## 1. Introduction

[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*use the following new dimensionless number:

_{e}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*) with some manipulation using equation (1) [

_{r}*Luce et al*., 2013]:

where *γ* = *ρ _{m}c_{m}*/(

*ρ*c

_{w}*) with*

_{w}*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*.

[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

[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.