Advancing our predictive capabilities of heat fluxes in streams and rivers is important because of the effects on ecology and the general use of heat fluxes as analogues for solute transport. Along these lines, we derived a closed-form solution that relates the in-stream temperature spectra to the responding temperature spectra at various depths in the sediment through a physical scaling factor including the effective thermal diffusivity and the Darcy flow velocity. This analysis considers the range of frequencies in temperature fluctuations that occur due to diurnal and meteorological variation both in the long and short term. This approach provides insight regarding the key frequencies for analysing temperature responses at different depths within the sediment and also provides a simple and accurate method that offers quantitative insight into heat transport and surface water interactions with groundwater. We demonstrate for Säva Brook, Sweden, how the values of effective thermal diffusivities can be estimated based on the observed in-stream and sediment temperature time series and explain the temporal scaling of the heat transport as a function of a dimensionless frequency number. We find that the lower limit of periods of significance for the analysis increases with depth, and we recommend further research regarding appropriate frequency windows.
 Because biogeochemical processes depend on temperature, understanding heat transfer in streams is key to better understanding stream ecology. Significant heat exchange within streams occurs at two primary interfaces with the atmosphere and the subsurface [Neilson et al., 2010; Webb et al., 2008]. Heat exchange at the stream-subsurface interface has been specifically used to investigate groundwater-surface water interactions [Conant, 2004; Evans and Petts, 1997]. Various techniques have been employed to quantify these exchanges, including the use of fibre optic cables to provide distributed temperature data along stream reaches [Selker et al., 2006] and temperature sensors placed vertically within the stream sediments [Constantz et al., 2002; Hatch et al., 2006; Keery et al., 2007]. The latter methods commonly assume one-dimensional (1D) convective and conductive heat transport through the sediments [Lautz, 2010; Rau et al., 2010]. These approaches provide estimates of seepage rates and effective thermal diffusivities based on the diurnal period of the instream temperature [e.g., Hatch et al., 2006]. However, the temperature time series for streams often exhibits a spectrum of different frequencies that reflect the variations in air mass temperature, solar radiation and precipitation. Vogt et al.  and Gordon et al. used real harmonic series to extract information from frequencies, amplitude and phase, but they rely on the diurnal period to evaluate thermal diffusivity and vertical fluxes. To further extract information from the coupled spectrum of temperature frequencies in the stream water and bed-sediments, we propose a physically based spectral transform that relates the in-stream temperature fluctuations to those at specific sediment depths. Such frequency spectra in the temperature fluctuations persist due to diurnal cycles, as well as both short and long-term variations in atmospheric conditions. Here we develop the appropriate theory and illustrate how the effective thermal diffusivity and Darcy flux can be evaluated from in-stream temperature data at various sediment depths and locations. These results are compared with results from laboratory tests as well as an optimized finite-element model.
2. Physical Spectral Transform of Temperature Time Series
 The temperature T(z,t) (°C) resulting from the thermal processes in the sediment bed can be expressed as a convolution of the boundary condition, i.e., the in-stream temperatureT0, and a unit or impulse response function ψδ of the form
where the unit response function ψδ = Tδ/A , Tδ is the solution to the process based governing heat transport equation valid for a unit pulse at the boundary T0,δ = T(z = 0, t) = Aδ(t), δ(t) is the Dirac delta function (s–1), A is the temperature integrated over a given period [s × °C], z is the depth coordinate (m) and t is the time (s). The Fourier transform of equation (1) becomes F[T] = F[T0] ⋅ F[ψδ], with time t replaced by the fundamental frequency ω = 2 π/λ, where λ is the time period of basic harmonics inherent to the time series (s). Because the power spectrum can be expressed as , where the over-bar denotes the complex conjugate,equation (1) yields the power spectrum as:
Consequently, the main implication of the suggested approach is that two temperature power spectra – at an upper boundary level (generally in-stream) (PT0) and at a lower sediment depth (PT) – are related through a physical scaling factor that represents the thermal processes in the sediment. This factor is key to the transformation of the time series spectra and can be referred to as the square of the modulus of the Fourier transform of the impulse response function [Miller, 1974, p. 133].
 This basic principle given by equation (2)can be applied to different situations for which the physical scaling factor is derived. The fundamental transport equation for such derivations is the heat conduction-convection model for saturated soil [Constantz et al., 2002; Keery et al., 2007]:
where κe is the effective thermal diffusivity (which includes the thermal dispersion and diffusion), q is the Darcy flow velocity (porosity multiplied by pore velocity), γ = (ρ c)/(ρfcf), (ρ c) is the product of the density and the specific heat capacity of the saturated sediment-fluid system [J/(K m3)] and (ρfcf) is the product of the density and the specific heat capacity of the fluid [J/(K m3)]. In the auxiliary material (section S1 of Text S1) we propose three possible cases for which the physical scaling factor is derived and which are physically motivated: 1) effective conduction-convection with semi-infinite depth (T(z = ∞, t) = 0), 2) effective conduction with semi-infinite depth and 3) thermal dispersion with limited depth, L (m). For the first case, one can derive the relationship between the temperature power spectra on the form of (section S1 ofText S1)
where the Peclet number Pe = (q/γ) z /κeis the ratio of the convective flux to the conductive flux and the conductive-frequency numberζ = z (ω/κe)0.5 is the ratio of the frequency to the inherent conduction frequency. Note that the exponential factor represents the solution to the physical scaling factor . The second case with pure effective conduction is obtained as a special case of equation (4) when Pe ≪ ζ, which yields . Rearrangement yields a direct estimate of κe as function of two observed power spectra in the form of
The third case with depth limited thermal dispersion is defined by the solution to equation (2)for a no-flux surface at a certain depth, which introduces the stratification-frequency number Ψ = L (ω/κe)0.5 (see section S1 of Text S1). This case applies for conditions with decreased mixing with depth in the hyporheic zone [Elliott and Brooks, 1997; Wörman et al., 2006; Bottacin-Busolin and Marion, 2010] or stratification of the subsurface soil with decreased thermal diffusivity below some depth. In the case of a no-flux surface vertical advection may be neglected, which leads to an additional transform equation for the temperature power spectra described in section S1 ofText S1.
 Prior to spectral analyses the time series were “de-trended” and Hamming windowing, generally acceptable for linear problems [Brillinger, 1981], was applied to improve the quality of the results. The numerical procedures are described in sections S2 and S3 of Text S1 and the optimization of the model parameters are described in section S5 of Text S1.
3. Interpretation of Heat Fluxes in Säva Brook
 Temperatures were measured at seven sites along 16 km of the Säva Brook in Uppland County, Sweden. Temperature sensors were used to record temperature at 5-minute intervals in the main-stream channel and in the sediment at depths ranging from 3 to 100 cm. The data collection period was 18 to 30 days, depending on the station location during May 2011, while the flow was mainly decreasing. Sediment samples were also obtained for laboratory measurements of thermal diffusivity. See section S4 ofText S1 for details on the data collection.
 The temperature time series obtained at Site 2 at different depths in the sediment show strong fluctuations (Figure 1). We also simulated the temperature time series by using COMSOL Multiphysics to indicate from this conventional approach to solve equation (3) that the temperature fluctuations at different depths can be accurately related to the physical description of the heat transport. This finite element solution considers a statistical optimization of an exponential distribution of effective thermal diffusivity (κe(z) = 8·10−8 + 3·10−7·(1-exp(−z/0.1)) as described in more detail in section S4 of Text S1.
Figure 2 (top) compares the observed power spectrum at 3 cm depth at Site 1 and equation (4) with the optimized values of the Darcy velocity and the effective thermal diffusivity, i.e., Pe and ζ (see section S5 of Text S1). When the optimization is limited to periods longer than 1 day (ω0.5 < 0.02 in Figure 2, bottom) the Darcy velocity is found to be close to zero (<10−10 m/s) and the effective thermal diffusivity is essentially the same as that obtained under the assumption of pure conduction (i.e., equation (5)). Hence, for ω0.5 < 0.02 the power spectrum in Figure 2 (bottom) follows the proportionality ln(PT) ∝ ω−0.5 appearing in equation (5), which indicates that the heat transport is controlled primarily by effective conduction (i.e., Pe ≪ ζ). For higher frequencies, both the observed power spectra and equation (4)indicates a non-linear behaviour. However, we find thatequation (4) cannot be well fitted to observations for plausible combinations of κe and the q/γ (see section S5 of Text S1) and, hence, the more constant behaviour of the power spectrum at high frequencies is probably not because of convection dominance. An explanation can be conceived from Figure 2 (top) in which, according to equation (5), the thermal diffusivity in the stream sediments is independent of the frequency of the heat input for all periods longer than 5–7 hours. Although, for shorter periods, the observed time series include increasing noise and a drift of κewith frequency. High-frequency events of energy flux can be caused by short-term variations in meteorological patterns (e.g., air mass temperature, wind and radiation), but their statistical significance seems to be low relative to the noise and the damping of temperature fluctuations with sediment depth. The noise can be reduced slightly by limiting the interval of the periods considered in the analysis and, especially, the lower limit of the significant periods tends to increase with depth (seesection 4.1). Some variability in this lower limit between sampling sites is attributed to site-specific factors (e.g., sediment properties, landscape and vegetation).
 Thus, for pure conduction (Pe ≪ ζ), the slope of the power spectrum of temperature scales exponentially with the conductive-frequency number . If Pe ≪ ζ, convection dominates the heat transport and the power function is independent of frequency. An optimization of the three derived power functions (conductive and conductive-convective with semi-infinite depth and conductive with limited depth) provides the same values of effective thermal diffusivity at all sampling sites. The stratification-frequency number Ψ = L (ω/κe)0.5 is also shown to be of minor importance for heat transport in Säva Brook sediments.
 With this understanding, the temperature signal at certain sediment depths can be used as a boundary condition for the response at a lower depth and one can evaluate the depth variation of the effective thermal diffusivity layer by layer for the Säva Brook data. Figure 3 (top) shows a comparison of the results of the spectral approach for different periods with the laboratory values of κefrom site-specific sediment core samples and the optimized results using COMSOL Multiphysics. The power spectrum results indicate that the thermal diffusivity at the depths of 50 cm and 100 cm decreases with an increase in the lower limit of the window of periods considered. We found that at these depths, periods up to at least 2 days had to be excluded to obtain a convergent result. The spatial variability of the estimated effective thermal diffusivity between sites is small (Figure 3, bottom), which is expected due to the relatively homogenous geological setting. The effective thermal diffusivity increases and asymptotically approaches a constant value with depth. A possible explanation for the relatively low thermal diffusivity at the sediment surface is the presence of gas [Cuthbert et al., 2010] that was qualitatively observed at the site (methane or nitrogen) with higher porosity and organic content in the shallow sediments. The results from sampling Site 3 diverge from this pattern, exhibiting a constant or even decreasing value of effective thermal diffusivity with depth (Figure 3, bottom).
 The Fourier spectral approach presented here can also be used to analyze how thermal properties change over time in a window moving along the time series as previously shown for analyses of runoff processes in watersheds [Wörman et al., 2010]. The analysis presented in section S6 of Text S1 illustrates that the effective thermal diffusivity for a period of a couple of days shows an increase with increasing flood stage. This increase may be due to the increased pumping occurring with increasing stream velocities [Elliott and Brooks, 1997], which would increase the thermal dispersion, or to the scour and deposition of the sediments that influences the location of the boundary condition.
4. Discussion of Implications
4.1. Feasibility of the Method
 The transform method developed here offers a simple way to relate one observed temperature power spectrum to another by multiplication with a physical scaling factor (cf. Matlab code in section S3 of Text S1). The basic scaling equation (2) applies to different heat transport models, whereas equation (4) provides a general spectral solution for the physical scaling factor based on 1D conduction-convection. The convection term can often be neglected (seesection 4.2 and section S1 of Text S1) and for conduction-dominated heat transport, effective thermal diffusivities can be obtained from the exact solution either separately for each frequency used in the transform (equation (5)) or fitted to the range of frequencies. A scale transform that accounts for depth-limited conduction is also included in section S1 ofText S1(equation (S12)). The applicability of these different transforms must be assessed using site-specific temperature data as subsequently demonstrated for Säva Brook.
 Thermal diffusivities obtained from Säva Brook using the spectral method, numerical solutions and laboratory measurements show excellent agreement. There is a lower limiting period that avoids noise and drift in the thermal diffusivity, which for Säva Brook was found to be 5–7 hours at sediment depths above about 10–20 cm. However, the required limit increases with depth and from 50–100 cm in depth we found drift in the thermal diffusivity for periods up to at least 2 days. Hence, a key finding in the evaluation of the thermal diffusivity using the spectral method is that at sediment depths deeper than 50 cm one may have to exclude the diurnal signal commonly used in heat transport studies.
4.2. Heat Mixing in Säva Brook Bed Sediments
 Temperature measurements have previously been used to identify up- or downward water fluxes in stream sediments [Silliman et al., 1995; Gordon et al., 2012]. However, in Säva Brook the conduction was found to be the dominant heat transport mechanism in the sediments. Previous studies by Hatch et al.  and Jensen and Engesgaard  stress the need to consider both amplitude and phase when estimating conduction and convection simultaneously. However, while the optimization based on the power spectrum gave a Darcy velocity close to zero (<10–10 m/s), optimization based on the phase spectrum resulted in finite, but still small velocities (<10–7 m/s). Even by varying the range of different frequencies considered and expressing the optimization based on a combination of power and phase spectra, the vertical advection was found to be insignificant. Use of the power transform, i.e., equation (4), or the phase transform, equation (S7) in Text S1, results in similar value of the effective thermal diffusivity.
 Previous studies on Säva Brook sediments, have indicated that hyporheic mixing of solutes is limited to about 10 cm [Wörman et al., 2002; Jonsson et al., 2003]. Consequently, we hypothesize that the physical properties of the stream sediment, which mainly consist of clay, restrict convective heat transport at depths greater than about 10 cm. Even if hyporheic mixing is likely caused by shallow flows in the bed sediment due to pumping [Thibodeaux and Boyle, 1987], the large-scale subsurface flows are sufficiently small and are not identified in this study as vertical 1D advection. The temporary increase in the effective thermal diffusivity identified for a couple of days in connection with increasing flood stage is likely caused by either small-scale convective/pumping processes confined to shallow bed-sediments and/or remobilisation of an upper layer of the stream-bed sediment. The latter would move the in-stream temperature fluctuations closer to the corresponding points in the sediment (i.e., the depth of the sensor would change) and result in an increase in the thermal dispersion. In these shallow sediments, there are also factors that tend to decrease the thermal diffusivity, such as the existence of gas and organic material (see section 3.2) or the variation of effective thermal diffusivity, which appears to be highest close to the bed surface (Figure 3, bottom).
4.3. Scaling of Heat Fluxes in Sediments
 The exact spectral solution enables the evaluation of the consistency in the physical model (e.g., the 1D convective-conductive equation) over the significant range of stream temperature fluctuations. Some of the shortest periods are not warranted because of a lack of periodicity or incomplete representation, but the general vertical heat responses in bed-sediments scales inequation (4) with the Peclet number, Pe = (q/γ) z/κeand the conductive-frequency number . Therefore, the physical scaling factor resulting from the spectral approach clearly separates scaling due to the Peclet number, which is frequency independent, from the conductive-frequency numbers, which are frequency dependent. In this paper we demonstrated how the different scaling numbers are related through the physical-mathematical formulation of the heat transport problem in bed-sediments. These relationships should help develop the understanding of heat transport in streams and assist in developing future observation and analysis methods.
 Part of the project was funded by the strategic research project STandUp for Energy (KTH), the Swedish Hydropower Centre (SVC) and The Utah Water Research Laboratory at Utah State University. Thanks also go to anonymous reviewers who helped improve this paper.
 The Editor thanks the anonymous reviewers for their assistance in evaluating this paper.