Advanced Inverse Modeling of Sediment Thermal Diffusion Processes : Reconstructing Temporal Variant Boundary Conditions for the One-Dimensional Heat Equation

Abstract

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, 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, on the other hand is ill-posed; the parameterized model is non-linear but low-dimensional. Different Newton-linke methods, as well as a linear fitting approach with Tikhonov minimization, and a Markov Chain Monte Carlo method are shown and their performances for this problem are compared. The algorithms work differently well on this problem, and regularising methods are not necessarily better. The heuristic linear fitting has the best accuracy in reasonable computing time, while the Markov Chain Monte Carlo method has proven convergence for enlarging ensembles.