Analytical solutions to the 1-D heat transport equation can be used to quantify surface-groundwater interactions directly from temperature time series data in streams and their streambeds. The solutions rely on three assumptions: purely vertical flow, no thermal gradient with depth in the streambed and sinusoidal temperature signals. Here numerical models of heat transport in streambeds are used to generate synthetic time series data under conditions that violate the aforementioned assumptions. These synthetic records are used to evaluate the impact of violations of model assumptions on vertical water velocity estimates. Analytical methods using the amplitude ratio to derive flux are less prone to error than methods that use lag time under nonideal field conditions. The greatest source of error is nonvertical flow in the streambed. Errors from analytical solutions for flux are comparable to or smaller than errors inherent to Darcy-based flux estimates and therefore the use of temperature data to quantify flux across the streambed is a promising alternative to more commonly used approaches, such as Darcy flux calculations.
 Exchanges of water, solutes, and heat across the stream water–groundwater (SW-GW) interface are critical components of a variety of water quality issues, including solute and contaminant transport, natural attenuation of water contaminants, ecological diversity, and nutrient dynamics [Boulton, 2007; Hancock, 2002; Hancock et al., 2005; Hayashi and Rosenberry, 2002]. In traditional investigations of these issues, streams and groundwater have been considered separate [Winter et al., 1998]. Recent advances indicate that interaction of surface and groundwater has important impacts on water quality [Jones and Mulholland, 2000], because the SW-GW interface acts as a filter for nutrients and other contaminants. SW-GW interaction increases stream residence times and the contact time between solutes, microbial communities, and geochemically reactive sediments in the streambed [Bencala, 2000]. Microbial processing of nutrients and other contaminants in the near-stream zone may be essential for maintaining surface water quality [Hancock, 2002]. SW-GW interactions also impact fish, macroinvertebrates, groundwater-dependent terrestrial fauna, and even rare taxa residing in groundwater aquifers [Hancock et al., 2005]. Although water fluxes between streams and their neighboring groundwater aquifers have significant impacts on biogeochemical and ecological processes occurring in streams [Boulton et al., 1998; Hayashi and Rosenberry, 2002], it has proven difficult to accurately characterize water fluxes between stream water and groundwater [e.g., Harvey and Wagner, 2000; Kalbus et al., 2006].
 There has been a renewed interest in the use of heat as a tracer in hydrologic systems, particularly to understand stream-aquifer interactions [e.g., Anderson et al., 2005; Constantz, 2008; Stonestrom and Constantz, 2004]. Recent advances in the application of heat as a tracer of SW-GW interaction have made the technique a cost-effective, accurate, and generalizable method for assessment of SW-GW interactions. In general, temperature patterns are different in the stream and groundwater end-members that bound the streambed. Stream temperatures fluctuate on daily and seasonal time scales while groundwater temperatures remain fairly constant. The propagation of the diurnal stream temperature signal into the streambed reflects the vertical flux rate of water to or from the stream [Conant, 2004; Lapham, 1989]. Characteristics of the streambed temperature record relative to the stream record, such as the amplitude of the diurnal signal and the lag between the stream and streambed maximum and minimum temperatures, can be used to model the flux of water between the stream and the subsurface [Goto et al., 2005; Hatch et al., 2006; Keery et al., 2007]. Additionally, snapshot temperature profiles at one point in time can be used to infer SW-GW flux [e.g., Conant, 2004; Constantz et al., 2003; Lapham, 1989; Schmidt et al., 2006].
 The use of heat transport theory to quantify SW-GW fluxes has a number of advantages over traditional methods, such as Darcy flux measurements based on hydraulic gradient and conductivity of bed sediments [e.g., Woessner and Sullivan, 1984]. Heat is a naturally occurring, nonreactive tracer [Constantz et al., 2003], and thermal properties of sediments, such as thermal conductivity, are much more tightly constrained than hydrologic properties, such as hydraulic conductivity, which can vary over orders of magnitude [Lapham, 1989]. The availability of inexpensive, accurate, high-capacity temperature data loggers is increasing [e.g., Hubbart et al., 2005; Johnson et al., 2005]. Just as data acquisition is becoming more powerful and cost-effective, straightforward analytical techniques are being developed to interpret detailed temperature records. Streambed seepage fluxes can be computed directly from temperature time series data using relatively simple analytical models [Hatch et al., 2006; Keery et al., 2007].
 Although temperature data acquisition and analytical modeling have become relatively straightforward, several assumptions of common 1-D analytical models, such as purely vertical flow and sinusoidal temperature fluctuations in the stream [Goto et al., 2005; Hatch et al., 2006], will be violated in the vast majority of field settings. Questions remain as to how differences between the model assumptions, and the actual field conditions affect results and cause error in the predicted seepage flux rates. Despite the potential limitations of 1-D analytical modeling, the ease of use and previous proof-of-concept investigations suggest further investigations are warranted [Fanelli and Lautz, 2008; Hatch et al., 2006; Keery et al., 2007]. Given the ease with which thermal records can be interpreted, the use of heat transport modeling of SW-GW interaction is likely to increase. The objectives of this work are (1) to evaluate the impact of various field conditions on stream thermal signal propagation into streambeds and (2) to assess the potential for error of SW-GW flux estimates derived from 1-D analytical models of streambed thermal signal amplitude and lag time.
2.1. Analytical Modeling of Streambed Heat Transport
where T is temperature (°C), t is time (s), κe is effective thermal diffusivity (m2 s−1), z is depth (m), q is vertical seepage flux (m s−1), and γ is the ratio of the heat capacity of the sediment-water matrix in the streambed to the heat capacity of the fluid.
 Solutions to equation (1) allow for seepage flux rates through the streambed to be quantified directly using the change in the amplitude of the diurnal temperature signal with depth in the bed or the lag time between the stream and subsurface temperature signals. This technique is advantageous because daily seepage fluxes can be computed directly over extended time periods (e.g., months) without calibrating fully distributed two- or three-dimensional models. Details on the analytical solution can be found in the work of Hatch et al. , and the method assumptions include (1) 1-D flux of water, vertically, (2) a sinusoidal diurnal temperature signal in the stream, and (3) no change in the average temperature with depth in the streambed.
 The analytical model by Hatch et al.  defines the relationships between streambed thermal properties, seepage flux, and the amplitude ratio (Ar, unitless) or the lag time (Δϕ, in hours) of the streambed temperature signal. Ar is the ratio of the amplitude of the diurnal temperature signal in the streambed to the amplitude of the diurnal temperature signal in the stream. Δϕ is the time difference between the arrival of the maximum or minimum temperature in the stream and the streambed. The derivation in the work of Hatch et al.  provides the following predictive equations for q, based on Ar or Δϕ and the thermal properties of the streambed:
where C and Cw are the heat capacity of the sediment water matrix and the water (J m−3 °C−1), respectively, Δz is the difference in elevation between two measurement points in the streambed (m), v is the velocity of the thermal front (m s−1), P is the period of the temperature signal in s, and α is defined by the following:
The heat capacity of the sediment water matrix is estimated as
where n is the sediment porosity, and Cw and Cs are the heat capacity of the water and sediment (J m−3 °C−1), respectively. Thermal diffusivity, κe, is estimated as
where λe is the effective thermal conductivity (J s−1m−1 °C−1) and β is the thermal dispersivity (m). Details on the mathematical development of this analytical model can be found in the work of Hatch et al.  and Goto et al. .
 We used forward analytical modeling with equation (1) to derive expected temperature signals in the streambed under various seepage fluxes. We also used inverse analytical modeling to derive seepage flux rates directly from Ar (equation (2)) and Δϕ (equation (3)) from numerical modeling results (see section 2.2) and field data (see section 2.3). The linear vertical velocity was then computed as q/n. We used the values of the model parameters given in Table 1.
Table 1. Model Parameters Used in the Analytical Solution to the Heat Transport Equation
J s−1 m−1 °C−1
Heat capacity of the sediment-water matrix
2.72 × 106
J m−3 °C−1
Heat capacity of the water
4.18 × 106
J m−3 °C−1
Heat capacity of the sediment
2.09 × 106
J m−3 °C−1
Period of temperature signal
 Ideally, temperature records should be filtered before analysis to isolate the diurnal temperature signal from other thermal signals, such as seasonal fluctuations [Hatch et al., 2006; Keery et al., 2007]. Such filtering is only appropriate for long temperature records (e.g., recorded continuously for several weeks) because filtering imposes edge effects that can impact the 3–4 days at the start and end of a temperature time series [Hatch et al., 2006]. Given the short duration of many temperature data sets, such filtering was not considered here. Rather, we were interested in whether short time series temperature data (e.g., <10 days) could still be used to obtain reasonable seepage flux rates, particularly when the duration of such records is too short for signal filtering [e.g., Fanelli and Lautz, 2008]. Of particular interest is whether such short duration records provide a relatively simple alternative to more traditional methods of measuring SW-GW flux, such as Darcy flux measurements.
2.2. Numerical Modeling
 To identify how nonvertical flow in the streambed, nonsinusoidal stream temperature signals and changes in average temperature with depth impact the amplitude ratio and lag time of the thermal signal with depth in the streambed, we simulated these different conditions using VS2DH. VS2DH is a U.S. Geological Survey finite difference model that solves Richards' equation for fluid flow and the advection-dispersion equation of energy transport in one or two dimensions [Healy and Ronan, 1996]. The energy transport equation solved by VS2DH is
where t is time (s), θ is volumetric moisture content (unitless), T is temperature (°C), λ(θ) is the thermal conductivity of the water and sediment matrix, as a function of porosity, (J s−1 m−1 °C−1), DH is hydrodynamic dispersion (m2 s−1), q is the rate of a fluid source (s−1), T* is the temperature of that fluid source (°C), and other variables are as defined previously. For fully saturated conditions, without sources or sinks internal to the model domain and no hydrodynamic dispersion, the terms can be combined and simplified to
 We first ran the VS2DH model under the assumptions given by the 1-D analytical model to verify that numerical and analytical model results for identical scenarios were the same. We then ran the 1-D model with (1) nonsinusoidal temperature signals at the stream boundary and (2) lower than average temperatures at the groundwater boundary. For each model scenario, the model was run five times to simulate a range of hydraulic gradients and associated water velocities through the model domain. We then used a 2-D model simulation to create nonvertical flow through the model domain.
 The details of the 1-D and 2-D models are given in Figure 1. The sediment properties for both models were selected to be consistent with the analytical modeling analysis (Table 2). We generated high-precision model output every 10 min to reduce error in the amplitude ratio or lag time as a result of low-thermal resolution output and to be consistent with the temporal resolution of typical field observations. Output was generated at the locations of the observation points indicated in Figure 1, all located 25 cm below the streambed interface. We ran all models for 10 days.
Table 2. Model Parameters Used in the VS2DH Simulations
Heat capacity of the sediment
2.09 × 106
J m−3 °C−1
Heat capacity of the water
4.18 × 106
J m−3 °C−1
J s−1 m−1 °C−1
 The 1-D models were 0.25 m in width and either 4 m (for ideal and nonsinousoidal stream temperature models) or 1 m (for thermal gradient models) in depth. The models had a grid spacing of 0.03 m in width by 0.01 m in depth. In the 1-D models, the vertical boundaries were simulated as no flow for both fluid and heat transport. The top and bottom boundaries were assigned constant pressure head values to force a specified hydraulic gradient through the model domain (ranging between 0.0 and 1.2). The model was run as a transient model with the stream temperature changing every 30 min. The temperature at the top boundary (the streambed interface) was imported as either a hypothetical sinusoidal temperature signal or an observed stream temperature signal, depending on the model run (Figure 1 and Table 3). The temperature at the bottom boundary was held constant at either the average stream temperature (17.8°C) or a constant colder temperature (12.8°C) depending on the model run. These temperatures were selected because they are representative of field observations of stream and subsurface temperatures used for comparison later in the paper.
 The 2-D model was 3 m in depth and 4 m in width with variable grid spacing. Grid cells were 0.01 cm in width and between 0.1 m (at the bottom) and 0.01 m (at the top) in depth. In the 2-D model, the vertical boundaries and bottom boundary were simulated as no-flow for both fluid and heat transport. The top of the model domain was assigned a variable pressure head to generate a number of flow paths with various flux rates and flow directions (predominantly vertical to predominantly horizontal). The pressure head ranged from 2.4 to 0.0 m. As in the 1-D case, the model was run as a transient model with the stream temperature changing every 30 min and the temperature at the top was imported from a synthetic sinusoidal temperature signal.
 We used VS2DH to generate synthetic time series temperature data in the streambed. The synthetic temperature records were used to derive daily Ar and Δϕ for various model conditions. During the first 2 days (48 h) of each simulation, the model was influenced by the initial conditions, so these 2 days of results were excluded from the analysis and only information from the last 8 days was considered. The amplitudes of the temperature signals in the stream and streambed were determined by taking half the difference between the maximum and minimum observed temperatures. Lag times were taken as the difference between the timing of the maximum or minimum temperatures observed in the stream and streambed.
 The Ar and Δϕ values derived from the synthetic temperature records from VS2DH were then used to directly compute vertical linear velocity based on the analytical model. We used changes in model boundary conditions to identify how violations of analytical model assumptions generate error in velocity estimates derived from the analytical model. A comparison of actual vertical velocity through the model and inversely modeled vertical velocity from the analytical model allows us to see the error introduced by violation of analytical model assumptions.
 The range of seepage fluxes and vertical velocities considered in this analysis were derived from Darcy's Law (q = KI), with the hydraulic conductivity of the streambed sediments, K, and the range of gradients, I, as given in Table 2.
2.3. Field Data Collection
 We compared vertical velocity estimates obtained from methods based on the Darcy equation (i.e., vz = KI/n) to vertical velocity estimates from thermal records at a field site near Lander, Wyoming, USA. These field results have been published previously by Fanelli and Lautz  and details on the site and methods can be found there. In summary, in July 2007 in-stream minipiezometers were installed in the streambed of a ∼15 m reach of Red Canyon Creek that has a 1.5 m tall dam located in the middle of the reach. Minipiezometers were installed in a grid pattern in two sections of the reach: a large, stagnant pool upstream of the dam (n = 9) and a plunge pool-riffle sequence downstream of the dam (n = 16). Minipiezometers were installed to a depth of 30 cm in the streambed and were constructed of 2.5 cm inner diameter PVC tubing with the bottom 10 cm screened. Vertical hydraulic gradients in the streambed were measured at least 24 h after initial installation and were measured as the difference between the water height inside the minipiezometer and the water height in the stream relative to the average depth of the screen, 25 cm.
 Minipiezometers were removed and temperature data loggers were immediately installed in their place. Temperature data loggers, iButtons (Dallas Semiconductor, Dallas, Texas) with an accuracy and precision of ±1°C and 0.125°C, were installed in the surface water column and at a depth of 25 cm in the streambed at each location by embedding the data loggers within a wooden dowel and inserting the dowel into the streambed. Thermal and hydraulic properties for the two sections of the reach were estimated based on the sediment properties of the streambed and guidelines presented by Lapham , as shown in Table 4 and discussed in the paper by Fanelli and Lautz .
Table 4. Thermal and Hydraulic Properties Used in the Analytical Model and Darcy Calculations for the Two Sets of Field Data
Heat capacity of the sediment-water matrix C (J m−3 °C−1)
3.03 × 106
2.72 × 106
Heat capacity of the water Cw (J m−3 °C−1)
4.18 × 106
4.18 × 106
Heat capacity of the sediment Cs (J m−3 °C−1)
2.09 × 106
2.09 × 106
Hydraulic conductivity (K) (m s−1)
 We monitored stream and streambed temperatures every 10 min for approximately 96 h. Two sites shown in the upstream pool in the paper by Fanelli and Lautz  were excluded because either the temperature record was missing due to a failed temperature data logger or no hydraulic gradient data was available. We used temperature records from 0:00 on 4 July to 23:59 on 6 July for this analysis. Temperature records could not be filtered due to the short duration of the records (e.g., 3 days). For each location and each day (4, 5, and 6 July), we used the temperature records to identify the minimum and maximum temperatures and the times of these maximum and minimum temperature observations. The difference between the observed minimum and maximum temperature in the streambed at each site, relative to the difference between the minimum and maximum temperature in the stream on the same day, was used to derive the amplitude ratio, Ar. The lag time between when the temperature maximum was observed in the stream versus the streambed was used to derive the lag time, Δϕmax. The timing of the minimum observed temperatures in the stream and streambed were used to derive a separate lag time, Δϕmin. The estimated vertical velocity reported for each site from the heat transport model is the average of the three values derived from the three individual days of record.
3.1. Numerical and Analytical Modeling Under Ideal Field Conditions
 The linear vertical velocities considered here range from 0 m/d (no flow) to 3.1 m/d and generate temperature signals in the streambed that have a broad range of amplitude ratios from 0.14 to 0.85 (Figure 2).
 Under ideal conditions (i.e., no thermal gradient in the streambed, stream temperatures following a sine curve and purely vertical flow), the temperature patterns from the VS2DH numerical model and the Hatch analytical model are virtually identical, particularly under low velocities (Figure 2). At higher velocities (>1 m/d), slight differences between the numerical and analytical modeling results can be seen, particularly around the time of the minimum streambed temperature. This is somewhat surprising given the two models are simulating identical field conditions. The small differences between the modeled temperatures are attributed to differences between the ways the models address temperature dependence of hydraulic conductivity. In contrast to the analytical model, VS2DH allows for the viscosity of water to vary with temperature, resulting in a temperature-dependent hydraulic conductivity and fluid velocity. Temperature is positively related to hydraulic conductivity, causing higher velocities at higher temperatures. As the temperature in the model domain fluctuates over time, the hydraulic conductivity fluctuates slightly and the rate of fluid flow through the model domain changes slightly. This causes a change in the transport of energy through the model domain over time. The analytical model does not account for viscosity differences or any other causes of variable velocities over time. Differences between VS2DH temperatures and analytical model temperatures are maximized at low velocity and low temperature (Figure 2). At the lowest temperatures in the VS2DH model simulation (i.e., 13.2°C), the hydraulic conductivity is 20% lower than the hydraulic conductivity at the highest temperature (i.e., 22.4°C). A 20% decrease in hydraulic conductivity will cause a corresponding 20% decrease in fluid velocity. The decrease in hydraulic conductivity and associated fluid velocity at lower temperatures is likely causing the lag of the temperature signal generated by VS2DH relative to the analytical model at the lowest temperatures (Figure 2).
 Despite slight differences between the predicted temperature time series from the analytical and numerical models, the Ar analytical model (equation (2)) correctly determined the velocity of water flow through the streambed in VS2DH under ideal conditions within 2%. Velocity errors from the Δϕmax and Δϕmin models (equation (3)) were slightly larger, but generally <8% (Figure 3). Note that percent errors for the no flow models (i.e., vz = 0.0 m/d) cannot be derived, as they require a denominator of zero.
 Under ideal field conditions, slightly larger errors occurred when estimating velocity from Δϕ versus Ar for two primary reasons. First, slightly inaccurate lag times may have been derived from the VS2DH results when the temperature changes in the streambed over time were relatively small, such as at low velocity. The accuracy of the lag time under such conditions is limited by both the temporal and thermal resolution of the time series data. Here we obtained VS2DH model output every 10 min, to be comparable to a typical field data set such as the data collected at Red Canyon Creek, and selected high-resolution temperature output with a resolution of 10−12°C, which is much higher than the resolution of typical field temperature data loggers (e.g., 0.125°C for the Red Canyon Creek field data). If thermal resolution were too low, two consecutive observation points in a time series could have identical temperatures that are the observed minima or maxima. In that situation, it would be unclear what time should be used as the maximum or minimum time and any error would translate to error in the estimated lag between the observed maximum or minimum in the stream and the corresponding maximum or minimum in the bed. Although the lag time may be in error under such circumstances, the value of the minimum or maximum temperature is still accurate and there is little or no impact on Ar. Alternatively, if temporal resolution is low, the same error is even more likely, as the temperature observations may not clearly distinguish smaller changes in temperature over short periods of time around the maximum and minimum observations. It is likely that the true minimum or maximum temperature falls at a time between recorded observations, making the accuracy of the lag time only within the time step recorded by the data logger (or the model output, in this case every 10 min).
 The second reason Δϕ estimates of velocity can be problematic is the Δϕ method imposes upper limits on the lag times that can be used to derive velocity from the model (Figure 4). The maximum lag time occurs under no flow conditions and increased vertical flow into the bed reduces the lag time. The range of plausible lag times given by the analytical model is dependent on the thermal properties of the streambed and the depth of the temperature observation point in the streambed. Slight errors in the observed lag times, attributed to the aforementioned reasons and particularly around the model limits, may cause lag times to be outside of the range acceptable for the Δϕ model and, under such circumstances, velocity cannot be determined. Under the conditions simulated here, the maximum lag time solvable for the Δϕ model is 7.5 h (Figure 4). From the VS2DH simulation under ideal conditions, the lag time at no flow was 7.67 h, only ∼10 min greater than the maximum allowed by the Δϕ model, due to errors obtaining accurate lag times at low flow, presumably due to temporal resolution of the time series data. The temporal resolution of the VS2DH output was 10 min to be consistent with results typically obtained in the field. Rather than obtaining a slightly erroneous velocity estimate under such conditions, the Δϕ method simply cannot be used. As a result, lag time errors, particularly at low or no flow, may make it impossible to use the Δϕ method to derive velocity, as was the case for our no flow model under ideal conditions (Figure 3).
 Lag time errors using the minimum versus maximum temperature lag time are different at the highest velocity of 3.1 m/d (−5.8% versus 1.0%, respectively). This is due to differences between results of VS2DH and the analytical model that are derived from the temperature dependence of K in the VS2DH model, as discussed above. The velocity of water flow through the model domain is different at the maximum and minimum temperatures due to the temperature dependence of K, so we derive different vertical velocities when using the lag time of the minimum or maximum temperature in VS2DH. In true field conditions, the K of the streambed will also be temperature dependent [Constantz, 1998] and differences in estimated velocity from the Δϕmax and Δϕmin models are likely to occur when there are large changes in temperature in the streambed over daily time scales.
3.2. Effects of Thermal Gradients in the Streambed
 Introducing a thermal gradient of 5°C per meter causes a shift in the observed temperature signal at a depth of 25 cm under very low or no flow conditions (Figure 5). Although Figure 5 shows that under no flow conditions the average temperature in the streambed is lower due to the thermal gradient, the amplitudes of the temperature signals in the bed are nearly identical between the ideal conditions model and thermal gradient model. When there is flow of water into the bed from the stream, the temperature signal is largely unaffected by the presence of the thermal gradient. Here we created a thermal gradient by imposing a relatively cold groundwater boundary temperature (12.8°C versus the stream average of 17.8°C) at a depth of 1 m. This is consistent with conditions at the Red Canyon Creek field site during July. A lower groundwater temperature closer to the observation site would potentially have a greater impact on the thermal records at 25 cm.
 Introducing a thermal gradient did not generate large (>10%) errors in estimated velocities from either the Ar or Δϕ methods (Figure 3). Using the Ar method, thermal gradients in the bed caused the largest percent error at lower velocities, although the percent error was still only 3%. This is in contrast to ideal conditions, which had <1% error at low or no flow. The slightly higher error is attributed to the lower temperatures in the streambed under low flow conditions, which slightly buffered the amplitude, reducing Ar and the associated velocity estimate. At the higher velocities, differences between Ar under ideal conditions and under a thermal gradient were minimal.
 Using the Δϕ method with a thermal gradient in the streambed, errors in estimated velocities were slightly larger than for Ar at the highest velocities (by up to 10%) and errors were different depending on whether the maximum or minimum temperature lag times were used (Figure 3). Lag time errors are attributed to variable K over time in the high velocity models. As velocity shifts with streambed temperature over time, the rate of heat transport between the stream and the streambed also shifts, causing differences in lag time between the minimum and maximum temperatures.
3.3. Effects of Nonsinusoidal Stream Temperature Signals
 A nonsinusoidal temperature signal in the stream translates to nonsinusoidal temperature fluctuations in the bed, regardless of the rate of fluid flow between the stream and streambed (Figure 5). The nonsinusoidal temperature patterns result in different lag times between the maximum and minimum temperatures in the stream and streambed on individual days. The nonsinusoidal temperature patterns also cause Ar, Δϕmax, and Δϕmin to vary day to day, even though the flux rate through the model remains constant. This is in contrast to the constant Ar and Δϕ values over time when the temperature signal in the stream follows a sine pattern.
 Because the stream and corresponding streambed temperature time series follow an irregular pattern, in contrast to the cyclical sine pattern in the other models, daily values of Ar and Δϕ and associated estimates of velocity were variable. Although daily estimated velocities varied each day, the true vertical velocity through the bed remained constant. Figure 3 shows the average estimated velocity for 8 days with a standard deviation reflecting the variability of the estimated velocity from day to day. Nonsinusoidal temperature patterns in the stream and streambed caused estimated velocity errors greater than for the ideal or thermal gradient models, regardless of the flow rate through the model or whether Ar, Δϕmax, or Δϕmin were used. Also, velocity estimates could not be derived when vz was <1.0 m/d because the lag times were outside the limit imposed by the analytical model, as discussed previously.
 Estimated velocity errors from Ar were between 5% and 10%, increasing with greater velocity (Figure 3). Although the average vertical velocity from Ar had the smallest percent error at lower flow rates (average error of only −5.4% at vz = 0.8 m/d versus 10.4% at vz = 3.1 m/d), the percent error of individual daily velocity estimates were highest at low flow. Individual daily estimates of velocity from Ar had errors as high as 31% at vz = 0.8 m/d and only up to 16% at vz = 3.1 m/d. When filtering of data is not possible due to the short time period of record, it is advisable to use multiple days and take an average to reduce the error of the velocity estimate. Errors from individual daily velocity estimates have a higher potential for error than average estimates from multiple days.
 Velocity errors for the nonsinusoidal model using Δϕ were much larger than for Ar, ranging from 5% to 55%. The standard deviation of average daily velocities was higher for the Δϕ method than the Ar method (Figure 3), indicating more variability of the daily velocity estimates, even when the average velocity was close to the true value. Due to the irregular temperature pattern, lag times of the maximum and minimum temperatures were not the same and errors derived from using the maximum or minimum lag time were very different. Velocity estimate errors from Δϕmax were much lower (between 5% and 11%) than for Δϕmin (between 23% and 55%). Lag times for the minimum temperature were much shorter than for the maximum temperature and caused all estimated velocities to be larger than the true velocity. This is attributed to the slower rate of temperature change over time leading up to the minimum temperature observation (Figure 5). The rate of temperature change leading up to the maximum was higher. A lower rate of temperature change over time allows the streambed temperature to respond more quickly to the slowly changing boundary condition, reducing the lag time. In a field setting, it may be difficult to determine whether the maximum or minimum temperature is better for deriving a lag time, so Δϕ should not be used when the lag times for the maximum and minimum temperature are different. It also appears that daily estimates of velocity are more variable when using Δϕ, so averages of multiple days are preferable to individual daily estimates of velocity.
3.4. Effects of Nonvertical Flow
 Previous investigations using streambed temperature data to model velocities of water flow through the streambed have not considered the impacts of nonvertical flow, although Keery et al.  suggested that this be explored by applying analytical modeling methods to synthetic temperature data, as we have done here. When flow through the streambed is not vertical, but rather in two dimensions, the estimated velocities from the analytical models are generally higher than the true vertical velocity, regardless of whether Ar, Δϕmax, or Δϕmin is used (Figure 6). Estimated vertical velocity errors ranged from 5% to 192% for the Ar method and from 15% to 60% for the Δϕ method. The higher velocity estimates from the Hatch methods are attributed to the additional transport of heat into the bed in directions oblique to the streambed interface. At each observation point in the model, the total velocity (v) was higher than the vertical velocity (vz) due to the horizontal component of flow (Table 5).
Table 5. Actual and Estimated Velocities for the 2-D Model, Including the Percent Error of Estimated Velocities
Estimated vz From Ar (m/d)
Estimated vz From Δmax (m/d)
Estimated vz From Δmin (m/d)
 The percent error of the estimated velocity from Ar was large (40% or greater) when the horizontal velocity was more than twice the vertical velocity (vz:vx < 0.4; Figure 6 and Table 5). When the vertical velocity was greater than the horizontal velocity, errors were generally below 20%. Caution should be used when estimating vertical velocity with Ar under 2-D flow conditions because the error of the velocity estimates varies with the ratio of vz to vx.
 The error from the two Δϕ methods was similar when vz:vx was greater than 0.3 (27–44% for Δϕmax and 15–25% for Δϕmin), and the estimated velocity was typically more representative of the vertical velocity vector, rather than the horizontal vector or the total velocity. At the lowest vertical velocity simulated (vz = 0.1 m/d), the Δϕmin could not be used to generate a velocity estimate because the lag time fell outside of the acceptable range for the Δϕ method, as discussed earlier. An advantage to using Δϕ over Ar under 2-D flow conditions is that the percent error of velocity estimates remains relatively constant, even as vz:vx changes.
3.5. Field Estimates of Seepage Flux From Darcy and Temperature-Based Methods
 The range of Ar observed in the field (0.01–0.54) was slightly smaller than the range of Ar simulated with the analytical and numerical models (0.00–3.10). The smaller range of Ar observed in the field generated a smaller range of vertical velocity estimates than generated from the numerical modeling results (vz = −0.2 to 1.1 m/d and 0.0 to 3.1 m/d, respectively). Although the ranges of Ar and vz are slightly different, these differences are negligible. Also, the depth of observation in the streambed (z = 0.25 m) is the same, and we therefore expect the error of velocity estimates from Ar for our field observations to be consistent with the modeling results. We did not use lag times to estimate velocity from field data because there were several lag time values that fell outside of the range acceptable for the model. Also, the minimum and maximum lag times were different due to the nonsinusoidal temperature records observed in the field.
 Estimated seepage flux rates from the Ar method (KI, rather than vertical velocities, KI/n) and Darcy flux calculations for this site have been previously reported elsewhere [Fanelli and Lautz, 2008]. Here we provide vertical velocity estimates (KI/n), and the velocity estimates given are for two distinct geomorphic units: a stagnant pool with large, negative hydraulic gradients and a plunge pool/riffle feature with small, negative hydraulic gradients. The stagnant pool is upstream of a small log dam, which traps fine sediments (fine sand, silt, and clay) and creates a bed with relatively low K (∼10−6 m/s). Flow in the plunge pool/riffle section is more turbulent and generates a coarse sand and gravel bed with a higher K (∼10−5 m/s). Details on morphology can be found in the work of Fanelli and Lautz  and Lautz and Fanelli .
 In the stagnant pool, we anticipate flow consistent with the thermal gradient and nonsinusoidal stream temperature models (i.e., predominantly vertical flow with a slight thermal gradient and nonsinusoidal stream temperatures). Therefore, we expect errors from Ar velocity estimates to be limited to ∼10% or less. For the Darcy flux calculations, gradients were relatively uniform (I = −1.00 ± 0.15, with the exception of one location where I = −0.23). Large differences in flux spatially would have to be attributed to spatial variability of K that we could not accurately characterize at the spatial scale of the minipiezometer installations. Stream minipiezometers were installed in a ∼1 m grid over a 3 m by 3 m feature that should have relatively uniform K. Although we assumed uniform K for our Darcy flux calculations, it is plausible that K varied by 50% or more. It is logistically difficult to capture those subtle changes in the field and the Ar method provides a means of obtaining quick and accurate velocity estimates without assessing K at each measurement point. Although the overall average vertical velocity estimate in the stagnant pool is similar for the Ar results (0.06 m/d) and the Darcy flux results (0.19 m/d), the Ar method is able to capture spatial variability that is not captured when K is assumed constant (Figure 7).
 In the plunge pool/riffle feature, we anticipate flow has both horizontal and vertical components and more closely resembles the modeling results from the 2-D case. Therefore, we anticipate errors as high as 50–75% within the range of estimated velocities we observed. Despite potentially large percent errors for the vertical velocity estimates, we expect the Ar method to distinguish areas of higher and lower rates of flow, relative to one another, as was the case for the modeling. Although we anticipate relatively high errors from the Ar method due to nonvertical flow, error due to variability of K is likely equally high. A similar range of velocity estimates is given for Ar (−0.15 to 1.12 m/d) and Darcy flux results (0.12 to 1.41 m/d), suggesting equal reliability, even under worst-case field conditions for the thermal modeling. Although the range of estimated vertical velocities for the plunge pool/riffle feature is similar, the spatial variability and scatter around the 1:1 line is large (Figure 7). We attribute the scatter to inherent error introduced by assuming constant K of the streambed in this unit. Vertical velocity estimates are logistically easier with the Ar method and errors are minimized when the vz component of flow is maximized.
 It is difficult to accurately measure the rate of water flux across the streambed interface using common field techniques, such as Darcy flux measurements, particularly due to the challenges of spatial variability of physical streambed characteristics, including hydraulic conductivity (K) [e.g., Cardenas and Zlotnik, 2003]. Streambed K values are highly variable [Calver, 2001] and using minipiezometers to measure K with slug tests can be impractical due to rapid recovery of the piezometer (e.g., <30 s) and problems installing automated water height data loggers in narrow diameter wells [Baxter et al., 2003]. Although repeated measurements of K using slug tests are recommended to improve accuracy, such repeated tests in the streambed can compromise the seal around the minipiezometer screen, creating a direct connection between water flow through the screen and the stream itself, compromising the slug test results.
 Given the challenges of accurately characterizing streambed hydraulic conductivity over space, a common approach is to make several measurements of hydraulic conductivity within a single morphological unit and assume the streambed within that unit can be represented by the geometric mean value of K [e.g., Lautz and Siegel, 2006; Wroblicky et al., 1998]. Although this approach is widely used, it has also been observed that K values can vary by several orders of magnitude over relatively small reaches [Cardenas and Zlotnik, 2003] and that this spatial variability of K can generate highly variable flux rates across the streambed interface [Cardenas et al., 2004; Woessner, 2000]. Here we assumed a uniform K value for individual morphological features. Although this may provide an accurate assessment of flux on average for the individual features, spatial variations in flux within the individual features are not accurately captured because the K distribution is not accurately represented. The use of streambed temperature data to estimate vertical velocities across the streambed interface in lieu of Darcy flux estimates has a greater potential for characterizing relative differences in flux with space (Figure 7). It is also logistically easier to collect the necessary data with the required spatial resolution. Even under nonideal field conditions, the analytical methods developed by Hatch et al.  outperformed expectations of Darcy calculations, suggesting that interpretation of temperature time series data to derive streambed flux is superior to Darcy flux measurements, even under nonideal conditions.
 Based on the results presented here, it is advisable to use the amplitude ratio to derive streambed flux rather than the lag time when it is possible that field conditions are different from the conditions assumed by the Hatch model. Interestingly, in all previous examples where this type of method has been applied to actual field data [Fanelli and Lautz, 2008; Hatch et al., 2006; Keery et al., 2007], only Ar has been used to derive streambed seepage rates, never Δϕ. The amplitude ratio model effectively predicted the velocity of water flow through the streambed even under conditions that violated the analytical model assumptions, including the presence of a thermal gradient in the streambed and nonsinusoidal stream temperatures (Figure 3). When the amplitude ratio method was used under conditions of vertical flow, the estimated velocities were generally within 10% of the true value. Although vertical flow proved to be a critical assumption of the model by Hatch et al. , when nonvertical fluid flow is present in the streambed, the estimated velocities from the amplitude ratios were typically within an order of magnitude and within 20% when vz > vx (Figure 6 and Table 5). Although the absolute value of the fluxes were off at low velocities, the Ar method was still useful for accurately comparing relative fluid velocity between points, which would be useful for within-site comparisons. The Ar method has the additional benefit of providing directionality of flux (i.e., upwelling or downwelling), whereas flux estimates from the lag time are simply absolute values [Hatch et al., 2006]. For example, a lag time of 7.41 h in the analytical model described here would provide a vertical velocity estimate of 0.33 m/d or −0.33 m/d (Figure 4). The directionality, or sign, of the SW-GW flux rate is not distinguished by the lag time.
 Errors of velocity estimates from lag time were generally larger and more variable (Figure 3). Also, the error of the velocity estimates depend on which point in the temperature time series is used to derive the lag time. A common choice is to identify the temperature maximum or minimum to derive the lag time, but when temperature signals are nonsinusoidal or flow is not vertical, the lag times of these different markers in the temperature signal are likely to be different. While one lag time may produce relatively small errors (such as the lag time of the maximum temperature in our nonsinusoidal model), it will be difficult to know which marker to use to minimize estimated velocity errors.
 Lag time estimates are also more prone to error due to logistical considerations in field measurements. If the temporal or thermal resolution of a temperature time series is somewhat low, such as temperature records that are recorded only every hour or only to the nearest 0.5°C, lag time estimates are prone to large errors, particularly under low or no flow conditions. Slight errors in lag time estimates, particularly at low-flow rates, can cause lag times to fall outside of the range solvable by the Δϕ model (Figure 4). Under such circumstances, no velocity estimate can be made. Using the amplitude ratio, there are no such limits.
 Although it is preferable to filter temperature time series to isolate the sinusoidal diurnal signal [e.g., Hatch et al., 2006; Keery et al., 2007], the amplitude ratio method proves robust to nonsinusoidal temperature signals. When only short records are available (e.g., 5–10 days) and filtering is not possible, velocity estimates from the amplitude ratio method are still considered reliable to within 10% when flow is vertical and to within an order of magnitude when flow is in two dimensions. To improve the accuracy of velocity estimates, multiday average velocity estimates should be used. Also, although the absolute value of velocity may be subject to error when flow is not vertical, results still distinguish relatively high and low rates of water flux, even without field estimates of hydraulic conductivity. This suggests that the deployment of temperature data loggers in the stream and streambed for even short periods of time can be used to derive reliable, relative estimates of streambed flux. Not only are results expected to have smaller errors than other measurements, such as Darcy flux estimates, but also the field data collection and interpretation are relatively simple. For this reason, estimates of streambed flux from temperature time series data are preferable to other methods.
 The use of temperature time series data to quantify water flux across the streambed interface is a promising alternative to more commonly used approaches, such as Darcy flux calculations. As has been observed elsewhere, uncertainty of parameters used in heat transport models, such as the Hatch method, are generally lower than the uncertainty of hydraulic properties, such as conductivity. Although the use of the Hatch model relies on three assumptions (no thermal gradient in the streambed, sinusoidal temperature changes and purely vertical flow), we have shown that the method is very robust to violations of these assumptions, particularly when flow is vertical. Darcy flux calculations that incorporate measurements of vertical hydraulic gradients in the streambed also assume vertical flow and do not capture the 2-D nature of flow found in many field settings. Although the vertical velocity estimates derived from the Hatch methods can be in error when flow is not vertical, they are equally reliable to Darcy flux calculations, particularly given the logistical challenges of obtaining accurate K values for the shallow streambed in the field. Temperature time series provides a potentially more accurate and certainly more consistent method of estimating streambed flux, even under nonideal field conditions, without filtering of signals.
 The amplitude ratio method is less prone to error than the lag time method under nonideal field conditions, and therefore should be used in preference to the lag time. Also, the amplitude ratio provides directionality, while the lag time inherently only provides the magnitude of flux [Hatch et al., 2006]. The amplitude ratio is particularly robust to thermal gradients in the streambed and nonsinusoidal temperature fluctuations. Although filtering of field data is recommended to isolate the sinusoidal diurnal signal in the temperature record, short records can be used to estimate velocity, even when filtering is not possible. Under such circumstances, several days of record should be used to obtain an average to minimize error.
 The greatest source of error for velocity estimates derived from amplitude ratios and lag times will be due to nonvertical flow in the streambed. Although the errors may be potentially high when horizontal velocities are much greater than vertical velocities, the relative velocities within a field site can be accurately compared. Therefore, temperature records may be a better alternative than Darcy flux calculations for capturing spatial variability in flux within sites. This can be particularly useful for relating rate of water exchange across the streambed interface with other parameters, such as geochemical setting or biodiversity indicators.
 This material is based upon work supported by the National Science Foundation under grant EAR-0901480. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. The quality of the manuscript was greatly improved by the input of two anonymous reviewers.