Convection methods and functional equations in order to solve inverse thermochronology problems

Abstract

Our aim is to give a complete account about convection methods which allow: • to get numerical simulations of both the simultaneous phenomena of population of nuclear tracks production and of thermal annealing of these tracks, giving rise, at the final state, to numerical histograms, which are similar to the ones which can be got, nowadays, by etching techniques applied to a sample of apatite and track length measurements. • to solve the corresponding inverse problem in which we have to go back from a given (computed or measured) histogram to a) the length and production time relation and b) to the thermal history, which is the main unknown of this kind of inverse problems. We deal with densities of probabilities of presence of tracks lying in given length intervals, which explain the use of models of functional equations both for the direct and inverse problem. We consider an integral mass conservation relation from which we derive two convection approaches. These approaches enable us to obtain functional equations in order to solve the inverse thermochronology problems for nearly all the main fading laws used in order to describe fission track annealing. These functional equations give rise to simple algebraic relation in the case of Bertagnolli's annealing law. This paper only concerns a mathematical modeling and the numerical aspects associated.