3.1. Winter Seepage Flux
 The solution to equation (1) presented by Bredehoeft and Papadopulos  assumes steady state thermal conditions within the streambed (∂T/∂t→0). The upper boundary corresponds to the interface of the surface water and the streambed (z = 0) while the lower boundary corresponds to the lowermost temperature measurement location (z = L, where L is the total length of the measured sediment zone, or 50 cm for our application). Boundary conditions at these locations can be summarized as [Bredehoeft and Papadopulos, 1965]:
where Tz is the temperature (°C) at any depth, z, T0 is the temperature at the upper boundary (°C), and TL is the lowermost temperature measurement (°C). The solution to equation (1) with boundary conditions (2) and (3) is:
where . The dimensionless parameter, ψ, is positive when q is positive (i.e., downward flow) and negative when q is negative (i.e., upward flow) and λ0 is the thermal conductivity of the sediment-fluid matrix (W m−1 °C−1).
 Weekly average temperatures for each monitoring location and depth were calculated from 10 December 2006 through 10 March 2007. Temperatures during this period were visually determined to approximate thermally steady state conditions (see Figure 4). To further refine acceptable temperatures used for flux calculations, if the total range in observed temperatures during the week was less than 0.6°C (±0.3°C, which is three times the sensor accuracy and allows for slight differences due to sensor precision), then conditions were treated as steady state. The Solver add-in for Microsoft Excel was used to solve equation (4) following the method presented by Arriaga and Leap . The lower limit of accepted velocities was taken as [Bredehoeft and Papadopulos, 1965]:
 Velocities with a magnitude less than that predicted from equation (5) are not reported. For nonunique solutions, which occurred when temperatures did not provide any gradient within the upper three sensors such that a unique downwelling or upwelling flux could be determined, the smallest possible seepage flux to fit the nonunique thermal profile was assigned to the sensor location for that given week. Thus, the calculated seepage flux is a conservative estimate for the seepage velocity when a nonunique temperature profile is observed.
3.2. Spring and Summer Seepage Flux
 The solution to equation (1), assuming sinusoidal variation in surface temperature and constant temperature at depth in the streambed, can be expressed as [Hatch et al., 2006; Keery et al., 2007; Stallman, 1965]:
where T0 is the average surface temperature (°C), A is the amplitude of the temperature fluctuations at the surface (°C) and ω is the angular frequency, defined as where P is the period of the temperature fluctuation (86,400 seconds for daily frequency). This solution can be applied with nonvertical (curvilinear) paths as long as fluxes in the directions orthogonal to the vertical do not result in a large divergence of energy fluxes within the space between the two sensors [Cuthbert and Mackay, 2013]. In equation (6), the a and b terms can be expressed as:
where the subscripts 1 and 2 refer to the shallower and deeper temperature sensors, respectively, and ϕ2 − ϕ1 is the phase shift (rad) associated with two temperature sensors. The newly presented term, η, relates a and b in one single metric [Luce et al., 2013]:
vt is the thermal front velocity associated with advective transport of thermal energy and is the diffusive phase velocity associated with the diffusive transport of thermal energy in the streambed. The symbol, η, the ratio of the logarithm of surface and depth amplitudes to the phase difference of surface and depth thermal time signals, relates to the dimensionless v* velocity, which contains both thermal front velocity (advective and diffusive) and phase velocity (diffusive) components representing different transport modes for the thermal signal. The dimensionless velocity, v*, is one-half of the Peclet number and provides direct information on the relative importance of advective and diffusive transport. Diffusion dominates where v*<<1 and advection dominates when v*≫1. The analytical explicit solution relating v* and η is [Luce et al., 2013]:
 For η in the range of 0 < η < 1, Darcian velocities are positive (i.e., downwelling). When η = 1, Darcian velocity is zero (i.e., only diffusive transport),and when η > 1, Darcian velocities are negative (i.e., upwelling).
 Equation (11) combines the amplitude ratio and time lag into one metric and allows the determination of a dimensionless velocity independent of information about the depths of the sensors. The additional degree of freedom obtained from using the sensor spacing and amplitude ratio (or phase difference) can be applied to estimate the thermal diffusivity of the streambed [Luce et al., 2013].
 Both equations provide a consistent estimate for the thermal diffusivity and therefore only one should be applied. Thermal diffusivity is related to the damping depth (depth at which the amplitude is 1/e times the surface amplitude) by .
 Calculation of spring and summer seepage flux, as well as determination of thermal diffusivity, requires the amplitude and phase angle associated with the diurnal forcing to be determined for each thermal time series. While short time series can be examined with graphical or visual methods, longer time series typically require the use of fitting functions, such as nonlinear least squares fitting [Swanson and Cardenas, 2010] or Fourier analysis, to isolate the amplitude and phase angle [Luce and Tarboton, 2010]. Dynamic harmonic regression (DHR), which is implemented within the MATLAB-based CAPTAIN toolbox [Taylor et al., 2007], was used to isolate the amplitude and phase information from thermal time series at Bear Valley Creek. DHR is a nonstationary extension to Fourier analysis and produces time-varying apparent amplitude and phase coefficients for a time series at a user-specified frequency (one day for this study). DHR has been used successfully in several studies to determine thermal time series characteristics and subsequently quantify streambed seepage flux [e.g., Briggs et al., 2012; Keery et al., 2007; Vogt et al., 2010]. DHR was implemented using the new VFLUX program [Gordon et al., 2012] as a front-end tool for applying a filter to the thermal time series, isolating the diurnal component (using DHR), and extracting the time-varying amplitude and phase information.
 The original thermal time series (Δt = 12 minutes) from 1 April 2007 through 10 July 2007 (the portion of the thermal time series that presented diurnally fluctuating surface and depth temperatures) was downsampled using a low-pass filter such that the time interval was 2 h. Using the low-pass filter eliminates much of the high-frequency noise that complicates fitting of the observed thermal time series. DHR was then implemented via the CAPTAIN toolbox within VFLUX to extract the time-varying apparent amplitude and phase of the temperature signal. The frequency was specified as one day (diurnal fluctuations). Following practices implemented by other DHR users, 2 days of the diurnal signal were removed from each end of the temperature record to eliminate possible edge effects due to filtering [Gordon et al., 2012; Keery et al., 2007].
 The time-varying amplitude and phase information was used to calculate a time-varying η parameter (equation (9)). Following the determination of η, v* was determined based on equation (11). Using equation (12a), thermal diffusivity was estimated for two 5 day periods within the period of record where the thermal time series was visually determined to approximate the idealized boundary conditions. The two 5 day periods selected were 8 May 2007 through 12 May 2007 and 2 July 2007 through 6 July 2007 (representing spring and summer conditions, respectively). The average of all thermal diffusivity values that demonstrated agreement between spring and summer estimates of thermal diffusivity was used to determine the value for bulk thermal diffusivity, which was used when calculating the streambed seepage flux for the entire spring/summer time period. The average thermal diffusivity value was checked to see if it fell within an appropriate range of 0.02 to 0.13 m2 d−1 as reported by Shanafield et al. .
 Since DHR provides a time series of apparent amplitude and phase angle, it is possible to calculate a time-varying thermal diffusivity value at each sensor location for the entire time series. Barring drastic changes to the streambed sediments over a relatively short time scale, thermal diffusivity values should remain relatively constant through time, and the Bear Valley Creek system is expected to be characterized by relatively constant thermal diffusivity values over time. Other researchers have applied spatially varying values of thermal diffusivity for highly heterogeneous sediment [Fanelli and Lautz, 2008] but that distinction without a priori knowledge of sediment heterogeneity was not warranted for this study. We used an average bulk diffusivity value calculated from two seasonally variant time periods to estimate an average thermal conductivity. The calculated value for thermal conductivity was used during the winter seepage flux calculations and so determining a value of thermal conductivity based on the calculated bulk diffusivity value provides continuity between the two methods and maintains consistent properties of the HZ for this study.
 Following the determination of the thermal diffusivity, the diffusive phase velocity was calculated for the Bear Valley Creek system. Equation (10) was then used to calculate the thermal front velocity. Thermal front velocity is related to the Darcian seepage flux by the γ term (ratio of the sediment–water matrix–specific heat and water-specific heat). Streambed seepage fluxes were calculated between the surface sensor and depth sensors (0 to 10 cm, 0 to 20 cm, and 0 to 50 cm) from 3 April 2007 through 8 July 2007. Thus, for each location, the calculations represent depth-averaged seepage flux between the surface and depth sensor at 10 cm, 20 cm, and 50 cm. For each of the depth-averaged sensor pairings, the downsampled frequency of 12 samples per day leads to 1164 estimates of time-varying seepage flux. It is important to note that the solutions for the winter, spring, and summer seepage fluxes represent only the vertical component of the seepage flux (both solutions solve the governing one-dimensional advection-diffusion equation). This one-dimensional simplification is commonly applied to HZ systems [e.g., Fanelli and Lautz, 2008; Hatch et al., 2010; Keery et al., 2007; Lautz, 2012]. The one-dimensional results are used to interpret aspects of three-dimensional system behavior.
 Three specific 5 day time windows were examined to determine the spatial and temporal characteristics of HZ fluxes. Two time windows correspond to spring runoff peak flows and the third window corresponds to low flow conditions on the receding limb of the hydrograph. The first time window (1 May 2007 through 5 May 2007: “Spring – Bankfull”) is slightly below bankfull flow in Bear Valley Creek as determined from the reconstructed hydrograph. The second time window (17 May 2007 through 21 May 2007: “Spring – Overbank”) corresponds to an overbank flow event, which is seasonally quite common for the Bear Valley Creek system due to the wide alluvial valley, low elevation floodplain, and local features (wood, partial channel-spanning beaver dams, etc.) causing increased water surface elevations. The last time window (2 July 2007 through 6 July 2007: “Summer – Low Flow”) corresponds to low flow conditions on the receding limb of the hydrograph. Absolute low flow conditions (typically late August or September) are not represented in the available thermal time series.
3.3. Seepage Flux Direction and Velocity Errors
 The error associated with η can be calculated with the following equation [Luce et al., 2013]:
where σA is the error associated with the amplitude measurement, σφ is the error associated with the time measurement, and ση is the absolute error on the η estimate. The error associated with v* can be calculated with the following equation:
where σv* is the absolute error on the v* estimate. For this study, amplitude errors were assumed to be approximately 0.05°C and phase errors were assumed to be 0.03 rad (6.87 minutes, or approximately Δt/2).