Published in Vadose Zone Journal 3:936-946 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
A Statistical Technique for Interpreting Streamflow Timing Using Streambed Sediment Thermographs
Kyle W. Blascha,b,*,
Ty P. A. Ferréb and
John P. Hoffmanna
a USGS, Tucson, AZ 85719
b Department of Hydrology and Water Resources, University of Arizona, Tucson, AZ 85721
* Corresponding author (kblasch{at}usgs.gov)
Received 16 October 2003.
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ABSTRACT
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A moving standard deviation (MSD) technique is developed to infer the onset and cessation of ephemeral streamflow using temperature data from the upper 2.25 m of streambed sediments. During periods of streamflow, shifting of the predominant thermal-transport mechanism within the sediments from conduction to advection produced changes in the amplitude of the vertically propagating diurnal temperature waves. Analytical expressions describing propagation of conductive and advective diurnal temperature waves through streambed sediments are presented for identifying depths with the largest changes in the diurnal temperature wave amplitude between periods of flow and no flow. The MSD statistical technique was developed to identify the thermal amplitude changes from bed sediment thermographs and to infer streamflow timing. The accuracy of the MSD technique is quantified using direct streamflow and streambed water content measurements. Accuracy of the technique was most sensitive to the MSD window length and the threshold parameter separating periods of conductive and advective heat transport. An alternative calibration procedure was developed using temperature measurements alone. The average error for streamflow timing was approximately 400 min for each event. The results show that temperature sensors may be deployed at a range of sediment depths depending on streamflow stage and soil thermal and hydraulic properties, and that the MSD procedure can provide an objective and repeatable means to quantify streamflow timing.
Abbreviations: MSD, moving standard deviation • TDR, time domain reflectometry
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INTRODUCTION
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RECORDING THE CONTINUAL PRESENCE and absence of streamflow within ephemeral channels in large semiarid and arid basins for use in water balance and hydrologic models requires an inexpensive and reliable method for broad distribution. Difficulties arise when monitoring streamflow timing and extent within ephemeral channels using traditional streamflow timing techniques because of the flashy nature of streamflow events and shifting elevations of the channel surface (Constantz and Thomas, 1997; Blasch et al., 2002). Consequently, measurement of streamflow presence using subsurface methods has been proposed (Constantz and Thomas, 1997). Because streambed temperature measurements are inexpensive and easy to obtain relative to other subsurface measurements, streambed thermographs have been introduced as a means to infer streamflow timing and extent in ephemeral streams (Constantz et al., 2001).
Constantz et al. (2001) placed thermal sensors at shallow depths (15 cm) to identify reductions in the amplitude of the diurnal temperature wave in the presence of streamflow. They visually identified changes in streambed temperature wave amplitudes to determine the start and end of streamflow events. Although this visual inspection technique has been successfully demonstrated in ephemeral streamflow settings for multiple day events, the technique is both subjective and time-consuming when analyzing large data sets. The technique can also be ineffective for identifying stream flow events that are <24 h in duration, which is typical of ephemeral streams in southern Arizona, for example. Finally, this method of analysis requires relatively shallow measurement depths to take advantage of reductions in the diurnal temperature wave amplitude that occur as heat is lost to the standing water above the sediments. Such shallow installed instruments are especially susceptible to removal by scour of the channel sediments.
The first objective of this investigation was to present a description of heat transport that considers the effects of both conductive and advective temperature transport. This description will expand the capability of this method to a greater number of ephemeral streams by allowing for monitoring of temperature at deeper depths where shallow temperature measurements are impractical, such as reaches that experience significant scour (Constantz et al., 2003). The second objective was to introduce an automated technique for identifying the onset and cessation of ephemeral streamflow based on a statistical analysis of streambed thermographs.
The following section presents two analytical expressions describing conductive and advective heat transport as well as the hydrological conditions necessary to use bed sediment thermographs for streamflow timing. We continue by introducing the moving standard deviation technique for analyzing bed sediment thermographs. Finally, we evaluate the moving standard deviation technique by using bed sediment thermographs collected from an array of temperature sensors within an ephemeral stream.
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OPTIMAL DEPTH SELECTION FOR PLACEMENT OF TEMPERATURE SENSORS
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When streamflow is absent from a channel, radiant heating of the surface sediments is transported into the sediment profile primarily through conduction (Fig. 1)
. The diurnal temperature, often represented as a sinusoidal function, propagates through the sediments with a decreasing amplitude and increasing time lag with depth (Van Wijk and De Vries, 1963). The magnitude of the one-dimensional diurnal temperature wave as a function of depth for a homogenous single layer is described by (Van Wijk and De Vries, 1963)
 | [1] |
where Tcond is the amplitude of the sediment temperature variation (°C) at depth z (m) below the land surface, To is the amplitude of the diurnal temperature wave (°C) at the sediment surface, and D is the damping depth (m) of the diurnal temperature waves, represented as
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where 
s is the thermal diffusivity (m2 s–1) and P is the period (s) of temperature oscillation at the sediment surface. At a depth D the relative temperature amplitude of the wave declines to e–1 (0.37), and at a depth D
the wave is 180° out of phase with the wave at the surface. Typical damping depth values for sandy soils are about 0.15 m and for clay soils are about 0.12 m (Van Wijk and De Vries, 1963).
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Fig. 1. Thermal responses from within a streambed divided into the case when streamflow is absent (primarily conductive heat transport through the sediments) and the case when streamflow is present (primarily advective heat transport through the sediments). Note the reduction in the thermal amplitude at the streambed surface caused by the transference of heat to the overlying water column and corresponding increase in thermal amplitude at depth caused by percolating water.
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When streamflow is present in the channel the amplitudes of the diurnal temperature fluctuations at the sediment surface are reduced (Fig. 1). During the daytime, a portion of the thermal energy from incoming solar radiation is used to raise the temperature of the water column and increase evaporation. The result is a lower maximum temperature at the sediment surface. Similarly, during the evening, the overlying water column insulates the sediment surface from the cooler air temperature, resulting in higher minimum temperatures. In addition, the presence of streamflow within a channel can produce infiltration and percolation of water through streambed sediments, increasing the amount of heat transported to deeper depths through a combination of advection and conduction. As a result, thermographs at deeper depths will show an increase in the diurnal temperature amplitude (Fig. 1). Stallman (1963)(1965) developed an analytical solution to the one-dimensional coupled heat and water transport equations assuming one-dimensional steady vertical flow in a single saturated layer and a sinusoidal temperature variation at the surface. The diurnal temperature wave amplitude for the combined advection and conduction equation using the Stallman solution is
 | [3] |
where Tadv is the amplitude of the sediment temperature variation (°C) at depth z (m) below the land surface, Ts is the amplitude of the temperature wave (°C) at the sediment surface, and a (m) is a coefficient based on the fluid flow and thermal properties of the sediment
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with
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and
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The variable c is the specific heat of the fluid and sediment in combination (J g–1 °C–1), 
is the density of the fluid and sediment in combination (g m–3), 
is the thermal conductivity of the fluid and sediment in combination (W m–1 °C–1), v is the fluid flux (m s–1), co is the specific heat of the fluid (J g–1 °C–1), and o is the density of the fluid (g m–3). If v is zero, then Eq. [3] reduces to the conduction equation and K = D–2, but as v increases, the transport of heat due to advection increases. The predominant heat transport mechanism within the sediments will depend on the infiltration rate, thermal parameters, and of course the depth of interest. For simplicity we will refer to the transport of heat during the presence of streamflow as advection even though conduction also contributes to the transport of heat.
Constantz et al. (2001) monitored temperature at the near surface (15 cm) to successfully infer the presence of streamflow. The method was appropriate for the streams the authors studied because two critical conditions for the success of this method were achieved. First, the diurnal temperature amplitude at the surface was sufficiently large so that minor temperature fluctuations, such as those caused by clouds, did not obscure the diurnal signal. Second, during the presence of streamflow the overlying water column produced a measurable reduction of the amplitude of the diurnal temperature wave at the streambed surface compared with the magnitude of the diurnal temperature wave at the streambed surface during the absence of streamflow (Fig. 1). If, however, either of these conditions had not been satisfied, then changes in the near surface diurnal temperature wave amplitude during streamflow would have been insufficient to infer the presence of streamflow.
Other circumstances noted by Constantz et al. (2001) that can preclude the use of streambed temperature for monitoring streamflow include precipitation, sudden changes in air temperature, and scour. Precipitation and sudden changes in air temperature associated with weather fronts can cause fluctuations in the sediment temperatures similar to those caused by the presence of streamflow. In general, precipitation-induced fluid fluxes and abrupt air temperature changes do not penetrate as deeply below the sediment surface as streamflow events and will have a greater influence on sensors near the surface. For these circumstances thermographs at deeper depths can be more suitable.
As an example of conductive (no flow) and advective (flow) heat transport, hydraulic and thermal parameters for coarse-grained sediments typically found in ephemeral streambeds were incorporated into Eq. [1] through [6]. The thermal and hydraulic parameters were used for both heat transport expressions and were assumed constant with depth (Table 1).
The conductive and advective diurnal temperature waves were plotted as a function of depth for three different streamflow cases (Fig. 2)
. The no dampening case assumes there is no reduction in the amplitude of diurnal temperature, and the 50% reduction case assumes the amplitude of diurnal temperature is reduced by 50% in the presence of streamflow. The fluid flux was reduced for the third case to demonstrate the dependence of the advective thermal amplitude on this variable. Comparison of the conductive (no flow) and advective (flow present) diurnal temperature waves as a function of depth shows that near the surface the conductive thermal amplitude is larger than the advective thermal amplitude (Fig. 2). It is also apparent that percolating water transports more heat to deeper depths than conduction alone during the absence of streamflow. Consequently, an important prerequisite for using thermographs at deeper depths to infer streamflow is that the hydrologic flux is sufficiently large to cause measurable changes in the amplitude of the diurnal temperature wave during the presence of streamflow compared with the absence of streamflow.
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Fig. 2. (A) Advective and conductive diurnal temperature wave amplitudes as a function of depth propagating through coarse-grained stream channel sediments. The radiant thermal amplitude at the surface is 1°C. The solid black line describes conductive transport during no-flow conditions. The no dampening case considers zero reduction of the diurnal thermal amplitude during the presence of streamflow. The 50% dampening case considers a 50% reduction during the presence of streamflow. The 50% dampening/50% fluid flux reduction considers a 50% diurnal thermal amplitude reduction and a reduction in the fluid flux from the previous two cases by 50%. (B) Difference between the advective and conductive diurnal temperature wave amplitudes as a function of depth. Intersections between the solid diurnal temperature wave amplitude segments and the dashed transition line represent depths corresponding to equivalent advective and diurnal temperature wave amplitudes during flow and no-flow conditions respectively.
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The conductive temperature wave amplitude (Tcond) at each depth was subtracted from the advective temperature wave amplitude (Tadv) to quantify the depth-dependent changes in amplitude of the temperature wave resulting from infiltration (Fig. 2). If there was no reduction in the amplitude of the temperature wave attributed to the presence of the overlying water column, then the difference in conductive and advective temperature wave amplitudes at the surface (0.0 m) was zero. At depth however, there was a change between the conductive and advective temperature wave amplitudes. If the presence of an overlying water column reduced the amplitude of the temperature wave at the sediment surface then there was a change in the amplitude of the temperature wave both at the surface and at depth. Thus, temperature monitoring at depth may be more dependable and informative than at the near surface because changes in the temperature wave amplitude are less sensitive to the overlying water column. This is especially true for ephemeral streams experiencing a range of stream stages.
Optimal measurement depths for inferring streamflow can be defined as depths where streamflow-induced infiltration causes the greatest measurable changes in temperature wave amplitudes compared with no-flow conditions. For this case the optimal depth is about 0.45 m, as shown in Fig. 2. Equations [1] through [6] can be used with knowledge of the sediment profile and streamflow stage to estimate the optimal depths. The least optimal depths are those where the thermal amplitude does not change due to the presence of streamflow (i.e., Tcond = Tadv). One depth that is not suitable is called the transition depth, zt, where the conduction and advection temperature wave amplitudes are equivalent during the presence and absence of streamflow. The transition depth also represents the point where the magnitude of the temperature wave amplitude during the presence of streamflow becomes larger than the temperature wave during the absence of streamflow. The transition depth is represented in Fig. 2 for the three different streamflow cases. The transition depth for each case is located where the lines representing the difference in the conductive and advective temperature amplitude intersect the vertical dashed line. For the no dampening case the combination of advection and conduction will transport more heat throughout the entire profile, so the transition depth does not occur. As dampening increases, the transition depth increases. For the 50% dampening case the transition depth occurs at approximately 0.11 m, and for the 50% dampening and fluid flux reduction case the transition depth is approximately 0.17 m. Additionally, depths closer to the surface are less suitable if they are not below the zone of scour or are influenced by temperature fluctuations induced by precipitation and weather fronts, as discussed above.
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STANDARD DEVIATION TECHNIQUE FOR DETECTING PERIODS OF STREAMFLOW
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Analysis of a streambed thermograph to infer streamflow timing is based on identification of the aforementioned temporal changes in the streambed thermograph (Fig. 1). As an example, two thermographs measured in a coarse-grained alluvial stream at depths above and below the transition depth are presented in Fig. 3
. When streamflow is present (indicated by the gray areas), as recorded by a stream gage, the diurnal temperature wave amplitudes in the thermograph above the transition depth decline, whereas the opposite is true of the thermograph below the transition depth. Streamflow timing can be inferred from the thermographs by identifying these changes in thermal amplitude either above or below the transition depth. This analysis can be conducted visually (Constantz et al., 2001) or using statistical tools (Stewart and Constantz, 1999; Stewart, 2003). Moving window averaging is a standard data smoothing technique. We apply a similar approach wherein the standard deviation of temperatures is determined within a defined time window (Fig. 3). For each window of time, 
, the number of measurements, n, is determined by the temperature sampling interval, i,
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Fig. 3. Thermograph for (A) a depth above the transition depth and (B) a depth below the transition depth. The gray areas denote observed periods of streamflow.
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The moving standard deviation, s, can be expressed as
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where
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is the moving average, and T is the total number of measurements. The analysis window is advanced in time, yielding a moving window of standard deviation for the entire thermograph (Fig. 4)
. Identification of streamflow is then based on measurable differences in the moving standard deviation during flow and no-flow periods. The advantage of using the moving standard deviation is that variations in the thermal record caused by streamflow infiltration are magnified in the standard deviation plots, increasing the ability to discern event timing. Additionally, the moving standard deviation removes longer time-scale fluctuations in the thermograph, which can obscure short-term variations.
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Fig. 4. Six-hour moving standard deviation window for temperature data measured at (A) a depth above the transition depth and (B) a depth below the transition depth. The gray areas denote observed periods of streamflow.
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Five parameters are required to design the moving standard deviation filter: the standard deviation window length, the reference time within the window, the flow–no-flow threshold, and two flow duration parameters. The window length, , is the time interval for which the standard deviation is calculated (Fig. 3). The reference time can be centered on the current time step, t, so that the interval spans either from (t – 0.5) to (t + 0.5), set to the beginning of the window so that the window spans from (t) to (t + ), or set to the end of the window so that the window spans from (t – ) to (t). The threshold parameter sm (°C) is interpreted as the magnitude of the standard deviation (i.e., temperature variability in the thermograph) separating advection-dominated from conduction-dominated thermal conditions. Periods of streamflow will appear either above or below the threshold standard deviation depending on the location of the monitoring depth relative to the transition depth (Fig. 4).
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For a controlled environment, the threshold standard deviation for each depth can be calculated using Eq. [1] through [6] and estimated surface temperatures. However, in practice the air and water temperatures during the presence of streamflow were not periodic and varied between events. Consequently, a threshold multiplier times the mean moving standard deviation for the period of record is used to calculate the threshold standard deviation value. The threshold multiplier is obtained during calibration. The duration parameters, measured in minutes, are used as filters to remove false interruptions in the prevailing flow conditions (Fig. 4). A minimum flow duration parameter, tmin, is defined as the shortest duration of a streamflow event likely to occur at a given location. This filter removes false positive streamflow identifications due to rapid air temperature changes that are associated with fluctuating weather conditions. These atmospheric events mimic the onset of streamflow, but are typically of shorter duration than streamflow events. Second, a minimum inter-event duration parameter, tint, is defined as the shortest interval likely to separate two consecutive streamflow events. This parameter is used to eliminate false negative flow identifications during streamflow. For example, during streamflow events surface heating may be reduced due to the presence of clouds, resulting in smaller diurnal temperature wave amplitudes at the streambed surface. For thermal monitoring below the transition depth this drop in diurnal heating can cause the standard deviation to drop below the threshold for short periods of time even though water is still percolating through the sediments.
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MATERIALS AND METHODS
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A field experiment was designed to determine the suitability of Eq. [1] through [6] for selection of optimal measurement depths and to determine the accuracy of the moving standard deviation technique. The study site (within Rillito Creek in Southern Arizona) is located 40 m upstream of the USGS Streamflow-Gaging Station 09485700 and experiences approximately 15 ephemeral streamflow events each year (Fig. 5)
. The channel reach is approximately 60 m wide, and the elevation of the stream channel over the cross section varies by <0.8 m. The study site is underlain by recent stream-channel deposits, which are underlain by basin-fill deposits. The recent deposits, consisting of more than 90% fine- to coarse-grained alluvium, are about 10 m thick (Hoffmann et al., 2002). The underlying basin-fill deposits generally are finer grained and extend to depths of several hundreds of meters. Depth from the channel surface to the water table is about 40 m.
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Fig. 5. Location of Rillito Creek study area and view of Rillito Creek within the Tucson Basin, Tucson, AZ.
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Rillito Creek is an ephemeral stream with three active seasons, summer North American monsoon (July–September), fall (October–November), and winter (December–March). Characteristic summer flows result from localized, short-duration convective storms, whereas longer-duration frontal storms and snowmelt produce winter flows. Flow events during all three seasons generally range in duration from several hours to <15 d at the study site.
A vertical array of thermocouple sensors was buried beneath the lowest part of the channel cross section. A 2.5-m-deep profile was excavated, and seven thermocouple temperature sensors were inserted into the side of the profile at depths of approximately 0.50, 0.75, 1.0, 1.25, 1.50, 2.0, and 2.5 m below the stream channel surface. A thermistor was placed at a depth of 0.15 m below the surface. The sediments were then returned to the profile. A second thermistor was placed on the south bank to record air temperature. All installed temperature sensors were programmed to measure temperature every 5 s and to record a time-averaged temperature at 5-min intervals with a precision of 0.1°C.
After each streamflow event the channel cross section was surveyed to record the level of deposition and scour. Immediately after installation of the sensors, a series of flow events scoured approximately 0.25 m of sediment from the site. The 0.15-m temperature sensor was installed after these scour evens. During the succeeding 24 events, sediment scour and deposition varied by <0.10 m and on average did not change. These 24 events were selected for analysis.
Time domain reflectometry (TDR) probes were installed adjacent to each thermocouple to provide independent measurements of the onset and cessation of streamflow. The TDR probes measured and recorded volumetric water content at 2-min intervals with a precision of approximately 0.03 cm3 cm–3. A Campbell Scientific (Logan, UT) TDR100 time domain reflectometer was used in conjunction with a Campbell Scientific CR10X datalogger to transmit, receive, and convert waveforms into water content. TDR probes were comprised of two stainless steel prongs 0.20 m in length, spaced 0.03 m apart. Probes were constructed and calibrated in-house.
The onset and duration of stream flow was identified by elevated volumetric water contents above 0.3 m3 m–3. At the onset of streamflow this TDR-based measurement will lag behind the onset of flow at the surface because of the time required for water to infiltrate to the instrument depth; however, it represents the closest measurement of onset and cessation times because the TDR and temperature probes are co-located. Furthermore, saturation of the sediment profile, measured using TDR, was achieved in <10 min at the onset of the streamflow events as determined by the stream gage. Similarly the cessation of streamflow inferred by the water content sensors lags behind the cessation at the surface.
Measurements were collected for a 365-d record (16 Sept. 2000–15 Sept. 2001), which included 24 streamflow events. Thermographs for the entire measurement period are shown in Fig. 6
, with streamflow events denoted by the shaded areas. The ranges of diurnal temperature amplitudes measured at a depth of 0.25 m were, on average, 14.4°C in the winter and 16.2°C in the summer. The variability in streamflow duration ranged from a few hours to several days, which was considered ideal for evaluation of the moving standard deviation technique. In addition to the variability in event duration, the period between streamflow events was also highly variable. The inter-event period range from approximately 6 h to 64 d. The onset of streamflow for five streamflow events occurred <16 h from the cessation of the previous event.
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Fig. 6. Thermograph from 16 Sept. through 15 Dec. 2000 for (A) a depth of 0.15 m and (B) a depth of 0.75 m. The gray areas denote observed periods of streamflow.
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Thermographs for depths of 0.15 and 0.75 m exhibit thermal amplitude responses for temperature sensors above and below the transition depth, respectively (Fig. 6). When streamflow is present, the thermal amplitude at a depth of 0.15 m is reduced, while the thermal amplitude at a depth of 0.75 m increases.
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RESULTS
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Initially, the sensitivity of streamflow timing accuracy to each of the five analysis parameters was examined using a continuous record of thermal measurements collected in Rillito Creek at all measured depths, from 16 Sept. 2000 through 15 Sept. 2001. The record was divided into approximately 105000 5-min intervals, and using the water content measurements to identify the presence of streamflow, each interval was identified as either a no-flow or flow period. A sequential parameter sensitivity analysis was performed using a range of likely parameter values. During this analysis a single parameter was varied, while the remaining parameters were held constant. The optimal value for this parameter was fixed, and a sequential parameter was varied. This continued until an optimal set of parameters was determined. The final stage of the analysis was to vary each of the parameters sequentially using this optimal set.
The standard deviation window length and the threshold were the most sensitive of the five parameters (Fig. 7)
. In general, shorter standard deviation windows are influenced by atmospheric temperature shifts, resulting in more frequent false identifications of streamflow. Longer windows reduce the number of false identifications of streamflow periods, but underestimated the duration of streamflow. The standard deviation window length was optimal at 1 h using 5-min data (n = 12) and from 1 to 6 h for 15-, 30-, and 60-min data (n = 4, 2, and 1, respectively). The threshold parameter is most sensitive at values below the mean moving standard deviation value and is comparatively insensitive between 1.25 and 1.75 times the mean moving standard deviation value for standard deviation window lengths from 1 to 12 h.
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Fig. 7. A sensitivity analysis for the moving standard deviation technique considering: (A) the moving standard deviation window length, (B) the threshold multiplier, (C) the minimum flow duration parameter, and (D) the minimum inter-event duration parameter. Timing error is presented as the percentage of time over a year the method incorrectly infers the presence or absence of streamflow. Generally this error occurs at the onset and cessation of flow by either overestimating or underestimating the period of streamflow.
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Location of the reference time within the standard deviation window is more important for the longer standard deviation window lengths. Otherwise, the error associated with this technique is relatively insensitive to the reference time location within the window. Centered locations produced the lowest timing error overall.
Selection of the minimum flow duration parameter is dependent on the influence of atmospheric temperature fluctuations. False indications of flow periods due to atmospheric temperature variations were more prevalent for temperature sensors near the surface. The lengths of false events caused by weather changes are usually <4 h, and this is exhibited in the sensitivity analysis with an increase in error for event durations less than about 240 min (Fig. 7). However, the timing error increased when the minimum duration of a flow event exceeded 240 min. It is important to note that the minimum flow duration parameter differentiates between short duration flow events and abrupt atmospheric temperature shifts. The parameter will be less important for streams that experience longer duration events than those recorded at Rillito Creek.
The minimum inter-event duration parameter is a balance between the length of flow interruptions caused by reduced surface heating and the separation of consecutive streamflow events (Fig. 7). These interruptions are depicted in Fig. 4, where the standard deviation values are below threshold during periods of streamflow. Five streamflow events in Rillito Creek were separated by <16 h. Optimized values for the inter-event duration parameter ranged from 6 to 12 h. Once again this parameter is not as important for streams experiencing fewer, longer duration events than Rillito Creek.
Following the sensitivity analysis, the standard deviation model was calibrated for each depth using the TDR data from a 76-d record (16 Sept. 2000–15 Dec. 2000), which included eight events. The optimal parameters were evaluated for a succeeding 250-d record (1 Jan. 2001–15 Sept. 2001), which included 16 events. The step-wise optimization was conducted by holding four of the parameters constant while a single parameter was varied. The parameter value producing the least error in streamflow timing (defined as the sum of the number of false indications of flow and false indications of no flow) was selected. Successive parameters were varied in the same manner. This process was repeated for all five parameters. The cycle of optimization was repeated until improvement in streamflow timing accuracy was <0.1%. The optimal parameter set was independently verified as the global minimum using an optimization software, PEST (Watermark Computing, 1994).
Results from the field experiment and standard deviation model indicate that the temperature sensor located at the 0.75-m depth was optimal for identifying streamflow timing. The optimal depth was about 0.3 m deeper than predicted by using the analytical expressions. This discrepancy is likely due to inaccurate estimates of the thermal and hydraulic parameters for the site. Specifically, the heat transport Eq. [1] through [6] assume steady-state infiltration. Transient infiltration fluxes at the onset of streamflow are typically higher than steady-state infiltration fluxes used in the analysis. Use of a higher infiltration flux that was an average of the transient and steady-state infiltration fluxes would reduce the discrepancy between the predicted and observed optimal depth. Sinusoidal temperature forcing at the onset of streamflow was not observed; instead, abrupt changes in temperature associated with weather fronts were observed. The abrupt changes were advantageous for identifying streamflow using standard deviation method but contributed partially to the underestimation of streamflow.
Identification of streamflow at shallower depths was less accurate than at 0.75 m because the amplitudes of the conductive and advective diurnal temperature waves were more similar than had been estimated by Eq. [1] through [6]. Additionally, the standard deviation method calculated many more false streamflow events using the thermographs from sensors above the transition depth than from sensors below the transition depth (Fig. 4). Thermographs from deeper depths were less accurate than the thermograph at a depth of 0.75 m because the ranges in amplitude of the diurnal temperature waves declined and at depths below 2.0 m were smaller than the precision of the temperature sensors.
Precipitation over the channel did not produce an increase in water content between 0.25 and 2.25 m. Thus rainfall did not produce sufficient percolation to advect heat to these depths as would a streamflow event. This is one of the advantages of monitoring temperature at deeper depths.
The optimized analysis parameters for this site were a 1-h standard deviation window, a centered reference time, a threshold approximately 1.5 times the mean standard deviation, a 300-min minimum flow duration parameter, and a 900-min minimum inter-event duration parameter. Using these analysis parameters, all of the events were identified, and no false events were identified. Timing errors arose primarily at the onset and cessation of streamflow. The standard deviation technique inferred streamflow on average 77 min before it was observed by TDR at the same depth. That is, temperature fluctuations were observed before the TDR probes indicated a change in soil water content. This onset timing error was primarily due to fluctuations in air temperature preceding 2 of 14 streamflow events. An example of the onset timing error is shown for the 10 Oct. 2000 event (Fig. 4). The standard deviation values calculated for a depth of 1.0 m rises above the threshold preceding the start of event as recorded by the TDR measurements and indicated by the shaded region. Excluding these two events, the standard deviation technique inferred flow on average 8 min before the onset of streamflow as identified by TDR. The TDR measurements lag approximately 2 to 4 min from the onset of streamflow at the surface to the sensor at 1.0 m.
The cessation of flow is more difficult to identify than onset because of the gradual shift in the dominant mechanism of heat transport from advection to conduction at the termination of flow. On average the optimized standard deviation technique inferred the cessation of flow for this site 257 min after the observed cessation of streamflow as determined by TDR. The error is evident for the 3 Oct. 2000 event show in Fig. 3 and 4. This gradual shift in temperature and error during analysis was fairly consistent, so in practice cessation times could be reduced to account for this overestimation.
Field Application
For most field applications, additional flow-monitoring devices will not be available on site to calibrate temperature-based methods for inferring streamflow timing. Consequently, the standard deviation method must exhibit a level of effectiveness suitable in a stand-alone fashion. To test the accuracy of the standard deviation method, parameters were selected without the benefit of alternative flow-monitoring information.
Parameters were selected by first employing a range of moving standard deviation window lengths to identify likely streamflow events (Fig. 8)
. As shown in Fig. 8, the contrast between the thermal amplitude during the presence and absence of streamflow was identifiable using a range of window lengths from 1 to 12 h. The 1-h standard deviation window was selected over the other window lengths as the most appropriate because of the greater contrast in the moving standard deviation values between the apparent flow and no-flow time periods. To improve accuracy, a rain gage near the study site and/or a temperature sensor placed on the dry bank can be used to differentiate between short duration flow events and false positives caused by weather fronts (Fig. 8). The reference time was arbitrarily centered within the standard deviation window because it was shown previously that streamflow timing accuracy is insensitive to this choice.
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Fig. 8. (A) Thermograph from 16 Sept. 2000 through 15 Sept. 2001 for a depth of 0.75 m, (B) 1-h, (C) 4-h, and (D) 12-h moving standard deviation windows for temperature data measured at a depth of 0.75 m. (E) Close-up of two identified events that provide an example of how moving standard deviation parameters can be selected. The gray area denotes periods of identified streamflow events.
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The second step required plotting the mean moving standard deviation temperature to provide a starting point for selecting the threshold parameter. The threshold parameter was set equal to the highest standard deviation value during the longest period of no flow as evidenced by the precipitation record.
Finally, the duration parameters were selected. General knowledge of monsoon season streamflow events within the basin indicated that the localized nature of precipitation produced many of the shortest duration events and events in succession with short inter-event periods (Fig. 6). Consequently, the monsoon time period was considered ideal for establishing the duration parameters. The shortest streamflow event identified during this period was used to define the minimum flow duration parameter (Fig. 8). The minimum inter-event duration parameter was estimated as the shortest time between two consecutive flow events inferred on the plots. The minimum flow duration and inter-event duration occurred during July 2001.
The optimized parameter set achieved using this calibration technique was a 1-h window length, a threshold of 1.35 times the mean standard deviation, a minimum flow duration of 150 min, and a minimum inter-event duration of 600 min. When evaluated against the validation period, the average timing error associated with these parameters was approximately 95 min at the onset of flow and 310 min at the cessation per event. Thus, even without the use of alternative flow detection devices it was possible to obtain a suitable parameter set for analysis that is as accurate as visual identification, with the benefit of being automated and repeatable.
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CONCLUSIONS
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Temperature sensors deployed in streambed sediments can provide a cost-effective alternative for monitoring the timing of streamflow on the basis of heat transport mechanisms through the water column and sediments. A technique was developed to identify the timing of streamflow by a statistical analysis of changes in the sediment diurnal temperature wave amplitude. The moving standard deviation technique requires the definition of five analysis parameters. The accuracy of streamflow identification is most sensitive to the standard deviation window length and a threshold parameter. Once the analysis parameters are established, either through calibration with independent flow timing measurements or through supporting climate information, identification of the presence or absence of flow is repeatable, objective, and easily automated to process large data sets. Furthermore, the thermal method and analysis technique allows for the use of deeper temperature measurements, which we demonstrated to be advantageous under some field conditions.
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ACKNOWLEDGMENTS
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The authors would like to acknowledge the counsel of Jim Constantz of the U.S. Geological Survey. This investigation was funded by the U.S. Geological Survey's Southwest Groundwater Resources Project.
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REFERENCES
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ABSTRACT
INTRODUCTION
