Surface temperature determination from Borehole measurements: Regularization and error estimates

Abstract

The authors consider the problem of determining the temperature distribution u(x,t) on the half-line x=0, t>0, from measurements at an interior point, for all t>0. As is well-known, this is an ill-posed problem Using the Tikhonov method, the authors give a regularized solution, and assuming the (unknown) exact solution is in Hα(ℝ), α>0. They give an error estimate of the order 1/(1n1/ ϵ )α for ϵ →0, where ϵ >0 is a bound on the measurement error.