The transport of solutes in river networks is controlled by the interplay of processes such as in-stream solute transport and the exchange of water between the stream channel and dead zones, in-stream sediments, and adjacent groundwater bodies. Transient storage models (TSMs) are a powerful tool for testing hypotheses related to solute transport in streams. However, model parameters often do not show a univocal increase in model performances in a certain parameter range (i.e. they are non-identifiable), leading to an unclear understanding of the processes controlling solute transport in streams. In this study, we increased parameter identifiability in a set of tracer breakthrough experiments by combining global identifiability analysis and dynamic identifiability analysis in an iterative approach. We compared our results to inverse modelling approaches (OTIS-P) and the commonly used random sampling approach for TSMs (OTIS-MCAT). Compared to OTIS-P, our results informed about the identifiability of model parameters in the entire feasible parameter range. Our approach clearly improved parameter identifiability compared to the standard OTIS-MCAT application, due to the progressive reduction of the investigated parameter range with model iteration. Non-identifiable results led to solute retention times in the storage zone and the exchange flow with the storage zone with differences of up to 4 and 2 orders of magnitude compared to results with identifiable model parameters respectively. The clear differences in the transport metrics between results obtained from our proposed approach and results from the classic random sampling approach also resulted in contrasting interpretations of the hydrologic processes controlling solute transport in a headwater stream in western Luxembourg. Thus, our outcomes point to the risks of interpreting TSM results when even one of the model parameters is non-identifiable. Our results showed that coupling global identifiability analysis with dynamic identifiability analysis in an iterative approach clearly increased parameter identifiability in random sampling approaches for TSMs. Compared to the commonly used random sampling approach and inverse modelling results, our analysis was effective at obtaining higher accuracy of the evaluated solute transport metrics, which is advancing our understanding of hydrological processes that control in-stream solute transport.
It is of crucial importance to understand how nutrients, solutes, and pollutants are transported in streams, since this process can drastically affect stream water quality along river networks (Smith, 2005; Krause et al., 2011; Rathfelder, 2016). A widely used technique to capture and study the processes controlling water transport downstream is via in-stream tracer injections. The measurement of the concentration over time of a tracer released in an upstream section (i.e. the breakthrough curve, BTC) reflects stream discharge (Beven et al., 1979; Butterworth et al., 2000) and longitudinal tracer advection and dispersion (Gooseff et al., 2008). A milestone in the study of solute transport was that in-stream solutes and water are exchanged with slowly moving channel waters, the dead zones (Hays, 1966), and with the saturated area that is physically influenced by water and solute exchange between the stream channel and the adjacent groundwater (i.e. the hyporheic zone, Triska et al., 1989; White, 1993; Cardenas and Wilson, 2007). This hydrologic exchange results in a skewed non-Fickian BTC with a pronounced tail, which makes the advection–dispersion equation (ADE) unable to correctly describe the observed tracer transport in stream channels (Bencala and Walters, 1983; Castro and Hornberger, 1991). Despite the large number of studies, the results of transient storage models (TSMs) offer numerous contradictory model interpretations (Ward and Packman, 2019), and model parameters are often non-identifiable, meaning that several parameter combinations return the same model performances (Ward et al., 2017). These outcomes raise the question of how informative such modelling results are (Knapp and Kelleher, 2020).
Considerable potential in reducing the uncertainty of the processes controlling solute transport in streams lies in modelling the tail of the BTC, since it contains information on the transient storage of the stream channels (Bencala et al., 2011). To simulate the retentive effect of dead zones on solute transport, Hays (1966) modelled the tail of the BTC by introducing a second differential equation in addition to the ADE. Following a similar approach, Bencala and Walters (1983) described the solute transport in streams as a pure advection–dispersion transport coupled with a hydrologic exchange term between the stream channel and a single, homogeneously mixed volume that delays the solute movement downstream (TSM). The estimation of model parameters often relies on the use of inverse modelling approaches via non-linear regression algorithms that return an estimation of model parameters with a narrow 95 % confidence interval (OTIS-P; Runkel, 1998). While this approach was widely applied in past decades, it does not allow a comprehensive assessment of parameter identifiability (Ward et al., 2017; Knapp and Kelleher, 2020). The term “identifiability” describes whenever good model performances are constrained in a relatively narrow parameter range (identifiable parameter) or spread (non-identifiable parameter) across the entire distribution of the possible parameter values (Ward et al., 2017), yet a good fit to observed data through inverse modelling does not provide information on performances and parameter identifiability over the entire feasible parameter range (Ward et al., 2017). Also, calibrated parameters obtained via inverse modelling approaches are not necessarily meaningful, as non-identifiable parameters can provide a good inverse model fit (Kelleher et al., 2019). These modelling uncertainties have led to a progressive abandonment of the search for a single best set of parameters and advocated the identification of “behavioural” parameter populations (i.e. parameter sets satisfying certain performance thresholds) via random sampling approaches in transient storage modelling (Wlostowski et al., 2013; Ward et al., 2017, 2018; Kelleher et al., 2019; Rathore et al., 2021).
Random sampling approaches provide information on parameter identifiability in the feasible parameter range; however, they rarely show identifiability for all the model parameters (Knapp and Kelleher, 2020). Kelleher et al. (2013) found that the parameters associated with the transient storage process are not identifiable for a large variety of stream reaches and experiments that they investigated. Other studies have shown that model parameters are often poorly identifiable (Camacho and González, 2008; Wlostowski et al., 2013; Ward et al., 2017; Kelleher et al., 2019; Knapp and Kelleher, 2020) and highly interactive, meaning that different parameters can produce similar modelled BTCs (Kelleher et al., 2013). This, in turn, hampers the ability to distinguish the role of a specific parameter in the shape of the simulated BTC (Wagener et al., 2002b; Ward et al., 2017).
The observed strong non-identifiability of model parameters in random sampling studies may have three causes. First, the parameters describing the
advection–dispersion process (streamflow velocity, cross-sectional area of the stream channel, and longitudinal dispersion) are known to be the best ones identifiable in the TSM (Ward et al., 2017). However, due to the known high interactivity among model parameters, it is generally not recommended to use a fixed value for a rather identifiable parameter, since this strategy may result in a misestimation of the other model parameters (Knapp and Kelleher, 2020). Constraining the values of the stream area and longitudinal dispersion proved to have a role in the identifiability of transient storage parameters (Lees et al., 2000; Kelleher et al., 2013; Ward et al., 2017). However, no study so far has evaluated the role of flow velocity in the identifiability of model parameters despite the velocity parameter often being considered known and thus fixed to equal the velocity of the arrival time of the BTC peak (i.e.
A robust assessment of transient storage parameters would not only improve the model fit of tracer transport and increase parameter identifiability, but it might also lead to a more robust interpretation of the physical processes controlling solute transport in streams. Model parameters are often used to calculate metrics on the solute exchange between the stream channel and the transient storage zone and the residence time of solutes in the coupled system (Thackston and Schnelle, 1970; Morrice et al., 1997; Hart et al., 1999; Runkel, 2002). These metrics are pivotal for addressing the potential for nutrient cycling, microbial activity, and the development of
hotspots in river ecosystems (Triska et al., 1989; Mulholland et al., 1997; Smith, 2005; Krause et al., 2017). However, no study so far has indicated and evaluated
Despite the increasing need for achieving parameter identifiability in TSMs,
only a few studies have explored the reliability of results obtained from
inverse modelling, and model interpretation is often based on a single set of parameters without testing their robustness (Knapp and Kelleher, 2020). We hypothesize that addressing the identifiability of model parameters in different sections of the BTC is key to increasing the identifiability of the parameters describing solute retention in streams. To address the enunciated TSM challenges, we have organized this contribution around three questions related to the key challenges of parameter identifiability in transient storage modelling.
How does the identifiability of model parameters change in the random sampling of TSMs when velocity is considered as a calibration parameter and when it is assumed fixed and equal to Does the identifiability analysis of specific sections of the BTC reduce the parameter non-identifiability in random sampling approaches of TSMs? How much does the identifiability of model parameters in random sampling approaches depend on the used parameter range and the number of parameter sets? How does the hydrologic interpretation of TSM results vary when model parameters are identifiable and when they are not?
With the outcomes of these questions, we will address the following question.
The studied stream reach (49
The one-dimensional Fickian-type advection–dispersion equation describes the combined effect of flow velocity and turbulent diffusion on solute transport
(Beltaos and Day, 1978; Taylor, 1921, 1954). The differential form of the ADE reads as
The TSM describes the solute transport in streams by combining the advection–dispersion process in the stream channel through a hydrologic exchange with an external storage zone. The model equations read as (Bencala
and Walters, 1983)
Several sampling approaches were previously used to estimate parameter
identifiability in TSMs, such as Monte Carlo sampling (Wagner and Harvey,
1997; Wagener et al., 2002b; Ward et al., 2013), Latin hypercube sampling (LHS, Ward et al., 2018; Kelleher et al., 2019), and a Monte Carlo approach coupled with a behavioural threshold (Kelleher et al., 2013; Ward et al., 2017). Here, we use LHS to sample from the selected parameter range due to LHS's higher efficiency compared to the classic Monte Carlo approach (Yin et al., 2011). A single combination of model parameters (
To obtain reliable TSM results, Ward et al. (2017) suggested a minimum number of parameter sets between 10 000 and 100 000. Thus, in each TSM iteration, we simulated 115 000 parameter sets. Results of each TSM iteration include RMSE values for the 115 000 parameter sets and results of identifiability analysis of the model parameters. The identifiability analysis includes parameter vs. RMSE plots (Wagener et al., 2003), parameter distribution plots (Ward et al., 2017), regional sensitivity analysis (Wagener and Kollat, 2007; Kelleher et al., 2019), and parameter distribution plots (Wagener et al., 2002a; Ward et al., 2017). Since the above-mentioned identifiability analysis refers to model performance (RMSE) evaluated on the entire BTC, we refer to it as a “global identifiability analysis”. Globally identifiable parameters satisfy the following criteria: a univocal peak of performance in parameter vs. RMSE plots and parameter distribution plots (Ward et al., 2017) and cumulative distribution functions (CDFs) corresponding to the best 0.1 % of the results deviating from the
The 100 best-performing parameter sets for each iteration were analysed with the DYNamic Identifiability Analysis (DYNIA, Wagener et al., 2002a) to address the role of model parameters in different sections of the BTC. Compared to the global identifiability analysis, the dynamic identifiability analysis evaluates the identifiability of a parameter on a moving window along the BTC. Following the approach of Wagener et al. (2002b), we used a window size of three time steps (
We simulated our tracer experiments with the ADE to avoid initial assumptions about advection–dispersion parameters that could affect the identifiability of transient storage parameters (Fig. 1). The RMSE value of the best-performing ADE parameter set is referred to as RMSE
Conceptual modelling workflow. The parameters have the following unit of measurement: velocity
Parameter names, abbreviations, and units together with a summary of publications that address the identifiability of model parameters with random sampling approaches. We reported the used number of parameter sets and the parameter ranges, while in parentheses we reported the method used for the parameter sampling. “Double step” indicates that the sampling procedure was divided into two steps. In the first step,
Similar to the Monte Carlo approach coupled with behavioural thresholds
(Kelleher et al., 2013; Ward et al., 2017) starting from the result of the
first TSM iteration, we simulated the three tracer experiments through a
stepwise approach with
While the first TSM iteration was conducted to investigate the identifiability of all the possible combinations in the feasible parameter
range reported in the literature and from the results of the ADE (Table 1), the successive iterations excluded pairs of
For each TSM iteration, we randomly extracted
We compared our results with both inverse modelling results (OTIS-P) and the
most common random sampling approach for TSMs (OTIS-MCAT). OTIS-P is an inverse modelling scheme that minimizes the residual sum of squares between the modelled and observed BTC. The OTIS-P model estimates the best-fitting
model parameter values and their identifiability via the 95 % confidence
interval. We carried out multiple OTIS-P iterations starting from different
initial parameter values to avoid a local minimum and interrupted the iterations when parameter values calibrated via OTIS-P changed by less than 0.1 % between subsequent runs (Runkel, 1998). OTIS-MCAT solves the TSM for
the selected number of parameter sets and addresses their identifiability
with a global identifiability analysis (Ward et al., 2017). Compared to our
approach, OTIS-MCAT considers Monte Carlo parameter sampling instead of LHS and velocity equal to
The model parameter sets obtained from OTIS-P, OTIS-MCAT, and the proposed
iterative TSM approach were used to compute some hydrologic metrics related to solute transport in streams. Here we computed the average distance a
molecule travels in the stream channel before entering the transient storage
zone (
The global identifiability analysis showed a clear peak of performance toward univocal values for
After the first TSM iteration, the global identifiability analysis indicated
that
Parameter values plotted against the corresponding RMSE values for the TSM results conducted for the tracer injections
The global identifiability of model parameters increased with increasing
iterations. In the TSM iterations where
The global identifiability of model parameters increased considerably through the iterative model approach, also when velocity was not considered a calibration parameter. After the third TSM iteration, the best-performing parameter sets approached unique parameter values (Fig. 3, blue dots), and the CDF corresponding to the best 0.1 % of the results deviated from the
Same as Fig. 2 but reporting TSM results when velocity was considered equal to
The dynamic identifiability analysis provided clearer insights into the
identifiability of the model parameters for different sections of the BTC compared to the global identifiability analysis (plots shown only for E1).
After the first TSM iteration,
Dynamic identifiability analysis of model parameters for E1 when
The dynamic identifiability analysis for the last TSM iteration showed that
the advection–dispersion parameters were important in controlling the rising limb and the tail of the BTC (Fig. 3k–p), while
After the first TSM iteration, the dynamic identifiability analysis indicated that
Same as Fig. 4 but reporting dynamic identifiability results for E1 when
The dynamic identifiability analysis for the last TSM iteration of E1 indicated that
When a rather wide parameter range was used (first TSM iteration, green dots, Fig. 2), the performance of the global identifiability analysis was strongly dependent on the chosen number of sampled parameter sets. This can be derived from the strong decrease in the mean and the standard deviation of the top model results with the number of sampled parameter sets
Mean (red lines, left axes) and standard deviation (blue lines, right axes) for RMSE values relative to the top 10 % of the modelling results as a function of the number of parameter sets used in the TSM. The results are reported for the
Our results showed that TSM results were poorly dependent on the sampled number of parameter sets when the model performance was studied for a narrow
parameter range around the peak of the performance (last TSM iteration, orange dots, Fig. 2). This was derived by the rather constant mean and standard
deviation of the top model results with the number of subset
Compared with results from our identifiability analysis, outcomes of OTIS-P were consistent with the best parameter sets obtained at the end of the iterative modelling approach (Table 2). Results from OTIS-P showed parameter
identifiability with a narrow 95 % confidence range for the
Summary of the TSM results. OTIS-MCAT results refer to the
case
The results of OTIS-MCAT showed low
The evaluated transport metrics showed high uncertainty as long as the model parameters were poorly identifiable or non-identifiable (Figs. 2 and 3, green and yellow
dots). This was particularly evident after the first and second TSM iterations, when the 100 best-performing parameter sets showed
Boxplots of the investigated transport metrics for the 100 best parameter sets for the three simulated experiments.
However, when the model parameters were identifiable, the transport metrics
converged toward constrained values and were consistent with OTIS-P results
(Fig. 7). This was achieved with a calibrated and fixed (as in the OTIS-MCAT model) streamflow velocity. Results of the last TSM iteration showed that the investigated transport metrics have low dispersion around the median and that the median almost coincides with the result of the best-performing parameter set (Fig. 7, red dots). When all model parameters were identifiable for each of the three tracer experiments, the transport metrics showed increasing
Our results showed that
Our results provide valuable guidance for future studies addressing parameter identifiability in TSMs. Specifically, our results support the current practice of considering velocity fixed and equal to
The assumption used in previous work of streamflow velocity equalling
The results of our dynamic identifiability analysis showed that both the advection and dispersion and the transient storage parameters control solute arrival time and solute retention in stream channels. This outcome is in contradiction with the common interpretation of model parameters, where it is assumed that the advection–dispersion parameters control the solute arrival time, while transient storage parameters are assumed to control the tail of the BTC (Bencala, 1983; Bencala and Walters, 1983; Runkel, 2002; Smith, 2005; Bencala et al., 2011). Following this common interpretation of the role of model parameters in the BTC, some authors decomposed the BTC into an advective part and a transient storage part (Wlostowski et al., 2017; Ward et al., 2019). This decomposition allowed them to quantify the role of advection and dispersion and transient storage embedded in the BTC. However, this modelling strategy also implicitly assumes a negligible role of advection–dispersion parameters in the tail of the BTC and of transient storage parameters in the rising limb and peak of the BTC, which is not consistent with our findings (Figs. 4, 5 and 8).
Qualitative plots of the TSM parameter influence different sections of the BTC.
Several studies addressed how different model parameters affect the shape of
the BTC and showed partly similar but also contrasting outcomes to our
findings (Fig. 8g–l, Wagner and Harvey, 1997; Wagener et al., 2002b; Scott et al., 2003; Wlostowski et al., 2013; Kelleher et al., 2013). Past studies found that the rising limb of the BTC was controlled by the stream channel area
The observed identifiability of model parameters in different sections of the BTC in past work and the differences compared to our findings (Fig. 8a, c and e) might be driven by different physical settings or discharge conditions of the study sites, by the methods used to account for parameter identifiability, by the parameter sampling procedure, or by the strategy used to obtain the best-fitting parameter sets (Wagner and Harvey; 1997;
Scott et al., 2003; Kelleher et al., 2013). For example, the identifiability
of the TSM to
Plots of the parameter values against the corresponding objective function
in Wagener et al. (2002b) and the regional sensitivity analysis in Wlostowski et al. (2013) do not indicate parameter identifiability for
Compared to our results, the different roles of the model parameters in
controlling the shape of the BTC in previous studies (e.g. Kelleher et al.,
2013) could be driven by the different approaches used for evaluating the
sensitivity (i.e. Sobol' sensitivity analysis). However, our results suggest
that the number of parameter sets (42 000) selected by Kelleher et al. (2013) might not have been sufficient to obtain identifiability of the model parameters with the rather wide parameter range chosen for their Monte Carlo sampling (Table 1). Results by Kelleher et al. (2013) are very similar to our TSM iterations for cases where
This study offers significant insights into understanding which model parameters influence the shape of the BTC, suggesting that only behavioural parameter sets should be considered in models aiming to understand the control of model parameters on the rising limb, peak, and tail of the BTC. Future work should address the interaction of model parameters on controlling different sections of the BTC for more complex model formulations (e.g. TSM with two or several transient storage zones, Choi et al., 2000; Bottacin-Busolin et al., 2011).
The applied iterative approach was effective in drastically improving parameter identifiability with the increase in TSM iterations. The identifiability of parameters in TSMs is commonly studied via random sampling approaches using between 800 and 100 000 parameter sets sampled from a parameter range spanning several orders of magnitude (Table 1). Despite a large number of parameter sets used in previous studies, model parameters were found to be identifiable only in a few studies (Ward et al., 2017, 2018), while at least one model parameter was found to be non-identifiable
in the majority of current TSM studies. Many authors found identifiable
The adopted iterative approach allowed us to achieve parameter identifiability and obtain physically realistic transport metrics. However,
this approach is based on the specific objective function used (RMSE) and on the subjective thresholds to control the refinement of the parameter range for successive iterations (top 10 % results for the global identifiability
analysis and information content
The applied iterative approach is foremost a tool for achieving parameter identifiability by investigating the entire range of feasible parameter values via existing identifiability tools (global identifiability analysis and dynamic identifiability analysis). The larger amount of time and computational power required by the adopted identifiability analysis compared to the rather straightforward application of OTIS-P paid off in terms of completeness of results and granted a more comprehensive view of the possible modelling outcomes on the feasible parameter range. Also, compared to the standard random sampling approach, the identifiability analysis used in the present work proved effective in iteratively constraining the parameter range to reduce the dimensionality of the model, eventually providing both identifiable model parameters and optimal parameter sets with model performances approaching (or even outperforming) calibrated results via inverse modelling (Table 2).
Our simulations with OTIS-P resulted in excellent model performances for the investigated BTCs, with low RMSE values and with calibrated model parameters comparable to the behavioural parameter populations obtained via our global identifiability analysis (Figs. 2 and 3). While the obtained performances of the OTIS-P calibration are certainly specific to the investigated BTCs, the use of OTIS-P alone would not have provided enough information to address the reliability of the obtained model parameters. This, in turn, would have raised concerns about the credibility of the transport metrics obtained, eventually compromising the robustness of the derived physical process involved at the study site. Compared to random sampling approaches coupled with global identifiability analysis, inverse modelling approaches are often considered not meaningful for interpreting modelling outcomes (Ward et al., 2013; Knapp and Kelleher, 2020). This is because parameters calibrated via inverse modelling might be non-identifiable despite an overall good model performance (Kelleher et al., 2019) and because identifiability analysis informs behavioural parameter sets, which is a preferable and more informative outcome for hydrological models than a single set of parameter values (Beven, 2001; Wagener et al., 2002a). Thus, our identifiability analysis over different investigated parameter ranges can offer an explanation about why in past studies identifiability analysis over a probably too large parameter range indicated non-identifiability and a lack of convergence with OTIS-P results (Ward et al., 2017).
Eventually, even if random sampling approaches are generally considered more
informative than the inverse modelling approach (Ward et al., 2013, 2017, 2018; Knapp and Kelleher, 2020), our results indicate that random sampling outcomes that show non-identifiability of transient storage parameters should not be used for process interpretation in TSMs. This was evident from TSM iterations showing non-identifiability of
Our results demonstrated that poor identifiability or non-identifiability of model parameters can result in a wrong hydrological interpretation of the processes controlling solute transport in streams. Additionally, our results showed that with increasing discharge conditions
However, if we had based the process interpretation on simulations before we reached the identifiability of model parameters, the conclusions
would have been different. The values for the transport metrics obtained
when
Our results also open developments for research seeking to increase the physical realism of the TSM and its results. Increased model complexity is associated with both a better analytical fitting to the observed BTC and an increased degree of freedom of the model with a consequent reduction of parameter identifiability (Knapp and Kelleher, 2020). Our approach offers a promising flexible tool to target parameter identifiability and physical interpretation also in TSM formulation with increasing complexity, such as multiple storage zone models (Choi et al., 2000), or for TSMs considering sorption kinetics (Gooseff et al., 2005) or different residence time distribution laws such as log-normal distribution (Wörman et al., 2002), exponential plus pumping distribution (Bottacin-Busolin et al., 2011), and power law distribution (Haggerty et al., 2002).
There is a clear need in stream hydrology to better identify parameters for
simulating solute transport in streams. Here we addressed the challenge of
parameter identifiability in TSMs by combining global identifiability analysis with dynamic identifiability analysis in an iterative modelling
approach. Our results showed that the value of stream velocity interacts with the transient storage parameters. That is, when stream velocity was a randomly sampled calibration parameter (within a physically reasonable range), we found non-identifiable
The modelling approach in this study constrained the parameter range iteratively. This strategy successfully reduced model dimensionality and allowed us to obtain identifiable model parameters for the three tracer experiments. As a complement to the existing body of literature, our work shows that the non-identifiability of model parameters in past studies might be related to the rather small number of sampled parameter sets compared to the investigated parameter range. The low uncertainty of the model parameters and the derived transport metrics were pivotal for obtaining a robust assessment of the hydrological processes driving the solute transport at the study site. In contrast, using non-identifiable model parameters or relying on OTIS-P results alone would have led to uncertain and rather different process interpretations at the study site.
Our study provides an enhanced understanding of the relevance of identifiable parameters of TSM models. We also provide insights into how parameter calibration without an assessment of their identifiability likely results in an unrealistic conceptualization of processes and unrealistic values for different solute transport metrics.
The interpretation of the parameter range is based on the sensitivity and
identifiability of the
Parameter vs. likelihood plots visualize the distribution of the investigated values of a certain parameter plotted against the corresponding values of
the objective function (Wagener et al., 2003; Wagener and Kollat, 2007). Identifiable parameters are described in parameter vs. likelihood plots by a univocal increase in model performances approaching a certain optimum value of the parameter (Fig. A1a). Non-identifiable parameters are described in parameter vs. likelihood plots by a non-univocal increase in performances of the model in a certain parameter range (Fig. A1b). Parameter distribution plots show the probability density function (PDF) divided by behavioural sets (from the top 20 % to the top 0.1 % of the results for the selected objective function) (Ward et al., 2017). Identifiable parameters are indicated by a narrow range of the PDF relative to the smaller behavioural sets (top 0.1 %, 0.5 % and 1 % of the results) compared to a wider range of the PDF relative to the larger behavioural sets (top 5 %, 10 % and 20 % of the results) (Fig. A1c). Non-identifiable parameters are defined by equally wide PDFs for the different investigated behavioural sets (Fig. A1d). Regional sensitivity analysis plots are obtained after dividing the population of the parameter by behavioural sets (from the top 10 % of the results to the top 1 % of the results with a 1 % step for the selected objective function: Ward et al., 2017; Kelleher et al., 2019). Each objective function population so obtained was transformed into cumulative distribution functions (CDFs) for equal size bins of the parameter range
(Kelleher et al., 2019; Wagener and Kollat, 2007). Sensitive parameters are identified by CDFs for the top 1 % of the results deviating from the CDF for the top 10 % of the results (Fig. A1e). If the CDFs lay on the
Examples of the four types of visualizations intended for parameter identifiability and sensitivity, with the plots in the first column
Dynamic identifiability analysis algorithm flowchart.
The plots used to address the global sensitivity analysis indicate parameter
identifiability and sensitivity on the entire observed BTC; however, they are unable to address whether the
Figure B1 shows the observed BTC for the three tracer experiments plotted against the top 100 simulated BTCs obtained using the proposed iterative approach. The observed poor visual fit on the tail of the BTC obtained at the end of the iterative modelling approach (Fig. B1d–f) is controlled by two factors: (i) the modelling structure of the TSM which assumes an exponential residence time distribution and (ii) the chosen objective function. By using alternative residence time distributions, TSM proved to have a more accurate fitting on the tail of the BTC (Haggerty et al., 2002; Bottacin-Busolin et al., 2011). Also, the RMSE could not be the best objective function for addressing a model fit on the tail of the BTC because it gives higher importance to the sections of the BTC with higher concentration values (peak of the BTC) compared to the sections of the BTC with low concentration values (on the tail of the BTC). As an example, the best-fitting BTC obtained at the end of the second TSM iteration (E1) shows a visually better fit on the BTC tail (Fig. B2) despite the large RMSE
(1.5197 mg L
Observed BTC (red line) together with the grey area comprised between the top 100 simulated BTCs and the best-fitting BTC (blue dashed
line) for
Observed BTC (red line) together with the grey area comprised between the top 100 simulated BTCs and the best-fitting BTC (blue dashed line) for the second TSM iteration (E1).
The software code used to obtain the identifiability of the transient storage model parameter can be freely obtained at
Solute breakthrough curve and metadata can be obtained at
EB, GB, and JK developed the concept of the manuscript; EB took care of the methodology, conducted the experiments, programmed the software, conducted the formal analysis, prepared the original draft of the manuscript, and implemented the revisions. GB and JK supervised the project, acquired the funding, and revised and edited the manuscript. JK administrated the project.
At least one of the (co-)authors is a member of the editorial board of
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors would like to thank three anonymous reviewers and editor Christa Kelleher for their supportive and constructive comments that improved the manuscript. The authors thank Ginevra Fabiani, Adnan Moussa, Laurent Pfister and Rémy Schoppach for their fruitful input and discussions during the preparation of the manuscript.
This research has been supported by the Fonds National de la Recherche Luxembourg (grant no. PRIDE15/10623093/HYDRO-CSI) and the Austrian Science Fund (grant no. DK W1219-N28).
This paper was edited by Christa Kelleher and reviewed by three anonymous referees.