamflow would have an average lag time of 30 min, the second hour's would be 60 min, and so forth, and therefore the daily average lag time would be 6 h. Thus if stream samples are time-integrated composites, the average lag time will be (k+0.25)Δt.

I now outline the fundamentals of the ensemble hydrograph separation approach to estimating transit time distributions. Conservation of water mass requires that the discharge at time step j equals the contributions from all lag times k (corresponding to all previous entry times i=j-k): Qj=∑k≥0qjk.

Because tracing contributions to streamflow from all previous time steps would be impractical, it will be necessary to truncate the summation in Eq. (31) at some maximum lag, which I will denote as m, and to combine the unmeasured older contributions in a water flux Qolderj: Qj=∑k=0mqjk+Qolderj,Qolderj=∑k=m+1∞qjk=Qj-∑k=0mqjk. Conservation of tracer mass requires that the tracer fluxes add up similarly, again with a catch-all flux QolderjColderj: QjCQj=∑k=0mqjkCPj-k+QolderjColderj=∑k=0mqjkCPj-k+Qj-∑k=0mqjkColderj.

Dividing Eq. (33) by Qj and rearranging terms directly yields CQj-Colderj=∑k=0mqjkQjCPj-k-Colderj, which readers will recognize as the multi-lag counterpart of Eq. (7).

Analogous to the approach in Sect. 2, here I account for the concentration of older inputs Colderj using the streamflow concentration at lag m+1, just beyond the longest lag m, with the goal of filtering out long-term patterns that could otherwise distort the correlations between CPj-k and CQj. Thus CQj-m-1 serves as a reference level for measuring fluctuations in precipitation and streamflow tracer concentrations, analogous to CQj-1 in Eq. (8). Adding a bias term α and an error term εj yields CQj-CQj-m-1=∑k=0mqjkQjCPj-k-CQj-m-1+α+εj, which almost looks like a conventional multiple linear regression equation, yj=∑k=0mβkxjk+α+εj, where yj=CQj-CQj-m-1andxjk=CPj-k-CQj-m-1, with the difference that the coefficients βk in Eq. (36) are constant over all exit times j and differ only as a function of the lag time k, whereas the qjk/Qj terms in Eq. (35) can differ among both lag times k and exit times j. Nonetheless, by analogy with the mathematical arguments in Appendix A and those at the end of Appendix B, one can expect that βk will closely approximate the average of the time-varying contributions qjk/Qj to streamflow over the ensemble of exit times j (please note that this is not the same as assuming that the transit time distribution is time-invariant). Substituting βk as an ensemble estimate of qjk/Qj, one obtains the ensemble hydrograph separation equation for estimating transit time distributions, CQj-CQj-m-1=∑k=0mβkCPj-k-CQj-m-1+α+εj.

When appropriately rescaled as described in Sect. 4.5–4.7 below, the coefficients βk in Eq. (38) – or more precisely, their regression estimates β^k – can be used to estimate the time-averaged (also sometimes called “marginal”) transit time distribution.

Using Y to represent the vector of reference-corrected streamflow tracer concentrations yj=CQj-CQj-m-1 and X to represent the matrix of reference-corrected input tracer concentrations xjk=CPj-k-CQj-m-1, we can rewrite Eq. (38) in the array form of a multiple regression equation: Y=∑k=0mβkXk+α+ε, where Xk is the kth column vector of X, and ε is the vector of the errors εj. The least-squares solution for multiple regressions like Eq. (39) can be expressed in matrix form as β^0β^1β^2⋮β^m=covX0,X0covX0,X1covX0,X2⋯covX0,XmcovX1,X0covX1,X1covX1,X2⋯covX1,XmcovX2,X0covX2,X1covX2,X2⋯covX2,Xm⋮⋮⋮covXm,X0covXm,X1covXm,X2⋯covXm,Xm-1covX0,YcovX1,YcovX2,Y⋮covXm,Y, where the regression coefficients β^k are the least-squares estimators of the true (but unknowable) coefficients βk. Equation (40) is the multidimensional counterpart to Eq. (10). The first term on the right-hand side of Eq. (40) is the inverse of the matrix of the covariances of the Xk at each lag with each other lag, and the second term is a vector of the covariances between Y and the Xk at each lag. Equation (40) is equivalent to the more widely known “normal equation” for solving multiple regressions, β^=XTX-1XTY, if one first normalizes Y and each of the Xk by subtracting their respective means; doing so has no effect on the estimates of the regression coefficients β^k. (The elements of the square matrix XTX are the covariances between the Xk's at each pair of lags, multiplied by the number of samples; likewise the elements of the column matrix XTY are the covariances between each of the Xk's and Y, multiplied by the number of samples.)

Astute readers will immediately notice a fundamental problem with applying Eqs. (40) or (41) in practice, namely that they require precipitation tracer concentrations CPj-k for all time steps j=1…n and lags k=0…m. In every practical case, many precipitation tracer concentrations will be missing, for two reasons. Some tracer concentrations will be missing due to sampling or measurement failures, and many more will be inherently missing because precipitation tracer concentrations cannot exist for time steps without precipitation. As we will see shortly, missing measurements that arise for these two different reasons must be handled in two different ways. But regardless of its origins, each missing tracer concentration CPi at time step i will create a diagonal line of missing values xj,k in the matrix X, causing a missing value in the first column (k=0) at j=i, and another in the second column (k=1) at j=i+1, and so on up to the last column (k=m) at j=i+m.

So-called “missing data problems” arise frequently in the statistical literature, and several approaches have been proposed for handling them (Little, 1992). One approach, termed “listwise deletion” or “complete-case analysis”, involves discarding all cases (meaning all rows j in the matrix X) in which any variables are missing and analyzing only the remaining (complete) cases. In our situation, this would mean analyzing only exit times tj that are preceded by unbroken series of rainy periods, up to the maximum lag m for which we want to estimate the coefficients β^k. Such ensembles of points would be mathematically convenient, but they would also be very strongly biased in a hydrological sense, because they would represent periods of unusually consistent rainfall (and thus unusually wet catchment conditions). Furthermore, if the maximum lag m is sufficiently long, records with continuous rainfall over all m+1 lags (k=0…m) will become impossible to find. For these reasons, complete-case analysis is not a feasible approach to our problem.

A second class of approaches to the missing data problem involves imputing values to the missing data (Little, 1992). In our case, however, many of the missing data are not simply unmeasured, but cannot exist at all (because rainless days have no rainfall concentrations), so it is not obvious how to impute the missing values.

A third approach, termed “pairwise deletion” or “available-case analysis”, first proposed by Glasser (1964), entails evaluating each of the covariances in Eq. (40) using any cases for which the necessary pairs of observations exist. Thus the covariances in Eq. (40) are replaced by covXk,Xℓkℓ=1nkℓ-1∑j∈kℓxjk-x‾kkℓxjℓ-x‾ℓkℓ and covXk,Yky=1nky-1∑j∈kyxjk-x‾kkyyj-y‾ky, where the notation (kℓ) indicates terms that are evaluated over all cases j for which both xjk and xjℓ exist (e.g., x‾kkℓ is the mean of the column vector Xk for rows j where neither xjk nor xjℓ is missing, and nkℓ is the number of such cases), and (ky) indicates terms that are evaluated over all cases j for which xjk and yj exist.

Glasser's approach can potentially handle the problem of tracer measurements that are missing at random due to sampling or analysis failures. However, it will not correctly handle the problem of tracer concentrations that are missing due to a lack of sufficient precipitation, because it assumes that the missing values occur randomly and therefore that Eqs. (42)–(43) are unbiased estimators of the covariances that one would obtain if no samples were missing. But when little or no precipitation falls on the catchment, it delivers little or no tracer to subsequent streamflow, and thus its contribution to the covariance between precipitation and streamflow concentrations will be nearly zero. Therefore different handling is required for precipitation tracer concentrations that are missing because they were not measured, versus those that are missing because they never existed at all (because no rain fell). As shown in Appendix B, periods without precipitation must be taken into account with weighting factors on the off-diagonal elements of the covariance matrix (because the tracer covariances will be less strongly coupled to one another, the less frequently precipitation falls). When the approach outlined in Appendix B is combined with Glasser's method for estimating each of the covariances, the end result is β^0β^1β^2⋮β^m=covX0,X0(0,0)nx0x1nx0covX0,X1(0,1)nx0x2nx0covX0,X2(0,2)⋯nx0xmnx0covX0,Xm(0,m)nx1x0nx1covX1,X0(1,0)covX1,X1(1,1)nx1x2nx1covX1,X2(1,2)⋯nx1xmnx1covX1,Xm(1,m)nx2x0nx2covX2,X0(2,0)nx2x1nx2covX2,X1(2,1)covX2,X2(2,2)⋯nx2xmnx2covX2,Xm(2,m)⋮⋮⋮nxmx0nxmcovXm,X0(m,0)nxmx1nxmcovXm,X1(m,1)nxmx2nxmcovXm,X2(m,2)⋯covXm,Xm(m,m)-1covX0,Y(0,y)covX1,Y(1,y)covX2,Y(2,y)⋮covXm,Y(m,y), where the covariance terms are defined by Eqs. (42)–(43), nxk is the number of time steps j for which precipitation fell at time i=j-k (whether or not that precipitation was sampled and analyzed), and nxkxℓ is the number of time steps j for which precipitation fell at both j-k and j-ℓ (again, whether or not those precipitation events were sampled and analyzed). As explained in Appendix B, the estimated coefficients β^k will closely approximate the average of the time-varying coefficients βj,k=qjk/Qj, averaged over times j for which precipitation fell at times i=j-k (but not over rainless periods, from which no streamflow can originate and thus βj,k=qjk/Qj must be zero). In practice, a single droplet of mist does not make a rainstorm, so there will be some threshold rate of precipitation below which there will be too little water to have any detectable effect on streamflow (and too little water to analyze). Thus nxk and nxkxℓ will be determined by counting the time steps that exceed this precipitation threshold: nxk=∑j=1n1:Pj-k≥Pthreshold0:Pj-k<Pthreshold,nxkxℓ=∑j=1n1:Pj-k≥PthresholdandPj-ℓ≥Pthreshold0:Pj-k<PthresholdorPj-ℓ<Pthreshold.

In the calculations presented here, I have assumed a precipitation threshold of 1 mm day-1, but expert judgment may lead to other values of Pthreshold in real-world situations. Note that some measurements will usually also be missing due to sampling or measurement failures in addition to precipitation intermittency. Thus nky and nkℓ in Eqs. (42)–(43), which account for both types of missing data, will typically be smaller than nxk and nxkxℓ in Eq. (44).

Gaps in the underlying data imply that, unlike covariance matrices in conventional multiple regressions, the covariance matrix in Eq. (40) is not guaranteed to be positive definite (and thus may not be invertible). Even when the covariance matrix is invertible, it may be ill-conditioned, making its inversion unstable. This issue arises frequently in inversion problems whenever different combinations of lagged inputs will have nearly equivalent effects on the output, making it difficult for the inversion to decide among them (this is the multidimensional analogue to nearly dividing by zero in Eq. 10). In minimizing the sum of squared deviations from the observations, inversions like Eq. (40) can potentially yield wildly oscillating solutions, with huge negative values of β^k at some lags delicately balancing huge positive values at other lags. Such results are not just unrealistic; they are also unstable, with tiny differences in the underlying data potentially having huge effects on the β^k estimates.

A standard therapy for this disease is Tikhonov–Phillips regularization (Phillips, 1962; Tikhonov, 1963). This technique (also known by many other names, including Tikhonov regularization, Tikhonov–Miller regularization, and the Phillips–Twomey method) is commonly used to solve ill-conditioned geophysical inversion problems (Zhadanov, 2015) but is less widely known in hydrology. Whereas conventional least-squares inversion finds the set of parameters β^k that will minimize the misfit between the predicted and observed yj, no matter how strange those β^k values may be, Tikhonov–Phillips regularization adds a second criterion that quantifies the strangeness of the β^k values themselves and finds the set of parameters β^k that will minimize the sum of both criteria. Phillips (1962) first showed how this joint minimization could be formulated as a simple extension of the normal matrix approach to solving linear inversion problems. This formulation, applied to our problem, is β^k=C+λH-1covXk,Y(ky), where C is the matrix of covariance terms in Eq. (44), and the parameter λ controls the relative weight given to the two criteria, namely the mean squared deviations of the predicted and observed yj values (controlled by the covariance matrix C) and the deviations from ideal behavior of the β^k values (controlled by the matrix H).

The form of H is determined by the criterion of reasonableness that is applied to the β^k. One possible criterion (among many that can be found in the literature) can be called “parsimony”: minimize the mean square of the β^k, thus penalizing solutions with large β^k values. Minimizing the functional β^k2 yields the identity matrix for H (Tikhonov, 1963): H=100⋯0010⋯0⋮⋱⋮0⋯0100⋯001.

This approach, also called “ridge regression” because it adds a “ridge” of extra weight along the diagonal of the covariance matrix, was Tikhonov's original regularization criterion and is widely used in geophysical inversions (including unit hydrograph estimation). In our case, however, it would have the undesirable effect of creating a systematic underestimation bias in our estimates of recent contributions to streamflow, by always making the β^k smaller than they would be otherwise.

A second possible criterion is consistency: minimize the variance of the β^k, thus penalizing solutions with individual β^k values that differ greatly from the mean of all the β^k. Minimizing the functional β^k-β^k2, where angled brackets indicate averages from k=0 to k=m, leads to an H matrix of the form (Press et al., 1992) H=1-1000⋯0-12-100⋯00-12-10⋯0⋮⋱⋮0⋯0-12-100⋯00-12-10⋯000-11.