Temperatures in marine sediments are driven by the geothermal heat flow from the Earth's crust and the evolution of the bottom water temperature. Mathematically, the temperature field can be modeled with the heat equation and a Robin boundary condition at the sediment-water interface and a Neumann condition at the lower boundary. Given the thermal properties of the sediment and a model for the bottom water temperature function the forward problem is well-posed.
The inverse problem, i.e. reconstructing the bottom water temperature function from measurements of the sediment temperature, is ill-posed; the parameterised model is non-linear but low-dimensional.
Different inversion algorithms work differently well on this problem, and regularising methods are not necessarily better. Different Newton-linke methods, as well as a linear fitting approach with Tikhonov minimising and a Markov Chain Monte Carlo method are shown and compared.