Continuous estimates of streamflow are challenging in ephemeral channels. The extremely transient nature of ephemeral streamflows results in shifting channel geometry and degradation in the calibration of streamflow stations. Earlier work suggests that analysis of streambed temperature profiles is a promising technique for estimating streamflow patterns in ephemeral channels. The present work provides a detailed examination of the basis for using heat as a tracer of stream/groundwater exchanges, followed by a description of an appropriate heat and water transport simulation code for ephemeral channels, as well as discussion of several types of temperature analysis techniques to determine streambed percolation rates. Temperature-based percolation rates for three ephemeral stream sites are compared with available surface water estimates of channel loss for these sites. These results are combined with published results to develop conclusions regarding the accuracy of using vertical temperature profiles in estimating channel losses. Comparisons of temperature-based streambed percolation rates with surface water-based channel losses indicate that percolation rates represented 30% to 50% of the total channel loss. The difference is reasonable since channel losses include both vertical and nonvertical component of channel loss as well as potential evapotranspiration losses. The most significant advantage of the use of sediment-temperature profiles is their robust and continuous nature, leading to a long-term record of the timing and duration of channel losses and continuous estimates of streambed percolation. The primary disadvantage is that temperature profiles represent the continuous percolation rate at a single point in an ephemeral channel rather than an average seepage loss from the entire channel.
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Rorabaugh  was the first to describe correlations between stream temperature and stream loss, subsequently proposing the use of temperature measurements as an indirect method for estimating stream losses beneath the Ohio River, OH. He indicated that a groundwater model capable of quantifying heat and water fluxes appeared to be a promising tool for using heat as a tracer of shallow groundwater flow beneath stream channels. Recently, heat as a tracer of stream/groundwater exchanges has been shown to be useful for quantifying exchanges in perennial streams in humid regions [e.g., Lapham, 1989; Silliman and Booth, 1993; Silliman et al., 1995]. Analysis of temperature profiles yielded point measurements of groundwater flow beneath perennial streams. In the present work, streambed temperature analysis is extended to ephemeral stream channels, where erratic streamflow patterns and a wide range of sediment moisture regimes create additional challenges in estimating shallow groundwater flow patterns. Currently continuous estimates of ephemeral streamflow are inhibited by the extremely transient nature of ephemeral flows, resulting in shifting channel geometry, and degradation of stream gage calibration (i.e., shifts in the stage-discharge rating curve). As a result, quantitative data concerning the nature of streamflow in ephemeral channels is sparse, with most data coming from intensive, short-duration field investigations. The extension of the use of heat as a shallow groundwater tracer to ephemeral channels has shown potential, due to the robust nature of temperature data acquisition.
Constantz et al.  demonstrated that large diurnal temperature variations within ephemeral channels created large diurnal variations in streambed infiltration rate, presumably due the temperature-dependence of the hydraulic conductivity in the shallow sediments. In response to short-duration streambed infiltration events, the timing and depth of streambed percolation was resolved using vertical temperature arrays installed beneath the ephemeral channels of Tijeras Arroyo, New Mexico [Constantz and Thomas, 1996]. A first attempt at temperature-based estimates of streambed percolation rates was performed beneath this channel, using a first approximation approach based on a heat-pulse/arrival time procedure [Constantz and Thomas, 1997]. These estimates of streambed percolation rates were encouraging, though not corroborated due to lack of available surface water monitoring techniques for flashy ephemeral channels. In the work of Ronan et al. , the multidimensional nature of shallow groundwater flow beneath ephemeral channels was examined using extensive cross-channel temperature profiles within Vicee Canyon, Nevada. This labor-intensive study relied heavily on elaborate instrumentation, with multiple streamflow monitoring locations, micrometeorological data to correct for evapotranspiration (ET), periodic stream reach cross-sectional surveys, and simulation model calibration to supplement temperature profile analysis. Our recent work has focused on the accuracy of the independent use of temperature profiles without reliance on surface water data and model calibration. Investigations at numerous sites in the arid Southwest utilized longitudinal surface-temperature arrays in ephemeral channels to successfully monitor the timing and duration of streamflows at discrete locations over extensive reaches of the ephemeral channels [Constantz et al., 2001].
 In the present work, temperature-based streambed percolation rates are compared with surface-based estimates of channel loss, where channel loss is defined as the streamflow loss in a given reach divided by channel area of the reach. A description of the analysis of temperature profiles for determination of streambed percolation is presented, followed by a description of the application of this analysis to several unpublished data sets from three ephemeral stream sites. Recently published temperature-based estimates of streambed percolation rates are combined with rates from these three sites, for comparison with concurrent surface water based estimates of channel loss for all sites.
2. Analysis of Temperature Profiles Beneath Streambeds
 The quantitative use of heat as a tracer of groundwater fluxes requires a heat and water transport model capable of estimating the simultaneous movement of heat and pore water. A physically based analysis of heat and water transport through porous materials was introduced by Philip and deVries . Their analysis resulted in a comprehensive mathematical description of the coupled process of liquid and vapor water transport simultaneous with the transfer of heat in the solid, liquid and vapor phases of unsaturated porous material. Application of their analysis has demonstrated that the transport of heat and water in the vapor phase is often significant in unsaturated soils, and generally dominates in dry environments [e.g., Scanlon and Milly, 1994]. As the degree of water saturation increases in sediments, heat transport in the vapor phase abruptly declines as the gas phase becomes discontinuous, and then vanishes as sediments approach saturation [e.g., Stonestrom and Rubin, 1989]. In general, streambed sediments beneath wetted channels are sufficiently saturated to ignore macroscopic vapor transport, such that the comprehensive approach developed by Philip and deVries is unnecessary for analysis of heat and water fluxes beneath stream channels during periods of streamflow. During these periods, a simple single-phase model can represent heat and water fluxes in the streambed.
Suzuki  and Stallman [1963, 1965] were the first to use this single-phase approach to predict water fluxes through saturated sediments, based on measured groundwater temperatures. Their work formed the basis for examination of flow in environments ranging from deep groundwater systems [Bredehoeft and Papadopulos, 1965] to humid hillslopes [Cartwright, 1974]. Though rarely used as a tool, these pioneers convincingly demonstrated that heat is a viable quantitative tracer of groundwater flow.
Stallman  presented a general equation describing the simultaneous flow of heat and fluid in the earth. He indicated that groundwater temperatures could be used to determine the direction and rate of water movement. He also indicated that temperatures in combination with hydraulic gradients could be used to estimate sediment hydraulic conductivity. Stallman's equation for the simultaneous transfer of heat and water through saturated sediments for the one-dimensional case of vertical flow (z direction) is as follows:
where Kt is the thermal conductivity of the bulk streambed sediments in W/m °C, T is temperature in °C, q is the liquid water flux through the sediments in m/s, Cw and Cs are the volumetric heat capacity of water and the bulk sediment in J/m3 °C, respectively, z is length in m, and t is time in s. The equation neglects dispersion, but has been successful in predicting numerous groundwater temperature data. The value of q is controlled by the Darcy's equation as the product of the hydraulic conductivity and the total head gradient. When q is zero the equation reduces to the Fourier equation for the transfer of heat by conduction, and when q is large, advection dominates the transfer of heat, as well as the change of temperature throughout the porous material. Thermal parameters can be estimated given some knowledge of streambed materials. The heat capacity of the sediments can be estimated by a summation of the product of the specific heat and density of each sediment constituent, weighted proportionally to the constituent's volumetric fraction in the sediment as the following:
where fs, fw, and fa are the volumetric fractions of the sediment, water, and air, respectively, cs, cw, and ca are specific heats in J/kg °C of the sediment, water, and air, respectively, and ρs, ρw, and ρa are the densities in kg/m3 of the sediment, water, and air, respectively. The product of the specific heat capacity and the density is the volumetric heat capacity, which is in the range of 0.8 × 106, 4.2 × 106, and .001 × 106 J/m3 °C for sediments, water and air, respectively [de Vries, 1963].
 A more general approach to describe simultaneous heat and water transport through sediments has been to utilize an energy transport approach via the convective-dispersion equation [Kipp, 1987]. These coupled heat and water flow equations are included here as equations (3), (4), and (5).
where θ is the volumetric fraction of the water content; ϕ is sediment porosity, dimensionless; Dh is the thermomechanical dispersion tensor, in m2/s; q is the water flux, in m/s, and Q is rate of fluid source, in s−1. The left side of the equation represents the change in energy stored in a volume over time. The first term on the right side describes the energy transport by heat conduction. The second term on the right side accounts for thermomechanical dispersion. The third term on the right side represents advective heat transport, and the final term on the right side represents heat sources and sinks to mass movement into or out of the volume. The themomechanical dispersion tensor is defined as [Healy, 1990]:
where αl , αt are longitudinal and transverse dispersivities, respectively, in m; δi,j is the Kronecker delta function; νi, νj are the ith and jth component of the velocity vector, respectively, in m/s.
 The familiar Buckingham–Richards flow equation is as follows:
where C(ψ, x) is specific moisture capacity, which is the slope of the water retention curve in m, ψ is the water pressure in m, h is the total head in m, x is length in m, t is time in s, and K is hydraulic conductivity in m/s [Buckingham, 1907; Richards, 1931].
 A critical difference between Kt and K is the level of uncertainty in assigning values for Kt versus K for the same material. Both parameters vary with texture and degree of saturation; however, for the typical case below a stream of saturated sediment of a given textural class, the uncertainty in Kt is relatively small compared with K. For example, the saturated Kt for a sand channel is likely to vary only between 1.0 and 2.0 W/m °C, so that the value of Kt can be estimated as 1.5 W/m °C ± 0.5 W/m °C [van Duin, 1963]. In contrast, the saturated hydraulic conductivity of sands may vary from 10−2 down to 10−6 m/s [Freeze and Cherry, 1979, p. 29], and as saturation decreases values of K have been measured to vary from 10−5 m/s down to 10−10 m/s [e.g., Constantz, 1982]. Often, Kt is assigned a value based on textural information, and identification of K becomes the primary focus.
 The two dimensional forms of equations (3), (4), and (5) are solved numerically in the computer simulation code, VS2DH [Healy and Ronan, 1996], specifically for use in stream environments. Currently, this code is one of several codes available for simulating simultaneous heat and groundwater transport. VS2DH is restricted to environments in which heat and water transport in the vapor phase are small relative to transport in the liquid phase. There are other heat and water transport simulation codes that are more suitable than VS2DH for specific environments. For example, SUTRA [Voss, 1984] is excellent in wet environments, while VS2DH generally handles the nonlinear behavior of parameters during infiltration better than SUTRA. TOUGH2 [Pruess et al., 1999] is excellent for environments where vapor transport of heat and water is significant. Using reasonable initial and boundary conditions, these simulation codes can be used to predict temperature patterns in stream sediments. Alternatively, an inverse approach is employed to determine K by matching simulated sediment temperatures to observed temperature data either by using a trial-and-error manual approach (TE), or by using a parameter estimation (PE) inverse-modeling approach. Stream channel parameters, such as stream stage and temperatures are monitored, as a basis for developing model initial and boundary conditions. Model simulation runs are performed to fit simulated sediment temperatures to observed (measured) temperatures. This fit is accomplished by using TE or PE through manual or automated adjustment of K for each consecutive simulations. The PE approach relies on a code designed to minimize the sum of the square deviations between model-predicted and observed temperatures PEST [Watermark Numerical Computing, 1998] and UCODE [Poeter and Hill, 1998] are two PE codes, that have proved successful in groundwater investigations. For 1-D problems, TE appears to be as efficient as PE; however, for multidimensional problems with significant uncertainty in several parameters, PE is superior.
 In the present work, TE was used at two sites, while PE was used at the third site by linking PEST to VS2DH in order to calibrate the model with respect to sediment temperatures. PEST uses the Gauss-Marquardt-Levenberg optimization algorithm to estimate model parameters. For the present study, the objective function may be written as the following:
where ϕ is the objective function value, i is measured sediment temperatures, and Yi is simulated sediment temperatures.
 This parameter estimation code matches the simulated temperature values to observed sediment-temperature values by optimizing hydraulic and thermal parameters to achieve the minimum differences between simulated and observed temperatures. As discussed above, the thermal conductivity possess significantly less variability compared with hydraulic conductivity for the same material. Therefore, thermal parameters can be specified within a narrow range, based on the literature for the particular textural class observed in the field, while the values of the hydraulic gradient and hydraulic conductivity are the primary variables available for optimizing the simulated sediment temperatures against the observed temperatures. Both hydraulic parameters are allowed to vary within physically reasonable ranges; the product of these parameters ( i.e., q) is determined during optimization of simulated and observed sediment temperatures. When a channel has saturated after an extended flow period, the hydraulic gradient can be determined using piezometers, such that simulation can be optimized varying only the value of K. Examples of both types of optimization are presented in discussions of field-site results.
3. Analysis of Temperature Profiles Beneath Three Ephemeral Channels
 In the present work, results from three field sites are reported to demonstrate the manner in which continuously acquired stream channel temperature profiles can be utilized to estimate the timing and magnitude of streambed percolation rates. A different approach of parameter measurement and/or data analysis is presented for each site, to provide a range of potential analysis options for ephemeral channels. Additionally, the complexity of analysis increases from the first case study through the third case study, such that additional options are added on earlier analysis options in a progressive manner. Finally for each site, temperature-based streambed percolation rates are compared with surface water-based estimates of streambed channel loss. Direct comparison of percolation rates with channel losses has limitations, especially for ephemeral channels. Percolation rate is defined as vertical downward transport of groundwater, while channel loss is defined as the streamflow loss over a given channel length divided by the channel area. As defined, channel losses are general significantly greater than percolation rates as a result of nonvertical transport of shallow groundwater during and immediately after streamflow events. Conversion of streamflow losses (m3/s) to channel loss (m/s) requires an estimate of the wetted stream channel area, which is typically difficult to obtain for ephemeral channels. Also, evapotranspiration (ET) losses are difficult to quantify for ephemeral channels, and thus ET losses not generally subtracted from streamflow loss when estimating channel loss. Nevertheless, a comparison of surface water based channel loss with temperature-based percolation rates provides measure of the timing, duration, and a general flux range that may be analyze to evaluate the utility of using vertical temperature profiles as a basis for estimating characteristics of stream losses from ephemeral channels. For convenience, channel loss and percolation rate are often expressed in units of both m/s and m/day.
3.1. Santa Clara River, California
 The Santa Clara River is located in southern California, flowing in a westerly direction from the San Gabriel Mountains for approximately 200 km to the Pacific Ocean. In the upper reaches, the gradient is steep, and the stream generally flows over shallow alluvium with a steady gain of groundwater. In the middle reaches, the streamflows in a wide sandy channel, resulting in large diurnal stream-temperature fluctuations, as well as significant potential for streamflow losses to the underlying sediments. A 17-km study section was defined in the middle reaches of the river, and a variety of hydrological parameters were monitored with a range of surface water and groundwater instrumentation. During October 1999, piezometers were installed by hand in the deepest sections of the stream channel at various locations along the channel, including the distal end of streamflows in the ephemeral reach (labeled as “SCR5” in this study). At SCR5, the streambed is composed of fine sands with negligible cobble, so the piezometer could be hand-driven. The piezometer was approximately 4 m in length with a 0.08 m internal diameter, and driven approximately 2.5 m into the streambed. Temperature was monitored between the streambed surface and the bottom of the piezometer by tethering four temperature microloggers inside the piezometer at 0.30, 0.61, 1.22, and 2.44 m below the streambed; another micrologger was tethered outside the piezometer to monitor surface water temperatures. Microloggers were programmed to sample at a 30-minute interval throughout the fall. In addition, a micrologger was inserted directly in the streambed sediment depth of 0.3 m using a steel insert tool. Data was compared with data from inside the piezometer at the same depth. After installation of all equipment, the hydraulic gradient at SCR5 was determined several times during the fall. Based on hydraulic gradients, SCR5 was a strongly losing reach during the fall of 1999. The observed gradient within the piezometer was generally observed to reside near 0.4 m/m during periodic manual measurements. The distal end of flows advanced upstream of SCR5 during one week of October, such that the piezometer was dry and nonfunctional during this week.
 VS2DH was used to simulate water and heat movement into the streambed by using a constant stage of 0.20 m, with observed (time-varying) surface water temperatures as the upper boundary conditions. Based on field inspection, the thermal parameters for a fine sand were assigned to VS2DH from the literature [de Vries, 1963; Van Duin, 1963]. The observed hydraulic gradient of 0.4 m/m was assigned to VS2DH. Simulated temperatures were fit using TE to observed sediment temperatures, by assigning a value of K to VS2DH and running a simulation, then assigning a new value of K and running a simulation until a minimal difference between simulated and observed sediment temperature was achieved for the period of record. The 1-D simulation grid spacing was 1 m wide by 10.5 m deep with no flow boundaries on the side. Grid size varied from 0.07 m at the top to 1.0 m at the bottom of the grid. The upper thermal boundary was the observed stream temperature and the lower thermal boundary condition was the constant regional groundwater temperature estimated to be 8.0°C.
Figure 1 compares observed and simulated sediment temperature at SCR5 during 13 days in November, 1999. Inspection of Figure 1 clearly shows an excellent fit of simulated to observed temperature data, with the capture of rapid changes in surface water temperature conditions through the period. The best fit TE match of simulated to observed sediment temperature resulted in a model-output-estimate for the percolation rate of 2.10 × 10−5 m/s (1.81 m/day). Using the observed gradient coupled with the temperature-derived percolation rate yielded a depth-averaged hydraulic conductivity of 5.25 × 10−5 m/s from Darcy's equation. This value of conductivity is in the range reported in the literature for fine sands [e.g., Freeze and Cherry, 1979], which as indicated above, was the field-estimated sediment texture at SCR5.
 Additional field testing of temperature equipment was performed, to determine how accurately temperatures observed within the piezometers represented temperatures in adjacent sediments under different conditions. Figure 2 shows results for SCR5 during October 1999 for observed compared with simulated sediment temperatures at three depths within the piezometer. Streamflow ceased at this location in the channel for a period in the middle of October, which is clearly seen in the observed data. VS2DH was used to simulate sediment temperature in an identical manner as for the November data set, i.e., same grid and boundary conditions, etc. Inspection of Figure 2 reveals that simulation results matched measured data very well during periods of ephemeral streamflow, but resulted in a poor match during the intermediate no-flow period. The poor match during the no-flow period is expected, since without streamflow water drained from inside the piezometer, suspending the micrologger in the air-filled interior of the piezometer. Consequently, the observed temperature failed to represent the adjacent sediment during the no-flow period. Once streamflow returned to this location in the stream, the microloggers were again submerged, and able to effectively represent sediment temperatures. For a concurrent test, Figure 3 gives results for a comparison of temperature logged in the streambed at 0.3 m versus temperature logged inside the piezometer at the same depth. The agreement between the temperatures is excellent when the streamflow is present, and as expected, the agreement is poor during the no-flow period as a result of draining of the piezometer, and thermal isolation of the logger inside the drained piezometer. In summary, during periods of streamflow, temperatures observed within piezometers provided an accurate estimate of temperatures in adjacent sediments for this site.
 An estimate of channel loss was determined from a series of streamflow measurements (also known as seepage runs) performed during the study period by USGS field office hydrological technicians. Also, measurements of stream channel wetted area in the distal reach were performed during the study period. Though significant daily streamflow variations were observed due to upstream ET and effluent discharge fluctuations, an average streamflow for much of the period was in the range of 0.11 m3/s. The wetted area between SCR5 and the distal end of any streamflows was estimated from field observations to be approximately 2750 m2. This estimate results in an average channel loss for the distal reach of streamflows to be 4.0 × 10−5 m/s (3.46 m/day). As reported above, the temperature-based percolation rate at this location during this period was estimated to be 2.10 × 10−5 m/s during this same period. The comparable magnitude of the values for percolation and channel loss is encouraging, since the value for channel loss represents both vertical and nonvertical seepage beneath the channel and was uncorrected for ET losses.
3.2. Santa Fe River, New Mexico
 A more extensive approach was attempted for estimating streamflow losses at the Santa Fe site; however, the challenges inherent in measuring streamflows in ephemeral channels limited the success of this approach. The Santa Fe River is a highly controlled stream which flows south from the Sangre de Cristo Range, then becomes ephemeral as it passes through Santa Fe, New Mexico. The study reach (also known as the St. Francis Reach) is characterized by a deepening sand-bottom channel and an increasing number of tributaries progressing down channel. These characteristics create technical problems in estimating streamflows and streamflow losses, because an unstable channel results in shifts in the streamflow rating curve, and tributary inflows result in uncertainties in loss estimates for the reach. In this study, multiple surface water and groundwater techniques for estimating channel loss and streambed percolation rates were compared during 1999.
 Vertical arrays of type-T thermocouples were installed between the streambed surface and about 3.0 m depth at sites along the study reach. Details of the installation are described by Thomas et al.  used in the La Bajada Reach of the Santa Fe River. Briefly, a truck-mounted drill rig was positioned in the channel, thermocouple wires were inserted down the length of a hollow-stem drill pipe, and the drill pipe was removed. For all sites on the Santa Fe River, the drill hole collapsed around the wires, eliminating the need for backfilling. The aerial (upper) lengths of the thermocouple wires were enclosed in plastic pipe and run horizontally out of the channel to an enclosure containing a data acquisition system. Streambed temperatures were monitored from March through June 1999. Figure 4 compares Santa Fe River streamflows from an upstream USGS stream gaging station and streambed sediment temperature responses at the temperature monitoring station. In Figure 4, there is a clear response in the shallower sediment temperature as a result of each streamflow event; but deeper sediment temperature appeared unaffected by these brief events. This suggests that only shallow percolation occurred during the periods of record.
 VS2DH was used to simulate water and heat movement into the streambed during periods of record containing streamflow by using a constant stage of 0.10 m and measured streambed surface temperatures as the upper boundary conditions. A 1-D simulation grid was constructed with 1 m width and a depth of 3.5 m. The grid size varied from 0.004 m at the top to 0.30 m at the bottom. The initial conditions were saturated material with 10°C at the bottom and the observed temperature at the top. Thermal parameters for a coarse sand were assigned to VS2DH from the literature [de Vries, 1963; Van Duin, 1963]. Simulated temperatures were fit using TE to observed sediment temperature by assigning a value of K for a simulation run, then assigning a new value for K until a minimum between simulated and observed sediment temperatures was obtained during the period of streamflow. Streamflow events were infrequent and brief during the monitoring period, with streambed temperatures responding abruptly and briefly to each streamflow event. As an example, the 0.67 m3/s streamflow event on 24 May 1999 was used to estimate streambed infiltration rates. Figure 5 depicts the best fit match of simulated temperature data using results from VS2DH compared with observed sediment temperatures for this streamflow event. Based on this match, model-output-estimate of the initial streambed percolation rate was 2.3 × 10−5 m/s (2.0 m/day), followed by a steady percolation rate of 1.2 × 10−6 m/s (0.1 m/day). As a second temperature procedure, the temperature arrival time method was used to estimate initial percolation rates, as described by Constantz and Thomas . Based on this simpler approach, the initial flux between the surface and a depth 0.03 m was calculated as 2.18 x 10−5 m/s, which compares well with the initial rate estimated from VS2DH best fit matches.
 A second USGS stream gaging station was installed at the lower end of a 4 km study reach, to compare surface water loss estimates of channel loss to temperature-based estimates of percolation rates. The second gage was operated from February to October of 1999, but problems inherent in ephemeral channels hampered accurate estimates of streamflow loss between the two USGS stream gages. Ungaged tributary inflows from a pair of tributaries were a source of additional water during high streamflows, and the shifting channel bed at each stream gage rapidly degraded the monthly stage-discharge rating curve for each site. For improved estimates of low-flow streamflow losses, parshall flumes were installed at the upper and lower ends of the study reach during the last period of the study by the Sangre de Cristo Water Agency. During this period, site inspection determined that tributaries were not contributing in streamflow generation to the study reach. During a subsequent streamflow in the study reach, the streamflow loss between the two flumes was 0.02 m3/s per km for the first day and stabilized at 0.008 m3/s per km after 3 days [Lewis, 2000]. (By comparison, the streamflow loss between the two gaging stations was determined to be 0.016 m3/s per km at the start of the flume test. During continued streamflow, the lower gage developed an erroneous record, possibly due to a shift in the channel morphology.) The wetted channel in the St. Francis Reach varied between 1 and 2 m in width during the study, with a mean width estimated as 1.7 m. Based on the flume streamflow losses and a wetted area of 1300 m2, initial streambed channel loss was 1.2 × 10−5 m/s (1.0 m/day), decreasing to a steady channel loss of 5.0 × 10−6 m/s (0.4 m/day). Thus, the surface water based steady channel loss is significantly higher than the final temperature-based percolation, i.e., 5.0 × 10−6 m/s compared with 1.2 × 10−6 m/s, respectively. As was the case for the Santa Clara study, this relative magnitude is reasonable considering that the value for channel loss represents both vertical and nonvertical seepage and was uncorrected for ET losses. Additional discussion is included in the summation section.
3.3. Bear Canyon, New Mexico
 The ungaged ephemeral stream in Bear Canyon is located in a roadless, mountain front watershed with virtually no data concerning streamflow characteristics beyond corollary information (such as the spatial pattern of specific types of vegetation). Consequently, this site was chosen for deployment of multiple temperature techniques to aid in tracking undocumented streamflow losses. The temperature profiling technique described in this paper was deployed to estimate channel loss, and a series of microloggers was deployed longitudinally down the channel to estimated the spatial and temporal pattern of streamflows (see Constantz et al.  for details of the technique). Additionally, temperature data from this site were chosen as a demonstration data set to examine the influence of the uncertainty in thermal parameters on simulated fluxes using Monte Carlo analysis. Also, the data set was selected for demonstrating the use of a PE approach to optimize simulations to observed sediment temperature based on equation (6).
 Bear Canyon is located on the eastern edge of Albuquerque, New Mexico. The small ephemeral stream within the canyon is a representative example of more than 100 small ephemeral streams draining from the western flanks of the Sandia and Manzano Mountains into the Middle Rio Grande Basin. These ephemeral streams possess the common characteristics of being bedrock-controlled in their upper gaining reaches, and alluvium-controlled in their lower losing reaches. The streamflow and stream loss patterns of these stream channels are poorly documented, but their cumulative streambed infiltration might contribute significantly to potential recharge to the basin. The study reach extends from the exposed bedrock at the mountain front down slope in a westward direction for about 3 km, at which point the stream channel is modified as a reach of surrounding urbanization. The stream is perennial east of the bedrock exposure at the mountain front, and streamflows rarely extend more than 1 km beyond the mountain front, though summer monsoons occasionally induce streamflow to the confluence with the Rio Grande, approximately 20 km to the west of the mountain front.
 Two temperature monitoring schemes were deployed within the stream channel from 1996 through 1999. Streambed surface temperatures were monitored at 8 locations between the mountain front and the location of urbanization of the channel, 3 km to the west of the mountain front, as described by Constantz et al. . Vertical temperature patterns were monitored at two locations within the 3 km study reach by using a series of thermocouple wires installed at depths between the streambed surface and about 3 m below the channel. After backfilling the installation holes, completed temperature nests monitored temperature at 0.40, 0.60, 1.10, 2.10, and 3.10 m below the streambed surface (see Thomas et al.  for installation description for a similar site). Temperatures were monitored at 15 minute intervals via a data logger in an enclosure near the stream channel from the summer of 1996 until the fall of 1999.
 Analysis of surface temperature patterns from the 8 locations resulted in characterization of the spatial and temporal pattern of streamflow in Bear Canyon. Analysis of surface-temperature patterns suggested that seasonal snowmelt resulted in a gradual progression of the distal end of streamflow down channel over several months in the spring, followed by streamflow recession up channel in early summer. These conclusions were substantiated by monthly site visits by USGS field staff. In addition, field staff determined that steady base flows generated a continuous discharge in the range of 0.25 cfs at the upstream location of the study reach (Fredrick Gebhardt, oral comm., U.S. Geological Survey, Albuquerque, New Mexico, 2000). This translates to approximately 0.01 m3/s, though there was observed geomorphic evidence of brief periods of streamflow in the range of an order of magnitude higher than this value during periods of rapid snowmelt at greater altitudes.
Figure 6 shows details of late stages spring streamflow at one vertical temperature nest in the channel. As expected, the greatest diurnal temperature variations occurred at the shallowest depths monitored (0.40 m), and the smallest diurnal temperature variations occurred a the greatest depth monitored (3.10 m). The abrupt retreat of streamflow up Bear Canyon is clearly detectable in the thermographic record. As streamflow retreated up channel of this site on 5 June 1999, the abrupt transition from advection-dominated heat transport to conduction-dominated heat transport is quite distinct. Reduced magnitudes in diurnal variations in streambed temperature result from the loss of advective heat transport with the cessation of streamflow into the streambed.
 During spring streamflow, the sediment temperature from vertical temperature arrays were used for inverse modeling with VS2DH. A 2-D simulation grid was constructed as shown in Figure 7. Grid size varied from 0.06 to 0.20 m in the vertical direction, and 0.15 to 0.60 m in the horizontal direction. PEST was used to determine the streambed sediment hydraulic conductivity from the best fit between the simulated streambed temperatures using VS2DH and measured sediment temperatures. Figure 8 shows sediment temperatures at four depths below the streambed at a vertical temperature site approximately 275 m west of the mountain front during June 1997. Figure 8 also shows the simulated best fit at 0.60 m using PEST. The measured streambed temperatures were applied as the upper thermal boundary condition, and measured stream stage was used for the upper hydraulic boundary condition. Saturated conditions existed below the stream channel bottom. This was determined by measuring water levels with minipiezometers set in the streambed. PEST optimized the simulated temperatures to the measured temperatures at depths of 0.4, 0.6, and 1.1 m, by adjusting the hydraulic conductivity with the thermal and hydraulic boundary conditions shown in Figure 7. For the period shown in Figure 8, an optimized percolation rate of 8.70 × 10−6 m/s (0.75 m/day) was generated in output from simulation runs, based on a PE fit to observed sediment temperatures. In the figure, the fit for a depth of 0.6 m is shown. (Fits for 0.4 and 1.1 m depths were comparable, but not shown in the figure for clarity of individual thermographs). For the entire seasonal streamflow, a PE fit to observed temperature resulted in an average vertical streambed percolation rate of 8.91 × 10−6 m/s (0.77 m/day).
 This extended record of observed data afforded the opportunity to evaluate some issues regarding uncertainty. In a similar fashion to Niswonger and Rupp , Monte Carlo analysis was used to evaluate the effects of uncertainty in measured sediment temperature, estimated heat capacity, and estimated thermal conductivity on predictions of streambed-percolation rates. A series of 400 inverse simulations were run (each with a new value of K), to examine the impact of temperature measurement errors. Each run was set to include an equally probable realization in error, using a normally distributed, random equipment error added to the temperature. (The total range in the error in temperature was based on equipment tested in the field.) Figure 9a presents a histogram of the simulated channel loss resulting from the error in temperature. The normally distributed channel loss shown in Figure 9a demonstrates the uniqueness of the solution for this parameter estimation problem. Figure 9b shows the running variance as a function of Monte Carlo realizations, resulting from the analysis of temperature measurement error. Each of the curves in Figure 9b represents a different Monte Carlo analysis, with a different standard deviation in the error distribution shown on the right of each curve. The trend in the curves in 9b is toward a horizontal line, which indicates convergence. These results are subject to the assumption that the error in temperature measurements follows a normal distribution, and all errors in the model simulation of the actual stream seepage process are accounted for in the Monte Carlo analysis. Figure 9c shows expected errors in flux predictions as a function of measured temperature standard deviation. The largest standard deviation considered was 0.2°C, resulting in a flux standard deviation of 0.029 m/day, as shown in Figure 9c. Thus, even for the largest expected error in temperature measurements, there is a 95% confidence (i.e., two standard deviations) that the percolation rate is 8.91 × 10−6 m/s, ±6.71 × 10−7 m/s (0.77 m/day, ±0.058 m/day).
 A similar analysis was performed to evaluate the effects of errors in the heat capacity and the thermal conductivity. Again, 400 inverse simulations were run, each with a new value of K and with an equally probable realization in possible error for the heat capacity and thermal conductivity. The parameter distributions were developed using a normal distribution, based on heat capacity and thermal conductivity ranges provide by Lapham . Generally, some sediment textural data is available, so that the ranges use in this analysis represent an extreme estimate of uncertainty. The optimal channel loss resulting from each of these inverse simulations were saved, and a running mean and variance in the channel loss were calculated. In this case, the Monte Carlo analysis resulted in a estimated percolation rate of 8.91 × 10−6 m/s, ±4.40 × 10−6 m/s at the 95% confidence interval. This indicates that poor estimates of heat capacity and thermal conductivity may result in as great as a 50% uncertainty in channel loss for the Bear Canyon experiments. Note that the temperature fits for all thermal parameter realizations were essentially equivalent. Thus, if a given realization of thermal parameters results in less heat conduction, the optimal hydraulic conductivity will be higher, which indicates that hydraulic and thermal parameters cannot be uniquely resolved based on measured sediment temperatures alone. Finally, when there is a lack of information concerning sediment texture, temperature measurement error contributed only a small uncertainty in the simulated percolation rates compared with the other two thermal parameters. Information concerning texture, leading to improved estimates of sediment heat capacity and thermal conductivity can significantly reduce the uncertainty in estimates of percolation.
 For Bear Canyon, a general comparison between temperature-derived estimates of percolation rates with streambed seepage loses was inhibited by limited surface water data from this site. Nevertheless, the surface-temperature analysis and the monthly field observation of flows permitted sufficient information to estimate a streamflow loss magnitude. As discussed above, the steady stream discharge flowing off the bedrock over the shallow alluvium of Bear Canyon was estimated as about 0.01 m3/s for the period depicted in Figure 8. Based on continuous surface-temperature monitoring of the spatial and temporal pattern of streamflow for Bear Canyon discussed by Constantz et al. , streamflows extended down-canyon from the bedrock contact for approximately 400 m during this period, resulting in an average wetted area of the channel of 600 m2. This would result in an average streambed channel loss of 1.67 × 10−5 m/s (1.44 m/day) compared with the temperature-based percolation rate of 8.91 × 10−6 m/s. As was the case for the two other studies, this trend is reasonable considering that the value for channel loss represents both vertical and nonvertical seepage and was uncorrected for ET losses.
Figure 10 displays a 1:1 plot for comparison of temperature-based estimates of streambed percolation rates versus surface water-based estimates of streambed channel loss, for the three sites reported on in this study. In addition, Figure 10 includes a comparison of percolation and channel loss for earlier work of a similar nature performed in Tijeras Arroyo, New Mexico [Constantz and Thomas, 1997], Vicee Canyon, Nevada [Ronan et al., 1998], and La Bajada, New Mexico [Thomas et al., 2000]. Examination of Figure 10 demonstrates that comparative results reside in three distinct regions on the 1:1 plot. The initial results for the Santa Fe site and the results for the Tijeras site reside in a region above the 1:1 line, indicating that estimates of percolation rates were higher than channel loss for these two comparisons. This was not expected, though in hindsight might be explained by differential time-duration of measurements for percolation compared with seepage estimates. The initially rapid percolation rates were determined by analysis of the abrupt temperature response during the inception of streambed infiltration at each site. Alternatively, channel loss were determined over a longer time period, as streamflow traveled to the lower streamflow gaging locations. This resulted in a channel loss that was composed of both initially high streambed infiltration rates at the distal end of flow and exponentially lower rates in the upper end of the reach.
 In Figure 10, the comparison for Vicee Canyon lies close to the 1:1 line. This is reasonable since the Vicee Canyon study was an intensive study, based on 2-D vertical temperature profiles across the stream channel, and calibrated to earlier flow events. Additionally, channel loss included compensation for field-determined ET losses, so that the net seepage was reduced below the total streamflow loss per unit area. Comparisons for the La Bajada site, Bear Canyon, the Santa Clara site, and the final rates at the Santa Fe site all reside in the lower region of the plot. For these four comparisons, the surface water based estimate for channel loss were consistently 2 to 3 times greater than temperature-based estimates of percolation rates. This difference is reasonable due to the multidimensional nature of streamflow channel loss, as well as the lack of compensation for ET losses. Comparing all these sites, the results suggest that percolation rates are initially rapid, and decrease to a value that approaches 30% to 50% of channel loss for these streams. Regarding the relative contribution of ET to the gross channel loss, field results from the both Vicee Canyon study and La Bajada study included estimates of ET during summer months. These studies concluded that ET represented approximately 10% of the total channel loss at these sites during this period. This would suggest that the primary difference between the magnitude in percolation rate versus channel loss is due to the nonvertical component of shallow groundwater flow in these ephemeral channels.
 In conclusion, streambed temperature profiles form the basis of a tool capable of: (1) estimating the timing and duration of streamflow channel losses, (2) determining the percolation rate, and (3) estimating the magnitude of channel loss at a point (or points) in the channel. Temperature profiles provide a robust and continuous record, creating a long-term record of streamflow patterns within an ephemeral channel. The primary disadvantage is the inability of a single or limited number of temperature profiles to reflect the multidimensional, spatially variable nature of streamflow losses beneath ephemeral channels. As a future application, temperature-based estimates of streamflow duration and percolation rate may lead to estimates of stream loss for an entire reach by assuming a value for channel loss of 2 or 3 times the point measurement of percolation rate, and multiplying this rate by an estimate of the channel area of the reach. Cumulative stream loss (m3) might then be estimated from the product of the stream loss and temperature-based estimate of flow duration for the reach. Though this approach possesses clear uncertainty due to spatial variability, challenges inherent in acquiring long-term streamflow patterns using current surface water-based methods suggest that this uncertainty may be acceptable for estimating stream losses beneath ephemeral channels.