[1] Work over the last decade has documented methods for estimating fluxes between streams and streambeds from time series of temperature at two depths in the streambed. We present substantial extension to the existing theory and practice of using temperature time series to estimate streambed water fluxes and thermal properties, including (1) a new explicit analytical solution to predict one-dimensional fluid velocity from amplitude and phase information; (2) an inverse function, also with explicit formulation; (3) methods to estimate fluid velocity from temperature measurements with unknown depths; (4) methods to estimate thermal diffusivity from the temperature time series when measurement depths are known; (5) methods to track streambed elevation between two sensors, given knowledge of the thermal diffusivity from (4) above; (6) methods to directly calculate the potential error in velocity estimates based on the measurement error characteristics ; and (7) methods for validation of parameter estimates. We also provide discussion and theoretical insights developed from the solutions to better understand the physics and scaling of the propagation of the diurnal temperature variation through the streambed. In particular, we note that the equations developed do not replace existing equations applied to the analysis, rather they are new equations representing new aspects of the process, and, as a consequence, they increase the amount of information that can be derived from a particular set of thermal measurements.Citation: Luce, C. H., D. Tonina, F. Gariglio, and R. Applebee (2013), Solutions for the diurnally forced advection-diffusion equation to estimate bulk fluid velocity and diffusivity in streambeds from temperature time series, Water Resour. Res., 49,

Two common refrains about using the one‐dimensional advection diffusion equation to estimate fluid fluxes and thermal conductivity from temperature time series in streambeds are that the solution assumes that (1) the surface boundary condition is a sine wave or nearly so, and (2) there is no gradient in mean temperature with depth. Although the mathematical posing of the problem in the original solution to the problem might lead one to believe these constraints exist, the perception that they are a source of error is a fallacy. Here we develop a mathematical proof demonstrating the equivalence of the solution as developed based on an arbitrary (Fourier integral) surface temperature forcing when evaluated at a single given frequency versus that derived considering a single frequency from the beginning. The implication is that any single frequency can be used in the frequency‐domain solutions to estimate thermal diffusivity and 1‐D fluid flux in streambeds, even if the forcing has multiple frequencies. This means that diurnal variations with asymmetric shapes or gradients in the mean temperature with depth are not actually assumptions, and deviations from them should not cause errors in estimates. Given this clarification, we further explore the potential for using information at multiple frequencies to augment the information derived from time series of temperature.