Corresponding author: A. M. McCallum, Connected Waters Initiative, University of New South Wales, 110 King St., Manly Vale, NSW, 2093, Australia. (email@example.com)
 In order to manage surface water (SW) and groundwater (GW) as a single resource, it is necessary that the interactions between them are understood and quantified. Heat, as a natural tracer of water movement, is increasingly being used for this purpose. However, analytical methods that are commonly used are limited by uncertainties in the effective thermal diffusivity of the sediments at the SW-GW interface. We present a novel 1-D analytical method. It utilizes both the amplitude ratio and phase shift of pairs of temperature time series at the SW-GW interface to estimate the Darcy velocity. This eliminates both the need to specify a value for effective thermal diffusivity and the need for iteration. The method also allows for an estimation of effective thermal diffusivity, which can indicate periods where assumptions to the analytical solution are violated. Riverbed temperature data from the Murray Darling Basin (Australia) are used to illustrate the method.
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 In water management globally, there is a move toward a conjunctive approach which recognizes that surface water (SW) and groundwater (GW) are a connected resource [Winter et al., 1998; Woessner, 2000]. For this to be successful, the interactions between SW and GW need to be understood and quantified.
 Heat as a natural tracer of water movement has increasingly become popular for the study of SW-GW interactions [Anderson, 2005; Constantz, 2008]. The principle behind this approach is that daily temperature fluctuations within a SW body (e.g., river) due to solar radiation lead to a temperature response in the sediments at the SW-GW interface (e.g., riverbed) due to conduction and convection. In particular, the sinusoidal components of the diel temperature fluctuations which propagate through the SW-GW interface have been exploited to quantify the Darcy velocity [Stallman, 1965; Hatch et al., 2006; Keery et al., 2007]. However, analytical methods that are commonly used are limited by uncertainties in the effective thermal diffusivity value [Shanafield et al., 2011].
 In this paper a novel method is presented which bypasses this issue by estimating the Darcy velocity from pairs of temperature time series without the need to specify the effective thermal diffusivity value. Estimations of the effective thermal diffusivity are also obtained from the time series. The presented method has the same basis, and inherent limitations, as the original Stallman solution. The method is illustrated using data from Maules Creek Catchment, Murray Darling Basin (Australia). The method would be equally applicable to other SW-GW settings such as streams, lakes, springs and channels.
2.1. Theoretical Background
 The one-dimensional (1-D) convection-conduction equation is
where T is temperature, t is time, z is depth, v is thermal front velocity, and D is effective thermal diffusivity [Suzuki, 1960].
 In this paper the convention used is that v is positive for gaining conditions (i.e., when the SW body gains water), and negative for losing conditions (i.e., when the SW body loses water).
 For saturated conditions, the thermal front velocity and effective thermal diffusivity are given by
where q is Darcy velocity; and are the density and heat capacity of water; is the bulk heat capacity; κ is bulk thermal conductivity; and β is thermal dispersivity (in the direction of fluid flow) [Rau et al., 2012].
 The first term in equation (3) is the thermal diffusivity and the second term the increase in thermal diffusivity due to thermal dispersivity; together they constitute the effective thermal diffusivity.
 The bulk heat capacity is defined as the arithmetic average of the phases
 As the solid phase is constituted of various minerals, can be approximated using
where Q is the quartz content of the total solids, is the solid thermal conductivity of quartz, and is the solid thermal conductivity of the fraction of other minerals [Johansen, 1975].
Equations (3) through (6) show that the effective thermal diffusivity is a function of a number of different parameters, each of which has a possible range, which leads to the uncertainty in its value.
 For a semi-infinite half space where the temperature of the upper surface is varying sinusoidally, the temperature at any depth and time is given by
Equation (7) is subject to the assumptions that (1) fluid flow is in the vertical direction only, (2) fluid flow is steady state, (3) fluid and solid properties are constant in both space and time, and (4) fluid and solid temperatures at any particular point in space are equal at all times [Stallman, 1965].
 Starting from equation (7), previous researchers [e.g., Hatch et al., 2006; Keery et al., 2007] have developed analytical methods to estimate the thermal front velocity. These utilize amplitude ratio or phase shift data from pairs of temperature time series recorded at different depths.
 The equation for use with amplitude ratio data is
where is the amplitude ratio (i.e., , see Figure 1).
 Similarly, the equation for use with phase shift data is
where is the phase shift of temperature maxima, minima, or zero crossings (i.e., ; see Figure 1).
 When using the convention v is positive for gaining and negative for losing conditions, a negative sign is required on the left hand side of equation (9). The absolute sign on the left-hand side ofequation (10) indicates that this equation provides the magnitude, but not the direction of the thermal front velocity.
 Given the assumptions of equation (7), the and results should be consistent. Equations (9) and (10) can therefore be combined and rearranged (done here using Mathematica, version 7) to give an expression for thermal front velocity as a function of and :
 Similarly, the effective thermal diffusivity can be expressed as a function of and :
 In this approach, v is a function of and but not D; the information about D is implicit in the and data. Likewise, D is a function of and but not v. Furthermore, v and D appear on the left hand side of the equations only, and so no iteration is required.
 It is thus possible to quantify the SW-GW interaction (i.e., Darcy velocity,q) using temperature time series by multiplying the estimated thermal front velocity (v) with a constant (i.e., equation (2) rearranged, γ):
 The value of γ is also uncertain as it depends on physical and thermal parameters that vary from site to site. Consequently, there is a range of possible Darcy velocities for a given value of thermal front velocity. To assess the uncertainty, , and , the 10th, 50th and 90th percentiles of γ, were computed using a Monte Carlo analysis of literature values (see Table 1). It was assumed that the porosity and solid heat capacity follow a normal distribution, with standard deviations equal to one fourth of the maximum range in values.
Table 1. Literature Values Used in the Monte Carlo Analysisa
Sources: Schön , Shanafield et al. , and Thermophysical Properties of Fluid Systems, Chemistry WebBook, National Institute of Standards and Technology, 2011, available at http://webbook.nist.gov/chemistry/fluid/.
J kg−1 °C−1
J kg−1 °C−1
2.3. Field Example
 To illustrate the method, data from Maules Creek Catchment, Murray Darling Basin (Australia), were collected and interpreted. A temperature array was designed consisting of temperature sensors (Onset HOBO Pro v2) at 0.05 and 0.23 m depth within a 30 mm PVC pipe. The sensors were separated by insulating spacers. At each measuring depth, the pipe was perforated to allow rapid thermal equilibrium. The array was installed vertically into the riverbed adjacent to a field site where the river and shallow aquifer levels (in a piezometer 56 m from the river) were monitored.
 Temperatures were logged every 15 min from October 2009 to July 2010. The resulting temperature time series were filtered, using a forward-backward filter, in order to create time series of amplitude ratio and phase shift data; the two-directional filter is required to prevent the introduction of errors into the phase [Hatch et al., 2006]. For filtering procedures, refer to Rau et al. . The amplitude ratio and phase shift data were then used to estimate the Darcy velocity and effective thermal diffusivity using equations (11) and (12) (with (13) and (14)).
 For the purpose of comparing methods, the data were also interpreted using equations (9) and (10) (with (13) and (14)). Effective thermal diffusivity values used were from the literature [e.g., Shanafield et al., 2011]. Furthermore, previously published data from another site within the same catchment were revisited. Rau et al.  had interpreted the data using equations (9) and (10) (with (13) and (14)). However, they observed that the velocity estimates from the two equations were inconsistent with one another in some cases. In this paper, the data are reinterpreted using the proposed method, i.e., equations (11) and (12) (with (13) and (14)).
3. Results and Discussion
 Shown in Figures 2a and 2b are the solutions to equations (9) and (10) (for = 0.18 m). These illustrate that there is a unique value of thermal front velocity for each (Figure 2a), while there are two possible values of thermal front velocity for each , symmetrical about 0 m d−1 (Figure 2b). Importantly, from these figures it can be seen that if and are used independently to estimate the thermal front velocity, the uncertainty in effective thermal diffusivity will cause uncertainty in the velocity calculation. For instance, = 0.3 corresponds to a range of velocities from −0.3 m d−1 (i.e., losing) to +0.4 m d−1 (i.e., gaining; see Figure 2a). Also, = 0.14 to 0.36 day all correspond to 0 m d−1 (i.e., neutral; see Figure 2b).
 Plotting the and results of Figures 2a and 2b against each other (which can be done since both are a function of thermal front velocity) leads to the conclusion that and are not independent variables but that a relationship exists between them (Figure 2c). The implications of this are significant: if the values of and are known, the value of effective thermal diffusivity can be known also, and hence the value of thermal front can be determined. The uncertainty in thermal front velocity normally associated with the uncertainty in effective thermal diffusivity is thus eliminated.
 The solution to equation (11) is shown in Figure 2d as a surface of thermal front velocities for different combinations of and . Combinations of and corresponding to reasonable values of effective thermal diffusivity (i.e., those shown in Figure 2c) have been used. Viewing this surface from either the versus v or versus v perspectives, demonstrates that equations (9) and (10) (see Figures 2a and 2b) are subsolutions of equation (11) (see Figure 2d).
3.2. Interactions in Maules Creek Catchment
 Collected data from the Namoi River for an 8 month period is shown in Figure 3. During the monitoring period there were two major flow events, consisting of a dam release (approximately 1 m change in river level) in early December 2009, followed by a natural flood (over 6 m increase in river level) in early January 2010 (Figure 3a, black line). These fluctuating river levels were reflected in the aquifer level data, although in a dampened and lagged manner (Figure 3a, gray line). The recorded temperature data ranged from approximately 10°C (winter) to 30°C (summer; Figure 3b). During summer the upper temperature sensor recorded higher temperatures than the lower sensor, while this reversed during the winter. Also, in the summer months the daily fluctuations were larger as compared to the winter months.
 The interpreted data is shown in Figure 4. The river-aquifer hydraulic gradient data (Figure 4a), demonstrate that during low flows prior to the major flow events, there was potential for slightly gaining conditions. This reversed to losing conditions due to the increased river stage during the events. This was then followed by a period of gaining conditions which likely reflects a return of bank storage to the river.
 In agreement with the gradient data, the Darcy velocities estimated using equation (11) (Figure 4b), show that during low flows there were slightly losing to slightly gaining conditions (−0.04 to 0.09 m d−1), while during high flows this reversed to losing conditions (to −0.58 m d−1), followed by a period of gaining conditions (to 0.17 m d−1). The Monte Carlo analysis showed that the uncertainty in γ is small ( = 0.68; = 0.77; = 0.84) thereby leading to a narrow band of possible Darcy velocities (Figure 4b, gray shading).
 The values for effective thermal diffusivity estimated using equation (12) (Figure 4c), generally fall within the expected literature range, and are slightly higher than the literature average, although it should be noted that these literature values assume β is negligible in equation (3). The effective thermal diffusivity results computed are inclusive of the effect of thermal dispersivity which increases with velocity. The estimated thermal diffusivities during low flows were slightly different before and after the major flow events. This may indicate that scouring and sedimentation/colmation processes are occurring. This may be either through reworking of the riverbed or movement of suspended material into the pores, resulting in changes in grain size and effective porosity.
 During the major flow events, however, the values for effective thermal diffusivity changed rapidly, and increased to unrealistic values (vertical gray shaded zone in Figure 4c). This was due to the transient conditions during the major flow events with velocities changing significantly on a time scale smaller than that used for individual and calculations. This violates a fundamental assumption of the method, as the analytical solution to the convection-conduction equation (i.e.,equation (7)) requires that fluid flow is at steady state within a period of diel temperature fluctuations. As the major departures in effective thermal diffusivity occur during the major flow events, which is likely to be the period of greatest SW-GW interaction, this is an inherent limitation of the proposed method, and of the existing methods [e.g.,Hatch et al., 2006; Keery et al., 2007].
3.3. Method Comparison
 The same data as that plotted in Figure 4b but reinterpreted using equations (9) and (10) (with (13) and (14)) are shown in Figure 5. For this reinterpretation, literature values for effective thermal diffusivity were used [e.g., Shanafield et al., 2011]. As the value of effective thermal diffusivity used increases, the velocities vary more and tend toward gaining conditions.
 Since equations (9) and (10) are two solutions of the one equation (i.e., equation (7)), they should give the same results. However, this is not the case and the results differ markedly depending on whether the or approach was used (Figure 5a versus Figure 5b). Furthermore, it was not possible to calculate the Darcy velocity with the approach when using the maximum effective thermal diffusivity value (i.e., = 0.13 m2 d−1) as solutions exist for only a narrow range of phase shift values (see Figure 2b). From these results, it is not possible to give reliable velocity estimates for the SW-GW interaction. Consequently, it is not even possible to conclude if the river is gaining or losing water. The deviations indicate, given that the assumptions behindequation (7) are valid, that incorrect values for effective thermal diffusivity are being used in the analysis; the effective thermal diffusivity is implicit in the obtained and data (see Figure 2c).
 The previously published data by Rau et al.  but here reinterpreted using equations (11) and (12) (with (13) and (14)) are shown in Figure 6. Previously, the Darcy velocity results for Horsearm Creek (HC) derived from the data differed from those derived from the data (Figure 6a, dashed lines). Similarly, the Darcy velocity results for Downstream Elfin Crossing (DEC) differed depending on the data used (Figure 6c, dashed lines). However, when both the and data were used in equation (11), a single velocity result was obtained (Figures 6a and 6c, solid lines). As there is uncertainty in the value of γ, bounds were added to the reinterpreted Darcy velocities. The two interpretations of the data are different and would lead to somewhat different conclusions about the system. For example, at HC the new interpretation suggests the systems is neutral to slightly losing, while the old interpretation suggested that perhaps the system was at times losing at rates of more than −0.5 m d−1.
 The corresponding values of D (i.e., from equation (12)) to the derived values of v (i.e., from equation (11)) are shown in Figures 6b and 6d. The effective thermal diffusivity data agree well for both locations and appear to be increasing with time. This could be due to a reduction in porosity over the monitoring period; Rau et al.  reported that colmation was taking place. Alternatively, it could illustrate that the assumptions behind equation (7) are being violated. Further testing of the proposed method for its reliability under different scenarios (e.g., lateral flow conditions) is required.
 Having proposed a method (i.e., Figure 2) for interpreting field data (i.e., Figures 3 and 4), and applied the existing methods to the data (i.e., Figure 5), as well as reinterpreting previously published data from another site (i.e., Figure 6), it is now possible to make the following observations.
 1. When equations (9) and (10) are used, a value for the effective thermal diffusivity has to be chosen. Shanafield et al.  have shown that uncertainties in the effective thermal diffusivity lead to uncertainties in estimated thermal front velocities. This issue is therefore highly significant. To overcome this problem, literature values of the relevant thermal parameters are frequently adopted [e.g., Fanelli and Lautz, 2008; Rau et al., 2010; Lautz, 2010; Jensen and Engesgaard, 2011]. Lapham is commonly referenced to justify this approach. However, this research reported a range of thermal parameters for fine-grained and coarse-grained sediment. The method proposed here bypasses this issue, and removes the uncertainty in thermal front velocity normally associated with uncertainty in the effective thermal diffusivity.
 2. When equations (9) and (10) are used the issue of what the value for thermal dispersivity should be is also raised. There is debate in the literature as to whether dispersivity should be included within the effective thermal diffusivity term, and if so, what the velocity relationship should be [e.g., de Marsily, 1986; Anderson, 2005]. Keery et al.  suggested that the disagreements should promote research into the matter. This has since been undertaken by Rau et al. ; see equation (3). Again, the proposed method bypasses this issue as it does not explicitly or implicitly set the dispersivity term to zero. Rather, the method computes the thermal front velocity using and data (i.e., equation (11)) where the effective thermal diffusivity (inclusive of dispersivity) is implicitly included in the computation.
 4. Since it is reasonable to assume that the effective thermal diffusivity does not change rapidly or to values outside a reasonable range, the results can indicate periods where the method assumptions are being violated. This provides a quality check on the thermal front velocity results. While a rapid departure from value of effective thermal diffusivity indicates that the method is invalid, it should be noted, however, that steadiness of estimated thermal diffusivity does not necessarily per se indicate times where all the model assumptions are valid. The steadiness of calculated effective thermal diffusivity implies steadiness of thermal front propagation, it does not imply that fluid fluxes are 1-D or that the thermal properties of the porous medium are constant in space and time. Nevertheless, the proposed method provides a new metric for detection of violations. This is particularly helpful for data where transient conditions may exist.
 5. Finally, when using equations (9) and (10), iterative routines are required to solve for thermal front velocity. This has recently been automated by Swanson and Cardenas  and Gordon et al. . However, the interactive routines can be numerically intensive. In addition, in the case of equation (10), they are potentially unstable, although limits can be imposed to help with stability. The method outlined in this paper circumvents this issue by solving for thermal front velocity as an explicit function of and without iteration.
 In summary, the proposed method overcomes five major issues by estimating the thermal front velocity using both and data without the need to specify a value for effective thermal diffusivity or the need for iteration (i.e., equation (11)). In addition, the method allows for the estimation of the effective thermal diffusivity (i.e., equation (12)). It should be noted though, in common with all methods based on Stallman , the proposed method has inherent limitations due to the assumptions behind the model, namely, (1) fluid flow is in the vertical direction only, (2) fluid flow is steady state, (3) fluid and solid properties are constant in both space and time, and (4) fluid and solid temperatures at any particular point in space are equal at all times.
 A novel method to estimate the SW-GW interaction (i.e., Darcy velocity), and effective thermal diffusivity, from pairs of temperature time series recorded at different depths at the SW-GW interface has been presented. The method does not require that the effective thermal diffusivity is specified, nor does it use iteration routines. Applying the method to data collected from Maules Creek Catchment in the Murray Darling Basin (Australia) demonstrated its usefulness in revealing SW-GW processes. In addition, the implementation of the method to data showed that while the riverbed Darcy velocities were reliably estimated at low river flows, the method had limitations under transient flows as seen in the estimated values of effective thermal diffusivity. This was due to a violation of the underlying model assumptions (i.e., that fluid flow is at steady state within a period of 1 day). The proposed method therefore also offers the ability to show periods when the model assumptions are violated (although not where the model assumptions are valid). It is expected that the method will be of wide usefulness in field studies of SW-GW interactions, particularly where the anticipated Darcy velocities are low and the thermal parameters are uncertain.
 Funding for the research was provided by the Cotton Catchment Communities CRC (projects 2.02.03 and 2.02.21). In-kind funding was provided by the National Centre for Groundwater Research and Training, an Australian Government initiative, supported by the Australian Research Council and the National Water Commission. Xavier Barthelemy helped with the derivation ofequations (11) and (12). Bayani Cardenas and three anonymous reviewers provided thoughtful comments on a draft of the paper. Rosemary Colacino reviewed the paper for readability.