Link: https://hess.copernicus.org/preprints/hess-2022-222/#discussion 

# Uncertainty in water transit time estimation with StorAge Selection functions and tracer data interpolation

This paper presents an analysis of uncertainty in transit time distributions estimated using SAS functions, including that arising from the interpolation of input tracer data, and from the SAS function parameterization. Uncertainty of each configuration of model and input data is assessed from the range of predictions made by the top 5% of monte-carlo sampled parameter sets ranked by goodness-of-fit (KGE). The fraction of young water $F_{yw}$ obtained from the method proposed by Kirchner (2016) is used to further constrain the behavioral set.

This paper aims to address an important gap in the literature. There is a need to better understand the uncertainty associated with SAS models, and how data can be best used to constrain them. However I think there are two major problems with the approach used here, and I think the resulting conclusions are unsupported as a result.

- I don't think it makes sense to use $F_{yw}$ to constrain the SAS model parameterizations

- I think the use of top 5% KGE to define behavioural parameter sets makes it impossible to meaningfully compare the uncertainty of each configuration

Unfortunately because of these I don't have confidence in the conclusions regarding the relative uncertainty of different parameterizations, nor of the value of $F_{yw}$ in constraining that uncertainty. More details are below.

# Major issues

## Use of $F_{yw}$ to constrain SAS models

- I do not think it makes sense to use the young water fraction obtained from the sine-wave ratio to constrain a SAS model. Kirchner's method for this is useful for obtaining rough estimates of the fraction of water that is roughly a quarter of a year old from tracer time series. The method might be robust (in some sense) but it isn't precise. SAS models are a more complex and sophisticated tool that have the *potential* to provide a much more precise estimate of water age distribution from the same data. It doesn't make sense to me to use the outputs of a rough-and-ready model to constrain the parameters of a more precise one.

- I believe the fact that the authors do find that $F_{yw}$ has power to constrain the SAS parameters is largely because the uncertainty in the associated age threshold $\tau_{yw}$ is not accounted for. The method that $F_{yw}$ relies on is based on a variety of assumptions, including that the inputs are sinusoidal and that the transit time distribution is approximately a gamma distribution. Two important *and distinct* sources of uncertainty here are:

    - The threshold age of the young water fraction $\tau_{yw}$ is not 75 days, as suggested by the authors. Rather it depends on the shape parameter of the assumed gamma distribution. As Figure 10 of Kirchner (2016) shows, for a shape parameter of 0.2 it is around 40 days, while for a shape parameter of 2 it is more like 100 days. This considerable uncertainty is not accounted for in the present paper.

    - The estimates of amplitudes $A_q$ and $A_p$ obtained from fitting sinusoids to the observed tracer timeseries are uncertain, and that uncertainty ought to be estimated and propagated into uncertainty in $F_{yw}$. The authors may have accounted for this (if I understand the brief statement on line 165) but they claim that in doing so they have also accounted for the uncertainty in $\tau_{yw}$, which is not the case. These errors are independent of each other. The errors obtained for $F_{yw}$ were only 0.07-0.08 (line 325), which I suspect contributes far less uncertainty than the 60 day window bracketing $\tau_{yw}$.

- Furthermore, the theory behind $F_{yw}$ and $\tau_{yw}$ rests on the assumption that flows through the system are steady, the transit time distribution is invariant, and that the input signal is a perfect sinusoid. These are not the case in general in real watersheds, which results in additional epistemic uncertainty into the estimates of $F_{yw}$ and $\tau_{yw}$. These particular sources of uncertainty do not necessarily apply to the SAS models, since they can allow for variable flows, variable transit time distributions, and make use of the observed input signal.

- In fact it is possible to reproduce the model used to justify Kirchner's method as a SAS model. This can be done by approximating the flows as constant, replacing the inputs concentrations with sinusoids, and choosing a SAS function whose corresponding steady-state TTD is a gamma. From this perspective $F_{yw}$ and $\tau_{yw}$ can be viewed as outputs of a particular SAS model parameterization run with degraded data. Why should the results of that parameterization be used to constrain other parameterizations run with the best available data?

## Use of top 5% KGE as the 'behavioural' parameter set

- The use of the top 5% KGE as the 'behavioural' parameter set makes it impossible to make meaningful comparisons between the different parameterizations (i.e. PLTI, PLTV, BETA). This is because the range of goodness-of-fit (i.e. the KGE) of each model's behavioral set depends on the size of the pool from which it was taken, in addition to how well it actually fits the data.  The range of KGE in the top 5% depends on the assumed prior distribution of the parameter set, since that determines what the 5% is a percentage of. Since each parameterization has fundamentally incommensurate parameters, there isn't an obvious way to normalize for this dependence across different parameter spaces. As a result each behavioral set would have a different total likelihood associated with it (if a formal likelihood were estimated). Comparing these different behavioral parameterizations therefore makes no sense, since they have been held to different standards.

- One consequence of effectively holding each parameterization to a different standard is that the error associated with the more flexible parameterizations (PLTV, BETA) is larger than that associated with the less flexible one (PLTI), when we would expect the opposite to hold. This is particularly true given that PLTI represents a special case of both PLTV and BETA (when $k_{Q1}=k_{Q2}=k$ and when $\alpha=k, \beta=1$ respectively). However, as seen in Figure 2 the behavioral sets of BETA (and to a lesser extent PLTV) seem to include models that are considerably worse fits to the data than the worst models in the behavioral set of PLTI.

- To make meaningful comparisons between different parameterizations the analysis would need to be redone with a standard for 'behavioral' that is consistent across the different parameterizations. This might be as simple as choosing a cutoff value of KGE to define the behavioral set, but it would likely change the resulting conclusions about the merits of each parameterization.

# Minor issues

- Line 57: The gamma distribution has also seen some use

- Line 64: I don't think that the statement that $F_{yw}$ is useful for short-term data is quite right, since the method does require data covering multiple cycles of sinusoidal variation to fit to reliably

- Line 111: $S_{T_0}$ is a function of age: $S_{T_0}(T)$

- Line 130: $k_{Q1}$ and $k_{Q2}$ 

- Table 2: Why are $k_{Q1}$ and $\alpha$ grouped together? Same with $k_{Q2}$ and $\beta$

- Line 188: Is $TT_{50}$ is the median of the *backward* transit time distribution $p_Q(T,t)$ as defined in equation (5)? In that case this statement is incorrect, and should be "the maximum time elapsed *since* the youngest 50% of the water in outflow first entered the catchment", or perhaps "the age that half the outflow is older than, and half younger than, as measured from the time it fell as precipitation".

- Figure 3: A legend explaining the colors and a reference to Table 1 would aid interpretation here

- Line 221: Parameters for the *SAS function* of $Q$...

Comments on “Uncertainty in water transit time estimation with StorAge Selection functions and tracer data interpolation” by Borriero et al.

In this study, the authors studied the uncertainty in transit time estimation. Two sources of uncertainty were considered: the assumed shape of the StorAge Selection (SAS) function (and the uncertainty in the associated parameters) and the interpolation scheme for the precipitation tracer data. The reported uncertainty is large, resulting in a  90% confidence interval between 286 – 895 days for the median transit time. The uncertainty was greater in dry conditions than in wet conditions. The uncertainty depended more on the SAS function parameterization and the temporal interpolation of the precipitation tracer data than the spatial interpolation of the precipitation tracer data. Importantly, the authors argued that it could be useful to utilize the young water fraction, F_yw, in estimating the SAS function parameters, as it could constrain the SAS function and reduce uncertainty.

Though understanding the uncertainty in the SAS function and transit time is important, it is unclear what readers could learn from this manuscript in its present form other than the summarized results above for the specific catchment. As the results were not discussed enough in detail, it is not easy to think about their implications (see major comment 1). The suggestion of using the young water fraction in the SAS function estimation is interesting, but the authors’ argument regarding using it must be more convincing (see major comment 2). In addition, some additional potential sources of uncertainty should be considered or mentioned explicitly (see my major comment 3). Recent advances in estimating transit should be mentioned (see my major comment 4). Thus, I think a significant revision is required before this manuscript can be considered for publication in HESS again.

1. Discussion of the results

I think it is necessary to discuss the results further to make the implications of this study clearer. The current manuscript focuses more on describing the results for the specific catchment and dataset than discussing the results, so it is not easy to think about those implications. For example, why is the uncertainty in the estimated TTD (or the median transit time) large? Why is the uncertainty greater under drier conditions? Why does the spatial interpolation method not substantially affect the water age simulation? Without discussing that type of question for each finding, it is not easy to truly understand the described results.

2. The use of F_yw

Overall, it needs to be clarified if the use of F_yw constrained the SAS function parameters in the right way.

The authors somehow decided to state that the estimated young water fraction indicates the fraction of water younger than τ_yw = 75 days (in L171). However, as the authors mentioned, for example in L171 and L256, the method of Kirchner (2016) does not provide a single value for τ_yw that can be utilized universally. Rather, it varies with the shape of TTD. While the authors argued that the arbitrary decision is okay since they considered the uncertainty in the estimated F_yw (in L 171-172), that argument was made without clear reasonings that support it.

Also, it needs to be clarified if the estimated F_yw based on the method of Kirchner is a good estimate that can be used to constrain the SAS function. The method of Kirchner is based on a set of assumptions. It seems like the authors want to argue that it's okay to use the estimate regardless of all assumptions because the estimated uncertainty of F_yw is low, but I think it would be better if the authors could provide more concrete arguments to convince readers why it's okay. Why is the uncertainty low? And how does the low uncertainty support that the estimate is a good estimate regardless of all the assumptions? In addition, the method approximates the precipitation and outflow tracer signal using sinusoidal functions, which was shown to be not a good approximation for the precipitation tracer data by the authors in this manuscript (e.g., in L305-308).

It also seems that the uncertainty in F_yw could depend on the temporal resolution of data (if the finer resolution data shows more deviation from the sinusoidal signal) and other properties that the authors mentioned in L348-350. Overall, based on the limitation discussed by the authors (in L348-350), I feel that the authors are unsure whether the utilization of F_yw will be useful for other datasets.

A minor comment related to F_yw

L163-164: The method used to estimate F_yw was described too briefly. For example, L163-165 is not enough for readers to understand the method.

3. Other sources of uncertainty

I believe that the uncertainty in precipitation, discharge, and evapotranspiration rates could propagate into the uncertainty in the estimated SAS function. The list of potential sources of uncertainty provided by the authors (L39-43) needs to include them. It would be helpful for readers if the authors provided a more concrete list of potential sources of uncertainty. Also, it would be essential to provide why this manuscript, where the authors consider only a few sources of uncertainty, is still useful.

4. Missing new methods of estimating TTDs

There have been some recent advances in the estimation of TTDs that are not discussed in this manuscript. The method of Kirchner (2019) and the method of Kim and Troch (2020) can estimate time-variable (or state-dependent) TTDs without assuming their form a priori. The estimated TTDs can be converted to the SAS functions. Thus, some descriptions of the motivation of this study, such as what is in L55-59, need to be revised.

References

Kirchner, J. W (2019) Quantifying new water fractions and transit time distributions using ensemble hydrograph separation: theory and benchmark tests, Hydrol. Earth Syst. Sci., 23, 303–349, https://doi.org/10.5194/hess-23-303-2019

Kim, M., & Troch, P. A. (2020). Transit time distributions estimation exploiting flow-weighted time: Theory and proof-of-concept. Water Resources Research, 56, e2020WR027186. https://doi.org/10.1029/2020WR027186

Minor comments

fSAS/rSAS: I think the authors should make it clear that they are discussing only the fSAS (fractional SAS) function. Another form of the SAS function, the rank SAS (rSAS) function, may have different uncertainty characteristics, especially because of the difference in how the storage is considered.

Naming of the “BETA” case: Better to name the case more clearly. While the beta distribution is used without any state dependency or time-variability in this study, several studies utilized state-dependent beta distribution that can consider the time-variable flow pathways (e.g., Van der Velde et al., 2015). Thus, it could confuse readers when the authors state something like “BETA could be appropriate where the catchment release scheme is expected to be relatively constant” (in L290-291). In this manuscript, the time-variability is mentioned explicitly in the case names only for the power law cases (PLTI and PLTV).

L11-12: Make it clear that this confidence interval is for the median transit time.

L113: V(t) (mm) “is”

L185: When using GLUE, the authors determined the behavioral parameters using an arbitrary criterion. The top 5% of the parameters, in terms of KGE, were selected as the behavioral parameters. However, there is no statement to support the decision regarding the criterion. It may be better for readers if the authors could explain why the criterion was chosen and whether this seemingly arbitrary choice is okay.

L198: “more positive”: I am not sure if “more positive” is the right expression here. The isotope ratios are always negative in the data.    

L188-190 I believe that the authors in general talked about the “backward” transit time distribution throughout this manuscript. However, the sentence in L188-190 is not clearly written if they talked about “forward” TTDs or the “backward” TTDs. Please revise.

Figure 2: I think it would be better to enumerate the subfigures using numbers (based on Table 1) instead of alphabets.

L290-291: Not clear if this trivial sentence is necessary.

L298-303: This paragraph is also trivial and not necessary for the manuscript.

L305: I think the figure illustrating the interpolation results is more important for this manuscript than some unnecessary paragraphs mentioned above.

L345: Typo: “ETET”

L360: Typo: “te modern”

L395: “smooth changes” are unclear.