### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results and Discussion
- 4. Conclusions
- Acknowledgment
- References

[1] 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.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results and Discussion
- 4. Conclusions
- Acknowledgment
- References

[2] 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.

[3] The work of *Luce et al*. [2013] 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, *κ*_{e}, 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 *κ*_{e} use the following new dimensionless number:

- (1)

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 *κ*_{e} can be quantified with the following equations, which could be also expressed as a function of ln(*A*_{r}) with some manipulation using equation (1) [*Luce et al*., 2013]:

- (2)

- (3)

where *γ* = *ρ*_{m}c_{m}/(*ρ*_{w} c_{w}) 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 Δ*z* is the sediment thickness between the two sensors. This distance, Δ*z*, 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].

[4] 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*.

- (4)

[5] So, while equation (3) allows the calculation of diffusivity if the thickness Δ*z* is known, we can equivalently find the thickness of sediment for a known diffusivity from

- (5)

[6] *Luce et al*. [2013] and *Gariglio et al*. [2013] 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 (Δ*z*(*t*_{1})) and deposition (Δ*z*(*t*_{0})) 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.

### 3. Results and Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results and Discussion
- 4. Conclusions
- Acknowledgment
- References

[14] Blindly applying equation (3) without accounting for changes in sediment thickness between sensors, Δ*z*, causes fictitious variations in *κ*_{e} values (Figure 2a, sensors T1, T2, and T3). The apparent values of *κ*_{e} 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 *κ*_{e}, 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 *κ*_{e} estimated with the probe C0 sensors at both 10 and 20 cm depths remains approximately constant. The estimated *κ*_{e} 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).

[15] When the actual experimentally imposed Δ*z* is used in equation (3) for the treatment probes, true *κ*_{e} 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 *κ*_{e} 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 *κ*_{e} 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 *κ*_{e} by the end of the experimental period.

[16] Because of the temporal changes in *κ*_{e} 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 *κ*_{e} 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 *κ*_{e} values averaged over the first week between sensor at C0 and those at T1, T2, and T3 at 10 cm to scale *κ*_{e} of C0 over the entire period, *R*_{i} = *κ*_{e}_{,}_{C}_{0}/*κ*_{e}_{,}_{Ti}, where *i* = 1, 2, and 3. Thus, the actual *κ*_{e} for each treatment sensor at depth 10 cm was estimated as *κ*_{e}_{,}_{Ti} = R_{i} κ_{e}_{,}_{C0}. The sensors at 20 cm depth on the treatment probes present a more complex change of *κ*_{e} over time due to the longer interval required to adjust after the disturbance induced by the installation. Thus, we used *κ*_{e} 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 *κ*_{e} 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.

[17] This pattern is consistent with the observations presented in *Gariglio et al*. [2013] 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 *κ*_{e} 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 *κ*_{e} values at the control and treatment probes. Consequently, the correction of *κ*_{e} with the control probe would not have been necessary, but we could have used the constant *κ*_{e} values reached at the end of the recovery period.

[18] Using the estimated values of *κ*_{e} 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 *κ*_{e} 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 TransitionRoot-Mean-Square Error (m) | Bias (m) |
---|

T1–10 | T2–10 | T3–10 | T1–20 | T2–20 | T3–20 | T1–10 | T2–10 | T3–10 | T1–20 | T2–20 | T3–20 |
---|

0.008 | 0.013 | 0.01 | 0.012 | 0.018 | 0.012 | −0.004 | 0.011 | −0.002 | −0.01 | 0.011 | −0.002 |

[19] 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* [2013]. 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 *κ*_{e} would be the spatially averaged thermal properties of the sediment along the flow path.

### 4. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Method
- 3. Results and Discussion
- 4. Conclusions
- Acknowledgment
- References

[20] 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.

[21] 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.

[22] 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.