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Sep 26, 1972 - Our study used inverse modeling to estimate parameters of the transient storage model OTIS. (One dimensional Transport with Inflow and ...
J. N. Am. Benthol. Soc., 2003, 22(4):492–510 q 2003 by The North American Benthological Society

Automated calibration of a stream solute transport model: implications for interpretation of biogeochemical parameters DURELLE T. SCOTT1,5, MICHAEL N. GOOSEFF2, KENNETH E. BENCALA3, ROBERT L. RUNKEL4

AND

US Geological Survey, 430 National Center, 12201 Sunrise Valley Drive, Reston, Virginia 20192 USA Department of Aquatic, Watershed, and Earth Resources, Utah State University, Logan, Utah 84322 USA 3 US Geological Survey, Mail Stop 439, Menlo Park, California 94025 USA 4 US Geological Survey, Denver Federal Center, Mail Stop 415, Denver, Colorado 80225 USA 1

2

Abstract. The hydrologic processes of advection, dispersion, and transient storage are the primary physical mechanisms affecting solute transport in streams. The estimation of parameters for a conservative solute transport model is an essential step to characterize transient storage and other physical features that cannot be directly measured, and often is a preliminary step in the study of reactive solutes. Our study used inverse modeling to estimate parameters of the transient storage model OTIS (One dimensional Transport with Inflow and Storage). Observations from a tracer injection experiment performed on Uvas Creek, California, USA, are used to illustrate the application of automated solute transport model calibration to conservative and nonconservative stream solute transport. A computer code for universal inverse modeling (UCODE) is used for the calibrations. Results of this procedure are compared with a previous study that used a trial-and-error parameter estimation approach. The results demonstrated 1) importance of the proper estimation of discharge and lateral inflow within the stream system; 2) that although the fit of the observations is not much better when transient storage is invoked, a more randomly distributed set of residuals resulted (suggesting nonsystematic error), indicating that transient storage is occurring; 3) that inclusion of transient storage for a reactive solute (Sr21) provided a better fit to the observations, highlighting the importance of robust model parameterization; and 4) that applying an automated calibration inverse modeling estimation approach resulted in a comprehensive understanding of the model results and the limitation of input data. Key words: solute transport, inverse modeling, transient storage, hyporheic zone, parameter estimation, tracer, OTIS, UCODE.

The transport and cycling of nutrients and metals within streams and the hyporheic zone is an active research topic (Kim et al. 1992, Tate et al. 1995, Harvey and Fuller 1998, Runkel et al. 1998, 1999, Chapra and Runkel 1999, Chapra and Wilcock 2000, Hinkle et al. 2001). These coupled environments (stream and hyporheic zone) are important regions of biological activity and nutrient uptake, which may significantly alter material cycling. For example, nutrient-rich stream water enters the hyporheic zone where microbial processes remove inorganic nutrients from hyporheic waters (e.g., through denitrification) leaving return flows to the stream depleted in inorganic nutrients compared to incoming stream water (Mulholland et al. 1997, A. R. Hill et al. 1998). Tracer studies are often used to examine the cycling of reactive solutes and identify and quantify biogeochemical processes occurring 5

E-mail address: [email protected]

within the coupled stream–hyporheic environments. This approach allows for quantification of potential storage zones within a stream, which includes the hyporheic zone. For example, D’Angelo et al. (1993) conducted several tracer experiments on 1st- to 3rd-order streams to examine the effects of transient storage on solute transport. Morrice et al. (1997) demonstrated the importance of the geological composition and alluvial characteristics within a catchment on nutrient processing within the hyporheic zone by doing several tracer experiments in geologically different catchments. Runkel (2002) used data collected from 53 tracer experiments to illustrate the effectiveness of a new transient storage metric based on median reach travel time. Other studies have used tracer additions to examine solute transport and instream reactions (Broshears et al. 1996, McKnight et al. 2001, 2002). For example, Tate et al. (1995) injected a conservative solute (LiCl) and radiolabeled PO432 to study the fate and transport of PO432

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in a small mountain stream. Their nonconservative transport and fate results were improved by considering hydrologic exchange and uptake in the storage zone. Stream solute models are based on physical processes that are difficult to measure in the field. In stream studies, the field observations from a tracer experiment are used to obtain parameter estimates with the goal of then examining hydrological and biogeochemical processes. The solute transport model parameters are calibrated against the observed conservative tracer concentrations. One approach used in calibrating solute transport models is trial and error (Bencala 1983, Bencala and Walters 1983, Bencala et al. 1990, D’Angelo et al. 1993, Morrice et al. 1997, Vallett et al. 1996). This approach involves adjusting the input parameter values to obtain a visual match of modeled output to field observations. Although 1 set of parameter values may be obtained that fit the observations, 3 common problems may exist: 1) the parameter set may be highly correlated, that is, individual parameters may not be independently identified, 2) a number of parameters may be unsupported by the field observations, requiring further examination of the modeled process, and 3) another parameter set provides an equal or better fit to the observations. These pitfalls are extremely important to stream ecologists because, if the underlying hydrologic model is flawed, further analysis using the model to examine a nonconservative solute may lead to erroneous conclusions. An alternate approach to calibrate a model is to use an inverse modeling procedure. We define inverse modeling as an automated calibration process where the relationships between the model, observations, and the parameters are mathematically quantified using techniques such as nonlinear least-squares regression to provide the best-fit estimates for the model and a robust, optimized parameter set. Our 1st objective in this paper is to demonstrate the benefits of using a formal inverse modeling procedure to calibrate solute transport models instead of the trial-and-error approach. We used UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling often used in groundwater applications (Mehl and Hill 2001), to parameterize 2 transient storage models. Benefits of inverse modeling include 1) obtaining best-fit parameter values, 2) assessing the qual-

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ity of model calibration and issues such as parameter correlation, and 3) the ability to quantitatively compare alternative models (Poeter and Hill 1997). To illustrate the use of UCODE, we used a well-published data set from a stream tracer experiment on Uvas Creek (Zand et al. 1976, Bencala and Walters 1983, Wagner and Gorelick 1986, Wo¨rman 1998). We chose to illustrate our main points using UCODE because this tool provides extensive information on the ability of the model to fit the data reliably through summary statistics and assessment of the influence of each observation on model fit. Calibration within UCODE is straightforward, which allows easy evaluation of different models (i.e., inclusion of transient storage). We compared previous nonautomated calibration approaches to the inverse modeling results, with particular attention to the transient storage parameters and associated metrics. Our 2nd objective is to investigate the importance of transient storage on conservative and reactive solutes. Thus, we calibrated the advection–dispersion model both with and without transient storage for a conservative solute. We then compared the 2 models using UCODE’s statistics and the resulting residual distributions, and applied the 2 calibrated models to Sr21 transport within the stream system, using UCODE to calibrate Sr21 sorption parameters. We addressed implications for other reactive solutes using these results and calculated transient storage metrics, such as the storage zone residence time (tS) (Thackston and Schnelle 1970), exchange length (LS) (Mulholland et al. 1994), and a combination of the 2, the hydrological retention factor (Rh) (Morrice et al. 1997). These metrics are useful in interpreting transient storage modeling results and comparing stream reach characteristics. Because these metrics depend on parameter estimates from the transient storage transport calibrations, optimal parameter values that provide a unique solution to the model and result in a best fit to the data are desired. Our results are intended to illustrate the value of a thorough model calibration, which is required to gain an understanding of the reliability and uncertainty of transient storage metrics. Methods Uvas Creek solute addition experiment Solute was added on 26 September 1972 to Uvas Creek, Santa Clara County, California

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(Zand et al. 1976). Uvas Creek is a small, steep, mountain stream with a slope of 0.03 for the experimental reach (Bencala and Walters 1983, Avanzino et al. 1984). A concentrated solution of dissolved NaCl and Sr21 was added to the stream at 50 mL/min for 3 h. Uvas Creek was close to annual baseflow conditions (12.5 L/s) at the addition site. Background stream Cl2 and Sr21 concentrations were 3.7 mg/L and 0.13 mg/L, respectively. Five stream sampling sites were established along the stream. These 5 sites delimited the 5 study reaches (Reach 1: 0–38 m; Reach 2: 38–105 m; Reach 3: 105–281 m; Reach 4: 281–433 m; Reach 5: 433–619 m; distances are in m downstream of the injection location). Details of sample location and solute analysis are provided in Avanzino et al. (1984). Solute transport We applied OTIS (One dimensional Transport with Inflow and Storage) (Runkel 1998), a 1-dimensional stream solute transport model with inflow and storage. The governing equations within OTIS account for transient storage, lateral inflow, advection, and dispersion as:

1

2

]C Q ]C 1 ] dC q 52 1 AD 1 L (CL 2 C) ]t A ]x A ]x dx A 1 a(CS 2 C) dCS A 5 2a (CS 2 C) dt AS

[1] [2]

where C 5 solute concentration in the stream (mg/L), Q 5 volumetric flow rate (m3/s), A 5 cross-sectional area of the main channel (m2), D 5 dispersion coefficient (m2/s); qL 5 lateral volumetric inflow rate (m3/s-m; equivalent units as m2/s), CL 5 solute concentration in lateral inflow (mg/L), CS 5 solute concentration in the storage zone (mg/L), AS 5 cross-sectional area of the storage zone (m2), a 5 stream storage exchange coefficient (/s), t 5 time (s), and x 5 distance downstream (m). Equations 1 and 2 are used to simulate the conservative transport of Cl2. We also modeled Sr21 dynamics within the OTIS framework by adding terms to transient storage model equations 1 and 2 to account for kinetic sorption, and introducing a 3rd mass balance for the sorbate concentration on the streambed sediment (Bencala 1983, Runkel

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1998). Model equations for a reactive solute with kinetic sorption are as follows: ]C 5 L(C) 1 rlˆ (Csed 2 Kd C) ]t dCS ˆ S 2 CS ) 5 S(CS ) 1 lˆ S (C dt dCsed 5 lˆ (Kd C 2 Csed ) dt

[3] [4] [5]

where L(C) and S(CS) represent physical processes in the main water column and the hyporheic zone (right-hand side of equations 1 and 2, respectively), CˆS 5 background storage zone solute concentration (mg/L), Csed 5 sorbate concentration on the streambed sediment (g/g), Kd 5 distribution coefficient (m3/g), lˆ 5 1st-order rate coefficient for sorption in the stream (/s), lˆ s 5 1st-order rate coefficient for sorption in the storage zone (/s), and r 5 mass of accessible sediment per volume of stream water (g/L). A detailed explanation of this formulation can be found in Bencala (1983). UCODE computations UCODE searches for optimal parameter values of the above models by calculating the weighted least-squares objective function (Table 1, Appendix). Minimizing the least-squares objective function is the goal of the modified Gauss–Newton algorithm used by UCODE. The maximum likelihood objective function (Table 1, Appendix) is also computed, and can be used to compare different models. UCODE allows the modeler to apply differential weighting to observations based on the modeler’s confidence or interest in particular data points. Sensitivity and parameter estimation statistics calculated by UCODE (Table 1, Appendix) can be used in model analysis and to evaluate the optimal parameter estimates. We discuss the dimensionless and composite scaled sensitivities (DSS, CSS), linear 95% confidence intervals (CI), Akaike information criterion (AIC) and Bayesian information criterion (BIC) statistics, and parameter correlation coefficients (cor). The relative magnitudes of the least-squares objective function, the AIC statistic, and the BIC statistic are dependent on the scale of the observation values and the associated weighting of each observation.

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TABLE 1. Summary statistics for the trial-and-error (Bencala and Walters 1983) and inverse modeling calibration approaches calculated using UCODE (Poeter and Hill 198), a computer code for universal inverse modeling. Variable definitions in Appendix. Statistic Objective functions Least-squares Maximum likelihood

Equation S(b ) 5 [y 2 y9(b )] Tv[y 2 y9(b )] S9(b ) 5 (ND 1 NPR)ln(2p) 2 ln(zvz) 1 S(b )

Source Hill 1998 Hill 1998

Sensitivity analysis parameters Parameter sensitivity Dimensionless scaled sensitivity (DSS) Composite scaled sensitivity (CSS) Statistics to assess model fit Akaike information criterion (AIC) Bayesian information criterion (BIC) 95% confidence interval Parameter correlation coefficient

]y9 Dc y9 y9(b 1 Db ) 2 y9(b 2 Db ) 5 (b 1 Db ) 2 (b 2 Db ) ]b Dc b

1 D b 2) zb zv (ss ) z CSS 5 O ND Dc y9i

ssij 5

j

j

c

[

ND

i51

ii

1/2

b

ij

2

]

Poeter and Hill 1998 Hill 1998, similar to Harvey et al. 1996

1/2

b

Hill 1992, 1998, M. C. Hill et al. 1998

AIC 5 S9(b9) 1 (2 3 NP)

Akaike 1974

BIC 5 S9(b9) 1 [NP 3 ln(ND 1 NPR)]

Akaike 1978

ns 2 ; x2U

ns 2 where s 2 5 S(b)/n x2L cov(i, j ) cor(i, j) 5 var(i)1/2var( j)1/2

Sensitivity of the model’s outputs to variations in parameter values provides information on model robustness. UCODE computes individual sensitivities for each observation with respect to each parameter, as well as a summary statistic for the influence of all observations on a single parameter. DSSs are similar to the normalized sensitivities computed by Harvey et al. (1996), and have the same application, representing the importance of particular observations to the estimation of a particular parameter. CSSs are the sum of the root mean square of DSS values and are a measure of the amount of information provided by the observations for the estimation of a single parameter. Larger CSS values for a particular parameter suggest that the observations contribute more information for the optimization of the particular parameter. Several statistics used to assess overall model fit, aside from the least-squares objective function, are also provided (Table 1, Appendix). Parameter 95% CIs are computed based on the calculated error variance of the entire model fit. The AIC statistic is calculated as the sum of the

Ott 1993, Hill 1998 Poeter and Hill 1997, Hill 1998

maximum likelihood objective function and 2 times the number of parameters. Similarly, the BIC statistic is also based on the maximum likelihood objective function (Table 1, Appendix). When the AIC and BIC statistics are compared among models, the model with lower values may be more suitable to the given scenario with the available observations, although other factors (such as specific observations to which the model adequately fits, or differences in observation weighting schemes) need to be considered together to make an appropriate choice. Model uniqueness is determined, to some extent, by parameter correlation coefficients. High correlations coefficients (cor . 0.85) suggest that the 2 parameters cannot be independently resolved. A unique solution with no correlated parameters is also the goal of an optimized, robust model. Model execution of OTIS/UCODE We constructed 4 models to illustrate the utility of inverse modeling for conservative and

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nonconservative transport. In these 4 models, we treated lateral inflow (qL) as a parameter to be estimated. The 1st model calibrated using the conservative solute (Cl2) was the simple advection–dispersion equation (1) without transient storage (a 5 0/s). Parameter estimation for qL, A, and D began in reach 1 using all of the observations within this reach. The fit of the simulated concentration values to the observations with this initial set of parameters was imperfect. Additional sets of parameters were used to generate additional sets of simulated concentration values. As the fit of the simulated concentration values to the observations improved, the model is said to have converged upon an optimal set of parameter estimates. The 2nd step was to estimate parameters for reach 2 using the observations for reach 2, while holding parameters for reach 1 constant. The 3rd step was to use the parameter estimates for reaches 1 and 2 as starting values of the parameters and all of the observations within these 2 reaches and repeating the calibration. This stepwise procedure was continued for each of the downstream reaches, where the 1st step was to estimate values for the specific reach while holding previous reach estimates constant and then calibrating for all of the parameters. The 2nd model considered transient storage. We performed a similar stepwise calibration procedure, beginning in reach 1. We 1st held qL, A, and D constant and estimated a and AS. The 2nd step involved estimating all 5 parameters in conjunction for the 1st reach. We then repeated this procedure for the remaining stream reaches. When we used this 2nd model, the successive sets of tested parameters did not converge upon an optimal parameter set for certain reaches. In these reaches, transient storage was omitted from the model of that reach and then an optimal parameter set was determined. The 3rd and 4th models we considered included kinetic sorption of Sr21 as described above (equations 3–5). As in most studies involving tracer experiments, the conservative solute is used to calibrate the hydrological processes of advection, dispersion, lateral inflow, and transient storage. These calibrated models (in our study models 1 and 2) are then used to simulate nonconservative transport of the reactive solute, Sr21. We used the Sr21 observations from the field experiment combined with laboratory estimated values of Kd (70 3 1026 m3/g) (Bencala

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1983) and a fixed value for CˆS (0.13 mg/L). Using the inverse modeling calibration approach, we estimated lˆ , lˆ s, and r for each reach without (model 3) and with (model 4) transient storage. Although we state these estimated sorption parameters, we included a reactive solute to discuss transient storage processes in relation to reactive solute dynamics rather than the specifics of kinetic sorption.

Calculation of lateral inflow rate and discharge Lateral inflow rates were determined using 2 different approaches. First, lateral inflow rates were specified as transport model parameters and estimated by the approaches described above. (Discharge values along the stream were then computed internally within the transport model from these rates.) The 2nd approach used calculations based directly upon the observed conservative tracer concentrations. Although the plateau-dilution technique for calculating discharge is most often used, plateau concentrations were not achieved in this field experiment. An approach that does not rely on steady state concentrations is through a mass-balance calculation of Cl2 within each of the 5 reaches over the course of the experiment (Kilpatrick and Cobb 1985). The amount of solute injected into the stream is computed, and the solute concentration is integrated with respect to time at each station using the following equation: Q5

E

Min j

[6]

CN (t) dt

where Q 5 stream discharge at a downstream station (m3/s), Minj 5 mass of conservative tracer injected into the stream (g), and CN(t) represents the solute breakthrough curve at the downstream station (g/m3), minus background concentrations. The flow rate can then be determined simply by dividing the total amount of mass computed at the upstream reach by the integrated value of concentration through time. The increase in flow is then used to determine an average lateral inflow rate along each reach. Alternatively, lateral inflow rate may be specified as a model parameter and estimated by inverse modeling as described above.

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Transient storage metrics We discuss 4 metrics pertaining to the storage zone that are commonly used in stream ecology to illustrate the importance of model parameter optimization to understanding solute dynamics (Runkel 2002). The storage zone residence time (Thackston and Schnelle 1970) is as follows: tS 5

AS aA

[7]

where tS is the mean storage zone residence time (s). The 2nd metric calculated in our study is an exchange length (‘‘turnover length’’ in Mulholland et al. 1994): LS 5

u a

[8]

where LS is the exchange length (m), and u is the mean velocity in the stream, equivalent to Q/A (m/s). This value is the distance traveled by an average water parcel in the water column before exchanging with the storage zone. The hydrological retention factor is also computed to illustrate the compounded differences between alternative models (Morrice et al. 1997). It is computed as: Rh 5

tS LS

[9]

where Rh has units of s/m and represents the hydrologic storage time per length of stream. The Damkohler number (DaI) (Wagner and Harvey 1997) is calculated as: DaI 5 a

(1 1 A/AS )L u

[10]

where L is the reach length (m). The DaI metric is a measure of the reliability of transient storage parameters. When DaI values are K1, very little solute enters the storage zone within the given reach. When DaI values are k1, the length of the study reach is so long (relative to the steam velocity) that there is nearly complete solute transfer between the main water column and the storage zone. Results Conservative solute calibration Inverse modeling vs trial-and-error calibration. Before a reactive solute is examined, the 1st step after a tracer experiment is to calibrate the phys-

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ical transport processes within the stream environment. We used UCODE to estimate parameters for a model without transient storage (1) and a model with transient storage (2). Inverse modeling resulted in an adequate match to the stream observations for both model 1 and 2 (Fig. 1A). Visually, the most evident difference between the 2 models occurred during the departure phase of Cl2, after the addition ended. This difference was most notable in reaches 4 and 5. Differences in fit between our inverse modeling results and the trial-and-error calibrations approach (Fig. 1A, B) were most noticeable during the arrival and departure of the tracer in reaches 3, 4, and 5. These differences are seen by comparing the simulated concentrations in Fig. 1A (inverse modeling results) and Fig. 1B (trialand-error modeling results) to the observations. Although the stream water simulations do not appear drastically different, the inverse modeling approach resulted in a distinctly different set of parameters (Table 2). Notable differences between the trial-and-error and inverse modeling approaches were in D, qL, AS, and a, which can have significant effects on storage zone concentrations. The predicted storage zone concentrations in reaches 3, 4, and 5 were much higher with the data set derived from inverse modeling (Fig. 1A, B). Two of the summary statistics, the least-squares objective function and the maximum likelihood objective function, were both lower using inverse modeling calibration (Table 3) compared to the trial-and-error approach, indicating a better match to the observed data. Another important finding in this comparison was that the trial-and-error parameter set had a large number of significant correlations (21 parameter sets .0.85, 15 of which were correlated .0.95), whereas the parameters derived from inverse modeling had no significant correlations (Table 4). We also calculated lateral inflow using a mass-balance approach because of the differences in lateral inflow rates between trial-and-error and inverse modeling approaches (Fig. 2). Our mass-balance-calculated increase in flow from the injection site agreed with both inverse modeling calibrated models. An increase in stream discharge often dilutes solute concentrations; thus, accurate estimates of lateral inflow are critical for examination of reactive solutes to ensure conservation of mass. Identification of the lack of sensitivity provid-

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FIG. 1. Observed Cl2 concentrations in 4 sampling reaches resulting from a steady 3-h salt addition into a stream. A.—Advection–dispersion model with transient storage calibrated using UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling. B.—Advection–dispersion model with transient storage calibrated through trial and error (Bencala and Walters 1983). TS 5 transient storage.

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TABLE 2. Parameter estimates for the trial-and-error approach (Bencala and Walters 1983) and inverse modeling calibration approaches calculated using UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling. Values in parentheses denote upper and lower 95% confidence intervals. A 5 cross-sectional area of the main channel (m2), D 5 dispersion coefficient (m2/s), qL 5 lateral volumetric inflow rate (m3/s-m; equivalent units as m2/s), AS 5 cross-sectional area of the storage zone (m2), a 5 stream storage exchange coefficient (/s).

Bencala and Walters (1983)

This study, without transient storage (model 1)

This study, with transient storage (model 2)

Reach 1 (0–38 m) D (m2/s) A (m2) a (/s) AS (m2) qL (m3/s-m)

0.12 0.3 0 0 0

0.012 (0.01–0.02) 0.31 (0.30–0.31) 0 0 0

0.01 (0.01–0.02) 0.31 (0.306–0.313) 0 0 0

Reach 2 (38–105 m) D (m2/s) A (m2) a (/s) AS (m2) qL (m3/s-m)

0.15 0.42 0 0 0

0.13 (0.11–0.14) 0.42 (0.41–0.43) 0 0 1.4 3 1026 (0.4–4.8 3 1026)

0.13 (0.11–0.14) 0.42 (0.41–0.43) 0 0 1.2 3 1026 (0.3–4.3 3 1026)

Reach 3 (105–281 m) D (m2/s) A (m2) a (/s) AS (m2) qL (m3/s-m)

0.24 0.36 3.00 3 1025 0.36 4.55 3 1026

0.04 (0.02–0.11) 0.32 (0.31–0.33) 0 0 1.5 3 1025 (1.4–1.7 3 1025)

0.07 (0.04–0.13) 0.33 (0.32–0.34) 3.0 3 1025 (2.0–4.4 3 1025) 0.054 (0.034–0.086) 1.3 3 1025 (1.1–1.5 3 1025)

Reach 4 (281–433 m) D (m2/s) A (m2) a (/s) AS (m2) qL (m3/s-m)

0.31 0.41 1.00 3 1025 0.41 1.97 3 1026

0.45 (0.40–0.51) 0.59 (0.57–0.61) 0 0 6.1 3 1026 (3.9–9.6 3 1026)

0.20 (0.15–0.27) 0.50 (0.49–0.52) 2.5 3 1025 (1.9–3.3 3 1025) 0.46 (0.22–0.97) 0

Reach 5 (433–619 m) D (m2/s) A (m2) a (/s) AS (m2) qL (m3/s-m)

0.4 0.52 4.50 3 1025 1.56 2.15 3 1026

0.10 (0.05–0.22) 0.70 (0.68–0.73) 0 0 3.2 3 1025 (2.9–3.6 3 1025)

0a 0.54 (0.49–0.58) 7.8 3 1025 (3.1–19.7 3 1025) 0.12 (0.09–0.16) 2.4 3 1025 (2.1–2.9 3 1025)

Parameter

a

In the model, dispersion was set to 1.00 3 1024 m2/s

ed by observations of individual parameters within UCODE is simple because UCODE’s output reports CSSs (Fig. 3). For example, in model 2, UCODE calculated low CSSs for the transient storage parameters (a, AS) in reaches 1 and 2. Low sensitivities highlighted the inability of UCODE to uniquely estimate these 2 parameters, so a was set to 0 in the first 2 reaches. Model selection. Another benefit of inverse modeling is the resulting model-fit statistics that can be used to choose the most appropriate

model (Table 3). The AIC and BIC statistics are commonly used to help choose between differing models, particularly those with differing numbers of parameters. In our study, high AIC and BIC statistics associated with model 2 indicated that the increased number of parameters did not justify the improvement in fit, probably because the improved fit in model 2 over model 1 occurred mainly in the relatively short rise and fall of the solute breakthrough curves. Although the AIC and BIC statistics indicated

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TABLE 3. Objective function values and statistics to assess model fit for the trial-and-error approach (Bencala and Walters 1983) and with use of UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling. AIC 5 Akaike information criterion, BIC 5 Bayesian information criterion (see Table 1).

Bencala and Walters (1983)

This study, without transient storage (model 1)

This study, with transient storage (model 2)

Objective function values Least-squares Maximum likelihood

53 702

10.3 659

7.9 657

Statistics to assess model fit AIC BIC

740 813

687 741

695 768

Statistic

model 1 was better than model 2, results from the residual analysis suggested the opposite. Cl2 residuals for both model 1 and model 2 are plotted as a function of time in Fig. 4. Model 2, which incorporates transient storage, resulted in a more evenly distributed set of residuals, especially after the Cl2 addition ceased, suggesting that it provided an overall better fit than model 1. TABLE 4. Correlation parameter pairs, based on correlation coefficients computed by UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling. Correlated parameter pairs for trial-and-error approach were determined in a UCODE sensitivity run (not an optimization run). Subscript numbers refer to reach number. A 5 cross-sectional area of the main channel (m2), qL 5 lateral volumetric inflow rate (m3/s-m; equivalent units as m2/s), AS 5 cross-sectional area of the storage zone (m2), a 5 stream storage exchange coefficient (/s).

Parameter correlation level (cor)

Bencala and Walters (1983)

This This study, study, without with transient transient storage storage (model 1) (model 2)

Another result of the inverse modeling procedure is demonstrated by examining the DSSs. Single observations and subsets of observations that had high DSSs are indicated on the Cl2 observations for reach 3 (Fig. 5). Most of the information for qL comes from observations in the phase of the tracer experiment during which concentrations are their most elevated. For D, A, and AS, observations occurring during the arrival and departure of Cl2 provided the most information for the calibration of these parameters. Observations supporting a occurred primarily after the tracer injection ceased. Reactive solute simulation–Sr21. To provide a calibration example for a reactive solute, Sr21 was modeled using the parameters for the hydrological processes derived from the Cl2 calibrations in model 1 and model 2 (Fig. 6). Our inverse modeling results confirmed that sorption in the storage zone was effectively instantaneous; thus, in the simulations, we set: lˆ s 5 1 (Table 5). The inverse modeling results provided a better fit to the observations compared to the trial-and-error model of Bencala (1983). Although differences between inclusion/exclusion of transient storage were not radically different, inclusion of transient storage and, thus, sorption in the transient storage zone resulted in a better match to the observations, as indicated by the lower least-squares objective function (0.51 vs 0.59; data not shown) and residual analysis. This result was most evident in reaches 3 and 5 during the departure phase of the conservative tracer and added Sr21 (Fig. 6).

0.95–1.0

AS-3–a3, AS-3–ql-3, AS-4–a4, AS-4–ql-4, AS-4–A4, AS4–A5, AS-5–a5, AS5–qL5, a3–qL3, a4–qL-4, a4–A4, a4–A5, a5–qL-5, qL-4–A4, qL-4–A5

None

None

0.90–0.95

AS-4–A3, a4–A3, qL-4–A3, A2–A3

None

None

Transient storage metrics

None

None

The minimum and maximum expected values of the transient storage metrics were calculated

0.85–0.90 A3–A4, A4–A5

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FIG. 2. Ratio of flow at distance downstream to flow at injection location based upon parameters calibrated through trial and error (Bencala and Walters 1983); calibrated using UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling, with transient storage; calibrated using UCODE without transient storage; and calculated from mass balance. TS 5 transient storage.

using the 95% CIs reported in Table 2 for the model parameter. The metric AS/A ranged from 0.1 to 0.3 in reach 3, and from 0.4 to 2.0 in reach 4 for the UCODE estimation (Table 6). The wider range exhibited in reach 4 was a result of the

wider 95% CIs on AS (Fig. 7). AS/A values were higher for the Bencala and Walters (1983) estimation than for the UCODE-calculated values. tS also had a wide range in reach 4 (3.5–28.9 h). For tS calculated from the Bencala and Walters

FIG. 3. Composite scaled sensitivities (CSSs) of each parameter estimated using inverse modeling parameter estimation with transient storage. In reaches 1 and 2, the CSSs for transient storage parameters (a and As) are 0, indicating that the transient storage process within the model was not supported by the data. A 5 crosssectional area of the main channel (m2), D 5 dispersion coefficient (m2/s), qL 5 lateral volumetric inflow rate (m3/s-m; equivalent units as m2/s), AS 5 cross-sectional area of the storage zone (m2), a 5 stream storage exchange coefficient (/s).

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(1983) estimation, reach 4 had a value of 27.9, whereas tS in reaches 3 and 5 were greater than the UCODE calculated values. LS varied by a factor of 2 to 20 for UCODE values, depending upon the reach (Table 6). In reach 5, LS ranged from 160 to 1340 m. The value of LS computed with the Bencala and Walters (1983) parameter estimates for reach 4 was 3317 m, which was much greater than the UCODE-calculated maximum estimate of 1629 m. The Rh values in reaches 3 and 5 were higher in Bencala and Walters (1983) (reach 3, Rh 5 27.2) compared to the UCODE-modeled Rh estimates (reach 3, Rh 5 2.2–5.8). Using the 95% CIs for the parameters, the range of UCODE-calculated DaI values for reaches 3 and 5 included 1.0, but was ,1.0 for reach 4. Discussion Benefits of inverse modeling

FIG. 4. Residuals (calculated by subtracting the measured observation from the simulated value) for inverse modeling parameter estimation with and without transient storage as a function of time of day for reaches 2 to 5. Observed concentrations are also represented to highlight the temporal locations of the residuals within the Cl-addition experiment. TS 5 transient storage.

Our primary objective was to highlight the benefits of using an inverse modeling calibration procedure for stream solute transport models. Currently, models are often calibrated using trial and error, which is labor intensive and can lead to model bias. We focused on applying UCODE to calibrate OTIS. We chose to apply UCODE as the calibration tool, but another inverse modeling tool known as OTIS-P (Runkel 1998), which incorporates OTIS, is also available. OTIS-P arrives at similar calibrated parameter values to UCODE (results not shown). The primary advantages of UCODE compared to OTIS-P include 1) the summary statistics provided by UCODE (Table 3); 2) the ability to easily plot residuals (Fig. 4), DSSs (Fig. 5), and other statistics against time; and 3) the ability to calibrate all of the parameters using the entire set of observations. A significant disadvantage of using UCODE is that it is not formally incorporated into OTIS. We believe successful model calibration is obtained using OTIS-P, although UCODE’s advantages permit a more thorough understanding of the model’s use of the data set and the potential shortcomings of both the model and the data set. Inverse modeling resulted in a set of best-fit parameters, a necessary requirement to compare different models and examine reactive solute transport. This conclusion is seen both in the summary statistics (Table 3) and through a com-

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FIG. 5. Regions of high dimensionless scaled sensitivities (DSSs) with respect to each parameter for reach 3 are highlighted as a function of time on the tracer data. For each parameter, regions of high DSS indicate observations that provide the most predictive ability for the calibration of the specific parameter. A 5 crosssectional area of the main channel (m2), D 5 dispersion coefficient (m2/s), qL 5 lateral volumetric inflow rate (m3/s-m; equivalent units as m2/s), AS 5 cross-sectional area of the storage zone (m2), a 5 stream storage exchange coefficient (/s).

parison of the conservative tracer observations and simulations (Fig. 1A). Comparison of the least-squares objective function and the maximum likelihood objective function between the trial-and-error approach, inverse modeling without transient storage (model 1), and inverse modeling with transient storage (model 2) illustrated that model 2 produced the best fits to the data for reaches 3 to 5. Visual comparison (Fig. 1A) showed that model 2 provided a better simulation than model 1 to the tail portion of each breakthrough curve. Another benefit of inverse modeling is the ability to assess parameter correlation. When 2 parameters are highly correlated, their calibrated values are dependent on one another. For example, if AS and a were highly negatively correlated, as AS increased, a would decrease proportionately. Our results showed that the transient storage parameters (AS and a) and lateral inflow selected in the trial-and-error approach were highly correlated (Table 4), compared to no significant correlation for the transient stor-

age parameters in our calibration of model 2. This result provided greater confidence in model 2 than the trial-and-error approach, and is a conclusion that would take many manual model runs to achieve for each set of parameters. CCSs provide a means of assessing the combined value of the observations in supporting each parameter. Parameters with low sensitivity values (,1% of the highest CSS) are difficult to calibrate because the observations do not provide enough information to derive unique parameter estimates. The results of model 2 illustrated this situation for the transient storage parameters in reaches 1 and 2 (Fig. 3), where the CSS values for AS and a were ,1%. Using UCODE or similar approaches, investigators can quickly arrive at the conclusion to not include transient storage in these reaches, saving considerable time in model optimization. The DSSs calculated by UCODE confirmed the importance of data collection at both the beginning and end of a tracer experiment, as suggested by Wagner and Harvey (1997) and Har-

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vey and Wagner (2000). Our plot of the Cl2 breakthrough curve in reach 3 highlights the areas of high DSS values for each parameter, and clearly shows the importance of fine temporal resolution in both the shoulder and tail portion of the breakthrough curve (Fig. 5). Our analysis demonstrated that DSS values identify specific points and/or subsets of points that influence calibration. This information could provide insight into the fate of reactive solutes. For example, a 1st-order decay term can be included within OTIS to examine the fate of a nonconservative solute. After calibrating the 1st-order decay parameter within UCODE, the DSS values can be plotted against time to determine which observations provide the greatest information for the 1st-order decay parameter. These types of plots may provide insight into biogeochemical cycling because the capability of plotting DSS against another variable (e.g., t, x) exists as a result of UCODE output. Another important benefit of inverse modeling is the ability to calculate 95% CIs. A question that commonly arises in modeling is ‘how confident are you in your calibration?’ One answer to this question is to simply state the 95% CIs of each parameter estimate. To illustrate this point, the best-fit estimates and the upper and lower 95% CIs were plotted for AS and a in reaches 3 to 5 from model 2 (Fig. 7). These results highlight the large range in AS values, especially in reach 4. The large uncertainty in these estimates would be overlooked if this information was not provided. Transient storage Model comparison. Incorporation of transient storage as a process into the advection–dispersion formulation defined the comparison of models 1 and 2. Although this discussion will focus on transient storage, the general process of model decision can also be applied to reaction processes involving reactive solutes. The fact that model 2 resulted in the lower least-squares objective func← FIG. 6. Observed and simulated Sr21 concentrations at 4 sampling reaches downstream from a 3-h constant Sr21 injection. Sr21 simulations were modeled by adding a kinetic sorption mechanism to the advection–dispersion model. TS 5 transient storage.

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TABLE 5. Parameter estimates in the simulation of Sr21 transport for the trial-and-error approach (Bencala 1983) and inverse modeling calibration approach calculated using UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling. Values in parentheses denote upper and lower 95% confidence intervals. lˆ 5 1st-order rate coefficient for sorption in the stream (/s), lˆ S 5 1st-order rate coefficient for sorption in the storage zone (/s), and r 5 mass of accessible sediment per volume of stream water (g/ L).

Bencala 1983

This study, with transient storage (model 4)

Reach 1 lˆ lˆ S r

5.6 3 1025 1.0 4.0 3 104

4.4 3 1025 (3.7–5.1 3 1025) 1.0 5.4 3 3 104 (4.7–6.3 3 104)

Reach 2 lˆ lˆ S r

5.6 3 1025 1.0 2.0 3 104

4.3 3 1024 (2.9–6.4 3 1024) 1.0 2.2 3 103 (1.7–2.9 3 103)

Reach 3 lˆ lˆ S r

5.6 3 1025 1.0 2.0 3 104

2.3 3 1024 (2.0–2.6 3 1024) 1.0 8.3 3 103 (7.4–9.4 3 103)

Reach 4 lˆ lˆ S r

5.6 3 1025 1.0 2.0 3 104

1.5 3 1025 (0.7–3.2 3 1025) 1.0 6.7 3 104 (3.5–12.7 3 104)

Reach 5 lˆ lˆ S r

5.6 3 1025 1.0 4.0 3 104

2.3 3 1024 (1.4–3.7 3 1024) 1.0 2.3 3 104 (1.8–2.9 3 104)

Parameter

FIG. 7. Parameter estimates and 95% confidence intervals for (A) cross-sectional area of the storage zone (AS, m2) and (B) stream storage exchange coefficient (a, /s) for inverse modeling parameter estimation with transient storage. Note the log scale for AS and a.

TABLE 6. Sensitivity of the ratio of the storage zone to stream cross-sectional area (AS/A), storage zone mean residence time (tS), turnover length (LS), hydraulic retention factor (Rh), and Damkohler number (DaI), using the 95% confidence intervals to compute the minimum and maximum expected values. AS/A

tS (h)

UCODE Reach B&Wa min–max b

3 4 5 a b

1.0 1.0 3.0

0.1–0.3 0.4–2.0 0.2–0.3

LS (m)

UCODE B&Wa min–max 9.3 27.9 18.5

0.6–3.8 3.5–28.9 0.2–3.0

Rh (s/m)

DaI

B&Wa

UCODE min–max

B&Wa

UCODE min–max

1232 3317 598

968–2360 830–1629 160–1340

27.2 30.3 111.4

2.2–5.8 15.2–63.9 4.5–8.1

b

b

b

Bencala and Walters (1983) Poeter and Hill (1998), a computer code for universal inverse modeling

UCODEb B&Wa min–max 0.1 ,0.1 0.1

0.3–1.8 0.1–0.4 0.8–4.0

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tion and maximum likelihood objective function than model 1 is not surprising because model 2 has 6 additional parameters (an AS and a for reaches 3–5) used to obtain a better fit to the observations. The UCODE estimated AS values were smaller than the Bencala and Walters (1983) estimated parameters, and the exchange rates (a values) were higher in the UCODE model, resulting in a higher simulated solute flux between the main water column and the storage zone, and greater storage zone conservative solute concentrations (Fig. 1). CSS values, AIC and BIC statistics, and the resulting residuals can all be analyzed together to help guide model selection. The fact that CSS values for the transient storage parameters were ,1% of the highest CSS value in reaches 1 and 2 indicated that transient storage within these first 2 reaches was not supported by the existing data. Comparison of the AIC and BIC statistics showed that incorporation of transient storage in reaches 3 to 5 resulted in slightly higher values because of the increase in the number of parameters. If this piece of information were taken alone, a decision might be to adopt model 1 and not incorporate transient storage into any of the reaches. In this case, and for most tracer experiments, residuals of the conservative solute can be plotted as a function of time (Fig. 4). Our results clearly showed that model 2 resulted in a more randomly distributed set of residuals, suggesting that no systematic errors were present. When transient storage was omitted, the residuals were significantly .0 during the tail portion of the breakthrough curve. These various pieces of information suggest that, on one hand, the improvement in model fit resulted from model over-parameterization (AIC and BIC statistics). However, the residual analysis clearly highlighted the benefits of including transient storage within model 2. CIs were also used to calculate DaI (Wagner and Harvey 1997). The DaI values were close to 1 for both reaches 3 and 5, a value indicated to be optimal for transient storage estimation (Wagner and Harvey 1997). The lower DaI values for reach 4 suggests that the transient storage parameter values are unreliable, which is consistent with our results (e.g., large 95% CIs on AS, low CSS for AS). DaI for the model calibrated by trial and error was significantly lower than for the inverse calibration models, further

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highlighting the importance of robust calibration. Estimating the DaI number after an experiment allows for one explanation of why the transient storage model had high CIs, assuming that the observations for the solute are well resolved in the rising and falling limb of the solute breakthrough curve, where the greatest DSSs for AS and a are found. In this experiment, a longer injection period would have allowed for a greater amount of solute to enter the storage zones, therefore allowing for a longer tail in the breakthrough curve as solute of higher concentration in the transient storage zone exchanged with water in the main water column. Simulated storage zone concentrations. Examination of the 2 approaches that include transient storage (Bencala and Walters 1983 and model 2) for the conservative solute revealed substantial differences in the storage zone concentrations. Higher Cl2 concentrations were calculated in the storage zone for model 2 than for the trialand-error method because of the smaller AS determined with UCODE (Fig. 1). Higher concentrations simulated in the storage zone are important when considering the fate of reactive solutes because many biogeochemical processes depend on solute concentrations. For Sr21, the parameterized model including transient storage calculated instantaneous sorption in the storage zone, resulting in no Sr21 within the storage zone. Future tracer studies aimed at understanding solute uptake in such areas of the stream system need to incorporate sampling strategies for solute in these zones (e.g., hyporheic water). These observations within the transient storage zone may then be used to further the model’s validity, including the calibration of transient storage parameters and the resulting transient storage solute concentrations. Transient storage metrics. Comparisons of the transient storage metrics tS and LS highlight the importance of a robust parameter estimation approach. Using inverse modeling, the calculated values of As and a and the associated 95% CIs were used to compute the range of tS, LS, and Rh values. The UCODE-optimized results indicated much lower tS and Rh values in reaches 3 to 5 than those computed with the trial-anderror parameterization (Table 5), which would result in a decrease in a solute’s contact time with sediments and attached microbial surfaces in storage zones. The Bencala and Walters (1983) computed LS fell within the UCODE val-

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ues, except in reach 4. Because of the large range of values associated with these 2 metrics, tracer experiments planned to study transient storage processes should include intensive sampling during the arrival and departure of the conservative solute to obtain narrower 95% CIs associated with the transient storage parameters. Use of the 95% CIs is also strongly suggested when reporting transient storage metric formulations and using them in hypothesis tests (CI calculations are available from both OTIS-P and UCODE output). Inverse models such as UCODE provide this information for any of the calibrated parameters, so a range of derived metrics can easily be calculated for comparisons among streams. The UCODE-estimated parameter set for conservative Cl2 transport did not produce a drastically different model fit (Fig. 1) compared to the trial-and-error approach, but the differences in the resulting transient storage parameter values substantially altered the estimated storage zone mean residence time and exchange lengths, 2 examples of ecosystem variables commonly used in stream comparisons (Fellows et al. 2001). Lateral inflow computations. We presented alternative methods for computing lateral inflows in each of the 5 reaches, through a mass balance and parameterization using trial-and-error estimates and UCODE-optimized values. The close agreement between the integration method and the UCODE-optimized values suggests that the data sets contain enough information to characterize dilution well. The integration method is not exact, but is certainly more accurate than measuring stream flow by wading or other methods currently available. Dismissing dilution at downstream sampling locations could result in insufficient model parameterizations in which either large transient storage areas are defined or perhaps 1st-order mass losses are inappropriately applied to compensate for diluted solute concentrations. In general, stream discharge can accurately be determined with tracer additions in which plateau concentrations are reached at all downstream sites (Stream Solute Workshop 1990). The work presented here, with a tracer addition that did not reach plateau, further demonstrates the importance of accurately determin ing lateral inflow in studying reactive solute transport. Reactive solute transport in transient storage. We modeled Sr21 transport by incorporating kinetic

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sorption in both the main water column and the storage zone, and calibrating the sorption parameters using UCODE. The modeling results highlight significant differences between the trial-and-error calibration (Bencala 1983) and the inverse modeling calibrations (model 3 [without transient storage] and model 4 [with transient storage]). Furthermore, simulations from inverse modeling calibrations (models 3 and 4, Fig. 6) illustrated that sorption alone in the main water column could not account for the observed Sr21 values at the end of each reach. For example, reactive Sr21 simulation without transient storage (model 3) over-predicted the Sr21 concentrations in the tail portion of the experiment in reach 3. This result implies that incorporation of transient storage into the classic advection–dispersion equation is even more important when considering reactive solutes. Our results also showed that lˆ s converged to 1/s, implying that sorption within the storage zone was instantaneous for Sr21. This situation is most likely not unique to Sr21; other reactive solutes are also greatly influenced by reactions occurring in storage zones. This result further confirms why a robust calibration is required to simulate reactive solutes that may undergo rapid processing in slow-moving pools and hyporheic zones that are suspected to be biogeochemically diverse. The higher estimated exchange parameter values from the inverse modeling calibrations (model 2 incorporated into model 4) than for the trial-and-error approach imply that biogeochemical processes occurring at the interface of the main water column and storage zone are greater than would be predicted through trial and error (Bencala 1983). Our Sr21 results support this hypothesis, because the large predicted a results in a higher delivery of Sr21 to the storage zone, and the lower AS values result in higher storage zone concentrations (reaches 3 and 5). This result is limited to Sr21, but the analysis further confirms the importance of parameter estimation when examining nonconservative solutes. Transient storage processes have been studied in systems ranging from small mountain streams (e.g., Morrice et al. 1997) to large rivers such as the Willamette River in Oregon (Laenen and Bencala 2001). When transient storage parameters are used to compare differing stream systems (Morrice et al. 1997, Mulholland et al.

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1997, Hall et al. 2002), the inferences made rely heavily on the model calibration. Without a robust calibration, the resulting model may either not be unique or may not be supported by the given observations. Future stream studies and solute addition experiments adopting rigorous calibration approaches will continue to improve the understanding of aquatic biogeochemistry, and produce insights into the importance of transient storage. Acknowledgements We thank Steffen Mehl, Steven Wondzell, Briant Kimball, and 2 anonymous reviewers for constructive comments that greatly improved the manuscript. This work was supported by the USEPA (grant no. 98-NCERQA-83), the NSF Office of Polar Programs (grant no. 9813061), the US Geological Survey National Research Program, and the US Geological Survey Toxic Substances Hydrology Program. Literature Cited AKAIKE, H. 1974. A new look at statistical model identification. Institute of Electrical and Electronics Engineers Transactions on Automatic Control AC-19:716–723. AKAIKE, H. 1978. Time series analysis and control through parametric models. Pages 1–25 in D. F. Findley (editor). Applied time series analysis. Academic Press, New York. AVANZINO, R. J., G. W. ZELLWEGER, V. C. KENNEDY, S. M. ZAND, AND K. E. BENCALA. 1984. Results of a solute transport experiment at Uvas Creek, September, 1972. U. S. Geological Survey Water-Resources Investigations Report 84–236. US Geological Survey, Menlo Park, California. BENCALA, K. E. 1983. Simulation of solute transport in a mountain pool-and-riffle stream with a kinetic mass transfer model for sorption. Water Resources Research 19:732–738. BENCALA, K. E., D. M. MCKNIGHT, AND G. W. ZELLWEGER. 1990. Characterization of transport in an acidic and metal-rich mountain stream based on lithium tracer injection and simulation of transient storage. Water Resources Research 26:989– 1000. BENCALA, K. E., AND R. A. WALTERS. 1983. Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model. Water Resources Research 19:718–724. BROSHEARS, R. E., R. L. RUNKEL, B. A. KIMBALL, D. M. MCKNIGHT, AND K. E. BENCALA. 1996. Reactive solute transport in an acidic stream: experimental

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pH increase and simulation of controls on pH, aluminum and iron. Environmental Science and Technology 30:3016–3024. CHAPRA, S. C., AND R. L. RUNKEL. 1999. Modeling impact of storage zones on stream dissolved oxygen. Journal of Environmental Engineering 125:415–419. CHAPRA, S. C., AND R. J. WILCOCK. 2000. Transient storage and gas transfer in a lowland stream. Journal of Environmental Engineering 126:708–712. D’ANGELO, D. J., J. R. WEBSTER, S. V. GREGORY, AND J. L. MEYER. 1993. Transient storage in Appalachian and Cascade mountain streams as related to hydraulic characteristics. Journal of the North American Benthological Society 12:223–235. FELLOWS, C. S., H. M. VALETT, AND C. N. DAHM. 2001. Whole-stream metabolism in two montane streams: contribution of the hyporheic zone. Limnology and Oceanography 46:523–531. HALL, R. O., E. S. BERNHARDT, AND G. E. LIKENS. 2002. Relating nutrient uptake with transient storage in forested mountain streams. Limnology and Oceanography 47:255–265. HARVEY, J. W., AND C. C. FULLER. 1998. Effect of enhanced manganese oxidation in the hyporheic zone on basin-scale geochemical mass balance. Water Resources Research 34:623–636. HARVEY, J. W., AND B. J. WAGNER. 2000. Quantifying hydrologic interactions between streams and their subsurface hyporheic zones. Pages 3–44 in J. A. Jones and P. J. Mulholland (editors). Streams and groundwaters. Academic Press, San Diego, California. HARVEY, J. W., B. J. WAGNER, AND K. E. BENCALA. 1996. Evaluating the reliability of the stream tracer approach to characterize stream-subsurface water exchange. Water Resources Research 32: 2441–2451. HILL, A. R., C. F. LABADIA, AND K. SANMUGADAS. 1998. Hyporheic zone hydrology and nitrogen dynamics in relation to the streambed topography of a N-rich stream. Biogeochemistry 42:285–310. HILL, M. C. 1992. A computer program (MODFLOWP) for estimating parameters of a transient, threedimensional, ground-water flow model using nonlinear regression. U. S. Geological Survey Open-File Report 91–484. US Geological Survey, Denver, Colorado. HILL, M. C. 1998. Methods and guidelines for effective model calibration. U. S. Geological Survey WaterResources Investigations Report 98–4005. US Geological Survey, Denver, Colorado. HILL, M. C., R. L. COOLEY, AND D. W. POLLACK. 1998. A controlled experiment in ground-water flow model calibration using nonlinear regression. Ground Water 36:520–535. HINKLE, S. R., J. H. DUFF, F. J. TRISKA, A. LAENEN, E. B. GATES, K. E. BENCALA, D. A. WENTZ, AND S. R. SILVA. 2001. Linking hyporheic flow and nitrogen cy-

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cling near the Willamette River—a large river in Oregon, USA. Journal of Hydrology 244:157–180. KILPATRICK, F. A., AND E. D. COBB. 1985. Measurement of discharge using tracers. Techniques of waterresources investigations of the United States Geological Survey, Book 3, Chapter A16. US Geological Survey, Denver, Colorado. KIM, B. K., A. P. JACKMAN, AND F. J. TRISKA. 1992. Modeling biotic uptake by periphyton and transient hyporheic storage of nitrate in a natural stream. Water Resources Research 28:2743–2752. LAENEN, A., AND K. E. BENCALA. 2001. Transient storage assessments of dye-tracer injections in rivers of the Willamette Basin, Oregon. Journal of the American Water Resources Association 37:367– 377. MCKNIGHT, D. M., G. M. HORNBERGER, K. E. BENCALA, AND E. W. BOYER. 2002. In-stream sorption of fulvic acid in an acidic stream: a stream-scale transport experiment. Water Resources Research 38:6-1–6-12 (American Geophysical Union Citation No. 1005, Digital Object Identifier: 10.1029/ 2001WR000269). MCKNIGHT, D. M., B. A. KIMBALL, AND R. L. RUNKEL. 2001. pH dependence of iron photoreduction in a rocky mountain stream affected by acid mine drainage. Hydrological Processes 15:1979–1992. MEHL, S. W., AND M. C. HILL. 2001. A comparison of solute-transport solution techniques and their effect on sensitivity analysis and inverse modeling results. Ground Water 39:300–307. MORRICE, J. A., H. M. VALETT, C. N. DAHM, AND M. E. CAMPANA. 1997. Alluvial characteristics, groundwater-surface water exchange and hydrological retention in headwater streams. Hydrological Processes 11:253–267. MULHOLLAND, P. J., E. R. MARZOLF, J. R. WEBSTER, D. R. HART, AND S. P. HENDRICKS. 1997. Evidence that hyporheic zones increase heterotrophic metabolism and phosphorus uptake in forest streams. Limnology and Oceanography 42:443– 451. MULHOLLAND, P. J., A. D. STEINMAN, E. R. MARZOLF, D. R. HART, AND D. L. DEANGELIS. 1994. Effect of periphyton biomass on hydraulic characteristics and nutrient cycling in streams. Oecologia (Berlin) 98:40–47. OTT, L. 1993. An introduction to statistical methods and data analysis. 4th edition. PWS-Kent Publishing Company, Boston. POETER, E. P., AND M. C. HILL. 1997. Inverse models: a necessary next step in ground-water modeling. Ground Water 35:250–260. POETER, E. P., AND M. C. HILL. 1998. Documentation of UCODE, a computer code for universal inverse modeling. U. S. Geological Survey Water-Resourc-

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es Investigation Report 98–4080. US Geological Survey, Denver, Colorado. RUNKEL, R. L. 1998. One-dimensional transport with inflow and storage (OTIS): a solute transport model for streams and rivers. U. S. Geological Survey Water-Resources Investigation Report 98– 4018. US Geological Survey, Denver, Colorado. (Available from: http://co.water.usgs.gov/otis) RUNKEL, R. L. 2002. A new metric for determining the importance of transient storage. Journal of the North American Benthological Society 21:529– 543. RUNKEL, R. L., B. A. KIMBALL, D. M. MCKNIGHT, AND K. E. BENCALA. 1999. Reactive solute transport in streams: a surface complexation approach for trace metal sorption. Water Resources Research 35:3829–3840. RUNKEL, R. L., D. M. MCKNIGHT, AND E. D. ANDREWS. 1998. Analysis of transient storage subject to unsteady flow: diel flow variation in an Antarctic stream. Journal of the North American Benthological Society 17:143–154. STREAM SOLUTE WORKSHOP. 1990. Concepts and methods for assessing solute dynamics in stream ecosystems. Journal of the North American Benthological Society 9:95–119. TATE, C. M., R. E. BROSHEARS, AND D. M. MCKNIGHT. 1995. Phosphate dynamics in an acidic mountain stream: interactions involving algal uptake, sorption by iron oxide, and photoreduction. Limnology and Oceanography 40:938–946. THACKSTON, E. L., AND K. B. SCHNELLE. 1970. Predicting effects of dead zones on stream mixing. Journal of Sanitary Engineering 96:319–331. VALETT, H. M., J. A. MORRICE, C. N. DAHM, AND M. E. CAMPANA. 1996. Parent lithology, surfacegroundwater exchange, and nitrate retention in headwater streams. Limnology and Oceanography 41:333–345. WAGNER, B. J., AND S. M. GORELICK. 1986. A statistical methodology for estimating transport parameters: theory and applications to one-dimensional advective-dispersive systems. Water Resources Research 22:1303–1315. WAGNER, B. J., AND J. W. HARVEY. 1997. Experimental design for estimating parameters of rate-limited mass transfer: analysis of stream tracer studies. Water Resources Research 33:1731–1741. WO¨RMAN, A. 1998. Analytical solution and timescale for transport of reacting solutes in rivers and streams. Water Resources Research 34:2703–2716. ZAND, S. M., V. C. KENNEDY, G. W. ZELLWEGER, AND R. J. AVANZINO. 1976. Solute transport and modeling of water quality in a small stream. Journal of Research: United States Geological Survey 4: 233–240. Received: 11 October 2002 Accepted: 27 August 2003

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APPENDIX. Parameter and variable definitions used within the diagnostic statistics calculated within UCODE (Poeter and Hill 1998), a computer code for universal inverse modeling. Variable or parameter y b b b9 y9(b) i j v ND NPR NP n

xU, xL

var ( ), cov ( )

Description Vector of observations Vector of parameter values at which the sensitivities are evaluated Parameter value for which the DSS is calculated Vector of optimal parameter values Vector of simulated values for b, a vector of parameter values Observation increment subscript Parameter increment subscript Observation weighting vector, userdefined in UCODE Number of observations being used in the regression Number of prior information assignments Number of parameters being estimated Number of degrees of freedom 5 ND 1 NPR 2 NP Upper (U) and lower (L) tail values of the x2 distribution with number of degrees of freedom, n Variance of a parameter, and covariance of 2 parameters, respectively

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