An evaluation of analytical streambank flux methods and connections to end-member mixing models : a comparison of a new method and traditional methods

Introduction Conclusions References


Introduction
Groundwater and surface water interactions are an important process in hydrologic systems (Winter, 1998).These interactions within and around streams and rivers impact decisions on municipal water supply extractions, water pollution, riverine habitat, and many others.To make better decisions on these impacts, the stream and ground Figures water interactions (streambank fluxes) need to be accurately quantified as stream losses and gains can account for a substantial proportion of the total flow and chemical load of a stream.In general, when people consider how to estimate the losses or gains along a stream reach they would take a discharge measurement upstream, a discharge measurement downstream, subtract the two values, and the result would be considered the gain or loss within the stream reach.Although this may be a relatively simple procedure to accomplish, the assumption that all flow within a stream reach must be either flowing into the stream or flowing out of the stream is in many cases an over simplification (Castro and Hornberger, 1991;Harvey and Bencala, 1993).Depending on local topography, geology, and the groundwater table, gains and losses into and out of the stream can be very dynamic even over short distances (Harvey and Bencala, 1993;Anderson et al., 2005;Payn et al., 2009).Consequently, what might have originally been estimated as a small gain to the stream from simply subtracting the upstream and downstream discharges might end up becoming a small loss out of the stream and a large gain into the stream.Without a proper method to estimate streambank fluxes, any attempt at estimating a water or nutrient mass balance would be difficult and laced with errors.Harvey and Wagner (2000) and many other researchers use a more realistic conceptual model of flow pathways within a stream (Fig. 1).These major flow pathways include initial (or upstream) discharge (Q init ), final (or downstream) discharge (Q final ), stream gains from groundwater (Q in ), stream losses to groundwater (Q out ), and hyporheic flow (Q hyp ).In this conceptual model, Q in is considered to be pure groundwater entering the stream, and Q out is stream water permanently leaving the stream.Hyporheic flow occurs when stream water temporarily leaves the stream into the surrounding groundwater (or more specifically the hyporheic zone), but returns again to the stream at some downstream location.During this temporary departure from the stream, additional biochemical reactions may occur that would not necessarily have occurred while in the stream itself.The mass is still retained in the stream and not lost (permanently) to the groundwater.Although the hyporheic flow pathways do occur and can be very im-Figures

Back Close
Full portant for stream ecosystems (i.e. the movement of oxygen into the hyporheic zone, nitrogen cycling, etc.), hyporheic flow will not be directly addressed in this study as the authors are most interested on fluxes that are permanently adding or removing mass over a significant length of stream.As hyporheic flows only temporarily leave the stream, the mass of the water is still retained over sufficient distances.
There are a number of methods to estimate gross stream gains and losses (Kalbus et al., 2006).The general categories are seepage meters, (heat or chemical) tracer tests, and hydraulic gradients derived from groundwater piezometers.Each has advantages and disadvantages.Seepage meters and groundwater piezometers are point measurements that can be accurate at a specific point, but in a heterogeneous system they may not represent the stream as a whole.On the other hand, chemical tracer tests are an aggregation of all fluxes along a stream reach, but do not represent any particular point along the stream.For this study, the focus is on the total aggregated flows over the stream reaches, so chemical tracer tests were found to be the most appropriate and inexpensive.Kalbus et al. (2006) and Scanlon et al. (2002) have a more thorough qualitative review of the different streambank flux methods.
Using chemical tracer tests for the source of data, the estimation of gross stream gains and losses is most frequently performed through numerical models like those similar to the OTIS model developed by the USGS (Runkel, 1998).While able to estimate fluxes in steady-state conditions, these types of models are primarily designed for non-steady-state conditions and provide many output parameters in addition to the inflow and outflow fluxes, and as a consequence require more input data than in steadystate conditions for estimating only streambank fluxes (i.e.stream cross-sectional area, flow advection, flow dispersion, etc).Additionally, the OTIS type models would require the estimation of parameters through a trial-and-error or an automated nonlinear least squares (NLS) procedure that are not directly measured.Under steady-state conditions, the data and parameter requirements for estimating only streambank fluxes are substantially lower requiring only discharge and tracer concentration measurements Introduction

Conclusions References
Tables Figures

Back Close
Full There are two traditional analytical methods to estimate streambank fluxes under steady-state conditions ignoring hyporheic flowpaths.These methods use simple mass balance equations to estimate both gains and losses within a stream reach and assume that the fluxes are independent and in a specific sequence.In this paper, a new analytical method has been developed using different assumptions on the spatial distribution of the inflowing and outflowing fluxes along the stream.
The goal of our study is to quantitatively evaluate the accuracy and sensitivity of the new method against the existing steady-state streambank flux tracer methods.This evaluation is performed through a combination of analytical comparisons and numerical stream simulations as described in the following sections.

Theoretical basis of the streambank flux tracer methods
All tracer based methods designed to estimate streambank fluxes start with the conservation of mass equations under steady-state conditions for both the tracer and the water flux and assume complete mixing of the individual flows: where Q final is the final discharge (in volume per unit time), C final is the final concentration (in mass per unit volume), Q init is the initial discharge, C init is the initial concentration, Q in is the discharge from the groundwater to the stream, C in is the concentration of Q in , Q out is the discharge from the stream to the groundwater, and C out is the concentration of Q out .Introduction

Conclusions References
Tables Figures

Back Close
Full The two traditional streambank flux estimation methods mentioned in the introduction make specific assumptions on the distribution of gains and losses throughout the reach (see Fig. 2).The first method, we call "Loss-Gain", assumes C out = C init , while the second method, we call "Gain-Loss", assumes C out = C final .In both variants, the methods assume that the mixing of Q in C in and Q out C out are mixed separately and are mixed in a sequence defined by the above assumptions.The Loss-Gain variant assumes that the mixing sequence begins with Q out followed by Q in , while Gain-Loss is vice-versa.
Combining Eqs. ( 1) and ( 2), the solution for Q out for Loss-Gain is: Similarly, the equation for Q out for Gain-Loss is: To get Q in for both methods, we need to include Eq. (2) into Eqs.( 3) and (4): If we use an artificial tracer (i.e.Bromide salt), we can safely assume C in ≈ 0 and the resulting equations are as follows:

Conclusions References
Tables Figures

Back Close
Full and Q in becomes: These methods can be applied conceptually along a stream length as illustrated in the left and central scheme of Fig. 2. Q init is the upstream discharge and Q final represents the downstream discharge.Depending on the equation variant, Q in is added or Q out is removed from Q init at the beginning of the stream and Q out is removed or Q in is added at the end of the stream resulting in a downstream discharge of Q final .As these methods make no assumptions about the exact location along the stream for Q in and Q out , they can occur over any length of the stream as long as they occur in sequence and independently.
From studies that tested multiple stream reaches for streambank fluxes, almost every stream reach had both gains and losses regardless of the method and of the reach length (Anderson et al., 2005;Ruehl et al., 2006;Payn et al., 2009;Covino et al., 2011;Szeftel et al., 2011).Additionally, studies that have tried to identify the spatial distribution of groundwater inflows and outflows to and from the stream have found a wide variety of diffuse flow locations throughout the stream and were not limited to one or two flow locations every several hundred meters (Malard et al., 2002;Wondzell, 2005;Schmidt et al., 2006;Lowry et al., 2007;Slater et al., 2010).This indicates that even short stream reaches typically have many instances of gains and losses to and from the stream and that limiting the flux instances to one flux each regardless of the stream length may not be the most accurate assumption.
Following this rationale, this paper presents a novel method based on a different assumption for the spatial distribution of streambank fluxes as compared to the Gain-Loss and Loss-Gain methods, namely that both Q in and Q out occur simultaneously and are constant throughout the entire stream section.This new method is denoted as Introduction

Conclusions References
Tables Figures

Back Close
Full "Simultaneous".Equations requiring the same input data as the Gain-Loss and Loss-Gain methods are derived in Sect.2.2 and length is integrated into the mass balance equation (Fig. 3).

Derivation of the method for simultaneous gains and losses
In this section, the fundamental equations of mass balance for the tracer and water flows will be applied on a control volume represented in Fig. 3 under the assumption of simultaneous and uniform gains and losses throughout the stream reach and stationarity in time in order to obtain the expressions predicting Q in and Q out as functions of Q init , C init , Q final , C final and C in .First, applying mass balance for discharge: where x is stream length, Q(x) is the discharge at length x, q in is the added discharge per unit length of stream, and q out is the lost discharge per unit of length.Both q in and q out are assumed constant for a given stream reach.In the one-dimensional and stationary case we can write

∂Q(x)
∂x dx = dQ.After rearranging and integrating from the beginning of the reach over an arbitrary length: which becomes: Then, applying mass balance for the tracer:

Conclusions References
Tables Figures

Back Close
Full where C(x) is the concentration at length x and C in is the concentration of q in .The inflowing concentration C in is assumed constant for a given stream reach.Again in the one-dimensional and stationary case, we can write

∂Q(x)
∂x dx = dQ and ∂C(x) ∂x dx = dC.Neglecting second order differentials and rearranging: Substituting Eqs. ( 12) and (13) for Q(x) and dQ respectively in Eq. ( 15), and rearranging: Simplifying and integrating from the beginning of the reach over an arbitrary length x: Evaluating Eq. ( 13) for x = L, where L represents the total length of the stream reach: Substituting Eq. ( 19) in Eq. ( 18) and evaluating for x = L:

Conclusions References
Tables Figures

Back Close
Full Calling Q in = q in • L and rearranging: and the solution for Q out is: where Q in,Sim and Q out,Sim are the Simultaneous equations for the streambank fluxes into and out of the stream, respectively.
As with the previous methods, if we use an artificial tracer (i.e.Bromide salt) we can safely assume C in ≈ 0 and the resulting equations are as follows: Naturally occurring tracers (i.e.Chloride salt) can also be applied to the Simultaneous equations with additional information about C in .As long as a quasi-steady-state condition applies and that Q in > 0, the only additional information to be collected would be 10428 Introduction

Conclusions References
Tables Figures

Back Close
Full where C init,prior is the upstream concentration prior to the tracer injection, C init is the upstream concentration from the tracer injection, C final,prior is the downstream concentration prior to the tracer injection, and C final is the downstream concentration from the tracer injection.The only main disclaimer to the application of this equation in the field is that the difference between C init,prior and C final,prior must be large enough to be statistically significant when estimated using available laboratory or field measurement techniques.The accuracy of the measurement techniques is a general problem for any chemical tracer test performed to estimate streambank fluxes.If the difference between the Q init and Q final is very small, much tracer may be needed to accurately measure a concentration difference between C init and C final .This issue will become more important with larger rivers as the proportion of the Q in and Q out to the Q init is substantially reduced.
It would also be possible to estimate C in from groundwater piezometers adjacent to the bank of the stream.As the intent of our study was to determine integrated values over a stream reach rather than point values, we preferred to use Eq. ( 25) as it is an integrated value of C in .
There might also be a need to estimate the groundwater concentration of other chemical compounds entering the stream in addition to the conservative tracer used to estimate the streambank fluxes.If other in-stream gains and losses in the new chemical compound can be neglected (i.e.without biochemical transformations), the only additional information needed would be the concentration of the new compound at the locations of Q init and Q final .The C in of the new chemical compound can be estimated Introduction

Conclusions References
Tables Figures

Back Close
Full using the following rearrangement of Eq. ( 21): where C in, new is the concentration of the new compound, C init, new is the upstream concentration of the new compound, and C final, new is the downstream concentration of the new compound.Any of the three streambank flux methods can be rearranged to calculate C in, new and they will all produce the same result.
The application of tracer methods to measure streambank fluxes in the field are typically performed by two different techniques: constant injection and slug injection.These two techniques have been well researched in the scientific community and will not be evaluated in this study (Wagner and Harvey, 1997;Payn et al., 2008).Both techniques can be used with the above streambank flux methods and provide very similar results.For simplicity, we will assume constant injection with steady-state conditions as the slug injection would require C final to be integrated over time.

Analytics
All three streambank flux methods were broken down analytically to better understand the dynamics of the equations of the methods.We wanted to know what caused the differences in the results of the three streambank flux methods and how these differences were related.The relative differences between the methods was accomplished by the ratio of one method's equation to another both analytically and illustratively.Introduction

Conclusions References
Tables Figures

Back Close
Full

Numerical simulations
Perfect measurements or estimates of streambank fluxes are impossible using any existing method.Arbitrarily comparing results of different methods using field collected data will only indicate that the different methods produce different results, and it will not indicate if one method is more accurate than another.Consequently, we thought that it would be appropriate to simulate an artificial stream with known streambank fluxes for comparisons.With streambank fluxes perfectly known, we could effectively evaluate the accuracy of the different methods.
We simulated the lateral inflows and outflows per unit length throughout a stream using an autoregressive integrated moving average (ARIMA) model performed using the arima.simpackage in the R statistical computing environment (R Development Core Team, 2011).The routine generates a variety of artificial time series with both a randomness and memory component.In an attempt to create realistic simulations of the streams, we tuned the ARIMA model to have spatial flux dynamics based on studies using distributed temperature sensing (DTS) of groundwater inflows within streams (Lowry et al., 2007;Westhoff et al., 2007;Briggs et al., 2012;Mwakanyamale et al., 2012).The quantitative surrogate we used for the spatial flux dynamics was the average length that the fluxes would switch from inflow to outflow or vice-versa within a stream reach.For example, if we simulate a stream with 1000 m total length and the fluxes in this stream oscillates between inflows and outflows 10 times then the av- The spatial discretization of the model was 1 m for all series and simulations.These four series of simulations were to test the effects of both length and intermittency on the stream flux methods.Without loss of generality, we defined C in = 0 for the simulations, which would be equivalent to the use of an artificial tracer (i.e.bromide salt) for the tracer test.
We tested two distinct assumptions when deciding on the appropriate streambank flux ARIMA model.One assumption was that both Q in and Q out can occur simultaneously at one point.For example, if the groundwater table is sloped perpendicular to the stream then water would be flowing into one side of the bank, while water would be flowing out of the other side of the bank.In this assumption, we created two separate and independent vectors of Q in and Q out along the stream.The second assumption was that both Q in and Q out cannot occur simultaneously at one point.In this assumption, only one vector of streambank flux was created that could oscillate between Q in and Q out .We decided to omit the option for simultaneity of Q in and Q out throughout the stream as this assumption coincides with the assumption in the Simultaneous method and consequently the Simultaneous method was vastly superior to the other streambank flux methods.To ensure a more rigorous evaluation against the Simultaneous method, we decided to omit the ARIMA model assumption of simultaneity and only use the non-simultaneity assumption for the simulations.
We attempted to simulate the stream with realistic dynamics of streambank fluxes, but we also tried to keep the model complexity as simple as possible.Although we did attempt to cover a wide range of streambank conditions when creating the many simulations, undoubtedly we did not cover all possible streambank flux conditions that could exist in nature.Realistically, the scientific community does not even know the full range of possibilities for natural streambank fluxes.We have also likely created simulations of streambank fluxes that do not exist in nature.Both issues are unavoidable when creating hydrologic simulations, particularly with the stochastic generation approach used in Introduction

Conclusions References
Tables Figures

Back Close
Full this paper.The hope is that the flux distributions of the simulations do closely represent reality for the purpose of our evaluation.
The statistical evaluation consisted of several methods and procedures.First, we took all of the simulated scenarios (5000 in our case) within an individual series and averaged the inflows and outflows for each simulation.This gave us an average inflow to the stream and outflow from the stream over the entire length of the stream for each scenario and served as our "true" values of the fluxes that the other streambank flux methods would be compared to.Next, we calculated the streambank fluxes of each scenario using the three streambank flux methods from the starting and end values of the scenarios.We did not include additional randomness in the input values for the streambank flux methods, which would equate to measurement error.This is due to the large variety of measurement devices and techniques that could be used in a tracer test, and each device and technique would have different measurement errors associated with them.Additionally, we calculated the net flux (we will call "Net") simply by subtracting Q init from Q final .We considered the Net as the upper error benchmark for the evaluation as the estimation of Net requires less information and should therefore perform worse than the other three streambank flux methods that require more information.
Once the streambank fluxes were calculated for all of the methods to be evaluated, we used as a performance measure the absolute normalized error for method m and for each simulation i, defined as: where Q m est,i is the estimated gross gain or loss value from the streambank flux method m and simulation i and Q true,i is the average flux from the ARIMA model at simulation i.The results of ε m i are two vectors (one for gross gains and one for gross losses) for each of the four methods.Each vector contains 5000 elements, one for each scenario.
To make an overall evaluation for each method, we simply took an average of all of the Introduction

Conclusions References
Tables Figures

Back Close
Full scenarios in each series for both vectors of gains and losses: where ε m is the mean absolute normalized error for each method m (either flux leaving the stream or entering the stream) and n is the total number of scenarios in each series (5000).This is a compound measure of relative bias and accuracy.
In addition to calculating the ε m for all of the streambank flux methods, we compared the ε m i within each of the streambank flux methods to determine how frequently one method outperformed another: where r m1,m2 is the frequency of m1 streambank flux method outperforming m2 streambank flux method.Once ε m i and ε m were estimated, we wanted to determine the causes of the errors in the individual methods.This was accomplished through a correlation of the ε m i to various combinations of the input parameters.

Analytics
When there is 0 flux of either Q in or Q out all three equations produce the same results.For example, if Q out = 0 then Eq. ( 2) becomes:

Conclusions References
Tables Figures

Back Close
Full As Q final and Q init are previously known, there is only one solution for Q in regardless of the other equations.Similarly, as the ratio of Q in to Q out grows to infinity or to 0, the results for the three equations will converge.Although somewhat obvious, if all of the assumptions are met for any of the streambank flux methods then the method will perfectly reproduce reality.For example, if there is only inflow to the stream from 1-100 m followed by only flow out of the stream from 101-1000 m then the Gain-Loss equation will estimate both fluxes perfectly.
If Q in > 0 then C final < C init assuming that C in = 0 from an artificial tracer injection.Subsequently, both C final and C init represent the end points of the concentration profile within the stream.As formulated in Eqs. ( 7)-( 10), the Loss-Gain and Gain-Loss equations are divided by the end point concentrations of the stream and will therefore represent the minimum and maximum values of fluxes within a stream reach.The Loss-Gain equations will always produce the minimum flux values, while the Gain-Loss equations will always produce the maximum flux values.Consequently, as Loss-Gain and Gain-Loss will have the minimum and maximum flux values, the flux values for the Simultaneous equations must be somewhere in between the two.
The Gain-Loss and Loss-Gain methods are very similar, and subsequently can be compared quite easily.Dividing the inflow and outflow equations for the two methods can show the rate of increase of one method over the other: and For both Q out and Q in , Gain-Loss grows from Loss-Gain at a rate proportional to the concentration ratio, and additionally Q in grows with load ratio.As Q out and Q in increase in a stream reach, Q final will change and C final will decrease.In the case of a lower C final Introduction

Conclusions References
Tables Figures

Back Close
Full caused by higher streambank fluxes, the ratio between the results of Gain-Loss and Loss-Gain grows larger (Fig. 4).
Unfortunately, the Simultaneous method does not simplify nearly as well as the others due to the non-linearity of the Simultaneous equations.For a better visual comparison, the three methods were plotted together with axes of concentration and discharge ratios (Fig. 4).As shown analytically in Eqs. ( 31) and ( 32), the ratio of Gain-Loss to Loss-Gain is insensitive to discharge for Q out and sensitive to both discharge and concentration for Q in .The ratios of Simultaneous to the other methods illustrate the non-linearity of the method.The methods' ratios for Q in show a surprising similarity in the distribution of the contours even though the magnitudes are different.

Numerical simulations
Figure 5 presents the major input and output parameter density distributions created by the ARIMA simulations for the inflow and outflow profiles.The parameter distributions for Q out , Q in , and Q net closely follow a normal distribution.As defined in the model, Q init and C init are equally distributed between 1 to 5 L s −1 and 20 to 150 mg L −1 , respectively.
The results of the numerical simulations are presented in Tables 1 and 2. Plots of the estimated gains and losses to the actual gains and losses for each of the methods for Series A are illustrated in Fig. 6.The plots for the other scenarios have similar patterns only with a greater or lesser degree of spread.The numerical simulations indicate that the Simultaneous streambank flux equation is on average the best performer when compared to the other two streambank flux methods with a 1 : 1 slope to the true value, the lowest ε m in every series, and the highest r m1,m2 in nearly every series.However, the Loss-Gain method has a slightly higher r m1,m2 to Net as compared to Simultaneous.Interestingly, simply using the net discharge between upstream and downstream (Net of ε m then ε m would be exactly 1. Gain-Loss and Simultaneous can have errors greater than 1 as they can have values larger than the true value, which is clearly exemplified by the Gain-Loss equation's high ε m in some series.
Figures 7 and 8 show the six simulations with the smallest ε m i for both Loss-Gain and Gain-Loss.Not surprisingly, they performed best when the assumptions of the individual methods were met. Figure 9 shows the six simulations with the smallest ε m i for Simultaneous.No obvious conclusion can be drawn from the simulations other than an evenly random spread between Q in and Q out with no clear spatial bias unlike the other methods.
The ratios of C init to C final and Q init C init to Q final C final show a strong correlation to the ε m i of the streambank flux methods (Fig. 10).They are the same ratios that were found during the analytical evaluation described by Eqs. ( 31) and ( 32).Both Loss-Gain and Gain-Loss have stronger correlations than Simultaneous.Simultaneous appears to have an error bias towards lower values rather than the full range of the correlation.
Loss-Gain and Gain-Loss also have a strong correlation to the midpoint concentrations and loads.Gain-Loss had a strong correlation to the ratios of C mid (the midpoint of the concentration profile of the stream) to C final and Q init C init to Q mid C mid (the midpoint of the load profile of the stream).Loss-Gain had a very strong correlation to the ratios of C mid to C init and Q final C final to Q mid C mid .

Bank flux methods evaluation
The streambank flux methods followed different patterns through the 4 series of the ε m .
Net was only slightly affected by both the switch length and the length of the stream reach.Alternatively, Simultaneous was not significantly affected by the switch length, but was affected by the stream length.Loss-Gain and Gain-Loss were affected by both the switch length and the stream length.Introduction

Conclusions References
Tables Figures

Back Close
Full While Simultaneous was the best performer in all categories, Gain-Loss performed substantially worse on the ε m overall and was only better than Net on the ε m for Series A and B and for the r m1,m2 .As described in the previous sections, the Gain-Loss equations can create results that can be many times larger than the other methods and consequently can be many times larger than the true value from the ARIMA model.
Although these circumstance may account for a small proportion of the total simulations, they can cause the average error to be very high.Net was clearly superior in Series C and D for the ε m , but Gain-Loss had a solid majority over Net in the r m1,m2 .In the Series A, Gain-Loss and Net had a similar ε m , but according to r m1,m2 Gain-Loss performed better almost 80 % of the time.Indeed, if the top 10 % of the simulations with the highest errors were removed from Series C then Gain-Loss and Net would have approximately the same ε m .Nevertheless, even with the help of removing 10 % or 20 % of the simulations with the highest errors, both Loss-Gain and Simultaneous performed substantially better than Gain-Loss.Most of the ε m i errors in Loss-Gain and Gain-Loss could be correlated by the ratio of the upstream and downstream concentrations for Q out and the ratio of the upstream and downstream loads for Q in .Q out,G-L had an especially strong correlation.Loss-Gain on the other hand had an especially strong correlation to the concentration and load midpoints along the stream (not shown in figures).As with much of the previous results, the midpoint correlations follows precisely the assumptions of the methods.Loss-Gain assumes that the Q out occurs at the beginning and if the ratio of C mid to C init does not follow a relationship that the method assumes then it will produce a larger error.At least in Loss-Gain, it appears that if the concentration ratio does not follow the predicted pattern by the time it reaches the midpoint then the method is more likely to create erroneous results.A similar pattern can be seen in Gain-Loss, but not nearly as strong as the upstream and downstream ratios.Unfortunately, Simultaneous did not have such clear correlations.There only appears to have an error trend towards smaller upstream and downstream ratios.

HESSD Introduction Conclusions References
Tables Figures

Back Close
Full There is much scientific literature on the estimation of streambank fluxes from chemical tracers.Many have preferred to use the well established OTIS numerical model, which effectively solves the differential equations with a finite difference model with similar spatial flux assumptions to our Simultaneous method.We found only one study that took the OTIS model and tested the three different assumptions that we also tested (Szeftel et al., 2011).However, the reasoning behind their test appeared to be precisely the opposite of ours.As they stated in the methods, they assumed that the simultaneous inflow and outflow at a single cell was unrealistic and implemented the Loss-Gain and Gain-Loss type scenarios to provide better alternatives.Although they did not test the accuracy of the three methods, they concluded that the spatial variability of the streambank fluxes had a significant impact on the output and that breakthrough curve (BTC) analysis is not sufficient to determine the spatial variability.
Like us, others have instead preferred to use the more simple analytical equations to estimate streambank fluxes (Harvey and Wagner, 2000;Payn et al., 2009;Covino et al., 2011).One of the earliest to hint at using tracers with analytical equations to determine streambank fluxes was Zellweger et al. (1989).The use of tracers with dilution gauging to estimate streambank fluxes was only mentioned in passing as an explanation for the differences in the estimation of discharge from a current meter and from dilution gauging.Later, Harvey and Wagner (2000) picked up on the idea of using dilution gauging with a current meter to estimate streambank fluxes.Their description for the procedure to estimate streambank fluxes was purely qualitative and did not fully explain the underlying assumptions in the method that they proposed (e.g. the spatial distribution of the fluxes).The dilution gauging method to estimate discharge was referenced back to Kilpatrick and Cobb (1985).Based on the dilution gauging method and the description provided by Harvey and Wagner (2000), they effectively proposed the use of the Gain-Loss method.Interestingly, Covino et al. (2011) also referenced back to Kilpatrick and Cobb, 1985, but  between the two different methods and correctly identified that the Loss-Gain and the Gain-Loss methods produce the maximum and minimum values for streambank fluxes, respectively.

Connections with end-member mixing models
End-member mixing models or end-member mixing analysis (EMMA) as they tend to be known is a method to estimate the relative contributions of defined upstream source waters from a downstream discharge measurement point.For example, EMMA can estimate the amount of groundwater contribution within a single hydrograph.
EMMA is used extensively for this precise purpose.Similarly to the bank flux methods, EMMA uses the mass balance equations with chemical tracers to formulate the model.The EMMA equations are well known and the equation for two end-members is the following: where Q gauge is the discharge at the stream measurement gauge, Q gauge,s1 is the part of the discharge of Q gauge from the first source, C gauge is the concentration of the tracer at the stream measurement gauge, C s1 is the concentration of the tracer in the first source, and C s2 is the concentration of the tracer in the second source.Equation ( 33) is strikingly similar to Eq. ( 5).Additionally, if we consider a bromide tracer test with the first source as upstream discharge and the second source as groundwater with C s2 ≈ 0, then the equation simplifies in the same way as Eq. ( 9).
Although they may look the same, they have different underlining assumptions and derivations.
The derivation of Eq. ( 33) is usually conceptualized by the conservative mixing of two sources in a large reservoir with only one outflow defined above as Q gauge .What if the equation for EMMA were derived in the context of a stream reach, with both Q in Introduction

Conclusions References
Tables Figures

Back Close
Full and Q out included in the derivation?Would Eq. ( 33) be different?Since these questions have not been directly addressed in the literature, we will provide the derivation setting up a scenario with the two sources similar to the streambank flux scenarios.It will have a first source equal to the upstream discharge with the upstream concentration (Q s1 = Q init , C s1 = C init ), a second source equal to the the diffuse groundwater entering the stream (Q s2 = Q in , C s2 = C in ), and the the gauging site equal to the downstream conditions (Q gauge = Q final , C gauge = C final ).First, we write the mass balance equation for the downstream discharge: where Q gauge,s1 and Q gauge,s2 are the respective parts of the discharge Q gauge from the first and second sources, i.e. upstream discharge and groundwater inflow.Separating Q out in their respective components and due to mass balance for the discharges gives us: and where Q s1 is the total discharge of the first source and Q s2 is the total discharge of the second source.We then write the mass balance for the tracer flows going out of the stream: where Q out,s1 is loss of water specifically from Q s1 and Q out,s2 is loss of water specifically from Q s2 .Then we apply the mass balance equation for the tracer between the upstream and downstream (similar to Eq. 1): Combining Eqs. ( 37) and ( 38) we get the following equation with some rearrangement.
The outflow components have completely fallen out and Eq. ( 40) has become Eq.( 33).
The end-member mixing equations are insensitive to any outflows from the stream system, and subsequently will provide the same result regardless of the streambank flux spatial dynamics occurring within the system upstream of the measurement gauge.Also, if we substitute the parameters in Eq. ( 40) with the relevant parameters used during the streambank flux derivations (e.g.Q init , Q final , and Q in ) then the right hand of the equation becomes identical to the right hand of Eq. ( 5).The fundamental difference between these two equations is the left hand of the equations: Q gauge,s2 and Q in are conceptually different, i.e. the amount of discharge downstream from the groundwater source does not need to be necessarily equal to the gross diffuse inflow throughout the reach.Nevertheless, it is important to realize that the estimation of Q in with the Loss-Gain assumptions leads to the same results as the EMMA estimate for Q gauge,s2 .
The similarity between Eq. ( 40) and the Loss-Gain Eq. ( 5) appears to have led some researchers to inadvertently apply the EMMA Eq. ( 40) to estimate the gross stream gains and losses (Covino and McGlynn, 2007).Although these streambank flux estimates are correct in terms of the Loss-Gain assumptions, it appears to be more of coincidence than deliberate, as little reasoning and background is given for the implicit Loss-Gain hypothesis on the spatial inflow and outflow dynamics of the EMMA equation when used for estimating streambank fluxes.Similarly, Briggs et al. (2012) referenced the above EMMA model (Kobayashi, 1985) to estimate streambank fluxes, but they instead used the Gain-Loss method without explanation.Introduction

Conclusions References
Tables Figures

Back Close
Full Both EMMAs and streambank flux analyses can be performed on a stream reach with the same input data to acquire several important hydrological aspects of surface water and groundwater interactions.They could be applied simultaneously to a given stream reach to estimate both the amount of discharge downstream from the groundwater source and the gross diffuse inflow throughout the reach, which will have different values if assumptions different to those of Loss-Gain are made.The streambank flux equations were derived for steady-state conditions and should be applied as such, while the EMMA derivation has no such critical steady-state requirement and subsequently can also be used in transient conditions (i.e.throughout a flooding event).While the streambank flux equations cannot be used dynamically throughout a flooding event, it would be possible to apply the streambank flux equations lumped over an entire flooding event.For example, the streambank flux equations could be applied by summing the total non-baseflow water in a flood hydrograph and the average concentration of a tracer at an upstream and downstream gauge.This use of the streambank flux equations would not be as accurate as in true steady-state conditions due to the additional assumption of a constant baseflow during the flooding event, which is certainly not going to be true.Nevertheless, as long as the changes in the baseflow water accounts for a small percentage of the total flow during an event, the equations should provide relatively accurate values.

Conclusions
A new streambank flux estimation method is presented and derived analytically with the assumptions of constant, uniform, and simultaneous groundwater inflow and outflow throughout a given stream reach.This novel method is confronted against the two traditional methods and presents the smallest error measures when applied to four different sets of generated scenarios.The main control of the model performance for all three cases is the spatial dynamics of the actual streambank fluxes in relationship with the assumptions for each method.Also for the same inputs, the different assumptions Introduction

Conclusions References
Tables Figures

Back Close
Full of each method can lead to values of gross stream gains and losses differing up to one order of magnitude between approaches.Estimating streambank fluxes using the proposed simple analytical method over numerical models solving full hydrodynamic sets of partial differential equations has the clear advantages of much less complexity and less parametrization.Although separate from the streambank flux methods, endmember mixing analysis can be used in conjunction with the streambank flux methods to acquire even more hydrologic information as both require the same type of input data.Nevertheless, these two approaches should not be conceptually mixed as they estimate different stream variables and are based on distinct derivations and assumptions.Introduction

Conclusions References
Tables Figures

Back Close
Full  Full A conceptual overview of the major inflows and outflows within a stream reach.Q init is the upstream discharge in volume per time, Q final is the downstream discharge, Q in is the groundwater entering the stream, Q out is the stream water leaving the stream to the groundwater, and Q hyp is the hyporheic flow water water that is temporarily leaving the stream into the hyporheic zone (reproduced after Harvey and Wagner, 2000).Introduction

Conclusions References
Tables Figures

Back Close
Full Fig. 2. The conceptualizations of the three streambank flux methods.The Loss-Gain method assumes Q in occurs in the first section followed by Q out in the last section.The Gain-Loss method assumes Q out occurs in the first section followed by Q in in the last section.Both the Loss-Gain and Gain-Loss methods assume that Q in and Q out occur in sequence and independently, although the lengths of the first and last sections are arbitrary and can be of any length that when summed together equal the total length.The Simultaneous method assumes that Q in and Q out are constant and occur simultaneously throughout the entire length of the stream reach.Introduction

Conclusions References
Tables Figures

Back Close
Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | upstream and downstream.If steady-state is appropriate, then analytical methods are sufficient.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | the C init and C final prior to the injection of the tracer.The equation to solve C in can be formulated from any of the streambank flux equations by setting itself equal to itself except replacing one side with the C init and C final prior to the injection of the tracer.
Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | erage length per switch would be 100 m.For our simulations, we used two different switch lengths of 100 m and 200 m and total stream lengths of 1000 m and 2000 m.The switch lengths had a strong linear relationship with the correlation lengths and resulted in correlation lengths of 40 m and 70 m for the switch lengths of 100 m and 200 m, respectively.Correlation length is commonly defined as the length at 1/e on the autocorrelation distribution (Blöschl and Sivapalan, 1995).The ARIMA model allowed us to create 5000 simulations of stream fluxes within a hypothetical stream.We ran four series of 5000 simulations.Series A had a 1000 m Discussion Paper | Discussion Paper | Discussion Paper | stream length and a 100 m average switch length, Series B had a 1000 m and a 200 m average switch length, Series C had a 2000 m and a 100 m average switch length, and Series D had a 2000 m and a 200 m average switch length.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) results in lower error values for ε m as compared to Gain-Loss in both 2000 m series.This is attributed to the fact that Net by definition cannot have an error of 1 or greater.Similarly, Loss-Gain also cannot have errors 1 or greater and must have errors less than those of Net.If 0 is used for all the values of Q in and Q out in the error assessment 10436 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | they instead used the Loss-Gain method.Payn et al. (2009) and Ward et al. (2013) estimated streambank fluxes using both the Loss-Gain and the Gain-Loss methods.They also found significant differences in the estimations Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | high resolution temperature observations, this is the publisher's final pdf, the published article is copyrighted by European Geosciences Union, available at: http://www.egu.eu/(last access: 27 June 2013), 2007.10431 Winter, T. C.: Ground water and surface water: a single resource, no.1139 in US Geological Survey circular, US Geological Survey, Denver, Colo, 1998.Discussion Paper | Discussion Paper | Discussion Paper | Fig. 1.A conceptual overview of the major inflows and outflows within a stream reach.Q init is the upstream discharge in volume per time, Q final is the downstream discharge, Q in is the groundwater entering the stream, Q out is the stream water leaving the stream to the groundwater, and Q hyp is the hyporheic flow water water that is temporarily leaving the stream into the hyporheic zone (reproduced afterHarvey and Wagner, 2000).
Discussion Paper | Discussion Paper | Discussion Paper |

Fig. 4 .Fig. 5 .
Fig. 4. Relative comparisons between the different methods due to changes in the input ratios.The rows are the ratios of two of the streambank flux methods for both Q out and Q in .For example, if the ratio of the input parameters C init and C final is 5 and the ratio of the input parameters Q init and Q final is 1 then the Simultaneous method will result in a Q in approximately 2 times larger than the Loss-Gain method.

Table 1 .
ε m : the average value of ε m i for each series and for both Q out and Q in .Sim is abbreviated for Simultaneous.