The linear regression in Eq. (1), like any least-squares technique, is potentially vulnerable to outliers. Because potential outliers are often present in environmental data, practical applications of ensemble hydrograph separation would benefit from a robust method for estimating new water fractions. Such a method should not only be insensitive to outliers; ideally it should also be statistically efficient (i.e., it should yield reasonable estimates from small samples), and it should be asymptotically unbiased (i.e., it should converge to the conventional regression results when outliers are absent, with a bias near zero for large samples).

Figure 2 shows ensemble hydrograph separation plots of the outlier-free benchmark data (Fig. 2a, estimated from the time series shown in Fig. 1b) and the outlier-corrupted benchmark data (Fig. 2b, estimated from the time series shown in Fig. 1c). On these axes – precipitation and streamflow tracer fluctuations on the x and y axes, respectively, each expressed relative to the streamflow tracer concentration in the previous time step – the regression slope estimates the event new water fraction QpFnew. Here we are interested in how outliers affect this regression slope. When outliers are absent (Fig. 2a), the regression slope (0.164 ± 0.006, estimate ± standard error) is consistent with the true event new water fraction QpFnew= (0.168 ± 0.005) calculated from water age tracking in the benchmark model.

By contrast, outliers substantially distort the ensemble hydrograph separation plot in Fig. 2b; they extend well beyond the range of the outlier-free data indicated by the gray rectangle and inflate the estimate of QpFnew by nearly a factor of 3. Outliers in precipitation tracer concentrations will be displaced left or right from the corresponding true values (in Fig. 2b, these outliers are displaced to the left because they are all negative). Precipitation outliers will thus tend to flatten the regression line. Outliers in streamflow concentrations will appear in two different ways. First, they will be displaced above or below the corresponding true values (in this case, they are only displaced below, because they are all negative). Secondly, they will also appear as strongly correlated deviations on both the x and y axes because streamflow concentrations at time j-1 are used as reference values for both precipitation concentrations (on the x axis) and streamflow concentrations (on the y axis) at time j. Unlike precipitation outliers, these correlated points will tend to artificially steepen the regression line. Thus, whether outliers steepen or flatten the regression relationships underlying ensemble hydrograph separation will depend on the relative abundance and size of the streamflow outliers and precipitation outliers (relative to each other, and relative to the variability in the true streamflow and precipitation tracer values). In the example shown in Fig. 2b, the outliers have the net effect of artificially steepening the fitted slope, yielding an apparent QpFnew of 0.430 ± 0.018 that is more than 2.5 times the true value of 0.168 ± 0.005 determined by age tracking in the benchmark model.

Many robust estimation methods will not be effective against outliers like those shown in Fig. 2b, which create points that have great leverage on the slope of the fitted line. This leverage can allow the outliers to pull the line close enough to themselves that they will not be readily detected as outliers. To address this problem, our robust estimation procedure has two parts. The first step is to identify extreme values of both precipitation and streamflow tracer concentrations at the outset and exclude them by setting them to NA (thus treating them as missing values). This will effectively prevent outliers from exerting strong leverage on the solution. Because the exclusion criterion must itself be insensitive to outliers, we define extreme values as those that lie farther from the median than 6 times MAD, the median absolute deviation from the median. The cutoff value of 6 times MAD was borrowed from the residual downweighting function used in locally weighted scatterplot smoothing (LOWESS: Cleveland, 1979). Any exclusion criterion may also eliminate points that are not outliers but simply extreme values. However, unless the underlying distribution has very long tails, the 6 ⋅ MAD criterion will exclude very few points that are not outliers. If the underlying data follow a normal distribution, for example, the chosen criterion will exclude only the outermost 0.005 % of that distribution.

As a second step, we use iteratively reweighted least squares (IRLS: Holland and Welsch, 1977) to estimate the regression slope and thus the event new water fraction QpFnew. IRLS iteratively fits Eq. (1) by linear regression, with point weights that are updated after each iteration. Points with unusually large residuals are given smaller weight. In this way, IRLS regressions follow the linear trend in the bulk of the data, giving less weight to points that deviate substantially from that trend. This behavior, which allows IRLS to down-weight outliers, can have undesirable effects in analyses of outlier-free data exhibiting divergent trends. In Fig. 2a, for example, higher flows have steeper trends, with the highest 20 % of flows (shown in red) exhibiting a much steeper trend than the rest of the data. Because IRLS gives these points relatively less weight, the robust estimate of QpFnew is 0.126 ± 0.004, 25 % less than the true value of 0.168 ± 0.005 determined from age tracking in the benchmark model. Thus, in this case, the robust estimation procedure is somewhat less accurate than ordinary least squares if the data are free of outliers. Conversely, however, the outliers in Fig. 2b have little effect on the robust estimation procedure, which returns a QpFnew estimate of 0.115 ± 0.005, within 10 % of the outlier-free value. This example demonstrates that like any robust estimation procedure, ours is highly resistant to outliers but at the cost of reduced accuracy when outliers are absent, particularly in cases, like Fig. 2a, that superimpose widely differing trends. Robust estimation is turned on by default, but users can turn it off if they are confident that their data are free of significant outliers.

In EHS_TTD, robust estimation of transit time distributions follows a multi-step approach that is analogous to that which is used in EHS_Fnew (described in Sect. 2.1 above). We first exclude extreme values of both precipitation and streamflow tracer concentrations using the 6 ⋅ MAD criterion. We then apply iteratively reweighted least squares (IRLS) to Eq. (2), without regularization; this yields a set of robustness weights, which down-weight points that lie far away from the multidimensional linear trend of the data. These robustness weights are then applied within the Tikhonov–Phillips regularized regression that estimates the transit time distribution. This robust estimation approach requires that we handle the missing data problem in a different way than the one that was documented in Sect. 4.3 of K2019. The necessary modifications are detailed in Appendix A.

This robust estimation procedure yields transit time distributions that are highly resistant to outliers (Fig. 5). The gray lines in Fig. 5 show the true transit time distributions of discharge (QTTD) and “forward” transit time distributions of precipitation (PTTD), as determined by age tracking in the benchmark model. These age tracking results are compared to transit time distributions calculated from the tracer time series using EHS_TTD, with and without robust estimation (dark and light symbols, respectively, in Fig. 5). When the tracer time series are outlier-free (Fig. 5a and c), both the robust and non-robust estimation procedures accurately estimate these TTDs (i.e., the light and dark blue points closely follow the gray lines). When the tracer time series are corrupted by outliers (Fig. 5b and d), the non-robust TTDs (light blue points) deviate substantially from the age tracking results (gray lines), but the robust TTDs (dark blue points) follow the gray lines nearly as well as with the outlier-free data.

The benchmark tests shown in Figs. 2–5 above, like most of those presented in K2019, are based on a benchmark model simulation that yields “L-shaped” TTDs, that is, those in which the peak occurs at the shortest lag. In this section we explore several phenomena associated with the analysis of distributions that are “humped”, that is, those that peak at an intermediate lag. Where tracer data are sufficient to constrain the shapes of catchment-scale TTDs, they suggest that humped distributions are rare (Godsey et al., 2010). They are also not expected on theoretical grounds, since precipitation falling close to the channel should reach it quickly and with little dispersion, leading to TTDs that peak at very short lags (Kirchner et al., 2001; Kirchner and Neal, 2013). Nonetheless, humped distributions could potentially arise in particular catchment geometries (Kirchner et al., 2001) or in circumstances where tracers are introduced far from the channel but not close to it. Thus we have re-run the benchmark model with parameters that generate humped TTDs (Su,ref= 50 mm, Sl,ref= 50 mm, bu= 5, bl= 2, and η= 0.01), driven by the same time series of precipitation rates and rainfall δ18O values used in Sects. 2–4.1 above.

Figure 6 shows both forward and backward humped transit time distributions, as estimated by EHS_TTD from the benchmark model daily tracer time series, with their standard errors. (Here, as in the other analyses presented in this note, ser_corr = true, so the standard errors account for serial correlation in the residuals.) It is visually obvious that the error bars, which represent a range of ±1 standard error, are much larger than either the differences in the TTD estimates themselves (the solid dots) between adjacent lags or the typical deviations of the TTD estimates from the true values determined from age tracking (the gray lines). In other words, the error bars greatly exaggerate the uncertainty or unreliability of the TTD estimates. If the TTD estimates are unbiased, and their standard error estimates are too, then the standard error should approximate the root-mean-square deviation between the estimate and the benchmark. If the errors are roughly normally distributed, the true value should lie within the error bars about 65 %–70 % of the time and outside the error bars about 30 %–35 % of the time. By contrast, the error bars in Fig. 6 are many times larger than the typical deviation of the TTD estimates from the true values. Figure 5 shows a milder form of this exaggeration of uncertainty; here too, the age tracking values almost never lie outside the error bars, whereas that should occur about one-third of the time if the error bars are estimated accurately.

Thus it appears that the TTD error estimates are generally conservative (i.e., they overestimate the true error), but with humped distributions the uncertainties are greatly exaggerated. Numerical experiments (Fig. 7) reveal that this problem arises from the nonstationarity of the transit times in the benchmark model (and, one may presume, in real-world catchment data as well). K2019 (Sect. 4 and Appendix B) showed that ensemble hydrograph separation correctly estimates the average of the benchmark model's nonstationary (i.e., time-varying) TTD, as one can also see in Figs. 6 and 8. When this (stationary) average TTD is used to predict streamflow tracer concentrations (which is necessary to estimate the error variance and thus the standard errors), however, it generates nearly the correct patterns of values but not with exactly the right amplitudes or at exactly the right times (see Fig. 7a). This is the natural consequence of estimating a nonstationary process with a stationary (i.e., time-invariant) statistical model. As a result, the residuals are larger, with much stronger serial correlations, than they would be if the underlying process were stationary (compare Fig. 7a and b), resulting in much larger calculated standard errors of the TTD coefficients. These tendencies are even stronger for humped TTDs, which introduce stronger serial correlations in the multiple regression fits that are used to estimate the TTD itself. Serial correlations in the residuals reduce the effective number of degrees of freedom by a factor of approximately (1-rsc)/(1+rsc), where rsc is the lag-1 serial correlation coefficient of the residuals, thus increasing the standard error by roughly a factor of [(1+rsc)/(1-rsc)]0.5. For the nonstationary case shown in Fig. 7a, rsc is 0.96 (thus increasing the standard error by a factor of roughly 7), whereas for the stationary case shown in Fig. 7b, rsc is 0.22 (thus increasing the standard error by only a factor of 1.25).

Since the exaggerated standard errors in Fig. 6 arise primarily from the nonstationarity in the benchmark model's transit times, one might intuitively suspect that this problem could be at least partly resolved by dividing the time series into separate subsets (representing, for example, wet conditions with shorter transit times and dry conditions with longer transit times) and then estimating TTDs for each subset separately using the methods described in Sect. 4.4 below. Benchmark tests of this approach were unsuccessful, however. This approach might theoretically work, if the “wet” and “dry” states persisted for long enough that tracers would both enter and leave the catchment while it was either “wet” or “dry”. Under more realistic conditions, however, many different precipitation events and many changes in catchment conditions will be overprinted on each other between the time that tracers enter in precipitation and leave in streamflow, making this approach infeasible.

A somewhat counterintuitive approach that shows more promise is to use lower-frequency tracer data to estimate humped TTDs. Figure 8 shows that if the same TTDs as those shown in Fig. 6 are estimated from weekly data rather than daily data, the standard errors more accurately approximate the mismatch between the TTD estimates and the true values (i.e., the difference between the blue dots and the gray curves). Weekly sampling yields much more reasonable standard errors in this case, because the multiple regression residuals are much less serially correlated (see Fig. 7c; rsc is 0.66, increasing the standard error by only a factor of 2.2). In addition, with daily data the TTD coefficients are estimated for a closely spaced mesh of lag times (with lag intervals of 1 d), and broad TTDs like the ones shown in Fig. 6 do not change much over such short lag intervals. Thus the individual TTD coefficients on such a closely spaced mesh are not well constrained; one could make the TTD stronger at the fifth daily lag and weaker at the sixth daily lag, for example, with little effect on the overall fit to the data. With weekly sampling, the TTD coefficients are more widely separated in time (with lag intervals of 1 week) and thus are less redundant with one another.

Figure 9 shows profiles of new water fractions (QFnew) and forward new water fractions (PFnew), analogous to those shown in Figs. 3–4, but based on model simulations yielding the humped distributions shown in Fig. 6. One can immediately see that the new water fractions are substantially overestimated and that this bias is particularly large for forward new water fractions associated with low rainfall rates (i.e., the left side of Fig. 9b). These artifacts arise because the random fluctuations in input tracer concentrations used in the benchmark model have a serial correlation of 0.5 between successive daily values. Thus the correlations between input and output tracer fluctuations at lag zero (and thus the new water fractions) are artificially inflated by leakage from the stronger correlations at longer lags, where the TTD is much stronger. Numerical experiments show that the bias in the new water fractions disappears when the short-lag serial correlation in the input tracers is removed, supporting this hypothesis for how the bias arises. Nonetheless, real-world precipitation tracer concentrations are often serially correlated (particularly in high-frequency measurements), so researchers should be aware of the bias that could potentially arise in new water fractions if transit time distributions are humped.

In principle the distortions arising from the correlations in the precipitation tracer data could potentially be alleviated by calculating TTDs for individual precipitation and discharge ranges using the methods outlined in Sect. 4.4 below and then estimating QFnew and PFnew from the lag-zero coefficients of QTTD and PTTD, respectively. Benchmark tests of this approach were not successful, however, possibly because the transit time distributions cannot be estimated reliably when the source data are split among so many narrow ranges of precipitation or discharge. (Indeed, in many cases users may seek to estimate new water fractions precisely because they lack sufficient data to reliably estimate transit time distributions.)

Instead, benchmark tests suggest that a practical cure for the biases shown in Fig. 9 may be, counterintuitively, to estimate profiles of QFnew and PFnew from lower-frequency measurements, similar to the estimation of humped TTDs. As Fig. 10 shows, the bias in Fig. 9 is effectively eliminated if the profiles of new water fractions are estimated from weekly samples instead of daily samples. This occurs because the input tracers are less correlated over weekly sampling intervals than over daily sampling intervals, and because the TTD is much stronger at short lags on weekly timescales (Fig. 8) than on daily timescales (Fig. 6). In real-world cases, biases like those shown in Fig. 9 may not be obvious, because the correct answer (shown here by the gray line, derived from benchmark model age tracking) will not be known. However, the behavior in Fig. 9b is implausible on hydrological grounds (Why should catchments quickly transmit a particularly large fraction of very small precipitation events to the stream?), and the PFnew profiles in Figs. 9b and 10b show strongly contrasting patterns. Thus observations like these may help in identifying biased new water fraction estimates, even in cases where the TTD itself has not been quantified.

Transit time distributions are typically constructed from the entire available tracer time series for a catchment, as in Figs. 5, 6, and 8. Such TTDs can be considered as averages of catchments' nonstationary transport behavior, as shown in Sect. 4.2 above. However, ensemble hydrograph separation can also be used to calculate TTDs for filtered subsets of the full catchment time series, focusing on either discharge or precipitation time steps that highlight particular conditions of interest. (In Appendix B we describe the new procedure that EHS_TTD uses for filtering precipitation time steps; this approach yields more accurate results than the one outlined in Sect. 4.2 of K2019.) TTDs from these filtered subsets of the full time series can yield further insights into catchment transport phenomena.

For example, we can map out the nonlinearities that give rise to catchments' nonstationary behavior, by comparing TTDs from subsets of the original time series that represent different catchment conditions (Fig. 11). Larger precipitation events in our benchmark model result in forward transit time distributions with peaks that are higher, earlier, and narrower (Fig. 11a). A similar progression in peak height, timing, and width is observed in forward TTDs (Fig. 11b) obtained from the benchmark tracer time series by setting the point filter Pfilter in EHS_TTD to focus on individual ranges of precipitation rates. The backward transit time distributions in the benchmark model (Fig. 11c) differ somewhat from the forward transit time distributions (Fig. 11a) but exhibit a similar shift to higher, earlier, and narrower peaks at higher discharges. This trend is also reflected in backward TTDs (Fig. 11d) obtained from the benchmark time series by setting Qfilter for the same discharge ranges used in Fig. 11c.

The ensemble hydrograph separation TTDs do not perfectly match the age tracking results shown by the dotted gray lines in Fig. 11b and d, particularly for the smallest fractions of the precipitation and discharge distributions, where fewer data points are available. Nonetheless, although the TTDs differ in detail from the age tracking results, they exhibit very similar progressions in peak height and shape, reflecting the nonlinearity in the benchmark model storages, which have shorter effective storage times at higher storage levels and discharges. Although the particular results shown in Fig. 11 are generated by a synthetic benchmark model, they illustrate how similar analyses could be used to infer nonlinear transport processes from real-world catchment data. Comparing TTDs representing different levels of antecedent catchment wetness, for example, could potentially be used to determine how much more precipitation bypasses catchment storage during wet conditions. Similarly, TTDs representing different levels of subsequent precipitation (over the following day or week, for example) could potentially be used to determine how effectively such precipitation mobilizes previously stored water. Thus Fig. 11 illustrates how TTDs from carefully selected subsets of catchment tracer time series can be used as fingerprints of catchment response and as a basis for inferring the mechanisms underlying catchment behavior.

Prospective users of ensemble hydrograph separation may naturally wonder what sample sizes and sampling frequencies are needed to estimate new water fractions and transit time distributions. The answers will depend on many different factors, including the timescales of interest to the user, the desired precision of the Fnew and TTD estimates, the logistical constraints on sampling and analysis, the frequency and intermittency of precipitation events, the variability of the input tracers over different timescales, and the timescales of storage and transport in the catchment itself (that is, what the TTD is and how non-stationary it is, which of course can only be guessed before measurements are available). Ideally one should sample at a frequency that is high enough to capture the shortest timescales of interest and sample much longer than the longest timescales of interest. One should also aim to capture many diverse transport events, spanning many different catchment conditions and precipitation characteristics.

Beyond these generalizations, it is difficult to offer concrete advice. We can, however, report our recent experience applying ensemble hydrograph separation to weekly and 7-hourly isotope time series at Plynlimon, Wales (Knapp et al., 2019). We were generally able to estimate TTDs out to lags of about 3 months based on 4 years of weekly sampling. The same 4 years of weekly samples yielded about 100 precipitation–discharge sample pairs (after samples corresponding to below-threshold precipitation were removed), which were sufficient to estimate weekly event new water fractions with an uncertainty of about 1 % (e.g., QpFnew∼8±1%). When these were split into four seasons, we could estimate event new water fractions with an uncertainty of about 2 %–3 % using 20–30 weekly precipitation–discharge pairs, and when they were split into 4–6 different ranges of precipitation and discharge, we could reasonably well constrain the profiles of new water response to catchment wetness and precipitation intensity (Fig. 10 of Knapp et al., 2019). We were able to estimate 7-hourly TTDs out to lags of 7 d based on about 17 months of 7-hourly isotope samples, including almost 1500 discharge samples and 540 above-threshold precipitation samples, and splitting these data sets in half allowed us to distinguish the TTDs for summer and winter conditions (Figs. 11 and 12 of Knapp et al., 2019). However, these numbers should not be uncritically adopted as rules of thumb for other catchments, since precipitation at Plynlimon is frequent and weakly seasonal, and the catchment is characterized by rapid hydrological response but relatively long storage timescales (Kirchner et al., 2000). All of these characteristics could potentially affect the sample sizes needed for estimating new water fractions and transit time distributions. As more experience is gained at more catchments, general rules of thumb may emerge. Until then, however, benchmark tests like those described here can potentially provide a more reliable site-specific guide to sample size requirements.

Another obvious question for users is the number of lags over which the TTD should be estimated. Here, too, there is no fixed rule; the answer will depend on the timescales of interest, the length of the available tracer time series, and the shape of the TTD itself (which of course will not be known in advance). An empirical approach is to compare the results for several different maximum lags m and see where the resulting TTDs are similar and different. Figure 12 shows this approach applied to daily and weekly tracer time series, yielding TTDs with contrasting shapes. The upper row (panels a and b) shows L-shaped TTDs estimated from the same synthetic tracer time series that underlie Figs. 1–5, and the lower row shows humped TTDs from the same benchmark model driven by the same inputs, but with different parameters as described above. In each panel, the shorter and longer TTDs (shown in different colors) are generally consistent with one another, except in the case of the four-lag TTD shown in blue in Fig. 12d. In that case, such a short TTD is evidently unable to capture the shape of the benchmark distribution, as indicated by its deviation from the TTDs of other lengths. One can also see that the last few lags of any TTD can diverge from the TTD shape defined by the other TTDs. In Fig. 12a the last few lags generally deviate downward, and in Fig. 12c they generally deviate upward; thus there appears to be no general rule except that the last few lags of any TTD estimates should be treated with caution and potentially excluded from analysis.

A further observation from Fig. 12 is that TTD estimates from weekly tracer data may be at least as accurate as, if not more so than, those calculated from daily tracer data. This may seem surprising, particularly because the time series underlying Fig. 12 are all 5 years long; thus the daily time series contain 7 times as many individual tracer measurements as the weekly time series. Nonetheless, for several reasons it is not surprising that in this case one could obtain more stable estimates from fewer data points. First of all, in these numerical experiments the precipitation tracer concentrations are serially correlated (as they also often are in the real world); thus there is more redundancy among the daily tracer inputs than among the weekly tracer inputs. Secondly, the precipitation volumes are less variable (in percentage terms) from week to week than they are from day to day, meaning that the weekly calculations use fewer input concentrations that are accompanied by very small water volumes (and that therefore could not have much influence on the real-world catchment). And thirdly, lower sampling frequencies entail TTDs with coefficients at more widely spaced lags, which are thus less redundant with one another and thus can be individually constrained better. Of course with lower-frequency sampling one loses the short-lag tail of the TTD, which may be of particular interest. But in cases where this information is not crucial – or where only lower-frequency data are available – it appears that TTDs can be reliably estimated from samples taken weekly and perhaps from samples taken even less frequently.