Extremal temperature bounds from surface gradient measurements

Abstract

Surface measurements of vertical temperature gradient do not uniquely specify the subsurface temperature distribution, both because of the incompleteness of the data and uncertainty in heat production by radioactivity. We derive a theory for discovering certain properties of the temperature function which, under several simplifying restrictions, can be recovered from the data if heat transfer is by steady-state conduction. In particular, with heat production constrained to lie within assigned bounds, we seek the greatest lower bound on the maximum temperature at any depth. Calculation of this bound as a function of depth allows us to deduce the maximum possible depth to a chosen isotherm. We demonstrate the use of linear programming in the computations. The method is applied to a two-data set in New England and the resulting optimal temperature bounds are compared with standard one-dimensional solutions.