Determination of a terrestrial heat flow from temperature measurements in bottom sediments

We consider the problem of the determination of a terrestrial heat flow from temperature measurements in bottom sediments. The problem is divided into two subproblems: first we solve the one-dimensional inverse problem of estimating the thermal conductivity $\lambda$, and second we compute the heat flow value by solving the direct stationary problem, based on the already-found value of $\lambda$. We develop a sweep method for solving the direct problem which differs form the standard one. An optimization approach is used for solving the inverse problem; explicit formulas are obtained for computing the gradient of the residual functional. We analyze factors causing errors in estimating the heat flow. We show that the main contribution to the errors is given by the presence of harmonics with periods exceeding the monitoring time interval in the temperature curves. We show that if the parameters of the harmonics are known then one can calculate corrections for the found value of the heat flow. The results were applied to the data of temperature measurements carried out at the bottom of Lake Teletskoye from June 2008 to September 2010. For finding long-period harmonics, we made use of meteorological data about bottom water temperature from 1968 to 2011. This allowed us to estimate the value of the heat flow through the bottom of Lake Teletskoye as well as the thermal diffusivity in the upper layer of the sediments.