In heat flow determinations, it is customary to treat the surface temperature variation as a finite sum of Fourier components. The medium is assumed to be homogeneous or horizontally stratified with each layer having a constant conductivity and diffusivity. This allows the effect of each periodic component to be calculated analytically. We extend this formulation to include cases where thermal conductivities in some layers of a stratified medium may vary linearly with depth as have been found in the sediments of some continental lakes. The application of this formalism to temperature measurements in Lake Greifensee and Lac Leman shows that even with excellent records of bottom temperature variations over several years, failure to take into account the conductivity variation leads to errors as high as 20% in heat flow density values, depending on the depth interval used. The combined effects of lack of detailed knowledge of conductivity structure and the use of too short and (or) inaccurate records of bottom temperature variations, leading to very significant errors, are also discussed, with particular reference to the problems arising from a lack of recognition of the existence of nonannual terms in the bottom temperature variation and the use of probes that do not penetrate the sediments deeply enough.
Minimizing finite sums with the stochastic average gradient
