Influence of constant rate versus slug injection experiment type on parameter identifiability in a 1-D transient storage model for stream solute transport

Authors

  • Adam N. Wlostowski,

    Corresponding author
    • Department of Civil and Environmental Engineering, The Pennsylvania State University, Pennsylvania, USA
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  • Michael N. Gooseff,

    1. Department of Civil and Environmental Engineering, The Pennsylvania State University, Pennsylvania, USA
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  • Thorsten Wagener

    1. Department of Civil and Environmental Engineering, The Pennsylvania State University, Pennsylvania, USA
    2. Department of Civil Engineering, University of Bristol, Bristol, UK
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Corresponding author: A. N. Wlostowski, Department of Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA 16802, USA. (anw5034@psu.edu)

Abstract

[1] One-dimensional solute transport modeling to simulate experimental tracer releases in rivers is common practice. However, few studies have investigated the effect of experiment type (slug injection versus constant rate injection) on model parameter sensitivity and identifiability. We conducted slug injection and constant-rate experiments in a low-gradient, alluvial, headwater tundra stream in northern Alaska. Each experimental data set was simulated with the one-dimensional transport with inflow and storage (OTIS) model and analyzed with Monte Carlo-based techniques to investigate differences in global sensitivity and time-varying identifiability of the model parameters. We found that the longitudinal dispersion parameter exhibited relatively high sensitivity for the slug injection data, while the storage zone area parameter exhibited relatively high sensitivity for the constant rate injection data. The storage zone area and storage zone – main channel exchange rate parameters show heightened sensitivity during rising and tailing portions of the constant rate injection breakthrough curve, whereas for the slug injection data, increased identifiability for these parameters is only observed during the tailing portion of the breakthrough curve. Results demonstrate that selection of slug injection or constant rate experiment type impacts parameter sensitivity and time-varying identifiability and that experimental data can easily be analyzed to understand the relative information content associated with each parameter of a 1-D transient storage model.

1. Introduction

[2] Stream solute transport models that account for advection, dispersion, and transient storage are commonly used to simulate observations from experimental releases of conservative tracers. Manual calibration or automated search algorithms can find a set of best-fit model parameters, which maximize the match of simulated to observed data. The best-fit parameter set may then be used to quantify the advective, dispersive, and transient storage properties of a given stream reach [Briggs et al., 2010; Harvey et al., 1996; Wondzell, 2006]. There are generally two types of solute addition experiments: (1) slug injections (SI), in which a relatively large mass of dissolved tracer is instantaneously released into the stream to label a single parcel of stream water, and (2) constant rate injections (CRI), in which a relatively small load of dissolved tracer is continuously introduced to the stream at a constant rate for a known duration of time, labeling (to a lesser extent than the SI approach) many parcels of stream water passing the injection location.

[3] Experimentalists are faced with many decisions prior to the execution of a tracer experiment, such as reach length, experiment timing, choice of tracer, and choice of experimental method (CRI or SI). Within this technical note, the authors use the term experimental design to refer to the fundamental choice of experiment type (i.e., SI or CRI), assuming that all other design variables (reach length, discharge, and solute choice) have been held reasonably constant. Payn et al. [2008] addressed this choice of experimental design (CRI or SI) by comparing SI and CRI experiments through nonparametric residence time distribution (RTD) analysis. They determined that both CRI and SI data have similar RTDs and hydrologic retention characteristics and concluded that linear transport models (e.g., OTIS) are appropriate for modeling both data sets. On the other hand, Wagner and Harvey [1997] compared both experiment types parametrically using synthetically generated concentration data and a 1-D transient storage model. They concluded that choice of experiment design has a profound effect on model results due to differences in local (i.e., around the optimum) parameter sensitivities and information content between SI and CRI data sets. However, to the best of our knowledge, there has not yet been a study that investigates transient storage model parameter sensitivity and time-varying identifiability using global sensitivity techniques for real-world experimental data.

[4] This note expands on a methodology presented by Wagener et al. [2002, 2003] to elucidate the influence of experimental design on parameter sensitivity and time-varying identifiability (i.e., parameter uniqueness), using two field data sets collected in injections of contrasting design (SI versus CRI). We also seek to add to previous work by Wagner and Harvey [1997] by evaluating a 1-D solute transport model with global sensitivity analysis strategies, rather than local, and applying these methods to real, rather than synthetic experimental data.

2. Methodology

[5] Experimental work was carried out on a 265 m reach of I8-Outlet, a low gradient, cobble bottom, alluvial tundra stream, located in the northern foothills of Alaska's Brooks Range. CRI and SI experiments were completed by 15 and 16 July 2011, respectively. Dissolved Cl was used as the conservative tracer: 12 kg NaCl dissolved in 113.5 L of stream water was near instantaneously injected for the SI, and 22.7 kg NaCl dissolved in 113.5 L of stream water was injected for approximately 4 h at approximately 470 mL/min for the CRI. Continuous specific conductance measurements, which were later transformed to continuous concentration (mg/L) estimates, were logged at 5 s intervals at the top and bottom of the reach throughout each experiment using a HOBO Conductivity Data Logger U24-001.

[6] Discharge was measured prior to each injection via dilution gauging methods [Kilpatrick and Cobb, 1985], and was 20% higher (133 L/s) during the SI than during the CRI (110 L/s) experiment. Discharge has been shown to be a partial control on transient storage in the same arctic tundra stream considered in this study [Zarnetske et al., 2007]. However, because the discharge variability observed in our study (110–130 L/s) is small relative to the variability observed by Zarnetske et al. [2007] (90–820 L/s), we can reasonably assume that the small difference in discharge observed between our two experiments will not overwhelmingly influence parameter estimation and parameter sensitivity.

[7] Tracer data were simulated with the one-dimensional solute transport with inflow and storage (OTIS) model (equations (1) and (2)) [Runkel, 1998].

display math(1)
display math(2)

where Q is stream discharge (m3/s), C is main channel solute concentration (mg/m3), A is channel area (m2), D is the dispersion coefficient (m2/s), qLIN is the lateral inflow discharge (m3/s m), CL is the storage zone solute concentration (mg/L), AS is the storage zone area (m2), and α is the main channel—storage zone exchange rate (1/s).

[8] This analysis is based on Monte-Carlo sampling of 2000 points in the feasible parameter space of A, D, AS, and α. Channel area A, D, and AS were uniformly sampled, while α was uniformly sampled from the log-transformed space because its feasible range spans several orders of magnitude. The performance of each parameter set was evaluated with a root-mean-squared error (RMSE) objective function (equation (3)),

display math(3)

where csim,i and cobs,i are the simulated and observed concentrations at the ith sample, and n is the total number of samples. The objective function was evaluated across the entire BTC. Smaller RMSE values indicate a better fit of simulated to observed data. Parameter sets were ranked by RMSE performance and the top 10% parameter sets were selected to discern behavioral (good performing) from nonbehavioral (poor performing) parameter sets (even though we accepted that more than 10% might provide acceptable simulations). Throughout the remainder of this note, we will be referring to “behavioral” samples as the best 10% of the total 2000 samples considered in the Monte Carlo analysis. We selected the 10% threshold because we are only investigating whether or not the top of the parameter distribution is variable across the feasible parameter range, i.e., whether a clear optimum can be identified or not (see discussion in Wagener et al. [2003]). For further information on the behavioral parameter sets, please refer to supporting information provided online.

[9] Sensitivity is a measure of a parameter's influence on a model's output. Parameters with higher sensitivity have a greater influence on the model output. We determined the sensitivity of the four model parameters using regional sensitivity analysis (RSA) [Freer et al., 1996; Spear and Hornberger, 1980; Wagener et al., 2001]. RSA compares the cumulative distribution functions (CDFs) of behavioral and nonbehavioral parameters. If the CDFs are similar in shape to the uniform CDF, then no particular range of values for this parameter is preferable to another, hence indicating a parameter that is not sensitive. However, if the CDF shapes differ, better parameter values are distributed across a narrower range of feasible values, indicating parameter sensitivity.

[10] A parameter is globally identifiable if it is uniquely locatable within the feasible parameter space [Kleissen et al., 1990]. In this note, we analyze the identifiability of each model parameter through time along the concentration-time profile with dynamic identifiability analysis (DYNIA) [Wagener et al., 2002, 2003]. The objective of the DYNIA framework is to locate times of high identifiability, and thus high information content, along the concentration profile (or some other time series) for each parameter. DYNIA is a direct extension of RSA as implemented by Freer et al. [1996] and Wagener et al. [2001]. Instead of generating a behavioral parameter set by evaluating RMSE across the entire concentration profile (RSA), DYNIA evaluates RMSE across a moving window. Here we chose a 4 min window size (including 48 time steps). So at each time step, the slope and 90% confidence interval of the behavioral CDF are evaluated for each parameter. Narrower confidence intervals and steeper gradients correspond to times of higher identifiability.

3. Results and Discussion

[11] The CDFs of the behavioral parameter populations for SI and CRI data are shown in Figure 1. The most sensitive parameter by far for both experimental designs is channel area A (Figure 1a), as shown by the greatest skew of the behavioral CDF. The median value for A, as indicated by dashed lines, lies between 7.5 and 8 m2 for the SI data and between 6.5 and 7 m2 for the CRI data. This difference is likely due to the higher discharge conditions and enhanced advection during the SI experiment. Dispersion D is slightly more sensitive using the SI data compared to the CRI data (Figure 1b). Storage zone area AS shows greater sensitivity and a smaller behavioral parameter range using the CRI data (Figure 1c). Storage zone – main channel exchange rate α is the least sensitive of all parameters for both the SI and CRI data sets (Figure 1d).

Figure 1.

Regional sensitivity analysis using the OTIS model structure for (a) parameters channel area A, (b) longitudinal dispersion D, (c) storage zone area AS, and (d) storage zone exchange rate α. CRI data results are displayed by red lines and SI data results area displayed as blue lines. Dashed lines in Figure 1a indicate the median A values for each experiment type. The uniform cumulative distributions are marked by solid thin black lines on each plot.

[12] To further interpret the data shown in Figure 1, the coefficients of variation (CV) were computed for the behavioral sets of each parameter, for both CRI and SI data sets. CV is defined as the standard deviation of the behavioral set of parameter i, divided by the mean of the behavioral set of parameter i (equation (4)),

display math(4)

where CV is the coefficient of variation value, μ(pi,behavioral) is the mean of the behavioral set of parameter i, and σ(pi,behavioral) is the standard deviation of the behavioral set of parameter i. Coefficient of variation values provide a way to compare parameter uncertainties between parameters values of very different magnitudes. CV values for both CRI and SI behavioral parameter values are shown in Table 1. Channel area, A, has the lowest CV value, nearly one order of magnitude beneath other values, for both SI and CRI data sets. The dispersion parameter has a lower degree of variation across the behavioral range with SI data compared to CRI data, and AS has a lower degree of variation across the behavioral range with CRI data compared to SI data. Parameter coefficient of variation results largely corroborates RSA, shown in Figure 1 and discussed above, where A was found to have the greatest sensitivity, D was slightly more sensitive with SI data compared to CRI data, and AS was slightly more sensitive with CRI data compared to SI data.

Table 1. Coefficient of Variation Values for Behavioral Parameters From Both CRI and SI Data Sets
 Slug InjectionConstant Rate Injection
D0.230.27
A0.050.08
AS0.440.39
α0.580.59

[13] The disproportionate sensitivity of A relative to all other parameters, in both experiment types, is partially a function of using the RMSE objective function, in which the error is squared and accumulated over the whole breakthrough curve. RMSE is more sensitive to timing errors than amplitude errors, because a slight difference in timing will create a significant error during the highest concentration periods, thus dominating the overall performance metric. Because A controls the advective component of transport, it has a strong influence on peak concentration (SI) and plateau (CRI) timing at the downstream monitoring location. Following the continuity equation, if A is too high for a particular discharge, the simulated breakthrough curve will arrive late at the downstream monitoring location; likewise, if A is too small, the simulated breakthrough curve will arrive early at the downstream monitoring location. Compared to A, other parameters are minimally sensitive for both experimental data sets, as shown by Figure 1 and Table 1, ultimately leading to unreliable parameter estimates for D, AS, and α for the river reach studied.

[14] Figure 2 shows the results of the DYNIA analysis for both SI and CRI data sets. The CDF gradient at each time step is shown as grayscale and 90% confidence intervals are displayed as dashed lines bounding the grayscale. Darker colors and narrower 90% confidence intervals correspond to periods of higher parameter identifiability. The vertical location of the darker colors indicate the range in which most of the behavioral parameters are located (the fraction of parameter sets found in this range is proportional to the gray shade—all sets in one grid cell would turn the cell black, no set would turn it white). The normalized concentration profile is superimposed on the plots to show changes in relative concentration through time.

Figure 2.

DYNIA analysis for OTIS model structure with SI (a, c, e, g) and CRI (b, d, f, h) data. The parameters are (a and b) channel area A, (c and d) longitudinal dispersion coefficient D, (e and f) storage zone area AS, and (g and h) storage zone exchange coefficient α. Bold dashed lines (Figures 2a and 2b) correspond to median best performing A values from RSA analysis. Bold boxes (1–12) highlight regions of increased parameter identifiability.

[15] For the SI data set, A shows a region of well-identified parameter values on the rising limb of the concentration profile through the peak concentration (Figure 2a, region 1). Shortly after the peak value and through the tail, parameter identifiability deteriorates and good-performing parameter values are widely distributed over the feasible range. The CRI data set shows well identified values of A on the rising limb (Figure 2b, region 2) and falling limb (Figure 2b, region 3) of the concentration profile, with more poorly identified parameter values through the leading plateau shoulder and concentration tail. The A values corresponding to regions of highest identifiability for both the SI and the CRI data sets, as shown by the dashed lines (Figures 2a and 2b), agree with the median best performing values from the RSA analysis (Figure 1a). This somewhat intuitive observation illustrates the connection of these two analyses, as DYNIA is simply a time-varying extension of RSA.

[16] Dispersion, D, shows well-identified values prior to the arrival of tracer for both SI (Figure 2c, region 3) and CRI (Figure 2d, region 4) experiments. Following the arrival of tracer the parameter exhibits a rapid deterioration in identifiability, where the 90% confidence intervals abruptly widen and more optimal parameter values are broadly distributed across the parameter space.

[17] Storage zone area AS, for the SI data set, is poorly identified throughout the arrival, peak, and early tail periods. However, during the late tail times (Figure 2e, region 6), better performing parameter values are distributed over a narrower portion of the parameter space. On the other hand, the CRI data set shows regions of increased AS identifiability on the rising and falling shoulder regions and tailing segments of the concentration-time profile. Similar to findings by Wagener et al. [2002], the CRI data set highlights an interaction between A and AS, where AS is best identified when A is most poorly identified—across the plateau shoulders and late tail times (Figure 2f, regions 7, 8, and 9). This interaction is forced by the contrasting functionality of A and AS. Channel area, A, controls the advective transport and thus heavily influences the timing of peak concentration values, whereas AS controls the late time release of solute from storage as well as the early-time filling of storage zones, hence the shape of the concentration profile tail.

[18] Storage zone – main channel exchange rate α is well identified in the tailing portions of both SI data (Figure 2g, region 10) and CRI data (Figure 2h, region 12). Thus, α has partial control on the late time release of tracer from storage zones to the main channel. The CRI data set also shows well-identified periods across the leading shoulder and early plateau times of the concentration profile (Figure 2h, region 11). This is likely due to the initial, early time, saturation of storage zones with tracer.

4. Conclusions

[19] This technical note aims to elucidate the influence of experimental design on transient storage model parameter sensitivity and time-varying identifiability, using Monte Carlo-based analysis methods, and experimental data from tracer experiments in a low gradient, alluvial headwater tundra stream in northern Alaska. Our conclusions are threefold: (1) experimental design has a profound influence on the global sensitivity and the time-varying identifiability of parameters in the OTIS model structure. (2) Data from the SI method are associated with increased model sensitivity to the D parameter, while data from the CRI technique are associated with increased model sensitivity to the AS parameter. (3) Data from the CRI technique reveal heightened identifiablility for the AS and α parameters during rising and tailing portions of the concentration-time profile, whereas slug injection data only show heightened identifiability to these parameters during the tailing portion of the concentration-time profile.

[20] The results found in this study largely corroborate findings summarized in Figure 5, from Wagner and Harvey [1997]. However, we used experimentally gathered tracer data in combination with robust global Monte Carlo sensitivity analyses to highlight basic differences in parameter sensitivity and identifiably between SI and CRI modeled tracer data. Given the use of real experimental tracer data, we believe the results shown in this note are important to support the designing of tracer experiments and to guide understanding of the model limitations/capabilities for field data from tracer experiments.

[21] These results are specific to the two injections simulated and to the characteristics of the reach analyzed. It was not our goal to propose a blanket assessment of which experimental design is better or worse, but rather to put forward a methodology for investigating parameter sensitivity and identifiability to experimental data sets. Wagener et al. [2002] came to similar conclusions based on slug injections in a low gradient UK stream, and Gooseff et al. [2005] came to similar conclusions based on a constant rate injection along Uvas Creek, CA. Scott et al. [2003] reported low sensitivities for transient storage parameters (α and AS) using CRI methods on 3 of 5 reaches on a small, steep mountain stream in CA. The techniques described here may be applied to comparative studies of breakthrough curve behavior to investigate the role of, for example, morphology on solute transport model parameter sensitivity, which would lead to more generalizable results. Our Monte Carlo-based approach is straightforward and can be applied to determine whether or not a parameter is sufficiently sensitive for solute transport modeling.

Acknowledgments

[22] The authors thank Toolik Field Station, William Bowden, Wilfred Wollheim, Kyle Whittinghill, Malcolm Herstand, Genna Woldvogel, Clarie Treat, Jon Herman, and Christa Kelleher for their field support, computational assistance, and ongoing encouragement. This material is based upon work supported by The National Science Foundations Office of Polar Programs under collaborative grants 0902029, 0902113, and 0902106. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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