In solving terrestrial heat flow problems, the complexity of the earth medium and boundary conditions calls for frequent use of numerical modeling techniques. The ill-posedness of the problems due to lack of perfect knowledge of the material properties and the boundary conditions requires that inverse theories be applied. Methods that incorporate both numerical techniques and inverse theories have therefore been gaining attentions in heat flow research.;In this study, an inverse finite element method is developed to solve 2-D steady state heat flow problems involving uncertain material properties and boundary conditions. The problems are first parameterized using an isoparametric finite element model, in which the field variables, the material properties and the boundary conditions are formulated as discrete parameters. Information on the parameters is described in the form of Gaussian probabilities. A nonlinear parameter estimation method of Bayesian type is then used to update our knowledge of the parameters. For computational efficiency, a gradient method is used in the parameter estimation procedure, and the gradients are derived analytically at the elemental level.;The method is applied to two types of conductive heat flow problems, namely the topographic correction and the downward continuation of heat flow data, and to the problem of coupled thermal and hydrological regimes of sedimentary basin scale. Numerical examples have shown that the method provides a rigorous treatment of uncertainities in these problems. In the case of the coupled problem, however, the power of the method is limited by the strong nonlinearity, and better a priori information is needed to constrain the solution
