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 Streambed seepage can be predicted using an analytical solution to the one-dimensional heat transport equation to take advantage of the relationship between streambed thermal properties, seepage flux, and the amplitude ratio and phase shift associated with streambed temperature signals. This paper explores the accuracy of streambed-seepage velocity estimates from this method when uncertainty in input parameters exists. Uncertainty in sensor spacing, thermal diffusivity, and the accuracy of temperature sensors were examined both individually and in combination using Monte Carlo analysis. The analytical solution correctly reproduced known thermal front velocities above 1.25 m d−1, using both the amplitude-ratio and phase-shift methods, despite introduced uncertainty in any of the variables. Noise in temperature measurements (because of sensor accuracy) caused erroneous prediction of velocity for gaining stream conditions using both the amplitude ratio and phase shift. Uncertainty in the thermal diffusivity and sensor spacing resulted in incorrect velocity, primarily under gaining conditions, when using the amplitude ratio and near-zero velocity using the phase shift. For a sensor accuracy of 0.15°C, we present combinations of parameters for which the resulting signal amplitude is sufficiently large for use with the Stallman equation.
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 As freshwater supplies become scarcer in many parts of the world and the need to quantify overall water availability grows more important, it has become increasingly clear that surface water and groundwater indeed represent one connected, finite, and dynamic source of water [Winter et al., 1998]. Therefore, it is important to understand where, and how much, water is moving between the aquifer and the surface water expressions above it. Several methods have been explored for this purpose, including seepage meters, differential-discharge measurements, shallow piezometers, tracer experiments, and temperature-tracer measurements. Several recent papers have compared the use of streambed temperatures with other methods to determine seepage [Anderson, 2005; Kalbus et al., 2006; Constantz, 2008; Rosenberry and LaBaugh, 2008] and found this method to produce acceptable results for a wide variety of conditions. Moreover, temperature measurements have the advantages of being comparatively cost-effective to collect with readily available data loggers, even over relatively long periods of time and with fine temporal resolution.
 Several methods of solving the coupled water and heat advection-dispersion equation have been employed to calculate streambed seepage using temperature measurements [Constantz, 2008; Anderson, 2005]. Stallman  proposed a one-dimensional (1-D) solution in which the change in the amplitude of the diel temperature signal, with depth and lag in the response time to temperature variation at the surface, is used to directly calculate seepage flux when all other parameters are known. This analytical solution assumes 1-D, uniform vertical flow, sinusoidal behavior of diel surface temperature, and no change in average temperature with depth. Several studies have used Stallman's approach to calculate seepage rates from thermal time series data [see, e.g., Goto et al., 2005; Hatch et al., 2006; Keery et al., 2007; Fanelli and Lautz, 2008; Lautz, 2010]. Hatch et al.  developed a useful, semiautomated set of Matlab routines to filter field data, derive amplitude and phase-shift information, and use these values to iteratively solve for time-varying seepage rates. Lautz  evaluated the impact of nonideal field conditions on flux estimates from this analytical model. Vogt et al.  used fine-resolution vertical temperature measurements to determine differences in velocity along a vertical profile. However, there has been little investigation as to the influence of uncertainty in input parameters on the accuracy of predicted seepage velocities and the range of applicability of this method.
 Monte Carlo simulation is a useful technique for testing uncertainty in seepage estimates from temperature measurements [Keery et al., 2005; Niswonger and Rupp, 2000; Constanz et al., 2002]. In this paper, we use a Monte Carlo analysis to explore the effects of sensor accuracy and uncertainty in the input parameters' thermal diffusivity (Ke) and sensor spacing on the predicted velocity estimates over a wide range of both gaining and losing streambed conditions. We explore the effects of uncertainty on velocity estimates for each parameter separately, as well as the cumulative effects of uncertainty in several parameters concurrently.
 Assuming that surface water temperatures vary temporally in a sinusoidal pattern in response to ambient atmospheric conditions and constant vertical flow of fluid between the surface and a homogeneous subsurface, the thermal response of saturated sediment below surface water systems was given by Stallman  as
where T is temperature (°C), z is depth below the streambed (m), A is amplitude of temperature variation at the upper boundary, v is thermal front velocity (m d−1), Ke is effective thermal diffusivity of the saturated sediment (m2 d−1), t is time in days, P is the period of temperature variations (always one day for this study), and (m2 d−2). Assuming optimal conditions of low heat conductivity, large surface temperature amplitude, and “careful measurements,” Stallman  predicted equation (2) could accurately predict velocities greater than 0.003 m d−1.
 Thermal diffusivity is dependent on the porosity, saturated thermal conductivity of the sediment, and the specific heat of the sediment grains [Ingebritsen and Sanford, 1998]:
where ρc is volumetric heat capacity of the saturated sediment-fluid system (J m−3 °C−1), is thermal dispersivity (m), vf is fluid velocity ( where is the ratio of heat capacity of the streambed to the fluid) (m d−1), and (W m−1 °C−1) is thermal conductivity in the absence of flow (, where f and g denote the fluid and grain properties, respectively, and n is porosity).
 When the thermal properties of the sediment are known, the ratio between the amplitude of temperature traces at two shallow depths in the subsurface (Ar = amplitude at lower sensor divided by the amplitude at upper sensor) can be used to solve for the velocity of the thermal front [Hatch et al., 2006]:
where is the spacing between the upper and lower sensors. Alternatively, the time lag (phase shift ) between the temperature peaks at the two depths can be used to determine velocity [Hatch et al., 2006]:
 The upper boundary condition can be satisfied by measuring the temperature of the surface water itself [see, e.g., Keery et al., 2007; Fanelli and Lautz, 2008; Lautz, 2010] or the temperature of the sediment just below the stream sediment interface [see, e.g., Hatch et al., 2006]. When the surface water temperature is used, it is possible that the depth of the lower sensor below the streambed will change over time because of erosion or accretion of sediment. An error in (likely up to 0.02 m) can also occur for both surface and subsurface boundary conditions as a result of incorrect installation of temperature sensors or piezometers. Here we have selected temperatures from depths of 0.10 and 0.35 m, and used an average of 0.25 m. We assumed that a change in streambed elevation (and therefore when the surface is used as the upper boundary condition) of up to 0.20 m is possible over time (standard deviation of 0.10 m; 1.96 ).
 Many different types of sensors can be used to collect temperatures over time. The accuracy of temperature sensors generally ranges between < 0.01°C [see, e.g., Goto et al., 2005] for thermistor probes connected to advanced data loggers to up to 1.0°C for some individually deployed ibutton® thermistors and loggers. To determine the impact of sensor accuracy in this study, we selected a midrange accuracy of 0.15°C, which corresponds with the Onset Stowaway and Hobo Tidbit sensors used by Hatch et al. . Errors because of the accuracy of the temperature sensors were assumed to be normally distributed with zero mean and a standard deviation of 0.077°C (i.e., 1.96 ). Although bias might also cause temperature sensor error, we do not consider this source of error because it can be easily quantified and corrected through sensor calibration.
 Temperature traces were created using equation (1) for a period of 150 days, assuming velocities in the range of −4–4 m d−1 (Figure 1). This length of simulation was chosen to ensure that enough daily amplitude and phase-shift information was available for averaging. Noise in , Ke, T, or all three was added to the temperatures using the averages and standard deviations. These traces were then passed through a band-pass filter, as described by Hatch et al. , after which daily temperature peaks were selected, amplitude ratios and phase shifts were calculated, and these were used to solve for velocity using equations (2) and (3). The 150 individual velocity estimates from each of the two equations were then averaged separately to determine the mean and standard deviation of the perturbed velocity. Within the predefined probability distributions, this process was repeated 1000 times for each perturbed variable. Each velocity was calculated iteratively because velocity-dependent variables appear on both sides of equations (2) and (3). In an effort to limit endless iteration on phase shift or amplitude ratios that are outside a reasonable range of sensitivity to changes in velocity, numerical limits were specified in a parameter file. Here as by Hatch et al. , for relatively flat slope(s) of dAr/dv ≤ 0.001 or amplitude-ratio and phase-shift values were flagged as outside the numerical limits, and then assigned a maximum or minimum velocity corresponding to this limit.
 In this paper, we denote downward flow (groundwater recharge) with positive velocities and upward flow (groundwater discharge) with negative velocities. Similarly, all velocities discussed in this paper are thermal front velocities and not fluid velocities, which are proportional to thermal front velocities but require knowledge of the heat capacities of the fluid and sediment.
 The Stallman solution correctly reproduced the input velocities for positive velocities (downward flow) greater than 1.5 m d−1 despite uncertainty in any of the parameters (Table 1, Figure 2), and, as discussed by Hatch et al. , the phase shift accurately predicted the magnitude of both positive and some negative seepage velocities, but not the sign (Figure 2). This is a result of velocity appearing under a square root sign when equation (4) is solved for the phase shift. Despite noise resulting from the accuracy of the temperature sensor, velocities were accurately reproduced down to −1.25 m d−1 using both the Ar and methods. Below −0.75 m d−1, the percent of model realizations that were within numerical limits decreased dramatically and the standard deviation increased (Figure 3a); therefore, this is the practical lower bound for estimates of velocity at this level of sensor accuracy. The amplitude ratio systematically overestimated velocities below −1.0 m d−1, likely because the temperature difference between the upper and lower sensors was very low (Figure 1b) and the sensor noise completely masked the difference between a constant groundwater temperature and a low amplitude diel signal above. Lack of a clear diel trend (as shown in Figure 1d) led the filter process to extract erroneous and random values, which then resulted in meaningless amplitude-ratio and phase-shift estimates (Figure 4).
Table 1. Summary of Results Showing Velocities for Which the Stallman Equation Reproduced Input Velocities With Less Than 10% Error and Greater Than 50% of Results Within Numerical Limits for Each Type of Parameter Noise
Temperature Sensor Accuracy (0.15°C)
Δz Uncertainty (0.20 m)
Ke Uncertainty (0.075 m2 d−1)
Uncertainty in all Three Parameters
>1.25, −0.75 to −2.5
>0.75, −0.75 to −2.5
>1.5, −0.75 to −1.0
 Of the three variables tested, uncertainty in of 0.20 m most limited the range over which this method accurately reproduced known velocities. For velocities less than 1.5 m d−1, the Ar method systematically underestimated known velocities and the standard deviation increased to very high values (Figures 2b and 3b). Although velocities less than −2.0 m d−1 appear to have lower standard deviation and are closer to the known values, predicted seepage values were increasingly outside of numerical limits and are therefore meaningless. Using the method, velocities greater than 0.5 m d−1, and absolute velocities between −0.75 and −2.5 m d−1, were correctly estimated. The impact of uncertainty on derived velocities was not unexpected because sensor spacing directly affects the lag time between daily temperature peaks.
 Similar to the pattern observed with certainty, noise imposed within the Ke variable caused velocities to be underpredicted when less than −1.25 m d−1 using the Ar method, and the percent of model realizations within numerical limits decreased drastically when velocities were less than −2.25 m d−1 (Figure 2c). The method correctly estimated velocities greater than 0.75 m d−1 and the magnitude, but not the sign, of velocities less than −0.75 m d−1, with results symmetrical about the y axis (Figure 2c). Although the standard deviation improved for velocities less than −2.25 m d−1 for the Ar method, the percent of model realizations within numerical limits decreased, and none of the 1000 model realizations produced an acceptable velocity less than −3.5 m d−1 for the method (Figure 3c). Therefore, −2.25 m d−1 represents the practical lower velocity limit for both Ar and methods when Ke uncertainty is present.
 When error was introduced for sensor accuracy Ke and , no synergistic effect was observed, although the range of correctly predicted velocities using was slightly less than for that observed when any of the variables were perturbed individually. Velocities greater than 1.5 m d−1 were correctly predicted for both methods, with velocities underpredicted for the Ar method and overpredicted by at lower velocities (Figure 2d). However, the absolute velocity (the magnitude but not the sign of the velocity) was correctly predicted by the for the narrow range between −0.75 and −1.0 m d−1. Standard deviation was extremely high for predicted velocities of less than 1.5 m d−1 for the Ar method (Figure 3d). Whereas standard deviation did not increase at low velocities for the , the percent of model realizations within numerical limits decreased when the velocities were less than −0.75 m d−1. Predicted velocities were consistently incorrect when less than −1.0 m d−1.
 The range of velocities over which the Ar and methods are accurate is not absolute, but it will instead depend on the amount of uncertainty present in each parameter, especially the accuracy and location of the temperature sensors. The amplitude of the weakest diel temperature signal (i.e., the lower boundary condition, where the diel signal is most attenuated) must be greater than the accuracy of the temperature sensors. From equation (3), the amplitude A of the diel signal at a specific depth can be computed as
 Therefore, given ranges in Ke, z, and v, a surface was developed below which the signal amplitude was greater than the accuracy of the sensor (Figure 5). This plane slopes upward as velocity increases, so this method becomes applicable under the whole range of Ke and a wide range of for most positive and slightly negative velocities; for simplicity, the positive velocities are therefore not shown. Equation (1) shows that this plane would be raised linearly as the magnitude of the amplitude at the upper boundary increases, whereas decreased sensor accuracy would result in a narrower range of Ke and values that will produce an accurate velocity. Further, this method will be applicable over a wider range when the lower sensor is placed closer to the surface, where the diel signal is likely to be stronger (Figure 6).
 The results of this study show that the Stallman solution precisely reproduced velocities for moderate rates of infiltration, as shown in Table 1, despite uncertainty in several input parameters. Therefore, this method is most suitable for estimating changes in seepage over long time periods for reaches that have been previously identified as moderately losing, i.e., through gauge differencing or using seepage meters. Stallman  estimated that the analytical solution would be insensitive to temperature variations at velocities less than 0.003 m d−1, whereas Silliman et al.  suggested 0.007 m d−1 as the minimum fluid velocity. Uncertainty in several parameters also affects the range over which this method is applicable.
 Uncertainty in both Ke and widened the range of insensitivity (Figures 2b and 2c), with low and negative velocities underestimated by the amplitude ratio when there was uncertainty in (Figure 2b). Hatch et al.  predicted a lack of sensitivity at zero velocity when using the phase shift, but a peak in sensitivity near zero velocity for the amplitude ratio. However, the location of this peak and the range of values to which the method is sensitive are dependent upon sensor spacing, with wider spacing resulting in higher sensitivity at negative velocities.
 For this study, a sensor spacing of 0.25 m was selected, consistent with the studies of Fanelli and Lautz  and Lautz , and similar to Goto et al.  and Hatch et al. . Sensor spacing is a trade-off between several factors. Low seepage velocities necessitate close probe spacing placed near to the surface to capture diel temperature variations propagating from the surface water. However, when probes are placed close together (i.e., m) or further below the surface, gaining conditions result in small differences in amplitude between temperatures at the two depths. The stream temperature provides a convenient, well-mixed, and advective upper boundary condition, but suffers from both the effects of solar warming of the surface sediments, as well as the effects of uncertainty in sensor spacing because of accretion or erosion of the stream-subsurface interface (e.g., Figure 3b). By placing the upper probe at a shallow depth below the streambed interface, the effects of solar warming and uncertainty in sensor spacing would be avoided, unless erosion at the surface exposed the upper probe over time. The optimal depth for the upper temperature sensor, as well as the most appropriate , will depend upon the conditions present in the streambed. Blasch et al.  reported that discerning periods of flow in an ephemeral stream require a larger of 0.45 m to differentiate the differences in thermal amplitude at depth during periods of flow and no-flow. However, as discussed by Silliman et al. , the magnitude of temperature response to diel surface temperature variations decreases as sensor depth increases, whereas the phase shift increases at depth. Smaller values also minimize potential errors in Ke because of heterogeneity of the sediment column between measurement depths. Figures 5 and 6 help to provide a guide for placement of temperature sensors based on expected velocity Ke and sensor accuracy.
Figure 5 suggests that the analytical solution may not be appropriate for most common Ke values when there is negative (upward) velocity, especially when using a sensor spacing of more than 0.20 m or temperature sensors with low-to-moderate accuracy. Under these conditions, the change in amplitude between sensor depths is not sufficient to overcome signal noise and parameter uncertainty (Figure 1). An example of the limit of this analytical solution for upward velocities may be seen in the riffles measured by Fanelli and Lautz , where Darcy methods predicted weakly gaining conditions, whereas the analytical solution predicted a weakly losing streambed. Alternatively, the low velocities predicted at those sites may have been within analytical uncertainty.
 The lower limits of applicability could be extended slightly by manually selecting peak temperature values from the data to correctly identify the amplitude ratio and the phase shift. In contrast, filtering of data prior to selection of peak temperatures is necessary when using automated routines when velocities are near the lower limits of this method and sensor noise is observed. The need for filtering is also described by Hatch et al.  and Keery et al. . Minimizing the sensor spacing increases the range of velocities for which the solution is appropriate (Figure 6). However, this would also increase sensitivity to small-scale variability in heat transport and may only represent a small portion of the vertical subsurface profile. Using fine-resolution temperature measurements, Vogt et al.  found that velocity was variable over a 2 m vertical profile. Therefore, wider sensor spacing may be needed when a general estimate of velocity in the shallow subsurface is desired.
 Although this study focused on uncertainty in three variables and their effect on velocity estimates using the Stallman solution, several other factors may influence how well the Stallman solution will accurately predict velocities from field data. Real-world data often violate the assumptions of sinusoidal boundary conditions and purely vertical flow. Lautz  investigated the impact of both of these situations on seepage estimates. Nonsinusoidal conditions were found to cause vertical fluid velocity errors of 5%–10% in estimates from Ar and 5%–55% in estimates from , and it was not possible to estimate fluid velocities below 1.0 m d−1. Nonvertical flow had an even greater impact on fluid-velocity estimates, with up to 192% error for using Ar and 60% for , depending on the amount of lateral flow introduced. When the fluid velocity (or seepage rate) is the primary variable of interest, the porosity of the streambed matrix must be given. The effect of uncertainty in porosity on the fluid velocities has not been investigated. Finally, the use of heat as a tracer, in general, also has inherent drawbacks that may add to error in seepage estimates, including drift in observed sensor readings, the possibility of heat conduction down into the well or circulation of water within the well, and difficulty in properly installing sensors in some stream environments.
 The analytical solution presented by Stallman  has been successfully used to estimate streambed infiltration in several studies [Goto et al., 2005; Hatch et al., 2006; Keery et al., 2007; Fanelli and Lautz, 2008; Lautz, 2010]. Some of the advantages of using the analytical solution to estimate seepage through the streambed are that it only requires temperature traces from two depths below the streambed and an understanding of local streambed thermal properties (i.e., no knowledge of hydraulic head or hydraulic conductivity is necessary), and it can provide estimates of temporal change in streambed seepage between the two sensors. However, the accuracy of velocity estimates is dependent upon the reliability of input parameters. The range of velocities over which this analytical solution correctly estimated seepage was identified for conditions in which uncertainty in thermal diffusivity, sensor error, and sensor spacing exists. When seepage was close to zero or velocities were negative, uncertainty in these input parameters resulted in incorrect estimated velocities. It is therefore imperative that the amplitude of the limiting temperature signal is greater than the accuracy of the temperature sensor itself. The effect of parameter uncertainty was minimized at higher values of thermal diffusivity and when sensor spacing was reduced; however, other methods may be more appropriate under these conditions.