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 We propose a new method based on temperature time series of surface and streambed pore waters to monitor local changes in streambed surface elevations at a nominally daily time scale. The proposed method uses the naturally occurring daily temperature signal changes in amplitude and phase between stream water and the water flowing within the streambed sediment. Application of the method in a fine-bedded stream predicts the timing and magnitude of a prescribed sequence of scour and deposition. This provides a new, effective, easy to use, and economic methodology to monitor the temporal evolution of erosion and depositional patterns in rivers.
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 Naturally occurring temporal variations of stream water temperature cause a complex thermal regime within streambed sediments [Constantz, 2008; Marzadri et al., 2012, 2013]. Heat transport within stream water and streambed matrix (pore water and sediment) stems from advection, due to intragravel flows, conduction through the sediment and fluid matrix and diffusion [Anderson, 2005; Constantz, 2008]. For the last several decades, temperature has been used as a tracer to quantify water fluxes into streambed sediments [Bredehoeft and Papadopulos, 1965; Constantz and Thomas, 1996, 1997; Constantz et al., 2001; Goto et al., 2005; Stallman, 1960, 1965] and hyporheic exchange [Gariglio et al., 2013; Gordon et al., 2012; Hatch et al., 2006; Keery et al., 2007; Rau et al., 2010; Swanson and Cardenas, 2010]. Temperature has also been used to investigate discharge losses in streams [Constantz and Thomas, 1996, 1997] and interaction with riparian vegetation [Constantz et al., 1994]. Here, we present a new application to monitor spatiotemporal variations of streambed surface elevations, i.e., scour and deposition at the daily time scale.
 The work of Luce et al.  presents a set of new analytical solutions to the one-dimensional heat transport equation with sinusoidal boundary condition, which represents the daily temperature oscillations of stream waters, at the water-sediment interface and constant temperature at the other lower depth boundary [Hatch et al., 2006; Keery et al., 2007]. The new solutions quantify both water fluxes, v, and the streambed sediment effective thermal diffusivity, κe, from paired water temperature time series measured at two depths in the streambed sediment. The former becomes the vertical flux component when fluxes are two-dimensional or three-dimensional [Cuthbert and Mackay, 2013; Hatch et al., 2006] and the later is the spatially averaged effective thermal diffusivity between the two sensors. The solutions of v and κe use the following new dimensionless number:
where the subscripts 1 and 2 indicate the sensor at the shallow and deep locations in the sediment and A and ϕ are the temperature signal amplitude and phase, respectively. This dimensionless number is only a function of measured quantities from which water flux, v, and κe can be quantified with the following equations, which could be also expressed as a function of ln(Ar) with some manipulation using equation (1) [Luce et al., 2013]:
where γ = ρmcm/(ρw cw) with ρ and c the density and the specific heat capacity of the sediment-water matrix, subscript m, and of the water, subscript w, respectively, ω=2π/P with P the period of the signal (1 day) and Δz is the sediment thickness between the two sensors. This distance, Δz, has historically been assumed to be known and time invariant [e.g., Hatch et al., 2006; Keery et al., 2007; Lautz, 2010]. However, previous research based on experiments with long-term stream and pore water temperature monitoring shows that scour and deposition may alter the thickness of sediment between sensors, thus it may change with time [Constantz et al., 2001; Gariglio et al., 2013].
 Most are familiar with the idea that temperature fluctuations on the inside of a wall have lower amplitude and are lagged compared to the fluctuations on the outside of the wall, and that the degree of damping and lag are functions of the thermal diffusivity of the wall and its thickness. The analogy between this and streambed sediment is fairly straightforward except for the movement of water carrying heat, which would seemingly confound the use of similar data to imply anything about the thickness or thermal properties of the bed. Rearrangement of equation (3), however, demonstrates the separability of the diffusive terms (thickness and diffusivity) from the measured quantities on the right, including the relative velocity information carried implicitly in η.
 So, while equation (3) allows the calculation of diffusivity if the thickness Δz is known, we can equivalently find the thickness of sediment for a known diffusivity from
Luce et al.  and Gariglio et al.  suggested that if diffusivity is known, uniform and taken to be time invariant over the daily time scale, temporal variations in the sediment thickness can be tracked via equation (5). Here, we test this hypothesis with a sequence of controlled scour (Δz(t1)) and deposition (Δz(t0)) in a small, low-gradient, fine-bedded stream (Figure 1). We suggest that the spatiotemporal changes in streambed surface elevation can be monitored with one temperature probe (e.g., sensor 0 in Figure 1b) measuring the in-stream water temperature and an array of sensors (e.g., sensor 1 in Figure 1b) embedded in the streambed sediment at given depths. This technique will provide four essential pieces of information about the streambed environment: (1) hyporheic vertical fluxes, (2) hyporheic thermal regime, (3) streambed effective thermal diffusivity, and (4) changes in streambed elevation.
2.1. Field Experiment Setup
 We built four temperature-monitoring probes with a 2.5 cm SDR 21 cold water PVC pipe perforated at each thermal measurement depth to house a temperature sensor, Onset StowAway TidBiT (measurement range between −4 and 38°C, resolution 0.15°C and accuracy 0.1°C). Each sensor was in direct contact with the surrounding water or sediment-water matrix (depending on whether the temperature sensor is above or below the streambed-water interface). TidBit sensors were set 10 cm apart separated by Styrofoam material sealed with silicon to prevent any preferential flows within the pipe. One probe, set as the control probe, C0, housed three sensors whereas the treatment probes, T1, T2, and T3, had two temperature sensors (Figure 1b). No treatment was applied at the control probe C0, and the streambed elevation was kept stationary, whereas we applied the treatment to probes T1, T2, and T3. TidBit sensors were calibrated before and tested after the experiment in a temperature-controlled bath. The post-experiment test did not show any divergence of the sensor measurements from the calibration values. Therefore, no corrections for thermal drift were applied.
 The probes were installed on a pool-riffle sequence of Warm Spring Creek, a small agricultural drainage channel and tributary of the Boise River, near the City of Boise, Idaho between 28 August and 5 October 2012 with a recording interval of 5 min (Figure 1a). The stream has low gradient (<0.001 m m−1) median grain size finer than 0.004 m, mean channel width of 1 m and mean water depth of 0.15 m. The probes were placed in the sediment by inserting a long narrow bladed soil sampling shovel in the streambed sediments and displacing the sediment laterally while the pipe was inserted. The shovel was then withdrawn, allowing sediment to settle around the pipe.
 The control probe C0 was buried approximately 0.25 m from the right stream bank between the pool-tail and riffle crest (Figure 1a). The topmost sensor was set flush to the water-sediment interface such that the sensors were monitoring water temperatures at 0 (in the stream), 10 and 20 cm below the streambed surface. The other probes T1, T2, and T3 were buried with the temperature sensors at 10 and 20 cm below the original streambed surface. T1 was placed at the same downstream location of probe C0 and 0.25 m from the left bank, T2 and T3 were set along the center of the stream near the riffle crest and at the head of the pool, respectively. The probes were approximately 50 cm apart from each other. The fine sediment material allowed for an accurate measurement of streambed surface and of the relative positions of the temperature sensors.
 We measured the streambed surface elevation around each probe and the water depth above the control probe weekly. The control probe did not have any measurable change of streambed surface elevation during the experiment. We imposed a sequence of scour and filling at probe T1, T2, and T3 as reported in Table 1. The “scour” was generated by removing the sediment around the probe within a radius of approximately 0.1 m by hand. We measured the relative change of streambed surface elevation from the top of each probe and we stopped digging once we lowered the streambed surface elevation by 0.05 m from the original elevation. Deposition was simulated by filling the scour hole to the initial streambed surface elevation with the removed material. Streambed elevation measurements showed negligible changes from the imposed treatment due to settling of the sediment or filling of the scour at the treatment probes.
Table 1. Sequence of Changes in Streambed Elevation Imposed at Each Temperature Probe (C0, T1, T2, and T3) and Water Stage Above the Control Probea
Change in Streambed Elevation (m)
Stream Water Stage (m)
Negative sign indicate lowering (scour) of the streambed surface elevation at the probe within approximately a 10 cm radius. Imposed change in elevation remains constant until the next change.
2.2. Data Analysis
 The vertical distance between two sensors Δz has historically been assumed to be entirely filled with streambed sediment (Δz(t0) in Figure 1b). However, this is not the case when erosion processes scour streambed sediment leaving a section of the separation between sensors unburied, Δz(t1) < Δz(t0), in Figure 1b. In this case, Δz referred to in equations (1)-(5) reduces to the thickness of the sediment remaining between the two sensors (Δz(t1) in Figure 1b), and equation (5) can be used to solve for Δz. Equation (5) assumes that the sediment thermal properties, κe, are known. The value of κe can be quantified during a period when the streambed elevation is stable and Δz is known, such as at the beginning of an experiment. Thermal properties would be time invariant as long as the porosity of the streambed does not change substantially. If porosity were expected to change over time as it ultimately did during our experiment, a pair of probes that both stay buried (constant Δz) could be used to track shifts in thermal properties.
 We paired each temperature sensor buried in the sediment with the in-stream sensor of probe C0 to quantify the amplitude ratio and the phase shift between surface and subsurface water temperatures.
 We calculated the amplitude and phase of the temperature signal with a discrete Fourier transform [described in Luce and Tarboton, 2010] using a 2 day window, but other techniques are available [e.g., Gordon et al., 2012; Keery et al., 2007; Swanson and Cardenas, 2010].
3. Results and Discussion
 Blindly applying equation (3) without accounting for changes in sediment thickness between sensors, Δz, causes fictitious variations in κe values (Figure 2a, sensors T1, T2, and T3). The apparent values of κe estimated from the treatment probes show abrupt increases when the scour treatment was imposed (Figure 2a). These apparent changes are larger at the shallow (10 cm) than deep (20 cm) sensors because of the larger fractional change in sediment thickness for the shallower sensor than for the sensor at greater depth (Figure 2a). Conversely, the sensors of the control probe C0 show small decreases in κe, which is due to the recovery of the streambed sediment from the disturbance caused by compacting and loosening the sediment when placing the temperature probes into the streambed. The effect of this disturbance last approximately 12 days after which κe estimated with the probe C0 sensors at both 10 and 20 cm depths remains approximately constant. The estimated κe values have approximately the same values at the shallower (10 cm) and deeper (20 cm) depths, indicating that thermal properties of the streambed are relatively constant within the surficial layer at the C0 sensor location (Figure 2b).
 When the actual experimentally imposed Δz is used in equation (3) for the treatment probes, true κe values do not change abruptly but show a trend similar to that of the control probe C0 (Figure 2b) as the sediment returned to its preinstrumentation condition. The recovery time lasts longer for the treatment sensor at the deep than shallow depth. A time invariant κe is reached after approximately 12 and 26 days for the treatment sensors at 10 and 20 cm depth, respectively. Note that the treatment operations themselves do not appear to be major contributors to true κe temporal variations but they may have prolonged the persistence of the effect induced by the installation operation for the deep sensors. All sensors except probe T2 predict a similar true κe by the end of the experimental period.
 Because of the temporal changes in κe for both control and treatment probes induced by their installation, we could not use a constant value evaluated during the first part of the experiment with constant streambed surface elevations. We used κe of the probe C0 as a reference value to account for temporal variation over the recovery period. Because the treatment sensors at a depth of 10 cm showed the same trend as the respective sensor of probe C0, we used the ratio of the κe values averaged over the first week between sensor at C0 and those at T1, T2, and T3 at 10 cm to scale κe of C0 over the entire period, Ri = κe,C0/κe,Ti, where i = 1, 2, and 3. Thus, the actual κe for each treatment sensor at depth 10 cm was estimated as κe,Ti = Ri κe,C0. The sensors at 20 cm depth on the treatment probes present a more complex change of κe over time due to the longer interval required to adjust after the disturbance induced by the installation. Thus, we used κe values averaged during the first, middle, and last week when the streambed surface was at the initial elevation to estimate the ratio between the control and the treatment sensors at 20 cm. We used a parabolic change of the ratio of κe over time between the first and the last week of the experiment with constant ratios at the beginning and at the end of the experiment.
 This pattern is consistent with the observations presented in Gariglio et al.  in a gravel bed river, who reported that the disturbance caused by probe installation persisted up to 30 days in some locations. Thus, the recovery period may last between 12 and 30 days depending on grain size distribution, location of the probes, and depth of the sensors. The length of this period can be estimated by monitoring κe over time during a period with no erosion and deposition. If we had waited 15 days before starting the treatment at probe T1, T2, and T3, we should have seen constant κe values at the control and treatment probes. Consequently, the correction of κe with the control probe would not have been necessary, but we could have used the constant κe values reached at the end of the recovery period.
 Using the estimated values of κe and equation (5) to quantify changes in streambed surface elevation provided a reasonable approximation of the imposed scour and fill treatments (Figure 3), both sequence and magnitude. Note that no further calibration or adjustment of κe was done to obtain these estimates of depth. The largest residual errors, difference between predicted and observed elevations, are at the time of treatment application (Figure 4). This is expected because the time scales of the applied erosion/deposition and the heat transport are different: the former is almost instantaneous in our case and the later has a daily time scale. Thus, the method should provide daily averaged scour/deposition information, which is much more continuous in nature than the typical annual scale measurements from driven or buried rods such as scour-chain or magnetic collars, which only provides maximum scour and no information on deposition. Other techniques such as transducers systems or scanning sonars are expensive and affected by other limitations, which may include water quality [Mueller, 1998]. Root–mean-square errors calculated with 12 h average are of the order of 1 cm, or about 20% error, which is good accuracy for daily scour/deposition predictions. The bias, which is the mean of the residuals, shows that there is a systematic error, which corrected may increase the accuracy to 10% of these measurements (Table 2). The predictions are more accurate with the shallow than the deep sensors. This different behavior could be due to a better estimation of the thermal properties with the former than latter sensors. Larger uncertainty could affect the deeper sensor more than the shallower sensor due to more pronounce curvilinear flow paths, which cause divergence-convergence of heat transport [Cuthbert and Mackay, 2013] and/or higher flow mechanical dispersivity, which could give rise to thermal dispersivity [Rau et al., 2012a, 2012b]. The lowest performance is for probe T2 at 20 cm but it has also the largest bias. Probe spatial location within the streambed does not affect the performance, which depends only on the depth of the sensors. The calculated intragravel velocities switched from upwelling to downwelling during the experiment as flow stage changed. Peclet numbers ranged between 0 and 2 over the different probes and flow stages, supporting the fact that the method applies when heat transport is dominated by either advection or diffusion. Additional research is needed to better define its accuracy under a range of intragravel flux and temperature variations, but here we provide proof of concept of the new method. The method can be extended to measure scour deeper than the distance between sensors by having a series of sensors approximately 15 cm apart [Constantz et al., 2001] along the vertical. The topmost sensor still buried would provide the signal for the estimation of the sediment thickness. We believe this method could be efficient in monitoring scour and deposition at bridge piers where scour could be several meters, by placing temperature sensors along the bridge pier. The approach is affordable even if the number of temperature sensors could be large because they are economical.
Table 2. Root-Mean-Square Error (RMSE) and Bias for Each Sensor Calculated on 12 h Averaged Values Excluding Values at Transition
Root-Mean-Square Error (m)
 The limitation of a 1-D heat transport approach in the presence of hyporheic flows, which are intrinsically two-dimensional [Elliott and Brooks, 1997] and three-dimensional [Marzadri et al., 2010; Tonina and Buffington, 2007, 2011], has been given a rigorous treatment in a recent publication of Cuthbert and Mackay . Their primary point is that strong curvature in the flow field, which in our experience is generally driven by surface topography curvature near the streambed surface, is necessary for significant departures between 1-D and 2-D approaches (as their setup was 2-D rather than 3-D) whereas nonvertical flow may have negligible effect as long as thermal dispersivity is small to negligible. Their results suggest errors on the order of 5–10% except as fluxes dropped toward zero between the 1-D and the 2-D approach. The 1-D thermal approach would provide us the vertical component of the velocity fluxes in the case of higher dimensionalities (2-D and 3-D) and κe would be the spatially averaged thermal properties of the sediment along the flow path.
 Our results show that time series analysis of paired in-stream and pore water temperatures can predict variations in streambed surface elevations at the daily time scale with accuracy of the order of 20% in our test. The largest errors occurred at the time of treatment application because the time scale of the method is linked to the daily temperature oscillations, which have a 1 day period and the almost instantaneous time scale of the applied treatment.
 This technique also provides a tool to quantify when the disturbance induced by placing the sensors in the streambed ceases to affect the sediment properties and hyporheic hydraulics and the system recovered to its preinstallation conditions.
 If we assume that stream flow is spatially uniform and not thermally stratified within a reach then we can use one single probe to measure in-stream water temperature coupled with an array of sensors buried in the streambed or continuously via optical cable [Selker et al., 2006]. This will allow spatial monitoring of temporal changes in streambed elevation. When maximum scour is unknown, multiple sensors could be deployed at different depths along the same vertical such that one sensor will be placed deeper than the maximum scour depth.