SOLUTE TRANSPORT DYNAMICS IN ALASKAN ARCTIC TUNDRA STREAMS

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SOLUTE TRANSPORT DYNAMICS IN ALASKAN ARCTIC TUNDRA STREAMS

ABSTRACT

(Chapter 1)

One-dimensional solute transport modeling to simulate experimental tracer releases in rivers is common practice. There are generally two experimental designs employed for tracer (1) slug injections (SI), where 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), where 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. However, relatively few studies have investigated the effect of experiment design 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 Onedimensional Transport with Inflow and Storage (OTIS) model, and analyzed with Monte Carlobased techniques to investigate differences in parameter sensitivity and time-varying identifiability. Slug injection data showed sensitivity to the longitudinal dispersion parameter, while constant-rate injection data exhibited sensitivity to the storage zone area parameter. Constant rate injection data shows heightened identifiability for the storage zone area and storage zone – main channel exchange rate parameters during rising and tailing portions of the breakthrough curve, whereas slug injection data only show heightened identifiability to these parameters during the tailing portion of the breakthrough curve. Results show that experimental design affects parameter sensitivity and time-varying identifiability, and that experimental data may be easily analyzed to understand information contents associated with each parameter in a

1D transient storage model.

(Chapter 2)

 

Advection, dispersion, and transient storage are three dominant solute transport processes in natural stream channels. One-dimensional numerical transient storage models are often used to simulate these processes along with stream flow gains and losses, in order to understand the relative spatial and temporal scales of each process.  Here we describe a new approach to determine the influence of advection, dispersion, transient storage, and longer-term exchanges by directly analyzing the solute data collected from stream tracer experiments, with no parametric modeling required.  The inherent challenges of using solute transport models include issues related to model appropriateness and parameter identification.  Using this new approach, we are able to parse the timescales and quantity of labeled water (i.e., tracer mass) that experience each process.  We are better able to discern the relative influence of advection/dispersion, transient storage, and long time-scale exchange across conditions in a single stream, and across different stream systems. We apply our approach to many slug injections of dissolved salt in lake-inlet and lake-outlet stream reaches in arctic Alaska, and are able to compare and contrast the characteristic transport processes of each reach.

(Chapter 3)

Lakes have been shown to alter basic geomorphic and hydrologic characteristics of outlet streams, compared to inlet streams. However, it is not well understood how lake-influenced differences in channel structure, open-channel hydrology, and subsurface hydrology affect solute transport mechanisms in inlet and outlet streams. Two arctic headwater streams, underlain by continuous permafrost, on Alaska’s North Slope were intensively monitored from June to September 2011. Sites were selected to focus on the influence of a single high arctic lake, known as I8-Lake. I8-Inlet, a 555m reach directly upstream of the lake is un-influenced by any upstream lakes, while I8-Outlet is a 386m reach located directly downstream of the lake. Width:depth ratio at I8-Outlet was 33 and 20 at I8-Inlet. I8-Outlet had consistently higher Manning’s n values at all discharge conditions. Water temperatures along I8 Outlet were consistently higher than water temperature along I8 Outlet. Outlet:inlet discharge ratio declined from 4 to 1 over a  4 month period (June through September of 2011) as lake storage from snowmelt drained throughout the thawed season. Shallow groundwater table dynamics at I8-Inlet were largely controlled by precipitation events, whereas I8-Outlet had a more stable shallow groundwater table, which rose quickly in the spring and remained relatively stable throughout the season. A non-parametric analysis based on objective breakthrough curve decomposition methods was used to analyze many small conservative slug injections from each stream to characterize solute transport differences. We found that more tracer mass was associated with the transient storage timescale on I8-Outlet compared to I8-Inlet (p = 0.005), while more tracer mass was associated with advection/dispersion timescales on I8-Inlet compared to I8-Outlet. we concluded that 1) a high arctic lake imposes measurable hydrologic and geomorphic changes along the down-valley river continuum, and 2) hydrogeomorphic differences amongst streams above and below a small arctic lake create significantly different solute transport environments. More specifically, the contribution of transient storage in I8 Outlet is greater than was observed in I8 Inlet.

 

 

 

 

 

TABLE OF CONTENTS

List of Figures ………………………………………………………………………………………………………….v

List of Tables ……………………………………………………………………………………………………………vi

Acknowledgements……………………………………………………………………………………………………vii

Chapter 1 The influence of experimental design on parameter identifiability in a 1D

transient storage model for stream solute transport …………………………………………………1

1.1 Introduction………………………………………………………………………………………………….1

1.2 Methodology………………………………………………………………………………………………..2

1.3 Results & Discussion …………………………………………………………………………………….5

1.4 Conclusions………………………………………………………………………………………………….9

Breakthrough curve decomposition: A framework for analyzing solute transport

processes in rivers, independent of numerical transport models………………………………..10

Chapter 2………………………………………………………………………………………………………………….10

2.1 Introduction………………………………………………………………………………………………….10

2.2 Methodology………………………………………………………………………………………………..12

2.3 Results and Discussion…………………………………………………………………………………..16

2.4 Conclusions………………………………………………………………………………………………….19

Chapter 3 Hydrogeomorphic contrasts between inlet and outlet streams of a high arctic

lake and subsequent solute transport implications…………………………………………………..20

3.1 Introduction………………………………………………………………………………………………….20

3.2 Study Site…………………………………………………………………………………………………….22

3.3Methodology…………………………………………………………………………………………………24

3.4 Results…………………………………………………………………………………………………………30

3.5 Discussion……………………………………………………………………………………………………36

3.6 Conclusions………………………………………………………………………………………………….43

References………………………………………………………………………………………………………………..45

Chapter 1  

 

The influence of experimental design on parameter identifiability in a 1D

transient storage model for stream solute transport

1.1 Introduction

Stream solute transport models that account for advection, dispersion, and transient storage are commonly used to simulate observations from experimental releases of conservative tracers. A set of best-fit model parameters, which maximize the match of simulated to observed data, can be found by manual calibration or using automated search algorithms. The best-fit parameter set may then be used to characterize the spatial and temporal extent of advective, dispersive, and transient storage properties of a given stream reach [Briggs et al., 2010; Harvey et al., 1996; Wondzell, 2006b]. There are generally two experimental designs used for solute additions: (1) slug injections (SI), where 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), where a relatively small load of dissolved tracer is continuously introduced to the stream at a constant rate for a known duration of time, labeling (probably to a lesser extent than the SI approach) many parcels of stream water passing the injection location.

Experimentalists are faced with a decision of which injection method to use – SI or CRI? Payn et al. [2008] addressed this question by comparing SI and CRI experiments through non – parametric 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 1D transient storage model. They concluded that experimental design has a profound affect on model results due to differences in 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 interrogates experimental data for transient storage model parameter sensitivity and time-varying identifiability using global sensitivity techniques.

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. We also seek to add to previous work by Wagner and Harvey [1997] by evaluating a 1D solute transport model with global sensitivity analysis strategies and applying these methods to real, rather than synthetic experimental data.

1.2 Methodology

Experimental work was carried out on a 265 m reach of I8-Outlet, a low gradient, alluvial tundra stream. This work was part of a larger study of hydroecological characteristics of tundra streams. However, during mid-simmer, tundra streams physically behave similarly to stream in temperate areas [Edwardson et al., 2003; Greenwald et al., 2008; Zarnetske et al., 2007]. Thus this the methodologies described in this note are extensible to other biomes and is not restricted to arctic environments.

CRI and SI experiments were completed July 15 and 16, 2011, respectively. Dissolved

NaCl was used as the conservative tracer: 12 kg for the SI, and 200 g/L injected for 3.5 hrs for the CRI. Discharge during the SI was 20% higher (133 L/s) than discharge during the CRI (110 L/s).

Continuous specific conductance measurements were logged (HOBO Conductivity Data Logger – U240-001) at five-second intervals at the top and bottom of the reach throughout each experiment.

Tracer data was simulated with the One-dimensional Solute Transport with Inflow and

Storage (OTIS) model (equations 1 and 2) [Runkel, 1998].

where Q is stream discharge (m3/s), C is main channel solute concentration (mg/m3), A is channel area (m2), Dx 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).

This analysis is based on Monte-Carlo sampling of 2000 points in the feasible parameter space, of A, Dx, AS, and α. A, Dx, 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),

(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. 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 non-behavioral (poor performing) parameter sets (even though we accepted that more than 10% might provide acceptable simulations). 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 (see discussion in Wagener et al., 2003).

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]. RSA compares the cumulative distribution functions (CDF) of behavioral and non-behavioral 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 shapes of the CDFs are very different, better parameters values are distributed across a narrower range of feasible values, indicating higher parameter sensitivities.

A parameter is globally identifiable if it is uniquely locatable within the parameter space, where a lack of identifiably makes it impossible to accept of reject constituent model hypotheses, given observed data [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 concentrationprofile (or some other time series) for each parameter. DYNIA is a direct extension of RSA as implemented by Freer et al. [1996]. 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 four-minute window size. So, at each time step the slope and 90% confidence interval of the behavioral CDF is evaluated for each parameter. Narrower confidence intervals and steeper gradients correspond to times of higher identifiability.

1.3 Results & Discussion

The CDFs of the behavioral parameter populations for SI and CRI data are shown in

Figure 1-1. The most sensitive parameter by far for both experimental designs was channel area A (Figure 1-1a), 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 Dx is slightly more sensitive using the SI data compared to the CRI data (Figure 1-1b). Storage zone area AS shows greater sensitivity and a smaller behavioral parameter range using the CRI data (Figure 1-1c). Storage zone – main channel exchange rate α is the least sensitive of all parameters for both the SI and CRI data sets (Figure 1-1d).

 

Figure 1-1: Regional Sensitivity Analysis using the OTIS model structure for parameters channel area A (a), longitudinal dispersion DX (b), storage zone area AS (c), and storage zone exchange rate α (d). CRI data results are displayed by grey lines and SI data results area displayed as black lines. Dashed lines on plot (a) indicate the median A values for each experiment type.

 

The disproportionate sensitivity of A relative to all other parameters, in both experiment types, is likely a function of the RMSE objective function, when the error is accumulated over the whole breakthrough curve. RMSE is far more sensitive to timing errors than amplitude errors, because a slight difference in timing will create a significant error during the highest concentration periods. 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. Thus, A is the most sensitive parameter. Compared to A, other parameters are minimally sensitive for both experimental data sets, ultimately leading to unreliable parameter estimates for Dx, AS, and α for the river reach studied.

 

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

Figure 1-2 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, or simply the vertical extent of the grayscale. Darker colors and narrower 90% confidence intervals correspond to periods of higher parameter identifiability. The vertical location of the darker colors indicates in what range most of the behavioral parameters are located (the number of parameter sets found in this range is proportional to the color – all sets in one grid cell would turn the cell black). The normalized concentration profile is superimposed on the plots to show changes in relative concentration through time.

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 1-2a, region 1). Shortly after the peak value and through the tail, parameter identifiability deteriorates and goodperforming parameter values are widely distributed over the feasible range. The CRI data set shows well identified values of A on the rising limb (Figure 1-2b, region 2) and falling limb (Figure 1-2b, 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 (Figure 1-2a and b), agree with the median best performing values from the RSA analysis (Figure 1-1a). This somewhat intuitive observation illustrates the connection of these two analyses, as DYNIA is simply a time-varying extension of RSA.

Dispersion, D, shows well-identified values prior to the arrival of tracer for both SI (Figure 1-2c, region 3) and CRI (Figure 1-2d, region 4) experiments. Following the arrival of tracer the analysis exhibits a rapid deterioration in parameter identifiability, where the 90% confidence intervals abruptly widen and more optimal parameter values are broadly distributed across the parameter space. A possible explanation for the early time identifiability of this parameter is that D mechanistically controls the spreading of the solute front. Thus, D will control the initial arrival time of tracer at the downstream monitoring location, given that A is appropriately identified.

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 1-3e, 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 1-2f, regions 7, 8,& 9). This interaction if 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.   Storage zone – main channel exchange rate α is well identified in the tailing portions of both SI data (Figure 1-2g, region 10) and CRI data (Figure 1-2h, 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 1-2h, region 11). This is likely due to the initial, early time, saturation of storage zones with tracer.

The results found in this study largely corroborate findings by 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, the results shown in this note can be better used for experimental design and to understand modeling limitations/capabilities of field data from tracer experiments.

1.4 Conclusions

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. We arrived at the following three conclusions: (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 Dx parameter, while data from the CRI method are associated with increased model sensitivity to the AS parameter. (3) Data from the CRI method show heightened identifiablility for 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.

These results are specific to the two injections simulated and to the characteristics of the reach analyzed. Wagener et al. [2002] came to similar conclusions based on slug injections in a low gradient UK stream and 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, USA.  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. 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.

SOLUTE TRANSPORT DYNAMICS IN ALASKAN ARCTIC TUNDRA STREAMS

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